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Comparing
Fractions

Learn reliable strategies to determine which fraction is greater, including common denominators, cross multiplication, and benchmark reasoning.

SSAT Middle Level: Quantitative

What Is a Fraction?

A fraction represents a part of a whole. It has two components: a numerator (top number) telling how many parts you have, and a denominator (bottom number) telling how many equal parts the whole is divided into.

\(3\)
\(4\)
 ← Numerator ← Denominator

Seeing Fractions as Parts of a Whole

The best way to understand fraction comparison is to visualize the fractions. Below, both bars represent the same whole, but they are divided into different numbers of equal parts.

\(\frac{2}{5}\)
\(\frac{3}{5}\)
Since 3 parts > 2 parts:
\(\frac{3}{5} > \frac{2}{5}\)

Why Does Comparing Fractions Matter?

On the SSAT, you will frequently need to order fractions, identify the greatest or least value, or decide between fractional quantities. Mastering comparison unlocks these skills.

Ordering
Arrange fractions from least to greatest or greatest to least.
Estimation
Quickly judge whether a fraction is closer to 0, \(\frac{1}{2}\), or 1.
Problem Solving
Choose the largest portion, compare rates, or evaluate answers.

Core Rule: Common Denominators

To compare fractions with different denominators, rewrite them as equivalent fractions that share the same denominator. Then compare the numerators directly.

\(\frac{2}{3}\)
 vs
\(\frac{3}{4}\)
\(\times \frac{4}{4}\)
\(\times \frac{3}{3}\)
\(\frac{8}{12}\)
\(\frac{9}{12}\)
 vs
8 < 9, so:
\(\frac{2}{3} < \frac{3}{4}\)

Shortcut: Cross Multiplication

Cross multiplication is a fast method for comparing two fractions without finding a common denominator. Multiply the numerator of each fraction by the denominator of the other.

\(3\)
\(5\)
 vs
\(5\)
\(8\)
\(3 \times 8 = 24\)
\(5 \times 5 = 25\)
24 < 25, so:
\(\frac{3}{5} < \frac{5}{8}\)

Benchmark Fractions

Benchmark fractions like 0, \(\frac{1}{2}\), and 1 help you quickly estimate which fraction is larger without any computation.

\(0\)
\(\frac{1}{2}\)
\(1\)
\(\frac{3}{8}\)
< \(\frac{1}{2}\)
\(\frac{5}{7}\)
> \(\frac{1}{2}\)

Same Numerator vs Same Denominator

Same Denominator
When denominators match, the fraction with the bigger numerator is greater.

\(\frac{5}{9} > \frac{2}{9}\) because 5 > 2.
Same Numerator
When numerators match, the fraction with the smaller denominator is greater.

\(\frac{3}{5} > \frac{3}{8}\) because fifths are larger than eighths.

Step-by-Step Comparison Strategy

1
Check denominators. If they are the same, compare numerators directly: bigger numerator wins.
2
Check numerators. If they are the same, compare denominators: smaller denominator wins.
3
Try benchmarks. If one fraction is above \(\frac{1}{2}\) and the other is below, you are done.
4
Cross multiply. Multiply each numerator by the other fraction's denominator and compare the products.
5
Verify. Double-check by converting to common denominators or decimals if time allows.

Example 1: Compare \(\frac{4}{7}\) and \(\frac{3}{7}\)

\(\frac{4}{7}\)
\(\frac{3}{7}\)
\(\frac{4}{7} > \frac{3}{7}\) ✓

Example 2: Compare \(\frac{5}{6}\) and \(\frac{7}{9}\)

Step 1: Denominators differ (6 and 9). Find the LCD: \(\text{LCM}(6, 9) = 18\).

\(\frac{5}{6} \times \frac{3}{3}\)
\(= \frac{15}{18}\)
\(\frac{7}{9} \times \frac{2}{2}\)
\(= \frac{14}{18}\)
 Compare numerators:
15 > 14
\(\frac{5}{6} > \frac{7}{9}\) ✓

Example 3: Compare \(\frac{7}{12}\) and \(\frac{5}{8}\)

Benchmark test: Is each fraction greater or less than \(\frac{1}{2}\)?

\(\frac{7}{12}\)
Half of 12 is 6. Since 7 > 6, this fraction is above \(\frac{1}{2}\).
\(\frac{5}{8}\)
Half of 8 is 4. Since 5 > 4, this fraction is also above \(\frac{1}{2}\).

Both are above \(\frac{1}{2}\), so benchmarks are not enough. Use cross multiplication:

Example 4: Comparing Three Fractions

Order from least to greatest: \(\frac{2}{3}\), \(\frac{3}{5}\), \(\frac{7}{10}\).

1
Find the LCD. \(\text{LCM}(3, 5, 10) = 30\).
2
Convert. \(\frac{2}{3} = \frac{20}{30}\), \(\frac{3}{5} = \frac{18}{30}\), \(\frac{7}{10} = \frac{21}{30}\).
3
Compare numerators: 18 < 20 < 21.
4
Result: \(\frac{3}{5} < \frac{2}{3} < \frac{7}{10}\).

Common Mistakes to Avoid

✕ Comparing Across
Comparing numerator of one fraction to denominator of the other. You cannot conclude \(\frac{2}{7}\) is greater than \(\frac{3}{5}\) just because you see the numbers 7 and 5. Always use a valid method like cross multiplication.
✕ Bigger Denominator = Bigger Fraction
A larger denominator means smaller pieces. \(\frac{1}{10}\) is much less than \(\frac{1}{3}\), even though 10 > 3.
✕ Forgetting to Simplify
Before comparing, check if fractions can be simplified. \(\frac{4}{8}\) is just \(\frac{1}{2}\): recognizing this saves time.

Comparison Methods Summary

Method When to Use Speed
Same Denominator Compare numerators Denominators are equal Instant
Same Numerator Compare denominators (flip!) Numerators are equal Instant
Benchmark Compare each to \(\frac{1}{2}\) One is above, one below \(\frac{1}{2}\) Fast
Cross Multiply Multiply across and compare Any two fractions Fast
Common Denom. Find LCD, convert, compare Three or more fractions Moderate

Question 1 of 5

Which fraction is greater: \(\frac{5}{9}\) or \(\frac{4}{7}\)?

Question 2 of 5

Which of the following is the least fraction?

Question 3 of 5

When comparing \(\frac{a}{b}\) and \(\frac{c}{d}\) using cross multiplication, you find that \(a \times d > c \times b\). What can you conclude?

Question 4 of 5

Which fraction is closest to \(\frac{1}{2}\)?

Question 5 of 5

Arrange these fractions from least to greatest: \(\frac{3}{4}\), \(\frac{2}{5}\), \(\frac{7}{10}\). Which order is correct?