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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

1. $16$
2. $13$
3. $10$
4. $7$

Let the total production and, per capita production of food gains be $\text{TP}$ and $PR$ respectively.

Let the initial population be $P$ and the final population be $P_1$

• $\text{TP} = PR \times P \quad \longrightarrow(1)$
• $\frac{140}{100}\text{TP} = \frac{127}{100}PR \times P_1 \quad \longrightarrow(2)$

Divide equation $(2)$ by equation $(1)$, we get

$\dfrac{\frac{140\text{TP}}{100}}{\text{TP}} = \dfrac{\frac{127PR}{100}\times P_1}{PR\times P}$

$\Rightarrow \dfrac{140}{100} = \frac{127P_1}{100P}$

$\Rightarrow \boxed{\frac{P_1}{P} = \frac{140}{127}}$

The percentage increase in population $= \left(\frac{P_1-P}{P}\right) \times 100\%$

$\qquad \qquad = \left(\frac{P_1}{P}-1\right)\times 100\%$

$\qquad \qquad = \left(\frac{140}{127}-1\right)\times 100\%$

$\qquad \qquad = \left(\frac{140-127}{127}\right)\times 100\%$

$\qquad \qquad = \frac{13}{127}\times 100\%$

$\qquad \qquad = 10.236\%$

$\qquad \qquad \cong 10\%$

$\therefore$  The percentage increase in population $= 10\%.$

Correct Answer $:\text{D}$

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