Unit 1: Ratio, Rate, & Proportion
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… |
|---|---|---|
| [latex]1:2=2:4[/latex] | [latex]\frac{1}{2}=\frac{2}{4}[/latex] | 1 is to 2 as 2 is to 4 |
| [latex]1:4=25:100[/latex] | [latex]\frac{1}{4}=\frac{25}{100}[/latex] | 1 is to 4 as 25 is to 100 |
| [latex]18:9=10:5[/latex] | [latex]\frac{18}{9}=\frac{10}{5}[/latex] | 18 is to 9 as 10 is to 5 |
| [latex]15:20=3:4[/latex] | [latex]\frac{15}{20}=\frac{3}{4}[/latex] | 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:
[latex]\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}[/latex]
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.
[latex]\dfrac{time}{distance}=\dfrac{time}{distance}[/latex]
Or it could be:
[latex]\dfrac{distance}{time}=\dfrac{distance}{time}[/latex]
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
To Solve a Proportion Problem Using Equivalent Ratios
- Step 1
Decide on the order of comparison and write a ratio that describes the information given in the problem. Write a proportion using words of the items that are being compared in fraction form. - Step 2
Write two more ratios with the numbers matching the words in the first ratio. The missing term (number) can be given a letter (ex. N). - Step 3
Mentally set the ratio with words (the first ratio) aside. - Step 4
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 teaspoons of baking powder for 6 cups of flour. Call this term N.
- Step 1
Ratio is [latex]\dfrac{\text{baking powder}}{\text{flour}}[/latex] - Step 2
[latex]\dfrac{\text{baking powder}}{\text{flour}}=\dfrac{1}{2}=\dfrac{\textit{N}}{6}[/latex] - Step 3
[latex]\dfrac{1}{2}=\dfrac{\textit{N}}{6}[/latex] - Step 4
[latex]\dfrac{1}{2}\times \left(\dfrac{3}{3}\right)= \dfrac{1 \times 3}{2 \times 3}=\dfrac{3}{6}[/latex], so [latex]\dfrac{1}{2}=\dfrac{3}{6}[/latex], so [latex]\textit{N}=3[/latex]
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.
- Step 1
Ratio is [latex]\dfrac{\text{people who will miss work}}{\text{all people at work}}[/latex] - Step 2
[latex]\dfrac{\text{people who will miss work}}{\text{all the people at work}}=\dfrac{3}{10}=\dfrac{\textit{P}}{1000}[/latex] - Step 3
[latex]\dfrac{3}{10}=\dfrac{\textit{P}}{1000}[/latex] - Step 4
[latex]\dfrac{3}{10}\times\left(\dfrac{100}{100}\right)= \dfrac{3 \times 100}{10 \times 100}=\dfrac{300}{1000}[/latex], so [latex]\dfrac{3}{10}=\dfrac{300}{1000}[/latex], so [latex]\textit{P}=300[/latex]
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.
- Three cups of flour to one teaspoon of yeast.
Answer: [latex]\dfrac{\textit{flour}}{\textit{yeast}}[/latex] - 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
- [latex]\dfrac{\text{oil}}{\text{gasoline}}[/latex]
- [latex]\dfrac{\text{centimetres}}{\text{kilometres}}[/latex]
- [latex]\dfrac{\text{grams}}{\text{dollars}}[/latex]
- [latex]\dfrac{\text{eggs}}{\text{milk}}[/latex]
- [latex]\dfrac{\text{men}}{\text{women}}[/latex]
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?[latex]\dfrac{\textit{sugar}}{\textit{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[/latex]
- 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.
- [latex]3:5=\textit{Y}:15[/latex]
- [latex]1:2=\textit{P}:8[/latex]
- [latex]5:7=10:\textit{N}[/latex]
- [latex]2:3=8:\text{W}[/latex]
- [latex]4:7=16:\textit{A}[/latex]
- [latex]1:3=2:\textit{N}[/latex]
- 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
- [latex]\textit{Y}=9[/latex]
- [latex]\textit{P}=4[/latex]
- [latex]\textit{N}=14[/latex]
- [latex]\textit{W}=12[/latex]
- [latex]\textit{A}=28[/latex]
- [latex]\textit{N}=6[/latex]
- [latex]\textit{N}=100 \text{mL}[/latex]
Using Cross-Multiplication to Solve a Proportion
Review cross products:
Multiply the numerator of each fraction with the denominator of the other fraction.
[latex]\dfrac{2}{5}\rlap{\nearrow}{\searrow}\dfrac{4}{10}[/latex]
[latex]\begin{equation} \begin{split} 2\times10 & =5\times4 \\ 20 & =20 \end{split} \end{equation}[/latex]
[latex]2\times10=20[/latex] and [latex]5\times4=20[/latex]
Therefore: [latex]\dfrac{2}{5}=\dfrac{4}{10}[/latex]
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
[latex]\dfrac{2}{3}=\dfrac{\textit{N}}{45}[/latex]
Cross multiply:
[latex]\begin{equation} \begin{split} 2\times45 &= 3\times\textit{N} \\ 90 &= 3\textit{N} \end{split} \end{equation}[/latex]
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.
[latex]\dfrac{90}{3}=\dfrac{3\textit{N}}{3}[/latex]
Solve by reducing the [latex]\frac{3}{3}[/latex] and dividing 90 by 3.
[latex]\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}[/latex]
Reducing the fraction [latex]\dfrac{3\textit{N}}{3}[/latex] to [latex]\dfrac{1\textit{N}}{1}[/latex] to N is also called cancelling. In math, a fraction can be cancelled when the numerator and denominator are the same number.
e.g. [latex]\dfrac{6\textit{P}}{6}=\dfrac{1\textit{P}}{1}=\textit{P}[/latex]
Example B:
[latex]\dfrac{6}{7}=\dfrac{24}{\textit{N}}[/latex]
Cross multiply:
[latex]\begin{equation}\begin{split} 6\times\textit{N} &= 7\times24 \\ 6\textit{N} &= 168 \end{split}\end{equation}[/latex]
Divide both sides by 6. The 6’s with the N will cancel (reduce), and the N will be alone.
[latex]\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}[/latex]
Check by cross-multiplying:
[latex]\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}[/latex]
Example C
[latex]\dfrac{8}{10}=\dfrac{\textit{N}}{80}[/latex]
Cross multiply:
[latex]\begin{equation}\begin{split} 8\times80 &= 10\times\textit{N} \\ 640 &= 10\textit{N} \end{split}\end{equation}[/latex]
Divide both sides by 10 so N will be alone.
[latex]\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}[/latex]
To Solve a Proportion Problem Using Cross-Multiplication
- Step 1
Write the first ratio using the information given. - Step 2
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. - Step 3
Write the proportion in the fraction form. (Try to simplify the ratio before you do all the calculations). - Step 4
Cross-multiply and set the cross-products equal to each other. - Step 5
Divide both sides of the equation by the number with the unknown term. - Step 6
Check by putting your answer back into the original proportion and cross-multiplying.
Exercise 4
Practise using cross-multiplying to find the missing term in these proportions.
- [latex]\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}[/latex]
- [latex]\dfrac{4}{\textit{N}}=\dfrac{24}{30}[/latex]
- [latex]\dfrac{12}{4}=\dfrac{18}{\textit{x}}[/latex]
- [latex]\dfrac{\textit{y}}{6}=\dfrac{20}{12}[/latex]
- [latex]4:15=8:\textit{N}[/latex]
- [latex]\textit{W}:100=6:50[/latex]
Answers to Exercise 4
- [latex]\textit{N}=5[/latex]
- [latex]\textit{x}=6[/latex]
- [latex]\textit{y}=10[/latex]
- [latex]\textit{N}=30[/latex]
- [latex]\textit{W}=12[/latex]
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
Rewrite the proportion:
[latex]\dfrac{2\frac{1}{4}}{3}=\dfrac{\textit{N}}{7}[/latex]
Cross-multiply:
[latex]\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}[/latex]
Exercise 5
Practise using cross-multiplying to find the missing term in these proportions.
- [latex]\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}[/latex]
- [latex]3\frac{1}{2}:2=\textit{N}:8[/latex]
- [latex]9:6=4\frac{1}{2}:\textit{N}[/latex]
- [latex]7.5:\textit{B}=10:20[/latex]
- [latex]3.75:5=9\textit{x}[/latex]
- [latex]4\frac{1}{8}:\textit{A}=3:6[/latex]
Answers to Exercise 5
- [latex]\textit{N}=14[/latex]
- [latex]\textit{N}=3[/latex]
- [latex]\textit{B}=15[/latex]
- [latex]\textit{x}=12[/latex]
- [latex]\textit{A}=8\frac{1}{4}[/latex] or [latex]8.25[/latex]
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 [latex]\dfrac{0.5}{10}=\dfrac{\text{cm given in chart}}{\text{actual distance in km}}[/latex]
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.
- How much fertilizer is needed for a 130 m2 lawn?
- 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 -
- 2.21 kg
- 1.275 kg
Topic C: Self-Test
Mark /20 Aim 17/20
- Solve these proportions.
(6 marks)- [latex]\textit{N}:14=28:56[/latex]
- [latex]3:11=\textit{N}:22[/latex]
- [latex]50:45=10:\textit{N}[/latex]
- [latex]4\frac{1}{5}:\textit{Y}=3:2[/latex]
- (14 marks)
- Get a map of BC, a map of Canada, and a map of your city or town.
- Find the scale on each map (usually at the bottom) and write down the ratio of map distance to the actual distance.
- 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
-
- [latex]\textit{N}=7[/latex]
- [latex]\textit{N}=6[/latex]
- [latex]\textit{N}=9[/latex]
- [latex]\textit{Y}=2\frac{4}{5}[/latex] or [latex]2.8[/latex]
- See your instructor.