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 Skip Navigation LinksMath Help > Basic Math > Word Problems and Basic Arithmetic > Adding and Subtracting Fractions

Adding and subtracting fractions is actually harder than multiplying them or dividing them, because of a special step that has to be done in adding called "finding a common denominator".

A common denominator is a common multiple of both (or all) denominators.

So, for example, if you are adding

1/2 + 2/3

Then you would recognize that 2 and 3 are different denominators, so you'll need to find a new denominator that is a multiple of both 2 and 3.  You should guess right away that 6 is a common multiple of 2 and 3.  If you ever have trouble finding a common multiple of two numbers, you should think about multiplying them together.  That will always give you a common multiple of the two numbers, but not always the least common multiple.

So now, you can express 1/2 as 3/6, and you can express 2/3 as 4/6.  After you do this, your addition problem becomes

3/6 + 4/6

Now that the denominators are the same, you just need to add the numerators.  So the answer is

3/6 + 4/6 = 7/6, or 1 1/6

Subtracting works the same way.  If you want to subtract

3/4 - 2/3

you will see that the two denominators are different.  A common multiple of 4 and 3 is 12, so you can convert 3/4 to twelfths by multiplying the numerator and denominator by 3, giving you 9/12.  You can convert 2/3 to twelfths by multiplying the numerator and denominator by 4, giving you 8/12.  Now the problem becomes

9/12 - 8/12 = 1/12

Many students find it very difficult to remember that you need to find a common denominator only when adding and subtracting and never when multiplying or dividing.  It might help you to think of slices of pizza.  If it's a big pizza, they usually slice it into eighths.  If it's really big, they sometimes slice it into twelfths.  So if it's really, really, big you can imagine them slicing it fifteen ways, right?  I mean you can really have any number of slices if you make 'em thin enough.  So if you need to add

2/3 + 1/5

You will realize that your 15-way pizza comes in handy.  2/3 of such a pizza is 10 slices, and 1/5 of that pizza is 3 slices.  So this addition problem is the same as

10/15 + 3/15

If you are going to add up 10/15 of a pizza and 3/15 of a pizza, then you can see that the result is 13/15 of the pizza, not 13/30, right?  I mean the number of slices that make up a whole pizza doesn't change when you add up 10 slices and 3 more slices; each slice is still 1/15 of a whole pizza.  So my advice to you is to always imagine a very big pizza (or, several pizzas, with improper fractions in which the numerator is larger than the denominator) that is divided into very small slices, so that all the fractions can work out to some whole number of slices.

Send me an email if you read this page carefully and still have questions about it.

Internet references

Math League: Fractions - Prime numbers, Greatest common factor, Least common multiple, Equivalent fractions, Comparing fractions, reducing fractions, Lowest terms, Improper fractions, Mixed numbers, Converting mixed numbers to/from improper fractions, Writing a fraction as a decimal, Rounding a fraction to the nearest hundredth, Adding and subtracting fractions, Adding and subtracting mixed numbers, Multiplying fractions and whole numbers, Multiplying fractions and fractions, Multiplying mixed numbers, Reciprocal, Dividing fractions, Dividing mixed numbers, Simplifying complex fractions, Repeating decimals

Related pages in this website

Multiplying and Dividing Fractions

 


The webmaster and author of this Math Help site is Graeme McRae.