Problem Solving

benjiboo 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?

A) (x+y+z)/3
B) (3xyz)/(xy+yz+zx)
C) (xyz)/(x+y+z)
D) (xy+yz+zx)/(x+y+z)
E) [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) = B

Cheers,
Brent

benjiboo 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?

A) (x+y+z)/3
B) (3xyz)/(xy+yz+zx)
C) (xyz)/(x+y+z)
D) (xy+yz+zx)/(x+y+z)
E) [3(x+y+z)]/(xyz)

[spoiler]Ans: B[/spoiler]

This is a tough one... GMAT experts want to give this one a try?

Hi benjiboo,

This is not a tough one. Many experts have put forth their approaches. Great ones. Here's mine.

Whenever the distance is the same and there are two or more speeds and we are asked to find out the average speed, do not take arithmetic mean of the speeds.

What you should do is the following.

1. Take the reciprocals of the speeds.
2. Take the arithmetic mean of the reciprocals.
3. Take the reciprocal of the arithmetic mean obtained in step 2.

Coming to the question...

We know that the distance is the same in each of the three occasions, but the speeds are different, so we can apply the above approach.

1. Take the reciprocals of the speeds => 1/x, 1/y, and 1/z
2. Take the arithmetic mean of the reciprocals.

Arithmetic mean = 1/3[1/x + 1/y + 1/z] = (xy + yz + xz)/3xyz

3. Take the reciprocal of the arithmetic mean obtained in step 2.

Average speed = Reciprocal of (xy + yz + xz)/3xyz = 3xyz / (xy + yz + xz)

The correct answer: B

Hope this helps!

Let us a see a simple question.

If Jack drives from city A to B at a uniform speed of 30 miles per and returns to A from B at a uniform speed of 40 miles per, what is his average speed for the entire journey?

Needless to state that the answer would not be the arithmetic mean of 30 and 40 = 35 miles per hour.

Apply the above steps.

1. Take the reciprocals of the speeds: 1/30 and 1/40
2. Take the arithmetic mean of the reciprocals: 1/2[1/30 + 1/40] = 1/2[7/120] = 7/240
3. Take the reciprocal of the arithmetic mean obtained in step 2: 240/7 = 34.3 miles per hour

Note that the correct answer 34.3 is less than the incorrect answer 35.

34.3 is called the harmonic mean, while 35 is the arithmetic mean. Note that harmonic mean is always less than arithmetic mean.

Hope this helps!

Relevant book: Manhattan Review GMAT Word Problems Guide

-Jay
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benjiboo 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?

A) (x+y+z)/3
B) (3xyz)/(xy+yz+zx)
C) (xyz)/(x+y+z)
D) (xy+yz+zx)/(x+y+z)
E) [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)

Answer: B