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The number of straight line miles traveled downriver in one

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The number of straight line miles traveled downriver in one [#permalink] New post 27 Dec 2012, 09:25
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Question Stats:

53% (03:30) correct 46% (03:47) wrong based on 76 sessions
The number of straight line miles traveled downriver in one hour by Lucy's boat is approximated within a linear range by 3n + 4, where n represents her fuel consumption in units/hr. Suppose that traveling x miles requires k hours at a fuel rate of 7 units/hr, but that increasing her fuel consumption by 5 units/hr would allow her to travel 40% further in 1 fewer hour. How far would she travel in k hours at a fuel rate of 10 units/hr?

A. 8
B. 200
C. 225
D. 236
E. 272
[Reveal] Spoiler: OA
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Re: The number of straight line miles traveled downriver in one [#permalink] New post 28 Dec 2012, 01:15
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danzig wrote:
The number of straight line miles traveled downriver in one hour by Lucy's boat is approximated within a linear range by 3n + 4, where n represents her fuel consumption in units/hr. Suppose that traveling x miles requires k hours at a fuel rate of 7 units/hr, but that increasing her fuel consumption by 5 units/hr would allow her to travel 40% further in 1 fewer hour. How far would she travel in k hours at a fuel rate of 10 units/hr?

A. 8
B. 200
C. 225
D. 236
E. 272


The wording of the question is a little complicated so take it one step at a time.

"The number of straight line miles traveled downriver in one hour by Lucy's boat "

miles driven in one hour gives you speed

"is approximated within a linear range by 3n + 4, where n represents her fuel consumption in units/hr."

So speed = 3n+4 where n is the fuel consumption

"Suppose that traveling x miles requires k hours at a fuel rate of 7 units/hr,"

Speed = x/k = 3*7 + 4 = 25
Distance traveled = 25k

"but that increasing her fuel consumption by 5 units/hr would allow her to travel 40% further in 1 fewer hour. "

Speed = 3*12 + 4 = 40
Distance traveled = 40(k-1) = (7/5)*25k (You multiply by 7/5 because 40% extra distance is traveled in this case)
k = 8

"How far would she travel in k hours at a fuel rate of 10 units/hr?"

Speed = 3*10 + 4 = 34
Distance traveled = 34*k = 34*8 = 272 miles
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Re: The number of straight line miles traveled downriver in one [#permalink] New post 27 Dec 2012, 09:56
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We need to find: traveled miles k hours and 10 units of fuel.
From the linear equation: Answer = k*(3*10+4) = 34k

Data:

x miles in k hours at 7 units/hr: equation: x =k*(3*7+4) = 25k
=> x=25k or k = x/25 (equation 1)

40% extra miles = 1.4 x
one hour less = k-1 hours
5 units/hr extra fuel = 7+5 = 12
Equation 1.4x=(k-1)*(3*12+4)
=> 14x = (k-1)*(40)*10
=> 14x= 400k-400
=> 14x = 16x - 400 (substitute k= x/25)
=> 2x = 400
=> x = 200
=> k = 200/25 = 8 (substitute back from equation 1: k=x/25)

Required Answer = 34k = 34*8 = 272


While solving the equation, you can either substitute x or k, depending on the numbers in hand and involved calculation.
I substituted k = x/25, because my numbers were becoming large and 400 was easily divisible by 25.
It was quicker to solve x first than k.
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Re: The number of straight line miles traveled downriver in one [#permalink] New post 30 Jan 2013, 03:26
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danzig wrote:
The number of straight line miles traveled downriver in one hour by Lucy's boat is approximated within a linear range by 3n + 4, where n represents her fuel consumption in units/hr. Suppose that traveling x miles requires k hours at a fuel rate of 7 units/hr, but that increasing her fuel consumption by 5 units/hr would allow her to travel 40% further in 1 fewer hour. How far would she travel in k hours at a fuel rate of 10 units/hr?

A. 8
B. 200
C. 225
D. 236
E. 272



The total distance traveled is K*(3n+4) and we know that n is 10 u/hr. So the total distance traveled is 34K or 2*17*K. Only 272 is divisible by 17. So, answer is E.

But here I got lucky as no other answer is divisible by 17.
Re: The number of straight line miles traveled downriver in one   [#permalink] 30 Jan 2013, 03:26
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