Ratio and proportion
Ratio
A ratio is a way of comparing two or more numbers. In this unit you learn to find equivalent ratios and to write a ratio in its simplest form.
What a ratio is
A ratio is a relationship between two or more numbers. A bowl of fruit with six apples and three oranges has a ratio of apples to oranges of six to three. We write this as 6:3. The order matters: apples to oranges is 6:3, but oranges to apples is 3:6.
Equivalent ratios
Equivalent ratios describe the same relationship. You make an equivalent ratio by multiplying or dividing BOTH numbers by the same number. For example, 6:3 is equivalent to 2:1, and 2:3 is equivalent to 12:18 (multiply both by 6).
Simplest form
A ratio is in its simplest form when both numbers have been divided by the largest number that goes into both — just like simplifying a fraction. For example, 21:14 both divide by 7 to give 3:2.
Using ratio to solve problems
In a ratio problem, first find the size of one 'part', then multiply. If the ratio of girls to boys is 3:2 and there are 42 girls, then one part is 42 ÷ 3 = 14, so the boys = 2 × 14 = 28.
Worked examples
Writing a ratio from a picture
- A bowl contains 6 apples and 3 oranges.
- Write the ratio of apples to oranges in the same order: 6 to 3.
- This is written as 6:3.
Answer: The ratio of apples to oranges is 6:3.
Simplest form of a ratio
- Simplify 21:14.
- The largest number that divides both 21 and 14 is 7.
- Divide both numbers by 7: 21 ÷ 7 = 3 and 14 ÷ 7 = 2.
Answer: 21:14 = 3:2 in its simplest form.
Finding a missing number in an equivalent ratio
- Complete 2:3 = 12:?
- The first number went from 2 to 12, which is × 6.
- Multiply the other number by 6 too: 3 × 6.
Answer: 2:3 = 12:18, so the missing number is 18.
Using equivalent ratios to find a total
- In a survey the ratio of apples to bananas was 3:5, and 45 people preferred bananas.
- 5 × 9 = 45, so multiply both numbers by 9: apples = 3 × 9 = 27.
- Add the two amounts: 27 + 45.
Answer: Total number of people = 27 + 45 = 72.
Sharing in a given ratio
- Share 24 counters in the ratio 3:5.
- Add the parts: 3 + 5 = 8 parts.
- One part is 24 ÷ 8 = 3 counters, then multiply for each share.
Answer: The 3-part share is 3 × 3 = 9 and the 5-part share is 5 × 3 = 15.
Where you see this in real life
- Mixing paint using 2 cans of blue for every 3 cans of yellow.
- Sharing a bag of counters between two friends in a given ratio.
- Buying beads in the ratio of white to coloured to make a necklace.
Key words
- Ratio
- A way of comparing two or more numbers, written with a colon, such as 3:5.
- Equivalent ratio
- A ratio that shows the same relationship, made by multiplying or dividing both numbers by the same number.
- Simplest form
- A ratio where both numbers have been divided down as far as they can go.