Multiplying Fractions Calculator

Multiply across, no common denominator needed

Multiplication is the one fraction operation that skips the shared denominator. Multiply the two tops, multiply the two bottoms, then reduce. Three quarters of two fifths is six twentieths, and that reduces to three tenths: 6/20 = 3/10 = 0.3.

Why it works: a fraction is a division waiting to happen. Multiplying by 3/4 means multiplying by 3 and dividing by 4, and the order of those two steps does not matter. Do that twice and you have multiplied by both tops and divided by both bottoms.

The result is smaller than either factor when both factors sit below 1, which surprises people who expect multiplication to grow a number. Two fifths of three quarters is three tenths, and 0.3 is below both 0.75 and 0.4.

The reduction step is where most errors enter, because 15/60 and 1/4 look nothing alike. The calculator prints the product before reducing, the greatest common divisor it used, and the final fraction, so the two ends of the reduction can be checked against each other.

Cancel first and the numbers stay small

Any top can cancel against any bottom, not only the top and bottom of the same fraction. In 14/15 times 20/21, the 14 and the 21 share a factor of 7, and the 20 and the 15 share a factor of 5. Cancel both before multiplying and the problem becomes 2/3 times 4/3 = 8/9.

Same product the long way: 14 x 20 = 280 and 15 x 21 = 315, then 280/315 divides by 35 to reach 8/9. The answer is identical, and only the size of the intermediate numbers changed. That size is the reason to cancel first when the fractions are large or when several of them are chained together.

Canceling works on factors, not on terms. In the sum (2 + 4)/8 the 4 and the 8 are not being multiplied, so they do not cancel; the sum is 6/8 = 3/4. Cancel the 4 against the 8 first and you get 2/2 = 1, which is wrong. This is the most common way a simplified answer comes out too large.

When a fraction is multiplied by a whole number, the whole number is a factor and can be canceled like any other. 12 x 5/8 cancels the 4 in 12 against the 8, leaving 3 x 5/2 = 15/2 = 7 1/2.

Mixed numbers: convert, multiply, convert back

A mixed number has to become an improper fraction before it can be multiplied, because the whole part and the fraction part are added together, and a product does not distribute across a sum the way people assume. Two and a half is 5/2, one and a third is 4/3, and their product is 20/6, which reduces to 10/3, or 3 1/3.

Multiplying the parts separately is the classic error: 2 x 1 = 2 for the wholes and 1/2 x 1/3 = 1/6 for the fractions, giving 2 1/6. The true answer is 3 1/3, more than a whole larger. The separate-parts method only lands on the right answer when both fraction parts are zero.

Whole numbers go over 1 without changing value. Five times three quarters is 5/1 x 3/4 = 15/4 = 3 3/4, and the conversion is a single line of arithmetic that removes the whole class of errors above. Going back at the end is a division: 15 divided by 4 is 3 with a remainder of 3, so the mixed number is 3 3/4.

The worked examples in OpenStax Prealgebra, section 4.2 follow the same order: convert, multiply, simplify. The page also shows the sign rule, which matters once a mixed number is negative.

The word of is a multiplication sign

Half of three quarters is 3/8, and that sentence is the multiplication 1/2 x 3/4 = 3/8 written in English. Two thirds of three quarters is 6/12, or 1/2. Whenever a word problem puts "of" after a fraction, multiply.

Recipes are where this shows up most. Three quarters of a batch that calls for 1 1/2 cups of flour needs 3/4 x 3/2 = 9/8 cups, which is 1 1/8 cups. The measuring cup set decides the form: 9/8 is exact, and 1.125 is the same number on a digital scale.

A cup of flour is not a cup of water by weight, so a recipe written in grams should be scaled in grams and left to a scale. The fraction arithmetic is the same either way, and scaling by 3/2 always means multiplying every amount by 3 and dividing by 2.

The same phrasing appears in discounts and probabilities: three quarters of a 2/3 discount is 1/2 off, and the chance of two independent events at 1/4 and 1/3 each is 1/12. In every case the multiplication is the same rule with a different story attached.

Fraction, mixed number, or decimal

The same product has three useful forms. 10/3 is exact, 3 1/3 is readable on a tape measure, and 3.3333 is what a calculator shows. Which one to report depends on what happens next: a carpenter wants the mixed number, a spreadsheet wants the decimal, and a dilution wants the exact fraction until the final step.

Rounding loses information that later steps may need. 35/48 is 0.7292 to four places, and 0.7292 times 48 is 35.0016, not 35. In a single problem that difference is invisible; in a chain of twenty steps it accumulates into a visible error.

Repeating decimals never end: 1/3 is 0.3333, 2/3 is 0.6667, and 10/3 is 3.3333. The fraction is the exact value and every decimal is an approximation with a chosen number of places, which is why school problems ask for the answer in lowest terms rather than as a decimal.

An improper fraction is not wrong. 15/4 is a complete answer, and turning it into 3 3/4 is a presentation choice. The calculator gives both, plus the decimal, so you can copy whichever form the next step needs.

Signs, zeros, and the edges

One negative sign makes the product negative: -3/4 x 2/5 = -6/20 = -3/10. Two negatives cancel: -2/3 x -9/4 = 18/12 = 3/2, or 1 1/2. The sign of the product is the product of the signs, and it can be settled before any multiplication happens.

A zero on top makes the whole product zero: 0/5 x 3/4 = 0. A zero on the bottom is not allowed, because division by zero has no value. The calculator asks for a bottom number above zero instead of returning a number.

Two proper fractions always multiply to something smaller than either one, which makes a fast sanity check: 7/8 x 5/6 = 35/48, about 0.729, below both 0.875 and 0.833. A product of two proper fractions that comes out larger than either factor means a denominator was added or a fraction was flipped by mistake.

A negative mixed number is the one case where the conversion needs care. -2 1/2 is -5/2, not -3/2: the whole part and the fraction part are on the same side of zero, so the improper form adds their magnitudes and keeps the sign. -2 1/2 x 2/5 is -5/2 x 2/5 = -10/10 = -1.

Checks that catch a wrong product in seconds

Estimate first. 7/8 is close to 1 and 5/6 is close to 1, so the product should sit a little under 1. 35/48 fits, and 35/40 would not. The estimate does not have to be exact to catch an answer that is off by a factor of ten.

Multiply by the reciprocal. Any fraction times its flipped self is 1: 4/5 x 5/4 = 20/20 = 1. If a simplified answer to that problem is anything else, the reduction step went wrong.

One more: run the problem through the calculator twice, once with the mixed numbers and once with the improper fractions. The same product means the conversion was right, and a different one points at the conversion rather than at the multiplication.

The method itself is short enough to write on one line, and Maths Is Fun's walkthrough uses the same three steps with pictures of the pieces. Pictures help the first time; after that the rule and the checks do the work.