Equivalent fractions are fractions that have the same value even though they look different. We use the equal sign (=) to show they are the same.

Key idea: When we multiply or divide both the numerator and denominator by the same number, the fraction keeps the same value.

Example 1

Write three fractions equal to $\frac{1}{2}$.

Solution

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

$\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

So, $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$

Example 2

Are $\frac{2}{5}$ and $\frac{18}{45}$ equal?

Solution

$\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}$

Yes! $\frac{2}{5} = \frac{18}{45}$

When fractions have different values, we use < (less than) or > (greater than) to compare them.

• < means "smaller than"
• > means "greater than"

Comparing Fractions with the Same Denominator

When denominators are the same, compare the numerators directly:

• Larger numerator = larger fraction
• Smaller numerator = smaller fraction

Example: Compare $\frac{3}{7}$ and $\frac{5}{7}$.

$\frac{3}{7} < \frac{5}{7}$ (because 3 is less than 5)

Comparing Fractions with the Same Numerator

When numerators are the same, compare the denominators:

• Larger denominator = smaller fraction
• Smaller denominator = larger fraction

Example: Compare $\frac{1}{4}$ and $\frac{1}{2}$.

$\frac{1}{4} < \frac{1}{2}$ (because 4 is greater than 2, so the parts are smaller)

Using Fraction Bars or Pictures

You can also compare fractions by drawing them:

Example: Which is greater, $\frac{1}{2}$ or $\frac{1}{4}$?

Draw two equal rectangles. Shade half of one and one-fourth of the other. The half rectangle has more shaded area.

$\frac{1}{2} > \frac{1}{4}$

Example 1

Compare $\frac{1}{4}$ and $\frac{2}{5}$.

Solution

Think: One-fourth is smaller than one-half. Two-fifths is also less than one-half but more than one-fourth.

$\frac{1}{4} < \frac{2}{5}$

Example 2

Compare $\frac{2}{3}$ and $\frac{3}{7}$.

Solution

$\frac{2}{3}$ is greater than $\frac{1}{2}$, while $\frac{3}{7}$ is less than $\frac{1}{2}$.

$\frac{2}{3} > \frac{3}{7}$

Example 3

Arrange from smallest to largest: $\frac{1}{3}, \frac{2}{3}, \frac{1}{2}$

Solution

$\frac{1}{2} < \frac{1}{3} < \frac{2}{3}$

In everyday life in Tanzania, we use fraction comparison when sharing things. For example, when two friends divide a cake—Mwanaasha gets $\frac{1}{4}$ of the cake and Fatuma gets $\frac{1}{3}$—Fatuma gets more cake because $\frac{1}{3} > \frac{1}{4}$. Similarly, when comparing prices at the market, like $\frac{1}{2}$ kilogram of rice at 1,000 TSH versus $\frac{1}{4}$ kilogram at 600 TSH, knowing which fraction is greater helps you find the better deal.