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

When we talk about food grain production in a village, there's a fundamental relationship we need to understand:

Total Production = Per Capita Production × Population

If each person produces a certain amount of food grains, and we know how many people there are, we can find the total production by multiplying these two quantities.

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Let's define our variables clearly:

$P$ = Total production of food grains

$C$ = Per capita production (production per person)

$N$ = Population of the village

So our fundamental equation is: $P = C \times N$

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Since we're dealing with percentage changes, we can assume any convenient initial values.

Let's assume:

Initial production = 1 unit

Initial per capita production = 1 unit

Initial population = 1 unit

This satisfies our equation: $1 = 1 \times 1$

This works because percentage changes are ratios, and ratios remain the same regardless of the actual numbers we start with.

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Now let's apply the percentage increases:

Production increased by 40%:

New production = $1 + 0.40 = 1.4$ units

Per capita production increased by 27%:

New per capita production = $1 + 0.27 = 1.27$ units

Population = ?

Let new population = $x$ units

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Using our fundamental relationship: Production = Per Capita × Population

$1.4 = 1.27 \times x$

$x = \dfrac{1.4}{1.27}$

$x = \dfrac{1.4}{1.27} = 1.102...$

$x \approx 1.10$

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Since our initial population was 1 unit and final population is 1.10 units:

Percentage increase = $\dfrac{\text{Final} - \text{Initial}}{\text{Initial}} \times 100\%$

Percentage increase = $\dfrac{1.10 - 1}{1} \times 100\%$

Percentage increase = $0.10 \times 100\% = 10\%$

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The population of the village increased by 10% during the same period.

When total production increases faster than per capita production, it means there are more people to account for the difference. This is exactly what we calculated!