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A certain logistics company’s fleet consisted of 120 Type A trucks at

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A certain logistics company’s fleet consisted of 120 Type A trucks at  [#permalink]

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New post 16 May 2018, 20:48
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A certain logistics company’s fleet consisted of 120 Type A trucks at the beginning of 1970. At the end of each year, starting with 1970, the company retired 3 of the type A trucks and acquired 6 new Type B trucks. How many years did it take before the number of Type A trucks left in the company’s fleet was less than 50 percent of the fleet?

A. 11
B. 12
C. 13
D. 14
E. 15
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A certain logistics company’s fleet consisted of 120 Type A trucks at  [#permalink]

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New post 17 May 2018, 02:10
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jfdelgado wrote:
A certain logistics company’s fleet consisted of 120 Type A trucks at the beginning of 1970. At the end of each year, starting with 1970, the company retired 3 of the type A trucks and acquired 6 new Type B trucks. How many years did it take before the number of Type A trucks left in the company’s fleet was less than 50 percent of the fleet?

A. 11
B. 12
C. 13
D. 14
E. 15


Since the answer options are between 11 and 15 years, we could do the calculations for 10 years.
Since 3 Type A trucks are retired and 6 Type B trucks are added every year after 1970.
10 years later, we would have 120 - 10(3) = 90 Type A trucks and 10*6 = 60 Type B trucks.

Let the remaining years after 10 years be x.
\(90 - 3x < 60 + 6x\) -> \(-9x < -30\) -> \(x > 10/3\)

The lowest possible integer value for x here is 4.
So when 14 years have passed, there are more type B trucks.
14 years later, there would be 90 - 12 = 78 type A trucks and 60 + 24 = 84 type B trucks.

Therefore, it takes 14 years(Option D) before the number of type A trucks is less than 50% of the trucks
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Re: A certain logistics company’s fleet consisted of 120 Type A trucks at  [#permalink]

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New post 17 May 2018, 02:48
jfdelgado wrote:
A certain logistics company’s fleet consisted of 120 Type A trucks at the beginning of 1970. At the end of each year, starting with 1970, the company retired 3 of the type A trucks and acquired 6 new Type B trucks. How many years did it take before the number of Type A trucks left in the company’s fleet was less than 50 percent of the fleet?

A. 11
B. 12
C. 13
D. 14
E. 15


Let n be the number of years. We have -
\(\frac{120 - 3n}{120-3n+6n} <\frac{1}{2}\)

n > 13.33 --> n =14
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Re: A certain logistics company’s fleet consisted of 120 Type A trucks at  [#permalink]

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New post 17 May 2018, 04:22
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Both resolutions are correct. But the equation to solve is simpler and obtained in a very straightforward way.

Number of Type A trucks after x years: 120 - 3x
Number of Type B trucks after x years: 6x

We simply want 120 - 3x < 6x, in other words 9x > 120.

120 / 9 = 40 / 3 = 13.3...

We need more than 13.3 years to meet the requirement which means 14 years.

Answer is D.
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Re: A certain logistics company’s fleet consisted of 120 Type A trucks at  [#permalink]

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New post 26 Jun 2018, 04:25
Let
A = Type A trucks
B = Type B trucks
Y = Time in year after 1970

Then
A = 120 -3Y
B = 6Y

We want to know when A is less than 50% of the fleet. If A was half the fleet, B would also be half, and A would equal B. Thus, A is less than half when A < B.

120 - 3Y < 6Y
→ 120 < 9Y
→ 13.33 < Y

after 14 years

D
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Re: A certain logistics company’s fleet consisted of 120 Type A trucks at &nbs [#permalink] 26 Jun 2018, 04:25
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