A village population Ist year `16(2)/(3)%` increase, IInd year `37(1)/(2)%` increase and IIIrd year `57(1)/(7)%` decrease. After IIIrd year population is `44000`. Find the present population .

Options

A. `62,000`
B. `44,000`
C. `64,000`
D. `74,000`

Answer

To find the present population of the village before the changes in population over the three years, we can follow these steps:

### Step 1: Understand the percentage increases and decrease
- The population increases by:
- First year: \(16 \frac{2}{3}\% = \frac{50}{3}\% = \frac{50}{3} \div 100 = \frac{50}{300} = \frac{1}{6}\)
- Second year: \(37 \frac{1}{2}\% = 37.5\% = \frac{75}{2} \div 100 = \frac{75}{200} = \frac{3}{8}\)
- Third year: \(57 \frac{1}{7}\% = \frac{400}{7}\% = \frac{400}{700} = \frac{4}{7}\)

### Step 2: Convert percentages to multipliers
- For the first year, the multiplier is \(1 + \frac{50}{300} = \frac{7}{6}\)
- For the second year, the multiplier is \(1 + \frac{75}{200} = \frac{11}{8}\)
- For the third year, since it's a decrease, the multiplier is \(1 - \frac{400}{700} = \frac{3}{7}\)

### Step 3: Calculate the overall multiplier
- The overall change in population can be calculated by multiplying these multipliers together:
\[ \text{Overall multiplier} = \frac{7}{6} \times \frac{11}{8} \times \frac{3}{7} \]

### Step 4: Simplify the multiplication
- Cancel out the \(7\) in the numerator and denominator:
\[ = \frac{11}{6 \times 8} \times 3 = \frac{11 \times 3}{48} = \frac{33}{48} = \frac{11}{16} \]

### Step 5: Set up the equation with the final population
- Let the present population be \(P\). After applying the overall change, we have:
\[ P \times \frac{11}{16} = 44000 \]

### Step 6: Solve for \(P\)
- Rearranging the equation gives:
\[ P = 44000 \times \frac{16}{11} \]
\[ P = 44000 \times \frac{16}{11} = 44000 \times 1.4545 \approx 64000 \]

### Final Answer
The present population of the village is approximately **64000**.

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