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:
Initial male population in 1970 = $M$ 
Initial female population in 1970 = $F$

From 1970 to 1980:

• Male population increases by 40% $\Rightarrow$ 1980 male population = $1.4M$
• Female population increases by 20% $\Rightarrow$ 1980 female population = $1.2F$

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

Given:
Total population increased by 25% from 1970 to 1980, so: 
$1.4M + 1.2F = 1.25(M + F)$

Expanding the right-hand side: 
$1.4M + 1.2F = 1.25M + 1.25F$

Rearranging terms: 
$1.4M - 1.25M = 1.25F - 1.2F$ 
$0.15M = 0.05F$ 
$\Rightarrow \boxed{F = 3M}$

From 1980 to 1990:

• Female population increases by 25%: 
  1990 female population = $1.25 \times 1.2F = 1.5F$
• Given: 1990 female population is twice the 1990 male population 
  $1.5F = 2 \times \text{Male}_{1990}$ 
  $\Rightarrow \text{Male}_{1990} = 0.75F$

Final Step:

Total population in 1970: $M + F = M + 3M = 4M$

Total population in 1990: $0.75F + 1.5F = 2.25F = 2.25 \times 3M = 6.75M$

Percentage increase: 
$\frac{6.75M - 4M}{4M} \times 100 = \frac{2.75M}{4M} \times 100 = \boxed{68.75\%}$

Answer 2

Step 1: From \(1970\) to \(1980\): 

• Male population increases by \(40\%\): \(M \rightarrow 1.4M\)
• Female population increases by \(20\%\): \(F \rightarrow 1.2F\)
• Overall population increases by \(25\%\): \(M + F \rightarrow 1.25(M + F)\)

Step 2: Setting up the equation based on the total population in 1980:

\(1.4M + 1.2F = 1.25(M + F)\)
Expand the right-hand side:
\(1.4M + 1.2F = 1.25M + 1.25F\)
Rearranging terms:
\(1.4M - 1.25M = 1.25F - 1.2F\)
\(0.15M = 0.05F\)
Solving for \(F\):
\(F = 3M\)

Step 3: From \(1980\) to \(1990\):

• Female population increases by \(25\%\):
  \(1.2F \rightarrow 1.25 \times 1.2F = 1.5F\)
• Given: Female population in 1990 is twice the male population
  \(\Rightarrow \text{Male population in 1990} = \frac{1.5F}{2} = 0.75F\)

Step 4: Substituting \(F = 3M\):

• \(0.75F = 0.75 \times 3M = 2.25M\)
• \(1.5F = 1.5 \times 3M = 4.5M\)

Step 5: Total population:

• In 1970: \(M + F = M + 3M = 4M\)
• In 1990: \(2.25M + 4.5M = 6.75M\)

Step 6: Percentage increase in population:

\(\frac{6.75M - 4M}{4M} \times 100 = \frac{2.75M}{4M} \times 100 = 68.75\%\)

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