# Increase and Decrease in Ratio

If the number of teachers in a college is increased from 50 to 60, then the ratio of new staff and old staff is:

$\begin{array}{*{20}{c}} {{\text{No}}{\text{. of new staff}}}&{\text{:}}&{{\text{No}}{\text{. of old staff}}} \\ {{\text{60}}}&{\text{:}}&{{\text{50}}} \\ {\text{6}}&{\text{:}}&{\text{5}} \end{array}$
$\frac{{{\text{no}}{\text{. of new staff}}}}{{{\text{no}}{\text{. of old staff}}}} = \frac{6}{5}$

We say that the number of teachers has been increased by the ratio 6 : 5. In other words, the number of new staff is $\frac{6}{5}$ times the number of old staff.

Hence $\boxed{{\text{no}}{\text{. of new staff = }}\frac{{\text{6}}}{{\text{5}}}{\text{ (no}}{\text{. of old staff)}}}$

Rule:

To increase a no. “$x$”  we multiply $x$ by an improper fraction.

$\boxed{{\text{no}}{\text{. of new staff = }}\frac{{\text{6}}}{{\text{5}}}{\text{ (no}}{\text{. of old staff)}}}$

Similarly to decrease a no.  “$x$”  we multiply $x$ by an proper fraction.

$\boxed{{\text{no}}{\text{. of old staff = }}\frac{{\text{6}}}{{\text{5}}}{\text{ (no}}{\text{. of new staff)}}}$

e.g. (1) Increase Rs. 20 by the ratio 6 : 5
New value = ${\text{20 x }}\frac{{\text{6}}}{{\text{5}}} = 24$

e.g. (2) Decrease 56 by the ratio 7 : 8
New value =  ${\text{56 x }}\frac{{\text{7}}}{{\text{8}}} = 49$