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How can I use the different formats?
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How to Deal with Math Anxiety

Emotions and Learning

Emotions, or what we feel about something, play a big part in how we learn. If we are calm, we learn well. If we are afraid or stressed, we do not learn as well.

Many people are afraid of math. They fear making a mistake. “Math anxiety” is the fear of math. People who suffer from math anxiety may get headaches, sick stomachs, cold hands, or they may just sweat a lot or just feel scared. Math anxiety can happen for a few different reasons:

Feeling anxious when writing tests
Negative experiences in a past math class
Embarrassment in a past math class
Social pressures and expectations to not like math or not do well in math
The want to get everything right
Negative self-message (“I don’t know how to do it,” or “I hate math”)

Math anxiety is a learned habit. If it is learned, it can be unlearned. Most math anxiety comes from bad memories while learning math. It may be from doing badly on a test or asking a question then being made fun of. These bad memories can make learning math hard.

Everyone can learn math. There is no special talent for math. There are some people who are better at math than others, but even these people had to learn to be good at math.

Do You Suffer from Math Anxiety?

Read the list below and put a check mark beside the ones you feel when thinking about or doing math.

Are your palms moist?
Is your stomach fluttering?
Do you feel like you can’t think clearly?
Do you feel like you would rather do anything else than learn math?
Are you breathing faster than normal?
Is your heart pounding?
Do you feel cold?
Do you feel sweaty?

If you answered yes to two or more of these items, you may have math anxiety.

If you have math anxiety, a first step to understanding it is to look at where it all started.

Make a list of your experiences with learning math. Think back to the first math experiences you had and write about them. Think about learning math in school from the younger grades to the higher grades and write about your experiences and feelings. Include this class and how you are feeling right now about learning math.

Beside each experience, write if it was a positive or negative experience.

Look at the examples below to give you an idea:

Positive or negative?	Math experience
Negative	My teacher in elementary school lined the whole class up in a row and made us play a multiplication game. I could see which question was mine, and I didn’t know the answer so I had to figure it out on my fingers before my turn came up. I got the answer right, but I was so nervous that I would be teased because I didn’t know the answer off the top of my head. I still don’t know my times tables.
Positive	In high school, I could use a calculator to figure out the simple multiplication problems, and then I could figure out the tougher problems without worrying about knowing my times tables.
Negative	Now that I am upgrading my math, I feel nervous every time I even think about opening the book. I want to get all the answers right, and I know that I won’t be able to. I really need everything to be right so that I know that I am getting it.

Positive or negative?

Math experience

Negative

My teacher in elementary school lined the whole class up in a row and made us play a multiplication game. I could see which question was mine, and I didn’t know the answer so I had to figure it out on my fingers before my turn came up. I got the answer right, but I was so nervous that I would be teased because I didn’t know the answer off the top of my head. I still don’t know my times tables.

Positive

In high school, I could use a calculator to figure out the simple multiplication problems, and then I could figure out the tougher problems without worrying about knowing my times tables.

Negative

Now that I am upgrading my math, I feel nervous every time I even think about opening the book. I want to get all the answers right, and I know that I won’t be able to. I really need everything to be right so that I know that I am getting it.

Once you have made a list of experiences, go over the stories with your instructor, or by yourself and try to find some common themes.

Can you see when you felt anxiety?
Can you see why you are now anxious about math?
Is there any experience you could use now to help you feel calmer about math?

Hopefully by examining the beginnings of the anxiety, you can feel more in control of it.

How to Deal with Math Anxiety

Anyone can feel anxiety that will slow down learning. The key to learning is to be the “boss” of your anxiety. Here are an overview of some strategies that may help deal with your anxiety:

Use breathing exercises
Think positive math messages
Know your textbook
Understand test-taking anxiety

Remember, learning to deal with your math anxiety may take some time. It took you a long time to learn math anxiety, so it will take some time to overcome it.

Use Breathing Exercises

One way to be the “boss” is to relax. Try this breathing exercise.

Breathing Exercise

Start by breathing slowly to the count of four. It may help to close your eyes and count.

Now hold your breath for four counts and then let your breath out slowly to the count of four.

The counting is silent and should follow this pattern: “Breath in, two, three, four. Hold, two, three, four. Breath out, two, three, four. Wait, two, three, four.”

With practice, the number of counts can be increased. This is an easy and good way to relax.

Now, try this exercise quietly and repeat it five times slowly.

Each time you feel anxious about learning, use the breathing exercise to help calm yourself. Ask yourself if what you tried worked. Do you feel calmer?

Think Positive Math Messages

Another way to be the “boss” is to give yourself positive math messages.

Read and think about the positive math messages listed below. Do you say any of those things to yourself?

If the answer is yes, then great, keep doing that.
If your answer is no, try to add this little mental trick to your day. The result will probably be that you start to see math as something you can do and that you may even like!

I like math.

I am good at math.

I understand math.

I can relax when I am studying math.

I am capable of learning math.

Math is my friend.

My math improves every day.

I am relaxed, calm and confident when I study math.

I understand math when I give myself a chance.

Math is creative.

Pick three statements that you like and say them to yourself as much as you can in each day. You can also write the statements out on paper and post them around your house so that you read them throughout the day.

Know Your Textbook

Look at the Table of Contents in the front of your textbook. It tells you what you will be learning. You may see some things that you already know, some things that you may have forgotten, and some things that are new to you.

Flip the pages. You can see that the textbook is split into units. Each unit is something to learn.

Each unit has exercises to do. Notice the answers are at the end of the exercise. You can check your answers as soon as you are done. You can also check your answer before moving on if are not sure if you are doing the question right.

At the end of each unit is a self-test. It is a chance for you to see how well you have learned the skills in the unit. If you do well, you can move on. If you don’t do well, you can go back and practice those skills.

Knowing your textbook gives you a good skill. If you get frustrated, you can use the Table of Contents to go back and find some help.

Understand Test-Taking Anxiety

There are four reasons people are anxious when writing tests. Any of the four reasons listed below might be the reason a person might feel anxious in a test-taking situation.

Not feeling prepared for the test
Not sure how to write the test in the best way
Feeling too much mental pressure
Poor health habits before writing a test

Here is an explanation of each reason and how to work your way out of the anxiety you may feel during tests.

1. Not feeling prepared for the test

Many students feel anxiety about taking math tests because they do not feel prepared for the test. To feel prepared, a student needs to have studied the work and know that they can do the problems they will be given. Get help from your classmates, friends, or your instructor to find out how you can improve your study habits.

Getting ready for a test starts on the first day of class. Everything you do in class and at home is part of that getting ready.

Always do as many exercises as you need to help you understand. Once you understand, do ten more questions, then you will know for sure that you really understand.
Always correct your exercises. It is good to know that you are understanding and getting the questions right. It is also good to know if you are not understanding and need some help.
Always do the self-tests. The self-tests can show things that you are not sure of.
Always do the review. Review is part of this book. It is a chance to go over all the things you have learned in a unit before moving on. It prepares you for what will be on the test.
Always do a practice test. A practice test gives you a chance to see how many questions and what kind of questions are on the test.

2. Not sure how to write the test in the best way

Here are some strategies students should know about how to write a test to do the best as possible on it:

Before the Test
1. Arrive early. Get out all the supplies you need to do the test (pencils, ruler, calculator, watch, etc.).
2. Be comfortable, but alert. Choose a good spot in the room, and make sure you have enough space to work. Maintain a comfortable posture in your seat, but don’t “slouch.”
3. Stay relaxed and confident. Keep a good attitude. If you find yourself anxious, take several slow, deep breaths to relax. Don’t talk about the test to other students just before entering the room: their anxiety can be contagious.
During the test.
1. Look over the test. Take a look at the whole test before starting. This takes very little time. Use a highlighter to highlight the questions that you know you can do easily, note key terms, mark the test with comments that come to mind. As you work, put a star beside any questions that you would like to go over again when you finish the test.
2. Relax. Before starting the test, imagine yourself somewhere where you are calm and confident. Go there in your mind. Focus on how good you feel and how in control you are. If you become anxious during the test, in your mind go to the calming place. Focus on how calm you feel. Then go back to your test.
3. Read the directions carefully. This may be obvious, but it will help you avoid careless errors.
4. Answer questions in a strategic order.
  - Answer the easy questions first. This will help to build confidence and score points. It may also help you make connections with more difficult questions.
  - Then answer the difficult questions. Work on these harder questions with all the energy of the easier ones.
5. Review your answers. Resist the urge to leave as soon as you are done writing. Spend as much time as you can going over your test to see if you:
  - Answered all the questions.
  - Wrote the answers in right.
  - Did not make simple mistakes.

3. Feeling too much mental pressure

There are many reasons why a student may feel mental pressure when writing a test. Listed below are a few main reasons:

Negative beliefs about one’s math abilities
Low self-esteem when it comes to math
Too high expectations of success
Fear that failure or low grades will affect the future
Feelings of pressure of not wanting to let down family members

When students feel this kind of pressure, it is very hard to feel calm and relaxed about a test. The key to success in a math test is to keep the anxiety at a manageable level. You can do this in two ways:

Change negative self-talk. Any time a negative thought creeps into your head, it will make it harder to stay positive and relaxed about your test. If you have a negative thought like “I can’t do it”, try to replace it with a positive thought like “I can do this”.
Use relaxing and calming techniques. Use the calming breathing mentioned earlier in this section. This will help you keep calm. Also, do not study in the last half hour before the test. You will be calmer by spending time relaxing and breathing deeply in that last half hour.

4. Poor health habits before writing a test

When your body and mind are healthy, you will have a better chance of doing well on a test. Eat well, drink plenty of water and get daily exercise. The better you feel, the better you can perform (and a test is a performance!).

Unit 1: Ratio, Rate, & Proportion

Topics

Unit 1 will introduce ratios, rates and proportions.

Ratios are used to compare quantities and illustrate the relationship between numbers in a simple way.
Rates are used when a ratio is comparing two different units.
Proportions are statements with two equal ratios that can be used in solving percent problems.

Topic A: Writing Ratios

Introduction to Ratios

Ratio is a comparison of one number or quantity with another number or quantity. Ratio shows the relationship between the quantities.

Ratio is pronounced “rā’shō” or it can be pronounced “rā’shēō.” Check out this YouTube video to listen to someone pronouce the word: How to Pronounce Ratio.

You often use ratios, look at these examples:

Scores in games are ratios. For example, the Penguins won 4 to 3 or the Canucks lost 1 to 5.
Directions for mixing can be ratios. For example, use 1 egg to each cup of milk or mix 25 parts gas to 1 part oil for the motorcycle.
Betting odds are given as ratios. For example, Black Jade is a 3 to 1 favourite or the heavyweight contender is only given a 2 to 5 chance to win.
The scale at the bottom of maps is a ratio. For example, 1 centimetre represents 10 kilometres.
Prices are often given as ratios. For example, 100 grams for $0.79 or 2 cans for $1.85.

For ratios to have meaning you must know what is being compared and the units that are being used. Read these examples of ratios and the units that are used. A general ratio may say “parts” for the units.

It rained four days and was sunny for three days last week. The ratio of rainy days to sunny days was $4:3$ . ( $4:3$ is properly read “4 is compared to 3” but is often read “4 to 3”).
The class has 12 men and 15 women registered. The ratio of men to women in the class is $12:15$ .
At the barbeque, 36 hot dogs and 18 hamburgers were eaten. The ratio of hot dogs eaten to hamburgers eaten is $36:18$ .
The class spends 3 hours on English and 2 hours on math each day. The ratio of time spent on English compared to math is $3:2$ .

Exercise 1

Write the ratios asked for in these questions using the $:$ symbol (for example, $4:1$ ). Write the units and what is being compared beside the ratio.

Powdered milk is mixed using 1 part of milk powder to 3 parts of water. Write a ratio to compare the milk powder to the water.
Answer: $1:3$ — 1 part of milk powder to 3 parts of water
One kilogram of ground beef will make enough hamburger for 5 people. Write a ratio to express the amount of ground beef for hamburgers to the number of people.
Seventy-five vehicles were checked by the police. Fifteen vehicles did not meet the safety standards, but 60 of them did. Write a ratio comparing the unsafe vehicles to the safe vehicles.
The recipe says to roast a turkey according to its weight. For every kilogram, allow 40 minutes of cooking. Write a ratio comparing time to weight.
The 4-litre pail of semi-transparent oil stain should cover 24 square metres of the house siding if the wood is smooth. Write the ratio comparing quantity of stain to the smooth wood surface area.
The same 4 L of stain will only cover 16 square metres of the house siding if the wood is rough. Write that ratio.

Answers to Exercise 1

$1:5$ 1 kg of beef to 5 people
$15:60$ 15 unsafe vehicles to 60 safe vehicles
$40:1$ 40 minutes to 1 kg of turkey
$4:24$ 4 L of stain to 24 m² of smooth wood
$4:16$ 4 L of stain to 16 m² of rough wood

The numbers that you have been using to write the ratios are called the terms of the ratio.

The order that you use to write the terms is very important. Read a ratio from left to right and the order must match what the numbers mean. For example, 3 scoops of coffee to 12 cups of water must be written $3:12$ as a ratio because you are comparing the quantity of coffee to the amount of water.

If you wish to talk about the amount of water compared to the coffee you have, you would say, “Use 12 cups of water for every 3 scoops of coffee” and the ratio would be written $12:3$ .

Ratios can be written 3 different ways:

Using the $:$ symbol — $2:5$
As a common fraction —
- The first number in the ratio is the numerator; the second number is the denominator.
- Ratios written as a common fraction are read as a ratio, not as a fraction. Say “2 to 5,” not “two-fifths.”
Using the word “to” — 2 to 5

Exercise 2

Use the ratios you wrote in Exercise 1 to complete the chart.

	$:$	Common fraction	to
A	$1:3$	$\frac{1}{3}$	1 to 3
B
C
D
E

Answers to Exercise 2

$1:5$ or $\frac{1}{5}$ or 1 to 5
$15:60$ or $\frac{15}{60}$ or 15 to 60
$40:1$ or $\frac{40}{1}$ or 40 to 1
$4:24$ or $\frac{4}{24}$ or 4 to 24
$4:16$ or $\frac{4}{16}$ or 4 to 16

Exercise 3

Use the diagram to write a ratio comparing the quantity of each shape, as asked.
Different shapes. There are 6 hearts, 7 suns, and 9 smilie faces.

Answers to Exercise 3

$9:6$ or $\frac{9}{6}$ or 9 to 6
$7:9$ or $\frac{7}{9}$ or 7 to 9
$6:9$ or $\frac{6}{9}$ or 6 to 9

Equivalent Ratios

Like equivalent fractions, equivalent ratios are equal in value to each other.

$10:100 = 1:10$

Ratios can be written as common fractions. It is convenient to work with ratios in the common fraction form.

You can then easily:

Find equivalent ratios in higher terms
Find equivalent ratios in lower terms
Find a missing term

Example A

Express $4:5$ in higher terms.

$4:5=\dfrac{4}{5}\longrightarrow \dfrac{4}{5} \times \left(\dfrac{2}{2}\right) \longrightarrow\left(\dfrac{4\times 2}{5\times 2}\right)\longrightarrow\dfrac{8}{10}$

$4:5$ is equivalent to $8:10$

Example B

Express $3:6$ in lower terms.

$3:6=\dfrac{3}{6}\longrightarrow \dfrac{3}{6} \div \left(\dfrac{3}{3}\right) \longrightarrow \left(\dfrac{3\div 3}{6\div 3}\right)\longrightarrow\dfrac{1}{2}$

$3:6$ is equivalent to $1:2$

To find equivalent ratios in higher terms, multiply each term of the ratio by the same number. To find equivalent ratios in lower terms, divide each term of the ratio by the same number.

Exercise 4

Write equivalent ratios in any higher term. You may want to write the ratio as a common fraction first. Ask your instructor to mark this exercise.

$5:6= \dfrac{5}{6} \times \left(\dfrac{3}{3}\right)=\left(\dfrac{5\times 3}{6\times 3}\right)=\dfrac{15}{18}=15:18$
$4:3$
$10:2$
$50:1$
$9:4$
$3:5$

Answers to Exercise 4
See your instructor.

Exercise 5

Write these ratios in lowest terms—that is, simplify the ratios.

$4:12=\dfrac{4}{12}\div\left(\dfrac{4}{4}\right)=\left(\dfrac{4\div4}{12\div4}\right)=\dfrac{1}{3}=1:3$
$10:5$
$7:21$
$20:5$
$6:14$
$2:4$
$6:3$
$16:8$

Answers to Exercise 5
Ratios written as a common fraction or using the word “to” will also be correct in this exercise. The terms must be the same.

$1:3$
$2:1$
$1:3$
$4:1$
$3:7$
$1:2$
$2:1$
$2:1$

Exercise 6

Using a colon, write a ratio in lowest terms for the information given.

In the class of 25 students, only 5 are smokers. Write the ratio of smokers to non-smokers in the class. (Note—you must first calculate the number of non-smokers.)
The police issued 12 roadside suspensions to drivers out of the 144 who were checked in the road block last Friday. Write the ratio of suspended drivers to the number checked.
Twenty-seven students registered for the course and 24 completed it. Write a ratio showing number of completions compared to number enrolled.
During an hour (60 minutes) of television viewing last night there were 14 minutes of commercials, so there were only 46 minutes of the actual program! Write the ratio of commercial time to program time.
For each pair of coins, write the ratio comparing the value. (Use cents.)
1. A nickel to a dime $5:10=1:2$
2. A nickel to a quarter
3. A nickel to a dollar
4. A dime to a nickel
5. A dime to a quarter
6. A dime to a dollar
7. A dollar to a dime

Answers to Exercise 6

$1:4$
$1:12$
$8:9$
$7:23$
1. $1:2$
2. $1:5$
3. $1:20$
4. $2:1$
5. $2:5$
6. $1:10$
7. $10:1$

Topic A: Self-Test

Mark /12 Aim 10/12

Write the definitions. (3 marks)
1. Ratio
2. Terms of the ratio
3. Equivalent ratios
Write the ratios asked for in lowest terms. Use the colon style like this: . Then write the ratio as it is read, like this: two to one. (4 marks)
1. The campground had three vacant campsites and 47 occupied sites. Write the ratio of occupied sites to vacant sites. Ratio: Read:
2. For every ten dogs in the city, only 2 have current dog licences. Write the ratio of licensed dogs to unlicensed dogs. (Find the number of unlicensed dogs first). Ratio: Read:
Simplify these ratios. (5 marks)
1. $9:12$
2. $6:4$
3. $500:1000$
4. $2:9$
5. $35:15$

Answers to Topic A Self-Test

Write the definitions.
1. A ratio is a comparison of one number or quantity with another number or quantity. Ratios show the relationship between the quantities or amounts.
2. Terms of a ratio are the numbers used in the ratio, the parts of the ratio.
3. Equivalent ratios are ratios of equal value to each other.
Write ratios in lowest terms.
1. $47:3$
  Read: “47 occupied sites to 3 vacant sites.”
2. $1:4$
  Read: “1 licensed dog to 4 unlicensed dogs.”
Simplify the ratios.
1. $3:4$
2. $3:2$
3. $1:2$
4. $2:9$
5. $7:3$

Topic B: Rates

When a ratio is used to compare two different units (e.g., apples and oranges, or metres and hours), it is called a rate. The denominator must be 1.

Example A

A car can drive 725 km on 55 L of gas. What is the rate in km per L? The ratio of this is $\dfrac{725\text{ km}}{55\text{ L}}$ . Find the rate by making the denominator 1.

Divide $\dfrac{725}{55} \div \left(\dfrac{55}{55}\right) = \dfrac{725\div55}{55\div55}=\dfrac{13.18}{1}=13.18$

The rate is 13.18 km/L.

Example B

Sue bought 10 lb of oranges for $4.99. What is the rate in cents per pound? The ratio is $\dfrac{$4.99}{10\text{ lb}}=\dfrac{499\text{ cents}}{10 \text{ lb}}$ . Find the rate by making the denominator 1.

Divide $\dfrac{499}{10} \div \left(\dfrac{10}{10}\right) =\dfrac{499\div10}{10\div10}=\dfrac{49.9}{1}=49.9$

The rate is 49.9 ¢/lb.

When talking about rate, use the word “per.”

In example A, say, “The fuel economy of the car is 13.18 kilometres per litre.”

In example B, say, “The oranges cost 49.9 cents per pound.”

Example C

It takes 60 ounces of grass seed to plant 30 m² of lawn. What is the rate in ounces per square metre (m²)? The ratio is $\dfrac{60\text{ oz}}{30\text{ m}^2}$ . Find the rate by making the denominator 1.

Divide $\dfrac{60}{30} \div \left(\dfrac{30}{30}\right) = \dfrac{60\div30}{30\div30}=\dfrac{2}{1}=2$

The rate is 2 oz/m², or 2 ounces per square metre.

Exercise 1

Write the following ratios as rates, comparing distance to time.

120 km, 3 hours
27 km, 9 hours
203 km, 29 seconds
444 km, 48 seconds

Answers to Exercise 1

40 km/hour
3 km/hour
7 km/second
9.25 km/second

Exercise 2

Write the following ratios as rates.

A leaky faucet can lose 52 litres of water in a week. What is the rate of litres lost per day? (round to two decimal places)
A ratio of distance travelled to time is called speed. What is the rate (speed) in kilometres per hour (km/h)?
1. 45 km, 3 hours
2. 129 km, 1.5 hours
3. 65 km, 13 hours
Vancouver Island has a population of 734,860, and a land mass of 32,134 square kilometres. What is the rate of number of people per square kilometre? (This is called population density.) Round your answer to the nearest whole number.
At rest, the heart beat of a mouse is 30,000 beats per 60 minutes. What is the rate of beats per minute?

Answers to Exercise 2

7.43 L/day
1. 15 km/hour
2. 86 km/hour
3. 5 km/hour
23 people/km²
500 beats/minute

Topic B: Self-Test

Mark /7 Aim 6/7

Write the definition. (1 mark)
1. Rate
Write the following ratios as rates. Round people to the nearest person. (6 marks)
1. 12 cups water, 3 cups sugar
2. 72 metres, 24 seconds
3. 1,365,000 people, 4,000 km²
4. 5,000 cars on the road, 250 bikes on the road
5. 12 cups of flour, 12 tsp. of baking powder
6. 8 litres of gas, 2 litres of oil

Answers to Topic B Self-Test

A rate is used when a ratio compares two different kinds of measure, and when the denominator is 1.
1. 4 cups of water/cup of sugar
2. 3 m/second
3. 341 people km²
4. 20 cars/bike
5. 1 cup flour/tsp baking powder
6. 4 litres gas/litre oil

Topic C: Proportion

A proportion is a statement that two ratios are equal or equivalent. Here are some proportions:

Proportion	Fraction Form	Read like this…
$1:2=2:4$	$\frac{1}{2}=\frac{2}{4}$	1 is to 2 as 2 is to 4
$1:4=25:100$	$\frac{1}{4}=\frac{25}{100}$	1 is to 4 as 25 is to 100
$18:9=10:5$	$\frac{18}{9}=\frac{10}{5}$	18 is to 9 as 10 is to 5
$15:20=3:4$	$\frac{15}{20}=\frac{3}{4}$	15 is to 20 as 3 is to 4

Proportions can be used to solve many math problems. You will soon learn to use proportions to solve problems involving percent. The techniques you practice in the next few pages are important for that problem solving work.

Problems often give incomplete information; that is, one of the terms is missing. To solve such problems, you first find the comparison or ratio that is given. It may be:

A quantity of one thing that is mixed with a larger quantity of something else
A scale of measurement given on a map such as 1 cm on the map represents 100 km distance on land
Cost for a certain number of items
Time to travel a certain distance

The problem will then give one term of the second ratio in the proportion. For example, if you have been told that 3 heads of lettuce cost $1.49, you may be asked to find the cost of 7 heads of lettuce.

The missing term is the second cost. The proportion will be:

$\begin{equation*}\begin{split} \dfrac{\text{number of heads of lettuce}}{\text{cost}} &= \dfrac{\text{number of heads of lettuce}}{\text{cost}} \\ \\ \dfrac{3}{$1.49} &= \dfrac{7}{?} \\ \\ 3:$1.49 &= 7: ? \end{split}\end{equation*}$

The most important thing to remember is to keep the order of comparison the same in the first and second ratios in a proportion. If the first ratio compares time to distance then the second ratio in the proportion must compare time to distance.

$\dfrac{\text{time}}{\text{distance}}=\dfrac{\text{time}}{\text{distance}}$

Or it could be:

$\dfrac{\text{distance}}{\text{time}}=\dfrac{\text{distance}}{\text{time}}$

Once you have decided on the order of comparison it is a simple matter to write the proportion using the numbers given in the problem. Use a letter to stand for the missing term.

How would you find a missing term?

You can use your skills with equivalent ratios (finding higher and lower terms)
You can use your fraction skills of cross-multiplying and then dividing to find the missing term

Using Equivalent Ratios to Solve Proportions

Solve a Proportion Problem Using Equivalent Ratios

Decide on the order of comparison and write a ratio that describes the information given in the problem. Write a proportion using words to represent the items that are being compared. Write the ratio in fraction form.
Write two more ratios with the numbers corresponding to the locations of the words in the first ratio. The missing term (number) can be represented by a letter (for example, N).
Mentally set the ratio with words (the first ratio) aside.
Multiply or divide the complete ratio to find the missing term.

Example A

Use 1 teaspoon of baking powder for every 2 cups of flour. If a recipe uses 6 cups of flour, how much baking powder is needed?

The missing term is the number of teaspoons of baking powder needed for 6 cups of flour. Call this term N.

Ratio is $\dfrac{\text{baking powder}}{\text{flour}}$
$\dfrac{\text{baking powder}}{\text{flour}}=\dfrac{1}{2}=\dfrac{\textit{N}}{6}$
$\dfrac{1}{2}=\dfrac{\textit{N}}{6}$
$\dfrac{1}{2}\times \left(\dfrac{3}{3}\right)= \dfrac{1 \times 3}{2 \times 3}=\dfrac{3}{6}$ , so $\dfrac{1}{2}=\dfrac{3}{6}$ , so $\textit{N}=3$

Use 3 teaspoons of baking powder for 6 cups of flour.

Example B

Reports suggest that 3 out of 10 people will at some time miss work due to back pain. If a company has 1,000 employees, how many can be expected to miss work due to back pain?

The missing term is the number of people out of 1,000 who will miss work due to back pain. Call this term P.

Ratio is $\dfrac{\text{people who will miss work}}{\text{all people at work}}$
$\dfrac{\text{people who will miss work}}{\text{all the people at work}}=\dfrac{3}{10}=\dfrac{\textit{P}}{1000}$
$\dfrac{3}{10}=\dfrac{\textit{P}}{1000}$
$\dfrac{3}{10}\times\left(\dfrac{100}{100}\right)= \dfrac{3 \times 100}{10 \times 100}=\dfrac{300}{1000}$ , so $\dfrac{3}{10}=\dfrac{300}{1000}$ , so $\textit{P}=300$

300 people out of 1,000 people may miss work due to back pain.

Exercise 1

In words, write a ratio to describe the information given:

three cups of flour to one teaspoon of yeast
$\dfrac{\text{flour}}{\text{yeast}}$
four parts oil, ten parts gasoline
one centimetre represents 100 kilometres
100 grams for $6.89
3 eggs for each cup of milk
5 men and 7 women

Answers for Exercise 1

$\dfrac{\text{oil}}{\text{gasoline}}$
$\dfrac{\text{centimetres}}{\text{kilometres}}$
$\dfrac{\text{grams}}{\text{dollars}}$
$\dfrac{\text{eggs}}{\text{milk}}$
$\dfrac{\text{men}}{\text{women}}$

Exercise 2

Use equivalent ratios to find the answers.

One cup of sugar and four cups of water will make great hummingbird food. How much sugar do you need for 8 cups of water?
$\dfrac{\text{sugar}}{\text{water}}\longrightarrow\dfrac{1}{4}=\dfrac{\textit{N}}{8}\longrightarrow\dfrac{1}{4}\times\left(\dfrac{2}{2}\right) = \dfrac{1 \times 2}{4 \times 2}=\dfrac{2}{8}\longrightarrow\textit{N}=2$
Reports show that, for every 100 vehicles checked by police, 20 vehicles do not meet the safety standard. If only 50 vehicles are checked, how many would not meet the safety standard?
Four litres of paint covers 24 square metres of wall. How much paint is needed to cover 72 square metres?
Powdered milk uses 1 part milk powder to 3 parts water. How much powder should be added to 9 parts water?

Answers for Exercise 2

10 cars would not meet the safety standards
12 litres of paint
3 parts milk powder

Exercise 3

Use equivalent ratios to find the missing term in these proportions.

$3:5=\textit{Y}:15$
$1:2=\textit{P}:8$
$5:7=10:\textit{N}$
$2:3=8:\text{W}$
$4:7=16:\textit{A}$
$1:3=2:\textit{N}$
The KX 250 motorcycle uses a mixture of one part oil to 30 parts of gasoline. How much oil must be added to 3,000 mL of gasoline?

Answers for Exercise 3

$\textit{Y}=9$
$\textit{P}=4$
$\textit{N}=14$
$\textit{W}=12$
$\textit{A}=28$
$\textit{N}=6$
$\textit{N}=100 \text{ mL}$

Using Cross-Multiplication to Solve a Proportion

Let’s review cross products, using this example problem:

$\dfrac{2}{5}\times\dfrac{4}{10}$

Multiply the numerator of each fraction by the denominator of the other fraction.

$\dfrac{2}{5}\rlap{\nearrow}{\searrow}\dfrac{4}{10}$

$\begin{equation*} \begin{split} 2\times10 & =5\times4 \\ 20 & =20 \end{split} \end{equation*}$

$2\times10=20$ and $5\times4=20$

Therefore: $\dfrac{2}{5}=\dfrac{4}{10}$

Remember that, when the cross products are the same, the fractions are equivalent.

When finding the missing terms in a proportion, cross-multiplication can be used. Follow the examples carefully.

Example A

$\dfrac{2}{3}=\dfrac{\textit{N}}{45}$

Cross-multiply:

$\begin{equation*} \begin{split} 2\times45 &= 3\times\textit{N} \\ 90 &= 3\textit{N} \end{split} \end{equation*}$

The idea is to have the unknown term N by itself on one side of the equal sign. To do that, remember these things that you already know:

Division and multiplication are opposite operations
Whatever is done to one side of an equation or proportion must be done to the other side to keep the equation equal

3N means N is multiplied by 3. To get rid of the 3, divide by 3.

You must also divide the other side of the equation by 3.

$\dfrac{90}{3}=\dfrac{3\textit{N}}{3}$

Solve by reducing the $\frac{3}{3}$ and dividing 90 by 3.

$\begin{equation*}\begin{split} \dfrac{90}{3} &= \dfrac{3\textit{N}}{3} \\ \\ \dfrac{90}{3} &= \dfrac{1\textit{N}}{1} \\ \\ \dfrac{90}{3} &= \textit{N} \\ \\ 90\div3 &= \textit{N} \\ \\ 30 &= \textit{N} \end{split}\end{equation*}$

Reducing the fraction $\dfrac{3\textit{N}}{3}$ to $\dfrac{1\textit{N}}{1}$ to N is also called cancelling. In math, a fraction can be cancelled when the numerator and denominator are the same number. For example:

$\dfrac{6\textit{P}}{6}=\dfrac{1\textit{P}}{1}=\textit{P}$

Example B

$\dfrac{6}{7}=\dfrac{24}{\textit{N}}$

Cross-multiply:

$\begin{equation*}\begin{split} 6\times\textit{N} &= 7\times24 \\ 6\textit{N} &= 168 \end{split}\end{equation*}$

Divide both sides by 6. The 6’s with the N will cancel (reduce), and the N will be alone.

$\begin{equation*}\begin{split} \dfrac{6\textit{N}}{6} &= \dfrac{168}{6}\\ \dfrac{\cancel{6}\textit{N}}{\cancel{6}} &= \dfrac{168}{6} \\ \\ \textit{N} &= 168\div6 \\ \\ \textit{N} &= 28 \\ \\ \dfrac{6}{7} &= \dfrac{24}{28} \end{split}\end{equation*}$

Check by cross-multiplying:

$\begin{equation*}\begin{split} \text{Is }6\times28 &= 7\times24? \\ 6\times28 &= 168 \\ 7\times24 &= 168 \\ \text{the cross product }168 &= \text{the cross product }168 \\ \text{Yes - }6:7 &= 24:28 \end{split}\end{equation*}$

Example C

$\dfrac{8}{10}=\dfrac{\textit{N}}{80}$

Cross-multiply:

$\begin{equation*}\begin{split} 8\times80 &= 10\times\textit{N} \\ 640 &= 10\textit{N} \end{split}\end{equation*}$

Divide both sides by 10 so N will be alone.

$\begin{equation*}\begin{split} \dfrac{640}{10} &= \dfrac{10\textit{N}}{10} \\ \dfrac{64\cancel{0}}{\cancel{10}} &= \dfrac{\cancel{10}\textit{N}}{\cancel{10}} \\ 64 &= \textit{N} \end{split}\end{equation*}$

Solve a Proportion Problem Using Cross-Multiplication

Write the first ratio using the information given.
Write the proportion, using a letter in place of the missing term. Be sure the order of comparison is the same in both the first and second ratios in your proportion.
Write the proportion in fraction form. (Try to simplify the ratio before you do all the calculations.)
Cross-multiply and set the cross products equal to each other.
Divide both sides of the equation by the number (coefficient) with the unknown term.
Check by putting your answer back into the original proportion and cross-multiplying.

Exercise 4

Practice using cross-multiplication to find the missing term in these proportions.

$\begin{equation*}\begin{split} \dfrac{5}{8} &= \dfrac{\textit{N}}{32} \\ \\ 5\times32 &= 8\times\textit{N} \\ \\ 160 &= 8\textit{N} \\ \\ \dfrac{160}{8} &= \dfrac{8\textit{N}}{8} \\ \\ 160\div8 &= \textit{N} \\ \\ 20 &= \textit{N} \end{split}\end{equation*}$
$\dfrac{4}{\textit{N}}=\dfrac{24}{30}$
$\dfrac{12}{4}=\dfrac{18}{\textit{x}}$
$\dfrac{\textit{y}}{6}=\dfrac{20}{12}$
$4:15=8:\textit{N}$
$\textit{W}:100=6:50$

Answers to Exercise 4

$\textit{N}=5$
$\textit{x}=6$
$\textit{y}=10$
$\textit{N}=30$
$\textit{W}=12$

The numbers in a ratio often are common fractions, decimals or mixed numbers. Follow exactly the same steps that you have been using to solve whole number proportions. The calculations will use your skills with fractions.

Example A

$2\frac{1}{4}:3=\textit{N}:7$

Rewrite the proportion:

$\dfrac{2\frac{1}{4}}{3}=\dfrac{\textit{N}}{7}$

Cross-multiply:

$\begin{equation*}\begin{split} 2\frac{1}{4}\times7 &= 3\times\textit{N} \\ \\ \frac{9}{4}\times\frac{7}{1} &= 3\times\textit{N} \\ \\ \frac{63}{4} &= 3\textit{N} \\ \\ \frac{63}{4}\div\frac{3}{1} &= \frac{3\times\textit{N}}{3} &\longrightarrow \frac{63}{4}\times\frac{1}{3} &= \textit{N} \\ \\ \frac{63}{12} &= \textit{N} &\longrightarrow 5\frac{3}{12}\longrightarrow5\frac{1}{4} &= \textit{N} \end{split}\end{equation*}$

Exercise 5

Practice using cross-multiplication to find the missing term in these proportions.

$\begin{equation*}\begin{split} 6.5:5 &= 13:\textit{A} \\ \\ \frac{6.5}{5} &= \frac{13}{\textit{A}} \\ \\ 6.5\textit{A} &= 65 \\ \\ \textit{A} &= 65\div6.5 \\ \\ \textit{A} &= 10 \end{split}\end{equation*}$
$3\frac{1}{2}:2=\textit{N}:8$
$9:6=4\frac{1}{2}:\textit{N}$
$7.5:\textit{B}=10:20$
$3.75:5=9:\textit{x}$
$4\frac{1}{8}:\textit{A}=3:6$

Answers to Exercise 5

$\textit{N}=14$
$\textit{N}=3$
$\textit{B}=15$
$\textit{x}=12$
$\textit{A}=8\frac{1}{4}$ or $8.25$

Exercise 6

Joanne can walk 18 km in 3 hours. How far can she walk, at the same rate in 5½ hours?
The taxes on the property valued at $300,000 are valued at $5,000. At the same rate of taxation, what would the taxes be on the smaller lot down the street which is valued at
$240,000?

One B.C. road map has a scale of 0.5 centimetres equal to 10 kilometres. Complete the chart by calculating actual driving distances in kilometres between some B.C. places.The proportions will be $\dfrac{0.5}{10}=\dfrac{\text{cm given in chart}}{\text{actual distance in km}}$

Places in B.C.	Number of cm between places on the map	Actual distance in kilometres
Kelowna and Vernon	2.5 cm
Burns Lake and Vanderhoof	5.5 cm
TaTa Creek and Skookumchuk	0.75 cm
Kitimat and Terrace	3.3 cm

The directions on the lawn fertilizer say to spread 1.7 kg over 100 m² of lawn.
1. How much fertilizer is needed for a 130 m² lawn?
2. How much fertilizer for a 75 m² lawn?

Answers to Exercise 6

33 km
$4,000

Places in B.C.	Number of cm between places on the map	Actual distance in kilometres
Kelowna and Vernon	2.5 cm	50 km
Burns Lake and Vanderhoof	5.5 cm	110 km
TaTa Creek and Skookumchuk	0.75 cm	15 km
Kitimat and Terrace	3.3 cm	66 km

1. 2.21 kg
2. 1.275 kg

Topic C: Self-Test

Mark /20 Aim 17/20

Solve these proportions. (6 marks)
1. $\textit{N}:14=28:56$
2. $3:11=\textit{N}:22$
3. $50:45=10:\textit{N}$
4. $4\frac{1}{5}:\textit{Y}=3:2$
(14 marks)
1. Get a map of BC, a map of Canada, and a map of your city or town.
2. Find the scale on each map (usually at the bottom) and write down the ratio of map distance to the actual distance.
3. With another student or an instructor, calculate actual distances between places by measuring the distance on the map and working out the proportion according to the scale given. Do at least three distance calculations on each map.
Ask your instructor to mark your work.

Answers to Topic C Self-Test

1. $\textit{N}=7$
2. $\textit{N}=6$
3. $\textit{N}=9$
4. $\textit{Y}=2\frac{4}{5}$ or $2.8$
See your instructor.

Unit 1 Review: Ratio, Rate, & Proportion

Questions

Write the ratios asked for in the questions. Reduce when able.
1. It snowed for two days and was sunny for five days last week. Write a ratio to compare snowy days to sunny days.
2. The class spends 6 hours a week on math and 8 hours on English. Write the ratio of hours spent on math to English.
3. One kilogram of pork will make enough sausages for 7 people. Write a ratio to express the amount of pork for sausages to the number of people.
4. A jam recipe calls for 8 cups of blue berries to 3 cups of sugar. Write a ratio to express the amount of berries to sugar.
Write the ratios in lowest terms.
1. $14:21$
2. $8:24$
3. $15:150$
4. $9:54$
Write the following ratios as rates.
1. 16 cups of water to 4 cups of sugar
2. 150 kilometres to 1.25 hours
3. 465,000 people to 3,000 square kilometres
4. $3.99 to 10 pounds of apples
Use cross-multiplication to solve the proportions. Round to 2 decimal places, if needed.
1. $\dfrac{5}{8}=\dfrac{\textit{P}}{24}$
2. $\dfrac{\textit{S}}{100}=\dfrac{8}{25}$
3. $\dfrac{4}{\textit{N}}=\dfrac{8}{40}$
4. $\dfrac{4}{14}=\dfrac{\textit{F}}{68}$
Use cross-multiplication to solve the proportions.
1. $\dfrac{2\frac{1}{4}}{5}=\dfrac{\textit{A}}{30}$
2. $\dfrac{11}{8}=\dfrac{4\frac{1}{8}}{\textit{P}}$
3. $\dfrac{3\frac{3}{4}}{5}=\dfrac{8}{\textit{X}}$
4. $\dfrac{2\frac{2}{3}}{3}=\dfrac{\textit{A}}{16}$

Answers to Unit 1 Review

1. $2:5$
2. $3:4$
3. $1\text{ kg}:7\text{ people}$
4. $8\text{ cups berries}:3\text{ cups sugar}$
1. $2:3$
2. $1:3$
3. $1:10$
4. $1:6$
1. 4 cups water per cup of sugar
2. 120 km/hour
3. 155 people per km²
4. $0.40 per pound of apples
1. $15$
2. $32$
3. $20$
4. $19.43$
1. $13\frac{1}{2}$
2. $3$
3. $10\frac{2}{3}$
4. $14\frac{2}{9}$

TEST TIME!

Ask your instructor for the Practice Test for this unit.

Once you’ve done the Practice Test, you need to do the Unit 1 test.

Again, ask your instructor for this.

Good luck!

Unit 2: Percent

Percents are another form of fractions and are used in many everyday situations. Interest rates, credit card charges, taxes, pay deductions, increases and decreases are all calculated with percent. Percents are a convenient way to express part of the whole thing because the unwritten denominator is always 100.

the denominator of 100 is not written or said
a percent sign % follows the number and is read as “percent”
the whole thing is 100%
$100\% = 1$

Remember that cent is the Latin word meaning a hundred or a hundredth. And per is a Latin word meaning by or for each. So, percent means for each hundred.

Topic A: Introducing Percent

Reading and Writing Percents

To write a percent:

Write the number in the usual way
Place the percent sign after the numerals
- $50\%$
- $5\frac{1}{2}\%$ or $5.5\%$
- $\frac{3}{4}\%$ or $0.75\%$

To read a percent:

Read the numbers in the usual way
Say “percent” after the number
- $16\%$ : say “sixteen percent”
- $4\frac{1}{2}\%$ : say “four and one-half percent”
- $0.25\%$ : say “twenty-five hundredths percent,” or “one-quarter percent,” or “point two five percent”

Exercise 1

Write these percents using numerals and a percent sign. Note that the mixed numbers may be expressed with common fractions or decimals.

thirty-four percent $34\%$
twelve percent
four-fifths percent
one hundred sixteen and three-tenths percent
thirteen percent
six and one-fifth percent
ninety-four and one-half percent

Answers to Exercise 1

$12\%$
$0.8\%$ or $\frac{4}{5}\%$
$116.3\%$ or $116\frac{3}{10}\%$
$13\%$
$6.2\%$ or $6\frac{1}{5}\%$
$94.5\%$ or $94\frac{1}{2}\%$

Exercise 2

Write these percents using words.

$62\%$ : sixty-two percent
$37\frac{1}{2}\%$
$202\%$
$\frac{3}{4}\%$
$18.3\%$
$14\frac{1}{2}\%$
$100\frac{1}{2}\%$

Answers to Exercise 2

thirty-seven and one-half percent
two hundred two percent
three-quarters percent
eighteen and three-tenths percent
fourteen and one-half percent
one hundred and one-half percent

Changing Decimals to Percents

Writing equivalent fractions is an important math skill.

Equivalent common fractions, decimals, and percents all represent the same amount.

Equivalent fractions, decimals, and percentages
Fractions	Decimals	Percentages
$\dfrac{1}{2}$	$0.5$	$50\%$
$\dfrac{3}{10}$	$0.3$	$30\%$

You need the skill of writing equivalent fractions for working with percents.

To change any number to a percent, multiply the number by 100% and place a percent sign % after the product.

Remember this shortcut for multiplying by 100?

$4.27\times100=427$
$0.287\times100=28.7$
$53\times100=5300$

The shortcut is: When multiplying by 100, move the decimal point two places to the right.

Example A

Change these numbers to a percent.

$\begin{equation*}\begin{split} &1 \ \ \ \ \ \ \ \ &1\times100\% \ \ \ \ &= \ \ \ \ &100\% \\ &0.25 \ \ \ \ \ \ \ &0.25\times100\% \ \ \ \ &= \ \ \ \ &25\% \\ &0.8 \ \ \ \ \ \ \ &0.8\times100\% \ \ \ \ &= \ \ \ \ &80\% \\ &0.375 \ \ \ \ \ \ \ &0.375\times100\% \ \ \ \ &= \ \ \ \ &37.5\% \\ \end{split}\end{equation*}$

So …

To change a decimal to a percent, move the decimal point two places to the right and then write the percent sign after the number.

Example B

Change each decimal to a percent.

$\begin{equation*}\begin{split} &\curvearrowright \\ 0.125=0&.125=12.5\% \end{split}\end{equation*}$

$\begin{equation*}\begin{split} &\curvearrowright \\ 1.375=1&.375=137.5\% \end{split}\end{equation*}$

If the decimal point moves to the end of the number, it is not necessary to write the decimal point. Remember that zeros at the beginning of a number are also not necessary.

$\begin{equation*}\begin{split} &\curvearrowright \\ 0.24=0&.24=\cancel{0.}24\%=24\% \end{split}\end{equation*}$

$\begin{equation*}\begin{split} &\curvearrowright \\ 0.05=0&.05=\cancel{0.0}5\%=5\% \end{split}\end{equation*}$

If the decimal is a tenth (one decimal place), it will be necessary to add a zero. If you are changing a whole number to a percent, add two zeros.

$\begin{equation*}\begin{split} &\curvearrowright \\ 0.4=0&.40=40\% \end{split}\end{equation*}$

$\begin{equation*}\begin{split} &\curvearrowright \\ 1.7=1&.70=170\% \end{split}\end{equation*}$

$\begin{equation*}\begin{split} &\curvearrowright \\ 2=2&.00=200\% \end{split}\end{equation*}$

Exercise 3

Change these decimals to percents.

	Decimal	× 100% Move decimal 2 places to right	= Percent
A.	$0.75$	$\begin{equation}\begin{split} &\curvearrowright \\ 0&.75 \end{split}\end{equation}$	$=75\%$
B.	$0.33$
C.	$0.1$
D.	$0.0025$
E.	$0.9$
F.	$0.325$
G.	$0.0625$
H.	$3$

Answers to Exercise 3

$33\%$
$10\%$
$0.25\%$
$90\%$
$32.5\%$
$6.25\%$
$300\%$

Changing Percents to Decimals

Review dividing by 100:

$\begin{align*} 47.39\div100&=0.4739 \\ 429\div100&=4.29 \\ 3.824\div100&=0.03824 \end{align*}$

To divide by 100, move the decimal point two places to the left.

Example A

Change each percent to a decimal or mixed number.

$\begin{align*} 58\% &&= &&58\div 100 &&=&& .58 &&=&& 0.58 \\ 20\% &&=&& 20\div 100 &&=&& .2\phantom{0} &&=&& 0.2\phantom{0} \\ 6\% &&=&& 6\div 100 &&=&& .06 &&=&& 0.06 \\ 110\% &&=&& 110\div 100 &&=&& 1.10 &&&& \end{align*}$

So …

To change a percent to a decimal, divide by 100 (move the decimal point two places to the left) and remove the percent sign.

Example B

Change each percent to a decimal.

$\begin{align*} 75\% &&=&& 75.0\% &&=&& 0.75\phantom{0} \\ 12\% &&=&& 12.0\% &&=&& 0.12\phantom{0} \\ 37.5\%&&=&&37.5\% &&=&&0.375 \\ 125\%&&=&& 125.0\%&&=&&1.25\phantom{0} \\ 5\%&&=&&5.0\% &&=&&0.05\phantom{0} \\ 4.6\%&&=&& 4.6\%&&=&&0.046 \end{align*}$

Some notes to remember:

If there is no decimal point in the percent, place the decimal point after the last numeral and then divide by 100.
$24\%=24.0\%=0.24$
It may be necessary to prefix zeros. (This means adding zeros in front of the number, if needed).
$6\%=6.0\%=0.06$
A zero at the right of a decimal is not needed and may be left off.
$40\%=40.0\%=0.40=0.4$

Exercise 4

Change each percent to its decimal equivalent.

	Percent	÷ 100% Move decimal 2 places to left	= Decimal
A.	$23\%$	$\begin{equation}\begin{split} &\curvearrowleft \\ & \ 23. \end{split}\end{equation}$	$=0.23$
B.	$1\%$
C.	$112\%$
D.	$10.3\%$
E.	$36\%$
F.	$147\%$

Answers to Exercise 4

$0.01$
$1.12$
$0.103$
$0.36$
$1.47$

To change a percent containing a common fraction to a decimal, do this:

Change the common fraction in the percent to a decimal in the percent.
Divide by 100 (move the decimal 2 places to the left).

Example C

Let’s convert these percents containing common fractions into decimals:

$\begin{align*} 3\tfrac{1}{2}\% &= 3.5\% &3.5\% \div 100 &=\phantom{0}.035\phantom{0}=0.035\phantom{0} \\ 37\tfrac{1}{2}\%&=37.5\% &3.75\% \div 100 &=\phantom{0}.375\phantom{0}=0.375\phantom{0} \\ \tfrac{1}{4}\%&=0.25\% &0.25\% \div 100 &=\phantom{0}.0025 =0.0025 \\ 17\tfrac{1}{3}\%&=17.\overline{3}\% & 17.\overline{3}\% \div 100 &= 0.17\overline{3} \end{align*}$

Exercise 5

Change each percent to its decimal equivalent.

$8\frac{4}{5}\%= \ \ \ \ \ 8.8\%=0.088$
$4\frac{1}{2}\%=$
$56\frac{3}{4}\%=$
$1\frac{3}{5}\%=$
$112\frac{1}{2}\%=$
$2\frac{3}{8}\%=$
$5\frac{1}{4}\%=$

Answers to Exercise 5

$0.045$
$0.5675$
$0.016$
$1.125$
$0.02375$
$0.0525$

Changing Common Fractions to Percents

To change any number to a percent, multiply the number by 100% and place the percent sign % after the product.

There are two methods you can use to change a common fraction to a percent.

Method One

To change a common fraction to an equivalent percent, multiply the common fraction by 100%.

Example A

$\dfrac{3}{4} = \,?\,\%$

$\dfrac{3}{4}$

Multiply by 100%

$\dfrac{3}{4} \times 100\%$

Convert 100% to a fraction.

$\dfrac{3}{4} \times \dfrac{100}{1}\%$

Simplify the fractions by dividing the numerator and denominator by 4.

$\dfrac{3}{\cancel{4}1} \times \dfrac{\cancel{100}25}{1}\%$

The 4 cancels and 100 is reduced to 25.

$\dfrac{3}{1} \times \dfrac{25}{1}\%$

Multiply 3 by 25%.

$3 \times 25\%$

$75\%$

$1\dfrac{1}{5} = \,?\,\%$

$1\dfrac{1}{5}$

Convert number to a fraction.

$\dfrac{6}{5}$

Multiply by 100%.

$\dfrac{6}{5} \times 100\%$

Convert 100% to a fraction.

$\dfrac{6}{5} \times \dfrac{100}{1}\%$

Simplify the fractions by dividing the denominator and numerator by 5.

$\dfrac{6}{\cancel{5}1} \times \dfrac{\cancel{100}20}{1}\%$

The 5 cancels and 100 is reduced to 20.

$\dfrac{6}{1} \times \dfrac{20}{1}\%$

Multiply 6 by 20%.

$6 \times 20\%$

$120\%$

Exercise 6

Multiply by 100% to change each common fraction to an equivalent percent.

$\dfrac{4}{5} \ \ \ \ \ \ \ \ \ \ \dfrac{4}{5}\times100\%=80\%$
$\dfrac{1}{5}=$
$\dfrac{9}{10}=$
$1\dfrac{1}{2}=$
$\dfrac{7}{10}=$
$3\dfrac{3}{4}=$
$\dfrac{1}{2}=$

Answers to Exercise 6

$20\%$
$90\%$
$150\%$
$70\%$
$375\%$
$50\%$

Method Two

To change a common fraction to an equivalent percent, first write the common fraction as a decimal. Then multiply the decimal by 100% (move the decimal point two places to the right).

Example A

$\dfrac{3}{8}=\, ? \,\%$
Use long division to write the fraction as a decimal.
$\dfrac{3}{8}=0.375$
Move the decimal two spots to the right.
$\begin{equation*}\begin{split}&\curvearrowright \\ 0&.375=37.5\end{split}\end{equation*}$

Convert $\dfrac{3}{8}$ to a percentage
	$\dfrac{3}{8} = \,?\,\%$
Use long division to write fraction as a decimal.
	$\dfrac{3}{8} = 0.375$
Move the decimal point two places to the right and add the percent sign.	$0.375 = 37.5\%$

Convert $\dfrac{1}{3}$ to a percentage
	$\dfrac{1}{3} = \,?\,\%$
Use long division to write fraction as a decimal.
	$\dfrac{1}{3} = 0.33\overline{3}$
Move the decimal point two places to the right and add the percent sign.	$0.33\overline{3}=33.\overline{3}\% \text{ also written as } 33\dfrac{1}{3}\%$

Convert $\dfrac{11}{12}$ to a percentage
	$\dfrac{11}{12} = \,?\,\%$
Use long division to write fraction as a decimal.
	$\dfrac{11}{12} = 0.91\overline{6}$
Move the decimal point two places to the right and add the percent sign.	$0.916\overline{6} = 91.\overline{6}\%$

Exercise 7

$\dfrac{1}{12}=0.08\overline{3} \ \ \ \ \ \ 0.08\overline{3}\times100\%=8.\overline{3}\%$
$\dfrac{1}{8}$
$\dfrac{5}{8}$
$\dfrac{7}{8}$
$\dfrac{2}{3}$
$\dfrac{5}{16}$
$\dfrac{5}{6}$
$\dfrac{4}{9}$

Answers to Exercise 7

$12.5\%$
$62.5\%$
$87.5\%$
$66.\overline{6}\%$
$31.25\%$
$83.\overline{3}\%$
$44.\overline{4}\%$

The method you use to change a common fraction to a percent will depend on the numbers you are working with. Choose whichever method seems easier for the situation. You will also memorize many equivalencies as you work with them. But you should definitely memorize:

$\dfrac{1}{3}=33\frac{1}{3}\%$
$\dfrac{2}{3}=66\frac{2}{3}\%$

Changing Percents to Common Fractions

You know that percents are a form of fraction with an unwritten denominator of 100. A % sign is used.

To change any percent to a decimal, common fraction, or mixed number, divide by 100 and remove the percent sign.

To change a percent to a common fraction:

Write the numerals in the percent as the numerator.
Write 100 as the denominator. (Remember that the line in a fraction can be a “divided by” sign, so $58\%=\dfrac{58}{100}$ is the same as $58\div100$ .)
Remove the % sign.
Simplify the fraction: $\dfrac{58}{100}=\dfrac{29}{50}$

Example A

Write each percent as a common fraction.

$38\%=\dfrac{38}{100}\div\left(\dfrac{2}{2}\right)= \dfrac{38 \div 2}{100 \div 2} =\dfrac{19}{50}$

$25\%=\dfrac{25}{100}\div\left(\dfrac{25}{25}\right)= \dfrac{25\div 25 }{100 \div 25}=\dfrac{1}{4}$

$3\%=\dfrac{3}{100}$

Note that percents greater than or equal to 100 become improper fractions, which will be rewritten as mixed numbers.

Here is an example:

$110\%=\dfrac{110}{100}=1\dfrac{10}{100}=1\dfrac{1}{10}$

$\begin{array}{r}1\\ 100\enclose{longdiv}{110}\\\underline{100}\\ 10\end{array}$

Here is another example:

$120\%=\dfrac{120}{100}=1\dfrac{20}{100}=1\dfrac{1}{5}$

$\begin{array}{r}1\\ 100\enclose{longdiv}{120}\\ \underline{100}\\ 20\end{array}$

Remember 100% is the whole thing: $100\%=1$ .

Exercise 8

Change each percent to a common fraction. Simplify to lowest terms.

$31\%=$
$11\%=$
$2\%=$
$20\%=$
$75\%=$
$100\%=$
$750\%=$

Answers to Exercise 8

$\dfrac{31}{100}$
$\dfrac{11}{100}$
$\dfrac{1}{50}$
$\dfrac{1}{5}$
$\dfrac{3}{4}$
$1$
$7\dfrac{1}{2}$

Percents Less Than 1%

Sometimes a percent smaller than 1% is used. For example, you will hear amounts such as $\tfrac{1}{4}\%$ or $\tfrac{1}{8}\%$ or $\tfrac{1}{2}\%$ on the news about the Bank of Canada rate and the rise and fall of inflation. These are small amounts. Sometimes the expression “½ of a percentage point” is used instead of $\tfrac{1}{2}\%$ .

What is $\dfrac{1}{4}\%$ ?

$\dfrac{1}{4}\%\text { is }\dfrac{1}{4}\text{ of }1\%$

$1\%=\dfrac{1}{100}\text{, so }\dfrac{1}{4}\text{ of }1\%=\dfrac{1}{4}\times\dfrac{1}{100}=\dfrac{1}{400}$

$\dfrac{1}{4}\%=0.25\%=0.0025$

What is $\dfrac{1}{2}\%$ ?

$\dfrac{1}{2}\%=\dfrac{1}{2}\text{ of }1\%=\dfrac{1}{2}\times\dfrac{1}{100}=\dfrac{1}{200}$

$\dfrac{1}{2}\%=0.5\%=0.005$

To work with percents less than 1%, change the percent to a decimal by dividing by 100 (move decimal point two places to the left):

$0.2\%=0.002$

$0.75\%=0.0075$

If the percent is expressed as a common fraction, do this:

Write the common fraction percent as a decimal percent.
Divide by 100 (move decimal point two places left).

$\begin{array}{rcrcl} \tfrac{1}{2}\% &= &0.5\% &=& 0.005 \\ \tfrac{1}{4}\% &= &0.0025\% &=& 0.0025 \\ \tfrac{1}{8}\% &=& 0.00125\% \ & =& 0.00125 \end{array}$

Exercise 9

Change each percent to an equivalent decimal.

$\dfrac{1}{2}\%=$
$0.6\%=$
$\dfrac{3}{10}\%=$
$\dfrac{3}{5}\%=$
$0.75\%=$
$\dfrac{3}{4}\%=$
$0.5\%=$
$\dfrac{1}{4}\%=$
$0.125\%=$
$\dfrac{5}{8}\%=$

Answers to Exercise 9

$0.005$
$0.006$
$0.003$
$0.006$
$0.0075$
$0.0075$
$0.005$
$0.0025$
$0.00125$
$0.00625$

16⅔%, 33⅓%, 66⅔%, 83⅓% …

These percents will become repeating decimals. For example:

$\begin{equation*}\begin{split} 33\tfrac{1}{3}\% &= 33.\overline{3}\% &= 0.33\overline{3} \\ \\ 66\tfrac{2}{3}\% &= 66.\overline{6}\% &= 0.66\overline{6} \end{split}\end{equation*}$

It is usually more convenient to use the common fraction equivalent of these percents. Memorize them, or make a note on a special paper and post it near your workspace.

$\begin{equation*}\begin{split} 33\tfrac{1}{3}\% &= 33\tfrac{1}{3}\div100 &= \dfrac{100}{3}\times\dfrac{1}{100} \ &= \dfrac{1}{3} \\ \\ 66\tfrac{2}{3}\% &= 66\tfrac{2}{3}\div100 &= \dfrac{200}{3}\times\dfrac{1}{100} \ &= \dfrac{2}{3} \end{split}\end{equation*}$

$\begin{equation*}\begin{split} 16\tfrac{2}{3}\% &= \dfrac{1}{6} \\ \\ 33\tfrac{1}{3}\% &= \dfrac{1}{3} \\ \\ 66\tfrac{2}{3}\% &= \dfrac{2}{3} \\ \\ 83\tfrac{1}{3}\% &= \dfrac{5}{6} \end{split}\end{equation*}$

Review of Equivalent Common Fractions, Decimals, and Percents

Complete this chart. These are equivalents that you will often use, so use this chart for reference. Memorize as many equivalents as you can. You may wish to put other equivalents on the chart.

Common Fraction	Decimal	Percent
$\frac{1}{4}$
	$0.5$
		$75\%$
$\frac{1}{8}$
	$0.375$
		$62.5\%$

Answers to Review of Equivalent Common Fractions, Decimals, and Percents

Common Fraction	Decimal	Percent
$\dfrac{1}{4}$	$0.25$	$25\%$
$\dfrac{1}{2}$	$0.5$	$50\%$
$\dfrac{3}{4}$	$0.75$	$75\%$
$\dfrac{1}{8}$	$0.125$	$12.5\%$
$\dfrac{3}{8}$	$0.375$	$37.5\%$
$\dfrac{5}{8}$	$0.625$	$62.5\%$

Topic A: Self-Test

Mark /15 Aim 13/15

Write these percents in numeral form. (2 marks)
1. sixty-two and one-half percent
2. one hundred six and one-fifth percent
Write these percents in words. (2 marks)
1. $72\%$
2. $\dfrac{3}{4}\%$
Change the percents to decimals. (5 marks)
1. $32\%$
2. $18.5\%$
3. $125\%$
4. $\dfrac{4}{5}\%$
5. $\dfrac{2}{3}\%$
Change these percents to common fractions in lowest terms. (4 marks)
1. $16\%$
2. $20\%$
3. $106\%$
4. $75\%$
Change these common fractions to equivalent percents. (2 marks)
1. $\dfrac{4}{5}$
2. $\dfrac{3}{8}$

Answers to Topic A Self-Test

1. $62.5\%$ or $62\tfrac{1}{2}\%$
2. $106.2\%$ or $106\tfrac{1}{5}\%$
1. seventy-two percent
2. three-quarters percent
1. $0.32$
2. $0.185$
3. $1.25$
4. $0.008$
5. $0.00\overline{6}$
1. $\dfrac{4}{25}$
2. $\dfrac{1}{5}$
3. $1\dfrac{3}{50}$
4. $\dfrac{3}{4}$
1. $80\%$
2. $37.5\%$

Unit 2 Review: Percent

Questions

Change these decimals to percents.
1. $0.75$
2. $0.156$
3. $0.03$
4. $0.0035$
5. $0.625$
6. $0.048$
7. $3.45$
Change each percent to a decimal.
1. $59\%$
2. $39.5\%$
3. $1\%$
4. $3\tfrac{1}{2}\%$
5. $1\tfrac{4}{5}\%$
6. $4\tfrac{3}{5}\%$
Change each fraction to an equivalent percent.
1. $1\tfrac{3}{4}$
2. $\tfrac{1}{5}$
3. $4\tfrac{3}{20}$
4. $\tfrac{9}{20}$
5. $2\tfrac{1}{3}$
Change each percent to a common fraction. Simplify your answer.
1. $16\%$
2. $25\%$
3. $110\%$
4. $95\%$
5. $650\%$
6. $284\%$
7. $\tfrac{1}{4}\%$
8. $\tfrac{1}{8}\%$
9. $0.125\%$
10. $\tfrac{5}{8}\%$
11. $\tfrac{3}{4}\%$
Change each percent to a common fraction. These few should be memorized.
1. $33\tfrac{1}{3}\%$
2. $66\tfrac{2}{3}\%$
3. $16\tfrac{2}{3}\%$
4. $83\tfrac{1}{3}\%$

Answers to Unit 2 Review

1. $75\%$
2. $15.6\%$
3. $3\%$
4. $0.35\%$
5. $62.5\%$
6. $4.8\%$
7. $345\%$
1. $0.59$
2. $0.395$
3. $0.01$
4. $0.035$
5. $0.018$
6. $0.046$
1. $175\%$
2. $20\%$
3. $415\%$
4. $45\%$
5. $233.3\%$
1. $\dfrac{4}{25}$
2. $\dfrac{1}{4}$
3. $1\dfrac{1}{10}$
4. $\dfrac{19}{20}$
5. $6\dfrac{1}{2}$
6. $2\dfrac{21}{25}$
7. $\dfrac{1}{400}$
8. $\dfrac{1}{800}$
9. $\dfrac{0.125}{100}=\dfrac{1}{800}$
10. $\dfrac{5}{800}=\dfrac{1}{160}$
11. $\dfrac{3}{400}$
1. $\dfrac{1}{3}$
2. $\dfrac{2}{3}$
3. $\dfrac{1}{6}$
4. $\dfrac{5}{6}$

TEST TIME!

Ask your instructor for the Practice Test for this unit.

Once you’ve done the Practice Test, you need to do the Unit 2 test.

Again, ask your instructor for this.

Good luck!

III

Unit 3: Working with Percent

Topics

In this unit, you will learn to solve three types of percent problems:

Finding a given percent of a number (finding the part).
Finding what percent one number is of another number (finding the %).
Finding a number when a percent of it is given (finding the whole).

Each type of percent problem can be solved using the following proportion:

$\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$

Both ratios in this proportion use the same order of comparison because in the ratio $\frac{\%}{100}$ , the % represents a part and 100 is the whole. That is, the % is a part of the whole.

Percent problems involve knowing three pieces of information:

the part (the “is” part)
the whole (the “of” part)
the percent

You will be given two pieces of information and you will find the third. That is, the problems will give two terms of the proportion, and you will solve for the missing term. Because these are problems of percent, the 100 is always known to you and will always be in the same position in the proportion.

Remember how to use cross multiplication to solve a proportion:

$\frac{\text{N}}{4}=\frac{6}{8}\longrightarrow4\times6=8\times\text{N}\longrightarrow24=8\text{N}\longrightarrow \frac{24}{8} = \frac{8N}{8}\longrightarrow\frac{24}{8}=\frac{\cancel{8}\text{N}}{\cancel{8}}\longrightarrow3=\text{N}$

Topic A: Finding a Percent of a Number

$\dfrac{\text{is (part)}}{\text{of (whole)}} = \dfrac{\%}{100}$

In problems in which you find a percent of a number, the missing term is the part. You will be given the %, which is always 100.

Example A

What is 25% of 40?

Part (is) = N
Whole (of) = 40
% = 25

$\dfrac{\text{is}}{\text{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{N}}{40}=\dfrac{25}{100}$

Solve the proportion.

Simplify if possible:
$\dfrac{\textit{N}}{40} = \dfrac{25}{100}$
$\dfrac{\textit{N}}{40} = \dfrac{\cancel{25} 1}{\cancel{100} 4}$
$\dfrac{\textit{N}}{40} =\dfrac{1}{4}$
Cross-multiply:
$\begin{equation*}\begin{split} \textit{N}\times4 &= 40\times1 \\ 4\textit{N} &= 40 \end{split}\end{equation*}$
Divide:
$\textit{N} = 40\div4=10$
$25\%\text{ of }40 = 10$

Example B

What is $20\%\text{ of }18$ ?

Part (is) = N
Whole (of) = 18
% = 20

$\dfrac{\text{is}}{\text{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{N}}{18}=\dfrac{20}{100}$

Solve the proportion:

$\begin{equation*}\begin{split} \dfrac{N}{18} &= \dfrac{20}{100} \\ \\ \dfrac{N}{18} &= \dfrac{\cancel{20} 1}{\cancel{100} 5} \\ \\ \dfrac{N}{18} &= \dfrac{1}{5} \\ \\ 5N &= 18 \\ \\ N &= 18\div5=3\tfrac{3}{5} \end{split}\end{equation*}$

$20\%\text{ of }18 = 3\tfrac{3}{5}$

The following examples all ask you to find a percent of a number. The missing term is the part (the “is” part). Look at the examples carefully so you’ll recognize the wording.

What is 14% of 60?
- $\dfrac{\text{is}}{\text{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{N}}{60}=\dfrac{14}{100}$
Find 10% of 27.
- $\dfrac{\text{is}}{\text{of}}=\dfrac{\%}{100}\longrightarrow\dfrac{\textit{P}}{27}=\dfrac{10}{100}$
5% of 15 is fill-in-the-blank.
- $\dfrac{\%}{100}=\dfrac{\text{is}}{\text{of}}\longrightarrow\dfrac{5}{100}=\dfrac{\textit{X}}{15}$
75% of 12 = fill-in-the-blank.
- $\dfrac{\%}{100}=\dfrac{\text{is}}{\text{of}}\longrightarrow\dfrac{75}{100}=\dfrac{\textit{K}}{12}$

In percent problems, the number after the word of usually represents the whole.

Exercise 1

Solve each problem by setting up the proportion $\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$ .

$20\%\text{ of }18 =$
$19\%\text{ of }200 =$
$25\%\text{ of }44 =$
$6\%\text{ of }110 =$
$3\%\text{ of }33 =$
$30\%\text{ of }64 =$
$50\%\text{ of }60$ is?
$30\%\text{ of }40$ is?
What is $72\%\text{ of }$425$ ?
What is $20\%\text{ of }85$ ?

Answers to Exercise 1

$3\tfrac{3}{5}$ or $3.6$
$38$
$11$
$6\tfrac{3}{5}$ or $6.6$
$\tfrac{99}{100}$ or $0.99$
$19\tfrac{1}{5}$ or $19.2$
$30$
$12$
$$306$
$17$

Percents Greater Than or Equal to 100%

Remember that $100\% = 1$ .

100% of anything is the whole thing. If you spend 100% of your paycheque, you spend the whole thing. If you get 100% on a test, you have the whole thing correct.

If you have more than 100%, you have more than the whole thing. If you spend 110% of your paycheque, you spent more than you earned, and you may be in trouble! It is hard to get more than 100% on a test unless the instructor has given bonus marks for extra questions. You may hear of percents more than 100% in increases, such as costs of housing or inflation. For example, “The Browns just sold their house and made a 200% profit.” This means they got back what they paid and two times more!

If a percent is less than (<) 100, it is less than the whole thing.

$90\%\text{ of }50 = 45$

If a percent is 100, it equals the whole thing.

$100\%\text{ of }50 = 50$

If a percent is more than (>) 100, it is more than the whole thing.

$120\%\text{ of }50 = 60$

Exercise 2

Look at the percent. Is it 100? Circle the correct answer for each question. Do not solve the problems.

is:
1. equal to 10
2. less than 10
3. greater than 10
is:
1. equal to 0.25
2. less than 0.25
3. greater than 0.25
is:
1. equal to 75
2. less than 75
3. greater than 75
is:
1. equal to 15
2. less than 15
3. greater than 15
is:
1. equal to 100
2. less than 100
3. greater than 100
is:
1. equal to 936
2. less than 936
3. greater than 936

Answers to Exercise 2

iii) greater than 10
ii) less than 0.25
ii) less than 75
ii) less than 15
i) equal to 100
iii) greater than 936

Exercise 3

Use the proportion method from Unit 1, Topic C: Proportion to solve these questions.

$16\tfrac{2}{3}\%\text{ of }12 =$
What is $60\%\text{ of }15$ ?
$75\%\text{ of }144$ is?
$30\%\text{ of }90 =$
What is $37\tfrac{1}{2}\%\text{ of }80$ ?
$25\%\text{ of }52$ is?
Find $8.2\%\text{ of }300$ .
$260\%\text{ of }45$ is?
What is $109\%\text{ of }200$ ?
$98.75\%\text{ of }50 =$

Answers to Exercise 3

$2$
$9$
$108$
$27$
$30$
$13$
$24.6$
$117$
$218$
$49.375$

Taxes

The amount of tax to be paid is calculated by finding a percent of a number. The tax rate is usually given as a percent. The basic proportion for these problems is:

$\dfrac{\text{tax (part)}}{\text{taxable amount (whole)}}=\dfrac{\% \text{ tax}}{100}$

The British Columbia Harmonized Sales Tax (HST) is 12%. In B.C., the provincial portion of the harmonized sales tax does not have to be paid on children’s clothes, food, books, gasoline and diesel fuel, and other special items.

Example A

How much HST (12%) will be charged on a new kitchen table that cost $125? Use proportion:

$\begin{array}{rcccl} \dfrac{\text{HST}}{$125} &=& \dfrac{12}{100}&& \\ \\ 100\text{ HST} &=& 12\times125 &=& $1,500 \\ \\ \text{HST} &=&$1,500\div100 &= &$15.00 \end{array}$

HST on a $125 table is $15.00.

Exercise 4

Find the total cost of each item. All are to be taxed with HST.

	Purchase Price	HST 12%	Total Cost
A	Clothes: $130
B	Washing Machine: $589
C	New Car: $10,000
D	Shoes: $59.99

Answers to Exercise 4

	Purchase Price	HST 12%	Total Cost
A	Clothes: $130	$15.60	$145.60
B	Washing Machine: $589	$70.68	$659.68
C	New Car: $10,000	$1,200	$11,200
D	Shoes: $59.99	$7.20	$67.19

Income tax is charged at different percentages according to the amount of a person’s taxable income. Note that different tax rules and charges than the ones mentioned in this book may apply in real situations.

Example B

Say that the first $28,000 of taxable income is taxed at 17%. If a person’s taxable income for the year is $23,400, what amount of income tax will that person pay?

The tax is the part. The income is the whole.

To use the proportion method, do this:

$\begin{align*} \dfrac{\text{tax}}{\text{income}}\longrightarrow\dfrac{T}{$23,400}&=\dfrac{17}{100}\\ \\ \dfrac{T}{$23,400}&\rlap{\nearrow}{\searrow}\dfrac{17}{100} \\ \\ 100T&=$397,800 \\ \\ \cancel{100}T&=$397,8\cancel{00} \end{align*}$

The income tax on $23,400 is $3,978.

Exercise 5

Calculate the income tax for the annual taxable earnings listed. These amounts are all under $28,000, so the tax rate is 17%.

$18,500
$27,620
$15,365
$25,900

Answers to Exercise 5

$3,145
$4,695.40
$2,612.05
$4,403

Cross-Border Shopping

The Canadian (CAN) and American (US) dollars are not equal in value. The exchange rate (the value of one Canadian dollar compared to a dollar from another country) changes often; the current rate is usually available from banks, on the news, in the newspapers and on a web site. In the winter of 2010, the Canadian dollar was around $0.92 of an American dollar (ratio is $$1.00\text{ CAN}:$0.92\text{ US}$ ), so CAN money was valued at 92% of US money.

To find the value of one US dollar in Canadian funds, use this proportion:

$\dfrac{$1\text{ CAN}}{$0.92\text{ US}}=\dfrac{\textit{N}\text{ CAN}}{$1\text{ US}}$

$N=$1\div$0.92=$1.086$ , so US money was valued at 109% of CAN money.

Therefore, $\dfrac{$1.086\text{ CAN}}{\$1\text{ US}}=\dfrac{\$109\text{ CAN}}{\$100\text{ US}}$

Note that the proportion changes as the exchange rate changes.

What if you buy in the United States?

Change the US cost to the Canadian equivalent (multiply by 109%).
If you have more than the purchases allowed (contact the Canada Border Services Agency for information), Canadian customs will charge duty on the Canadian value of your purchases. The percent of the duty (the rate) varies according to what the item is, where it was made, and the duty rates of the day. For example, at the time of writing, the duty on poultry is 12.5%, on non-U.S. cotton is 25%, and on liquor is 110%!
HST (12%) is charged on the duty and on the Canadian value of the purchases (which includes any US sales taxes).

Look at this example (assume $1.00 CAN = $0.92 US).

$\begin{equation*}\begin{split} \text{Men's leather shoes, US price} &\hspace{2cm} &$64.80 \\ \\ \text{US sales tax }6\% &\hspace{2cm} &3.89 \\ \text{US total cost} &\hspace{2cm} &68.69 \\ \\ \text{Equivalent cost in CAN funds (US cost }\times 109\%\text{)} \\ \\ \dfrac{\textit{N}}{68.69}=\dfrac{109}{100} &\hspace{2cm} &$74.87 \\ \\ \text{Duty on leather shoes is }22.8\% &\hspace{2cm} &$17.07 \\ \\ \text{HST (}12\%\text{) on CAN value plus duty} \\ \\ 12\%\text{ of }$74.19\text{ and }16.92\text{ together} &\hspace{2cm} &$11.03 \end{split}\end{equation*}$

The total cost of a pair of leather shoes priced at $64.80 in the United States will be the American price in Canadian funds + duty + HST = $102.97 CAN.

Exercise 6

For each item, do the calculations using the duty and tax rates given. Assume $1.00 US is $1.09 CAN.

Groceries, US price $75.
1. 3% US sales tax
2. US total cost
3. Equivalent value in CAN funds
4. Duty at 10.2% (No HST on food)
5. Total cost in Canadian funds

Answers to Exercise 6

1. $2.25
2. $77.25
3. $84.20
4. $8.59
5. $92.79

Increases and Decreases, Discounts and Markups

Increases (amount changing to more) and decreases (amount changing to less) are often given as a percent. For example,

The Insurance Corporation of B.C. increased car insurance rates by 3.3% in March 2007.
The number of acute care beds at the local hospital has decreased by 28% in the last year.
The new work contract provides a 4% wage increase in the first year and a 2½% increase in the second year.

The amount of an increase or decrease is calculated by finding a percent of a number. When the percent of an increase or decrease is given, the proportion is:

$\dfrac{\text{amount of increase or decrease}}{\text{whole amount}}=\dfrac{\text{increase or decrease }\%}{100}$

Discounts are a form of decrease. The discount is the amount taken off a price; it is the price reduction.

Sale prices (discounted prices) may be advertised as:

“All items 20% off.”
“Everything in stock reduced by 25% to 50%.”
“33⅓% savings!”
“2% discount for cash.”

Decrease and discount problems may need to be solved in several steps. Sometimes the problems ask for:

the amount of the decrease (may be called the “saving”) (1 step).
the amount left after the decrease (2 steps).

Example A

The sign says, “All winter coats 40% off.” How much money will you save on a coat originally priced at $128.99? What is 40% of $128.99?

One-step problem

$\begin{align*} \dfrac{\text{decrease}}{\text{original price}}\longrightarrow\dfrac{D}{\$128.99}&=\dfrac{40}{100}\\ \\ 100D&=40\times\$128.99 \\ \\ D&=\$5,519.60\div 100=\$51.596 \\ \\ \text{round to nearest cent}&=\$51.60 \end{align*}$

You will save $51.60.

Example B

The couch and chair are advertised in a 33⅓% price reduction sale. How much will you pay for a couch and chair originally priced at $798?

Two-step problem

First: Find the amount of savings (the decrease).

Recall that it’s best to substitute ⅓ for 33⅓%.

$\dfrac{\text{saving}}{\text{full cost}}\longrightarrow\dfrac{\textit{S}}{798}=\dfrac{1}{3}$

Second: Subtract the savings from the original amount.

$\text{original price} - \text{savings} = \text{sale price}$

Find the amount of savings.

$\begin{equation*}\begin{split} \dfrac{\text{S}}{798} &= \dfrac{1}{3} \\ \\ 798\times\dfrac{1}{3} &= \text{S} \\ \\ \dfrac{798}{3} &= \text{S} \\ \\ 266 &= \text{S} \end{split}\end{equation*}$

Find the sale price.

$\overset{\text{original price}}{\$798}-\overset{\text{savings (decrease)}}{\$266}=\overset{\text{sale price}}{\$532}$

The couch and chair will cost $532 on sale (plus sale tax, of course, but you do not have to calculate tax for this problem).

Exercise 7

Solve these problems. Round all answers to the nearest cent.

The employees agreed to take a 5% pay cut (reduction) so no one will be laid off. If the pay rate was $15.50 per hour, how much less per hour will the workers earn?
“All shoes 25% off,” says the sign. What will be the sale price of a pair of dress shoes originally priced at $69.98?
The workforce at the factory has to be reduced by 16⅔% over the next two years. Early retirements, attrition (not replacing people who leave) and some layoffs will be used. The workforce is 3,000 people right now. What is the planned size of the workforce in two years?

Answers to Exercise 7

$0.78
$52.48
2,500 people

Increases and markups are calculated in the same way as decreases and discounts. However, an increase or markup is added to the original amount.

Example C

The auto insurance rate increased 19%. The basic insurance rate for Don’s car was $550 before the increase. What is the basic insurance after the increase?

Calculate the amount of the increase. Find 19% of $550.
$\begin{align*} \dfrac{\text{amount of increase}}{\text{present cost}}\longrightarrow\dfrac{N}{550}&=\dfrac{19}{100}\\ \\ 100N&=550\times19 \\ \\ N&=104.5 \end{align*}$
Add the amount of the increase to the original amount.
$\overset{\text{original amount}}{\$550}+\overset{\text{increase}}{\$104.50}=\overset{\text{new insurance cost}}{\$654.50}$

Don’s new basic insurance is $654.50.

Markups are the amount added to the cost price before an item is resold. Many factors must be considered when businesses decide on the percent of the markup:

all costs of operating a business
the profit wanted
the community the business is in
the competition the business has

For example, the markup on leather shoes may be 45%, but on running shoes it may be 60%. Kitchen appliances might have a 42% markup, while lawn mowers might have a 55% markup.

Example D

A shoe seller pays $40.00 per pair of running shoes from the factory. The shoe seller makes the markup 75%. What is the selling price of the shoes? What is 75% of $40.00?

$\begin{align*} \dfrac{\text{markup cost}}{\text{original cost}}\longrightarrow\dfrac{N}{40}&=\dfrac{75}{100}\\ \\ 100N&=3,000 \\ \\ N&=30 \end{align*}$

75% of $40.00 is $30.00.

Add the markup to the original cost to get the selling price of the shoes:

$40 + $30 = $70.00

Having a wage increase at work is always a good thing! Often, the raise will be given as a percentage. That means that everyone will see more money on their paycheque, but they will each have a different amount because they all get paid a different amount to start with.

Example E

The boss at A-1 House Painting will give a 1.5% wage increase to the 10 employees. Three staff — Employee Group A — are paid the minimum wage of $13.85 an hour, and the other 7 staff — Employee Group B — are paid $21.00 an hour.

Calculate the wage increase for Employee Group A. What is 1.5% of $13.85?

$\begin{align*} \dfrac{\text{increase}}{\text{present wage}}\longrightarrow\dfrac{I}{\$13.85}&=\dfrac{1.5}{100} \\ \\ 100I&=\$13.85\times 1.5 \\ \\ I&= \$0.21 \end{align*}$

The new wage will be the old wage plus the increase:

$13.85 + $0.21 = $14.06 per hour for Employee Group A

Calculate the wage increase for Employee Group B. What is 1.5% of $21.00?

$\begin{align*} \dfrac{\text{increase}}{\text{present wage}}\longrightarrow\dfrac{I}{\$21}&=\dfrac{1.5}{100} \\ \\ 100I&=\$21\times 1.5 \\ \\ I&= \$0.32 \end{align*}$

The new wage will be the old wage plus the increase:

$21.00 + $0.32 = $21.32 per hour for Employee Group B

Exercise 8

Solve the problems. Round money to the nearest cent.

If the markup on the craft supplies was set at 75%, calculate the selling price for these items and complete the chart.

Cost Price to Business Markup (75%) Selling Price

Silk Flowers: $1.48 $\dfrac{\text{N}}{1.48}=\dfrac{75}{100}=$1.11$ $$1.48+$1.11=$2.59$

Stuffing: $4.50/bag

Beads: $3.20/dozen
The population of the town has increased 30% since the pulp mill was built. The population before the pulp mill was 8,436 people. What is the population now?
(Round to the nearest person.)
The wage contract gave the workers a 4% increase in the first year and a 2½% increase in the second year. If the hourly rate of pay was $12.45 before the new contract, calculate the following:
1. The amount of the increase per hour in the first year.
2. The hourly pay rate in the first year. (old pay + increase = first year rate)
3. The amount of pay increase in the second year. (Note—use the new hourly pay rate from the first year to calculate the increase for the second year.)
4. The hourly pay rate in the second year. (first year rate + increase = second year rate)

Cost Price to Business	Markup (75%)	Selling Price
Silk Flowers: $1.48	$\dfrac{\text{N}}{1.48}=\dfrac{75}{100}=$1.11$	$$1.48+$1.11=$2.59$
Stuffing: $4.50/bag
Beads: $3.20/dozen

Answers to Exercise 8

Cost Price to Business Markup (75%) Selling Price

Silk Flowers: $1.48 $1.11 $2.59

Stuffing: $4.50/bag $3.38 $7.88

Beads: $3.20/dozen $2.40 $5.60
10,967 people
1. $0.50
2. $12.95
3. $0.32
4. $13.27

Cost Price to Business	Markup (75%)	Selling Price
Silk Flowers: $1.48	$1.11	$2.59
Stuffing: $4.50/bag	$3.38	$7.88
Beads: $3.20/dozen	$2.40	$5.60

Commission and Tips

Salespeople may receive a commission as part or all of their pay. The business owner pays the salesperson an agreed-upon percent of the selling price of the product.

Real estate agents are paid by commission on their sales.
Car and truck salespeople may be paid a small salary per month, but their main income is the commission on the vehicles they sell.

Tips are appreciation payments for service. The customer gives tips directly to the worker. Taxi drivers, waiters, bellhops and chambermaids in hotels often receive a minimal hourly wage. A large part of their earnings is from tips. In restaurants, expect to leave at least a 15% tip for adequate service.

To calculate the amount of a commission (or a tip), find the percentage of the total amount using the proportion:

$\dfrac{\text{commission }\square}{\text{total amount}}=\dfrac{\text{commission }\%}{100}$

The commission is the part. The total amount is the whole.

Commission problems often have several steps. You may have to:

add together several items to find the total of sales
subtract a base amount for which salespeople do not receive a commission
add the amount of commission to the basic wage to find out how much the person earned

Example A

The bill for an excellent dinner at a restaurant was $56.40. The service was very good, and the waiter was very pleasant, so Bob and Diane wanted to leave at least a 15% tip.

To calculate the tip, find 15% of $56.40. The tip is the part. The bill is the whole.

$\begin{align*} \dfrac{\text{tip}}{\text{bill}}\longrightarrow \dfrac{N}{\$56.40}&=\dfrac{15}{100} \\ \\ 100N&=\$56.40\times 15 \\ \\ N&= \$8.46 \end{align*}$

Bob will round this amount to $8.50. How much will he pay for the meal and the tip?

$\overset{\text{cost of dinner}}{\$56.40}+\overset{\text{tip}}{\$8.50}=\overset{\text{total paid}}{\$64.90}$

Bob will pay $64.90 total for dinner.

In a real situation, we would probably round the amount of the bill to the nearest dollar and then calculate the tip.

Example B

The salespeople at XW Ford receive a monthly salary of $1,000. They also receive a 12% commission on any sales over $35,000 in a month. This means they are expected to sell $35,000 worth of vehicles every month to earn the $1,000 salary.

If a saleswoman made $54,000 in sales one month, what would her gross earnings be?

You are asked to find the gross monthly earnings. What do you know?

The saleswoman earns $1,000 per month.
She earns 12% commission on sales over $35,000.
She was responsible for $54,000 in sales.

Find the amount of commissionable sales. Subtract the base amount for which she will not earn a commission from her total sales:

$\overset{\text{total sales}}{\$54,000}-\overset{\text{base amount}}{\$35,000}=\overset{\text{commissionable sales}}{\$19,000}$

Calculate the commission. What is 12% of $19,000?

$\begin{align*} \dfrac{\text{commission}}{\text{commissionable sales}}\longrightarrow \dfrac{X}{\$19,000}&=\dfrac{12}{100} \\ \\ 100X&=\$19,000\times 12 \\ \\ 100X &= \$228,000 \\ \\ X&= \$2,280 \end{align*}$

Add the salary and commission to find gross earnings:

$1,000 + $ 2,280 = $3,280

The saleswoman earned $3,280.

Exercise 9

A clerk sold $18,000 worth of clothes last year. He was paid a 15% commission. How much was his commission?
Mr. Green receives a weekly salary of $325 plus a commission of 10% on all sales he makes over $1,500. Last week Mr. Green sold $3,500 worth of merchandise. How much money did he earn last week?
The final bill at the restaurant is $160 and you want to leave a 15% tip.
1. What amount of tip should you leave?
2. What is the total cost (bill and tip)?

Answers to Exercise 9

$2,700
$525
1. $24.00
2. $184.00

More Problems for Finding a Percent of a Number

To successfully pass the course, the student must get at least 80% on the test. The test is out of 125. What mark will give the student 80%?
George made a 12½% down payment on a new car which cost $3,200. How much was the down payment?
In B.C., employers are required to pay 6% holiday pay to all employees. Holiday pay is added to regular salary if a paid vacation is not taken. The young grocery clerk who earns $8.00 an hour worked 25 hours last week.
1. What amount of holiday pay is he eligible for?
2. His employer pays holiday pay on each cheque to part-time employees. What are his total earnings for the week? (salary + holiday pay)
Nutrition experts recommend that no more than 30% of the food calories a person consumes should be from fats. Foods such as fatty meat, dairy products with lots of butter fat, cooking oils, margarine, some salad dressings, and nuts contain a high percentage of fat. If a person’s daily intake of calories is 2,560, what is the most number of calories that should be from fat?

Answers to More Problems for Finding a Percent of a Number

100 marks
$400
Answers:
1. $12.00
2. $212.00
768 calories

Topic A: Self-Test

Mark /11 Aim 9/11

Answer the following. (4 marks)
1. $6\%\text{ of }30 =$
2. $\tfrac{1}{4}\%\text{ of }48$ is?
3. Find $131\%\text{ of }400$ .
4. Write the general proportion that can be used to solve percent problems.
Solve the problems. (2 marks each, except iii) (7 marks)
1. The $125,000 house is insured for 85% of its value against fire damage. How much money should the owner receive if the house is destroyed by fire?
2. Charlotte sells leisure clothes in her small town for a large national company. She receives $500 a month, and a 20% commission on all monthly sales over $1 000. What are her monthly earnings if her total sales are $2,300?
3. The barbeque was originally priced at $599 but Jack bought it during a 35% off, end-of-season sale.
  1. What was the sale price? (1 mark)
  2. Calculate the harmonized sales tax (12%). (1 mark)
  3. Give the total cost of Jack’s barbeque. (1 mark)

Answers to Topic A Self-Test

Answers:
1. $1.8$
2. $0.12$
3. $524$
4. $\dfrac{\text{part}}{\text{whole}}=\dfrac{\%}{100}$ , or $\dfrac{\text{is}}{\text{of}}=\dfrac{\%}{100}$
Answers:
1. $106,250
2. $760.00
3. Answers:
  1. $389.35
  2. $46.72
  3. $436.07

Please note that the tax rates used in the questions in this book are for the year 2010 and are subject to change.

As of January 2026, Canada, the United States, and Mexico have a free trade agreement that eliminates duty on many goods sold by one country and imported to another.

Unit 3 Review: Working with Percent

Questions

Solve each problem by setting up a proportion.
1. $150\%\text{ of }7$
2. $12.5\%\text{ of }48$
3. $20\%\text{ of }16$
4. $18\%\text{ of }40$
5. $\tfrac{1}{4}\%\text{ of }20$
6. $0.5\%\text{ of }122$
Use the proportion method to solve these questions. Round to 2 decimal places, if needed.
1. $75\%\text{ of }12$
2. $3\tfrac{1}{4}\%\text{ of }200$ is?
3. What is $16\%\text{ of }34$ ?
4. $90\%\text{ of }75$ is
5. Find $10\%\text{ of }27$
6. $16\tfrac{2}{3}\%\text{ of }163$
Find the total cost of each item, using 12% HST.

Item Cost HST (12%) Total Cost

A T-shirt $14.99
$\begin{align*} \dfrac{X}{\$14.99}&=\dfrac{12}{100} \\ \\ X&= \$1.80 \end{align*}$
$$1.80 + $14.99 = $16.79$

B 40″ flat screen TV $699.00

C tent $168.79

D drill and bit set $248.99

E shoes $79.98
Solve these percent problems.
1. All pants are 25% off said the sign in the window. What will be the sale price of a pair of pants originally priced at $59.78?
2. 30% of the forest will be cut down in the next three years. The forest is 150 acres large. How many acres will be left after the cut?
3. All winter clothes are on sale for 15% off. A set of gloves cost $18.00. How much will you save?
4. Rent will increase 10% on Oliver’s apartment on January 1. His rent was $460.00 per month. What will his new rent be?
5. The French Bread Bakery staff will get a one-time 15% pay increase on their Christmas pay check. If Manon works 63 hours at $12.78/hour for that pay check, how much will she get paid? (Figure out the regular pay first and then add on the 15% increase.)
6. The bill at the restaurant was $38.00. Add a 20% tip. What did the customer pay in total?
7. The mark up on a pair of fancy running shoes is 45%. The shoes cost the buyer $38.00. How much will they be sold for after mark-up?

	Item	Cost	HST (12%)	Total Cost
A	T-shirt	$14.99	$\begin{align} \dfrac{X}{\$14.99}&=\dfrac{12}{100} \\ \\ X&= \$1.80 \end{align}$	$$1.80 + $14.99 = $16.79$
B	40″ flat screen TV	$699.00
C	tent	$168.79
D	drill and bit set	$248.99
E	shoes	$79.98

Answers to Unit 3 Review

1. $10.5$
2. $6$
3. $3.2$
4. $7.2$
5. $0.05$
6. $0.61$
1. $9$
2. $6.5$
3. $5.44$
4. $67.5$
5. $2.7$
6. $27.17$

	Item	Cost	HST (12%)	Total Cost
A	T-shirt	$14.99	$1.80	$16.79
B	40″ flat screen TV	$699.00	$83.88	$782.88
C	tent	$168.79	$20.25	$189.04
D	drill and bit set	$248.99	$29.88	$278.87
E	shoes	$79.98	$9.60	$89.58

1. $44.83
2. 105 acres
3. $2.70
4. $506.00
5. $925.91
6. $45.60
7. $55.10

TEST TIME!

Ask your instructor for the Practice Test for this unit.

Once you’ve done the Practice Test, you need to do the Unit 3 test.

Again, ask your instructor for this.

Good luck!

Unit 4: More Working with Percent

Topic A: Finding What Percent One Number is of Another

$\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$

In problems where you must find what percent one number is of another, the missing term is the percent. You will be told the part (is) and the whole (of), you know the 100, and you solve for the missing percent.

Example A

4 is what percent of 5?

4 is the part (is)
5 is the whole (of)
% is unknown (call it P)

Write the proportion:
$\dfrac{4}{5} = \dfrac{\textit{P}}{100}$
Cross-multiply to solve:
$\begin{equation*}\begin{split} 4\times100 &= 5\times\textit{P} \\ \\ 400 &= 5\textit{P} \\ \\ 400\div5 &= \textit{P} \\ \\ 80\% &= \textit{P} \end{split}\end{equation*}$

4 is 80% of 5. (Be sure to write the percent sign: %.)

In percent problems, the number after “of” usually is the whole. The number close to “is” usually is the part. You may find it helpful to think “is over of.”

Like this: $\dfrac{\text{is}}{\text{of}}$ .

An equal sign (=) can substitute for “is.”

Consider: 12 is what percent of 15?

$\dfrac{\text{is}}{\text{of}}$ will help to find $\dfrac{\text{part}}{\text{whole}}$ .

$\dfrac{12\text{ (is)}}{15\text{ (of)}}=\dfrac{\textit{N}}{100}$

Example B

What percent of 85 is 60? Round your answer to the nearest tenth.

60 = the part (close to “is”)
85 = the whole (after “of”)
% = the unknown

Set up the proportion:
$\dfrac{\text{is}}{\text{of}}$
$\dfrac{60}{85} = \dfrac{\textit{P}}{100}$
Simplify:
$\dfrac{\cancel{60}12}{\cancel{85}17} = \dfrac{\textit{P}}{100}$
$\dfrac{12}{17} = \dfrac{\textit{P}}{100}$
Cross-multiply to solve:
$12\times100 = 1,200$ , and $17\times\textit{P}=17\textit{P}$
$\begin{equation*}\begin{split} 1,200 &= 17\textit{P} \\ \\ 1,200\div17 &= \textit{P} \\ \\ 70.588\% &= \textit{P} \end{split}\end{equation*}$
Round to 70.6%.

Exercise 1

The following examples ask you to find what percent one number is of another. The missing term is the percent. Look carefully at the wording and decide which number is the part (close to “is”) and which number is the whole thing (after “of”). Write the proportion but do not solve the problem.

3 is what % of 6?
12 is blank% of 5?
What % of 27 is 9?
What % of ½ is ¼?
blank% of 50 is 25?
blank% of 64 = 48

Answers to Exercise 1

$\dfrac{3}{6}=\dfrac{\textit{N}}{100}$
$\dfrac{\textit{F}}{100}=\dfrac{12}{5}$
$\dfrac{9}{27}=\dfrac{\textit{X}}{100}$
$\dfrac{\tfrac{1}{4}}{\tfrac{1}{2}}=\dfrac{\textit{P}}{100}$
$\dfrac{\textit{X}}{100}=\dfrac{25}{50}$
$\dfrac{48}{64}=\dfrac{\textit{N}}{100}$

Exercise 2

Solve each question by first setting up the proportion $\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$ . Review Unit 1, Topic C: Proportion as necessary.

3 is what percent of 60?
3 is what percent of 4?
1 is what percent of 3?
What % of 50 is 35?
What percent of 350 is 42?
15 is blank% of 12.
14 is blank% of 700.
What percent of 96 is 12?
2 is blank% of 125.
46 is blank% of 40.

Answers to Exercise 2

$5\%$
$75\%$
$33.\overline{3}\%$
$70\%$
$12\%$
$125\%$
$2\%$
$12.5\%$
$1.6\%$
$115\%$

Exercise 3

Solve the following by setting up the proportion.

16 is blank% of 64.
17 is blank% of 85.
What % of 52 is 13?
What percent of 125 is 75?
1 is blank% of 200.
36 = blank% of 12.
blank% of 72 = 27.
blank% of 48 = 18.
125 = blank% of 75.
What % of 18 is 24?

Answers to Exercise 3

$25\%$
$20\%$
$25\%$
$60\%$
$0.5\%$
$300\%$
$37.5\%$
$37.5\%$
$166.\overline{6}$ or $166\tfrac{2}{3}\%$
$133.\overline{3}$ or $133\tfrac{1}{3}\%$

Finding the Percent of an Increase or Decrease

You learned in Unit 3, Topic A: Finding a Percent of a Number to find the amount of an increase (gain) or decrease (loss) when given the percent of the increase or decrease.

Now you are going to find the percent of the increase or decrease when you are given the amounts. This is called the rate of the increase or decrease.

Problems that ask you to find the percent of increase or decrease often involve two steps:

Find the amount of change (either increase or decrease) by finding the difference between the two amounts given. Subtract to find the difference.
Find the percentage of increase/decrease. Always compare the change (amount of increase or decrease) to the amount before the change (the original amount) using the following proportion:

$\dfrac{\text{amount of increase or decrease}}{\text{original amount}}=\dfrac{\textit{P}}{100}$

Example A

The rent went from $375 a month to $427.50 a month. What is the percent of the increase?

Find the change (the amount of increase) by finding the difference between the amounts:

$427.50 − $375 = $52.50

The amount of increase is $52.50.

Then, find the % of increase.

The amount of increase is $52.50. The original amount (the amount before the increase) is $375. What % of $375 is $52.50?

$\begin{equation*}\begin{split} \dfrac{\textit{X}}{100} &= \dfrac{52.50}{375} \\ \\ 375\textit{X} &= 5250 \\ \\ \textit{X} &= \dfrac{5250}{375} \\ \\ \textit{X} &= 14\% \end{split}\end{equation*}$

The rent increase is 14%.

Example B

The hours of operation at the college were reduced from 35 hours a week to 30 hours a week. What is the percent of this cut in operations?

Find the amount change (a decrease) by finding the difference between the amounts:

35 hours − 30 hours = 5 hours

The amount of decrease is 5 hours.

Find the % of increase.

The decrease is 5 hours. The original amount is 35 hours. What percent of 35 is 5?

$\begin{equation*}\begin{split} \dfrac{5}{35} &= \dfrac{\textit{P}}{100} \\ \\ 500 &= 35\textit{P} \\ \\ \dfrac{500}{35} &= \textit{P} \\ \\ 14\tfrac{2}{7}\% &= \textit{P} \end{split}\end{equation*}$

The hours of operation at the college were cut $\boldsymbol{14\frac{2}{7}\%}$ .

Exercise 4

Solve the following problems. Round decimal answers to the nearest tenth.

Ms. Lister’s bi-weekly unemployment cheque increased from $405 to $435. What percent increase is this?
Joan weighed 72 kg before she went on a programme of strict exercise and careful eating. She now weighs 60 kg. What is the percent of her weight loss?
The car dealership gives a special deal if the customer does not have a trade-in and pays cash. The dealers will only charge $10,650 for a car listed at $12,000. What is the percent savings in this deal?
A regular toilet uses 20 litres of water per flush. By purchasing a new low flow toilet, the water use is 6 litres per flush. What is the percent savings of water per flush if the new tank is used?

Answers to Exercise 4

7.4% increase
16.7% decrease
11.25% savings
70% savings

School Grades

When looking at test results, the number of questions you got right out of the total number of questions shows how you did on the test. Sometimes, it can be helpful to see your mark as a percentage, too.

Example A

If you get $\tfrac{7}{10}$ on a test, you know you got 7 answers right and 3 answers wrong. But how do you find the percentage of answers you got right?

By solving for $N$ , the percentage can be found:

$\begin{equation*}\begin{split} \dfrac{7}{10}&= \dfrac{\textit{N}}{100} \\ \\ 7 \times 100 &= \textit{N} \times 10 \\ \\ \dfrac{\cancel{700}70}{\cancel{10}}&= \dfrac{\cancel{10}\textit{N}}{\cancel{10}} \\ \\ \dfrac{70}{1} &= \dfrac{\textit{N}}{1} \\ \\ 70&=\textit{N} \end{split}\end{equation*}$

So, a test mark of $\frac{7}{10} = 70\%$ .

Example B

If someone earned a test result of $\dfrac{15}{43}$ , what percent did they score on the test?

$\begin{equation*}\begin{split} \dfrac{15}{43} &= \dfrac{\textit{N}}{100} \\ \\ 15\times100 &= \textit{N}\times43 \\ \\ \dfrac{1500}{43} &= \dfrac{\cancel{43}\textit{N}}{\cancel{43}} \\ \\ \dfrac{1500}{43} &= \dfrac{\textit{N}}{1} \\ \\ 34.88\% &= \textit{N} \end{split}\end{equation*}$

Rounding to the nearest percentage point, $\textit{N} = 35\%$ .

Example C

Find the percent of the following grade: $\dfrac{89}{97}$ .

$\begin{equation*}\begin{split} \dfrac{89}{97} &= \dfrac{\textit{P}}{100} \\ \\ 89\times100 &= \textit{P}\times97 \\ \\ \dfrac{8900}{97} &= \dfrac{\cancel{97}\textit{P}}{\cancel{97}} \\ \\ \dfrac{8900}{97} &= \dfrac{\textit{P}}{1} \\ \\ 91.75\% &= \textit{P} \end{split}\end{equation*}$

$\textit{P} = 91.75\%$ or $92\%$ , if rounding to the nearest percentage point.

Exercise 6

Find the percents for the following test grades. Round your answer to the nearest percent.

$\dfrac{33}{42}$
$\dfrac{24}{40}$
$\dfrac{90}{120}$
$\dfrac{100}{110}$
$\dfrac{10}{20}$

Answers to Exercise 6

$79\%$
$60\%$
$75\%$
$91\%$
$50\%$

Topic A: Self-Test

Mark /12 Aim 10/12

Solve to find the missing percents. (4 marks)
1. 12 is blank% of 60.
2. 15 is blank% of 75.
3. 8.2 = blank% of 32.8.
4. What percent of 64 is 48?
Solve the following word problems: (4 marks)
1. A $140 jacket was on sale for $126. What percent is the savings?
2. The rent on an apartment went from $320 a month to $400 a month. What percent is this rent increase?
Find the percents for the following grades on tests. Round your answer to the nearest whole percent. (4 marks)
1. $\dfrac{12}{15}$
2. $\dfrac{10}{19}$
3. $\dfrac{71}{92}$
4. $\dfrac{132}{140}$

Answers to Topic A Self-Test

1. $20\%$
2. $20\%$
3. $25\%$
4. $75\%$
1. 10% savings
2. 25% increase
1. $80\%$
2. $53\%$
3. $77\%$
4. $94\%$

Topic B: Finding a Number when a Percent of it is Given

$\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$

In problems that provide a certain percentage of a number, the missing term is the whole. You will be told the % and the part (is), and asked to find the whole (of), which is 100%.

Example A

20% of what number is 14?

The part = 14
The whole (“what number”) = unknown (call it N)
The percent = 20%

Write the proportion:

$\dfrac{14}{\textit{N}} = \dfrac{20}{100}$

Solve the proportion:

Simplify:

$\dfrac{14}{\textit{N}} = \dfrac{\cancel{20}1}{\cancel{100}5}$

$\dfrac{14}{\textit{N}} = \dfrac{1}{5}$

Cross-multiply to solve:

$\begin{equation*}\begin{split} 14\times5 &= \textit{N}\times1 \\ \\ 70 &= \textit{N} \end{split}\end{equation*}$

14 is 20% of 70

Check by finding 20% of 70. The answer should be 14.

$\begin{equation*}\begin{split} \dfrac{20}{100} &= \dfrac{\textit{N}}{70} \\ \\ 1400 &= 100\textit{N} \\ \\ 14 &= \textit{N} \end{split}\end{equation*}$

Example B

33⅓% of blank is 60.

part = 60
whole = N
percent = 33⅓% = ⅓

Set up proportion:

$\begin{equation*}\begin{split} \dfrac{60}{\textit{N}} &= \dfrac{1}{3} \\ \\ 3\times60 &= \textit{N}\times1 \\ \\ 180 &= \textit{N} \end{split}\end{equation*}$

33⅓% of 180 is 60.

To check the answer, find 33⅓% of 180. The answer should be 60.

Exercise 1

Set up the proportion. Do not solve the question.

18 is 50% of what number?
24 is 15% of what number?
200% of what number is 86?
66⅔% of what number= 500?

Answers to Exercise 1

$\dfrac{18}{P}=\dfrac{50}{100}$
$\dfrac{24}{N}=\dfrac{15}{100}$
$\dfrac{86}{P}=\dfrac{200}{100}$
$\dfrac{500}{P}=\dfrac{66\tfrac{2}{3}}{100}$ or $\dfrac{2}{3}$

Exercise 2

Solve the following. Check your answers to see if you set up the proportion correctly.

60 is 75% of what number?
950 is 95% of:
125 = 33⅓% of what number?
120% of blank is 6.
270 is 100% of what number?

Answers to Exercise 2

$\dfrac{60}{X}=\dfrac{75}{100},\hspace{0.5em}80$
$\dfrac{950}{P}=\dfrac{95}{100},\hspace{0.5em}1,000$
$\dfrac{125}{N}=\dfrac{33\tfrac{1}{3}}{100},\hspace{0.5em} 375$
$\dfrac{120}{100}=\dfrac{6}{J}, \hspace{0.5em}5$
$\dfrac{100}{100}=\dfrac{270}{N},\hspace{0.5em}270$

Exercise 3

Solve the questions.

480 is 66⅔% of blank.
40% of blank is 50.
33⅓% of what number is 99?
3 is 150% of what number?
122 is 80% of blank.

Answers to Exercise 3

$720$
$125$
$297$
$2$
$152.5$

Solving Problems when the Percent of a Number is Given

Read the problems carefully. More than one step may be needed. Look at the wording so you will recognize problems missing the whole and be able to tell them from problems missing the part.

Exercise 4

Solve the problems. Round money to the nearest cent.

When a business was declared bankrupt, all the creditors (people owed money by the business) were paid 55% of the money owed them.
1. If a creditor received $8,000 from the bankrupt business, how much money had he really been owed?
2. How much money did the creditor lose on this business?
In telethons and other fundraising events, records show that about 80% of the money pledged is actually collected. The local telethon organizers need to raise $12,000. To actually raise $12,000, their goal for pledges should be what amount?
The shoe store sold all merchandise at 25% off in a huge clearance sale. They took in $3,500 in the first day of the sale. If the same shoes had been sold at the regular price, how much money would they have taken in? (Note: this problem has 2 steps. The merchandise was 25% off, so it sold for 100% − 25% = 75% of the original price.)
A common rate of commission earned by real estate agents is 3½%. If an agent had a gross income of $63,000 from commissions in one year, what was the value of the houses they sold?

Answers to Exercise 4

Answers:
1. $14,545.45
2. $6,545.45
$15,000.00
$4,666.67
$1,800,000.00

Topic B: Self-Test

Mark /8 Aim 6/8

Solve these questions. (4 marks)
1. 2.5% of what number is 160?
2. 5 is $\frac{4}{5}\%$ of fill-in-the-blank.
3. 180 is 90% of what number?
4. 28 is 35% of what number?
Solve these problems. (4 marks)
1. If a bank insists that new home buyers need a cash down payment of 12%, what house price can a couple afford if they have saved a $15,000 down payment?
2. Jim has really cut down on his smoking. He now smokes 7 cigarettes a day, which he says is only 20% of what he used to smoke. How many cigarettes a day did Jim smoke before he started cutting down?

Answers to Topic B Self-Test

Answers:
1. $6,400$
2. $625$
3. $200$
4. $80$
Answers:
1. $125,000
2. 35 cigarettes per day

Unit 4 Review: More Working with Percent

Review

You have been practicing three types of percent problems. You have learned that one proportion can be used to solve all the problems:

$\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}$

Finding a percent of a number
- You are given the percent and the whole.
- The missing term is the part (call it N).
  $\dfrac{N}{\text{whole}}=\dfrac{\%}{100}$
Finding what percent one number is of another
- You are told the part and the whole.
- The missing term is the percent (call it P).
  $\dfrac{\text{part}}{\text{whole}}=\dfrac{P}{100}$
Finding a number when a percent of it is given
- You are given the part and the percent.
- The missing term is the whole (call it W).
  $\dfrac{\text{part}}{W}=\dfrac{\%}{100}$

Real-life situations and real math problems often require several steps to collect and organize all the information. Look for those extra steps in the problems that follow. When you read the problems, look for the part, the whole and the percent. Decide which term is missing. Once you know which term is missing, the problem can be solved by using the proportion $\dfrac{\text{part}}{\text{whole}}=\dfrac{\%}{100}$ or the appropriate short method.

Questions

Solve these problems using proportion. Write all your work with the problem so your instructor can help you, should you have any difficulty. Remember to check that the answer makes sense and to write a full-sentence answer. For these problems, round your answers this way:

Percents to the nearest tenth of a percent
Money to the nearest cent
Decimal fractions to the nearest thousandth

You need a minimum of 80% on the test, which means you must get at least 36 marks. What are the total possible marks on the test?
Ann and Joe bought their home twelve years ago and have paid 45% of the principal amount of their mortgage. They have paid $18,000 towards the principal of the mortgage. What was the principal amount of the mortgage to start with?
The waiters at the restaurant must contribute money to be shared among the cocktail servers and kitchen staff. Each waiter contributes 4% of their total food and drink sales. Craig’s total sales were $645 in his 5-hour shift.
1. How much money must he contribute to the kitchen and cocktail servers?
2. Craig made 12% in tips on his sales tonight. How much money did he earn in tips?
The cost of hydroelectric power for our home last year was 210% of what it was six years ago. Last year, our power bill totalled $960. How much was it six years ago?
16⅔% of the tickets for the rock concert were sold in the first hour that tickets were available for sale. In that hour, 2,500 tickets were sold. What was the total number of tickets available for the concert?
The total student enrolment in the school district has increased from 18,506 students to 19,724 students in the last year. What is the percent of this increase in student enrolment?
Al bought his secondhand off-road motorcycle for $1,500 and sold it three years later for $1,175. By what percent did his motorcycle depreciate (decrease in value)?
Pat operates a street-vendor’s cart selling hot dogs, sausages on a bun, and soft drinks. The basic pay is $100 per week and 28% commission on all sales over $450 in a week. Pat sold $1,244 of food and soft drinks last week. Calculate Pat’s earnings from the street-vending cart for the week.
The 1,500 blouses purchased by the large retail chain of ladies’ clothing stores cost the company a total of $24,000. The blouses were then priced to sell at $45 each. What is the percent of the mark-up on these blouses? (Hint: First calculate the company’s initial purchase price per blouse.)
Maureen was happy to see that she got 27 out of 30 on her English essay. What percent did she get?
The ski jackets were on the summer clearance rack, marked “45% off.”
1. What is the sale price of a jacket priced regularly at $229.95?
2. What is the total cost of this jacket with HST (12% tax)?
Tuition fees at the university have increased from $49 per credit hour to $62 per credit hour in the last three years. What is the percent of the tuition increase?
Jill bought 3 bottles of liquor in the U.S. The bill, including U.S. sales tax, was $28.50. Assume the American dollar was $1.08 Canadian. Calculate:
1. The value of the liquor in Canadian funds.
2. Duty at 110%.
3. HST at 12% (on Canadian value + duty).
4. Total cost in Canadian funds.

The consignment store will sell your lightly used women’s clothing for you. The store owners take a percentage of the selling price as their fee for service. The consignment charges (the % the store owners keep) are as follows:

$\begin{equation*}\begin{split} &\text{Coats} &\hspace{3cm} &45\% \\ &\text{Dresses and skirts} &\hspace{3cm} &33\tfrac{1}{3}\% \\ &\text{Bridal and evening gowns} &\hspace{3cm} &50\% \\ &\text{Blouses, jeans, and slacks} &\hspace{3cm} &25\% \end{split}\end{equation*}$

Lisa and her daughters did a huge closet cleanout and had the following items sold at the consignment store. For each category, calculate the amount for the store fee and the amount Lisa and the girls received.

	Item	Selling Price
a	Wedding dress	$275
b	3 dresses at $40 each	3 @ $40 = $120
c	Lisa’s winter coat at $120	$120
d	4 pairs of outgrown jeans at $10 each	4 @ $10 = $40

You have just completed four units of the five units in this book. What percent of the book have you completed?!?!?

Answers to Unit 4 Review

45 marks
$40,000.00
Answers:
1. $25.80
2. $77.40
$457.14
15,000 tickets
6.6% increase
21.7% depreciation
$322.32
181.3% mark-up
90% correct
Answers:
1. $126.47
2. $141.65
26.5% increase
Answers:
1. $30.78
2. $33.86
3. $7.76
4. $72.40

	Item	Selling Price	Store Fee	Amount for Lisa & Her Daughters
a	Wedding dress	$275	$137.50	$137.50
b	3 dresses at $40 each	3 @ $40 = $120	$40.00	$80.00
c	Lisa’s winter coat at $120	$120	$54.00	$66.00
d	4 pairs of outgrown jeans at $10 each	4 @ $10 = $40	$10.00	$30.00

80%. Well done!

TEST TIME!

Ask your instructor for the Practice Test for this unit.

Once you’ve done the Practice Test, you need to do the Unit 4 test.

Again, ask your instructor for this.

Good luck!

Unit 5: Statistics

Introduction and Vocabulary

The word graph comes from a Greek word meaning to write or draw.

Graphs are a special type of drawing or picture showing how numbers relate to each other. Graphs are a convenient way to organize numbers. You may know the old saying, “One picture is worth a thousand words.” Graphs give us a general picture of the information to look at first. The details of the information can then be read from the graph.

In this unit you will practice reading five types of graphs: line graphs, bar graphs, histograms, picture graphs, and circle graphs. You will also learn about reading charts and tables.

Study this vocabulary:

Axis: An axis is any straight line used for measuring or as a reference. In graphs, we talk about two axes (axes is the plural of axis).
- The horizontal axis goes across the page. (You will learn in later math courses that the horizontal axis is the x-axis.)
- The vertical axis goes up and down the page. (The vertical axis is the y-axis.)

Here is a trick to remember the difference between horizontal and vertical:

- The horizon is that line of earth you see as you look out at a view.
- The word horizontal comes from the root word horizon. The horizon goes from left to right, so does the horizontal line in a graph.

A row of trees on the far side of a field and blue sky behind them.

Titles: Titles of a graph give important information about the graph and often tell more about what the scale means.
Scale: Each axis has a scale. The scale is a series of numbers; beside or below the numbers will be a few words telling what the numbers represent. The numbers we want to place on the graph can be read from the scales. The horizontal scale describes the positions on the horizontal (x) axis. The vertical scale describes the positions on the vertical (y) axis.
Source: Graphs often give information that has been collected from other reports or publications. The source of the information should be written under the graph. Knowing where the information has come from helps you decide the accuracy of the information in the graph.
Trend: The trend is the general direction of events, the general idea of changes that are occurring. For example, there is an upward trend in the cost of houses.

[Image Description]

Image Descriptions

Bar Graph Image: A bar graph showing the favourite fruits of upgrading students. The graph is labelled with the following:

Title: Favourite Fruits of Upgrading Students April 2010
Vertical axis title: Number of students
Vertical axis scale: The numbers 0 to 9
Horizontal axis title: Types of Fruit
Horizontal axis: Apples, oranges, bananas, pears, strawberries, grapefruit, grapes
Source: Comox Upgrading Survey

[Return to Image]

Image Attributions

“Row of Trees on the Horizon” by Martin Vorel is licensed under a CC0 Licence.

Topic A: Line Graphs

Line graphs are used to show changes that happen over a period of time. Line graphs easily show trends and patterns.

The most common way to set up a line graph is to put time on the horizontal ( $x$ ) axis. Whatever is being measured is then put on the vertical ( $y$ ) axis.

Graph 1

A labelled line graph. Image description linked in caption.

[Image Description of Graph 1]

Leah has taken up cycling and jogging to improve her cardiovascular fitness and for weight control. She takes her pulse every Monday morning before she gets out of bed; that is her basal heart rate. Graph 1 records Leah’s heart rate for the first 12 weeks of her exercise program.

What is the title of the graph?
What is being measured on the vertical axis?

On this particular graph, the vertical scale is one heartbeat per line. The scale on the horizontal axis is one week per line.

What was Leah’s basal heart rate on the fourth week of her exercise program?
- Find Week 4 on the horizontal ( $x$ ) axis.
- Look straight up from Week 4 until you come to the point in the graphed line.
- Now look at the scale on the vertical ( $y$ ) axis. Lay a ruler or straight piece of paper across the graph to help you read the scale at the point for Week 4.
Leah’s basal heart rate was 74 beats/min in Week 4.
Find her basal heart rate in Week 7.
What does the graph show us happened between Week 9 and Week 10?
What trend does this graph show? The graph shows Leah’s basal heart rate is “going down” or decreasing.

Often, you need to estimate the value of the point in the graphed line. Look at Graph 2, which has Leah’s same heart rates recorded.

Answers to Graph 1

Leah’s Weekly Basal Heart Rate
Heartbeats per minute
Given: 74 beats/min
71 beats/min
Leah’s heart rate went up from 68 to 69 beats/min
Given: Leah’s basal heart rate is “going down” or decreasing

Graph 2

[Image Description of Graph 2]

Now the scale on the vertical axis is five heartbeats per line. Use a straightedge (ruler or paper) across the graph to help you read the vertical scale.

Give Leah’s heart rate in Week 5.
Give Leah’s heart rate in Week 10.
What was Leah’s heart rate in Week 12?
Using the information on the graphs, state how much Leah’s basal heart rate decreased (in beats per minute) from Week 1 to Week 12.

Answers to Graph 2

75 beats/min
69 beats/min
67 beats/min
Decreased 11 beats/min (78 − 67 = 11)

The person drawing a graph decides how to label it and how to write the scale on the axes, depending on the information to be shown on the graph. The graphs about Leah focus on the range of her heart rate. There is no need to make the heart rate scale lower than 55 or higher than 80 for Leah.

Often, information about several things are illustrated on the same graph. Such graphs are very useful for making comparisons. Look for a legend or key that explains what each graphed line represents. The legend may be printed right by the graphed information or it may be beside or below the graph.

Graph 3

Leah’s husband, John, decided to exercise as well. He had a rapid heart rate at the beginning of the exercise program. Since his heart rate is higher than 85, we must increase the numbers on the vertical scale so we can graph John’s heart rate on the same graph as Leah’s.

[Image Description of Graph 3]

What was John’s heart rate in Week 6?
What was John’s heart rate in Week 10?
How much lower was Leah’s heart rate than John’s in Week 8?
How much did John’s heart rate drop in the 12-week exercise program?
What was the amount of change in John’s heart rate between Week 3 and Week 7? Was it an increase (+) or decrease (−)?
Compare the two graphed lines.
1. How is the slant of the lines different?
2. How are the lines the same?
3. You can tell from looking at this graph that John and Leah had a similar trend in the change in their basal heart rates. Did their heart rates increase or decrease?

Answers to Graph 3

88 beats/min
83 beats/min
17 beats/min
12 beats/min
Decrease 4 beats/min
Answers:
1. John’s goes up at Week 8, Leah’s goes up at Week 10, Leah stays the same from Week 11 to 12 while John’s goes down.
2. Similar, because both decrease at a similar slant and both show an increase in two separate weeks.
3. Both heart rates decreased.

If the graphed lines have about the same slant, the rate of change is the same for the information being graphed. By looking at graphed lines, you can tell if one has increased or decreased more quickly. You can easily compare changes and tell when the changes occurred.

Steps to Follow When Reading Line Graphs

Read all the titles so you know what the graph is about.
Look at the information on the vertical and horizontal scales.
Decide what the graphed lines represent. Look for a legend and be sure you know which line is which.
Interpret (read) the information on the graph.
- First, get a general look at what is on the graph.
- Second, look for the detailed information that you need.

Graph 4

[Image Description of Graph 4]

What is the general trend in the Crosses’ average monthly cost for electricity?
What was the average monthly cost of electricity in 1998?
How much more did they pay per month in 2003 than in 2000?
What happened to the average cost between 2005 and 2006?
By how much did the Crosses’ average monthly electricity cost increase between 1998 and 2010?

Answers to Graph 4

Increased
$40
Approximately $23 ($77 – 54)
Decreased
$58

Line graphs can also be used to graph two different types of related information on the same chart. For example, we may have wanted to put Leah and John’s weight changes and heart rate changes on the same graph.

When we graph different types of information:

A subtitle (written in smaller print under the title) usually explains the two types of information.
The horizontal axis is the same.
The vertical axis on the right side of the graph has the scale for the second set of information. For example, on the graph for Leah and John, the right vertical axis would be marked in kilograms for weight changes.

Steps to Follow When Reading Line Graphs with Two Types of Information

Read the titles and subtitles.
Look at the legend to identify the graphed lines and check where the scale for each line is written (on the left or on the right vertical axis).
Go up from the horizontal axis in the same way. Read the appropriate scale on the left or right vertical axis for each graphed line.

Graph 5

[Image Description of Graph 5]

What is the title of this graph?
What is the subtitle?
Look at the legend. What does the graphed line for “Kilowatt Hours Used” look like?
Where is the scale for kilowatt hours?
How is the scale for kilowatt hours labelled?
in thousands means that each figure in the scale is to be multiplied by 1,000. For example, “32” represents “32,000.”
In which two years was the use of kWh the greatest?
In which year was the least kWh used?
Look at 1999 and 2000.
1. Find the amount of kWh used (in thousands) in 1999 and the cost of electricity in 1999.
2. Compare to the amount of kWh used in 2000 and the cost of electricity in 2000.
3. What can you conclude by comparing the cost and the kWh used in 1999 and 2000?
Now look at 2001 and 2002.
1. Compare the use of kWh.
2. Compare the cost.
3. What can you conclude?
Between 2009 and 2010, the use of kWh went up or down? by fill-in-the-blank kWh.The cost of electricity went up or down? by $fill-in-the-blank.

Answers to Graph 5

Cross Family’s Electricity Costs
Average Cost/month and Annual Kilowatt Hours Used
Dotted line
Right side of graph
kWh used in thousands
1998 + 2000
2002
Answers:
1. ~36,000 kWh and $41.
2. ~40,000 kWh and $54.
3. The price per kWh increased.
Answers:
1. 2001: ~34,500 kWh; 2002: ~33,500 kWh
2. 2001: $69; 2002: $69
3. You can conclude the price/kWh increased.
The use of kWh went up by ~500 kWh. Cost went up by $3.

The steepness of the slant of a graphed line gives you an idea of the rate of change. The steeper the slant, the greater the change.

This double graph shows that the average monthly cost of electricity has increased, while the annual use of kilowatt hours has shown an overall decrease.

Image Descriptions

Graph 1 (Line Graph)

A line graph shows the change in Leah’s weekly basal heart rate over 12 weeks of exercise.

The vertical axis is heartbeats per minute and has the numbers 65 to 80 in increments of 1.
The horizontal axis is weeks of exercise and has the numbers 1 to 12 in increments of 1.

The line graph data is represented in the following table:

Leah’s Weekly Basal Heart Rate
Weeks of Exercise (Horizontal Axis)	Heartbeats per Minute (Vertical Axis)
1	78
2	78
3	76
4	74
5	75
6	73
7	71
8	70
9	68
10	69
11	67
12	67

[Return to Graph 1]

Graph 2 (Line Graph)

A line graph showing Leah’s weekly basal heart rate.

The vertical axis is heartbeats per minute and has the numbers 60 to 80 in increments of 5.
The horizontal axis is weeks of exercise and has the numbers 1 to 12 in increments of 1.

The line graph data is represented in the following table:

Leah’s Weekly Basal Heart Rate
Weeks of Exercise (Horizontal Axis)	Heartbeats per Minute (Vertical Axis)
1	78
2	78
3	76
4	74
5	75
6	73
7	71
8	70
9	68
10	69
11	67
12	67

[Return to Graph 2]

Graph 3 (Line Graph)

A line graph showing Leah and John’s weekly basal heart rate.

The vertical axis is heartbeats per minute and has the numbers 60 to 95 in increments of 5.
The horizontal axis is weeks of exercise and has the numbers 1 to 12 in increments of 1.
The legend refers to two different graphed lines on the line graph. One refers to Leah’s heartbeats per minute, and the other refers to John’s heartbeats per minute.

The line graph data is represented in the following table:

Leah and John’s Weekly Basal Heart Rate
Weeks of Exercise (Horizontal Axis)	Leah’s Heartbeats per Minute (Vertical Axis)	John’s Heartbeats per Minute (Vertical Axis)
1	78	92
2	78	92
3	76	90
4	74	89
5	75	90
6	73	88
7	71	86
8	70	87
9	68	85
10	69	83
11	67	81
12	67	80

[Return to Graph 3]

Graph 4 (Line Graph)

A line graph showing the Cross family’s electricity costs.

The vertical axis is the average monthly cost in dollars, and contains the numbers 0 to 120 in increments of 20.
The horizontal axis is years and contains the years 1998 to 2010 in increments of one year.

The line graph data is represented in the following table:

Cross Family’s Electricity Costs
Year (Horizontal Axis)	Average Monthly Cost in $ (Vertical Axis)
1998	40
1999	41
2000	54
2001	69
2002	69
2003	77
2004	81
2005	85
2006	78
2007	79
2008	87
2009	95
2010	98

[Return to Graph 4]

Graph 5 (Line Graph)

A line graph showing the Cross family’s electricity costs, displaying both average cost per month and annual kilowatt hours (kWh) used.

The left vertical axis is the monthly cost in dollars and contains the numbers 35 to 99 in increments of 2.
The right vertical axis is the kWh used in the thousands and contains the numbers 30 to 42 in increments of 2.
The horizontal axis is years and contains the years 1998 to 2010 in increments of one year.
The legend refers to two different graphed lines on the line graph. One refers to the monthly cost, and the other refers to the kWh used.

The line graph data is represented in the following table:

Cross Family’s Electricity Costs: Average Cost/Month and Annual Kilowatt Hours (kWh) Used
Year (Horizontal Axis)	Monthly Cost in $ (Left Vertical Axis)	kWh used in Thousands (Right Vertical Axis)
1998	40	~40
1999	41	~36
2000	54	~40
2001	69	~34.5
2002	69	~33.5
2003	77	~36
2004	81	~37.5
2005	85	~36.5
2006	78	~34.5
2007	79	~36
2008	87	~37
2009	95	~34.5
2010	98	~35

[Return to Graph 5]

Topic B: Bar Graphs

Bar graphs compare quantities. Bar graphs are commonly used to illustrate information in newspapers, in magazine articles, and so on. Bar graphs may be written with the bars arranged vertically or horizontally. Graph One is shown both ways – first with vertical bars and second with horizontal bars.

Steps to Follow When Reading a Bar Graph

Read the title and subtitles so you know what you are looking at.
Read the information on the vertical and horizontal axes. Notice that each bar represents a different item.
Look carefully at the scale. What unit of measure is being used? The unit of measure will be the same for each bar so that you can compare them.
Compare the length or height of each bar to find the information that you want.

Graph 1

[Image Description]

How many rivers are shown on this graph?
What is the title of the graph?
What is the unit of measure?
Look at the scale for kilometres. How many kilometres are represented by each division on the page?
Which river is the longest? What is its length?
Which river is the shortest? What is its length?
Name two rivers which are approximately the same length.
Compare the Columbia River and the Fraser River.
1. Which is the longer?
2. Give the approximate difference in their lengths.
Give the approximate length of the North Thompson, South Thompson, and Thompson Rivers combined.

Answers to Graph 1

12 rivers
Lengths of some British Columbia Rivers
Kilometres
250 km
Columbia; ~1,950 km
Quesnel; ~100 km
North Thompson & South Thompson
1. Columbia
2. ~600 km (1,950 – 1,350)
~750 km

Graph 2

[Image Description]

Give the source of the information for this graph.
What is the unit of measure for the population scale?
What number of people is represented by each section on the scale?
1. Which country had the largest population?
2. What was the approximate population of this country?
Name the two countries which were the closest in population size.
What was the approximate population of Bangladesh?
What was the approximate population of India?

Answers for Graph 2

United Nations
In the Millions
50 million
1. China
2. 1,354,000,000
Nigeria and Bangladesh
164,000,000
1,214,000,000

Bar graphs can show more than one type of information for each item. These graphs are useful for making comparisons. The bars are usually shaded or coloured differently and a legend will be placed near the graph. The bar graphs must still all use the same unit of measure.

Graph 3

[Image Description]

What is the subtitle?
Look at the legend.
1. For which year do the solid blue bars represent each country’s population?
2. For which year do the grey patterned bars represent each country’s population?
What trend does the graph show?
Consider these questions:
1. Which country had the largest increase in population? (In other words: which country’s population went up by the highest number?)
2. About how much was that increase?
Consider these questions:
1. Which country had the least change in population?
2. About how much was that change?

Answers for Graph 3

Year 2010 and Year 1950
Answers:
1. 2010
2. 1950
That countries around the world are growing in population
Answers:
1. India
2. The increase was 843 million
Answers:
1. Pakistan
2. The increase was 108 million

Image Descriptions

Graph 1.1 (Bar Graph)

A bar graph showing the lengths of some British Columbia rivers.

The vertical axis is kilometres and has the numbers 0 to 2,000 in increments of 250.
The horizontal axis is the names of the following British Columbia rivers: Fraser, North Thompson, South Thompson, Thompson, Quesnel, Nechako, Chilcotin, Lillooet, Kootenay, Columbia, Peace, and Laird.

The bar graph data is represented in the following table:

Lengths of Some British Columbia Rivers
British Columbia Rivers (Horizontal Axis)	Kilometres (Vertical Axis)
Fraser	~1,350
N. Thompson	~300
S. Thompson	~300
Thompson	~150
Quesnel	~100
Nechako	~400
Chilcotin	~250
Lillooet	~200
Kootenay	~675
Columbia	~1,950
Peace	~1,675
Laird	~1,100

[Return to Image]

Graph 1.2 (Bar Graph)

A bar graph showing the lengths of some British Columbia rivers.

The vertical axis is the names of the following British Columbia rivers: Fraser, North Thompson, South Thompson, Thompson, Quesnel, Nechako, Chilcotin, Lillooet, Kootenay, Columbia, Peace, and Laird.
The horizontal axis is kilometres and has the numbers 0 to 2,000 in increments of 250.

The bar graph data is represented in the following table:

Lengths of Some British Columbia Rivers
British Columbia Rivers (Vertical Axis)	Kilometres (Horizontal Axis)
Fraser	~1,350
N. Thompson	~325
S. Thompson	~325
Thompson	~150
Quesnel	~100
Nechako	~400
Chilcotin	~250
Lillooet	~200
Kootenay	~675
Columbia	~1,950
Peace	~1,675
Laird	~1,100

[Return to Image]

Graph 2 (Bar Graph)

A bar graph showing the population of the world’s most populated countries in 2010.

The vertical axis is population in millions, and has the numbers 0 to 1,400 in increments of 50.
The horizontal axis is countries, and contains the following: China, India, United States, Indonesia, Brazil, Bangladesh, Nigeria, and Pakistan.

The bar graph data is represented in the following table:

Population of the World’s Most Populated Countries in 2010
Country (Horizontal Axis)	Population in Millions (Vertical Axis)
China	1,354
India	1,214
United States	287
Indonesia	232
Brazil	174
Bangladesh	164
Nigeria	158
Pakistan	149
Source: United Nations, 2010

[Return to Image]

Graph 3 (Bar Graph)

A bar graph showing the population of the world’s most populated countries in 2010 and 1950.

The vertical axis is population in millions, and has the numbers 0 to 1,400 in increments of 50.
The horizontal axis is countries, and contains the following: China, India, United States, Indonesia, Brazil, Bangladesh, Nigeria, and Pakistan.
The legend denotes that each country is associated with two different bars – one referring to its population in 2010, and the other referring to its population in 1950.

The bar graph data is represented in the following table:

Population of the World’s Most Populated Countries: Year 2010 and Year 1950
Country (Horizontal Axis)	Population in Millions in 2010 (Vertical Axis)	Population in Millions in 1950 (Vertical Axis)
China	1,354	544
India	1,214	371
United States	287	157
Indonesia	232	77
Brazil	174	53
Bangladesh	164	43
Nigeria	158	36
Pakistan	149	41
Source: United Nations, 2010

[Return to Image]

Topic C: Picture Graphs

Picture graphs are similar to bar graphs. Picture graphs show comparisons between quantities. A little picture represents a certain amount. Look for the legend to find out that amount. Picture graphs will give fractions of a picture also. For example, if the picture represents 100 things, half a picture would be 50.

Graph 1

[Image Description]

As a quick first impression when you look at this graph, which type of vehicle is most in use?
What does each picture represent according to the legend?
Out of every 1,000 vehicles, how many are:
1. cars?
2. bikes and motorcycles?
3. large trucks?
4. RVs and small trucks?
Look for other examples of picture graphs in the newspaper and in magazines. Television programs often display picture graphs to illustrate statistics.

Answers for Graph 1

car
100 vehicles
1. 600
2. 100
3. 100
4. 200

Image Descriptions

Graph 1 (Picture Graph)

A picture graph showing the results of an informal survey of vehicles per 1,000 used in summer months.

The vertical axis is vehicles, and contains the following types of vehicles: cars, bikes & motorcycles, large trucks, and RVs & small trucks.
The horizontal axis is pictures of vehicles, each of which stands for 100 vehicles of the corresponding type according to the legend.

The picture graph data is represented in the following table:

Informal Survey of Vehicles per 1,000 Used in Summer Months
Type of Vehicle (Vertical Axis)	Number of Pictures (Horizontal Axis)
Cars	6
Bikes & Motorcycles	1
Large Trucks	1
RVs and Small Trucks	2

[Return to Image]

Topic D: Circle Graphs (“Pie Graphs”)

Circle graphs show how the parts of something compare to each other. Circle graphs also give a good picture of each part compared to the whole thing. In a circle graph or pie graph, the complete circle is the whole thing. The parts of a circle graph may be identified with a percentage. The total of the parts must be 100%.

Graph 1

The circle represents each dollar the government spends. The information for the graph was sourced from the Department of Finance, April 2010, at Where Your Tax Dollar Goes – 2007–2008 (Department of Finance Canada).

The parts are shown as cents of the dollar.

[Image Description]

What is the biggest expense of the federal government?
How much of each federal dollar is spent in actually operating the government business?
What part of the federal dollar is spent on defence?
How much of each dollar is spent on Provincial Payments? Write this amount as a percent.
What is the smallest expenditure of the federal government? Write this amount as a percent.

Answers to Graph 1

Payments to Persons
20¢
7¢
20¢; 20%
Budgetary Surplus; 4%

Graph 2

2004 Nanaimo Regional Landfill Solid Waste Composition

[Image Description]

What makes up the largest part of the waste in the landfill site?
What four categories contribute equal weight to the landfill site?
In a municipality of 139,000 people, the amount of waste going to a landfill site in one day is 150 tonnes.
1. What is the mass of plastics?
2. What is the mass of yard waste?
3. What is the mass of construction/demo waste?
4. If all the food waste was composted, how many tonnes of waste would not end up in the landfill each day?
The plastics category can be separated into these categories:
- 6% Non-recyclable mixed plastics
- 4% film plastic
- 3% recyclable rigid food containers
If all the 3% recyclable rigid food containers were actually recycled, how many tonnes of waste would not end up in the landfill?

Answers to Graph 2

Food waste
- Diapers, Personal Hygiene
- Glass
- Bulky Goods
- HHW (Household Hazardous Waste
1. 19.5 tonnes
2. 10.5 tonnes
3. 24 tonnes
4. 34.5 tonnes
4.5 tonnes

Image Descriptions

Graph 1 (Circle Graph)

A circle graph showing the Canadian tax dollar was spent in the 2007-2008 fiscal year.

The whole circle represents one dollar of the Canadian budgetary expenditure.
Each part reflects one of the specific expenses, reflected in cents to the dollar, ¢. The included parts are (clockwise from top): Public Debt, Provincial Payments, Grants and Contributions, Payments to Persons, National Defense, Operating Costs, and Budgetary Surplus.

The circle graph data is represented in the following table:

How Your Federal Tax Dollar is Spent
Expense	Expenditure, reflected in cents to the dollar, ¢
Public Debt	14
Provincial Payments	20
Grants and Contributions	11
Payments to Persons	24
National Defense	7
Operating Costs	20
Budgetary Surplus	4
Source: Where Your Tax Dollar Goes – 2007-2008 (Department of Finance Canada)

[Return to Image]

Graph 2 (Circle Graph)

A circle graph showing the solid waste composition of the Nanaimo Regional Landfill in 2004.

The whole circle represents the total composition of solid waste in the Nanaimo Regional Landfill in 2004.
Each part reflects one element of the total composition. The included elements are (clockwise from top): Food Waste, Yard Waste, Compostable Paper, Construction/Demo Waste, Plastic, Mixed Paper, Metal, Textiles & Carpet & Tires, Diapers & Personal Hygiene, Glass, Bulky Goods, HHW, and Other.

The circle graph data is represented in the following table:

2004 Nanaimo Regional Landfill Solid Waste Composition
Element of Waste	Percentage of Total
Food Waste	23%
Yard Waste	7%
Compostable Paper	4%
Construction/Demo Waste	16%
Plastic	13%
Mixed Paper	8%
Metal	6%
Textiles, Carpet, Tires	10%
Diapers, Personal Hygiene	2%
Glass	2%
Bulky Goods	2%
HHW	2%
Other	5%
Source: 2004 Nanaimo Regional Landfill Solid Waste Composition

[Return to Image]

Topic E: Histograms

A histogram is a special bar graph that shows how a frequency (the number of times something happens) relates to a class interval (a range of numbers). A histogram is useful when looking at how many times something happens. It is useful to look at monthly or yearly temperatures, at test scores and groups of people based on age.

In the following graph, the height of each bar relates to how many days a temperature was between the listed temperatures in the horizontal axis.

This graph is created by counting how many days the temperatures were:

between 0°C and −5°C (1 day)
between −6°C and −10°C (6 days)
between −11°C and −15°C (8 days)
between −16°C and −20°C (5 days)
between −21°C and −25°C (9 days)
between −26°C and −30°C (2 days)

Then the information is put into graph form.

Graph 1

[Image Description]

How many degrees in temperature change is in each bar?
What is the source of the information?
Which temperature was the most common in the month of January?
Which community does this graph represent?
Which temperature was felt the least in Atlin in January, 2010?

Answers to Graph 1

Five degrees
Environment Canada
−21°C to −25°C
Atlin, BC
−1°C to −5°C

Graph 2

In this Fundamentals Math course, students’ marks were collected from the final test. The graph shows the results.

$A labelled histogram. Image description linked in caption.$

[Image Description]

How many students took the test?
How many students scored 80% to 89% on the test?
In order to pass, students must get 80% or over on the test. How many students will have to re-write the test?
How many more students got 80% to 89% than 90% to 100%?

Answers to Graph 2

Image Descriptions

Graph 1 (Histogram)

A histogram showing the daily minimum temperatures of Atlin, British Columbia in the month of January, 2010.

The horizontal axis contains the following temperature ranges, in degrees Celsius: −1°C to −5°C, −6°C to −10°C, −11°C to −15°C, −16°C to −20°C, −21°C to −25°C, and −26°C to −30°C.
The vertical axis is numbers of days, and has the numbers 0 through 10.

The histogram data is represented in the following table:

Daily Minimum Temperatures in the Month of January, 2010: Atlin, BC
Temperature Range in °C (Horizontal Axis)	Number of Days (Vertical Axis)
−1°C to −5°C	1
−6°C to −10°C	6
−11°C to −15°C	8
−16°C to −20°C	5
−21°C to −25°C	9
−26°C to −30°C	2
Source: Environment Canada

[Return to Image]

Graph 2 (Histogram)

A histogram showing the scores on the final math test for Fundamentals Math students.

The horizontal axis contains the following ranges of test scores, in percentages: 60%–69%. 70%–79%, 80%–89%, and 90%–99%.
The vertical axis is numbers of students, and has the numbers 0 through 14 in increments of 2.

The histogram data is represented in the following table:

Scores on Final Math Test: Fundamentals Math Students
Test Score Range, in percentages (Horizontal Axis)	Number of Students (Vertical Axis)
60%–69%	1
70%–79%	2
80%–89%	12
90%–99%	10

[Return to Image]

Topic F: Tables

Tables are an everyday way of organizing information, or one’s own work.

Graph 1

The following table is from the BC Ferries website: BC Ferries Current Conditions

This table shows the departure times from each community.

Comox – Powell River (Little River-Westview). Crossing Time: 1 hour 20 minutes. Distance: 17 nautical miles
Leave Comox (Little River)	Leave Powell River (Westview)
6:30 am Daily except Dec 25 and Jan 1	8:10 am Daily except Dec 25 and Jan 1
10:10 a.m.	12:00 p.m.
3:15 p.m.	5:15 p.m.
7:15 p.m.	8:45 p.m.

How many ferry runs go from Powell River to Comox each day?
On what days does the ferry not run at 6:30 a.m. and 8:10 a.m.?
How long is a crossing time?
When will the first ferry from Comox arrive in Powell River?
When will the last ferry from Powell River arrive in Comox?
How many nautical miles are covered by the ferry’s course?

Answers to Graph 1

four
Dec 25 and Jan 1
1 hour and 20 minutes
7:50 a.m.
10:05 p.m.
17 nautical miles

Graph 2

A cereal recipe explains the quantities to use when making hot cereal.

Recipe for Hot Cereal
Servings	Salt	Water	Cereal
1	¼ tsp	1.5 cups	¼ cups
4	1 tsp	6 cups	1 cup

How much salt should be added for one serving?
How much water is needed when making 4 servings of hot cereal?
How many cups of cereal should be used for making 1 serving of hot cereal?
To make ten servings of hot cereal, how much salt should be used?

Answers to Graph 2

¼ tsp
6 cups
¼ cups
2.5 tsp

Unit 5 Review: Statistics

Refer back to each lesson on graphs and explain when to use any three types of graph.

Line graph:
Bar graph:
Picture graph:
Circle graph:
Histogram:
Table:

TEST TIME!

Ask your instructor for the Practice Test for this unit.

Once you’ve done the Practice Test, you need to do the Unit 5 test.

Now that you have completed the whole book, you will need to do the Final test.

Please see your instructor for the Practice test and the Final Test.

Good luck!

Book 6 Review

You will now practice all the skills you learned in Book 6. You can use this as a review for your final test.

If you can’t remember how to do a question, go back to the lesson on this topic to refresh your memory. The unit and topic for where each question came from is listed next to the question.

Example: 1A means Unit 1, Topic A

Unit 1

1-A

Write the ratios asked for.
1. Lillian biked for 6 hours, and covered a total of 45 km. What is the ratio of kilometers to hours?
2. Nine hundred cars were lined up at the ferry terminal. 300 hundred cars got on the next sailing. Write a ratio of how many cars were left behind to how many cars got on the first sailing.

1-B

Simplify these ratios.
1. $9:12$
2. $50:5$
3. $56:7$
4. $100:120$
Write the following ratios as rates.
1. 110 kilometres to 2 hours
2. 9 cups of flour to 3 tablespoons of yeast
3. 240,000 people to 300 square kilometers

1-C

Solve these proportions.
1. $1:3 = \textit{N}:12$
2. $25:\textit{N} = 20:4$
3. $\textit{N}:49 = 14:98$
4. $4\tfrac{1}{2}:6 = \textit{N}:3.6$
5. The dose for cough syrup is 20 millilitres for each 100 pounds of body weight. How much should be given to a 34 pound child? Round to the nearest millilitre.

Unit 2

2-A

Write these percents using numerals and the percent sign.
1. seventy-two percent
2. three-fourths percent
3. one hundred two percent
Write these percents in words.
1. $12\%$
2. $\tfrac{1}{5}\%$
Change the percents to equivalent decimals.
1. $17\%$
2. $98\tfrac{1}{2}\%$
3. $\tfrac{1}{3}\%$
Write the decimals as percents.
1. $0.45$
2. $4.75$
3. $0.099$
Change each percent to an equivalent common fraction. Put the fraction in lowest terms.
1. $33\tfrac{1}{3}\%$
2. $14\%$
3. $250\%$
Write the percent equivalent
1. $\tfrac{1}{5}$
2. $\tfrac{2}{3}$
3. $\tfrac{1}{4}$

Unit 3

3-A

Find the answers.(Express percents rounded to the nearest tenth, money to the nearest cent and decimals to the nearest thousandth. Please show all your work. Use proportion.)
1. $13\%\text{ of }52 =$
2. $\tfrac{9}{10}\%\text{ of }2,400$ is .
3. What is $135\%\text{ of }1,080$ ?
Solve these problems. Be sure to show all your work.
1. Marianne is renovating her kitchen, and she is ordering everything from her local hardware store. She is getting a sink for $204.79, a dishwasher for $524.95, a counter for $949.99, flooring for $719.95, and a fridge for $579.49.
  1. Calculate the HST (12%).
  2. Calculate the total cost, including the taxes.
2. Shane sold a home for $340,500.00 for a client. He earned 6% commission. How much money did Shane make?
3. A love seat is originally priced at $904.00, it is offered at 45% off. What is the discount price?
4. Calculate the total cost in Canadian dollars of this purchase made in the United States. Assume $1.00 Canadian = $0.92 U.S.
  .
  1. Price in Canadian dollars
  2. Duty at 13.5%
  3. Total of Canadian value + duty
  4. HST (12%) on Canadian value + duty
  5. Total cost in Canadian dollars

Unit 4

4-A

Find the answers.
1. 34 is what percent of 85?
2. What % of 150 is 114?
3. 33⅓% of what number is 60?
4. 32 is 20% of what number?
5. 75% of what number is 675?
6. 3.75 is 1¼% of?
Solve these problems. Be sure to show all your work.
1. A printer is priced at $399. It is marked 30% off.
  1. What is the sale price of the printer?
  2. What is the cost of this printer with HST (12%)?
2. The Vancouver Fire and Rescue Service has 797 uniformed personnel, about 0.8% are women. About how many women are in uniform in the Vancouver Fire and Rescue Service? Answer to the nearest whole number (since you cannot have a fraction of a person).
3. Jake is a computer salesperson. He receives a monthly salary of $1,055 plus 15% on all his sales over $5,500. What was his total monthly earnings if his sales were
  $12,400 in one month?
4. The local ski hill sold 3,800 season’s passes in 2009. The 2010 sales are down 10.5%. Find the number of season’s passes sold in 2010.

Unit 5

5-A

Line graph.

[Image Description]
1. Which month has the highest temperature in Smithers?
2. Which month has the lowest temperature in Smithers?
3. Between the months of January to July, is there an increase or decrease in temperature?
4. What is the difference between the monthly temperature for August and the monthly temperature for October?
5. What is the trend of the temperature in Smithers?

5-B

Bar graph.

[Image Description]
1. What is the height of the tallest mountain in BC?
2. How many mountains are over 4,000 metres and under 5,000 metres in height?
3. Which two mountains in this chart are very similar in height?
4. What is the difference (approximately) in height between Fairweather Mountain and Asperity Mountain?

5-C

Picture graph.

[Image Description]
1. How many students in grade 2 ride bikes to school?
2. Which classes have the most bikers?
3. Which class has the least bikers?
4. How many more bikers are in grade 7 than kindergarten?
5. How many students bike in total?
6. What percent of the school bikes each day?

5-D

Circle graph.

[Image Description]
1. Which is the most favourite ice cream flavour?
2. Which is the least liked ice cream flavour?
3. How many people (out of 1,000) like fruit-flavoured ice cream?
4. What percentage of people like vanilla over chocolate?

Answers to Book 6 Review

1. $45:6$
2. $600:300$
1. $3:4$
2. $10:1$
3. $8:1$
4. $5:6$
1. 55 km/hr or $55 \text{km}:1 \text{hr}$
2. 3 cups of flour to 1 tbsp of yeast
3. 800 people to 1 square km
1. $4$
2. $5$
3. $7$
4. $2.7$
5. $7\text{ mL}$
1. $72\%$
2. $\tfrac{3}{4}\%$
3. $102\%$
1. Twelve percent
2. One-fifth percent
1. $0.17$
2. $0.985$
3. $0.00\overline{3}$
1. $45\%$
2. $475\%$
3. $9.9\%$
1. $\tfrac{1}{3}$
2. $\tfrac{7}{50}$
3. $2\tfrac{1}{2}$
1. $20\%$
2. $66.\overline{6}\%$ or $66\tfrac{2}{3}\%$
3. $25\%$
1. $6.76$
2. $21.6$
3. $1,458$
1. 1. $357.50
  2. $3,336.67
2. $20,430.00
3. $497.20
4. 1. $345.63
  2. $46.66
  3. $392.29
  4. $47.07
  5. $439.36
1. $40\%$
2. $76\%$
3. $180$
4. $160$
5. $900$
6. $300$
1. 1. $279.30
  2. $312.82
2. 6
3. $2,090
4. 3,401
1. July
2. January
3. Increase
4. Approximately 9 degrees
5. The temperature goes up from January to July, and goes down from July to December
1. Approximately 4,650 metres
2. 3
3. Combatant Mountain and Mount Columbia
4. Approximately 950 metres
1. 20
2. Grades 2, 5, 7
3. Kindergarten
4. 15
5. 110
6. Approximately 19.5%
1. Vanilla
2. Cake and cookie
3. 180
4. 20%

Image Descriptions

Graph 1 (Line Graph)

A line graph displays the average temperature in Smithers, BC each month.

The horizontal axis lists each month of the calendar year.
The vertical axis is temperature in degrees Celsius, and contains the numbers -15 to 20 in increments of 5.

The line graph data is represented in the following table:

Average Temperatures for Smithers, BC
Month (Horizontal Axis)	Temperature in Degrees Celsius (Vertical Axis)
January	~−9
February	~−5
March	~0
April	~5
May	~9
June	~12.5
July	~15
August	~14
September	~10
October	~5
November	~−2.5
December	~−8

[Return to Image]

Graph 2 (Bar Graph)

A bar graph displays the height of the highest mountains in British Columbia in metres.

The horizontal axis lists the following mountains: Fairweather Mountain, Mount Quincy Adams, Mount Waddington, Mount Robson, Mount Root, Mount Tiedemann, Combatant Mountain, Mount Columbia, and Asperity Mountain.
The vertical axis is height in metres, and contains the numbers 5,000 in increments of 100.

The bar graph data is represented in the following table:

Highest Mountains in British Columbia
Mountain (Horizontal Axis)	Height in metres (Vertical Axis)
Fairweather Mountain	~4,650
Mount Quincy Adams	~4,100
Mount Waddington	~4,000
Mount Robson	~3,950
Mount Root	~3,900
Mount Tiedemann	~3,850
Combatant Mountain	~3,750
Mount Columbia	~3,750
Asperity Mountain	~3,700
Source: Statistics Canada, 2010

[Return to Image]

Graph 3 (Picture Graph)

A picture graph displays the number of students at Marmot Elementary School who bike to school each day by grade.

The vertical axis lists the following grades at Marmot Elementary School: Grade 7, Grade 6, Grade 5, Grade 3, Grade 2, Grade 1, and Kindergarten.
The horizontal axis contains pictures of bicycles, with each bicycle representing 5 bikers according to the legend.
The total school population is 563 students.

The picture graph data is represented in the following table:

Number of Students who Bike to School each School Day: By grade at Marmot Elementary School
Grade (Vertical Axis)	Pictures of Bikes (Horizontal Axis)
Grade 7	4
Grade 6	3
Grade 5	4
Grade 3	3
Grade 2	4
Grade 1	3
Kindergarten	1

[Return to Image]

Graph 4 (Circle Graph)

A circle graph displays the favourite ice cream flavours per 1,000 people.

The entire graph represents all 1,000 people surveyed for their favourite ice cream flavour.
Each part represents an ice cream flavour and its popularity as a percent of the whole. The flavours are (clockwise from top): Vanilla, Fruit, Nut, Candy, Chocolate, and Cake and Cookie.

The circle graph data is represented in the following table:

Favourite Ice Cream Flavours: per 1000 people
Flavour	Percentage of Total
Vanilla	31%
Fruit	18%
Nut	16%
Candy	14%
Chocolate	11%
Cake and Cookie	10%

[Return to Image]

Source: Internal Ice Cream Association, 2010

Unit	Date of Test A	Test A	Date of Test B	Test B
Example	September 4, 2020	25/33	September 7, 2020	25/33
1
2
3
4
5
Final Test

Version	Date	Change	Details
1.00	2014	Book published
1.01	January 2016	Book updated.
2.00	February 17, 2022	2nd Edition published.
2.01	January 26, 2023	Minor edits for consistency of ALF Math series.	Created a “How to Deal with Math Anxiety” front matter section, which is now standardized across all ALF Math books. Added acknowledgments for the 1st edition to the back matter.
2.02	March 24, 2025	Corrected errors and formatting.	Corrected content and formatting errors in all units, including incorrect answers, erroneous formatting, and typographical errors.
2.03	February 20, 2026	Corrected errors and formatting.	Corrected content and formatting errors in all units, including incorrect answers, erroneous formatting, and typographical errors.

Weeks of Exercise (Horizontal Axis)	Leah’s Heartbeats per Minute (Vertical Axis)	John’s Heartbeats per Minute (Vertical Axis)
1	78	92
2	78	92
3	76	90
4	74	89
5	75	90
6	73	88
7	71	86
8	70	87
9	68	85
10	69	83
11	67	81
12	67	80

Weeks of Exercise (Horizontal Axis)	Leah’s Heartbeats per Minute (Vertical Axis)	John’s Heartbeats per Minute (Vertical Axis)
1	78	92
2	78	92
3	76	90
4	74	89
5	75	90
6	73	88
7	71	86
8	70	87
9	68	85
10	69	83
11	67	81
12	67	80

Adult Literacy Fundamental Mathematics: Book 6 - 2nd Edition

Adult Literacy Fundamental Mathematics: Book 6 - 2nd Edition

Contents

Accessibility Statement

Accessibility of This Textbook

Known Accessibility Issues and Areas for Improvement

Let Us Know if You are Having Problems Accessing This Book

For Students: How to Access and Use This Textbook

Tips for Using This Textbook

About BCcampus Open Education

To the Learner

How to Use This Book

Grades Record

How to Deal with Math Anxiety

Emotions and Learning

Do You Suffer from Math Anxiety?

How to Deal with Math Anxiety

Use Breathing Exercises

Think Positive Math Messages

Know Your Textbook

Understand Test-Taking Anxiety

1. Not feeling prepared for the test

2. Not sure how to write the test in the best way

3. Feeling too much mental pressure

4. Poor health habits before writing a test

Unit 1: Ratio, Rate, & Proportion

Topic A: Writing Ratios

Introduction to Ratios

Equivalent Ratios

Topic A: Self-Test

Answers to Topic A Self-Test

Topic B: Rates

Topic B: Self-Test

Answers to Topic B Self-Test

Topic C: Proportion

Using Equivalent Ratios to Solve Proportions

Solve a Proportion Problem Using Equivalent Ratios

Using Cross-Multiplication to Solve a Proportion

Solve a Proportion Problem Using Cross-Multiplication

Topic C: Self-Test

Answers to Topic C Self-Test

Unit 1 Review: Ratio, Rate, & Proportion

Questions

TEST TIME!

Unit 2: Percent

Topic A: Introducing Percent

Reading and Writing Percents

Changing Decimals to Percents

Changing Percents to Decimals

Changing Common Fractions to Percents

Method One

Method Two

Changing Percents to Common Fractions

Percents Less Than 1%

16⅔%, 33⅓%, 66⅔%, 83⅓% …

Review of Equivalent Common Fractions, Decimals, and Percents

Topic A: Self-Test

Unit 2 Review: Percent

Questions

Answers to Unit 2 Review

TEST TIME!

Unit 3: Working with Percent

Topic A: Finding a Percent of a Number

Percents Greater Than or Equal to 100%

Taxes

Cross-Border Shopping

Increases and Decreases, Discounts and Markups

Commission and Tips

Topic A: Self-Test

Unit 3 Review: Working with Percent

Questions

Answers to Unit 3 Review

TEST TIME!

Unit 4: More Working with Percent

Topic A: Finding What Percent One Number is of Another

Finding the Percent of an Increase or Decrease

Other Problems

School Grades

Topic A: Self-Test

Topic B: Finding a Number when a Percent of it is Given

Weeks of Exercise (Horizontal Axis)	Leah’s Heartbeats per Minute (Vertical Axis)	John’s Heartbeats per Minute (Vertical Axis)
1	78	92
2	78	92
3	76	90
4	74	89
5	75	90
6	73	88
7	71	86
8	70	87
9	68	85
10	69	83
11	67	81
12	67	80