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Multiply two proper fractions

Multiplication of two positive proper fractions represents finding a fractional part of a fractional amount, such as ab\frac{a}{b} of cd\frac{c}{d}, and can be modeled with area diagrams or partitioned wholes. The product is found by multiplying numerators and denominators, then simplifying when appropriate; because each factor is less than one, the product is less than either factor. This understanding supports proportional reasoning and later fraction operations.

Detailed Explanation: Multiply two proper fractions

To multiply two proper fractions, multiply the numerators, multiply the denominators, and simplify the result if possible.

Example:

23×34\frac{2}{3}\times\frac{3}{4}

This means finding 23\frac{2}{3} of 34\frac{3}{4}.

  1. Multiply the numerators:

2×3=62\times 3=6

  1. Multiply the denominators:

3×4=123\times 4=12

So,

23×34=612\frac{2}{3}\times\frac{3}{4}=\frac{6}{12}

  1. Simplify 612\frac{6}{12}. Both 6 and 12 can be divided by 6:

6÷612÷6=12\frac{6\div 6}{12\div 6}=\frac{1}{2}

Therefore,

23×34=12\boxed{\frac{2}{3}\times\frac{3}{4}=\frac{1}{2}}

You can picture a whole divided into 3 rows and 4 columns, making 12 equal parts. Shading 34\frac{3}{4} one way and 23\frac{2}{3} another way leaves 6 overlapping parts out of 12, which is 612=12\frac{6}{12}=\frac{1}{2}. Since both factors are less than 1, the product 12\frac{1}{2} is less than either factor.

Learn by doing: Multiply two proper fractions

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