sandy wrote:
Town A has a population of 160,000 and is growing at a rate of 20% annually. Town B has a population of 80,000 and is growing at a rate of 50% annually.
Quantity A |
Quantity B |
The number of years until town B’s population is greater than that of town A |
3 |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Let's keep track of each population for 3 years
TOWN A
Present Population: 160 (we'll drop the 000's at the end for both populations)
Population in 1 year: (160)(1.2)
Population in 2 years: (160)(1.2)(1.2)
Population in 3 years: (160)(1.2)(1.2)(1.2) = (160)(1.2)³TOWN B
Present Population: 80
Population in 1 year: (80)(1.5)
Population in 2 years: (80)(1.5)(1.5)
Population in 3 years: (80)(1.5)(1.5)(1.5) = (80)(1.5)³Now that we know the populations after 3 years, we can check whether or not 3 years is enough time for Town B's population to exceed Town A's population.
So, let's examine the fraction:
(160)(1.2)³/
(80)(1.5)³There are 3 possible cases:
case a)
(160)(1.2)³/
(80)(1.5)³ = 1, in which case, the 2 populations are EQUAL
case b)
(160)(1.2)³/
(80)(1.5)³ < 1, in which case,
Town A's population is LESS THAN
Town B's populationcase c)
(160)(1.2)³/
(80)(1.5)³ > 1, in which case,
Town A's population is GREATER THAN
Town B's population160/80 = 2 and (1.2)³/(1.5)³ = (1.2/1.5)³ = (12/15)³ = (4/5)³ = 64/125
So,
(160)(1.2)³/
(80)(1.5)³ = (2)(64/125) = 128/125
Since 128/125 > 1, we have a case c situation.
So, after 3 years,
Town A's population is still GREATER THAN
Town B's populationSo, it will take MORE THAN 3 YEARS for Town B's population to exceed Town A's population
We get:
Quantity A: MORE THAN 3 YEARS
Quantity B: 3 years
Answer: A
Cheers,
Brent
_________________
Brent Hanneson - founder of Greenlight Test Prep