To find the sum of money (principal) given that the sum of annual compound interest and semi-annual compound interest for 2 years at the rate of 20% per annum is Rs. 18,082, we need to follow a systematic approach integrating both kinds of compounding.
Step 1: Understanding the Problem
- Annual Interest: This is the interest calculated with the compounding done once per year.
- Semi-Annual Interest: This is the interest calculated with the compounding done twice per year.
- Time Period: 2 years.
- Interest Rate: 20% per annum.
Step 2: Making Use of Compound Interest Formula
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
- \( A \) is the amount after time \( t \).
- \( P \) is the principal.
- \( r \) is the annual interest rate.
- \( n \) is the number of times interest is compounded per year.
- \( t \) is the time in years.
For annual compounding (\( n = 1 \)):
[tex]\[ A_{annual} = P \left(1 + \frac{0.20}{1}\right)^{1 \times 2} = P \left(1.20\right)^2 \][/tex]
For semi-annual compounding (\( n = 2 \)):
[tex]\[ A_{semi} = P \left(1 + \frac{0.20}{2}\right)^{2 \times 2} = P \left(1.10\right)^4 \][/tex]
Step 3: Calculating the Compound Interest
1. Annual Interest:
[tex]\[ CI_{annual} = A_{annual} - P \][/tex]
[tex]\[ CI_{annual} = P \left(1.20\right)^2 - P \][/tex]
[tex]\[ CI_{annual} = P \left(1.44 - 1\right) \][/tex]
[tex]\[ CI_{annual} = 0.44P \][/tex]
2. Semi-Annual Interest:
[tex]\[ CI_{semi\text{-}annual} = A_{semi} - P \][/tex]
[tex]\[ CI_{semi\text{-}annual} = P \left(1.10\right)^4 - P \][/tex]
[tex]\[ CI_{semi\text{-}annual} = P \left(1.4641 - 1\right) \][/tex]
[tex]\[ CI_{semi\text{-}annual} = 0.4641P \][/tex]
Step 4: Setting up the Equation
Given the total interest:
[tex]\[ CI_{annual} + CI_{semi\text{-}annual} = 18082 \][/tex]
[tex]\[ 0.44P + 0.4641P = 18082 \][/tex]
Step 5: Solving for \( P \)
Combining like terms:
[tex]\[ 0.9041P = 18082 \][/tex]
Solving for \( P \):
[tex]\[ P = \frac{18082}{0.9041} \][/tex]
[tex]\[ P = 20000 \][/tex]
Thus, the sum of money (principal) is Rs. 20,000.
