Can I always use cross-multiplication to compare any two fractions?

Yes, as long as both denominators are positive. In standard fractions, they always are. The rule works because multiplying both sides of a comparison by positive numbers preserves the direction of the inequality.

What if the two fractions turn out to be equal?

If ad = bc, then the two fractions are equivalent (the same value). For example, \frac{2}{3} and \frac{4}{6} both give a product of 12 when cross-multiplied (2 \times 6 = 12 and 3 \times 4 = 12), so they are equal.

Is cross-multiplication the only method to compare fractions?

No. You can find a common denominator and compare numerators, convert to decimals, compare each fraction to a reference point like \frac{1}{2}, or use the Equivalent Fractions method. Cross-multiplication is often the fastest, especially for larger denominators.

Why do negative denominators change how comparison works?

If a denominator is negative, multiplying by it flips the inequality (just like multiplying an inequality by -1 does). For example, 2 -3. That is why the rule explicitly requires b, d > 0.

Comparing Fractions

Compare two fractions to find which is larger or smaller without finding a common denominator—use when you need a quick size comparison.

\frac{a}{b} > \frac{c}{d} \iff ad > bc \quad (b, d > 0)

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What each symbol means

What Comparing Fractions takes

Symbol	Meaning
$a$	The numerator of the first fraction; if you read it as the denominator instead, you reverse which fraction is bigger.
$b$	The denominator of the first fraction; must be positive, and confusing it with $a$ will flip the inequality.
$c$	The numerator of the second fraction; mixing it with $d$ causes you to compare the fractions backwards.
$d$	The denominator of the second fraction; must be positive, as the rule requires both denominators to be greater than zero.

When to use it

Use this when you need to determine whether one fraction is bigger, smaller, or equal to another.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Compare two simple fractions with small integers

Problem

Which is larger,

\frac{3}{5}

\frac{2}{4}

$3 \cdot 4 = 12$
Cross-multiply: multiply the numerator of the first fraction by the denominator of the second.
$5 \cdot 2 = 10$
Multiply the denominator of the first fraction by the numerator of the second.
$12 > 10$
Compare the products: since $12 > 10$ , the first fraction is larger by the cross-multiplication rule.

Answer: $\frac{3}{5} > \frac{2}{4}$

Cross-multiplication gives us the answer without finding a common denominator. Since $12 > 10$ , we know $\frac{3}{5}$ is the larger fraction.

2. Compare improper fractions where the answer is counterintuitive

Problem

\frac{5}{3}

greater than

\frac{7}{4}

$5 \cdot 4 = 20$
Cross-multiply: numerator of the first fraction times denominator of the second.
$3 \cdot 7 = 21$
Denominator of the first fraction times numerator of the second.
$20 < 21$
Since $20$ is not greater than $21$ , the first fraction is not greater than the second.

Answer: $\frac{5}{3} < \frac{7}{4}$

Both fractions are improper (greater than 1), but cross-multiplication still applies. The fact that $20 < 21$ tells us $\frac{5}{3}$ is actually smaller than $\frac{7}{4}$ , which might surprise you if you only looked at the integers.

3. Compare savings rates in a word problem

Problem

An investment grew by

\frac{5}{12}

of its value in year one and

\frac{3}{7}

in year two. Which year had the larger growth rate?

$5 \cdot 7 = 35$
Cross-multiply: the numerator of the first growth rate times the denominator of the second.
$12 \cdot 3 = 36$
The denominator of the first growth rate times the numerator of the second.
$35 < 36$
Since $35 < 36$ , the first year's growth was smaller than the second year's.

Answer: $\frac{5}{12} < \frac{3}{7}$

Year two had the larger growth rate. Cross-multiplication lets us compare without converting to decimals, saving time when dealing with real-world data.

Common mistakes

Where Comparing Fractions usually goes wrong

The mistake	Why it is wrong	The fix
Writing $3 > 2$ , so $\frac{3}{5} > \frac{2}{4}$ .	You only compared numerators and completely ignored the denominators, which determine the size of each piece.	Cross-multiply: $3 \times 4 = 12$ and $5 \times 2 = 10$ ; since $12 > 10$ , then $\frac{3}{5} > \frac{2}{4}$ .
Thinking that $\frac{3}{5}$ is bigger than $\frac{2}{4}$ because $3 + 5 = 8$ and $2 + 4 = 6$ .	Adding the numerator and denominator has no meaning in fraction comparison and is unrelated to the cross-multiplication rule.	Use cross-multiplication correctly: $3 \times 4 = 12$ and $5 \times 2 = 10$ , then compare the products $12 > 10$ .
Assuming that if $\frac{a}{b} > \frac{c}{d}$ , then both $a > c$ and $b > d$ .	The cross-multiplication rule only compares the products $ad$ and $bc$ ; there is no requirement for both numerators and both denominators to follow the same order.	Apply the rule $ad > bc$ , not $a > c$ or $b > d$ . For instance, $\frac{5}{3} < \frac{7}{4}$ even though $5 > 4$ , because $5 \times 4 = 20 < 21 = 3 \times 7$ .

Tips and when to use something else

When both fractions have the same denominator, just compare numerators—no cross-multiplication needed.
If cross-multiplication feels tedious, use Fraction to Decimal to convert and compare decimals instead.
Or find a common denominator using Equivalent Fractions and compare numerators—an alternative method to cross-multiplication.
For improper fractions, converting to mixed numbers may help you visualize and estimate the answer.

Related formulas

Equivalent FractionsMultiply or divide both parts of a fraction by the same non-zero number to create an equivalent fraction with the same value.Simplifying FractionsSimplify a fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor.Fraction to DecimalConvert a fraction to its decimal form by dividing the numerator by the denominator, useful for comparing fractions directly.Adding FractionsAdd fractions with different denominators by finding a common denominator, then combining the numerators to get your answer.

Frequently asked questions

Can I always use cross-multiplication to compare any two fractions?: Yes, as long as both denominators are positive. In standard fractions, they always are. The rule works because multiplying both sides of a comparison by positive numbers preserves the direction of the inequality.
What if the two fractions turn out to be equal?: If $ad = bc$ , then the two fractions are equivalent (the same value). For example, $\frac{2}{3}$ and $\frac{4}{6}$ both give a product of $12$ when cross-multiplied ( $2 \times 6 = 12$ and $3 \times 4 = 12$ ), so they are equal.
Is cross-multiplication the only method to compare fractions?: No. You can find a common denominator and compare numerators, convert to decimals, compare each fraction to a reference point like $\frac{1}{2}$ , or use the Equivalent Fractions method. Cross-multiplication is often the fastest, especially for larger denominators.
Why do negative denominators change how comparison works?: If a denominator is negative, multiplying by it flips the inequality (just like multiplying an inequality by $-1$ does). For example, $2 < 3$ becomes $-2 > -3$ . That is why the rule explicitly requires $b, d > 0$ .

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Reviewed 2026-09-18