A man buys oranges at $18$ for ₹$135$ and sells them at $11$ for ₹$90$. How many oranges must he sell to gain ₹$150$?

First, let's find the Cost Price ($\text{CP}$) and Selling Price ($\text{SP}$) per orange.$\text{CP}$ of $18$ oranges = ₹$135$ $
$\text{CP of 1 orange}=\frac{135}{18}=\frac{15}{2}=\text{₹}7.50$ $
$\text{SP}$ of $11$ oranges = ₹$90$ $
$\text{SP of 1 orange}=\frac{90}{11}$$Now,findtheprofitgainedfromsellingasingleorange:$$\text{Profit per orange}=\text{SP}-\text{CP}=\frac{90}{11}-\frac{15}{2}$ $
$$\text{Profit per orange}=\frac{180-165}{22}=\frac{15}{22}$$
We want the total profit to be ₹$150$. Let the total number of oranges to be sold be $N$:
$$N\times \text{Profit per orange}=150$$
$$N\times \frac{15}{22}=150$$
$$N=150\times \frac{22}{15}$$
$$N=10\times 22=220$$
Thus, he must sell $220$ oranges to achieve a gain of ₹$150$.