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4
Welcome to Adult Literacy Fundamental Mathematics: Book 6.
You have the skills you need to be a strong student in this class. Your instructor knows this because you have passed the Adult Literacy Fundamental Mathematics Level 5 class, or you have been assessed into this level.
Adult math learners have many skills. They have a lot of life experience. They also use math in their everyday lives. This means that adult math learners may already know some of what is being taught in this book. Use what you already know with confidence!
This textbook has:
You have also been given a sheet to write down your grades. After each test, you can write in the mark. This way you can keep track of your grades as you go through the course. This is a good idea to use in all your courses.
Unit | Practice Test | 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 |
5
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:
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.
Read the list below and put a check mark beside the ones you feel when thinking about or doing math.
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. |
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.
Hopefully by examining the beginnings of the anxiety, you can feel more in control of it.
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:
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.
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?
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?
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.
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.
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.
Here is an explanation of each reason and how to work your way out of the anxiety you may feel during tests.
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.
Here are some strategies students should know about how to write a test to do the best as possible on it:
There are many reasons why a student may feel mental pressure when writing a test. Listed below are a few main reasons:
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:
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!).
I
Topics
Unit 1 will introduce ratios, rates and proportions.
1
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:
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.
Exercise 1
Write the ratios asked for in these questions using the symbol (for example, ). Write the units and what is being compared beside the ratio.
Answers to Exercise 1
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 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 .
Ratios can be written 3 different ways:
Exercise 2
Use the ratios you wrote in Exercise 1 to complete the chart.
Common fraction | to | ||
---|---|---|---|
A | 1 to 3 | ||
B | |||
C | |||
D | |||
E |
Answers to Exercise 2
Exercise 3
Use the diagram to write a ratio comparing the quantity of each shape, as asked.
Answers to Exercise 3
Like equivalent fractions, equivalent ratios are equal in value to each other.
Ratios can be written as common fractions. It is convenient to work with ratios in the common fraction form.
You can then easily:
Example A
Express in higher terms.
is equivalent to
Example B
Express in lower terms.
is equivalent to
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.
Answers to Exercise 4
See your instructor.
Exercise 5
Write these ratios in lowest terms—that is, simplify the ratios.
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.
Exercise 6
Using a colon, write a ratio in lowest terms for the information given.
Answers to Exercise 6
Mark /12 Aim 10/12
2
When a ratio is used to compare two different kinds of measure (e.g. apples and oranges, or meters 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 . Find the rate by making the denominator 1.
Divide
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 . Find the rate by making the denominator 1.
Divide
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 . Find the rate by making the denominator 1.
Divide
The rate is 2 oz/m2, or 2 ounces per square metre.
Exercise 1
Write the following ratios as rates, comparing distance to time.
Answers to Exercise 1
Exercise 2
Write the following ratios as rates.
Answers to Exercise 2
Mark /7 Aim 6/7
3
A proportion is a statement that two ratios are equal or equivalent. Here are some proportions:
Proportion | Fraction Form | Read like this… |
---|---|---|
1 is to 2 as 2 is to 4 | ||
1 is to 4 as 25 is to 100 | ||
18 is to 9 as 10 is to 5 | ||
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:
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:
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.
Or it could be:
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?
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 teaspoons of baking powder for 6 cups of flour. Call this term N.
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 1000 who will miss work due to back pain. Call this term P.
300 people out of 1,000 people may miss work due to back pain.
Exercise 1
Write the ratio of the words to describe the information given.
Answers for Exercise 1
Exercise 2
Use equivalent ratios to find the answers.
Answers for Exercise 2
Exercise 3
Use equivalent ratios to find the missing term in these proportions.
Answers for Exercise 3
Review cross products:
Multiply the numerator of each fraction with the denominator of the other fraction.
and
Therefore:
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
Cross multiply:
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:
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.
Solve by reducing the and dividing 90 by 3.
Reducing the fraction to to N is also called cancelling. In math, a fraction can be cancelled when the numerator and denominator are the same number.
e.g.
Example B:
Cross multiply:
Divide both sides by 6. The 6’s with the N will cancel (reduce), and the N will be alone.
Check by cross-multiplying:
Example C
Cross multiply:
Divide both sides by 10 so N will be alone.
Exercise 4
Practise using cross-multiplying to find the missing term in these proportions.
Answers to Exercise 4
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
Exercise 5
Practise using cross-multiplying to find the missing term in these proportions.
Answers to Exercise 5
Exercise 6
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 |
Answers to Exercise 6
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 |
Mark /20 Aim 17/20
Ask your instructor to mark your work.
4
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!
II
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.
5
To write a percent:
To read a 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.
Answers to Exercise 1
Exercise 2
Answers to Exercise 2
Writing equivalent fractions is an important math skill.
Equivalent common fractions, decimals, and percents all represent the same amount.
Fractions | Decimals | Percentages |
---|---|---|
You need the skill of writing equivalent fractions for working with percents.
Remember this shortcut for multiplying by 100?
The shortcut is: When multiplying by 100, move the decimal point two places to the right.
Example A
So…
Example B
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.
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.
Exercise 3
Change these decimals to percents.
Decimal | × 100% Move decimal 2 places to right | = Percent | |
---|---|---|---|
A. | |||
B. | |||
C. | |||
D. | |||
E. | |||
F. | |||
G. | |||
H. |
Answers to Exercise 3
Review dividing by 100:
To divide by 100, move the decimal point two places to the left.
Example A
Change each percent to a decimal or mixed number.
So…
Example B
Change each percent to a decimal.
Some notes to remember:
Exercise 4
Change each percent to its decimal equivalent.
Percent | ÷ 100% Move decimal 2 places to left | = Decimal | |
---|---|---|---|
A. | |||
B. | |||
C. | |||
D. | |||
E. | |||
F. |
Answers to Exercise 4
To change a percent containing a common fraction to a decimal, do this:
Example C
Exercise 5
Change each percent to its decimal equivalent.
Answers to Exercise 5
There are two methods you can use to change a common fraction to a percent.
To change a common fraction to an equivalent percent, multiply the common fraction by 100%.
Example A
Multiply by 100% | |
Convert 100% to a fraction. | |
Simplify the fractions by dividing the numerator and denominator by 4. | |
The 4 cancels and 100 is reduced to 25. | |
Multiply 3 by 25%. | |
Convert number to a fraction. | |
Multiply by 100%. | |
Convert 100% to a fraction. | |
Simplify the fractions by dividing the denominator and numerator by 5. | |
The 5 cancels and 100 is reduced to 20. | |
Multiply 6 by 20%. | |
Exercise 6
Multiply by 100% to change each common fraction to an equivalent percent.
Answers to Exercise 6
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
Use long division to write fraction as a decimal. | |
Move the decimal point two places to the right and add the percent sign. |
Use long division to write fraction as a decimal. | |
Move the decimal point two places to the right and add the percent sign. |
Use long division to write fraction as a decimal. | |
Move the decimal point two places to the right and add the percent sign. |
Exercise 7
Answers to Exercise 7
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:
You know that percents are a form of fraction with an unwritten denominator of 100. A % sign is used.
To change a percent to a common fraction:
Example A
Write each percent as a common fraction.
Note that percents greater than or equal to 100 become improper fractions which will be rewritten as mixed numbers.
Remember 100% is the whole thing. .
Exercise 8
Change each percent to a common fraction. Simplify to lowest terms.
Answers to Exercise 8
Sometimes a percent smaller than 1% is used. For example, you will hear amounts such as or or on the news about the Bank of Canada rate and the rise and fall of inflation. These are small amounts. Sometimes the expression “” is used instead of “”.
What is ?
is
, so
What is ?
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).
If the percent is expressed as a common fraction, do this:
Exercise 9
Change each percent to an equivalent decimal.
Answers to Exercise 9
These percents will become repeating decimals. For example:
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 work space.
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 |
---|---|---|
Answers to Review of Equivalent Common Fractions, Decimals, and Percents
Common Fraction | Decimal | Percent |
---|---|---|
Mark /15 Aim 13/15
Answers to Topic A Self-Test
6
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
Topics
In this unit you will learn to solve three types of percent problems:
Each type of percent problem can be solved using the following proportion:
Both ratios in this proportion use the same order of comparison because in the ratio , 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:
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:
7
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 ?
Solve the proportion.
Example B
What is ?
Solve the proportion:
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.
In percent problems, the number after the word of usually represents the whole.
Exercise 1
Solve each problem by setting up the proportion .
Answers to Exercise 1
Remember that .
100% of anything is the whole thing. If you spend 100% of your pay cheque, 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.
If a percent is 100, it equals the whole thing.
If a percent is more than (>) 100, it is more than the whole thing.
Exercise 2
Look at the percent. Is it 100? Circle the correct answer for each question. Do not solve the problems.
Answers to Exercise 2
Exercise 3
Review p. 70 first and use the proportion method to solve these questions.
Answers to Exercise 3
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:
Please note that the tax rates used in the questions in this book are for the year 2010 and are subject to change.
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:
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. The first $28,000 of taxable income is taxed at 17%. Note that other tax rules and charges may apply in real situations.
Example B
If a person’s taxable income for the year is $23,400, what amount of income tax will that person pay? To use the proportion method, do this:
The tax is the part. The income is the whole.
Solve for $3,978.
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%.
Answers to Exercise 5
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 ), so CAN money was valued at 92% of US money.
To find the value of one US dollar in Canadian funds, use this proportion:
, so US money was valued at 109% of CAN money.
Note that the proportion changes as the exchange rate changes.
What if you buy in the United States?
Look at this example (assume $1.00 CAN = $0.92 US).
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.
Answers to Exercise 6
Increases (amount changing to more) and decreases (amount changing to less) are often given as a percent. For example,
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:
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:
Decrease and discount problems may need to be solved in several steps. Sometimes the problems ask for:
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
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 from p. 70 that it’s best to substitute ⅓ for 33⅓%.
Second: Subtract the savings from the original amount.
Exercise 7
Solve these problems. Round all answers to the nearest cent.
Answers to Exercise 7
Increases and mark-ups are calculated in the same way as decreases and discounts. However, an increase or mark-up 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?
Mark-ups 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 mark-up:
For example, the mark-up on leather shoes may be 45%, but on running shoes it may be 60%. Kitchen appliances might have a 42% mark-up, while lawn mowers might have a 55% mark-up.
Example D
A shoe seller pays $40.00 per pair of running shoes from the factory. The shoe seller makes the mark up 75%. What is the selling price of the shoes?What is 75% of $40.00?
75% of $40.00 is $30.00.
Add the mark up to the original cost to get the selling price of the shoes:
.
Wage Increase
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 pay cheque, 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.
Exercise 8
Solve the problems. Round money to the nearest cent.
Cost Price to Business | Mark-up (75%) | Selling Price |
---|---|---|
Silk Flowers: $1.48 | ||
Stuffing: $4.50/bag | ||
Beads: $3.20/dozen |
Answers to Exercise 8
Cost Price to Business | Mark-up (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 |
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.
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:
The commission is the part.
The total amount is the whole.
Commission problems often have several steps. You may have to:
Example A
The bill for the excellent dinner at the restaurant was $56.40. The service had been good and the waiter very pleasant so Bill and Diane wanted to leave at least a 15% tip.
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 earning be?You are asked to find the gross monthly earnings. What do you know?
Exercise 9
Answers to Exercise 9
More Problems for Finding a Percent of a Number
Answers to More Problems for Finding a Percent of a Number
Mark /11 Aim 9/11
Answers to Topic A Self-Test
8
Item | Cost | HST (12%) | Total Cost | |
---|---|---|---|---|
A | T-shirt | $14.99 | ||
B | 40″ flat screen TV | $699.00 | ||
C | Tent | $168.79 | ||
D | Drill and bit set | $248.99 | ||
E | Shoes | $79.98 |
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 |
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!
IV
9
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?
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: .
An equal to sign (=) can substitute for “is”.
12 is what percent of 15?
will help to find .
Example B
What percent of 85 is 60?
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.
Answers to Exercise 1
Exercise 2
Solve each question by first setting up the proportion . Review p. 70 as necessary.
Answers to Exercise 2
Exercise 3
Solve the following by setting up the proportion.
Answers to Exercise 3
You learned in 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 which ask you to find the percent of increase or decrease often involve two steps:
Example A
The rent went from $375 a month to $427.50 a month. What is the percent of the increase?
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?
The hours of operation at the college were cut 14²⁄₇%.
Exercise 4
Solve the following problems.
Answers to Exercise 4
Many situations compare one number to another.
These numbers are often more easily thought about if written as a percent.
The following problems ask you to find what percent one number is of another. Often several steps are involved to calculate the part or to calculate the whole (as in question e) You may be asked to use the % after you find it. Remember the whole thing = 100%.
Exercise 5
Solve the following problems.
Answers to Exercise 5
When looking at test results, the mark shows how you did on the test.
If you get on a test, you know you got 7 answers right, and 3 answers wrong.
Sometimes it is also helpful to see your mark as a percentage.
Example A
By solving for N, the percentage can be found:
So, .
Now, you can see that the test mark of equals .
Example B
The test result was , what was the percent on the test?
If you round, .
Not such a great mark!
Example C
Find the percent of the following grade: .
or .
Exercise 6
Find the percents for the following test grades. Round your answer to the nearest percent.
Answers to Exercise 6
Mark /12 Aim 10/12
Answers to Topic A Self-Test
10
In problems when a certain percentage of a number is given, 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?
Check by finding 20% of 70. The answer should be 14.
Example B
33⅓% of is 60.
Set up proportion:
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.
Answers to Exercise 1
Exercise 2
Solve the following. Check your answers to see if you set up the proportion correctly.
Answers to Exercise 2
Exercise 3
Solve the questions.
Answers to Exercise 3
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.
Answers to Exercise 4
Mark /8 Aim 6/8
Answers to Topic B Self-Test
11
You have been practising three types of percent problems. You have learned that one proportion can be used to solve all the problems:
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 or the appropriate short method.
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 sentence answer. For these problems, round your answers this way:
Item | Selling Price | Store Fee | Amount for Lisa & Her Daughters | |
---|---|---|---|---|
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 |
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 |
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!
V
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:
Here is a trick to remember the difference between horizontal and vertical:
Bar Graph Image: A bar graph showing the favourite fruits of upgrading students. The graph is labelled with the following:
12
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
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 One records Leah’s heart rate for the first 12 weeks of her exercise program.
On this particular graph the vertical scale is one heartbeat per line. The scale on the horizontal axis is one week per line.
Leah’s basal heart rate was 74 beats/min in week four.
Often, you need to estimate the value of the point in the graphed line. Look at Graph Two which has Leah’s same heart rates recorded.
Answers to Graph 1
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.
Answers to Graph 2
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.Graphs often show information about several things 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 he would 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.
Answers to Graph 3
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.
Graph 4
Answers to Graph 4
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,
Graph 5
Answers to Graph 5
The steepness of the slant of a graphed line gives you a picture 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.
A line graph shows the change in Leah’s weekly basal heart rate over 12 weeks of exercise.
The line graph data is represented in the following table:
Weeks of Exercise (Horizontal Axis) | Heart Beats 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 |
A line graph showing Leah’s weekly basal heart rate.
The line graph data is represented in the following table:
Weeks of Exercise (Horizontal Axis) | Heart Beats 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 |
A line graph showing Leah and John’s weekly basal heart rate.
The line graph data is represented in the following table:
Weeks of Exercise (Horizontal Axis) | Leah’s Heart Beats per Minute (Vertical Axis) | John’s Heart Beats 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 |
A line graph showing the Cross family’s electricity costs.
The line graph data is represented in the following table:
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 |
A line graph showing the Cross family’s electricity costs, displaying both average cost/month and annual kilowatt hours (kWh) used.
The line graph data is represented in the following table:
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 |
13
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.
Graph 1
Answers to Graph 1
Graph 2
Answers for Graph 2
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
Answers for Graph 3
A bar graph showing the lengths of some British Columbia rivers.
The bar graph data is represented in the following table:
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 |
A bar graph showing the lengths of some British Columbia rivers.
The bar graph data is represented in the following table:
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 |
A bar graph showing the population of the world’s most populated countries in 2010.
The bar graph data is represented in the following table:
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 |
A bar graph showing the population of the world’s most populated countries in 2010 and 1950.
The bar graph data is represented in the following table:
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 |
14
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
Answers for Graph 1
A picture graph showing the results of an informal survey of vehicles per 1,000 used in summer months.
The picture graph data is represented in the following table:
Type of Vehicle (Vertical Axis) | Number of Pictures (Horizontal Axis) |
---|---|
Cars | 6 |
Bikes & Motorcycles | 1 |
Large Trucks | 1 |
RVs and Small Trucks | 2 |
15
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
Answers to Graph 1
Graph 2
2004 Nanaimo Regional Landfill Solid Waste Composition
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
A circle graph showing the Canadian tax dollar was spent in the 2007-2008 fiscal year.
The circle graph data is represented in the following table:
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) |
A circle graph showing the solid waste composition of the Nanaimo Regional Landfill in 2004.
The circle graph data is represented in the following table:
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 |
16
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:
Then the information is put into graph form.
Graph 1
Answers to Graph 1
Graph 2
In this Fundamentals Math course, students’ marks were collected from the final test. The graph shows the results.
Answers to Graph 2
A histogram showing the daily minimum temperatures of Atlin, British Columbia in the month of January, 2010.
The histogram data is represented in the following table:
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 |
A histogram showing the scores on the final math test for Fundamentals Math students.
The histogram data is represented in the following table:
Test Score Range, in percentages (Horizontal Axis) | Number of Students (Vertical Axis) |
---|---|
60%–69% | 1 |
70%–79% | 2 |
80%–89% | 12 |
90%–99% | 10 |
17
Tables are an everyday way of organizing information, or one’s own work.
Graph 1
It shows the departure times from each community.
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 am | 12:00 pm |
3:15 am | 5:15 pm |
7:15 am | 8:45 pm |
Answers to Graph 1
Graph 2
A cereal recipe explains the quantities to use when making hot cereal.
Servings | Salt | Water | Cereal |
---|---|---|---|
1 | ¼ tsp | 1.5 cups | ¼ cups |
4 | 1 tsp | 6 cups | 1 cup |
Answers to Graph 1
18
Refer back to each lesson on graphs and explain when to use any three types of graph.
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!
1
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.
A line graph displays the average temperature in Smithers, BC each month.
The line graph data is represented in the following table:
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 |
A bar graph displays the height of the highest mountains in British Columbia in metres.
The bar graph data is represented in the following table:
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 |
A picture graph displays the number of students at Marmot Elementary School who bike to school each day by grade.
The picture graph data is represented in the following table:
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 |
A circle graph displays the favourite ice cream flavours per 1,000 people.
The circle graph data is represented in the following table:
Flavour | Percentage of Total |
---|---|
Vanilla | 31% |
Fruit | 18% |
Nut | 16% |
Candy | 14% |
Chocolate | 11% |
Cake and Cookie | 10% |
2
The numbers to be added together in an addition question. In 3 + 5 = 8, the addends are 3 and 5.
Any straight line used for measuring or as a reference.
Balance has many meanings. In money matters, the balance is the amount left. It might be the amount left in a bank account (bank balance) or it might be the amount you still must pay on a bill (balance owing).
A cheque that has been cashed. The cheque is stamped, or cancelled, so it is no longer negotiable.
The distance around a circle; the perimeter of a circle.
Salespeople may be paid a percentage of the money made in sales. The commission is part or all of their earnings.
e.g., ⅔, ³⁄₇ , ⁴⁹⁄₅₀
In a proportion, multiply the numerator of the first fraction times the denominator of the second fraction. Then multiply the denominator of the first fraction times the numerator of the second fraction. In a true proportion, the products of the cross multiplication are equal.
The bottom number in a common fraction; tells into how many equal parts the whole thing has been divided.
The distance across a circle through its centre.
The result of a subtraction question, the answer. Subtraction gives the difference between two numbers.
Any of the ten numerals (0 to 9) are digits. This term comes from our ten fingers which are called digits. The numerals came to be called "digits" from the practice of counting on the fingers!
An amount taken off the regular cost. If something is bought "at a discount" it is bought at less than the regular price.
To separate into equal parts.
The number or quantity to be divided; what you start with before you divide.
The number of groups or the quantity into which a number (the dividend) is to be separated.
The same as
A mathematical statement that two quantities are equal. An equation may use numerals with a letter to stand for an unknown quantity. 6 + Y = 9
Equal in value; equivalent numbers (whole or fractions) can be used interchangeably; that is, they can be used instead of each other.
Make an approximate answer. Use the sign ≈ to mean approximately equal.
The numbers or quantities that are multiplied together to form a given product. 5 × 2 = 10, so 5 and 2 are factors of 10.
Part of the whole; a quantity less than one unit.
In a flat position, e.g. we are horizontal when we lie in a bed. A horizontal line goes across the page.
A common fraction with a value equal to or more than one.
Without end, without limit.
To turn upside down.
With the same denominators.
When the terms of a common fraction or ratio do not have a common factor (except 1), the fraction or ratio is in lowest terms (also called simplest form).
The first number in a subtraction question.
A whole number and a decimal fraction. 1.75
A whole number and a common fraction. 1 ¾
If a certain number is multiplied by another number, the product is a multiple of the numbers. Think of the multiplication tables. For example, 2, 4, 6, 8, 10, 12, 14... are multiples of 2.
The number to be multiplied.
The number you multiply by.
Something which can be cashed, that is, exchanged or traded as money.
Numbers represent the amount, the place in a sequence; number is the idea of quantity or order.
The digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 are also called numerals. These ten digits are combined to make infinite numerals. Digits are like letters, numerals are like words, and numbers are the meaning.
The top number in a common fraction; the numerator tells how many parts of the whole thing are being considered.
If the value of the cheques or money taken from a bank account is higher than the amount of money in the account, then the account is overdrawn. The account is "in the hole" or "in the red" are expressions sometimes used.
Two objects or lines side by side, never crossing and always the same distance from each other. Railway tracks are parallel, the lines on writing paper are parallel.
For every one hundred.
The distance around the outside of a shape.
We understand numbers by the way the digits (numerals) are arranged in relationship to each other and to the decimal point. Each position has a certain value. Our number system is a decimal system. The place value is based on ten.
A number that can only be divided evenly by itself and 1.
The result of a multiplying question, the answer.
A common fraction with a value less than one.
Generally, proportion is a way of comparing a part of something to the whole thing. E.g., his feet are small in proportion to his height. In mathematics, proportion is used to describe two or more ratios that are equivalent to each other.
The result of a division question; the quotient tells how many times one number is contained in the other.
The distance from the centre of a circle to the outside of the circle.
The relationship between two or more quantities. E.g., the ratio of men to women in the armed forces is 10 to 3 (10:3)
A number, when multiplied by its reciprocal, equals 1. To find the reciprocal of a common fraction, invert it. ⅗ × ⁵⁄₃ = 1
Write a common fraction in lowest terms. Divide both terms by same factor.
The amount left when a divisor does not divide evenly into the dividend. The remainder must be less than the divisor.
In mathematics, a symbol that tells what operation is to be performed or what the relationship is between the numbers.
+ plus, means to add
− minus, means to subtract
× multiplied by, "times"
÷ divided by, division
= equal, the same quantity as
≠ not equal
≈ approximately equal
< less than
> greater than
≤ less than or equal to
≥ greater than or equal to
See reduce.
The amount that is taken away in a subtraction question.
The result of an addition question, the answer to an addition question.
A written or printed mark, letter, abbreviation etc. that stands for something else.
a) A definite period of time, such as a school term or the term of a loan.
b) Conditions of a contract; the terms of the agreement.
c) In mathematics, the quantities in a fraction and in a ratio are called the terms of the fraction or the terms of the ratio. In an algebra equation, the quantities connected by a + or − sign are also called terms.
The amount altogether.
One piece of business. A transaction often involves money. When you pay a bill, take money from the bank or write a cheque, you have made a transaction.
Any fixed quantity, amount, distance or measure that is used as a standard. In mathematics, always identify the unit with which you are working. E.g., 3 km, 4 cups, 12 people, $76, 70 books, 545 g.
The price for a set amount. E.g., price per litre, price per gram.
Fractions which have different denominators.
In an up and down position, e.g., we are vertical when we are standing up. On a page, a vertical line is shown from the top to the bottom of the page.
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Version | Date | Change | Details |
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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. |
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