Multiplying Fractions Formula

Multiplying Fractions: to multiply fractions, multiply the numerators together and the denominators together: a/b × c/d = a × c/(b × d).

The Formula

ab×cd=acbd

When to use: 23×34 means 'two-thirds of three-quarters.' Take 34 of something, then take 23 of that result.

Quick Example

23×34=2×33×4=612=12

Notation

ab×cd — multiply numerators and denominators straight across

What This Formula Means

To multiply fractions, multiply the numerators together and the denominators together: ab×cd=a×cb×d. Simplify the result by cancelling common factors.

23×34 means 'two-thirds of three-quarters.' Take 34 of something, then take 23 of that result.

Formal View

ab×cd=a⋅cb⋅d where b,d≠0

Worked Examples

Example 1

easy
Multiply 23×57.

Answer

1021

First step

1
Multiply the numerators: 2×5=10.

Full solution

  1. 2
    Multiply the denominators: 3×7=21.
  2. 3
    The product is 1021, which is already in simplest form since 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 49×38.

Example 3

easy
Multiply 27×34 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 — 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?" — rather than by familiar numbers — 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: ab×cd=a×cb×d. Simplify the result by cancelling common factors.

How do you use the Multiplying Fractions formula?

23×34 means 'two-thirds of three-quarters.' Take 34 of something, then take 23 of that result.

What do the symbols mean in the Multiplying Fractions formula?

ab×cd — 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 — 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?" — rather than by familiar numbers — 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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