To find the compound annual growth rate (CAGR) of Riya's investment, we need to use the CAGR formula. The CAGR formula is given by:
[tex]\[ \text{CAGR} = \left( \frac{\text{Final Value}}{\text{Initial Investment}} \right)^{\frac{1}{\text{Number of Years}}} - 1 \][/tex]
Here's a step-by-step solution for this problem:
1. Identify the Initial Investment and Final Value:
- Initial Investment (P) = ₹ 20,000
- Final Value (A) = ₹ 32,000
2. Determine the Number of Years:
- Riya invested the amount in 2016 and the final value is considered in 2021.
- Number of Years (n) = 2021 - 2016 = 5 years
3. Calculate the Ratio of the Final Value to the Initial Investment:
[tex]\[ \frac{\text{Final Value}}{\text{Initial Investment}} = \frac{32000}{20000} = 1.6 \][/tex]
4. Express the formula for CAGR:
[tex]\[ \text{CAGR} = (1.6)^{\frac{1}{5}} - 1 \][/tex]
5. Utilize logarithmic values provided:
- We are given that [tex]\(\log(1.6) = 0.2041\)[/tex].
- We can use this to find the intermediate exponent value.
6. Compute exponent:
- Since we need [tex]\((1.6)^{\frac{1}{5}}\)[/tex], we can write this as [tex]\(\exp\left(\frac{1}{5} \cdot \log(1.6)\right)\)[/tex].
- Given that [tex]\(\frac{\log(1.6)}{5} = \frac{0.2041}{5} = 0.04082\)[/tex].
7. Find the Antilog (exponentiation) value:
- We are given that [tex]\(\text{antilog}(0.04082) = 1.098\)[/tex].
- Therefore, [tex]\((1.6)^{\frac{1}{5}} = 1.098\)[/tex].
8. Subtract 1 from the component to find CAGR:
[tex]\[ \text{CAGR} = 1.098 - 1 = 0.098 \][/tex]
9. Express CAGR as a percentage:
[tex]\[ \text{CAGR} = 0.098 \approx 9.8 \% \][/tex]
So, the compound annual growth rate (CAGR) of Riya's investment is approximately 9.8% per annum.
