EQN-06-03

Comparing Fractions, Decimals & Percentages

Decide which quantity is larger without full computation — cross-multiply, benchmark, and estimate.

~5 min read•Difficulty 2/5

Put both in the same form

The fastest comparison converts both quantities to the same type — usually decimals — or uses one of the shortcuts below.

Cross-multiply two fractions

To compare a/b and c/d (positive), compare $a \times d$ with $c \times b .$

5/8 vs 7/11 \to 5 \times 11 = 55 vs 7 \times 8 = 56 \to 7/11 is larger

Benchmark against ½ and 1

A fraction is greater than ½ when the numerator is more than half the denominator; greater than 1 when the top exceeds the bottom. Often that alone settles the comparison.

Percent of vs percent more

"20% of 50" $(= 10)$ is very different from "50 increased by 20%" $(= 60) .$ Read the wording before computing — a classic comparison trap.

Estimate, don't over-compute

You only need the relationship. Round both quantities; if the gap is clear after rounding, you're done.

Worked examples

Example 1 (difficulty 2/5)

Quantity A: 3/7. Quantity B: 0.45. Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Quantity A: 30% of 60. Quantity B: 20% of 90. Which is greater?

◎ Predict the answer first:

Tips

Cross-multiply to compare two positive fractions without finding a common denominator.
Benchmark against ½ and 1 — it often settles the comparison in one glance.
Distinguish "percent OF" from "percent MORE/LESS THAN" before computing.

Common mistakes

Comparing fractions by their numerators (or denominators) alone

Why: The bigger top looks like the bigger fraction.

Fix: Cross-multiply or convert to decimals; both parts matter.

Reading "increased by 20%" as "20% of"

Why: The words look similar.

Fix: "Increased by 20%" multiplies by 1.20; "20% of" multiplies by 0.20.

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