A man promises to his wife a birthday present, given her each year a number of rupee equal to the number of years in her age. If her birthday falls on August 8, what sum must be placed at simple interset at 7 % on January 1 before she is 63 (non leap year) in order to raise the required sum ?

Options

A. Rs 1600
B. Rs 1550
C. Rs 1500
D. Rs 1450

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the total amount to be given as a birthday present
The man promises to give his wife a birthday present equal to her age in rupees when she turns 63. Therefore, the total amount he needs to have is:
- **Total Amount (A) = 63 rupees**

### Step 2: Calculate the time period from January 1 to August 8
We need to find out how many days there are from January 1 to August 8 in a non-leap year.

1. **January:** 31 days
2. **February:** 28 days (non-leap year)
3. **March:** 31 days
4. **April:** 30 days
5. **May:** 31 days
6. **June:** 30 days
7. **July:** 31 days
8. **August:** 8 days

Now, let's add these days:
- Total days = 31 (Jan) + 28 (Feb) + 31 (Mar) + 30 (Apr) + 31 (May) + 30 (Jun) + 31 (Jul) + 8 (Aug)
- Total days = 31 + 28 + 31 + 30 + 31 + 30 + 31 + 8 = 219 days

### Step 3: Convert days into years
To convert the total days into years, we divide by the total number of days in a year (365):
- **Time (T) = 219 days / 365 days = 219/365 years = 3/5 years**

### Step 4: Use the Simple Interest formula
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest (63 rupees)
- P = Principal amount (unknown)
- R = Rate of interest (7%)
- T = Time in years (3/5 years)

### Step 5: Set up the equation
We know that the Simple Interest is 63 rupees, so we can set up the equation:
\[ 63 = \frac{P \times 7 \times \frac{3}{5}}{100} \]

### Step 6: Solve for P
1. Rearranging the equation:
\[ 63 = \frac{P \times 7 \times 3}{500} \]
2. Multiply both sides by 500:
\[ 63 \times 500 = P \times 7 \times 3 \]
3. Calculate \( 63 \times 500 = 31500 \):
\[ 31500 = P \times 21 \]
4. Divide both sides by 21 to find P:
\[ P = \frac{31500}{21} = 1500 \]

### Conclusion
The sum that must be placed at simple interest at 7% on January 1 before she is 63 is **1500 rupees**.

---