A city present population is `512000` and `20%` decrease every year then find the 3 year before population and after 3 year population ?

Options

A. `262144`
B. `242144`
C. `262144`
D. `262544`

Answer

To solve the problem of finding the population of a city three years ago and three years in the future given a current population of 512,000 and a 20% decrease in population each year, we can follow these steps:

### Step 1: Understand the current population and the decrease
The current population of the city is 512,000. Each year, the population decreases by 20%. This means that 80% of the population remains after each year.

### Step 2: Calculate the population after 3 years
To find the population after 3 years, we can use the formula for exponential decay:
\[ \text{Future Population} = \text{Current Population} \times (1 - \text{decrease rate})^{\text{number of years}} \]
Here, the decrease rate is 20% or 0.20, so:
\[ \text{Future Population} = 512,000 \times (0.80)^3 \]

Calculating this step by step:
- First, calculate \( (0.80)^3 \):
\[ (0.80)^3 = 0.512 \]
- Now, multiply by the current population:
\[ \text{Future Population} = 512,000 \times 0.512 = 262,144 \]

### Step 3: Calculate the population 3 years ago
To find the population three years ago, we need to reverse the decrease. Since the population decreases by 20% each year, we can find the population three years ago by using the formula:
\[ \text{Past Population} = \text{Current Population} \div (1 - \text{decrease rate})^{\text{number of years}} \]
So:
\[ \text{Past Population} = 512,000 \div (0.80)^3 \]

Calculating this step by step:
- We already calculated \( (0.80)^3 = 0.512 \).
- Now, divide the current population by this value:
\[ \text{Past Population} = 512,000 \div 0.512 = 1,000,000 \]

### Final Results
- The population three years ago was **1,000,000**.
- The population three years in the future will be **262,144**.