Problem Solving

A certain airline's fleet consisted of 60 type A planes at the beginning of 1980. At the end of each year, starting with 1980, the airline retired 3 of the TYPE A planes and acquired 4 new type B plans. How many years did it take before the number of type A planes left in the airline's fleet was less than 50 percent of the fleet?

A. 6
B. 7
C. 8
D. 9
E. 10

Answer: D
Source: GMAT prep

BTGModeratorVI wrote: ↑

Wed Jul 29, 2020 2:53 pm

A certain airline's fleet consisted of 60 type A planes at the beginning of 1980. At the end of each year, starting with 1980, the airline retired 3 of the TYPE A planes and acquired 4 new type B plans. How many years did it take before the number of type A planes left in the airline's fleet was less than 50 percent of the fleet?

A. 6
B. 7
C. 8
D. 9
E. 10

Answer: D
Source: GMAT prep

Let \(x =\) number of years. Each year we lose \(3\) Type A planes and gain \(4\) Type B planes.

Since we start off with \(60\) type A planes and \(0\) Type B planes, the following equation would determine the point in time where Type A planes \(=\) Type B planes

\((60-3x) = 4x \ldots \) this comes to \(8 \frac{4}{7}\) years.

Since we are looking for the number of years (rounded to the nearest whole number) where \(< 50\%\) of the planes are of Type A, this must be \(> 8\) years.

Therefore, \(9\) years.

120 - 6n < 60 + n
-7n < -60
n > -60/-7
n > 8 4/7

Answer: D

Airline's fleet consisted of 60 type A planes at the beginning of 1980.
Pre 1980, A = 60 and B = 0

Starting with 1980, the airline retired 3 of the TYPE A planes and acquired 4 new type B plans.
1980, A = 57 and B = 4

If we look carefully then the total number of planes will be increasing by 1 every year as they are removing 3 of A type and adding 4 of B type.

All these sequences can form an Arithmetic Progression.
Starting with 1980, the total number of planes every year is: 61, 62 , 63 ... i.e. 61+ (n - 1) , where n is the number of years after 1980.

Similarly, the number of B type planes since 1980 would be: 4, 8 ,12 ... i.e. 4+(n-1)4 , where n is the number of years after 1980.

Now we need to find the number of years it took "before" A was <50% of fleet.

Let us first find the number of years it took for A to be < 50% of total
OR the number of years it took for B to be > 1/2 of total, i.e.

4 + (n-1)4 > (1/2) [ 61 + (n-1)4]
=> n > 9

But we know that is the number of years after 1980. But we need to include 1980 since the addition and subtraction of planes started from that year.

So number of years taken = 9 + 1 = 10
But we need to find the number of years 'before' , so 9

Answer D


Thanks,
Ignite