Multiplication as Scaling

When we multiply a number, we are often used to the result getting bigger. However, when working with fractions, multiplication acts as a way of scaling a number. Depending on the fraction you multiply by, the number can stretch (get larger), shrink (get smaller), or stay exactly the same.

By understanding multiplication as scaling, you can compare numbers and predict outcomes without doing any actual computing!

The Three Rules of Scaling

When you multiply a number by a fraction, look at the size of the fraction to know what will happen to your starting number:

1. Multiplying by a number greater than 1 makes it larger. If the fraction is an improper fraction (where the top is bigger than the bottom, like $\frac{5}{4}$ or $\frac{7}{5}$), the value is greater than $1$. Multiplying by it will scale the number up.

2. Multiplying by a number less than 1 makes it smaller. If the fraction is a proper fraction (where the top is smaller than the bottom, like $\frac{1}{2}$ or $\frac{3}{4}$), the value is less than $1$. Multiplying by it will scale the number down.

3. Multiplying by exactly 1 keeps it the same. If the top and bottom of the fraction are the same (like $\frac{3}{3}$ or $\frac{8}{8}$), the fraction equals $1$. The starting number will not change.

Comparing Without Computing

Let's use these rules to solve some problems without doing any math calculations.

Example 1: Is $\frac{3}{4} \times 8$ greater or less than $8$? Look at the fraction: $\frac{3}{4}$. Since the numerator ($3$) is less than the denominator ($4$), the fraction is less than $1$. Therefore, multiplying $8$ by something less than $1$ shrinks it. Answer: $\frac{3}{4} \times 8$ is less than $8$.

Example 2: Is $5 \times \frac{7}{5}$ greater or less than $5$? Look at the fraction: $\frac{7}{5}$. Since the numerator ($7$) is greater than the denominator ($5$), the fraction is greater than $1$. Multiplying $5$ by something greater than $1$ stretches it. Answer: $5 \times \frac{7}{5}$ is greater than $5$.

Example 3: Compare $6 \times \frac{2}{3}$ with $6$. Look at the fraction: $\frac{2}{3}$. Because $\frac{2}{3}$ is less than $1$, scaling $6$ by this fraction will make it smaller. Answer: $6 \times \frac{2}{3} < 6$.

Summary

To quickly compare multiplication expressions:

• Factor $> 1 \rightarrow$ Result is larger.
• Factor $< 1 \rightarrow$ Result is smaller.
• Factor $= 1 \rightarrow$ Result is the same.