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A certain airline's fleet consisted of 60 type A planes at

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A certain airline's fleet consisted of 60 type A planes at [#permalink]

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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

I solved this method using a chart interested in the algebraic approach
[Reveal] Spoiler: OA

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 09 Jan 2013, 21:46
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Algebraic approach:
Consider x as number of years when both planes A & B will reach towards their break-even and then type B just exceeds type A planes.

i.e. in x years, The number of Type B planes with annual rate of 4 will be just greater than (existing inventory of 60 planes minus Type A inventory reduced at annual rate 3 in x years)
i.e. \(4x\geq{60 - 3x}\)
i.e. \(7x\geq{60}\)
i.e. \(x\geq{8.57}\)
Round to next integer \(x = 9 years\)

Hence choice(D) is the correct answer.

PS: This problem is similar to water tank problem - one pipe is filling the tank and another one draining at different rate.
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Last edited by PrashantPonde on 09 Jan 2013, 23:51, edited 3 times in total.

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 09 Jan 2013, 21:48
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fozzzy 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

I solved this method using a chart interested in the algebraic approach


60 - 3*n = 4*n , n = 60/7 = 8.5

So, in 8.5 years, there would be equal number of planes A & B(which is of course a hypothetical situation).

So, in 9 years the number of B would be more.

Answer is D.
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 09 Jan 2013, 21:55
PraPon wrote:
Algebraic approach:
Consider x as number of years when both planes A & B will reach towards their break-even and then type B just exceeds type A planes.

i.e. in x years, Number Type A with annual rate 4 will be just greater than (existing inventory of 60 planes minus Type A inventory reduced at annual rate 3 in x years)
i.e. \(4x\geq{60 - 3x}\)
i.e. \(7x\geq{60}\)
i.e. \(x\geq{8.57}\)
Round to next integer \(x = 9 years\)

Hence choice(D) is the correct answer.

PS: This problem is similar to water tank problem - one pipe is filling the tank and another one draining at different rate.



It should by TYPE B, isnt it?

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 09 Jan 2013, 23:29
bhavinshah5685 wrote:
PraPon wrote:
Algebraic approach:
Consider x as number of years when both planes A & B will reach towards their break-even and then type B just exceeds type A planes.

i.e. in x years, Number Type A with annual rate 4 will be just greater than (existing inventory of 60 planes minus Type A inventory reduced at annual rate 3 in x years)
i.e. \(4x\geq{60 - 3x}\)
i.e. \(7x\geq{60}\)
i.e. \(x\geq{8.57}\)
Round to next integer \(x = 9 years\)

Hence choice(D) is the correct answer.

PS: This problem is similar to water tank problem - one pipe is filling the tank and another one draining at different rate.



It should by TYPE B, isnt it?


Thanks. You are right. Its a typo. I have updated the thread.
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 10 Jan 2013, 08:04
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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....this comes to 8 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. So 9 years.

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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fozzzy 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

I solved this method using a chart interested in the algebraic approach


Very nice problem fozzy, congrats!
Now this is basically an algebraic translation and that's all we need to do so let's go and nail this

60-3x < 1/2(60+x)

Answer is D

Cheers!
J :)

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 08 Jun 2014, 01:04
I think the best and fastest approach here is to back solve
start with C as usual

No of Years Type A Type B
8 60-8*3= 36 8*4 = 32 --> We need more years
9 60-9*3= 33 9*4 = 36 --> We found answer

Ans: D
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 07 Dec 2014, 15:02
Hi

I actually just plugged in the numbers and got 8. Can someone please clarify?

End of year 1980 -- 57 A and 4 B.
End of year 1981 -- 54 A and 8 B... and so on

End up year 1988 -- 33 A and 36 B

Since that's 8 years, I picked C. It tips over at the end of year 8, so it's still in year 8. Why do we choose 9?

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 08 Dec 2014, 03:31
russ9 wrote:
Hi

I actually just plugged in the numbers and got 8. Can someone please clarify?

End of year 1980 -- 57 A and 4 B.
End of year 1981 -- 54 A and 8 B... and so on

End up year 1988 -- 33 A and 36 B

Since that's 8 years, I picked C. It tips over at the end of year 8, so it's still in year 8. Why do we choose 9?


From the beginning of 1980 to the end of 1988 there are 8 years and 364 days so 9 years.
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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So, I also created a table, since for me this was faster and safer than venturing an equation. True, in many cases an equation is the only way to reach an answer in good time, but there is no reason to torture yourself with finding one if you have another option and you cannot readily come up with the correct equation! Right?? :)

______A:60 - 3 per year____B: 0 + 4 per year
Year1______57_______________4
Year2______54_______________8
Year3______51_______________12
Year4______48_______________16
Year5______45_______________20
Year6______42_______________24
Year7______39_______________28
Year8______36_______________32
Year9______33_______________36

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 21 Nov 2015, 21:02
I used an equality and back-solving:
y=years

60-3y = 4y

and then plugging in answer choices starting with E)10
60-30< 40 Could Work since 30 is less than half of 70

So trying C)
60-24 > 32
36 > 24 Nope too high but close

So without even doing the math I picked 9 Ans: D

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 25 Nov 2015, 16:03
Hi All,

When dealing with questions that can be solved with simple arithmetic and a bit or brute force, it's often best/fastest to just put pen-to-pad and 'map out' the solution. The prompt mentions that planes retired and acquired at the END of each year. You can create a quick table to track what you're looking for:

Start: 60 A and 0 B
Yr 1: 57 A and 4 B
Yr 2: 54 A and 8 B
Yr 3: 51 A and 12 B
Yr 4: 48 A and 16 B
Yr 5: 45 A and 20 B
Yr 6: 42 A and 24 B
Yr 7: 39 A and 28 B
Yr 8: 36 A and 32 B
Yr 9: 33 A and 36 B

After 9 years, the type A planes were less than 50% of the total.

Final Answer:
[Reveal] Spoiler:
D


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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 25 Nov 2015, 17:41
pacifist85 and Rich demonstrate the utility (and power) of simply creating a table of values.
For more on using tables to solve quantitative questions, watch the following free video - http://www.gmatprepnow.com/module/gmat- ... /video/929

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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fozzzy 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


We are given that the fleet started with 60 type A planes. We are also given that the airline retired 3 of the type A planes and acquired 4 new type B planes each year. We need to determine how many years it took for the number of type A planes left in the airline's fleet to be less than 50 percent of the fleet. We can let n = the number of years it will take for this to occur.

The number of type A planes in the fleet can be expressed as 60 - 3n (because the fleet loses 3 type A planes each year). The total number of planes in the fleet is 60 - 3n + 4n, which takes into account the loss of 3 type A planes and addition of 4 type B planes each year.

We are interested in the number of years it will take until the number of type A planes is less than ½ the total planes in the fleet, as illustrated in the following equation:

60 - 3n < (60 - 3n + 4n) x ½

Multiplying the entire equation by 2, we have:

120 - 6n < 60 + n

-7n < -60

n > -60/-7

n > 8 4/7

Since n > 8 4/7, the smallest integer value n can be is 9.

Answer: D
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]

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New post 01 Jul 2017, 12:20
fozzzy 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

I solved this method using a chart interested in the algebraic approach


I go the answer as D - 9,, however, I used the crude way of solving, where I reduced A by 3 every year and simultaneously added 4 for B.

After seeing some posts I can see there are some short cut ways of solving this.

See some similar posts in the way I solved.
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Re: A certain airline's fleet consisted of 60 type A planes at   [#permalink] 01 Jul 2017, 12:20
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