The simple interest on a sum of money for 2 years at 4% per annum is Rs 340. Find :
the compound intersect on this sum for one year payble half-hearly at the same rate.

Answer

To solve the problem step by step, we will first find the principal amount using the given simple interest, and then we will calculate the compound interest for one year payable half-yearly at the same rate.

### Step 1: Find the Principal Amount (P)

Given:
- Simple Interest (SI) = Rs 340
- Rate (R) = 4% per annum
- Time (T) = 2 years

We use the formula for Simple Interest:
\[ SI = \frac{P \cdot R \cdot T}{100} \]

Substituting the known values:
\[ 340 = \frac{P \cdot 4 \cdot 2}{100} \]

This simplifies to:
\[ 340 = \frac{8P}{100} \]

Multiplying both sides by 100:
\[ 34000 = 8P \]

Now, divide by 8:
\[ P = \frac{34000}{8} = 4250 \text{ Rs} \]

### Step 2: Calculate the Compound Interest for 1 Year Payable Half-Yearly

Since the interest is compounded half-yearly, we need to adjust the rate and time:
- Half-yearly Rate (r) = \(\frac{4\%}{2} = 2\%\)
- Time (T) for half-yearly compounding = \(1 \text{ year} \times 2 = 2 \text{ half-years}\)

Now, we use the formula for Compound Interest:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal amount
- \(r\) = Rate of interest per period
- \(n\) = Number of compounding periods

Substituting the values:
\[ A = 4250 \left(1 + \frac{2}{100}\right)^2 \]

Calculating:
\[ A = 4250 \left(1 + 0.02\right)^2 = 4250 \left(1.02\right)^2 \]
\[ A = 4250 \times 1.0404 = 4421.70 \text{ Rs} \]

### Step 3: Calculate the Compound Interest (CI)

Now we find the Compound Interest (CI):
\[ CI = A - P \]
Substituting the values:
\[ CI = 4421.70 - 4250 = 171.70 \text{ Rs} \]

### Final Answer
The compound interest on the sum for one year payable half-yearly at the same rate is **Rs 171.70**.

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