Ratio and Proportions

Lessons

Senior 1 · Mathematics

Ratio and Proportions

5 lessons

Lesson 1 of 5

Ratios: simplifying and sharing

22 min

Let's begin

A builder making mortar for a wall uses 1 bucket of cement for every 4 buckets of sand. For a small repair she uses 2 buckets of cement and 8 of sand; for a whole house, 50 bags of cement and 200 of sand. The amounts change, but the comparison stays the same: always four times as much sand as cement. That fixed comparison is a ratio.

Definition

Ratio: a comparison of two or more quantities of the same kind, measured in the same unit. The ratio of $a$ to $b$ is written $a : b$ and read " $a$ to $b$ ". It can also be written as the fraction $\frac{a}{b}$ .

Example: cement to sand is $1 : 4$ ; sand to cement is $4 : 1$ . The order matters.

Definition

Equivalent ratios: ratios that make the same comparison, such as $1 : 4 = 2 : 8 = 50 : 200$ .

Simplest form: an equivalent ratio of whole numbers with no common factor except 1. Example: the simplest form of $12 : 18$ is $2 : 3$ .

1. Simplifying a ratio

Rule. Multiplying or dividing every part of a ratio by the same non-zero number gives an equivalent ratio. To simplify, divide every part by the highest common factor (HCF).

Why. A ratio is a fraction in disguise: $2 : 8$ is $\frac{2}{8}$ , and $\frac{2}{8} = \frac{1}{4}$ because the top and bottom were both divided by 2. Making the batch of mortar twice as big doubles both amounts, so the comparison does not change.

Before simplifying:

write both quantities in the same unit;
clear decimals or fractions by multiplying every part by the same number.

Worked example

Write each ratio in its simplest form

Ratio	Working	Simplest form
$24 : 36$	HCF is 12; divide both parts by 12	$2 : 3$
$750\ \text{g} : 2\ \text{kg}$	$750 : 2,000$ in grams; divide by 250	$3 : 8$
$45\ \text{min} : 2\ \text{h}$	$45 : 120$ in minutes; divide by 15	$3 : 8$
$1.5 : 2$	multiply by 2 to get $3 : 4$	$3 : 4$
$\frac{1}{2} : \frac{1}{3}$	multiply by 6, the LCM of 2 and 3	$3 : 2$
$6 : 9 : 15$	HCF is 3	$2 : 3 : 5$

2. Sharing in a given ratio

Important

To share a total in the ratio $a : b : c$ :

Add the parts: $a + b + c$ equal parts in all.
Find one part: $\text{one part} = \dfrac{\text{total}}{a + b + c}$ .
Multiply: the first share is $a \times \text{one part}$ , and so on.
Check that the shares add up to the total.

In one formula, the first share is $\dfrac{a}{a + b + c} \times \text{total}$ .

Worked example

Three members of a maize cooperative invested 200,000 Frw, 300,000 Frw and 500,000 Frw. They share a profit of 1,500,000 Frw in the ratio of their investments. How much does each receive?

Ratio of investments: $200,000 : 300,000 : 500,000 = 2 : 3 : 5$ (divide by 100,000).
Total parts: $2 + 3 + 5 = 10$ .
One part: $\dfrac{1{,}500{,}000}{10} = 150{,}000$ Frw.
Shares: $2 \times 150{,}000 = 300{,}000$ ; $3 \times 150{,}000 = 450{,}000$ ; $5 \times 150{,}000 = 750{,}000$ Frw.
Check: $300,000 + 450,000 + 750,000 = 1,500,000$ . ✓

Worked example

Uwase and Gatete share some money in the ratio $3 : 5$ . Uwase receives 9,000 Frw. How much does Gatete receive, and how much was shared?

This time you know one share, not the total. Uwase's 3 parts are 9,000 Frw, so one part is $9{,}000 \div 3 = 3{,}000$ Frw. Gatete gets $5 \times 3{,}000 = 15{,}000$ Frw, and the total is $8 \times 3{,}000 = 24{,}000$ Frw.

Worked example

In a class the ratio of girls to boys is $5 : 4$ , and there are 6 more girls than boys. How many students are in the class?

The difference between the parts is $5 - 4 = 1$ part, and that is 6 students. So one part is 6. The class has $5 + 4 = 9$ parts, which is $9 \times 6 = 54$ students: 30 girls and 24 boys.

Warning

$2 : 3$ does not mean "2 out of 3". In a class where girls to boys is $2 : 3$ , there are $2 + 3 = 5$ parts, so the girls are $\frac{2}{5}$ of the class, not $\frac{2}{3}$ . And keep the order: if the question asks for boys to girls, the answer is $3 : 2$ .

Warning

Same units first. $50\ \text{cm} : 2\ \text{m}$ is not $50 : 2 = 25 : 1$ . Change to centimetres: $50 : 200 = 1 : 4$ .

Tip

Draw the bar model. One box for each part turns every ratio problem into "find the value of one box", whether you are given the total, one share, or the difference.

Corbettmaths: "Ratio sharing the total". Watch how the parts are added first, then one part is found, exactly as in the cooperative example.

✦

Practice

Give ratios in their simplest form.

Simplify $18 : 27$ .
Simplify $400\ \text{m} : 1.2\ \text{km}$ .
Simplify $2.5 : 4$ .
Mortar is mixed in the ratio cement : sand $= 1 : 4$ . How many buckets of sand are needed with 3 buckets of cement?
Share 42,000 Frw in the ratio $2 : 5$ .
Share 36 kg of fertiliser between three plots in the ratio $1 : 2 : 3$ .
Mugisha and Keza share sweets in the ratio $4 : 7$ . Keza gets 21 sweets. How many does Mugisha get, and how many were there in all?
In a club the ratio of boys to girls is $3 : 5$ , and there are 10 more girls than boys. How many members does the club have?

Show answersHide answers

$2 : 3$
$400 : 1,200 = 1 : 3$
Multiply by 2: $5 : 8$
$3 \times 4 = 12$ buckets
One part $= 42{,}000 \div 7 = 6{,}000$ . Shares: 12,000 Frw and 30,000 Frw.
One part $= 36 \div 6 = 6$ kg. Shares: 6 kg, 12 kg and 18 kg.
One part $= 21 \div 7 = 3$ . Mugisha gets 12; there were 33 sweets.
The difference is 2 parts $= 10$ , so one part is 5. Total $= 8 \times 5 = 40$ members.

Sources and credits

Grade 6, Unit 2: Introducing Ratios, IM 6–8 Math by Illustrative Mathematics, originally developed by Open Up Resources (© 2017–2019 Open Up Resources; adaptations © 2019 Illustrative Mathematics), provided by Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: Ratio sharing the total, corbettmaths, embedded from YouTube.

The mortar, cooperative and classroom examples, the bar models, the practice items and the quiz questions were written by Ganzaa.

11. Simplifying a ratio

22. Sharing in a given ratio

1. Simplifying a ratio

2. Sharing in a given ratio