The population of Patna city is 96,000, the decreasing rate of population is 800 per year. Pop ulation of Patliputra is 68,000, the increasing rate of population is 1200 per year. In how many years will the population of both places be equal?

Options

A. 10
B. 14
C. 18
D. 22

Answer

To find out in how many years the population of Patna and Patliputra will be equal, we can set up an equation based on the information given.

### Step-by-step Solution:

1. **Identify the initial populations and rates of change:**
- Population of Patna (P_A) = 96,000 (decreasing by 800 per year)
- Population of Patliputra (P_B) = 68,000 (increasing by 1,200 per year)

2. **Set up the equations for the populations after 'x' years:**
- The population of Patna after 'x' years:
\[ P_A = 96000 - 800x \]
- The population of Patliputra after 'x' years:
\[ P_B = 68000 + 1200x \]

3. **Set the two equations equal to find when the populations will be the same:**
\[ 96000 - 800x = 68000 + 1200x \]

4. **Rearrange the equation to isolate 'x':**
- Move all terms involving 'x' to one side and constant terms to the other side:
\[ 96000 - 68000 = 1200x + 800x \]
\[ 28000 = 2000x \]

5. **Solve for 'x':**
- Divide both sides by 2000:
\[ x = \frac{28000}{2000} = 14 \]

6. **Conclusion:**
- The populations of Patna and Patliputra will be equal in **14 years**.