rupees 12,000 is invested for ` 1 (1)/(2)` years at C.I annually. If Rs 15,972 is received at the end of this period, find the rate of interest per annum.

Answer

To solve the problem step by step, we will use the formula for compound interest and manipulate it to find the rate of interest per annum.

### Step 1: Identify the Given Values
- Principal (P) = ₹12,000
- Amount (A) = ₹15,972
- Time (t) = 1.5 years (which can be converted to a fraction as \( \frac{3}{2} \) years)

### Step 2: Write the Compound Interest Formula
The formula for the amount \( A \) in compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = Final amount
- \( P \) = Principal amount
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 3: Substitute the Known Values into the Formula
Substituting the known values into the formula:
\[ 15,972 = 12,000 \left(1 + \frac{r}{100}\right)^{\frac{3}{2}} \]

### Step 4: Isolate the Compound Interest Term
Divide both sides by 12,000 to isolate the compound interest term:
\[ \frac{15,972}{12,000} = \left(1 + \frac{r}{100}\right)^{\frac{3}{2}} \]
Calculating the left side:
\[ \frac{15,972}{12,000} = 1.331 \]
So we have:
\[ 1.331 = \left(1 + \frac{r}{100}\right)^{\frac{3}{2}} \]

### Step 5: Remove the Exponent
To eliminate the exponent \( \frac{3}{2} \), raise both sides to the power of \( \frac{2}{3} \):
\[ (1.331)^{\frac{2}{3}} = 1 + \frac{r}{100} \]

### Step 6: Calculate \( (1.331)^{\frac{2}{3}} \)
Calculating \( (1.331)^{\frac{2}{3}} \):
\[ (1.331)^{\frac{2}{3}} \approx 1.1 \]
So we have:
\[ 1.1 = 1 + \frac{r}{100} \]

### Step 7: Solve for \( r \)
Now, isolate \( r \):
\[ \frac{r}{100} = 1.1 - 1 = 0.1 \]
Multiplying both sides by 100 gives:
\[ r = 0.1 \times 100 = 10 \]

### Step 8: Conclusion
Thus, the rate of interest per annum is:
\[ \boxed{21\%} \]