The simple interset on a sum for 8 years is Rs. 47500. The rate of interset for the first 5 years is 10 % per annum and for the next 3 years is 15 % per annum. What is the value (in Rs.) of sum ?

Options

A. 50000
B. 60000
C. 45000
D. 62500

Answer

To find the principal amount (P) based on the given simple interest (SI) over 8 years with varying interest rates, we can follow these steps:

### Step 1: Understand the Given Information
- Total Simple Interest (SI) for 8 years = Rs. 47500
- Rate of Interest for the first 5 years (R1) = 10% per annum
- Rate of Interest for the next 3 years (R2) = 15% per annum
- Time for the first part (T1) = 5 years
- Time for the second part (T2) = 3 years

### Step 2: Write the Formula for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal amount
- R = Rate of Interest
- T = Time in years

### Step 3: Calculate Simple Interest for Each Time Period
1. For the first 5 years at 10%:
\[ SI_1 = \frac{P \times 10 \times 5}{100} = \frac{50P}{100} = \frac{P}{2} \]

2. For the next 3 years at 15%:
\[ SI_2 = \frac{P \times 15 \times 3}{100} = \frac{45P}{100} = \frac{9P}{20} \]

### Step 4: Combine the Simple Interests
The total Simple Interest for 8 years is the sum of the two parts:
\[ SI = SI_1 + SI_2 \]
Substituting the values we calculated:
\[ 47500 = \frac{P}{2} + \frac{9P}{20} \]

### Step 5: Find a Common Denominator and Solve for P
To combine the fractions, we need a common denominator. The least common multiple of 2 and 20 is 20:
\[ \frac{P}{2} = \frac{10P}{20} \]
Now we can rewrite the equation:
\[ 47500 = \frac{10P}{20} + \frac{9P}{20} \]
Combine the fractions:
\[ 47500 = \frac{19P}{20} \]

### Step 6: Solve for P
To isolate P, multiply both sides by 20:
\[ 47500 \times 20 = 19P \]
\[ 950000 = 19P \]
Now divide both sides by 19:
\[ P = \frac{950000}{19} = 50000 \]

### Final Answer
The principal amount (P) is Rs. 50000.

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