Number · Ratios and Proportion · Lesson 1

Ratios and the Language of Ratio

Masons, cooks, farmers and market women compare quantities all the time: one headpan of cement to three of sand, two cups of water for every cup of rice. A ratio is the mathematician's way of making such comparisons exact. In this lesson you will write ratios, simplify them and describe them in clear words like "twice as many" and "for every".

NaCCA B7.1.4.1.1 ⏱ about 45 min

By the end you will be able to

✔Write the ratio of two quantities in the correct order using the notation a : b.
✔Simplify a ratio to its lowest terms by dividing both parts by their HCF.
✔Change quantities to the same unit before writing them as a ratio.
✔Describe a ratio in words such as "for every", "twice as", "half of" and as a fraction.

Key words

Ratio: A comparison of two or more quantities of the same kind, written a : b and read "a to b".
Terms of a ratio: The numbers in a ratio. In 3 : 5, the first term is 3 and the second term is 5.
Simplest form: A ratio whose terms are whole numbers with no common factor except 1, e.g. 1 : 2 instead of 60 : 120.
HCF: Highest common factor: the biggest number that divides both terms exactly.
Ratio language: Words that describe a ratio, such as "for every", "twice as many", "three times as old" and "half as much".

Curriculum reference

Content standard: B7.1.4.1 Demonstrate an understanding of the concept of ratios and its relationship to fractions and use it to solve problems that involve rates, ratios, and proportional reasoning.

Indicator: Find ratio and use ratio language to describe relationship between two quantities.

Starter

"One to three, my brother!"

Walk past any building site in Kasoa or Ejisu and listen to the mason. "One to three!" he shouts, and his helper pours one headpan of cement for every three headpans of sand. If the helper gets it the wrong way round, three of cement to one of sand, the mixture wastes money and the wall may crack. That little phrase "one to three" is a ratio, and today you will learn to read, write and simplify ratios like a professional.

1. What is a ratio?

A ratio compares two quantities by showing how many times one contains the other, or how many of one there are for every certain number of the other. We write the ratio of 1 headpan of cement to 3 headpans of sand as 1 : 3 and read it as "one to three".

Think of the bird house at the Kumasi Zoo. Every bird has 2 wings and 1 beak. So the ratio of wings to beaks is 2 : 1, because for every 2 wings there is 1 beak. It does not matter whether there are 5 birds or 50 birds: the ratio stays 2 : 1.

However many birds there are, the ratio of wings to beaks simplifies to 2 : 1.
Number of birds	Wings	Beaks	Wings : Beaks
1	2	1	2 : 1
5	10	5	10 : 5 = 2 : 1
20	40	20	40 : 20 = 2 : 1

2. Simplifying a ratio

A ratio works exactly like a fraction: you can divide both terms by the same number and the comparison does not change. A ratio is in its simplest form when the only number that divides both terms is 1. The quickest way is to divide both terms by their HCF.

There are 60 boys and 120 girls in a school. Boys : girls = 60 : 120. The HCF of 60 and 120 is 60, so divide both terms by 60 to get 1 : 2. Notice this is the same as the fraction $\frac{60}{120} = \frac{1}{2}$ . You can also simplify in smaller steps, as the animation shows.

▶ Each step divides BOTH terms by the same number. Notice the ratio gets simpler but still says the same thing.

Same units first. A ratio compares quantities of the same kind, so both must be in the same unit before you simplify. To compare 50 pesewas with GH₵2, change GH₵2 to 200 pesewas: 50 : 200 = 1 : 4. The answer has no units, because the units cancel, just like in a fraction.

Quantities	Same unit	Divide by HCF	Simplest form
12 : 18	12 : 18	÷ 6	2 : 3
50p : GH₵2	50 : 200	÷ 50	1 : 4
45 min : 2 h	45 : 120	÷ 15	3 : 8
400 g : 1.2 kg	400 : 1,200	÷ 400	1 : 3
1.5 : 2.5	15 : 25 (× 10)	÷ 5	3 : 5

3. Speaking the language of ratio

A good mathematician can say what a ratio means in plain words. Alhasan is 50 years old and his son Musa is 25 years old. Musa's age : Alhasan's age = 25 : 50 = 1 : 2. We can describe this in several correct ways:

Alhasan is twice as old as his son.
Musa is half the age of his father.
For every 1 year of Musa's age, Alhasan has lived 2 years.
Musa's age is $\frac{1}{2}$ of Alhasan's age.

A ratio also tells you what fraction of the whole each part is. In the school with boys : girls = 1 : 2, think of the pupils in groups of 1 boy and 2 girls. Each group has 1 + 2 = 3 pupils, so the boys are $\frac{1}{3}$ of the school and the girls are $\frac{2}{3}$ . The tape diagram below shows this.

▶ Count the equal boxes: 3 altogether. Boys fill 1 of the 3 boxes, so boys are one third of the school.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Boys and girls in a school

There are 60 boys and 120 girls in a school. Find the ratio of boys to girls in its simplest form.

Show the solution

Step 1Write the ratio in the order asked: boys : girls = 60 : 120.
Step 2HCF of 60 and 120 is 60.
Step 3Divide both terms by 60: 60 ÷ 60 = 1 and 120 ÷ 60 = 2.
Step 4As a fraction check: $\frac{60}{120} = \frac{1}{2}$ ✓

Answer: 1 : 2

Example 2

Wings and beaks

A cage at the Kumasi Zoo holds 5 birds. Write the ratio of wings to beaks and explain what it means.

Show the solution

Step 1Wings = 5 × 2 = 10; beaks = 5 × 1 = 5.
Step 2Wings : beaks = 10 : 5. Divide both by 5: 2 : 1.
Step 3Meaning: for every 2 wings there is 1 beak.

Answer: 2 : 1

Example 3

Changing to the same unit

Express 75 pesewas to GH₵3 as a ratio in its simplest form.

Show the solution

Step 1Change GH₵3 to pesewas: 3 × 100 = 300 pesewas.
Step 2Ratio = 75 : 300.
Step 3HCF of 75 and 300 is 75. Divide both by 75: 1 : 4.

Answer: 1 : 4

Example 4

Describing ages with ratio language

Alhasan is 50 years old and his son Musa is 25. Write the ratio of Musa's age to Alhasan's age and describe it in two ways.

Show the solution

Step 1Musa : Alhasan = 25 : 50.
Step 2Divide both by 25: 1 : 2.
Step 3In words: Alhasan is twice as old as Musa; Musa is half the age of his father.

Answer: 1 : 2 — Alhasan is twice as old as Musa

Example 5

Decimals in a ratio

A trader mixes 1.5 kg of groundnuts with 2.5 kg of maize. Find the ratio of groundnuts to maize in its simplest form.

Show the solution

Step 1Ratio = 1.5 : 2.5.
Step 2Multiply both terms by 10 to remove decimals: 15 : 25.
Step 3HCF of 15 and 25 is 5. Divide both by 5: 3 : 5.

Answer: 3 : 5

Summary

A ratio a : b compares quantities of the same kind; read it "a to b".
Write the terms in the same order as the words in the question.
Simplify by dividing both terms by their HCF; change to the same unit first and clear decimals.
Ratio language: 1 : 2 means "for every 1 there are 2", "twice as many" or "half as many".
In a ratio a : b, the first part is $\frac{a}{a + b}$ of the whole.

"One to three, my brother!"

1. What is a ratio?

2. Simplifying a ratio

3. Speaking the language of ratio

Worked examples

Boys and girls in a school

Wings and beaks

Changing to the same unit

Describing ages with ratio language

Decimals in a ratio

Summary

Practice: 15 questions