Multiplying Fractions Calculator

Multiplication is the easiest of the four fraction operations: there is nothing to line up, nothing to borrow and no common denominator to find. This calculator applies the rule and prints every intermediate line, so the answer can be checked rather than trusted.

The formula and where it comes from

Multiplying by a fraction means taking a part of a part. Half of two thirds is a rectangle cut into three columns and two rows: two of the three columns are shaded, one of the two rows is taken, and the piece that is shaded twice covers 2 of the 6 cells — that is 1/2 × 2/3 = 2/6 = 1/3. The denominators multiply because they count how many pieces the whole was cut into; the numerators multiply because they count how many of those pieces are taken. That is also why a common denominator is never needed: nothing is being counted together, one cut is simply applied to another.

Step-by-step guide

1. Simplify any fraction that can be simplified. 3/6 becomes 1/2 — the value does not change, the numbers just stay small.
2. Write both numbers as fractions. A whole number goes over 1 (4 = 4/1); a mixed number becomes an improper fraction (2 3/4 = (2 × 4 + 3)/4 = 11/4).
3. Multiply the numerators together — that is the numerator of the answer.
4. Multiply the denominators together — that is the denominator of the answer.
5. Reduce the result by the greatest common factor of the two parts, and write it as a mixed number if the numerator is larger than the denominator.

Common fraction multiplications

Where fraction multiplication is used

• Cooking: baking two thirds of a recipe that calls for 3/4 cup of flour needs 2/3 × 3/4 = 1/2 cup.
• Shopping: a third off a price already cut by a half is 1/2 × 2/3 of the original — the discounts multiply, they do not add.
• Woodworking and sewing: five pieces of 3/8 inch each need 5 × 3/8 = 15/8 = 1 7/8 inches of material.
• Probability: two independent events with chances 1/2 and 1/3 happen together with chance 1/2 × 1/3 = 1/6.
• Scale drawings: a model at 1/8 scale of a part that is already 1/2 of the original is 1/16 of it.

Limits and things to watch

• Multiplying does not always make a number bigger. Multiplying by a fraction smaller than 1 makes it smaller: 6 × 2/3 = 4. Only a factor greater than 1 increases the value.
• Order does not matter. 1/2 × 4 and 4 × 1/2 are the same question with the same answer, because multiplication is commutative — unlike division, where 3/4 ÷ 2 and 2 ÷ 3/4 differ.
• The decimal form is often not exact. 2/3 × 2/3 = 4/9 is 0.4444… forever; whenever the reduced denominator has a prime factor other than 2 or 5, the decimal repeats and the page marks it with ≈. The fraction is the exact answer.
• An unreduced input is the same question. 3/6 × 2 and 1/2 × 2 both equal 1; the page reduces first and says so, so the two pages agree instead of contradicting each other.