Title: Adding, subtracting, multiplying and dividing fractions
Course: UGQuants (SSPS) > Section: Maths Self Test

Adding, subtracting, multiplying and dividing fractions

Adding and subtracting fractions expressed as decimals is quite straightforward: we employ exactly the same principles as with whole numbers, and all we need to do is make sure the decimal point is in the right place. E.g. just as

16 + 27 = 43, 0.16 + 0.27 = 0.43 (and 0.27 – 0.16 = 0.11).

Multiplying and dividing fractions produces the opposite outcomes to those we are used to when using whole numbers and numbers greater than 1. i.e., when we multiply by a fraction this results in a lower number.  (if you find this confusing just think of it as ‘a half of a half is a quarter’). Conversely when we divide by a fraction this results in a larger number.

E.g. 1 ÷ 0.25 = 4

(again if you find this confusing just think of it as ‘we need four lots of 0.25 to make 1); or

0.4 ÷ 0.8 = 0.5.

Adding, subtracting, multiplying and dividing ‘common’ fractions is a bit more complicated because different common fractions have different denominators, and hence it is difficult to intuitively add together even quite simple fractions: e.g. what is the sum of ½ and 1/3?

To resolve this problem we find what is called the lowest common denominator (a phrase that has now acquired a whole new meaning in popular as opposed to mathematical language!). For the denominators 2 and 3, this turns out to be 6: we can express ½ as 3/6 and 1/3 as 2/6. Hence, because 3/6 + 2/6 is 5/6, the answer to our original question is that ½ + 1/3 = 5/6. Exactly the same principle can be applied to subtraction: hence ½ - 1/3 = 1/6.

To multiply common fractions we simply multiply the numerators and the denominators, and then (if necessary) express the product using the lowest possible denominator (we simplify the fraction). E.g.:

½ x ⅔ = [(numerator 1 x denominator 2) + (numerator 2 x denominator 1)]/(denominator 2 x denominator 2) = [(1 x 3) + (2 x 2)] / (2x3)= (3+4) / 6 = 7/6

To divide common fractions we follow a similar procedure to multiplication but we first invert the fraction by which we want to divide and then multiply the numerators and the denominators. The inverse of any number is 1 divided by that number, but with common fractions it is simpler to find the inverse by swapping the top (numerator) and bottom (denominator) parts of the fraction: e.g. the inverse of 2/3 is 3/2. So e.g. to divide two-thirds by three-quarters:

½ ÷ ⅔ = ½ x 3/2 = 0.5 x 1.5 = 0.75

This video will help you understand what fractions are and how to add and subtract them:

This video will demonstrate how to multiply and divide fractions: