Adding Fractions

How much is ? (one half + one quarter).

Either halves or quarters – which is more convenient? Should we state how many quarters equal one half or vice versa? The former, of course: 1 half is 2 quarters . So, the becomes , and the above task becomes , meaning 2 quarters + 1 quarter = 3 quarters, i.e. .

In the above sum, 1 half + 1 quarter, the different types were associated with the different denominators (dividers), so the problem was solved by a transformation that led to identical denominators. Let us now do this for the more general case:

(where we have more than one of each type, namely: 2 thirds and 5 twelfths).

Again, we want the same denominators under both, either 3 or 12. Which shall we choose? We must remember that the values of the fractions must remain unchanged, so, if we change the denominator, we must also change the numerator (the divided). If we choose the 3 as the common denominator, we must change the 12 in the second term into a 3, that is, make the denominator 4 times smaller. To keep the value of the fraction unchanged, we must then also make the 5 on top (the numerator) 4 times smaller, resulting in 5/4 as the numerator. This is inconvenient because it creates another fraction above the dividing line.

Instead, we can choose the second denominator, the 12, as the common one. We leave the as it is and change the first denominator, the 3, to 12. This we do by increasing the 3, by multiplying it by 4. Then, to protect the value of this fraction, we must also make the numerator (the 2) 4 times bigger, this time giving a manageable whole number, 8. Note that this resulted from choosing the larger of the two denominators as the common one. And so, the above now becomes , 8 twelfths + 5 twelfths = 13 twelfths, i.e. .

But what about ?

To get the first (smaller) denominator (the 3) to be the same as the second denominator 4, the 3 has to be multiplied by 1.33…, and the same must then be done to the 2 above it, which results in a messy 2.66…! So we now need a common denominator which is neither 3 nor 4. In principle, we could choose anything to put equally at the bottom of both fractions. The only problem is that we need to adjust the numerators (to keep the values of the given fractions unchanged) and as we found, we may only do this by multiplying the numerators by whole numbers. Form this follows that the denominators, too, can be changed only by multiplying by whole numbers.

The task becomes: By what whole number do we multiply the 3 (the denominator), and by what (different) whole number do we multiply the 4 (the other denominator) so that in both cases we get the same result (namely, the same common, denominator)? The nice trick for this is to multiply the first denominator (3) by the second denominator (4) and the second denominator (4) by the first (3) ! (always, of course, multiplying the numerator by the same as the denominator).

So this is what was done:

Generalizing: