Question

The total of male and female populations in a city increased by 25% from 1970 to1980.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

• A. 68.75
• B. 68.50
• C. 68.25
• D. 69.25

Answer 1

Answer: (A)

68.75

Answer 2

Let the initial male population in 1970 be $M$ and the initial female population in 1970 be $F$.

Given:
Total population in 1970 = $M + F$ From 1970 to 1980:
Male population increased by 40% $\to$ Male population in 1980 =$1.4M$
Female population increased by 20% $\to$ Female population in 1980 = $1.2F$

Total population in 1980 = $1.4M + 1.2F$

Given that the total population increased by 25% from 1970 to 1980:
$1.4M + 1.2F = 1.25(M + F)$
$1.4M + 1.2F = 1.25M + 1.25F$
$0.15M = 0.05F$
$3M = F$
This gives the relation between the male and female populations in 1970.

From 1980 to 1990:
Female population increased by 25% $\to$ Female population in 1990 =$1.25 \times 1.2F = 1.5F$
Given that in 1990, the female population is twice the male population:
$1.5F = 2 \times \text{Male population in 1990}$
$\text{Male population in 1990} = 0.75F$

Since no information is provided about how the male population changed from 1980 to 1990, we'll use the relation from the previous decade:

Male population in 1980 = $1.4M$ Given that $M = \frac{F}{3}$ Male population in 1980 = $1.4 \times\frac{ F}{3}$= $0.4667F$
Now, comparing male populations from 1980 and 1990:
$0.4667F \to 0.75F$

The male population increased by $0.2833F$ or approximately 60.7% of the male population in 1980.
Total population in 1990 = $1.5F + 0.75F = 2.25F$

The total population in 1970 = $M + F$ = $\frac{F}{3} + F = \frac{4F}{3}$

Percentage increase from 1970 to 1990:
$\frac{(2.25F - \frac{4F}{3})}{\frac{4F}{3}} \times 100$

$\frac{6.75F - 4F}{4F} \times 100$

$\frac{2.75F}{4F} \times 100$

$= 68.75\%$

Thus, the percentage increase in the total of male and female populations in the city from 1970 to 1990 is 68.75