In a factory, the production of cycles rose to 48,400 from 40,000 in 2 years. The rate of growth per annum is

Options

A. 0.09
B. 0.08
C. 0.105
D. 0.1

Answer

To find the rate of growth per annum of cycle production in the factory, we can follow these steps:

### Step 1: Identify the Given Values
- Initial production (Principal) = 40,000 cycles
- Final production (Amount) = 48,400 cycles
- Time period (n) = 2 years

### Step 2: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \( A \) = Final amount (48,400)
- \( P \) = Principal amount (40,000)
- \( R \) = Rate of growth per annum (unknown)
- \( n \) = Time period in years (2)

### Step 3: Substitute the Values into the Formula
Substituting the known values into the formula gives:
\[ 48,400 = 40,000 \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Simplify the Equation
Divide both sides by 40,000:
\[ \frac{48,400}{40,000} = \left(1 + \frac{R}{100}\right)^2 \]
This simplifies to:
\[ 1.21 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: Take the Square Root
To eliminate the square, take the square root of both sides:
\[ \sqrt{1.21} = 1 + \frac{R}{100} \]
Calculating the square root gives:
\[ 1.1 = 1 + \frac{R}{100} \]

### Step 6: Solve for R
Subtract 1 from both sides:
\[ 1.1 - 1 = \frac{R}{100} \]
This simplifies to:
\[ 0.1 = \frac{R}{100} \]

### Step 7: Multiply by 100
To find R, multiply both sides by 100:
\[ R = 0.1 \times 100 \]
This gives:
\[ R = 10 \]

### Conclusion
The rate of growth per annum is **10%**.

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