Recognizing and Testing Ratio Equivalence: Dedicated Practice Set (Grade 8)

Position in Rea

Use this record after studying:

• rea.m08.number.ratios-rates-proportions.equivalent-ratios.recognizing-equivalence.overview
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.recognizing-equivalence.worked-examples

For broader or adjacent practice, see:

• rea.m08.number.ratios-rates-proportions.ratio-concepts.equivalent-ratios.practice-set
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.scaling-ratios.practice-set

This set is narrower: every question is about recognizing or testing whether ratios are equivalent, and explaining why.

How to use this set

For each pair of ratios:

1. Choose a method: simplify, scale, or compare multiplicative relationships.
2. Decide whether the ratios are equivalent.
3. If they are equivalent, state why.
4. If they are not equivalent, identify what fails.

Unless a question asks for a specific method, use the method that is quickest and clearest.

Difficulty tags

• Mild: direct simplification or obvious scaling
• Moderate: less obvious numbers, mixed forms, or short explanations
• Stretch: larger numbers, missing values, or contextual interpretation
• Challenge: multi-step reasoning, distractors, or explanation-heavy items

Practice problems

A. Direct recognition

1. Mild Are 2:3 and 4:6 equivalent?
2. Mild Are 5:8 and 15:24 equivalent?
3. Mild Are 3:7 and 9:20 equivalent?
4. Mild Are 6/11 and 18/33 equivalent?
5. Mild Are 12:16 and 3:4 equivalent?
6. Mild Are 14:21 and 2:3 equivalent?

B. Decide and justify

7. Moderate Are 8:12 and 10:15 equivalent? Give one sentence of justification.
8. Moderate Are 9:14 and 18:28 equivalent? Give one sentence of justification.
9. Moderate Are 15:25 and 21:35 equivalent? Give one sentence of justification.
10. Moderate Are 16/24 and 20/32 equivalent? Show the simplification that supports your answer.
11. Moderate Are 7:9 and 35:44 equivalent? Explain briefly.
12. Moderate Are 18:27 and 30:45 equivalent? Explain briefly.

C. Mixed forms and missing values

13. Moderate Is 4:5 equivalent to 24:30? Is it equivalent to 28:35? Answer both.
14. Moderate Fill in the blank so the ratios are equivalent: 3:8 = 12:x.
15. Moderate Fill in the blank so the ratios are equivalent: y:14 = 5:7.
16. Stretch Fill in the blank so the ratios are equivalent: 18:42 = x:7.
17. Stretch A student says 6:10 and 9:15 are not equivalent because the differences are different (10 - 6 = 4, 15 - 9 = 6). Is the student correct? Explain.
18. Stretch A student scales 4:9 to 12:18 and claims the ratios are equivalent because both numbers got larger. Identify the error and give the correct decision.

D. Context and reasoning

19. Stretch One recipe uses 6 cups of flour for 9 cups of oats. Another uses 10 cups of flour for 15 cups of oats. Do the recipes use the same flour-to-oats ratio?
20. Stretch In two classes, the ratio of red notebooks to blue notebooks is 8:12 in Class A and 14:21 in Class B. Are the ratios equivalent?
21. Challenge A map uses a legend where 3 cm represents 8 km. Another map uses 9 cm to represent 24 km. Are the two map scales equivalent as ratios of cm:km?
22. Challenge A team wins 12 of 18 games. Another team wins 20 of 32 games. Do the teams have equivalent win-to-total-game ratios?
23. Challenge Which of the following are equivalent to 5:12? Explain each choice.

• 10:24
• 15:36
• 20:45
• 25:60

24. Challenge Find all values of n that make the statement true: 6:n is equivalent to 18:24.

Answer key

1. Yes
2. Yes
3. No
4. Yes
5. Yes
6. Yes
7. Yes
8. Yes
9. Yes
10. No
11. No
12. Yes
13. Yes to 24:30; yes to 28:35
14. x = 32
15. y = 10
16. x = 3
17. The student is not correct; the ratios are equivalent
18. The student used different scale factors; the ratios are not equivalent
19. Yes
20. No
21. Yes
22. No
23. Equivalent: 10:24, 15:36, 25:60; not equivalent: 20:45
24. n = 8

Full solutions for the hardest third

17. Difference reasoning error

Question: A student says 6:10 and 9:15 are not equivalent because the differences are different. Is the student correct?

Solution:

• Ratio equivalence depends on multiplicative comparison, not additive difference.
• Simplify 6:10 by dividing both terms by 2: 6:10 = 3:5.
• Simplify 9:15 by dividing both terms by 3: 9:15 = 3:5.
• Both simplify to 3:5, so they are equivalent.
• The student's error is using subtraction instead of checking whether both terms are related by the same scale factor or simplify to the same ratio.

Answer: The student is not correct. 6:10 and 9:15 are equivalent.

18. Incorrect scaling

Question: A student scales 4:9 to 12:18 and claims the ratios are equivalent because both numbers got larger.

Solution:

• To keep a ratio equivalent, both terms must be multiplied or divided by the same non-zero factor.
• From 4 to 12, the scale factor is 3.
• From 9 to 18, the scale factor is 2.
• Because the scale factors are different, the new ratio is not equivalent.
• Check by simplifying 12:18: divide both terms by 6 to get 2:3.
• The original ratio 4:9 is already in simplest form and is not 2:3.

Answer: The error is using different scale factors. 4:9 and 12:18 are not equivalent.

19. Recipe comparison

Question: One recipe uses 6 cups of flour for 9 cups of oats. Another uses 10 cups of flour for 15 cups of oats. Do they use the same ratio?

Solution:

• Compare 6:9 and 10:15.
• Simplify 6:9 by dividing by 3: 6:9 = 2:3.
• Simplify 10:15 by dividing by 5: 10:15 = 2:3.
• Since both simplify to 2:3, the recipes use the same flour-to-oats ratio.

Answer: Yes, the ratios are equivalent.

20. Notebook ratios

Question: In two classes, the ratio of red notebooks to blue notebooks is 8:12 in Class A and 14:21 in Class B. Are the ratios equivalent?

Solution:

• Simplify 8:12 by dividing by 4: 8:12 = 2:3.
• Simplify 14:21 by dividing by 7: 14:21 = 2:3.
• Both simplify to 2:3.

Answer: Yes, the ratios are equivalent.

Note: This item is tagged Stretch because the numbers may look less familiar at first glance, but the reasoning is still direct.

21. Map scale comparison

Question: A map uses 3 cm for 8 km. Another uses 9 cm for 24 km. Are the scales equivalent as cm:km ratios?

Solution:

• Compare 3:8 and 9:24.
• From 3 to 9, multiply by 3.
• From 8 to 24, also multiply by 3.
• The same scale factor is applied to both terms, so the ratios are equivalent.
• Alternatively, simplify 9:24 by dividing by 3 to get 3:8.

Answer: Yes, the map scales are equivalent.

22. Win ratios

Question: A team wins 12 of 18 games. Another team wins 20 of 32 games. Do they have equivalent win-to-total-game ratios?

Solution:

• Compare 12:18 and 20:32.
• Simplify 12:18 by dividing by 6: 12:18 = 2:3.
• Simplify 20:32 by dividing by 4: 20:32 = 5:8.
• Since 2:3 and 5:8 are different, the ratios are not equivalent.

Answer: No, the teams do not have equivalent win ratios.

23. Equivalent to 5:12

Question: Which of the following are equivalent to 5:12?

Solution:

• 10:24:
  • 5 x 2 = 10 and 12 x 2 = 24.
  • Same scale factor, so equivalent.
• 15:36:
  • 5 x 3 = 15 and 12 x 3 = 36.
  • Same scale factor, so equivalent.
• 20:45:
  • 5 x 4 = 20, but 12 x 4 = 48, not 45.
  • Not equivalent.
  • Also, 20:45 simplifies to 4:9, not 5:12.
• 25:60:
  • 5 x 5 = 25 and 12 x 5 = 60.
  • Same scale factor, so equivalent.

Answer: 10:24, 15:36, and 25:60 are equivalent to 5:12. 20:45 is not.

24. Find the value of n

Question: Find all values of n that make 6:n equivalent to 18:24.

Solution:

• First simplify 18:24.
• Divide both terms by 6: 18:24 = 3:4.
• So 6:n must also simplify to 3:4.
• Since 6 is double 3, multiply 4 by 2 as well.
• n = 8.
• Check: 6:8 simplifies by dividing by 2 to 3:4, which matches.

Answer: n = 8.

Common misconception repairs

• Misconception: Equivalent ratios must have the same difference between terms.
  • Repair: Ratios compare multiplicatively, not additively.
• Misconception: If both numbers increase, the ratio stays equivalent.
  • Repair: Both terms must be scaled by exactly the same non-zero factor.
• Misconception: You must always use one method.
  • Repair: Simplifying, scaling, and comparing multiplicative structure are all valid; choose the clearest method.

Suggested mastery target

A learner is ready to move on when they can:

• correctly decide equivalence in routine pairs,
• justify the decision using simplification or scaling,
• repair common errors in another student's reasoning,
• solve missing-value equivalence problems without guessing.