A certain sum of money amounts to $\text{Rs. }1,680$ in $3$ years and to $\text{Rs. }1,800$ in $5$ years. Find the sum and the rate of simple interest.

Let the principal sum be $P$ and the annual rate of simple interest be $R\%$.
Given:

$$\text{Amount in }5\text{ years}=\text{Rs. }1,800$$

$$\text{Amount in }3\text{ years}=\text{Rs. }1,680$$

Since simple interest is constant every year:

$$\text{Simple Interest (SI) for }2\text{ years (from year 3 to 5)}=1800-1680=\text{Rs. }120$$

$$\text{SI for }1\text{ year}=\frac{120}{2}=\text{Rs. }60$$

$$\text{SI for }3\text{ years}=60\times 3=\text{Rs. }180$$

Now, find the principal sum $P$:

$$P=\text{Amount in }3\text{ years}-\text{SI for }3\text{ years}=1680-180=\text{Rs. }1,500$$

Now, calculate the rate of interest $R$:

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

$$60=\frac{1500\times R\times 1}{100}$$

$$60=15R⟹R=\frac{60}{15}=4\%$$

Therefore, the sum is $\text{Rs. }1,500$ and the rate is $4\%$.