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

[spoiler]OA: D. [/spoiler]

If 'x' is the number of years after 1980,
Number of type A planes after x years, a = 60 - 3*x (The initial number of planes minus 3 planes every year)
Number of type B planes after x years, b = 4*x (4 planes are added every year)

The problem requires:
a < 50% of fleet
which will happen when
a < b
60 - 3x < 4x
x > 60/7
x > 8.6

Since the number of years is an integer, this inequality will first hold true after 9 years.

[spoiler](D)[/spoiler] is the answer.

rahulvsd wrote: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

Hi! You could solve this question very quickly by brute force - just make a chart and keep track of what happens each year. Remember, on the GMAT you get points for picking the right answer, not for HOW you arrive at that answer. Sometimes brute force is quicker than algebra.

Year A planes B planes
1980 60 0
+1 57 4
+2 54 8
+3 51 12
+4 48 16
+5 45 20
+6 42 24
+7 39 28
+8 36 32
+9 33 36 DING DING DING! Choose D!

Now, the really smart test taker would have saved a bunch of time by using the answer choices to our advantage. We know that the smallest choice is 6, so let's jump right to +6 and ignore the first 5 years.

Even better, a great GMAT strategist would have backsolved - plugged in the answers to see which one gave us slighty more Bs than As. When you backsolve, generally start with B or D. Here we want the first time the Bs outnumbered the As, so let's start with the smaller choice.

B) 7

After 7 years, we have -21 A and +28B, giving us a 39/28 split - too many As. Eliminate choices (A) and (B).

D) 9

After 9 years, we have -27A and +36 B, giving us a 33/36 split - more than 50% Bs! Further, since the difference is less than 7 (-3As and +4Bs means B gains 7/year), we know that this is the smallest number of years that will work. Choose D!

Hi Stuart,

I did try this problem the brute force way but got 8 as the answer. Per the question, the number of fleet at the beginning of 1980 will be 60 in type A and 0 in type B, the number of fleet will start decreasing by end of 1980 itself, so in this case won't type B reach more than 50 % of fleet by 1988 (8 years)?

rahulvsd wrote:Hi Stuart,

I did try this problem the brute force way but got 8 as the answer. Per the question, the number of fleet at the beginning of 1980 will be 60 in type A and 0 in type B, the number of fleet will start decreasing by end of 1980 itself, so in this case won't type B reach more than 50 % of fleet by 1988 (8 years)?

You can think of it that way, but you have to count the actual number of years starting from the beginning of 1980. I guess technically you'd hit your target 8 years and 364 days after you start, but that's still 9 years, not 8. (If you start on Jan 1st 1980, you have your 9th change on Dec 31st 1988).

Hi Rahul,
just chipping in here.

The question says 'at the beginning of 1980', which could be 00:00:01 hours on Day 1 of 1980 or even better a really really small fraction of a second past 00:00 hours on day-1, there were 60 type A planes. 'At the end of 1980', which could be 23:59:59 hours on Day-last of 1980 or better a really tiny fraction of a second before the end of the year, 3 planes were retired. So the entire year, minus a fraction of a second maybe, was used in 1980 and thus should be included in 'the number of years it took for type B to exceed type A'.

On a different note, and with all due respect to Stuart, I think solving this question using Inequalities is a less time-consuming method (as I've shown above) than calculating the number of planes at the end of every year and finding when B crosses A. You might be a little uncomfortable in adapting to the Inequalities method, but once you've got a hang of it your life in GMAT QA will become really really easy.

let's plug in answer choices:

start in the middle, choice (C): 3x8 = 24 (type A retired)--> 60-24= 36 (type A left). 4X8 = 32 (type B added).

At this point we can see that 36 is slightly greater than 50% of the fleet population -- we want an answer choice just less than 50% -- We can safely pick (D) as it would definitely be less than 50% of the fleet population.


(I hope I am not missing something here because all the other solutions are pretty detailed)

Hi, I faced the same problem as Rahul did. Stuart may be right that it is 9 years not 8 but question asks that 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?


Year A planes B planes
1980 60 0
+1 57 4
+2 54 8
+3 51 12
+4 48 16
+5 45 20
+6 42 24
+7 39 28
+8 36 32 This is the year BEFORE A<B so we should ring DING DING DING here I believe?
+9 33 36

Please clarify, Thanks.[/u]

rahatali wrote:Hi, I faced the same problem as Rahul did. Stuart may be right that it is 9 years not 8 but question asks that 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?


Year A planes B planes
1980 60 0
+1 57 4
+2 54 8
+3 51 12
+4 48 16
+5 45 20
+6 42 24
+7 39 28
+8 36 32 This is the year BEFORE A<B so we should ring DING DING DING here I believe?
+9 33 36

Please clarify, Thanks.[/u]

Ah, good question! "Before" is here synonymous with "until", so you're looking for the number of years UNTIL this happens, not the number of years UNTIL the year BEFORE this happens. (Really annoying, I know! That said, you should probably ask yourself whether a question would really ask you for the year BEFORE something happens rather than the year DURING WHICH that thing happens.)