Equivalent Ratios and Simplification: Extended Worked Examples

Position in the topic

This record supports Grade 8 learners studying Equivalent Ratios and Simplification. Use it with:

• rea.m08.number.ratios-rates-proportions.ratio-concepts.equivalent-ratios.lesson for the core concept, visual intuition, and formal rule.
• rea.m08.number.ratios-rates-proportions.ratio-concepts.worked-examples for broader ratio-comparison examples.
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.worked-examples for a parallel extended set with more emphasis on proportions.
• rea.m08.number.ratios-rates-proportions.ratio-concepts.unit-mastery-quiz after study and practice.

This record does not reteach the whole lesson. Instead, it shows how to choose a sensible method and how to check that the result still represents the same multiplicative comparison.

Fast method guide

When working with equivalent ratios, ask:

1. Can I scale both terms by the same factor?
2. Can I simplify by dividing both terms by a common factor?
3. If a value is missing, what multiplication or division connects the known terms?
4. Does my answer preserve the same ratio?

A reliable check is to compare both ratios as fractions, such as a/b and c/d, or to verify that both terms were multiplied or divided by the same non-zero number.

Worked examples

Example 1: Write two equivalent ratios

Problem. Write two ratios equivalent to 3:5.

Solution. Equivalent ratios are made by multiplying both terms by the same non-zero factor.

• Multiply by 2: 3:5 = 6:10
• Multiply by 4: 3:5 = 12:20

Answer. 6:10 and 12:20

Reasoning commentary. This is the most direct case: nothing is missing and no simplification is needed. The only choice is the scale factor. Small whole-number factors are best when the goal is fluency and pattern recognition.

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Example 2: Decide whether two ratios are equivalent

Problem. Are 8:12 and 10:15 equivalent?

Solution. Simplify each ratio.

• 8:12 divides by 4 to give 2:3
• 10:15 divides by 5 to give 2:3

Since both simplify to 2:3, the original ratios are equivalent.

Answer. Yes, they are equivalent.

Reasoning commentary. There are two common ways to test equivalence: simplify both ratios, or compare 8/12 and 10/15. Simplifying is usually easier here because both pairs have obvious common factors.

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Example 3: Simplify a ratio fully

Problem. Simplify 18:30.

Solution. We want the ratio in lowest terms, so divide by the greatest common factor of 18 and 30.

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Greatest common factor is 6

Now divide both terms by 6:

18:30 = 3:5

Answer. 3:5

Reasoning commentary. Learners often divide by a common factor such as 2 and stop too early: 18:30 = 9:15. That ratio is equivalent, but not fully simplified. Using the greatest common factor gets to lowest terms in one step.

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Example 4: Fill in a missing term by scaling up

Problem. Complete the equivalent ratio: 4:7 = 20:x

Solution. Look at the first terms: 4 became 20.

4 x 5 = 20

So the scale factor is 5. Apply the same factor to the second term:

7 x 5 = 35

So x = 35.

Answer. x = 35

Reasoning commentary. This is the cleanest missing-value situation because the scale factor is obvious. The key idea is consistency: if one term was multiplied by 5, the other must also be multiplied by 5.

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Example 5: Fill in a missing term by scaling down

Problem. Complete the equivalent ratio: 36:54 = x:9

Solution. This time it is easier to look at the second terms:

54 became 9

That means we divided by 6, because 54 / 6 = 9.

So divide the first term by 6 too:

36 / 6 = 6

So x = 6.

Answer. x = 6

Reasoning commentary. Good strategy is to scale in the direction that makes the factor easiest to see. Going from 54 to 9 is much clearer than guessing a relation from 36 to x.

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Example 6: Use a ratio in context

Problem. A paint mix 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?

Solution. The blue part is 3, and in the actual mixture it is 18.

So the scale factor is:

18 / 3 = 6

Now scale the red part:

2 x 6 = 12

So 12 cups of red paint are needed.

Answer. 12 cups

Reasoning commentary. In context problems, first identify which quantity matches which part of the ratio. Here 2:3 means red:blue, so 18 must match the blue part, not the red part.

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Example 7: Distinguish equivalent from non-equivalent ratios

Problem. Which ratio is equivalent to 9:14?

A. 18:28

B. 15:20

C. 27:40

D. 36:58

Solution. Test each option by checking whether both terms come from the same scale factor.

• A: 9 x 2 = 18 and 14 x 2 = 28 so this works.
• B: 9 to 15 is not the same factor as 14 to 20.
• C: 9 x 3 = 27, but 14 x 3 = 42, not 40.
• D: 9 x 4 = 36, but 14 x 4 = 56, not 58.

Only A is equivalent.

Answer. A. 18:28

Reasoning commentary. When several choices are offered, you do not need to simplify every option. A faster test is to check whether one consistent scale factor works across both terms.

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Example 8: Simplify a part-to-part ratio from a word problem

Problem. In a club, 24 students play soccer and 36 play basketball. Write the soccer-to-basketball ratio in simplest form.

Solution. Start with the ratio in the correct order:

soccer:basketball = 24:36

Now simplify. The greatest common factor of 24 and 36 is 12.

24 / 12 = 2

36 / 12 = 3

So the simplified ratio is 2:3.

Answer. 2:3

Reasoning commentary. The main choice here is setting up the ratio in the requested order. Reversing it to 36:24 would describe a different comparison, even though it uses the same numbers.

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Example 9: Multi-step missing-value problem

Problem. Complete the ratio table.

A	B
5	8
15	?
?	40

Solution. The base ratio is 5:8.

For the row with 15:?:

• 5 became 15, so the scale factor is 3
• Multiply 8 by 3: 8 x 3 = 24

So the missing value is 24.

For the row with ?:40:

• 8 became 40, so the scale factor is 5
• Multiply 5 by 5: 5 x 5 = 25

So the missing value is 25.

Completed table:

A	B
5	8
15	24
25	40

Answer. The missing values are 24 and 25.

Reasoning commentary. A ratio table is useful because every row must represent the same multiplicative comparison. Each row can use a different scale factor, but within one row the factor must be consistent across both columns.

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Example 10: Challenge problem with two valid methods

Problem. Tickets for a school event were sold in the ratio 7:9 for student tickets to adult tickets. If 128 tickets were sold altogether, how many were student tickets and how many were adult tickets?

Solution. The ratio parts add to:

7 + 9 = 16

So the 128 tickets represent 16 equal parts.

Each part is:

128 / 16 = 8

Now find each quantity:

• Student tickets: 7 x 8 = 56
• Adult tickets: 9 x 8 = 72

Check:

56 + 72 = 128

So the values are consistent.

Answer. 56 student tickets and 72 adult tickets

Reasoning commentary. This problem is harder because the total is given, not one part of the ratio directly. The correct choice is to interpret 7:9 as 16 equal parts in total. A common mistake is to subtract 9 - 7 or divide 128 by 7 or 9 separately without using the total number of parts.

Common misconceptions and repairs

Misconception 1: “Equivalent means add the same amount to both terms.”

Incorrect idea. From 3:5, making 5:7 by adding 2 to both terms.

Repair. Equivalent ratios preserve multiplicative comparison, not additive difference. Check as fractions:

• 3/5 = 0.6
• 5/7 ≈ 0.714

So they are not equivalent.

Misconception 2: “Any common factor gives the final simplified form.”

Incorrect idea. 18:30 = 9:15, so the simplification is finished.

Repair. 9:15 is equivalent, but not fully simplified because both terms still divide by 3. Keep simplifying to 3:5, or divide by the greatest common factor first.

Misconception 3: “Order does not matter in ratios.”

Incorrect idea. Treating boys:girls = 2:5 as the same as girls:boys = 2:5.

Repair. A ratio is tied to the order named in the problem. Reversing the order changes the meaning.

Short independent practice

Try these after studying the examples.

1. Write one ratio equivalent to 4:9.
2. Simplify 16:24.
3. Complete 6:11 = 24:x.
4. Are 14:21 and 8:12 equivalent?
5. A recipe uses flour and sugar in the ratio 5:2. If there are 15 cups of flour, how much sugar is needed?

Answers

1. 8:18 (many answers are possible)
2. 2:3
3. x = 44
4. Yes, both simplify to 2:3
5. 6 cups

What this record prepares you for

After this set, a learner should be ready to:

• test whether two ratios are equivalent,
• simplify ratios to lowest terms,
• solve one-step missing-value ratio problems,
• interpret ratio language correctly in context,
• move on to fuller proportion problems and end-of-unit assessment.