Discover Mathematics P6: Term 2, Unit 9, Lessons 4–6
GoalCompare two ratios by making one term equal, and find missing values and mistakes.
New words
compare · to decide which is bigger or strongercommon term · the same number made in both ratios so they can be comparedreasoning · the explanation of why a method works or does not
Explanation
To compare two ratios, make one of their terms the same by finding equivalent ratios, and then compare the other terms. For two lemonades with lemon to water ratios of 3 : 4 and 5 : 8, we make the water the same: 3 : 4 = 6 : 8, and the other is 5 : 8. With the same 8 cups of water, the first has 6 cups of lemon and the second has 5, so the first is stronger. A tape diagram or a table can show the same thing. When we check someone’s reasoning, we look for the common mistake of adding instead of multiplying, or of comparing the differences instead of the ratios.
★Remember
To compare: make one term the same, then compare the other term. A ratio is not the difference: 2 : 3 is not 12 : 13.
Worked examples
Which squash is stronger? Squash A has 2 cups of concentrate to 5 cups of water, and squash B has 3 to 7. Make the water 35 in both: A = 2 : 5 × 7 = 14 : 35 and B = 3 : 7 × 5 = 15 : 35. With the same 35 cups of water, B has more concentrate (15 against 14), so B is a little stronger.
Team A has 3 wins for every 2 losses, and Team B has 4 wins for every 3 losses. Make the losses equal to 6: A = 9 : 6 and B = 8 : 6. Team A has more wins for the same 6 losses, so Team A has the better record.
Check the reasoning. Mina says 2 : 3 and 12 : 13 are equivalent because in both the second number is 1 more. But 2 : 3 × 6 = 12 : 18, not 12 : 13. Equivalent ratios come from multiplying, not from equal differences.
Class activity
Which is stronger? Make two tea or juice mixes at home with different ratios (for example 1 spoon of sugar to 3 cups of tea and 2 spoons to 5 cups). Make the cups equal and decide which is sweeter. Explain it with a table.
Practice
1
Cocoa to milk is 3 : 5 in mix A and 2 : 3 in mix B. Which mix has the stronger cocoa taste?
Make the milk 15: A = 9 : 15 and B = 10 : 15. Mix B is stronger.
2
Are 4 : 6 and 6 : 9 equivalent?
Yes. Both simplify to 2 : 3.
3
Find the missing number: 7 : 9 = 21 : ?
× 3: 9 × 3 = 27.
4
Which team has the better record: 3 wins to 2 losses or 4 wins to 3 losses?
Make the losses 6: 9 : 6 and 8 : 6. The first team (3 : 2) is better.
5
Amal says 4 : 5 = 8 : 9 because she added 4 to both numbers. What is wrong?
Ratios are scaled by multiplying. 4 : 5 × 2 = 8 : 10.
