Grade 8 Practice Set: Equivalent Ratios and Simplification

Position in the unit

Use this set after the main instruction on equivalent ratios and after broader ratio fluency work. For wider mixed practice across ratio meaning and comparison, see rea.m08.number.ratios-rates-proportions.ratio-concepts.practice-set. For proportion-focused practice across a wider skill range, see rea.m08.number.ratios-rates-proportions.equivalent-ratios.practice-set. For the fraction analogue of the same simplifying idea, compare rea.m08.number.fraction-operations.equivalence.practice-set.

This record does not reteach the full topic. It concentrates on disciplined practice, answer-checking, and error control.

Quick reminders

• A ratio is in simplest form when its two terms have no common factor greater than 1.
• Two ratios are equivalent if both terms are multiplied or divided by the same non-zero number.
• To test equivalence, you can simplify both ratios or compare cross-products.
• Keep ratio order fixed: a:b is not the same as b:a.

Common error checks

• Do not add or subtract the same number to both terms to make an equivalent ratio.
• Do not simplify only one term.
• In word problems, make sure the ratio compares matching quantities in matching order.
• If units differ, rewrite so the compared quantities are stated consistently before deciding about equivalence.

Graded practice problems

A. Simplify ratios

1. Simplify 12:18. [easy]
2. Simplify 21:28. [easy]
3. Simplify 45:60. [easy]
4. Simplify 14:49. [easy]
5. Simplify 36:54. [medium]
6. Simplify 48:72. [medium]

B. Make equivalent ratios

7. Write two different ratios equivalent to 3:5. [easy]
8. Write two different ratios equivalent to 4:7. [easy]
9. Complete: 6:9 = __:15. [medium]
10. Complete: 8:12 = 10:__. [medium]
11. Complete: 15:20 = __:28. [medium]
12. Complete: 18:27 = 14:__. [medium]

C. Decide whether ratios are equivalent

13. Are 8:12 and 10:15 equivalent? Explain briefly. [medium]
14. Are 14:21 and 18:27 equivalent? Explain briefly. [medium]
15. Are 9:24 and 15:40 equivalent? Explain briefly. [medium]
16. Are 16:20 and 12:15 equivalent? Explain briefly. [medium]

D. Missing-value ratio equations

17. Solve: 5:8 = x:24. [hard]
18. Solve: 7:9 = 35:y. [hard]
19. Solve: 12:15 = z:35. [hard]
20. Solve: 18:30 = 27:w. [hard]

E. Context problems

21. A paint mixture uses red and blue in the ratio 2:3. If 18 cups of blue paint are used, how many cups of red paint are needed? [hard]
22. A trail map uses a scale ratio of 1:25 000. If two towns are 8 cm apart on the map, what real distance does that represent in centimetres and in metres? [hard]
23. In a class, the ratio of students wearing runners to boots is 5:2. If 14 students are wearing boots, how many are wearing runners? How many students are represented altogether? [hard]
24. A recipe uses flour and sugar in the ratio 4:1. A student says that 18 cups of flour and 5 cups of sugar keep the ratio because both numbers are “close.” Is the student correct? If not, give one correct sugar amount for 18 cups of flour. [hard]

Answer key

1. 2:3
2. 3:4
3. 3:4
4. 2:7
5. 2:3
6. 2:3
7. Examples: 6:10, 9:15
8. Examples: 8:14, 12:21
9. 10
10. 15
11. 21
12. 21
13. Yes
14. Yes
15. Yes
16. Yes
17. x = 15
18. y = 45
19. z = 28
20. w = 45
21. 12 cups red
22. 200 000 cm, which is 2 000 m
23. 35 runners, 49 students total
24. No; correct sugar amount is 4.5 cups

Full solutions for the hardest third

17. Solve: 5:8 = x:24

The second term changed from 8 to 24.

24 ÷ 8 = 3

So multiply the first term by 3 as well:

x = 5 x 3 = 15

Answer: x = 15

18. Solve: 7:9 = 35:y

The first term changed from 7 to 35.

35 ÷ 7 = 5

So multiply the second term by 5:

y = 9 x 5 = 45

Answer: y = 45

19. Solve: 12:15 = z:35

First simplify the known ratio:

12:15 = 4:5

Now compare with z:35.

From 5 to 35 is multiplying by 7, so multiply 4 by 7 too:

z = 28

You can also check by cross-products:

12/15 = z/35

12 x 35 = 15z

420 = 15z

z = 28

Answer: z = 28

20. Solve: 18:30 = 27:w

Simplify 18:30 first:

18:30 = 3:5

Now compare 3:5 with 27:w.

From 3 to 27 is multiplying by 9, so multiply 5 by 9:

w = 45

Answer: w = 45

21. Paint mixture 2:3

The ratio red:blue is 2:3.

Blue is 18 cups, so find the scale factor:

18 ÷ 3 = 6

Multiply the red part by 6:

2 x 6 = 12

Answer: 12 cups of red paint

22. Map scale 1:25 000

A scale of 1:25 000 means:

1 cm on map = 25 000 cm in real life

For 8 cm on the map:

8 x 25 000 = 200 000 cm

Now convert centimetres to metres:

200 000 cm ÷ 100 = 2 000 m

Answer: 200 000 cm or 2 000 m

23. Runners to boots 5:2

Boots correspond to the 2 part. There are 14 boots.

14 ÷ 2 = 7

So the scale factor is 7.

Runners:

5 x 7 = 35

Total students:

35 + 14 = 49

Answer: 35 runners and 49 students altogether

24. Recipe ratio 4:1

The ratio flour:sugar must stay 4:1.

Check the student's claim with 18:5.

If 18:5 were equivalent to 4:1, then simplifying should give 4:1, but it does not.

Use cross-products:

18 x 1 = 18

5 x 4 = 20

Since 18 != 20, the ratios are not equivalent.

Now find the correct sugar amount for 18 cups of flour.

18 ÷ 4 = 4.5

So sugar must be:

1 x 4.5 = 4.5

Answer: The student is not correct; a correct sugar amount is 4.5 cups.

Self-check and next step

If you missed mostly Questions 1-8, return to simplification and ratio form. If you missed mostly Questions 13-20, focus on equivalence tests and scale factors. If the main difficulty was context interpretation in Questions 21-24, move next to broader proportional applications in rea.m08.number.ratios-rates-proportions.equivalent-ratios.practice-set before attempting rea.m08.number.ratios-rates-proportions.ratio-concepts.unit-mastery-quiz.