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Question:

The total of male and female populations in a city increased by 25% from 1970 to 1980. During the same period, the male population increased by 40% while the female population increased by 20%. From 1980 to 1990, the female population increased by 25%. In 1990, if the female population is twice the male population, then the percentage increase in the total of male and female populations in the city from 1970 to 1990 is

Explanation:

Let the male and female population be ‘m’ and ‘f’ in 1970. 

Using alligation

$\frac{m}{f}$ = $\frac{25-20}{40-25}$ = $\frac{1}{3}$

∴ f = 3m

Total population in 1970 = f + m = 4m

⇒ Number of females in 1980 = 1.2f = 3.6m and number of males in 1980 = 1.4m

Population in 1990
Number of females = 1.25 × 3.6m = 4.5m
Number of males = ½ × 4.5m = 2.25m
Total population = 4.5m + 2.25m = 6.75m

⇒ Total population increases from 4m in 1970 to 6.75m in 1990.

∴ Percentage increase = 2.75m/4m×100 = 68.75%.

Alternately,
Let the number of males and females in 1970 be 'm' and 'f' respectively.

∴ In 1980, number of males = 1.4m, females = 1.2f and total population = 1.25(m + f)
⇒ 1.4m + 1.2f = 1.25(m + f)
⇒ 0.15m = 0.05f
⇒ m : f = 1 : 3

∴ In 1990, number of females = 1.25 × 1.2f = 1.5f, number of males = 1/2 × 1.5f = 0.75f 
∴ Total population in 1990 = 1.5f + 0.75f = 2.25f

Also, total population in 1970 = m + f = f + f/3 = 4f/3

⇒ Required percentage = $\left(\frac{2.25f-4f/3}{4f/3}\right)$ × 100% = $\frac{11}{16}$ × 100% = 68.75%

Hence, option (c).