Use Common Multiples to Express Fractions with the Same Denominator

## What you need to know

Things to remember:

• To add fractions, the denominators have to be the same.
• If we want to add, or subtract, two fractions with different denominators, we have to make them the same first.
• To make the denominators the same we have to find common multiples.
• We can find a common denominator by multiplying the two denominators together.

We have seen adding and subtracting fractions when the denominators are the same, all we do is add or subtract the numerators; the denominators stay the same.

$$\frac{1}{5}+\frac{2}{5}=\frac{3}{5}$$

$$\frac{9}{11}-\frac{4}{11}=\frac{5}{11}$$

But, how do we add fractions like $\frac{2}{5}+\frac{1}{3}$? Well, we need the denominators to be the same. We do this by looking for common multiples of them.

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

Multiples of 5: 5 10 15 20 25 30 35 40 45 50

Multiples of 2: 3 6 9 12 15 18 21 24 27 30

The common multiples in these two lists are 15 and 30, so we need to make the denominators one of these. If we look at the numbers above the multiples, it tells us what we have to multiply them by.

$$5\times3 = 15$$

$$3\times5 = 15$$

NOTE: Here, we have just multiplied the denominators together to find a common multiple!

$$5\times6 = 30$$

$$3\times10 = 30$$

Now that we’ve found the common multiples and what we multiply by, we do the same to the tops!

$$\frac{2}{5}=\frac{2\times3}{5\times3}=\frac{6}{15}$$

$$\frac{1}{3}=\frac{1\times5}{3\times5}=\frac{5}{15}$$

$$\frac{2}{5}+\frac{1}{3}= \frac{6}{15}+\frac{5}{15}=\frac{11}{15}$$

$$\frac{2}{5}=\frac{2\times6}{5\times6}=\frac{12}{30}$$

$$\frac{1}{3}=\frac{1\times10}{3\times10}=\frac{10}{30}$$

$$\frac{2}{5}+\frac{1}{3}= \frac{12}{30}+\frac{10}{30}=\frac{22}{30}$$

Hmmmmmmmmmmmm, these answers are different. Or are they? Remember, we can often simplify fractions.

$$\frac{22}{30} =\frac{22\div2}{30\div2}=\frac{11}{15}$$

Actually, these are the same! It doesn’t matter which common multiple we use, they’ll always simplify down to be the same answer!

So, really, we did this in four steps:

Step 1: Find a common multiple of the denominators.

Step 2: Multiply the fractions so that the denominators are the same.

Step 3: Add or subtract the new fractions.

Step 4: See if you can simplify.

What is $\frac{3}{4}-\frac{2}{7}$?

Step 1: Find a common multiple of the denominators.

HINT: The easiest way to do this is multiply the denominators together.

$$4\times7=28$$

So, we are making the denominators into 28.

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{3}{4}=\frac{3\times7}{4\times7}=\frac{21}{28}$$

$$\frac{2}{7}=\frac{2\times4}{7\times4}=\frac{8}{28}$$

Step 3: Add or subtract the new fractions.

$$\frac{3}{4}-\frac{2}{7}=\frac{21}{28}-\frac{8}{28}=\frac{13}{28}$$

Step 4: See if you can simplify.

We can’t simplify, so this is our final answer!

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## Example Questions

Step 1: Find a common multiple of the denominators.

HINT: The easiest way to do this is multiply the denominators together.

$$2\times9=18$$

So, we are making the denominators into 18.

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{1}{2}=\frac{1\times9}{2\times9}=\frac{9}{18}$$

$$\frac{3}{9}=\frac{3\times2}{9\times2}=\frac{6}{18}$$

Step 3: Add or subtract the new fractions.

$$\frac{1}{2}+\frac{3}{9}=\frac{9}{18}+\frac{6}{18}=\frac{15}{18}$$

Step 4: See if you can simplify.

We can simplify this one!

$$\frac{15}{18} =\frac{15\div3}{18\div3}=\frac{5}{6}$$

$$\frac{1}{2}+\frac{3}{9}= \frac{5}{6}$$

Step 1: Find a common multiple of the denominators.

HINT: The easiest way to do this is multiply the denominators together.

$$7\times12=84$$

So, we are making the denominators into 84.

Step 2: Multiply the fractions so that the denominators the same.

$$\frac{5}{7}=\frac{5\times12}{7\times12}=\frac{60}{84}$$

$$\frac{1}{12}=\frac{1\times7}{12\times7}=\frac{7}{84}$$

Step 3: Add or subtract the new fractions.

$$\frac{5}{7}-\frac{1}{12}=\frac{60}{84}-\frac{7}{84}=\frac{53}{84}$$

Step 4: See if you can simplify.

This can’t be simplified, so Step 3 gave us our answer.

$$\frac{1}{2}+\frac{3}{9}= \frac{5}{6}$$

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