A sum of money was lent at simple interest at a certain rate for 3 years. Had it been lent at 2.5 %per annum higher rate, it would have fetched Rs 540 more. The money lent was :

Options

A. Rs 6400
B. Rs 6472
C. Rs 6840
D. Rs 7200

Answer

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

**Simple Interest (SI) = (Principal × Rate × Time) / 100**

Let's denote:
- Principal = P
- Original Rate = R%
- Time = 3 years
- Higher Rate = (R + 2.5)%

According to the problem, the difference in interest between the two rates over 3 years is Rs. 540.

### Step 1: Write the formula for Simple Interest at the original rate
The Simple Interest for the original rate (R%) for 3 years is:
\[ SI_1 = \frac{P \times R \times 3}{100} \]

### Step 2: Write the formula for Simple Interest at the higher rate
The Simple Interest for the higher rate (R + 2.5)% for 3 years is:
\[ SI_2 = \frac{P \times (R + 2.5) \times 3}{100} \]

### Step 3: Set up the equation based on the difference in interest
According to the problem, the difference in interest between the two rates is Rs. 540:
\[ SI_2 - SI_1 = 540 \]

Substituting the formulas from Steps 1 and 2:
\[ \frac{P \times (R + 2.5) \times 3}{100} - \frac{P \times R \times 3}{100} = 540 \]

### Step 4: Simplify the equation
Factor out the common terms:
\[ \frac{P \times 3}{100} \left( (R + 2.5) - R \right) = 540 \]
This simplifies to:
\[ \frac{P \times 3 \times 2.5}{100} = 540 \]

### Step 5: Solve for P
Now, we can solve for P:
\[ \frac{P \times 7.5}{100} = 540 \]
Multiply both sides by 100:
\[ P \times 7.5 = 54000 \]
Now, divide both sides by 7.5:
\[ P = \frac{54000}{7.5} \]
Calculating this gives:
\[ P = 7200 \]

### Final Answer
The money lent was Rs. 7200.

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