A sum of money becomes three times at $5\%$ per annum simple interest. At what rate $\%$ will it become six times in the same time?

Let the Principal amount borrowed be $P$.
Case 1:
The sum becomes $3$ times itself $⟹{\text{Amount}}_1=3P$.

$${\text{Simple Interest (SI)}}_1={\text{Amount}}_1-P=3P-P=2P$$

$${\text{SI}}_1=\frac{P\times R_1\times T}{100}⟹2P=\frac{P\times 5\times T}{100}$$

$$2=\frac{5T}{100}⟹T=\frac{200}{5}=40\text{ years}$$

Case 2:
The sum needs to become $6$ times itself $⟹{\text{Amount}}_2=6P$ in the same time ($T=40\text{ years}$).

$${\text{Simple Interest (SI)}}_2={\text{Amount}}_2-P=6P-P=5P$$

$${\text{SI}}_2=\frac{P\times R_2\times T}{100}⟹5P=\frac{P\times R_2\times 40}{100}$$

$$5=\frac{40R_2}{100}⟹R_2=\frac{500}{40}=12.5\%$$

Alternative Shortcut Method:
When the time period $T$ is constant, we can use the formula:

$$\frac{N_1-1}{N_2-1}=\frac{R_1}{R_2}$$

$$\frac{3-1}{6-1}=\frac{5}{R_2}⟹\frac{2}{5}=\frac{5}{R_2}$$

$$2R_2=25⟹R_2=12.5\%$$