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# Jacob drove from Town A to Town B at an average rate of x miles per ho

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Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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14 Feb 2016, 00:50
24
00:00

Difficulty:

55% (hard)

Question Stats:

67% (02:04) correct 33% (02:01) wrong based on 527 sessions

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Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$
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Posts: 1877
Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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14 Feb 2016, 02:54
4
4
Nez wrote:
Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$

The best way to go about in these question is to find out the total distance covered and the total time taken.

Assume the distance for one trip = d
Total distance covered = 3d.
Time take for 1st trip = d/x
Time take for 2nd trip = d/y
Time take for 3rd trip = d/z

Average speed = total distance/ total time

Average speed = 3d/ (d/x + d/y + d/z) = 3 / (1/x + 1/y + 1/z) = 3xyz / (yz + zx + xy)
Option B
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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14 Feb 2016, 07:41
2
3
Nez wrote:
Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$

Average speed = (total distance traveled)/(total travel time)
= (total distance)/(time of 1st journey + time of 2nd journey + time of 3rd journey)

Let d = the distance between Town A and Town B
So, total distance traveled = 3d

Time = distance/speed
time of 1st journey = d/x
time of 2nd journey = d/y
time of 3rd journey = d/z

Total time = d/x + d/y + dz
To simplify, rewrite with common denominator: dyz/xyz + dxz/xyz + dxy/xyz
So, total time = (dyz + dxz + dxy)/xyz

Average speed = (total distance)/(total time)
= 3d/[(dyz + dxz + dxy)/xyz]
= (3dxyz)/(dyz + dxz + dxy)
Divide top and bottom by d to get: (3xyz)/(yz + xz + xy)

Cheers,
Brent
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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14 Feb 2016, 22:18
2
Nez wrote:
Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$

For this and other formulas, check: http://www.veritasprep.com/blog/2015/02 ... -the-gmat/
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Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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02 Apr 2017, 22:05
Nezdem wrote:
Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$

3/[(1/x)+(1/y)+(1/z)]➡
3/[(xy+yz+zx)/xyz]➡
3xyz/(xy+yz+zx)
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GMAT 1: 650 Q49 V30
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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28 Apr 2017, 14:31
pretty time consuming...
we have the same distance.
say D.
time taken for first leg: D/x
time taken for second leg: D/y
time taken for third leg: D/z
total time: D/x + D/y + D/z => after re-arranging - we get: D(yz+xz+xy)/xyz
total distance: 3D.
3D divide by D(yz+xz+xy)/xyz
we get:
3xyz/(yz+xz+xy)
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Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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Updated on: 03 Jul 2017, 07:43
Also use some simple values. x=1, y=2, z=3. Let distance d = 6.
$$Avg speed = \frac{total distance}{total time}$$
= $$\frac{6*3}{(6+3+2)} = \frac{18}{11}$$

Plugging in values for x,y,z and d

1) $$\frac{x+y+z}{3} = \frac{6}{3} = 2$$......... wrong
2) $$\frac{3xyz}{(xy+yz+zx)} = \frac{18}{11}$$...............correct

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Originally posted by sasyaharry on 02 Jul 2017, 14:10.
Last edited by sasyaharry on 03 Jul 2017, 07:43, edited 1 time in total.
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Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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02 Jul 2017, 15:48
sasyaharry wrote:
Also use some simple values. x=1, y=2, z=3. Let distance d = 6.
$$Avg speed = \frac{total distance}{total time}$$
= $$\frac{6*3}{(6+3+2)} = \frac{9}{5}$$

Plugging in values for x,y,z and d

1) $$\frac{x+y+z}{3} = \frac{6}{3} = 2$$......... wrong
2) $$\frac{3xyz}{(xy+yz+zx)} = \frac{9}{5}$$...............correct

Your fraction should be $$\frac{18}{11}$$, not $$\frac{9}{5}$$. With your numbers for d, x, y, and z, answer B does indeed yield the correct answer. Cheers!
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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03 Jul 2017, 07:43
genxer123 wrote:
sasyaharry wrote:
Also use some simple values. x=1, y=2, z=3. Let distance d = 6.
$$Avg speed = \frac{total distance}{total time}$$
= $$\frac{6*3}{(6+3+2)} = \frac{9}{5}$$

Plugging in values for x,y,z and d

1) $$\frac{x+y+z}{3} = \frac{6}{3} = 2$$......... wrong
2) $$\frac{3xyz}{(xy+yz+zx)} = \frac{9}{5}$$...............correct

Your fraction should be $$\frac{18}{11}$$, not $$\frac{9}{5}$$. With your numbers for d, x, y, and z, answer B does indeed yield the correct answer. Cheers!

gracias. Corrected.
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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12 Dec 2017, 17:15
Ekland wrote:
Jacob drove from Town A to Town B at an average rate of x miles per hour, then returned along the same route at y miles per hour. If he then drove back to Town B at z miles per hour along the same route, what was Jacob’s average rate of speed for the entire trip, in miles per hour?

$$\frac{(x+y+z)}{3}$$

$$\frac{3xyz}{(xy+yz+zx)}$$

$$\frac{xyz}{(x+y+z)}$$

$$\frac{(xy+yz+zx)}{(x+y+z)}$$

$$\frac{3(x+y+z)}{xyz}$$

To solve, we can use the formula for average rate:

average rate = total distance/total time

We can let distance from Town A to Town B (or vice versa) = d, and since he went from A to B, then from B to A, and then from A to B, his total distance traveled was 3d. Recall that time = distance/rate, so the time to get from Town A to Town B = d/x, the time it takes to get from Town B back to Town A = d/y, and the time it takes to go back to Town B = d/z. Thus:

average rate = 3d/(d/x + d/y + d/z)

To combine the three fractions in the denominator, we use the common denominator xyz:

average rate = 3d/(yzd/xyz + xzd/xyx + xyd/xyz)

average rate = 3d/[(yzd + xzd + xyd)/xyz]

average rate = 3d * xyz/(yzd + xzd + xyd)

average rate = 3d * xyz/[d(yz + xz + xy)]

The ds cancel and we are left with:

3xyz/(yz + xz + xy)

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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho  [#permalink]

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25 Dec 2018, 14:52
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Re: Jacob drove from Town A to Town B at an average rate of x miles per ho   [#permalink] 25 Dec 2018, 14:52
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