The simple interest on a certain sum for 8 months at 4% per annum is Rs 129 less than the simple interest on the same sum for 15 months at 5% pr annum. The sum is :

Options

A. Rs 2,580
B. Rs, 2400
C. Rs, 2529
D. Rs 3600

Answer

To solve the problem step by step, we need to find the sum (principal amount) based on the information given about simple interest.

### Step 1: Understand the formula for Simple Interest
The formula for calculating simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (the sum we need to find)
- \( R \) = Rate of interest (in percentage)
- \( T \) = Time (in years)

### Step 2: Set up the equations based on the problem
We have two scenarios given in the problem:

1. Simple Interest for 8 months at 4% per annum:
- Convert 8 months to years: \( T_1 = \frac{8}{12} = \frac{2}{3} \) years
- The interest can be expressed as:
\[ SI_1 = \frac{P \times 4 \times \frac{2}{3}}{100} = \frac{8P}{300} = \frac{2P}{75} \]

2. Simple Interest for 15 months at 5% per annum:
- Convert 15 months to years: \( T_2 = \frac{15}{12} = \frac{5}{4} \) years
- The interest can be expressed as:
\[ SI_2 = \frac{P \times 5 \times \frac{5}{4}}{100} = \frac{25P}{400} = \frac{5P}{80} \]

### Step 3: Set up the equation based on the relationship given
According to the problem, the simple interest for 8 months at 4% is Rs 129 less than the simple interest for 15 months at 5%. This can be expressed as:
\[ SI_1 + 129 = SI_2 \]
Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{2P}{75} + 129 = \frac{5P}{80} \]

### Step 4: Clear the fractions by finding a common denominator
The least common multiple of 75 and 80 is 600. Multiply the entire equation by 600 to eliminate the denominators:
\[ 600 \left( \frac{2P}{75} \right) + 600 \times 129 = 600 \left( \frac{5P}{80} \right) \]
This simplifies to:
\[ 16 \times 2P + 77400 = 7.5 \times 5P \]
Which can be rewritten as:
\[ 32P + 77400 = 37.5P \]

### Step 5: Rearrange the equation to isolate P
Rearranging gives:
\[ 37.5P - 32P = 77400 \]
\[ 5.5P = 77400 \]

### Step 6: Solve for P
Now, divide both sides by 5.5:
\[ P = \frac{77400}{5.5} = 14000 \]

### Conclusion
The sum (principal amount) is Rs 14000.