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Comparison of Ratios

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Introduction

In aptitude tests, you are often asked to compare two ratios. For example: Is 3 : 4 greater than 5 : 7? To answer such questions, we need to bring both ratios to a common form and then compare.

This is very similar to comparing fractions. Once you understand the method, it becomes very simple.

Pattern: Comparison of Ratios

Pattern: Comparison of Ratios

To compare a : b and c : d, write them as fractions → a/b and c/d.

Find the LCM of denominators and bring both to a common denominator.

Then compare the numerators to decide which ratio is greater.

Step-by-Step Example

Question

Compare the ratios 3 : 4 and 5 : 7. Which is greater?

Solution

  1. Step 1: Write ratios as fractions.

    3 : 4 = 3/4, and 5 : 7 = 5/7

  2. Step 2: Find a common denominator.

    Denominators are 4 and 7 → LCM = 28

  3. Step 3: Convert fractions.

    3/4 = (3 × 7)/(4 × 7) = 21/28 5/7 = (5 × 4)/(7 × 4) = 20/28

  4. Step 4: Compare numerators.

    21/28 > 20/28 → 3/4 > 5/7

  5. Step 5: Final Answer.

    3 : 4 is greater than 5 : 7.

  6. Step 6: Quick Check.

    Convert to decimals: 3/4 = 0.75, 5/7 ≈ 0.714. Since 0.75 > 0.714, the answer is correct ✅

Question

Compare the ratios 7 : 9 and 11 : 15.

Solution

  1. Step 1: Write as fractions.

    7 : 9 = 7/9, and 11 : 15 = 11/15

  2. Step 2: Find a common denominator.

    Denominators are 9 and 15 → LCM = 45

  3. Step 3: Convert fractions.

    7/9 = (7 × 5)/(9 × 5) = 35/45 11/15 = (11 × 3)/(15 × 3) = 33/45

  4. Step 4: Compare numerators.

    35/45 > 33/45 → 7/9 > 11/15

  5. Step 5: Final Answer.

    7 : 9 is greater than 11 : 15.

  6. Step 6: Quick Check.

    Convert to decimals: 7/9 ≈ 0.778, 11/15 ≈ 0.733. Since 0.778 > 0.733, the answer is correct ✅

Quick Variations

Sometimes the problem asks: “Which ratio is smaller?” → just reverse the answer.

Another type: Compare more than two ratios (e.g., 2 : 3, 3 : 5, 5 : 7). In that case, convert all ratios into fractions with a common denominator and arrange in order.

Trick to Always Use

  • Step 1: Convert ratios into fractions.
  • Step 2: Find LCM of denominators.
  • Step 3: Convert fractions to the common denominator.
  • Step 4: Compare numerators directly.
  • Step 5: For a quick check, convert to decimals.

Summary

The Comparison of Ratios pattern is solved by treating ratios like fractions.

  • a : b = a/b, c : d = c/d
  • Find LCM of denominators.
  • Convert to equivalent fractions.
  • Compare numerators to decide greater/smaller.
  • Quick check: convert to decimals for confirmation.

With practice, comparing ratios becomes a fast mental calculation!

Practice

(1/5)
1. Compare the ratios 3 : 4 and 5 : 6. Which is greater?
easy
A. 3 : 4 is greater
B. 5 : 6 is greater
C. Both equal
D. Cannot be determined

Solution

  1. Step 1: Convert both ratios to fractions

    3/4 = 0.75, 5/6 ≈ 0.833.

  2. Step 2: Compare values

    Since 0.833 > 0.75, 5/6 is larger.

  3. Final Answer:

    5 : 6 is greater → Option B

  4. Quick Check:

    Cross multiply: 3 × 6 = 18, 4 × 5 = 20 → 20 > 18, confirming 5 : 6 is larger.
Hint: Use cross multiplication for instant comparison: a/b vs c/d → compare ad and bc.
Common Mistakes: Comparing only numerators or only denominators instead of full ratios.
2. Which statement is true for the pair: 6 : 9 and 2 : 3?
easy
A. 6 : 9 is greater
B. 2 : 3 is greater
C. Cannot be determined
D. Both equal

Solution

  1. Step 1: Simplify both ratios

    6 : 9 → 2 : 3, and the second ratio is already 2 : 3.

  2. Final Answer:

    Both ratios are equal → Option D

  3. Quick Check:

    6/9 = 2/3 and 2/3 = 0.6667 → equal.
Hint: Always reduce both ratios before comparing.
Common Mistakes: Comparing unsimplified ratios directly.
3. Compare the ratios 7 : 9 and 11 : 15. Which is greater?
medium
A. 7 : 9 is greater
B. 11 : 15 is greater
C. Both equal
D. Cannot be determined

Solution

  1. Step 1: Convert to fractions

    7/9 ≈ 0.777..., 11/15 ≈ 0.733....

  2. Step 2: Compare values

    0.777... > 0.733..., so 7 : 9 is greater.

  3. Final Answer:

    7 : 9 is greater → Option A

  4. Quick Check:

    Cross multiply: 7 × 15 = 105, 9 × 11 = 99 → 105 > 99.
Hint: Cross-multiplication gives an exact integer comparison.
Common Mistakes: Over-rounding decimals and making the wrong comparison.
4. Which is larger: 12 : 17 or 8 : 11?
medium
A. 12 : 17 is greater
B. 8 : 11 is greater
C. Both equal
D. Cannot be determined

Solution

  1. Step 1: Convert to fractions

    12/17 ≈ 0.7059, 8/11 ≈ 0.7273.

  2. Step 2: Compare values

    0.7273 > 0.7059 → 8 : 11 is larger.

  3. Final Answer:

    8 : 11 is greater → Option B

  4. Quick Check:

    Cross multiply: 8 × 17 = 136, 11 × 12 = 132 → 136 > 132.
Hint: Convert to fractions or immediately cross-multiply to avoid rounding errors.
Common Mistakes: Switching ratio order during comparison.
5. Compare 9 : 14 and 13 : 20. Which is greater?
medium
A. 9 : 14 is greater
B. Both equal
C. 13 : 20 is greater
D. Cannot be determined

Solution

  1. Step 1: Convert to fractions

    9/14 ≈ 0.6429, 13/20 = 0.65.

  2. Step 2: Compare values

    0.65 > 0.6429 → 13 : 20 is larger.

  3. Final Answer:

    13 : 20 is greater → Option C

  4. Quick Check:

    Cross multiply: 9 × 20 = 180, 14 × 13 = 182 → 182 > 180.
Hint: Use exact cross multiplication when decimals are close.
Common Mistakes: Incorrect rounding leading to wrong comparison.