A sum of $\text{₹}6900$ was lent partly at $5\%$ and the rest at $8\%$ simple interest. The total interest received after $3$ years from both was $\text{₹}1359$. What was the ratio of money lent at $5\%$ and $8\%$?

Total interest received in $3$ years = $\text{₹}1359$.Therefore, total interest received per year:
$$\text{Interest per year}=\frac{1359}{3}=\text{₹}453$$
Let us find the overall effective annual rate of interest ($R$) on the total sum of $\text{₹}6900$:
$$\text{Simple Interest}=\frac{P\times R\times T}{100}$$
$$453=\frac{6900\times R\times 1}{100}$$
$$453=69\times R⟹R=\frac{453}{69}=\frac{151}{23}\%$$
Using the method of alligation:Rate of first part ($R_1$) = $5\%$ Rate of second part ($R_2$) = $8\%$ Mean rate ($R_m$) = $\frac{151}{23}$$Theratioofthesumsinvestedinthetwopartsisgivenby:$$\text{Ratio}=(R_2-R_m):(R_m-R_1)$ $
$$\text{Ratio}=\left(8-\frac{151}{23}\right):\left(\frac{151}{23}-5\right)$$
$$\text{Ratio}=\left(\frac{184-151}{23}\right):\left(\frac{151-115}{23}\right)$$
$$\text{Ratio}=33:36=11:12$$