During the first year, the population of a village is increased by 5% and the second year, it is diminished by 5%. At the end of the seond year, its population was 47880. What was the population at the beginning of the first year ?

Options

A. 45500
B. 48000
C. 43500
D. 53000

Answer

To find the initial population of the village at the beginning of the first year, we can follow these steps:

### Step 1: Understand the problem
We know that the population increases by 5% in the first year and then decreases by 5% in the second year. At the end of the second year, the population is 47,880. We need to find the initial population at the beginning of the first year.

### Step 2: Set up the equations
Let the initial population at the beginning of the first year be \( P \).

After the first year, the population increases by 5%. Therefore, the population at the end of the first year is:
\[ P_1 = P + 0.05P = 1.05P \]

In the second year, the population decreases by 5%. Thus, the population at the end of the second year is:
\[ P_2 = P_1 - 0.05P_1 = 0.95P_1 = 0.95 \times 1.05P = 0.9975P \]

### Step 3: Set up the equation with the final population
We know that at the end of the second year, the population is 47,880:
\[ 0.9975P = 47880 \]

### Step 4: Solve for \( P \)
To find \( P \), we can rearrange the equation:
\[ P = \frac{47880}{0.9975} \]

Calculating this gives:
\[ P = 48000 \]

### Step 5: Conclusion
Thus, the initial population at the beginning of the first year was **48,000**.

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