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Skier Lindsey Vonn completes a straight 300-meter downhill

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Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 26 Aug 2011, 13:26
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Difficulty:

  75% (hard)

Question Stats:

66% (03:21) correct 34% (03:29) wrong based on 187 sessions

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Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

Spoiler: :: Solution
Time taken for the ride down = 300/(x+10)
Time taken for the ride up = 300/(x-8)

Now 300/(x+10) +135 = 300/(x-8)
20/(x+10) +9 = 20/(x-8)

x=10 clearly satisfies this

speed for the downhill journey = x+10 = 20 m/sec

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Re: Skier Lindsey Vonn  [#permalink]

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New post 31 Dec 2013, 11:20
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MBAhereIcome wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

Spoiler: :: Solution
Time taken for the ride down = 300/(x+10)
Time taken for the ride up = 300/(x-8)

Now 300/(x+10) +135 = 300/(x-8)
20/(x+10) +9 = 20/(x-8)

x=10 clearly satisfies this

speed for the downhill journey = x+10 = 20 m/sec


Backsolving is the preferred approach for this one

300/(x-8) - 300/(x+10) = 135

We are asked for the value of x+10

So we can start with C = X+10 = 20
So x = 10
300/2 - 300 / 20 = 135

It satisfies

Hence answer is C

Hope it helps!
Cheers!
J :)
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Re: Skier Lindsey Vonn  [#permalink]

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New post 26 Aug 2011, 22:35
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x+10?

300/(x+10) + 135 = 300/(x-8)

solving this we get x^2+2x-120 =0

=> x = 10

=> x+10 = 20.

Answer is C.
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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 30 Jul 2015, 19:31
jlgdr wrote:
MBAhereIcome wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

Spoiler: :: Solution
Time taken for the ride down = 300/(x+10)
Time taken for the ride up = 300/(x-8)

Now 300/(x+10) +135 = 300/(x-8)
20/(x+10) +9 = 20/(x-8)

x=10 clearly satisfies this

speed for the downhill journey = x+10 = 20 m/sec


Backsolving is the preferred approach for this one

300/(x-8) - 300/(x+10) = 135

We are asked for the value of x+10

So we can start with C = X+10 = 20
So x = 10
300/2 - 300 / 20 = 135

It satisfies

Hence answer is C

Hope it helps!
Cheers!
J :)



This is way more simpler than the solution offered by MGMAT! Thanks
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Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 01 Dec 2018, 05:49
Spidy001 wrote:
x+10?

300/(x+10) + 135 = 300/(x-8)

solving this we get x^2+2x-120 =0

=> x = 10

=> x+10 = 20.

Answer is C.



can anyone explain step by step how from this 300/(x+10) + 135 = 300/(x-8) we got this x^2+2x-120 =0
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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 03 Dec 2018, 01:33
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dave13, this is a great example of patience ( Delaying the simplification by opening brackets)

\(\frac{300}{t} = (x + 10)\) ... this is the distance time speed relation from her run downhill.

\(\frac{300}{(t+135)} = (x - 8)\) ... this second relation comes from her chairlift uphill.

We need to solve for x and hence from the above two equations we need to eliminate t.

Let's substitute the value of \(t = \frac{300}{(x+10)}\) in the second equation. Yes, it looks scary but things will get better now :-)

\(\frac{300*(x + 10)}{(300+(x+10)*135)} = x - 8\)
\(300*(x+10) = 300*(x-8) + 135*(x+10)*(x-8)\) Let's not start opening brackets yet as things will reduce.
\(60*(x+10) = 60*(x-8) + 27*(x+10)*(x-8)\) 5 is common in all numbers hence it has been cancelled out
\(20*(x+10) = 20*(x-8) + 9*(x+10)*(x-8)\) Now 3 is common'

Now we are ready to open brackets and simplify

\(20x + 200 = 20x - 160 + 9(x+10)(x-8)\)
\(360 = 9*(x+10)(x-8)\) 9 can still cancel from both sides
\(40 = x^2 + 2x - 80\) Finally opening the brackets
\(x^2 + 2x - 120 = 0\)

Easy quadratic to solve now :-)

\(x^2 + 12x - 10x - 120 = 0\)
\(x(x+12)-10(x+12) = 0\)
\((x-10)*(x+12) = 0\)

Hence x = 10 discarding the negative root

Hence the speed downhill was ( x + 10) or 20 /s

Key takeaway here is : smart calculation saves a lot of time

Best,
Gladi

dave13 wrote:
can anyone explain step by step how from this 300/(x+10) + 135 = 300/(x-8) we got this x^2+2x-120 =0

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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 06 Dec 2018, 02:40
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dave13 wrote:
Spidy001 wrote:
x+10?

300/(x+10) + 135 = 300/(x-8)

solving this we get x^2+2x-120 =0

=> x = 10

=> x+10 = 20.

Answer is C.



can anyone explain step by step how from this 300/(x+10) + 135 = 300/(x-8) we got this x^2+2x-120 =0


I also got stuck with the algebra....its not difficult but it is LONG.
my takeaway: if you are sure of the basic setup of the equation AND it looks that it will get ugly, then it is faster and CLEANER to plug the answer choices than to solve for roots.

this is something i have to remind myself each time i start a question....to get into the habit.
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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 21 Mar 2019, 08:02
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MBAhereIcome wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

Spoiler: :: Solution
Time taken for the ride down = 300/(x+10)
Time taken for the ride up = 300/(x-8)

Now 300/(x+10) +135 = 300/(x-8)
20/(x+10) +9 = 20/(x-8)

x=10 clearly satisfies this

speed for the downhill journey = x+10 = 20 m/sec


The ride up the mountain took 135 seconds longer than her run down the mountain
Start with a word equation: (time going UP mountain) = (time going DOWN mountain) + 135
time = distance/speed
We can now write: 300/(x - 8) = 300/(x + 10) + 135
Multiply both sides by (x - 8) to get: 300 = 300(x - 8)/(x + 10) + 135(x - 8)
Multiply both sides by (x + 10) to get: 300(x + 10) = 300(x - 8) + 135(x - 8)(x + 10)
Divide both sides by 5 to get: 60(x + 10) = 60(x - 8) + 27(x - 8)(x + 10)
Divide both sides by 3 to get: 20(x + 10) = 20(x - 8) + 9(x - 8)(x + 10)
Expand both sides to get: 20x + 200 = 20x - 160 + 9x² + 18x - 720
Rearrange and simplify to get: 9x² + 18x - 1080 = 0
Divide both sides by 9 to get: x² + 2x - 120 = 0
Factor to get: (x + 12)(x - 10) = 0
So, EITHER x = -12 OR x = 10
Since x can't be the speed, we know that x = 10

What was her average speed, in meters per second, during her downhill run?
Her downhill speed = x + 10
Since x = 10, her downhill speed = 10 + 10 = 20

Answer: C

Cheers,
Brent

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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 23 Mar 2019, 23:57
The conventional explanation involving quadratic equation is quite long. Fortunately, in this particular problem, there is a much quicker way using the answer choices.
Lindsay's uphill speed is 18 m/s less than her downhill speed so the latter must be more than 18 m/s otherwise her uphill speed will be negative which is not possible. So options (A) and (B) are out. Let's consider option (C):
Her downhill speed is 20 m/s; therefore, her uphill speed is 2 m/s and the time she takes for her downhill run is 300/20=15 seconds. The time taken for her uphill trip is thus (15+135)=150 seconds. So we see that option (C) checks out because she can cover travelling 150 seconds at 2 m/s she covers the 300 meters of her ski run.
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Re: Skier Lindsey Vonn completes a straight 300-meter downhill  [#permalink]

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New post 24 Mar 2019, 18:19
MBAhereIcome wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30



We can let the time going down = 300/(x +10) and the time going up as 300/(x - 8), thus:

300/(x +10) + 135 = 300/(x - 8)

Multiplying by (x + 10)(x - 8), we have:

300(x - 8) + 135(x + 10)(x - 8) = 300(x + 10)

20(x - 8) + 9(x + 10)(x - 8) = 20(x + 10)

20x - 160 + 9(x^2 + 2x - 80) = 20x + 200

20x - 160 + 9x^2 + 18x - 720 = 20x + 200

9x^2 + 18x - 1,080 = 0

x^2 + 2x -120 = 0

(x + 12)(x - 10) = 0

x = -12 or x = 10

Since x can’t be negative, x must be 10. Therefore, the speed for the downhill run was 10 + 10 = 20 meters per second.

Answer: C
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Re: Skier Lindsey Vonn completes a straight 300-meter downhill   [#permalink] 24 Mar 2019, 18:19
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