Maths in a minute: Adding fractions (the easy way)

Maths in a minute: Adding fractions (the easy way)

Ruler

Adding fractions is probably the first difficult bit of maths we come across at school. For example, to work out $\frac{5}{6} + \frac{7}{10}$ you first need to figure out that the lowest common multiple of 6 and 10 is 30, and that in order to get 30 in the denominator of both fractions you need to multiply the numerator 5 by 5 and the numerator 7 by 3. This gives

$\frac{5}{6} + \frac{7}{10} = \frac{(5 \times 5)}{30} + \frac{(7 \times 3)}{30} = \frac{25}{30} + \frac{21}{30} = \frac{(25+21)}{30} = \frac{46}{30}.$

You then need to get rid of the common factors of 46 and 30, giving the final result $\frac{23}{15},$ which bears no resemblance whatsoever to the original two fractions. Doing this as a ten-year-old who has never seen it before is pretty tough.

Here is an alternative recipe that always works and doesn't involve faffing around with lowest common denominators. Writing "top" for numerator and "bottom" for denominator, the idea is to do:

(top left x bottom right + top right x bottom left) / (bottom left x bottom right).

Applied to our example this gives:

  \[  \frac{(5 \times 10 + 7 \times 6)}{(6 \times 10)} = \frac{(50 + 42)}{60} = \frac{92}{60} = \frac{23}{15}.  \]    

The difference to the standard way of adding fractions is that you are not bothered with finding the lowest common denominator. You simply use the product of the two denominators as a common denominator. Then, in order to bring both fractions on that common denominator you only need to multiply the numerator of each by the denominator of the other. Easy!

Apparently this is how Vedic mathematicians in ancient India added up fractions. If you happen to speak German, you can also explore this method in musical form in this maths rap by DorFuchs. And even if you don't speak German, it's cute!

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