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.
- Cross-multiply to compare any two fractions: compute a×d and b×c; the larger product belongs to the larger fraction. No common denominator needed.
- Same numerator shortcut: the fraction with the smaller denominator is always larger - fewer, bigger parts.
- Same denominator shortcut: just compare numerators directly - no formula required.
- Negative fractions flip the comparison: larger absolute value means smaller (more negative) number. -3/4 < -2/3 even though 3/4 > 2/3.
- Ordering 3 or more fractions: find the LCD of all denominators, convert each fraction, then rank by numerator.
How to use our fractions comparing 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).
Explained to a beginner
Imagine comparing two discounts: "save 3 out of every 4 dollars" vs "save 2 out of every 3 dollars." To decide which is better, you need a common reference. If everything cost $12, the first deal saves $9 (3/4 of 12) and the second saves $8 (2/3 of 12). So 3/4 is the better deal. But finding that $12 reference takes work.
Cross-multiplication skips the $12 step entirely: multiply 3×3=9 and 2×4=8. Those products are exactly what the numerators become over the common denominator - so you get the comparison without computing it. Larger product, larger fraction: 9 > 8, so 3/4 > 2/3.
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 for finding the greater fraction
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 |
When I compare fractions by hand, I always scan for the same-numerator or same-denominator shortcut before reaching for cross-multiplication. Cross-multiplication works universally, but spotting that 5/9 vs 7/9 is just a numerator comparison takes one second; cross-multiplying takes five.
On a test or under time pressure, pattern recognition beats formula application every time.
Comparing 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.
Negative fractions are where fraction comparison intuition breaks down most reliably. I've seen people correctly find |−3/4| > |−2/3| and then conclude −3/4 > −2/3 because they're still thinking in absolute-value terms.
The number-line check fixes this: which number sits further left? Further left means more negative means smaller. One mental picture prevents dozens of sign errors.
How to order 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 when comparing fractions
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.
Quiz: how well do you know comparing fractions?
1. What does cross-multiplication compare to determine which fraction is larger?
2. Using cross-multiplication, compare 5/8 and 3/5. What are the two products?
3. When two fractions have the same numerator but different denominators, which fraction is larger?
4. According to the reference table on the page, what is the correct comparison for 2/5 vs 3/7?
5. Which of the following correctly compares -3/4 and -2/3?