Adding and Subtracting Fractions with Related Denominators

When we want to add or subtract fractions, they must have the same denominator (the bottom number). But what happens if the denominators are different?

Sometimes, the denominators are related, meaning one denominator is a multiple of the other. For example, in halves and fourths ($2$ and $4$), $4$ is a multiple of $2$. In this case, you only need to change one of the fractions to make the denominators match.

Steps to Solve

1. Identify the related denominators: Find the smaller denominator and figure out what number you need to multiply it by to get the larger denominator.
2. Create an equivalent fraction: Multiply both the top (numerator) and bottom (denominator) of the fraction with the smaller denominator by that number.
3. Add or subtract: Now that the denominators are the same, just add or subtract the numerators. Keep the denominator exactly the same!

Example 1: Addition

Let's add $\frac{1}{2}$ and $\frac{3}{4}$.

First, look at the denominators: $2$ and $4$. Since $2 \times 2 = 4$, we can turn halves into fourths.

Multiply the top and bottom of $\frac{1}{2}$ by $2$:

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

Now, add the fractions together:

$\frac{2}{4} + \frac{3}{4} = \frac{5}{4}$

Example 2: Subtraction

Let's subtract $\frac{5}{6} - \frac{1}{3}$.

Look at the denominators: $6$ and $3$. Since $3 \times 2 = 6$, we can turn thirds into sixths.

Multiply the top and bottom of $\frac{1}{3}$ by $2$:

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

Now, subtract the numerators:

$\frac{5}{6} - \frac{2}{6} = \frac{3}{6}$

(Note: $\frac{3}{6}$ can be simplified to $\frac{1}{2}$, but finding the common denominator is the main goal!)

Example 3: Tenths and Hundredths

Let's add $\frac{3}{10} + \frac{7}{100}$.

Here, $10 \times 10 = 100$. We need to change tenths into hundredths. Multiply the top and bottom of $\frac{3}{10}$ by $10$:

$\frac{3 \times 10}{10 \times 10} = \frac{30}{100}$

Now add them together:

$\frac{30}{100} + \frac{7}{100} = \frac{37}{100}$