To multiply fractions, multiply the numerators to obtain the new numerator and multiply the denominators to obtain the new denominator. Unlike addition, no common denominator is needed. The rule follows naturally from taking a fraction of another fraction, and an area model makes that meaning visible.
An area model explains why both parts multiply
Shade two thirds of a rectangle in one direction, then three fifths in the other. The overlap occupies six of the fifteen equal small rectangles, so the product is 6/15, which simplifies to 2/5. Change the four values below to see how numerator and denominator affect the overlap.
Build a product of two fractions
The shaded overlap represents the product before and after reduction.
Direct multiplication followed by reduction
For 7/12 times 8/21, a direct product gives 56/252. Dividing both terms by their greatest common divisor, 28, produces 2/9. The unreduced and reduced fractions represent exactly the same number.
Always check that denominators are nonzero. A negative sign may be placed in the numerator or in front of the fraction; an odd number of negative factors makes the product negative, while an even number makes it positive.
Cross-cancellation keeps the arithmetic small
Before multiplying, divide a numerator and a denominator from opposite fractions by a common factor. This does not change either product because it removes the same factor from the overall numerator and denominator.
Cross-cancellation is optional, not a different rule. It is particularly useful with large values because it reduces the risk of an arithmetic error and often leaves the final fraction already in lowest terms.
Whole numbers and mixed numbers need one extra step
Write an integer n as n/1. Convert a mixed number to an improper fraction by multiplying the whole part by the denominator, adding the numerator, and keeping the denominator. Then apply the usual multiplication rule.
For example, two and one third becomes 7/3. Multiplying it by three fifths gives 21/15, which reduces to 7/5 or one and two fifths. Convert to a decimal only when the context needs it; the fraction preserves the exact value. Estimation provides a useful check: 2 1/3 times 3/5 should be a little greater than 1, so 1 2/5 is plausible.
Working with numerator-by-numerator and denominator-by-denominator multiplication after any valid cancellation
Before calculating, identify numerator-by-numerator and denominator-by-denominator multiplication after any valid cancellation. Write the known values with their units or dimensions and state what the result must represent. This prevents a correct-looking operation from answering the wrong question.
A frequent failure is finding a common denominator, canceling two numerators, or leaving a mixed number unconverted. To avoid it, keep each transformation on its own line, preserve exact values during intermediate steps, and explain why every factor, exponent, or sign is permitted. If two methods are available, use the second as a check rather than as unexplained repetition.
A reliable check using a reduced result, a nonzero denominator, the expected sign, and an estimate of size
- Start with numerator-by-numerator and denominator-by-denominator multiplication after any valid cancellation; it determines what the answer must represent.
- Rule out finding a common denominator, canceling two numerators, or leaving a mixed number unconverted before rounding.
- Use a reduced result, a nonzero denominator, the expected sign, and an estimate of size as an independent check, not as a repetition of the same calculation.
- Read the result in the context of scaling recipes, probability, areas, and taking a fractional part of a quantity, keeping only the precision supported by the data.
This procedure is useful in scaling recipes, probability, areas, and taking a fractional part of a quantity. The same formula may produce a numerical answer in each setting, but interpretation still depends on what was measured, which assumptions were made, and how much precision the data supports. A calculator or interactive diagram can reveal patterns, but it does not replace the compatibility, domain, and reasonableness checks.
