Compare Fractions Calculator

Comparing Two Fractions with Cross-Multiplication

When two fractions share the same denominator, comparing them is easy — whichever has the larger numerator is the larger fraction. When the denominators are different, the most reliable method is cross-multiplication. To compare \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) (with both denominators positive), multiply diagonally across the equals sign and compare the two products:

$$a \times d \quad \text{vs.} \quad c \times b$$

Whichever cross-product is larger tells you which fraction is larger — because multiplying both sides of an inequality by the same positive number (here, \(b \times d\)) never changes the direction of the inequality. If the two cross-products are equal, the fractions represent the exact same value, even if they don't look identical.

Worked Example

Compare \(\dfrac{3}{4}\) and \(\dfrac{5}{7}\):

$$3 \times 7 = 21 \qquad 4 \times 5 = 20$$

Since \(21 > 20\), we know \(\dfrac{3}{4} > \dfrac{5}{7}\). No common denominator or decimal conversion was needed — just two quick multiplications.

Ordering Three or More Fractions with a Common Denominator

Cross-multiplication works great for two fractions at a time, but once you have three or more fractions to sort, it's faster to rewrite every fraction over the same Least Common Denominator (LCD). Once all the fractions share a denominator, you can simply order them by their numerators — largest numerator is the largest fraction, smallest numerator is the smallest fraction.

Worked Example

Order \(\dfrac{1}{2}\), \(\dfrac{3}{5}\), and \(\dfrac{5}{8}\) from least to greatest. The LCD of 2, 5, and 8 is 40:

$$\dfrac{1}{2} = \dfrac{20}{40} \qquad \dfrac{3}{5} = \dfrac{24}{40} \qquad \dfrac{5}{8} = \dfrac{25}{40}$$

Comparing the numerators 20, 24, and 25 gives the final order:

$$\dfrac{1}{2} < \dfrac{3}{5} < \dfrac{5}{8}$$

The Order Multiple Fractions tab above automates this exact process for anywhere from 3 to 6 fractions, showing the LCD, the converted table, and the final order using the fractions the way you originally typed them.

Related Calculators

Need to work with fractions in a different way? Try these related tools:

• Fraction Calculator — for adding, subtracting, multiplying, or dividing fractions
• Simplify Fraction — to reduce a fraction to lowest terms
• LCM and GCF Calculator — to find the least common denominator directly
• Equivalent Fractions Calculator — to generate a list of equivalent fractions for a given value

Frequently Asked Questions

How do I compare fractions with different denominators?

Use cross-multiplication: for \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\), compare \(a \times d\) against \(c \times b\). The larger product corresponds to the larger fraction. This works without ever finding a common denominator.

What if two fractions are equal but written differently?

Fractions like \(\dfrac{1}{2}\) and \(\dfrac{2}{4}\) represent the same value even though they're written differently. The cross-multiplication test catches this automatically: \(1 \times 4 = 4\) and \(2 \times 2 = 4\) — since the products match, the calculator reports the fractions as "equal to" one another rather than picking a winner.

Can I compare negative fractions?

Yes. Enter a minus sign in front of either fraction, such as -3/4. The cross-multiplication method still works correctly because both denominators stay positive — a more negative numerator simply means a smaller (more negative) value overall, e.g. \(-\dfrac{3}{4} < -\dfrac{2}{3}\).

How do I put a list of fractions in order?

Switch to the "Order Multiple Fractions" tab, enter between 3 and 6 fractions, choose Ascending or Descending, and click Order. The calculator converts every fraction to a common denominator so they can be compared directly, then lists them in the fraction notation you originally typed.

What's the difference between comparing and simplifying?

Comparing tells you which of two (or more) fractions is larger, smaller, or equal. Simplifying reduces a single fraction to its lowest terms without changing its value at all. If you just need to reduce a fraction, use the Simplify Fraction calculator instead.