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Comparing
Fractions

Learn how to decide which fraction is greater, which is smaller, and when two fractions are equal.

ISEE Lower Level

What Is a Fraction?

A fraction shows part of a whole. It has two numbers: the top number (numerator) tells how many parts you have, and the bottom number (denominator) tells how many equal parts the whole is split into.

3 shaded parts out of 8

Numerator and Denominator

Numerator (top)
The number of parts you actually have. In \( \frac{3}{4} \), the numerator is 3.
Denominator (bottom)
The number of equal parts in the whole. In \( \frac{3}{4} \), the denominator is 4.

Why Comparing Matters

On the ISEE, you will often be asked which fraction is greatest or least. Comparing fractions also helps in everyday life: sharing food, measuring, and reading scores.

\( \frac{1}{3} \) of pizza \( \frac{1}{4} \) of pizza

Rule 1: Same Denominator

When two fractions have the same denominator, the whole is split into the same number of parts. So the fraction with the larger numerator is greater.

\( \frac{2}{5} \) \( \frac{4}{5} \) \( \frac{2}{5} < \frac{4}{5} \)

Rule 2: Same Numerator

When two fractions have the same numerator, you have the same number of parts. But a smaller denominator means each part is bigger, so that fraction is greater.

\( \frac{1}{3} \) \( \frac{1}{6} \) \( \frac{1}{3} > \frac{1}{6} \)

Don't Mix Up the Two Rules

It is easy to confuse the two simple rules. Remember which number is the same first, then apply the right rule.

Same Denominator
Compare numerators. Bigger numerator = bigger fraction. Example: \( \frac{5}{8} > \frac{3}{8} \).
Same Numerator
Compare denominators. Smaller denominator = bigger fraction. Example: \( \frac{2}{3} > \frac{2}{7} \).

When Everything Is Different

If the numerators and denominators are different, rewrite both fractions with a common denominator. Then compare the numerators.

\( \frac{2}{3} \) vs \( \frac{3}{4} \) Common denominator = 12 \( \frac{8}{12} \) \( < \) \( \frac{9}{12} \)

Step-by-Step Approach

1
Check the denominators. If they match, compare numerators directly.
2
Check the numerators. If they match, the smaller denominator is the bigger fraction.
3
Otherwise, find a common denominator. Rewrite both fractions, then compare the new numerators.
4
Use benchmarks. Compare each fraction to \( \frac{1}{2} \) or \( 1 \) for a quick estimate.

The Cross-Multiply Shortcut

A fast way to compare two fractions: cross-multiply. Multiply each numerator by the other fraction's denominator, then compare the products.

\( \frac{3}{5} \) ? \( \frac{5}{8} \) \( 3 \times 8 = 24 \) \( 5 \times 5 = 25 \) \( 24 < 25 \Rightarrow \frac{3}{5} < \frac{5}{8} \)

Example 1: Same Bottom

Which is greater, \( \frac{3}{7} \) or \( \frac{5}{7} \)?

1
Denominators match (both 7), so just compare the tops.
2
\( 5 > 3 \), so the fraction with numerator 5 is greater.

Example 2: Same Top

Which is greater, \( \frac{4}{9} \) or \( \frac{4}{5} \)?

1
Numerators match (both 4), so compare the denominators.
2
The smaller denominator gives bigger pieces. Since \( 5 < 9 \), the fraction with denominator 5 is greater.

Example 3: Different Both

Which is greatest: \( \frac{2}{3} \), \( \frac{3}{5} \), or \( \frac{7}{10} \)?

1
Find a common denominator. A number that 3, 5, and 10 all divide into is 30.
2
Rewrite: \( \frac{2}{3} = \frac{20}{30} \), \( \frac{3}{5} = \frac{18}{30} \), \( \frac{7}{10} = \frac{21}{30} \).
3
Compare numerators: \( 21 > 20 > 18 \).

Common Mistakes

✕ Bigger bottom = bigger
A larger denominator does NOT mean a larger fraction. \( \frac{1}{8} \) is smaller than \( \frac{1}{2} \).
✕ Comparing tops only
You can only compare numerators directly when the denominators are the same.
✕ Forgetting to convert
When bottoms differ, you must rewrite both fractions before comparing.

Quick Summary

What to Check Rule
Same denominator Compare numerators Bigger top wins
Same numerator Compare denominators Smaller bottom wins
Both different Common denominator Compare new tops
Two fractions only Cross-multiply Compare products

Question 1 of 5

Which fraction is greater: \( \frac{2}{9} \) or \( \frac{5}{9} \)?

Question 2 of 5

Which fraction is greater: \( \frac{3}{4} \) or \( \frac{3}{8} \)?

Question 3 of 5

Which of these fractions is the greatest?

Question 4 of 5

Using cross-multiplication, which is greater: \( \frac{4}{7} \) or \( \frac{5}{9} \)?

Question 5 of 5

Which of these fractions is the least?