Multiplying Fractions Formula

Multiplying fractions are to multiply fractions, multiply the numerators together and the denominators together: a/b x c/d = a x c/b x d.

The Formula

abร—cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

When to use: 23ร—34\frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take 34\frac{3}{4} of something, then take 23\frac{2}{3} of that result.

Quick Example

23ร—34=2ร—33ร—4=612=12\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}

Notation

abร—cd\frac{a}{b} \times \frac{c}{d} โ€” multiply numerators and denominators straight across

What This Formula Means

To multiply fractions, multiply the numerators together and the denominators together: abร—cd=aร—cbร—d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}. Simplify the result by cancelling common factors.

23ร—34\frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take 34\frac{3}{4} of something, then take 23\frac{2}{3} of that result.

Formal View

abร—cd=aโ‹…cbโ‹…d\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} where b,dโ‰ 0b, d \neq 0

Worked Examples

Example 1

easy
Multiply 23ร—57\frac{2}{3} \times \frac{5}{7}.

Answer

1021\frac{10}{21}

First step

1
Multiply the numerators: 2ร—5=102 \times 5 = 10.

Full solution

  1. 2
    Multiply the denominators: 3ร—7=213 \times 7 = 21.
  2. 3
    The product is 1021\frac{10}{21}, which is already in simplest form since gcdโก(10,21)=1\gcd(10, 21) = 1.
To multiply fractions, multiply numerator by numerator and denominator by denominator. No common denominator is needed, unlike addition.

Example 2

medium
Compute 49ร—38\frac{4}{9} \times \frac{3}{8}.

Example 3

easy
Multiply 27ร—34\frac{2}{7} \times \frac{3}{4} and simplify if possible.

Common Mistakes

  • Finding a common denominator before multiplying - multiplication goes straight across, no matching needed.
  • Multiplying only the numerators and keeping one denominator - multiply both tops and both bottoms.
  • Expecting the product to be bigger - a proper fraction times a proper fraction is smaller than both.

Why This Formula Matters

Multiplication is the operation where fractions stop needing a common denominator, and where 'multiplying makes smaller' first appears โ€” taking a part of a part shrinks it. It powers fraction-of-a-number, scaling, area, and probability of independent events. Recognizing it by "Am I taking a part of a part, multiplying tops and bottoms straight across?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from adding fractions with unlike denominators and dividing fractions and fraction of a number in a mixed problem set.

Frequently Asked Questions

What is the Multiplying Fractions formula?

To multiply fractions, multiply the numerators together and the denominators together: abร—cd=aร—cbร—d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}. Simplify the result by cancelling common factors.

How do you use the Multiplying Fractions formula?

23ร—34\frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take 34\frac{3}{4} of something, then take 23\frac{2}{3} of that result.

What do the symbols mean in the Multiplying Fractions formula?

abร—cd\frac{a}{b} \times \frac{c}{d} โ€” multiply numerators and denominators straight across

Why is the Multiplying Fractions formula important in Math?

Multiplication is the operation where fractions stop needing a common denominator, and where 'multiplying makes smaller' first appears โ€” taking a part of a part shrinks it. It powers fraction-of-a-number, scaling, area, and probability of independent events. Recognizing it by "Am I taking a part of a part, multiplying tops and bottoms straight across?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from adding fractions with unlike denominators and dividing fractions and fraction of a number in a mixed problem set.

What do students get wrong about Multiplying Fractions?

The procedure for multiplying fractions is the easy part; the trap is finding a common denominator before multiplying. Asking "Am I taking a part of a part, multiplying tops and bottoms straight across?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Multiplying Fractions formula?

Before studying the Multiplying Fractions formula, you should understand: fractions, multiplication.

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