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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]

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New post 27 Dec 2012, 10:25
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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
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Re: The number of straight line miles traveled downriver in one [#permalink]

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New post 27 Dec 2012, 10:56
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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]

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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]

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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.
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Re: The number of straight line miles traveled downriver in one [#permalink]

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New post 08 Feb 2018, 16:48
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 first sentence of the problem really is saying that the speed of the boat is approximated by 3n + 4, where n is the fuel consumption in units/hr. Thus we are saying that at a fuel rate of 7 units/hr,

x/k = 3(7) + 4

x/k = 25

x = 25k

At a fuel rate of 7 + 5 = 12 units/hr,

1.4x/(k - 1) = 3(12) + 4

1.4x/(k - 1) = 40

Since x = 25k, we have:

1.4(25k)/(k - 1) = 40

35k = 40(k - 1)

35k = 40k - 40

40 = 5k

8 = k

Since we know now k = 8, let’s determine m, the number of miles the boat travels in 8 hours at a fuel rate of 10 units/hr:

m/8 = 3(10) + 4

m/8 = 34

m = 272

Answer: E
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Re: The number of straight line miles traveled downriver in one   [#permalink] 08 Feb 2018, 16:48
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