Simplifying Fractions (A)

The simplest form of a fraction is the form of that fraction where the numerator and denominator do not have any common factors apart from 1. If we are given a fraction, how do we find the simplest form?

I. Simplifying a fraction

To simplify a fraction is to divide the numerator and the denominator by the same positive integer. This integer must be a common factor of the numerator and the denominator.


Example: Can $\frac{150}{400}$ be written in a simplified form?


We can see that 150 and 400 are multiples of 10. Therefore, we can write: $\frac{150}{400}=\frac{150÷10}{400÷10}=\frac{15}{40}$.
We say that we have simplified the fraction $\frac{150}{400}$ by a factor of 10.


Now we can see that 15 and 40 are multiples of 5. Therefore we have: $\frac{15}{40}=\frac{{\displaystyle 15÷5}}{{\displaystyle 40÷5}}=\frac{3}{8}$.
We say that we have simplified the fraction $\frac{15}{40}$ by a factor of 5.
Finally, we can write: $\frac{150}{400}=\frac{3}{8}$. Now we can say that we have simplified the fraction $\frac{150}{400}$.


Note: If we had started by simplifying by 5 and then by 10, we would still have obtained $\frac{3}{8}$ (and also if we had started with 2, then 5, then 5 again).

II. Simplifying to obtain the simplest form of a fraction

Among all the different fraction forms of a number, there is one that cannot be simplified. We say that it is the simplest form of the number. How do we find it?


There are several methods:
Perhaps, as is the previous example, we could just make obvious simplifications, however we might stop simplifying too soon;
instead, we could look for the greatest common factor of the numerator and the denominator, and then divide the numerator and denominator of the fraction by this number.


Example: How do we find the simplest form of the fraction $\frac{24}{42}$ ?


The factors of 24 are: 1; 2; 3; 4; 6; 8; 12; and 24.


The factors of 42 are: 1; 2; 3; 6; 7; 14; 21; and 42.


1; 2; 3; and 6 are the common factors of 24 and 42. 6 is the greatest.


We can write: $\frac{24}{42}=\frac{24÷6}{42÷6}=\frac{4}{7}$


 $\frac{4}{7}$ is the simplest form of $\frac{24}{42}$. (4 and 7 do not have any common factors apart from the number 1.)


Note:

Some calculators have a button (Simp or d/c) that allows you to simplify fractions.


For example, the input sequences:

3 2 3 / 1 5 3 = Simp


3 2 3 a$a/c$ 1 5 3 d/c


respectively return 19/9 and 19_|9. (The calculator has simplified by 17.)

III. Applications of fraction calculations

Before launching into a complicated calculation, it might be wise to first simplify any fractions.


Example: Calculate $\frac{55}{77}-\frac{32}{48}$.


If we look for a common multiple for 77 and 48, we find (at best): 3,696.


Whereas: $\frac{55}{77}=\frac{{\displaystyle 55÷11}}{{\displaystyle 77÷11}}=\frac{{\displaystyle 5}}{{\displaystyle 7}}$ and $\frac{32}{48}=\frac{{\displaystyle 32÷16}}{{\displaystyle 48÷16}}=\frac{2}{3}$.


Finally: $\frac{55}{77}-\frac{{\displaystyle 32}}{{\displaystyle 48}}=\frac{{\displaystyle 5}}{{\displaystyle 7}}-\frac{2}{3}=\frac{{\displaystyle 5\times 3}}{{\displaystyle 7\times 3}}-\frac{2\times 7}{3\times 7}=\frac{15}{21}-\frac{14}{21}=\frac{1}{21}$.


Subtracting or adding the simplified forms of the fractions is easier.