Multiply fractions and mixed numbers

Multiplication of fractions represents scaling one quantity by another, including finding a fractional part of a quantity; area models and numerical expressions show why numerators and denominators are multiplied, followed by simplifying and checking whether the product is larger or smaller than the factors. Mixed numbers are converted to improper fractions before multiplying, including products involving whole numbers, while estimation helps distinguish multiplication from addition; symbolic rational expressions and more advanced generalizations are outside this scope.

Detailed Explanation: Multiply fractions and mixed numbers

To multiply fractions and mixed numbers:

Convert each mixed number to an improper fraction.
Multiply the numerators.
Multiply the denominators.
Simplify the answer.
Check whether the answer makes sense.

Example: Find

1\frac{1}{2}\times\frac{2}{3}.

This means “find $\frac{2}{3}$ of $1\frac{1}{2}$ .”

Step 1: Convert the mixed number

Convert $1\frac{1}{2}$ to an improper fraction:

1\frac{1}{2}=\frac{1\times2+1}{2}=\frac{3}{2}.

Now the problem is

\frac{3}{2}\times\frac{2}{3}.

Step 2: Multiply numerators and denominators

Multiply across:

\frac{3\times2}{2\times3}=\frac{6}{6}.

The numerator tells how many fractional parts are included, and the denominator tells the size of those parts. In an area model, the side lengths $\frac{3}{2}$ and $\frac{2}{3}$ create a total area of $\frac{6}{6}$ .

Step 3: Simplify

Since $\frac{6}{6}=1$ ,

1\frac{1}{2}\times\frac{2}{3}=\boxed{1}.

Check the answer

Because $\frac{2}{3}$ is less than $1$ , multiplying $1\frac{1}{2}$ by $\frac{2}{3}$ should make it smaller. The answer $1$ is smaller than $1\frac{1}{2}$ , so it makes sense. Also, multiplication is different from addition: $1\frac{1}{2}+\frac{2}{3}$ would be greater than either number, but the product is $1$ .

Learn by doing: Multiply fractions and mixed numbers

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Fraction Multiplication - Mixed

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