The simple interest on a certain sum at 5% per annum for 3 years and 4 years differs by Rs. 82. Find the sum.

We have, 

Simple Interest gained from 3rd to 4th year = Rs 82
Time (4th year – 3rd year) = 1 year

Rate = 5% p.a.

Using the formula : $\mathrm{SI}=\frac{\mathrm{P}\times \mathrm{R}\times \mathrm{T}}{100}$, we can calculate the principal amount

The principal amount is given as

$\mathrm{P} = \frac{\mathrm{SI} \times 100}{\mathrm{R} \times \mathrm{T}}$

$\mathrm{P}= \frac{82 \times 100}{5 \times 1} = \mathrm{Rs} 1640$

$\therefore \mathrm{Required} \mathrm{sum} =\mathrm{Rs} 1640$

Hence, the correct answer is Rs 1640.