The population of a town is increases by `16(2)/(3)%` in Ist year, but decreases by 37.5% in 2nd year, increases by `57(1)/(7)%` in 3rd year. Then find the present population if after 3 years the population will become 275,000 ?

Options

A. 240000
B. 250000
C. 200000
D. 225000

Answer

To find the present population of the town given the population after 3 years is 275,000, we will work backwards through the percentage changes over the three years.

### Step-by-Step Solution:

1. **Identify the final population after 3 years:**
- The population after 3 years is given as 275,000.

2. **Calculate the population before the third year:**
- The population increases by \(57\frac{1}{7}\%\) in the third year.
- Convert \(57\frac{1}{7}\%\) to an improper fraction:
\[ 57\frac{1}{7} = \frac{400}{7}\% \]
- This means the population at the end of the third year is \( \frac{1100}{700} \) times the population at the beginning of the third year.
- Let \( P_2 \) be the population at the beginning of the third year:
\[ 275,000 = P_2 \times \frac{1100}{700} \]
- Rearranging gives:
\[ P_2 = 275,000 \times \frac{700}{1100} \]
- Simplifying:
\[ P_2 = 275,000 \times \frac{7}{11} = 175,000 \]

3. **Calculate the population before the second year:**
- The population decreases by \(37.5\%\) in the second year.
- This means the population at the end of the second year is \(62.5\%\) of the population at the beginning of the second year.
- Let \( P_1 \) be the population at the beginning of the second year:
\[ 175,000 = P_1 \times \frac{62.5}{100} \]
- Rearranging gives:
\[ P_1 = 175,000 \times \frac{100}{62.5} = 280,000 \]

4. **Calculate the present population:**
- The population increases by \(16\frac{2}{3}\%\) in the first year.
- Convert \(16\frac{2}{3}\%\) to an improper fraction:
\[ 16\frac{2}{3} = \frac{50}{3}\% \]
- This means the population at the end of the first year is \( \frac{350}{300} \) times the present population \( X \):
\[ 280,000 = X \times \frac{350}{300} \]
- Rearranging gives:
\[ X = 280,000 \times \frac{300}{350} = 240,000 \]

### Final Answer:
The present population of the town is **240,000**.

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