The population of a village is 10,000. In one year, the male population…

Concept

When two parts of a total grow at different percentage rates, represent one original part by a variable and the other as the total minus that variable.

Apply each growth multiplier separately: a rise of r% changes a quantity P to P(1 + r/100). Then add the changed parts and solve the resulting linear equation.

Application

Let x be the original male population. The original female population is 10,000 - x.

1. After one year, the male population is 1.06x and the female population is 1.05(10,000 - x).

2. Use the final total: 1.06x + 1.05(10,000 - x) = 10,540.

3. Expand: 1.06x + 10,500 - 1.05x = 10,540.

4. Combine like terms: 0.01x + 10,500 = 10,540.

5. Subtract 10,500: 0.01x = 40.

6. Divide by 0.01: x = 4,000.

Cross-check

The original female population is 6,000. After the increases, the two populations are 4,240 and 6,300; their sum is 10,540, the stated final population.

Therefore, the original male population was 4,000.