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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]  09 Jan 2013, 19:42
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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]  09 Jan 2013, 20:46
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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, 22:51, edited 3 times in total.
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]  09 Jan 2013, 20:48
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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.

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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]  09 Jan 2013, 20:55
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]  09 Jan 2013, 22:29
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]  10 Jan 2013, 07:04
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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]  20 Nov 2013, 06:53
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)

Cheers!
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Re: A certain airline's fleet consisted of 60 type A planes at [#permalink]  08 Jun 2014, 00:04
I think the best and fastest approach here is to back solve

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]  07 Dec 2014, 14:02
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]  08 Dec 2014, 02:31
Expert's post
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]  22 Jan 2015, 08:43
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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
Re: A certain airline's fleet consisted of 60 type A planes at   [#permalink] 22 Jan 2015, 08:43
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