Guide

How to Multiply and Divide Fractions

Multiplication of fractions does not need a common denominator. You multiply numerators, multiply denominators, and simplify. Division uses that product after you replace the divisor with its reciprocal. A multiplying fractions calculator shows the unsimplified product and the cancelled form so you can see both layers.

Multiplying numerators

For a/b × c/d the new numerator is a × c. 2/3 × 3/5 has numerator 2 × 3 = 6. The multiplication is ordinary integer arithmetic. Negative signs follow the usual rules: one negative factor makes a negative product; two negatives make a positive product.

Multiplying denominators

The new denominator is b × d. In the same example, 3 × 5 = 15. The unsimplified product is 6/15. You do not search for a common denominator here. That hunt belongs to addition.

Cross-cancellation

Before multiplying, you may cancel a common factor of a numerator and the other fraction’s denominator. gcd(2, 5) = 1, so nothing cancels on that diagonal. gcd(3, 3) = 3, so the 3 in the first denominator cancels the 3 in the second numerator. The product becomes 2/1 × 1/5 = 2/5. Cancellation is optional for correctness if you simplify at the end, but it keeps intermediate BigInt values smaller, which is why the engine reduces factors before multiplying when it can.

Simplifying products

6/15 has GCD 3, so 2/5. 1/2 × 1/2 = 1/4, already simplest. 0/3 × 5/7 = 0. An integer result such as 8/4 after simplification is 2, displayed as an integer rather than 2/1. Check with the multiply tool.

Worked multiplication example

2/3 × 3/5. Numerators 6, denominators 15, GCD 3, result 2/5. Reverse check: 2/5 ÷ 3/5 should return 2/3 if division is implemented as a reciprocal product. That is a useful way to check answers without converting to decimals.

Reciprocal fractions

The reciprocal of c/d, when c is not zero, is d/c. The reciprocal of 2/5 is 5/2. The reciprocal of 4 is 1/4. The reciprocal of 0 is undefined, which is why a zero divisor is rejected. Signs move with the numerator in canonical form: the reciprocal of −2/5 is −5/2.

Dividing fractions

a/b ÷ c/d = a/b × d/c. You identify the divisor, flip it, and multiply. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. Mixed number 1 7/8. 1/2 ÷ 1/4 = 1/2 × 4/1 = 2. The dividing fractions calculator writes those steps and will not print Infinity.

Worked division example

Divide 3/4 by 2/5. Reciprocal 5/2. Product 15/8. 15 ÷ 8 = 1 remainder 7, so 1 7/8. Open the division desk with those values. A second check: 15/8 × 2/5 should recover 3/4 after simplification.

Zero divisors

3/4 ÷ 0/5 is undefined. The numerator of the divisor is zero, so there is no reciprocal. The calculator shows an error. 0/3 ÷ 5/7 is defined and equals 0. Zero as a dividend is ordinary; zero as a divisor is not.

Improper fractions

Improper fractions multiply and divide with the same rules. 5/4 × 6/5 = 30/20 = 3/2. You do not have to convert to mixed numbers first. If you start from mixed numbers, convert them to improper fractions, then multiply. See mixed numbers.

Negative fractions

−2/3 × 3/5 = −2/5. −2/3 × −3/5 = 2/5. −3/4 ÷ 2/5 = −15/8. The sign is part of the numerator after canonicalization. Do not drop it when you write the mixed number: −15/8 is −1 7/8, meaning the opposite of 1 7/8.

Checking answers

Three checks help. Multiply the quotient by the divisor and recover the dividend. Convert a simple case you already know, such as 1/2 × 1/2 = 1/4. Avoid checking only with rounded decimals: 2/3 × 3/5 is exactly 2/5, not “about 0.4” as the primary claim. Decimal display on this site is labeled as exact only when the expansion terminates in the shown places. Related: main calculator, addition and subtraction.