Shubham invests ₹5000 in a bank at a certain rate of interest, compounded annually. The ratio of the investment's value after 2 years to its value after 4 years is $25:49$. What is the minimum number of years for the value to exceed ₹25000?
Answer and solution
Correct answer: 5
Solution
Ratio after 2 years to 4 years: $\dfrac{P \cdot r^2}{P \cdot r^4} = \dfrac{25}{49} \Rightarrow r^2 = \dfrac{49}{25} \Rightarrow r = \dfrac{7}{5}$ (rate 40%).
Value after $n$ years $= 5000 \times \left(\dfrac{7}{5}\right)^n$. Need this to exceed 25000:
Year 4: $5000 \times \left(\dfrac{7}{5}\right)^4 = 5000 \times \dfrac{2401}{625} = 19208 < 25000$.
Year 5: $19208 \times \dfrac{7}{5} = 26891.20 > 25000$.
Minimum number of years $= \mathbf{5}$.
Value after $n$ years $= 5000 \times \left(\dfrac{7}{5}\right)^n$. Need this to exceed 25000:
Year 4: $5000 \times \left(\dfrac{7}{5}\right)^4 = 5000 \times \dfrac{2401}{625} = 19208 < 25000$.
Year 5: $19208 \times \dfrac{7}{5} = 26891.20 > 25000$.
Minimum number of years $= \mathbf{5}$.