☰ Year plan · Mathematics

Ratios, ratio tables and equivalent ratios

Week 18 · What a ratio is, how to show it, and equivalent ratios · 3 lessons
3

Equivalent ratios from a recipe table

Discover Mathematics P6: Term 2, Unit 9, Lessons 1–3
GoalMake equivalent ratios by multiplying or dividing both terms by the same number, simplify a ratio and test two ratios.
New words
equivalent ratios · ratios that describe the same relationship, such as 2 : 3 and 4 : 6term · one of the two numbers of a ratiosimplest form · a ratio that cannot be divided any more by a common factorscale · to multiply or divide both terms by the same number
Explanation

Equivalent ratios are made by multiplying or dividing both terms by the same number. A recipe that uses 3 cups of flour for every 2 cups of water can be made bigger (6 : 4, 9 : 6) or smaller. To make a ratio as simple as possible we divide both terms by their greatest common factor: 6 : 8 becomes 3 : 4. To test whether two ratios are equivalent, simplify both or scale one to match the other. Adding the same number to both terms does not give an equivalent ratio: 3 : 4 is not 6 : 7.

★Remember
Multiply or divide BOTH terms by the same number. Never add the same number to both terms.
Worked examples
Batches1234
Flour (cups)36912
Water (cups)2468
A dough uses 3 cups of flour for every 2 cups of water. Multiply both terms: ×2 gives 6 : 4, ×3 gives 9 : 6, ×4 gives 12 : 8. All of them are equivalent to 3 : 2.
Simplify by dividing both terms by the same number: 12 : 18 ÷ 6 = 2 : 3, and 6 : 8 ÷ 2 = 3 : 4. Test 4 : 10 and 6 : 15: 4 : 10 ÷ 2 = 2 : 5 and 6 : 15 ÷ 3 = 2 : 5. They are equivalent.
A banner has 4 red stripes for every 6 white stripes. A small banner with 2 red stripes needs 3 white stripes (÷ 2). A large banner with 20 red stripes needs 30 white stripes (× 5). Not equivalent: 3 : 5 and 6 : 9, because 3 : 5 × 2 = 6 : 10, not 6 : 9.
Class activity

Recipe scaling. Choose a simple recipe you know (for example 2 cups of rice for 3 cups of water). Write what you need for 2, 3, 5 and 10 batches in a ratio table. Then check by dividing: is each column equivalent to the first?

Practice
1
Complete: 2 : 5 = 4 : ? = ? : 15.
2
Write 15 : 20 and 18 : 24 in simplest form.
3
Are 5 : 8 and 15 : 24 equivalent?
4
A recipe uses 2 cups of rice for every 3 cups of water. How much water is needed for 10 cups of rice?
5
Hana says 3 : 4 = 6 : 7 because she added 3 to both. What is wrong, and what is 3 : 4 doubled?