A shepherd has n goats in the year 2000. In 2001 the no. of goats increased by `40%` . In 2002 the no. of goats declined to `70%` . In 2003 the no, of goats grew by`30%` . In 2004, he sold `10%` goats, then he had only 34,398 goats. The percentage increase of the no. of goats in this duration was:

Options

A. `14.66%`
B. `16.66%`
C. `20%`
D. `33.33%`

Answer

To find the percentage increase in the number of goats over the years, we will follow these steps:

### Step 1: Initial Number of Goats
Let the initial number of goats in the year 2000 be \( n \).

### Step 2: Increase in 2001
In 2001, the number of goats increased by 40%.
- The number of goats after the increase = \( n + 0.4n = 1.4n \).

### Step 3: Decline in 2002
In 2002, the number of goats declined to 70% of the number in 2001.
- The number of goats after the decline = \( 0.7 \times 1.4n = 0.98n \).

### Step 4: Growth in 2003
In 2003, the number of goats grew by 30%.
- The number of goats after the growth = \( 0.98n + 0.3 \times 0.98n = 1.3 \times 0.98n = 1.274n \).

### Step 5: Selling Goats in 2004
In 2004, the shepherd sold 10% of the goats.
- The number of goats after selling = \( 1.274n - 0.1 \times 1.274n = 0.9 \times 1.274n = 1.1466n \).

### Step 6: Setting Up the Equation
According to the problem, after selling, the number of goats is 34,398.
- Therefore, we have the equation:
\[ 1.1466n = 34,398 \]

### Step 7: Solving for \( n \)
To find \( n \), we will divide both sides of the equation by 1.1466:
\[ n = \frac{34,398}{1.1466} \]
Calculating this gives:
\[ n \approx 30,000 \]

### Step 8: Finding the Percentage Increase
Now, we need to find the percentage increase from \( n \) to the final number of goats (34,398).
- The increase in the number of goats = \( 34,398 - n = 34,398 - 30,000 = 4,398 \).
- The percentage increase is calculated as:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original}} \right) \times 100 = \left( \frac{4,398}{30,000} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Increase} \approx 14.66\% \]

### Final Answer
The percentage increase in the number of goats over the duration is approximately **14.66%**.