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


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 =  $${\text{56 x }}\frac{{\text{7}}}{{\text{8}}} = 49$$