Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 4: Fractions

Lesson 6: Comparing Fractions

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra, students learn how to compare fractions by finding a common denominator, including the least common denominator using the least common multiple of the denominators. The lesson covers comparing fractions with the same denominator, rewriting fractions with a common denominator by multiplying numerator and denominator by the same factor, and using the number line as a visual tool. Part of Chapter 4 on Fractions, this lesson builds directly on fraction simplification skills to develop reliable comparison strategies for AMC 8 problem solving.

Section 1

Benchmark Fractions

Property

The fractions $\frac{1}{2}$ , $\frac{1}{4}$ , and $\frac{3}{4}$ are called benchmark fractions because we have a good intuitive feel for their size. We can estimate other fractions by comparing them to the benchmarks. To compare a fraction to a benchmark, write the benchmark fraction with the same denominator.

Examples

Is the fraction $\frac{7}{20}$ closer to $\frac{1}{4}$ or $\frac{1}{2}$ ? With a denominator of 20, $\frac{1}{4} = \frac{5}{20}$ and $\frac{1}{2} = \frac{10}{20}$ . Since 7 is closer to 5 than to 10, $\frac{7}{20}$ is closer to $\frac{1}{4}$ .

A team won 14 out of 30 games. Is this fraction, $\frac{14}{30}$ , closest to $\frac{1}{4}$ , $\frac{1}{2}$ , or $\frac{3}{4}$ ? We know $\frac{1}{2}$ of 30 is 15. Since 14 is very close to 15, the fraction is closest to $\frac{1}{2}$ .

Section 2

Comparing Fractions with Same Denominators

Property

When two fractions have the same positive denominator, the fraction with the larger numerator is the greater fraction: If $\frac{a}{c}$ and $\frac{b}{c}$ where $c > 0$ , then $\frac{a}{c} > \frac{b}{c}$ if and only if $a > b$ .

Examples

Section 3

Writing Equivalent Fractions

Property

To write an equivalent fraction with a larger denominator:

Divide the old denominator into the desired denominator. This gives you the 'building factor'.
Use that factor to multiply the old numerator.

This process is shown as:

\frac{\text{old numerator}}{\text{old denominator}} \times \frac{\text{building factor}}{\text{building factor}} = \frac{\text{new numerator}}{\text{new denominator}}

Examples

To write $\frac{2}{5}$ with a denominator of 15, the building factor is $15 \div 5 = 3$ . We multiply to get $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Benchmark Fractions

Property

Examples

Is the fraction $\frac{7}{20}$ closer to $\frac{1}{4}$ or $\frac{1}{2}$ ? With a denominator of 20, $\frac{1}{4} = \frac{5}{20}$ and $\frac{1}{2} = \frac{10}{20}$ . Since 7 is closer to 5 than to 10, $\frac{7}{20}$ is closer to $\frac{1}{4}$ .

A team won 14 out of 30 games. Is this fraction, $\frac{14}{30}$ , closest to $\frac{1}{4}$ , $\frac{1}{2}$ , or $\frac{3}{4}$ ? We know $\frac{1}{2}$ of 30 is 15. Since 14 is very close to 15, the fraction is closest to $\frac{1}{2}$ .

Section 2

Comparing Fractions with Same Denominators

Property

Examples

Section 3

Writing Equivalent Fractions

Property

To write an equivalent fraction with a larger denominator:

Divide the old denominator into the desired denominator. This gives you the 'building factor'.
Use that factor to multiply the old numerator.

This process is shown as:

\frac{\text{old numerator}}{\text{old denominator}} \times \frac{\text{building factor}}{\text{building factor}} = \frac{\text{new numerator}}{\text{new denominator}}

Examples

To write $\frac{2}{5}$ with a denominator of 15, the building factor is $15 \div 5 = 3$ . We multiply to get $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Benchmark Fractions

Property

Examples

Is the fraction $\frac{7}{20}$ closer to $\frac{1}{4}$ or $\frac{1}{2}$ ? With a denominator of 20, $\frac{1}{4} = \frac{5}{20}$ and $\frac{1}{2} = \frac{10}{20}$ . Since 7 is closer to 5 than to 10, $\frac{7}{20}$ is closer to $\frac{1}{4}$ .

A team won 14 out of 30 games. Is this fraction, $\frac{14}{30}$ , closest to $\frac{1}{4}$ , $\frac{1}{2}$ , or $\frac{3}{4}$ ? We know $\frac{1}{2}$ of 30 is 15. Since 14 is very close to 15, the fraction is closest to $\frac{1}{2}$ .

Section 2

Comparing Fractions with Same Denominators

Property

Examples

Section 3

Writing Equivalent Fractions

Property

To write an equivalent fraction with a larger denominator:

Divide the old denominator into the desired denominator. This gives you the 'building factor'.
Use that factor to multiply the old numerator.

This process is shown as:

\frac{\text{old numerator}}{\text{old denominator}} \times \frac{\text{building factor}}{\text{building factor}} = \frac{\text{new numerator}}{\text{new denominator}}

Examples

To write $\frac{2}{5}$ with a denominator of 15, the building factor is $15 \div 5 = 3$ . We multiply to get $\frac{2 \times 3}{5 \times 3} = \frac{6}{15}$ .