Quick answer
To compare two fractions, use cross-multiplication: multiply the numerator of each fraction by the denominator of the other. The fraction whose product is larger is the greater fraction. For equal products, the fractions are equivalent.
How to use this calculator
Enter the numerator and denominator of each fraction. The calculator instantly shows which fraction is greater (or that they are equal), the decimal value of each fraction, the cross-multiplication steps, and both fractions converted to a common denominator (LCD form).
Comparing fractions formula
The cross-multiplication rule for two fractions a/b and c/d:
$$\frac{a}{b} \text{ vs } \frac{c}{d}: \quad \text{compare } a \times d \text{ with } c \times b$$
If a × d > c × b, then a/b > c/d. If a × d < c × b, then a/b < c/d. If a × d = c × b, then a/b = c/d. This works because it is equivalent to finding a common denominator — without actually computing it.
Cross-multiplication method
Cross-multiplication is the fastest method for comparing exactly two fractions. No common denominator computation is required:
$$\frac{3}{4} \text{ vs } \frac{2}{3}: \quad 3 \times 3 = 9, \quad 2 \times 4 = 8 \implies \frac{3}{4} > \frac{2}{3}$$
Since 9 > 8, the first fraction 3/4 is greater than 2/3. The method derives from the fact that both fractions can be expressed over b × d, making the comparison equivalent to comparing numerators directly.
Cross-multiplication is valid only when both denominators are positive. If a denominator is negative, move the negative sign to the numerator first (e.g. 3/−4 = −3/4).
LCD method
An equivalent approach: convert both fractions to the same denominator (the LCD), then compare numerators directly:
$$\frac{a}{b} = \frac{a \times \frac{\text{LCD}}{b}}{\text{LCD}}, \quad \frac{c}{d} = \frac{c \times \frac{\text{LCD}}{d}}{\text{LCD}}$$
For 3/4 vs 2/3 — LCD(4, 3) = 12:
$$\frac{3}{4} \text{ vs } \frac{2}{3}: \text{LCD}(4,3) = 12 \implies \frac{9}{12} > \frac{8}{12}$$
Comparing numerators: 9 > 8, so 3/4 > 2/3. The LCD method is more informative when you need to see both fractions on the same scale, not just know which is larger.
Worked examples
Example 1: simple comparison
Compare 5/8 and 3/5. Cross-multiply: 5 × 5 = 25 and 3 × 8 = 24. Since 25 > 24, we have 5/8 > 3/5. As decimals: 0.625 > 0.6.
Example 2: same numerator
Compare 2/5 and 2/7. When numerators are equal, the fraction with the smaller denominator is always larger. 2/5 > 2/7 because dividing into fewer parts gives larger parts. Cross-check: 2 × 7 = 14 > 2 × 5 = 10. ✓
Example 3: same denominator
Compare 5/9 and 7/9. Equal denominators mean equal part sizes, so compare numerators directly: 7 > 5, therefore 7/9 > 5/9.
| Fraction A | Fraction B | a × d | b × c | Result |
|---|---|---|---|---|
| 1/2 | 1/3 | 3 | 2 | 1/2 > 1/3 |
| 3/4 | 2/3 | 9 | 8 | 3/4 > 2/3 |
| 2/5 | 3/7 | 14 | 15 | 2/5 < 3/7 |
| 4/6 | 2/3 | 12 | 12 | 4/6 = 2/3 |
| 5/8 | 7/12 | 60 | 56 | 5/8 > 7/12 |
| 3/11 | 4/15 | 45 | 44 | 3/11 > 4/15 |
Negative fractions
Cross-multiplication carries the sign through the products automatically:
$$\frac{-3}{4} \text{ vs } \frac{-2}{3}: \quad -3 \times 3 = -9, \quad -2 \times 4 = -8 \implies \frac{-3}{4} < \frac{-2}{3}$$
−9 < −8, so −3/4 < −2/3. This matches the intuition that −3/4 = −0.75 and −2/3 ≈ −0.667, and more negative means smaller.
A useful rule: for two negative fractions, the one with the larger absolute value is the smaller (more negative) number. So |−3/4| = 3/4 > |−2/3| = 2/3, which confirms −3/4 < −2/3.
Ordering 3 or more fractions
To order a list of fractions, find the LCD of all denominators and convert each fraction. Then rank by numerator:
$$\frac{1}{2}, \frac{2}{5}, \frac{3}{7}: \text{LCD}(2,5,7)=70 \implies \frac{35}{70}, \frac{28}{70}, \frac{30}{70} \implies \frac{2}{5} < \frac{3}{7} < \frac{1}{2}$$
General process: (1) find the LCM of all denominators, (2) convert every fraction to that denominator, (3) sort numerators. For a short list, pairwise cross-multiplication also works but takes more steps.
Common mistakes
Comparing numerators when denominators differ
3/7 vs 3/5 — the numerators are equal, so students sometimes conclude the fractions are equal. They are not: 3/7 ≈ 0.429 and 3/5 = 0.6. Equal numerators with different denominators means the fraction with the larger denominator is always smaller.
Assuming a larger denominator means a larger fraction
1/10 vs 1/3 — 10 > 3, yet 1/10 < 1/3. A larger denominator means each part is smaller, so the fraction itself is smaller (for equal numerators).
Forgetting sign rules for negatives
Applying the positive-number reasoning to negatives: −1/2 vs −1/3. In positive terms, 1/2 > 1/3. But with negatives, the larger absolute value means the number is further left on the number line, so −1/2 < −1/3.
Frequently asked questions
How do you compare fractions with different denominators?
Use cross-multiplication: multiply the numerator of each fraction by the other fraction's denominator. The fraction whose product is larger is the greater fraction.
Can you compare fractions by converting to decimals?
Yes — divide numerator by denominator for each fraction and compare. This is reliable but may introduce rounding error with repeating decimals. Cross-multiplication is exact with no rounding.
Which fraction is bigger: 3/4 or 2/3?
3/4. Cross-multiply: 3 × 3 = 9 and 2 × 4 = 8. Since 9 > 8, the fraction 3/4 is larger. As decimals: 3/4 = 0.75 and 2/3 ≈ 0.667.
How do you compare negative fractions?
Cross-multiplication carries the sign automatically. Remember: the fraction with the larger absolute value is the smaller (more negative) number — it lies further left on the number line.
How do you order more than two fractions from least to greatest?
Find the LCD of all denominators, convert every fraction to an equivalent fraction with that denominator, and sort by numerator in ascending order.