Lesson 12: Compare Without Multiplying

Property

To compare a fraction $\frac{a}{b}$ to the benchmarks $0$, $\frac{1}{2}$, and $1$:

• If the numerator $a$ is very small compared to the denominator $b$, the fraction is close to $0$.
• If the numerator $a$ is about half of the denominator $b$ (i.e., $a \approx \frac{b}{2}$), the fraction is close to $\frac{1}{2}$.
• If the numerator $a$ is very close to the denominator $b$, the fraction is close to $1$.

Examples

Property

When comparing two fractions on the same side of a benchmark, determine their distance from that benchmark.
The fraction with the smaller distance is closer.
A larger denominator creates a smaller unit fraction, which represents a smaller distance (e.g., a distance of $\frac{1}{12}$ is smaller than a distance of $\frac{1}{8}$).

Examples

Property

For any positive number $a$:

• Multiplying $a$ by a fraction greater than 1 results in a product greater than $a$. ($a \times \frac{b}{c} > a$ if $\frac{b}{c} > 1$)
• Multiplying $a$ by a fraction less than 1 results in a product less than $a$. ($a \times \frac{b}{c} < a$ if $\frac{b}{c} < 1$)
• Multiplying $a$ by a fraction equal to 1 results in a product equal to $a$. ($a \times \frac{b}{c} = a$ if $\frac{b}{c} = 1$)

Examples

• Is $8 \times \frac{5}{4}$ greater than, less than, or equal to 8?

Since $\frac{5}{4} > 1$, the product is greater than 8.

• Is $\frac{7}{8} \times \frac{1}{2}$ greater than, less than, or equal to $\frac{7}{8}$?

Since $\frac{1}{2} < 1$, the product is less than $\frac{7}{8}$.

• Which is greater: $15 \times \frac{2}{3}$ or $15 \times \frac{3}{4}$?

Since $\frac{3}{4} > \frac{2}{3}$, multiplying 15 by the larger fraction results in a larger product. So, $15 \times \frac{3}{4}$ is greater.

Explanation

This skill involves reasoning about the size of a product without performing the actual multiplication. This is also known as scaling. When you multiply a number by a fraction greater than 1, you are scaling it up, so the result is larger. Conversely, multiplying by a fraction less than 1 scales the number down, making the result smaller. This allows you to quickly compare products by analyzing the value of the fraction you are multiplying by.