In a village, the production of food grains increased by 40% and the per capita production of food grains increased by 27% during a certain period. The percentage by which the population of the village increased during the same period is nearest to

Options

A. 16
B. 13
C. 10
D. 7

Answer

To solve the problem, we need to find the percentage increase in the population of the village given the increase in food grain production and per capita production.

### Step-by-Step Solution:

1. **Define Initial Values:**
- Let the initial production of food grains be \( P = 100 \) units.
- Let the initial population be \( N = 100 \) people.

2. **Calculate New Production:**
- The production of food grains increased by 40%.
- Therefore, the new production \( P' \) is:
\[ P' = P + 0.40 \times P = 100 + 40 = 140 \text{ units} \]

3. **Calculate New Per Capita Production:**
- The per capita production increased by 27%.
- The initial per capita production \( \text{PC} \) is:
\[ \text{PC} = \frac{P}{N} = \frac{100}{100} = 1 \text{ unit per person} \]
- The new per capita production \( \text{PC}' \) is:
\[ \text{PC}' = \text{PC} + 0.27 \times \text{PC} = 1 + 0.27 = 1.27 \text{ units per person} \]

4. **Relate New Per Capita Production to New Population:**
- The new population \( N' \) can be calculated using the new production and new per capita production:
\[ N' = \frac{P'}{\text{PC}'} = \frac{140}{1.27} \]
- Calculating \( N' \):
\[ N' \approx 110.24 \text{ people} \]

5. **Calculate the Increase in Population:**
- The increase in population is:
\[ \text{Increase} = N' - N = 110.24 - 100 = 10.24 \text{ people} \]

6. **Calculate the Percentage Increase in Population:**
- The percentage increase in population is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{N}\right) \times 100 = \left(\frac{10.24}{100}\right) \times 100 \approx 10.24\% \]

7. **Final Calculation:**
- Since we need the nearest percentage, we can round it to:
\[ \text{Percentage Increase} \approx 10\% \]

### Conclusion:
The percentage by which the population of the village increased during the same period is approximately **10%**.