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

This problem involves tracking population changes over two decades with specific growth rates and constraints.

Given Information:

1970 to 1980: Total population increased 25%, Male increased 40%, Female increased 20%

1980 to 1990: Female increased 25%

1990 constraint: Female population = 2 × Male population

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Let's assume the total population in 1970 was 100 (this makes percentage calculations easier).

If we call the male population in 1970 as $m$ and female population as $f$, then:

$m + f = 100$

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From the growth rates:

Male in 1980 = $m \times 1.4 = 1.4m$

Female in 1980 = $f \times 1.2 = 1.2f$

Total in 1980 = $100 \times 1.25 = 125$

This gives us our second equation:

$1.4m + 1.2f = 125$

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We have two equations:

1. $m + f = 100$
2. $1.4m + 1.2f = 125$

From equation 1: $f = 100 - m$

Substituting into equation 2:

$1.4m + 1.2(100 - m) = 125$

$1.4m + 120 - 1.2m = 125$

$0.2m = 5$

$m = 25$

Therefore: $f = 75$

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Now we can find the exact populations in 1980:

Male in 1980 = $25 \times 1.4 = 35$

Female in 1980 = $75 \times 1.2 = 90$

Total in 1980 = $35 + 90 = 125$

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Female population in 1990:

Female increased by 25% from 1980 to 1990

Female in 1990 = $90 \times 1.25 = 112.5$

Male population in 1990:

We're told that in 1990, female population = 2 × male population

Therefore: $112.5 = 2 \times \text{(male population in 1990)}$

Male in 1990 = $\tfrac{112.5}{2} = 56.25$

Total population in 1990:

Total = $112.5 + 56.25 = 168.75$

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Percentage increase from 1970 to 1990:

$= \tfrac{\text{Final value} - \text{Initial value}}{\text{Initial value}} \times 100\%$

$= \tfrac{168.75 - 100}{100} \times 100\%$

$= \tfrac{68.75}{100} \times 100\%$

$= 68.75\%$

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Year	Male	Female	Total
1970	25	75	100
1980	35	90	125
1990	56.25	112.5	168.75

Answer: 68.75%

The key insight is using the assumption technique of setting the base year total as 100 to simplify percentage calculations, combined with the constraint that female population equals twice the male population in 1990 to determine the final values.