Lesson 13: Compare to 1

Property

For any positive number $n$:

• $n \times \frac{a}{b} < n$ if $\frac{a}{b} < 1$
• $n \times \frac{a}{b} = n$ if $\frac{a}{b} = 1$
• $n \times \frac{a}{b} > n$ if $\frac{a}{b} > 1$

Examples

• Is $5 \times \frac{3}{4}$ greater than or less than 5? Since $\frac{3}{4} < 1$, the product is less than 5.
• Is $12 \times \frac{5}{5}$ greater than or less than 12? Since $\frac{5}{5} = 1$, the product is equal to 12.
• Is $8 \times \frac{7}{6}$ greater than or less than 8? Since $\frac{7}{6} > 1$, the product is greater than 8.

Explanation

This skill involves comparing the size of a product to the original number without actually calculating the product. When you multiply a positive number by a fraction, you are scaling it. If you multiply by a fraction less than 1, the number gets smaller. If you multiply by a fraction greater than 1, the number gets larger.

Property

To compare the product of two fractions to 1 without calculating, examine each fraction.
If both fractions are less than 1, their product will also be less than 1.
If one fraction is greater than 1 and the other is less than 1, their product could be greater than, less than, or equal to 1.

Examples

• Is $\frac{3}{4} \times \frac{7}{8}$ greater than or less than 1? Since $\frac{3}{4} < 1$ and $\frac{7}{8} < 1$, their product is less than 1.
• Is $\frac{2}{5} \times \frac{4}{3}$ greater than or less than 1? We cannot tell without calculating because $\frac{2}{5} < 1$ and $\frac{4}{3} > 1$. In this case, $\frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$, which is less than 1.
• Is $\frac{5}{2} \times \frac{3}{4}$ greater than or less than 1? We cannot tell without calculating because $\frac{5}{2} > 1$ and $\frac{3}{4} < 1$. In this case, $\frac{5}{2} \times \frac{3}{4} = \frac{15}{8}$, which is greater than 1.

Explanation

This skill helps you estimate the size of a product involving fractions by comparing it to the benchmark of 1. When you multiply a number by a fraction less than 1, the result is smaller than the original number. Therefore, multiplying two fractions that are both less than 1 will always result in a product that is even smaller and thus less than 1. However, if one factor is greater than 1 and the other is less than 1, you must perform the calculation to determine how the product compares to 1.