Divide X 2602 between X and Y, so that the amount of X after 7 yr is equal to the amount of Y after 9 yr, the interest being compounded at 4% pa.

Options

A. X 1352, X 1250
B. X 1400, X 1350
C. X 1415, X 1300
D. X 1500, X 1450

Answer

To solve the problem of dividing ₹2602 between X and Y such that the amount of X after 7 years is equal to the amount of Y after 9 years, with an interest rate of 4% per annum compounded annually, we can follow these steps:

### Step 1: Define Variables
Let the amount invested by X be \( P_X = x \) and the amount invested by Y be \( P_Y = 2602 - x \).

### Step 2: Write the Formula for Compound Interest
The formula for the amount \( A \) after \( n \) years with principal \( P \) and rate \( r \) is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]

### Step 3: Calculate Amount for X after 7 Years
For X, the amount after 7 years will be:
\[ A_X = x \left(1 + \frac{4}{100}\right)^7 = x \left(1.04\right)^7 \]

### Step 4: Calculate Amount for Y after 9 Years
For Y, the amount after 9 years will be:
\[ A_Y = (2602 - x) \left(1 + \frac{4}{100}\right)^9 = (2602 - x) \left(1.04\right)^9 \]

### Step 5: Set the Amounts Equal
According to the problem, the amount of X after 7 years is equal to the amount of Y after 9 years:
\[ x \left(1.04\right)^7 = (2602 - x) \left(1.04\right)^9 \]

### Step 6: Simplify the Equation
We can divide both sides by \( \left(1.04\right)^7 \):
\[ x = (2602 - x) \left(1.04\right)^2 \]

### Step 7: Expand and Rearrange
Now, let's expand and rearrange the equation:
\[ x = (2602 - x) \cdot 1.0816 \]
\[ x = 2602 \cdot 1.0816 - x \cdot 1.0816 \]
\[ x + x \cdot 1.0816 = 2602 \cdot 1.0816 \]
\[ x(1 + 1.0816) = 2602 \cdot 1.0816 \]
\[ x \cdot 2.0816 = 2816.1632 \]

### Step 8: Solve for x
Now, divide both sides by \( 2.0816 \):
\[ x = \frac{2816.1632}{2.0816} \approx 1352 \]

### Step 9: Calculate Y's Amount
Now, we can find Y's amount:
\[ P_Y = 2602 - x = 2602 - 1352 = 1250 \]

### Final Answer
Thus, the amounts are:
- Amount for X: ₹1352
- Amount for Y: ₹1250

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