Understanding the Problem
To solve this problem, we need to invest ₹ 8829 into three different sectors such that the amounts after 5, 6, and 8 years at 4% simple interest are equal.
Formula for Simple Interest
The formula for calculating the total amount \( A \) using simple interest is:
\[
A = P + (P \times r \times t)
\]
Where:
- \( P \) = Principal amount
- \( r \) = Rate of interest (as a decimal)
- \( t \) = Time in years
Defining Variables
Let:
- \( P_1 \) = Amount invested for 5 years
- \( P_2 \) = Amount invested for 6 years
- \( P_3 \) = Amount invested for 8 years
We know that:
\[
P_1 + P_2 + P_3 = 8829
\]
Calculating Amounts
The amounts for each sector after the respective years will be:
- For 5 years:
\[
A_1 = P_1 + (P_1 \times 0.04 \times 5) = P_1(1 + 0.20) = 1.2P_1
\]
- For 6 years:
\[
A_2 = P_2 + (P_2 \times 0.04 \times 6) = P_2(1 + 0.24) = 1.24P_2
\]
- For 8 years:
\[
A_3 = P_3 + (P_3 \times 0.04 \times 8) = P_3(1 + 0.32) = 1.32P_3
\]
Setting these amounts equal gives us:
\[
1.2P_1 = 1.24P_2 = 1.32P_3
\]
Solving the Equations
From the equal amounts, express \( P_1, P_2, \) and \( P_3 \) in terms of a common variable \( x \):
- \( P_1 = \frac{1.24}{1.2}x \)
- \( P_2 = x \)
- \( P_3 = \frac{1.24}{1.32}x \)
Substituting these into the sum equation:
\[
\frac{1.24}{1.2}x + x + \frac{1.24}{1.32}x = 8829
\]
Solving this will provide the values for \( P_1, P_2, \) and \( P_3 \).
Conclusion
This approach allows you to distribute the total investment into three parts while ensuring that the amounts grow to be equal after their respective durations at 4% interest.