by Ron Kurtus (revised 18 February 2009)

When you add two fractions together, the denominator (number on the bottom) of each fraction must be the same to complete the addition. You then add the numerators (numbers on the top).

If the denominators are different, you must change them to a common denominator. An easy way to do this is by the cross multiplying method. If one denominator is a multiple of the other, you can use the larger as the lowest common denominator.

Questions you may have include:

• How do you add fractions if the denominators are the same?
• What if the denominators are different?
• How do you find the lowest common denominator?

This lesson will answer those questions.

## Denominators are the same

When you add two fractions where the denominators are the same, you simply add the numerators.

For example, so solve:

1/5 + 2/5

(1 + 2)/5 = 3/5

Another example:

1/13 + 7/13 + 2/13 = (1 + 7 + 2)/13 = 10/13

### Exercises

3/5 + 1/5

2/11 + 5/11 + 3/11

1/3 + 1/3

4/5, 10/11, 2/3

## When denominators different

When the denominators are different, you must make them the same by finding a common denominator and multiplying the numerator and denominator in each fraction by 1.

1/3 + 2/5

Since the denominators 3 and 5 are different, you must find a common denominator in order to do the addition. The easiest way is to multiply the two denominators together and use that number:

3 × 5 = 15.

Then multiply 1/3 by a fraction equal to 1 that will result in a denominator equal to 15.

Since 5 × 3 = 15 and 5/5 = 1:

1/3 × 5/5 = 5/15

Likewise, multiply 2/5 by a fraction equal to 1 that will result in a denominator equal to 15. Since 3 × 5 = 15 and 3/3 = 1:

2/5 × 3/3 = 6/15

Thus

1/3 + 2/5 = 5/15 + 6/15 = 11/15

### Cross multiplying

A quick way or shortcut is by what is called cross multiplying. To add 1/3 + 2/5, you multiply across: 1 × 5 = 5, 2 × 3 = 6 and 3 × 5 = 15 and insert those numbers in the proper places to get 5/15 + 6/15. This is a method when you get good enough to do things in your head.

Another example of using the cross multiplying method is:

2/9 + 5/7

Cross multiply the numerator of the first fraction times the denominator of the second: 2 × 7 = 14

14/9 + 5/7

Cross multiply the numerator of the second fraction times the denominator of the first: 9 × 5 = 45

14/9 + 45/7

Multiply the denominators together: 9 × 7 = 63

14/63 + 45/63

14/63 + 45/63 = 59/63

It is a pretty tricky method, once you get the hang of it.

### Exercises

2/7 + 1/3

3/8 + 2/5

2/11 + 3/7

13/21, 31/40, 47/77

## Lowest common denominator

In some cases, there is a common denominator that is less than the product of the two denominators.

Suppose you wanted to add 1/3 + 7/12. You could find a common denominator of 3 × 12 = 36. But note that 12 is a multiple of 3: 12 = 4 × 3. It would be easier to change 1/3 to 4/12 by multiplying it by 4/4. This would save a step and make the multiplication easier.

1/3 + 7/12 = 4/12 + 7/12 = 11/12

That is better than working with 36ths.

If you notice that the larger denominator is a multiple of the smaller denominator, you should use that fact to simplify your addition. For example, if you were adding the fractions 3/7 + 2/35, you should notice that 35 = 7 × 5. Thus,

3/7 + 2/35 = 15/35 + 2/35 = 17/35

1/9 + 1/3

2/5 + 3/25

2/7 + 1/14

4/9, 13/25, 5/14

## Summary

When you add two fractions together, the denominator of each must be the same to complete the addition, and then you can add the numerators. If the denominators are different, you must change them to a common denominator. An easy way to do this is by the cross multiplying method. If one denominator is a multiple of the other, you can use the larger as the lowest common denominator.

Work smart, not hard

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