Out of a sum of Rs. 625, a part was lent at 5% SI and the other at 10% SI. If the interest on the first part after 2 years is equal to the interest on the second part after 4 years , then the second sum (in Rs. ) is :

Options

A. 250
B. 300
C. 125
D. 275

Answer

To solve the problem step by step, we will follow the given information and apply the formula for Simple Interest (SI).

### Step 1: Define the Variables
Let:
- \( x \) = the amount lent at 5% SI
- \( 625 - x \) = the amount lent at 10% SI

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

### Step 3: Calculate the Interest for Each Part
1. **Interest on the first part (at 5%) after 2 years:**
\[ SI_1 = \frac{x \times 5 \times 2}{100} = \frac{10x}{100} = \frac{x}{10} \]

2. **Interest on the second part (at 10%) after 4 years:**
\[ SI_2 = \frac{(625 - x) \times 10 \times 4}{100} = \frac{40(625 - x)}{100} = \frac{40(625 - x)}{100} \]

### Step 4: Set the Interests Equal
According to the problem, the interest on the first part after 2 years is equal to the interest on the second part after 4 years:
\[ \frac{x}{10} = \frac{40(625 - x)}{100} \]

### Step 5: Simplify the Equation
Cross-multiply to eliminate the fractions:
\[ 100x = 10 \times 40(625 - x) \]
\[ 100x = 400(625 - x) \]

### Step 6: Expand and Rearrange
Expanding the right side:
\[ 100x = 250000 - 400x \]
Now, add \( 400x \) to both sides:
\[ 100x + 400x = 250000 \]
\[ 500x = 250000 \]

### Step 7: Solve for \( x \)
Divide both sides by 500:
\[ x = \frac{250000}{500} = 500 \]

### Step 8: Find the Second Part
Now, substitute \( x \) back to find the second part:
\[ 625 - x = 625 - 500 = 125 \]

### Final Answer
The second sum lent at 10% SI is **Rs. 125**.

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