Comparing Ratios: Extended Worked Examples (Grade 8)

Grade level: Grade 8 Number
Topic: Ratios, Rates and Proportions → Ratio Concepts and Comparison → Comparing Ratios

Use rea.m08.number.ratios-rates-proportions.ratio-concepts.ratio-comparison.overview for the main concept development and unit structure. Use rea.m08.number.ratios-rates-proportions.ratio-concepts.worked-examples for broader ratio setup and interpretation, rea.m08.number.ratios-rates-proportions.equivalent-ratios.worked-examples for scaling and equivalence fluency, and rea.m08.number.ratios-rates-proportions.proportional-modeling.worked-examples for richer decision problems after this skill is secure.

This record is narrower: it is about deciding whether one ratio is greater than another or whether two ratios are equal, and about choosing a comparison method that matches the numbers and the context.

Quick decision rule

Before comparing two ratios, ask:

1. Do they describe the same kind of relationship?
   Part-to-part and part-to-whole ratios cannot be compared as if they mean the same thing.
2. Is the order the same?
   red:blue is not the same as blue:red.
3. What comparison method is cleanest here?
   Use one of these:
   • compare directly if one term already matches;
   • simplify first if the ratios reduce nicely;
   • scale to a common first or second term;
   • use a unit rate if “per 1” is natural;
   • for a compact challenge method, compare cross-products when the ratio meanings match.

Worked Example 1: Same second term already given

Problem. Compare 3:8 and 5:8. Which ratio is greater?

Step 1: Check what matches.
The second term is already the same: both ratios are compared to 8.

Step 2: Compare the first terms.
5 > 3, so for the same second quantity, 5:8 represents more of the first quantity.

Answer: 5:8 is greater than 3:8.

Reasoning commentary.
No scaling is needed because the ratios are already on a common basis. This is the simplest kind of ratio comparison: same second term, so compare the first terms directly.

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Worked Example 2: Same first term already given

Problem. Compare 7:12 and 7:15. Which ratio is greater?

Step 1: Check what matches.
The first term is the same in both ratios: 7.

Step 2: Interpret the meaning.
Each ratio tells how much first quantity there is for some second quantity.

• 7:12 means 7 for every 12.
• 7:15 means 7 for every 15.

For the same first quantity, needing a smaller second quantity gives a larger ratio.

Step 3: Compare the second terms.
Since 12 < 15, we have 7:12 > 7:15.

Answer: 7:12 is greater.

Reasoning commentary.
Students sometimes think the larger second number makes the ratio larger. Here it does the opposite. With the same first term, a larger second term spreads that first quantity more thinly.

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Worked Example 3: Simplify first, then compare

Problem. Compare 18:24 and 21:28. Are they equal, or is one greater?

Step 1: Simplify each ratio.
18:24 = 3:4 because both terms divide by 6.

21:28 = 3:4 because both terms divide by 7.

Step 2: Compare the simplified forms.
Both simplify to 3:4.

Answer: The ratios are equal.

Reasoning commentary.
Simplifying was the best choice because both pairs had obvious common factors. Once both ratios reduce to the same simplest form, they represent the same relationship.

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Worked Example 4: Scale to a common second term

Problem. Compare 4:7 and 6:10. Which ratio is greater?

Step 1: Look for an easy comparison method.
The ratios do not already have a matching term, and they do not simplify to the same form.

Step 2: Build a common second term.
A common multiple of 7 and 10 is 70.

Scale each ratio:

4:7 = 40:70

6:10 = 42:70

Step 3: Compare the first terms.
For the same second term 70, compare 40 and 42.

Since 42 > 40, 6:10 is greater.

Answer: 6:10 is greater.

Reasoning commentary.
Matching the second term makes the comparison fair. This method is reliable when simplification alone does not settle the question.

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Worked Example 5: Scale to a common first term

Problem. Compare 9:14 and 12:20. Which ratio is greater?

Step 1: Decide on a basis.
This time we will match the first term.

A common multiple of 9 and 12 is 36.

Step 2: Scale each ratio.
9:14 = 36:56

12:20 = 36:60

Step 3: Interpret the result.
Now both ratios have first term 36.

• 36:56
• 36:60

For the same first quantity, the ratio with the smaller second term is greater.

So 36:56 > 36:60, which means 9:14 > 12:20.

Answer: 9:14 is greater.

Reasoning commentary.
Either common-first-term or common-second-term scaling would work. The important choice is to create one shared reference point, then compare the unmatched terms carefully.

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Worked Example 6: Part-to-whole comparison in context

Problem. Which club has the larger fraction of Grade 8 students?

• Art Club: 14 Grade 8 students out of 20 students
• Music Club: 18 Grade 8 students out of 30 students

Step 1: Write the part-to-whole ratios.
Art Club: 14:20

Music Club: 18:30

Step 2: Simplify.
14:20 = 7:10

18:30 = 3:5

Step 3: Put them on a common whole.
3:5 = 6:10

Now compare:

• Art Club: 7:10
• Music Club: 6:10

Since 7:10 > 6:10, Art Club has the larger fraction of Grade 8 students.

Answer: Art Club.

Reasoning commentary.
Because this is part-to-whole, using a common whole is especially natural. The question is not “who has more Grade 8 students?” but “who has a larger share of Grade 8 students?”

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Worked Example 7: Check comparability before computing

Problem. A student says these two ratios can be compared directly because they use the same numbers:

• Class A: girls to boys = 12:8
• Class B: girls to students = 12:8

Is that comparison valid?

Step 1: Check the ratio types.
12:8 in Class A is part-to-part.

12:8 in Class B is claimed to be girls to students, which is part-to-whole.

Step 2: Test the second statement.
If Class B has 12 girls and 8 students total, that is impossible, because the whole cannot be smaller than one part.

If the intended data were 12 girls and 8 boys, then girls to students would be:

12 : (12 + 8) = 12:20 = 3:5

Answer: The comparison is not valid as stated.

Reasoning commentary.
This is a setup check, not a calculation contest. Before comparing ratios, make sure they describe the same kind of relationship. Many ratio errors happen before any arithmetic begins.

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Worked Example 8: Compare using unit rates

Problem. Which runner had the better pace in terms of distance per minute?

• Runner A: 5 km in 8 min
• Runner B: 7 km in 11 min

Compare the ratios 5:8 and 7:11.

Step 1: Decide whether “per 1” is useful.
Because the second quantity is time, a unit rate is natural.

Step 2: Find distance per minute.
Runner A: 5/8 = 0.625 km/min

Runner B: 7/11 ≈ 0.636 km/min

Step 3: Compare the unit rates.
0.636... > 0.625

So Runner B ran farther per minute.

Answer: Runner B had the better pace.

Reasoning commentary.
Unit rates are just ratio comparisons rewritten with second term 1. This is often the clearest method when the context already suggests a “per 1” meaning, such as per minute or per item.

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Worked Example 9: Equivalent or not in a mixture context

Problem. Which juice mixture is more concentrated in juice?

• Mix A: 15 cups juice to 25 cups water
• Mix B: 18 cups juice to 32 cups water

Step 1: Write the ratios.
Mix A: 15:25

Mix B: 18:32

Step 2: Simplify if possible.
15:25 = 3:5

18:32 = 9:16

They are not immediately equal.

Step 3: Compare on a common second term.
A common multiple of 5 and 16 is 80.

3:5 = 48:80

9:16 = 45:80

Step 4: Interpret.
For the same amount of water, Mix A has 48 parts juice while Mix B has 45 parts juice.

Answer: Mix A is more concentrated.

Reasoning commentary.
The simplification step made the later scaling easier. In mixture problems, always keep the interpretation visible: you are comparing strength, not total size.

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Worked Example 10: Challenge comparison by cross-products

Problem. Compare 11:17 and 13:20 without converting to decimals. Which ratio is greater?

Step 1: Recognize that neither ratio simplifies nicely.
A common-term scaling method would work, but it would use large numbers.

Step 2: Compare cross-products.
To compare 11:17 and 13:20, compare:

11 x 20 = 220

13 x 17 = 221

Since 221 > 220, we get:

13/20 > 11/17

So 13:20 > 11:17.

Answer: 13:20 is greater.

Why this works.
Comparing 11/17 and 13/20 by cross-products is a compressed version of making a common denominator:

11/17 = 220/340

13/20 = 221/340

Now the denominators match, and 221/340 > 220/340.

Reasoning commentary.
This is a good challenge method when the numbers do not scale neatly. It is still the same core idea as the earlier examples: create a fair common basis, then compare.

Common mistakes and repairs

• Mistake: Comparing first numbers only.
  Repair: A ratio is a relationship between both numbers, so both matter.
• Mistake: Assuming bigger numbers mean a bigger ratio.
  Repair: 3:4 is greater than 30:50, even though 30 and 50 are larger numbers.
• Mistake: Comparing part-to-part with part-to-whole.
  Repair: Check the wording before calculating.
• Mistake: Reversing the order in one ratio.
  Repair: Keep labels attached: juice:water must stay juice:water.

Takeaway

A ratio comparison is fair only when both ratios are rewritten to describe the same kind of relationship on the same basis. The specific arithmetic method can change, but the mathematical goal stays the same: compare the relationships, not just the raw numbers.