Comparing Ratios

Grade level: Grade 8 Number

Place in the unit

This lesson develops one core skill from the Grade 8 unit on ratios: deciding whether two ratios are equal, or which one is larger.

Use these linked records for surrounding context rather than treating this lesson as the whole unit:

• Ratio Concepts and Comparison — Grade 8 Unit Overview
• Ratios, Rates and Proportions (Grade 8, Number)
• Grade 8 Practice Set: Ratio Concepts and Comparison
• Grade 8 Unit Mastery Quiz: Ratio Concepts and Comparison

This record focuses specifically on comparing ratios.

Learning goal

By the end of this lesson, you should be able to:

• decide whether two ratios are equal or not equal,
• determine which of two ratios is greater,
• justify your comparison using more than one method,
• choose a comparison method that fits the numbers.

1. Intuition: what does it mean for one ratio to be larger?

A ratio compares two quantities in a fixed order.

For example, in the ratio 3:5, you can think:

• 3 of one thing for every 5 of another, or
• 3/5 when comparing the first quantity to the second.

When comparing two ratios, you are asking:

Which situation has more first quantity per same amount of second quantity?

Visual idea

Suppose one mixture has 2 cups of juice concentrate for 5 cups of water, and another has 3 cups of concentrate for 5 cups of water.

Because the water amount is the same, the second mixture is stronger: 3:5 > 2:5.

Now suppose the water amounts are different:

• Mix A: 2:3
• Mix B: 3:5

You cannot compare just the first numbers (2 and 3) or just the second numbers (3 and 5). You need to rewrite the ratios so they describe the comparison on a common basis.

A useful picture is a strip model:

• 2:3 means 2 shaded parts for 3 unshaded parts.
• 3:5 means 3 shaded parts for 5 unshaded parts.

To compare fairly, make the same number of comparison parts, or convert both to fractions/decimals.

2. Precise notation

If a ratio compares quantity a to quantity b, write it as:

• a:b
• a to b
• a/b (when thinking of it as a multiplicative comparison)

For comparing ratios, order matters:

• a:b is generally not the same as b:a.

If two ratios are equivalent, write:

• a:b = c:d
• equivalently, a/b = c/d, provided b and d are not 0.

If one ratio is larger, write for example:

• a:b > c:d
• meaning a/b > c/d.

3. Main comparison methods

There are three main methods in Grade 8.

Method A: Rewrite as equivalent ratios with a common term

Make the second quantities the same, or make the first quantities the same.

Example idea:

• 2:3 and 4:5
• Make the first terms both 4: 2:3 = 4:6
• Compare 4:6 and 4:5
• Since for the same first amount, less second amount means a larger ratio, 4:5 > 4:6, so 4:5 > 2:3.

Method B: Convert each ratio to a fraction, decimal, or percent

Treat:

• a:b as a/b

Then compare the values.

Example idea:

• 3:4 = 3/4 = 0.75
• 5:8 = 5/8 = 0.625
• so 3:4 > 5:8

Method C: Use cross-products

To compare a:b and c:d, compare a/b and c/d by checking:

• a x d and b x c

Then:

• if a x d > b x c, then a:b > c:d
• if a x d < b x c, then a:b < c:d
• if a x d = b x c, then a:b = c:d

This works because both fractions are being compared on a common product basis.

4. When should you use each method?

Use equivalent ratios when:

• the numbers scale easily,
• you want to preserve visual meaning,
• the problem is contextual.

Use fractions/decimals when:

• the ratios are easy to divide,
• you want a direct numerical comparison,
• percent language is helpful.

Use cross-products when:

• the numbers do not scale nicely,
• division gives awkward decimals,
• you need a fast exact comparison.

5. Step-by-step procedure

Procedure 1: Compare by equivalent ratios

1. Write both ratios in the same order.
2. Choose one term to match: first term or second term.
3. Scale one or both ratios to create a common term.
4. Compare the other terms.
5. State which original ratio is greater, or that they are equal.

Procedure 2: Compare by fractions/decimals

1. Write each ratio as a fraction: a:b becomes a/b.
2. Convert each fraction to a decimal or percent, if useful.
3. Compare the values.
4. Translate the result back into ratio language.

Procedure 3: Compare by cross-products

To compare a:b and c:d:

1. Compute a x d.
2. Compute b x c.
3. Compare the two products.
4. Conclude:
   • a x d > b x c means a:b > c:d
   • a x d < b x c means a:b < c:d
   • equal products mean equal ratios.

6. Fully worked examples

Example 1: Same second term

Compare 4:7 and 5:7.

Thinking: both ratios compare something to 7, so compare the first terms directly.

• In 4:7, there are 4 for every 7.
• In 5:7, there are 5 for every 7.

Since 5 > 4,

• 5:7 > 4:7

Conclusion: 5:7 is the larger ratio.

Example 2: Equivalent ratios with a common term

Compare 2:3 and 3:4.

We will make the first terms the same.

• 2:3 can be multiplied by 3 to get 6:9
• 3:4 can be multiplied by 2 to get 6:8

Now compare 6:9 and 6:8.

For the same first quantity 6:

• needing only 8 of the second quantity gives a larger ratio than needing 9 of the second quantity.

So:

• 6:8 > 6:9
• therefore 3:4 > 2:3

Check with fractions:

• 2:3 = 2/3 ≈ 0.667
• 3:4 = 0.75
• yes, 0.75 > 0.667

Conclusion: 3:4 is the larger ratio.

Example 3: Cross-products for awkward numbers

Compare 5:12 and 7:15.

Write them as fractions:

• 5/12 and 7/15

Use cross-products:

• 5 x 15 = 75
• 12 x 7 = 84

Compare the products:

• 75 < 84

So:

• 5/12 < 7/15
• therefore 5:12 < 7:15

Conclusion: 7:15 is the larger ratio.

Example 4: Determining equality

Are 6:9 and 10:15 equal?

Method 1: simplify.

• 6:9 simplifies by dividing both terms by 3: 6:9 = 2:3
• 10:15 simplifies by dividing both terms by 5: 10:15 = 2:3

Since both simplify to the same ratio,

• 6:9 = 10:15

Method 2: cross-products.

• 6 x 15 = 90
• 9 x 10 = 90

Equal products confirm the same result.

Conclusion: the ratios are equal.

Example 5: Context problem with interpretation

Class A has 12 students who prefer soccer out of 20 surveyed. Class B has 15 students who prefer soccer out of 24 surveyed. Which class has the greater ratio of soccer preference?

Write the ratios:

• Class A: 12:20
• Class B: 15:24

Simplify both:

• 12:20 = 3:5
• 15:24 = 5:8

Now compare 3/5 and 5/8.

Convert to decimals:

• 3/5 = 0.6
• 5/8 = 0.625

Since 0.625 > 0.6,

• 15:24 > 12:20

Interpretation: a greater fraction of students in Class B prefer soccer.

Conclusion: Class B has the greater ratio.

Example 6: Decimal ratios

Compare 1.2:2 and 3:5.

Write as fractions:

• 1.2/2 = 0.6
• 3/5 = 0.6

Since both are 0.6,

• 1.2:2 = 3:5

Conclusion: the ratios are equal.

7. Common misconceptions and repairs

Misconception 1: Comparing only the first numbers

A student says 4:9 > 3:5 because 4 > 3.

Why this is wrong: the second numbers are different, so the comparisons are not on the same basis.

Repair: rewrite as fractions or equivalent ratios.

• 4/9 ≈ 0.444
• 3/5 = 0.6
• so actually 3:5 > 4:9

Misconception 2: Reversing one ratio accidentally

A student compares 2:7 and 7:2 as if they are similar.

Why this is wrong: ratio order matters.

Repair: say the meaning aloud.

• 2:7 means 2 for every 7
• 7:2 means 7 for every 2 These describe very different relationships.

Misconception 3: Simplifying only one term

A student changes 8:12 into 4:12.

Why this is wrong: a ratio stays equivalent only if both terms are multiplied or divided by the same nonzero number.

Repair:

• 8:12 = 2:3 by dividing both terms by 4.

Misconception 4: Using cross-products but reading the result backward

A student compares 2:5 and 3:7.

• 2 x 7 = 14
• 5 x 3 = 15 Then says 2:5 is greater because 14 is “close”.

Repair: compare exactly.

• Since 14 < 15, it follows that 2/5 < 3/7
• so 2:5 < 3:7

8. Practice with answers

Try these before checking.

1. Compare 3:8 and 4:8.
2. Compare 2:5 and 3:7.
3. Are 9:12 and 6:8 equal?
4. Compare 14:21 and 5:8.
5. Compare 0.9:1.5 and 3:5.

Answers

1. 4:8 > 3:8
2. 2:5 < 3:7 because 2 x 7 = 14 and 5 x 3 = 15
3. Yes. 9:12 = 3:4 and 6:8 = 3:4
4. 14:21 = 2:3 ≈ 0.667, and 5:8 = 0.625, so 14:21 > 5:8
5. Equal, because 0.9/1.5 = 0.6 and 3/5 = 0.6

9. Summary of the lesson

To compare ratios well:

• keep the order consistent,
• compare on a common basis,
• use equivalent ratios, decimals/fractions, or cross-products,
• justify your conclusion, not just state it.

A ratio is larger when it represents more of the first quantity for the same amount of the second quantity.

10. What to study next

After this lesson, consolidate with:

• the linked practice set for fluency,
• the unit mastery quiz for assessment,
• later lessons on unit rates and proportional reasoning, where ratio comparison becomes even more useful.