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Addition and Subtraction of Fractions

We will consider only the addition and subtraction of two fractions here, see the help on Complex Calculations after reading this section for details on the addition and subtraction of more than two fractions.

If mixed fractions are involved then we can either perform the addition (or subtraction) on the whole numbers and fraction parts separately or we can convert the mixed fractions to improper fractions and perform the addition (or subtraction) on the improper fractions.

The Fraction Calculator can do either, the user selects the 'Use whole numbers and fractions' or 'Use fractions only' radio button.

The addition of improper fractions is exactly the same as the addition of proper fractions so we will talk here about the addition of mixed fractions by performing the addition (or subtraction) on the whole numbers and fraction parts separately.

Perform the addition (or subtraction) on the whole numbers (ignore the fraction parts) and save the result, we will combine it with the result of the addition (or subtraction) on the fraction parts later.

The secret of addition and subtraction of fractions is not to add or subtract the fractions we have been asked to subtract but to find equivalent fractions to the two fractions we have been given and subtract them. The important thing the equivalent fractions must have is the same denominator (bottom)  as each other.

I hope you are now asking how these two equivalence's were found. Well there is a simple method to find the smallest equivalence's  (there are many possible solutions, it is best to always use the smallest equivalence's). Take the denominators (bottoms) of both fractions and find all the factors  (See Factoriser) of both denominators. Write a list of all the factors in the denominator (bottom) of the first fraction and then remove from this list any of these factors that are also factors of the second fractions denominator (bottom). The factors we have removed are called the common factors, note if a factor occurs more than once in the list of factors of the first fractions denominator (bottom) only remove one occurrence for each time it occurs as a factor of the second fractions denominator (bottom). Now take the factors that remain in the list and multiply them all together, then multiply the result by the denominator (bottom) of the second fraction, this will be the denominator (bottom) of both equivalent fractions we are looking for (if there were no factors left in list then the denominator (bottom) of second fraction is what we are looking for) . This sounds a little complicated so lets do it for the above example:

The factors of 6 are 1,2,3.  The factors of 15 are 1,3,5. 1 and 3 are factors of both numbers (common factors) , removing them from the factors of 6 leaves just 2 we multiply this by 15 to give us 30.

What we have found is called the 'Common Denominator', we now need to use this to find the two equivalent fractions, which is simply. Take the first fractions denominator and divide it into the common denominator (it will go an exact number of times, if it does not you have made an error calculating the common denominator), the number of times it divides into the common denominator is multiplied by the fractions numerator (top) and this gives the numerator (top) of the first equivalent fraction we require. Repeat this with the second fraction to obtain the numerator (top) of the second equivalent fraction.

We will continue with our example, 6 divides into 30 exactly 5 times, the resulting 5 is then multiplied by 1 to give 5

next divide 15 into 30, it goes 2 times which multiplied by 2 gives 4 so the second

Now we have two fractions with the same denominators (bottoms) we can add or subtract the numerators (tops) and express the result as a single fraction. Continuing our example:

should always be done.

Finally if there were any whole numbers that we added or subtracted at the start we can combine the whole number we set aside with our fraction to make a mixed fraction.

Subtraction and addition are the same except for one complication that can occur with subtraction. This can best illustrated with an example.

Perform the subtraction on the whole numbers i.e. 5 - 2 = 3.

This is not a correct mixed fraction, we correct it by taking 1 of the 3 and expressing the 1 we have taken as an equivalent fraction with the same denominator (bottom) as the fraction i.e.

So if you end up with a negative fraction and have some whole numbers to play with, take one and turn it into a fraction that the negative fraction can be subtracted from.

Multiplication and Division of Fractions

Fraction Calculator and Cheat Homepage

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