5.1: Writing Ratios

Kim Tamblyn

5.1: Writing Ratios

Chapter 5: Learning Outcomes

By the end of this chapter, you should be able to:

Read, write, interpret and compare ratios.
Read, write and identify proportions and use them to solve problems.
Use ratio and proportion to solve problems involving:
- Finding percent when part and whole are known
- Finding part when percent and whole are known
- Finding whole when part and percent are known

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

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

How To Pronounce Ratio – Pronunciation Academy [22 s] (Pronunciation Academy, 2015).

You often use ratios. For example:

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 being used. Read these examples of ratios and the units used. A general ratio may say “parts” for the units.

It rained for 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 as “4 is compared to 3” but is often read as “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.

Example: 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 the 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.

Exercise 1 Answers

$1:5$ 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 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; the order must match what the numbers mean. For example, 3 scoops of coffee to 12 cups of water must be written as $3:12$ 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 three different ways:

Using the $:$ symbol — $2:5$
As a common fraction — 25
- 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

Using the ratios you wrote in Exercise 1, write the ratios in the three ratio forms:

:
common fraction
to

Exercise 2 Answers

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

Exercise 3

Use the diagram in Figure 5.1.1 to write a ratio comparing the quantity of each shape, as asked.

Different shapes. There are 6 hearts, 7 suns, and 9 smilie faces. — **Figure 5.1.1**

Exercise 3 Answers

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

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
equivalent ratios in lower terms
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.

Example: $5:6= \dfrac{5}{6} \times \left(\dfrac{3}{3}\right)=\left(\dfrac{5\times 3}{6\times 3}\right)=\dfrac{15}{18}=15:18$

$4:3$
$10:2$
$50:1$
$9:4$
$3:5$

Exercise 4 Answers

See your instructor.

Exercise 5

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

Example: $4:12=\dfrac{4}{12}\div\left(\dfrac{4}{4}\right)=\left(\dfrac{4\div4}{12\div4}\right)=\dfrac{1}{3}=1:3$

$10:5$
$7:21$
$20:5$
$6:14$
$2:4$
$6:3$
$16:8$

Exercise 5 Answers

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 the 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 roadblock 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 the number of completions compared to the 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
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

Exercise 6 Answers

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

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

5.1: Writing Ratios

Chapter 5: Learning Outcomes