The comound interest on a certain sum for 2 years at 10% per annum is 1260. The simple interest on the same sum for double the time at half the rate per cent per annum is

Options

A. 1200
B. 1160
C. 1208
D. 1175

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the Principal Amount
We know that the compound interest (CI) for 2 years at 10% per annum is 1260. The formula for compound interest is:

\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]

Where:
- \(CI\) = Compound Interest
- \(P\) = Principal Amount
- \(r\) = Rate of Interest
- \(n\) = Number of years

Given:
- \(CI = 1260\)
- \(r = 10\%\)
- \(n = 2\)

Substituting the values into the formula:

\[ 1260 = P \left(1 + \frac{10}{100}\right)^2 - P \]

Calculating \( \left(1 + \frac{10}{100}\right)^2 \):

\[ \left(1 + 0.1\right)^2 = 1.1^2 = 1.21 \]

Now substituting back into the equation:

\[ 1260 = P \cdot 1.21 - P \]

This simplifies to:

\[ 1260 = P(1.21 - 1) = P(0.21) \]

Now, solving for \(P\):

\[ P = \frac{1260}{0.21} = 6000 \]

### Step 2: Calculate Simple Interest for Double the Time at Half the Rate
Now we need to find the simple interest (SI) on the same principal amount for double the time and half the rate.

- Double the time = \(2 \times 2 = 4\) years
- Half the rate = \(\frac{10}{2} = 5\%\)

The formula for simple interest is:

\[ SI = \frac{P \times r \times t}{100} \]

Where:
- \(SI\) = Simple Interest
- \(P\) = Principal Amount (6000)
- \(r\) = Rate of Interest (5%)
- \(t\) = Time (4 years)

Substituting the values into the formula:

\[ SI = \frac{6000 \times 5 \times 4}{100} \]

Calculating:

\[ SI = \frac{6000 \times 20}{100} = \frac{120000}{100} = 1200 \]

### Final Answer
The simple interest on the same sum for double the time at half the rate per annum is **1200**.

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