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

The Correct Option is A

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

Answer 2

From \(1970\) to \(1980,\)
The male population increased by \(40\%\).
The female population increased by \(20\%\).
The overall population increased by \(25\%\).
\(⇒1.4M + 1.2F = 1.25(M + F)\)
\(⇒1.4M + 1.2F = 1.25M + 1.25F\)
\(⇒1.4M - 1.25M = 1.25F - 1.2F\)
\(⇒0.15M = 0.05F\)
\(⇒F = 3M\)
From \(1980\) to \(1990\),
Female population increased by \(25\%\).
Hence, the female population in \(1990= 1.25 × 1.2F = 1.5F\)
Since, the female population in \(1990\) is twice the male population,
Male population in \(1990= \frac {1.5F}{ 2} = 0.75F\)
Since \(F = 3M,\)
Male population in \(1990 = 2.25M\)
Total population in \(1970 = M + F = M + 3M = 4M\)
Total population in \(1990 = 2.25M + 1.5F = 2.25M + 4.5M = 6.75M\)
The percentage increase in population \(=\frac {6.75M−4M}{4M}×100\)
\(= \frac {2.75}{4}×100\)
\(= 68.75\%\)

So, the correct option is (A): \(68.75\%\)