 Fractions with Different Denominators and Mixed Numbers

## What you need to know

Things to remember:

• Answer may be improper fractions or mixed numbers.
• When adding and subtracting mixed numbers, split them into “whole” parts and “fractions” parts, then add or subtract separately. Put these pieces back together at the end.
• Be careful when subtracting a fraction. Try borrowing 1 from the whole number.

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.

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

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

We have also seen how to add fractions when the denominators are different.

What is $\dfrac{1}{2}+\dfrac{3}{9}$?

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!

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

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

But, what happens if we put a whole number in front of the fraction? What happens if we’re adding or subtracting mixed numbers?!?!?!?! Well, we do it in four steps.

What is $3\dfrac{3}{5}+7\dfrac{3}{9}$?

Step 1: Add or subtract the whole numbers

$$3+7=10$$

Step 2: Add or subtract the fractions

$$\frac{3}{5}+\frac{3}{9}=\frac{3\times9}{5\times 9}+\frac{3\times5}{9\times 5}=\frac{27}{45}+\frac{15}{45}=\frac{42}{45}$$

Step 3: See if you can simplify.

This fraction does simplify.

$$\frac{42}{45}=\frac{14}{15}$$

Step 4: Add the whole number from Step 1 and the fraction from Step 3 together.

$$3\frac{3}{5}+7\frac{3}{9}=10\frac{14}{15}$$

We need to be careful if we have to subtract a fraction from a whole number in Step 4.

For example, what is $3-\frac{5}{8}$?

Let’s look at this as steps.

Step 1: Borrow a 1 from the whole number.

$$3-\frac{5}{8} = 2 + 1 -\frac{5}{8}$$

Step 2: Turn the 1 into a fraction with the same denominator.

$$2 + \frac{8}{8} -\frac{5}{8}$$

Step 3: Subtract.

$$2 + \frac{8}{8} -\frac{5}{8} =2 + \frac{3}{8}$$

Step 4: Add the whole number and the fraction together.

$$2 + \frac{3}{8} =2\frac{3}{8}$$

$$3-\frac{5}{8} =2\frac{3}{8}$$

## Example Questions

Step 1: Add or subtract the whole numbers

$$4+2=6$$

Step 2: Add or subtract the fractions

$$\frac{1}{3}+\frac{1}{7}=\frac{1\times7}{3\times 7}+\frac{1\times3}{7\times 3}=\frac{7}{21}+\frac{3}{21}=\frac{10}{21}$$

Step 3: See if you can simplify.

This fraction doesn’t simplify.

Step 4: Add the whole number from Step 1 and the fraction from Step 3 together.

$$4\frac{1}{3}+2\frac{1}{7}=6\frac{10}{21}$$

Step 1: Add or subtract the whole numbers

$$10-9=1$$

Step 2: Add or subtract the fractions

$$\frac{4}{11}-\frac{7}{12}=\frac{4\times12}{11\times12}-\frac{7\times11}{12\times 11}=\frac{48}{132}-\frac{77}{132}=-\frac{29}{132}$$

Step 3: See if you can simplify.

This fraction doesn’t simplify.

Step 4: Add the whole number from Step 1 and the fraction from Step 3 together.

$$1-\frac{29}{132}=\frac{132}{29}-\frac{29}{132}=\frac{103}{132}$$

$$10\frac{4}{11}-9\frac{7}{12}=\frac{103}{132}$$

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