Jaipur city population Ist year `16(2)/(3)%` increase, Iind year `20%` increase and IIIrd year `40%` decrease. If decrease in population is `20000` in 3 year then find the present population ?

Options

A. `105000`
B. `115000`
C. `905000`
D. `155000`

Answer

To solve the problem, we need to find the present population of Jaipur city after a series of percentage increases and a decrease in population over three years. Let's break it down step by step.

### Step 1: Understand the Percentage Increases and Decrease
- In the first year, the population increases by \(16 \frac{2}{3}\%\).
- In the second year, the population increases by \(20\%\).
- In the third year, the population decreases by \(40\%\).

### Step 2: Convert Percentages to Fractions
- \(16 \frac{2}{3}\% = \frac{50}{3}\% = \frac{1}{6}\) (as a fraction).
- \(20\% = \frac{1}{5}\).
- \(40\% = \frac{2}{5}\).

### Step 3: Let the Initial Population be \(P\)
- After the first year, the population becomes:
\[ P + \frac{50}{3}\% \text{ of } P = P + \frac{1}{6}P = \frac{7}{6}P \]

### Step 4: Calculate the Population After the Second Year
- After the second year, the population becomes:
\[ \frac{7}{6}P + 20\% \text{ of } \frac{7}{6}P = \frac{7}{6}P + \frac{1}{5} \cdot \frac{7}{6}P = \frac{7}{6}P \left(1 + \frac{1}{5}\right) = \frac{7}{6}P \cdot \frac{6}{5} = \frac{7}{5}P \]

### Step 5: Calculate the Population After the Third Year
- After the third year, the population decreases by \(40\%\):
\[ \frac{7}{5}P - 40\% \text{ of } \frac{7}{5}P = \frac{7}{5}P - \frac{2}{5} \cdot \frac{7}{5}P = \frac{7}{5}P \left(1 - \frac{2}{5}\right) = \frac{7}{5}P \cdot \frac{3}{5} = \frac{21}{25}P \]

### Step 6: Set Up the Equation with the Decrease Information
- We know that the decrease in population is \(20,000\):
\[ \frac{7}{5}P - \frac{21}{25}P = 20000 \]

### Step 7: Simplify the Equation
- To solve for \(P\), we first need a common denominator:
\[ \frac{35}{25}P - \frac{21}{25}P = 20000 \]
\[ \frac{14}{25}P = 20000 \]

### Step 8: Solve for \(P\)
- Multiply both sides by \(\frac{25}{14}\):
\[ P = 20000 \cdot \frac{25}{14} = \frac{500000}{14} \approx 35714.29 \]

### Step 9: Calculate the Present Population
- The present population after the three years:
\[ \text{Present Population} = \frac{21}{25}P = \frac{21}{25} \cdot 35714.29 \approx 30000 \]

### Final Answer
The present population of Jaipur city is approximately **30,000**.