Two partners A and B together lend ₹ 84100 at 5% compounded annually. The amount which A gets at the end of 3 years is the same as what B gets at the end of 5 years. Determine the ratio of shares of A and B.

Options

A. `21:20`
B. `441: 400`
C. `1:4`
D. `5:21`

Answer

To solve the problem, we need to determine the ratio of the shares of partners A and B based on the compound interest they receive after different time periods. Here’s a step-by-step solution:

### Step 1: Define the variables
Let:
- A's investment = ₹ X
- B's investment = ₹ Y
- Total investment = ₹ 84100

Thus, we have:
\[ X + Y = 84100 \]

### Step 2: Write the formula for compound interest
The formula for the amount (A) after compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( P \) = principal amount (initial investment)
- \( r \) = rate of interest
- \( n \) = number of years

### Step 3: Calculate the amount for A and B
For partner A, the amount after 3 years at 5% interest:
\[ A_A = X \left(1 + \frac{5}{100}\right)^3 = X \left(1.05\right)^3 \]

For partner B, the amount after 5 years at 5% interest:
\[ A_B = Y \left(1 + \frac{5}{100}\right)^5 = Y \left(1.05\right)^5 \]

### Step 4: Set the amounts equal
According to the problem, the amounts received by A and B are equal:
\[ X \left(1.05\right)^3 = Y \left(1.05\right)^5 \]

### Step 5: Simplify the equation
Dividing both sides by \( \left(1.05\right)^3 \):
\[ X = Y \left(1.05\right)^{5-3} \]
\[ X = Y \left(1.05\right)^2 \]

### Step 6: Express X in terms of Y
Now we can express X in terms of Y:
\[ X = Y \left(1.05\right)^2 \]
\[ X = Y \cdot 1.1025 \]

### Step 7: Substitute into the total investment equation
Substituting \( X \) in the total investment equation:
\[ Y \cdot 1.1025 + Y = 84100 \]
\[ Y(1.1025 + 1) = 84100 \]
\[ Y(2.1025) = 84100 \]

### Step 8: Solve for Y
Now, solve for Y:
\[ Y = \frac{84100}{2.1025} \]
\[ Y \approx 40000 \]

### Step 9: Find X
Now substitute back to find X:
\[ X = Y \cdot 1.1025 \]
\[ X = 40000 \cdot 1.1025 \]
\[ X \approx 44100 \]

### Step 10: Find the ratio of X to Y
Finally, the ratio of shares of A and B is:
\[ \frac{X}{Y} = \frac{44100}{40000} = \frac{441}{400} \]

### Conclusion
Thus, the ratio of shares of A and B is:
\[ \text{Ratio of A to B} = 441:400 \]