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ALPHAVILLE
Population
Same as
BETAVILLE
Problem of the Week
Problem E and Solution
The Same Again, Again
BETAVILLE
Population
Same as
ALPHAVILLE
Problem
The populations of Alphaville and Betaville were equal at the end of 2010. The population of
Alphaville decreased by 2.8% during 2011, then increased by 8.5% during 2012. The population
of Betaville increased by r% in each of the two years. That is, each year the population of
Betaville increased by r% of the previous year’s population. If the populations of the towns
were equal again at the end of 2012, determine the value of r correct to one decimal place.
Solution
Let p be the population of Alphaville at the end of 2010. Since Alphaville and Betaville have
the same population size, then p is also the population of Betaville at the end of 2010.
The populationof Alphaville
decreased by 2.8% in 2011, so the population at the end of 2011
2.8
2.8
p= 1−
p = 0.972p.
is p −
100
100
The population of Alphaville
by 8.5% during 2012, so the population at the end
then increased
8.5
8.5
of 2012 is 0.972p +
(0.972p) = 1 +
(0.972p) = 1.085(0.972p) = 1.05462p.
100
100
The population
increased by r% in 2011, so the population at the end of 2011 is
of Betaville
r r
p= 1+
p.
p+
100
100
The population
of Betavillethen increased
so the population
at the end of
by r%r during
2012,
r r
r r r 2
2012 is 1 +
p+
1+
p= 1+
p 1+
= 1+
p.
100
100
100
100
100
100
Since the populations of Alphaville and Betaville are equal at the end of 2012, we have
r 2
1+
p = 1.05462p
100
r 2
1+
= 1.05462 dividing both sides by p, since p > 0
100
r
1+
≈ 1.02695 taking the square root of both sides
100
r
≈ 0.02695
100
r ≈ 2.7
Therefore, correct to one decimal place, r = 2.7.