Problem 17 · 2015 Math Kangaroo
Hard
Fractions, Decimals & Percents
percent-multiplierratio
In the past 20 years the population of Arnberg has increased by 40%. In the same time span the population of Berghausen has increased by 60%. In total the population of the two villages has increased by 54%. What was the ratio of the populations 20 years ago?
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Answer: C — 3 : 7
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Hint 1 of 2
The combined 54% growth is a weighted average of 40% and 60%.
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Hint 2 of 2
Let the old populations be a and b and set up the weighted-average equation.
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Approach: weighted average of the two growth rates
- With old populations a and b: (1.4a + 1.6b)/(a+b) = 1.54.
- So 1.4a + 1.6b = 1.54a + 1.54b, giving 0.06b = 0.14a, i.e. a:b = 6:14 = 3:7.
- The ratio 20 years ago was 3:7 (C).
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