Garima borrowed 40,000 at 10% p.a. simple interest. She immediately inverted this money at 10% p.a., compounded half-yearly. Calculate Garima's gain in 18 months.

Answer

To solve the problem step by step, we will calculate the simple interest Garima pays on the borrowed amount, the amount she earns from investing that money at compound interest, and finally, her gain.

### Step 1: Calculate Simple Interest (SI)

**Formula for Simple Interest:**
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount = 40,000
- \(R\) = Rate of interest = 10%
- \(T\) = Time in years = 18 months = \(\frac{18}{12} = 1.5\) years

**Calculation:**
\[ SI = \frac{40000 \times 10 \times 1.5}{100} \]
\[ SI = \frac{40000 \times 15}{100} = \frac{600000}{100} = 6000 \]

### Step 2: Calculate Total Amount after Simple Interest

**Total Amount (A) after Simple Interest:**
\[ A = P + SI \]
\[ A = 40000 + 6000 = 46000 \]

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

**Formula for Compound Interest (compounded half-yearly):**
\[ A = P \left(1 + \frac{R}{200}\right)^{n} \]
Where:
- \(P\) = Principal amount = 40,000
- \(R\) = Rate of interest = 10%
- \(n\) = Number of compounding periods = 18 months = 3 half-years

**Calculation:**
\[ A = 40000 \left(1 + \frac{10}{200}\right)^{3} \]
\[ A = 40000 \left(1 + 0.05\right)^{3} \]
\[ A = 40000 \left(1.05\right)^{3} \]
Calculating \(1.05^3\):
\[ 1.05^3 = 1.157625 \]
Now substituting back:
\[ A = 40000 \times 1.157625 = 46305 \]

### Step 4: Calculate Garima's Gain

**Garima's Gain:**
\[ \text{Gain} = \text{Amount from Compound Interest} - \text{Amount paid as Simple Interest} \]
\[ \text{Gain} = 46305 - 46000 = 305 \]

### Final Answer
Garima's gain in 18 months is **305**.

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