The ratio between the number of boys and girls in a school is $4:5$. If the number of boys are increased by $30\%$ and the number of girls increased by $40\%$, then what will the new ratio of boys and girls in the school.

Find the new ratio of boys and girls in the school:

Let the number of boys and girls in the school be $4x$ and $5x$ respectively.

The number of boys is increased by $30\%$

So, The number of boys after getting increased is:

$\begin{array}{rcl}& & \\ \mathrm{New}\mathrm{Number}\mathrm{of}\mathrm{boys}& =& 130\%\mathrm{of}4\mathrm{x}\\ & =& \frac{130}{100}\times 4\mathrm{x}\\ & =& \frac{26\mathrm{x}}{5}\end{array}$

The number of girls increased by $40\%$

So, The number of girls after getting increased is:

$\begin{array}{rcl}\mathrm{New}\mathrm{Number}\mathrm{of}\mathrm{girls}& =& 140\%\mathrm{of}5\mathrm{x}\\ & =& \frac{140}{100}\times 5\mathrm{x}\\ & =& \frac{35\mathrm{x}}{5}\end{array}$

Thus, the required ratio is calculated as follows:

$$\begin{gathered} \frac{\mathrm{New}\mathrm{number}\mathrm{of}\mathrm{boys}}{\mathrm{New}\mathrm{number}\mathrm{of}\mathrm{girls}}=\frac{{\displaystyle \frac{26x}{5}}}{{\displaystyle \frac{35x}{5}}} \\ \frac{\mathrm{New}\mathrm{number}\mathrm{of}\mathrm{boys}}{\mathrm{New}\mathrm{number}\mathrm{of}\mathrm{girls}}=\frac{26}{35} \end{gathered}$$

Hence, the new ratio of boys and girls in the school is $26:35$.