If the rate of interest is 10% p.a. and Rs. 12000 lent at the compound interest, half yearly. What is the equivalent simple rate of interest for first year ?

Answer

To find the equivalent simple rate of interest for the first year when Rs. 12000 is lent at a compound interest of 10% per annum, compounded half-yearly, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for compound interest (CI) is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = principal amount (the initial amount of money).
- \( r \) = annual interest rate (decimal).
- \( n \) = number of times that interest is compounded per year.
- \( t \) = the number of years the money is invested or borrowed.

### Step 2: Identify the Values
In this problem:
- \( P = 12000 \)
- \( r = 10\% = 0.10 \)
- \( n = 2 \) (since the interest is compounded half-yearly)
- \( t = 1 \)

### Step 3: Calculate the Amount after 1 Year
Substituting the values into the compound interest formula:
\[ A = 12000 \left(1 + \frac{0.10}{2}\right)^{2 \times 1} \]
\[ A = 12000 \left(1 + 0.05\right)^{2} \]
\[ A = 12000 \left(1.05\right)^{2} \]
\[ A = 12000 \times 1.1025 \]
\[ A = 13230 \]

### Step 4: Calculate the Compound Interest
Now, we can find the compound interest (CI) for the first year:
\[ CI = A - P \]
\[ CI = 13230 - 12000 \]
\[ CI = 1230 \]

### Step 5: Find the Equivalent Simple Interest (SI)
The equivalent simple interest for the first year can be calculated using the formula:
\[ SI = P \times \frac{r_{equiv}}{100} \times t \]
Where \( r_{equiv} \) is the equivalent simple interest rate we need to find.

Since we are looking for the equivalent rate for 1 year:
\[ 1230 = 12000 \times \frac{r_{equiv}}{100} \times 1 \]
\[ r_{equiv} = \frac{1230 \times 100}{12000} \]
\[ r_{equiv} = \frac{123000}{12000} \]
\[ r_{equiv} = 10.25\% \]

### Final Answer
The equivalent simple rate of interest for the first year is **10.25%**.

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