Acertain sum in certain time become Rs 10,000 at the rate 24% per annum SI and the same sum amounts to Rs 4000 at the rate of 8% per annum SI in the same duration. Find the sum and time:

Options

A. a)Rs 800, 35 yr
B. b)Rs 600, 75yr
C. c)Rs 1600, 55 yr
D. d)Rs 1000, 37.5yr

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI):

\[ \text{SI} = \frac{P \times R \times T}{100} \]

where:
- \( P \) = Principal amount (the sum we need to find)
- \( R \) = Rate of interest (given in percentage)
- \( T \) = Time (in years)

### Step 1: Set up the equations based on the information provided.

1. From the first case:
- The amount becomes Rs 10,000 at 24% per annum.
- The Simple Interest (SI) can be expressed as:
\[ \text{SI}_1 = 10,000 - P \]
- Using the SI formula:
\[ 10,000 - P = \frac{P \times 24 \times T}{100} \quad \text{(Equation 1)} \]

2. From the second case:
- The amount becomes Rs 4,000 at 8% per annum.
- The Simple Interest (SI) can be expressed as:
\[ \text{SI}_2 = 4,000 - P \]
- Using the SI formula:
\[ 4,000 - P = \frac{P \times 8 \times T}{100} \quad \text{(Equation 2)} \]

### Step 2: Rearranging the equations.

From Equation 1:
\[ 10,000 - P = \frac{24PT}{100} \]
Rearranging gives:
\[ 10,000 - P = 0.24PT \quad \text{(1)} \]

From Equation 2:
\[ 4,000 - P = \frac{8PT}{100} \]
Rearranging gives:
\[ 4,000 - P = 0.08PT \quad \text{(2)} \]

### Step 3: Subtract Equation 2 from Equation 1.

\[ (10,000 - P) - (4,000 - P) = 0.24PT - 0.08PT \]
This simplifies to:
\[ 10,000 - 4,000 = (0.24 - 0.08)PT \]
\[ 6,000 = 0.16PT \]

### Step 4: Solve for \( PT \).

Rearranging gives:
\[ PT = \frac{6,000}{0.16} \]
Calculating this:
\[ PT = 37,500 \]

### Step 5: Find the values of \( P \) and \( T \).

Now we have \( PT = 37,500 \). We can substitute \( P \) from either equation into this equation to find \( P \) and \( T \).

Using Equation 1:
\[ 10,000 - P = 0.24PT \]
Substituting \( PT = 37,500 \):
\[ 10,000 - P = 0.24 \times 37,500 \]
Calculating:
\[ 10,000 - P = 9,000 \]
Thus:
\[ P = 10,000 - 9,000 = 1,000 \]

### Step 6: Find \( T \).

Now substituting \( P = 1,000 \) into \( PT = 37,500 \):
\[ 1,000 \times T = 37,500 \]
\[ T = \frac{37,500}{1,000} = 37.5 \text{ years} \]

### Final Answer:
- The principal amount \( P \) is Rs 1,000.
- The time \( T \) is 37.5 years.

---