A man bought a house for Rs. $5$ lakhs and rents it. He puts $12\frac{1}{2}\%$ of each month’s rent aside for repairs, pays Rs. $1660$ as annual taxes and realises $10\%$ on his investment thereafter. The monthly rent of the house is

Let the monthly rent of the house be Rs. $R$.
Annual rent received $=12\times R$

The amount put aside for repairs is $12\frac{1}{2}\%=\frac{25}{2}\%=\frac{1}{8}$ of the rent.
Remaining rent after repairs $=12R\times \left(1-\frac{1}{8}\right)=12R\times \frac{7}{8}=\frac{21R}{2}$

After paying annual taxes of Rs. $1660$, the net amount realized is:

$$\text{Net Realization}=\frac{21R}{2}-1660$$

He realizes a $10\%$ return on his investment of Rs. $5,000,000$:

$$\text{Net Realization}=\frac{10}{100}\times 500000=50000$$

Equating the two expressions:

$$\frac{21R}{2}-1660=50000$$

$$\frac{21R}{2}=51660$$

$$21R=103320$$

$$R=\frac{103320}{21}=4920$$

Thus, the monthly rent of the house is Rs. $4920$.