A doctor borrowed $\text{Rs. }3,825$ and he returned the money in two equal yearly instalments. If the annual rate of compound interest (compounded annually) is $4\%$, then find the value of each instalment.

Let each equal yearly instalment be $X$.
The formula for the principal value of compound interest instalments for 2 years is:

$$P=\frac{X}{1+\frac{R}{100}}+\frac{X}{{\left(1+\frac{R}{100}\right)}^2}$$

Given:
Principal sum borrowed, $P=3825$
Rate of interest, $R=4\%$

First, simplify the term $\left(1+\frac{R}{100}\right)$:

$$1+\frac{4}{100}=1+\frac{1}{25}=\frac{26}{25}$$

Substitute this into the equation:

$$3825=\frac{X}{\frac{26}{25}}+\frac{X}{{\left(\frac{26}{25}\right)}^2}$$

$$3825=\frac{25X}{26}+\frac{625X}{676}$$

Take the common denominator, which is $676$ (since $26\times 26=676$):

$$3825=\frac{25\times 26\times X+625X}{676}$$

$$3825=\frac{650X+625X}{676}$$

$$3825=\frac{1275X}{676}$$

Solve for $X$:

$$X=\frac{3825\times 676}{1275}$$

Notice that $1275\times 3=3825$:

$$X=3\times 676=2028$$

Thus, the value of each instalment is $\text{Rs. }2,028$.