Multiplying Fractions by Whole Numbers

Multiplying a fraction by a whole number is a lot like multiplying regular numbers. When you multiply a whole number by a fraction, you are simply adding that fraction to itself multiple times.

Understanding the Concept

Think of multiplication as repeated addition. If you have the problem $5 \times \frac{3}{4}$, it means you have $5$ groups of $\frac{3}{4}$.

You can write this out as addition:

$5 \times \frac{3}{4} = \frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4} + \frac{3}{4}$

When we add fractions with the same denominator, we just add the numerators:

$\frac{3 + 3 + 3 + 3 + 3}{4} = \frac{15}{4}$

The Quick Rule

Instead of writing out long addition sentences, you can use a simple shortcut:

1. Multiply the whole number by the numerator (the top number of the fraction).
2. Keep the denominator (the bottom number) exactly the same.

Let's try another example: $4 \times \frac{2}{3}$

Multiply the whole number $4$ by the numerator $2$:

$4 \times 2 = 8$

Keep the denominator $3$ the same. Your answer is an improper fraction:

$4 \times \frac{2}{3} = \frac{8}{3}$

(Optional: You can convert $\frac{8}{3}$ into a mixed number, which is $2\frac{2}{3}$.)

Multiplying Whole Numbers by Mixed Numbers

Sometimes you will need to multiply a whole number by a mixed number, like $3 \times 2\frac{1}{5}$. There are two easy ways to do this.

Method 1: Break it Apart

Multiply the whole number parts together, then multiply the fraction parts, and add them up.

1. Multiply the whole numbers: $3 \times 2 = 6$
2. Multiply the fraction: $3 \times \frac{1}{5} = \frac{3}{5}$
3. Combine them: $6 + \frac{3}{5} = 6\frac{3}{5}$

Method 2: Use Improper Fractions

Change the mixed number into an improper fraction first.

1. Convert $2\frac{1}{5}$ to an improper fraction: $2 \times 5 + 1 = 11$, so it becomes $\frac{11}{5}$.
2. Multiply by the whole number: $3 \times \frac{11}{5} = \frac{33}{5}$.
3. Convert back to a mixed number: $\frac{33}{5} = 6\frac{3}{5}$.