Average Speed Tough DS

[late4thing]

A cyclist rode up and down a mountain along the same route each way. What was his average speed for the full trip?

(1) His uphill speed averaged 22 kilometers per hour, and his downhill speed averaged 48 kilometers per hour.

(2) The route is 16 kilometers each way.


Source: VeritasPrep

[Brent@GMATPrepNow]

A cyclist rode up and down a mountain along the same route each way. What was his average speed for the full trip?

(1) His uphill speed averaged 22 kilometers per hour, and his downhill speed averaged 48 kilometers per hour.

(2) The route is 16 kilometers each way.


Source: VeritasPrep

Target question: What was the average speed for the full trip?

Statement 1: His uphill speed averaged 22 kilometers per hour, and his downhill speed averaged 48 kilometers per hour.
Average speed = (TOTAL distance)/(TOTAL time)
Let D = distance traveled in each direction.

Since time = distance/speed, we can say that:
Time going UPhill = D/22
Time going DOWNhill = D/48

So, Average speed = (TOTAL distance)/(TOTAL time)
= (D + D)/(D/22 + D/48)

IMPORTANT: at this point, we not actually solve this equation. We need only determine whether or not it yields a definitive answer to the target question.

So, (D + D)/(D/22 + D/48) = (2D)/[50D/(22)(48)]
= (2D)[(22)(48)/(50D)]
= (2)(22)(48)/50
STOP!
At this point, we can see that we no longer have any variables.
We COULD evaluate (2)(22)(48)/50, but it's unnecessary to do since we are already certain that we have a unique answer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The route is 16 kilometers each way.
This isn't enough information. Consider these two cases:
Case a: The average uphill speed is 5 kmh and the average downhill speed is 5 kmh, which means the average OVERALL speed is 5 kmh
Case b: The average uphill speed is 10 kmh and the average downhill speed is 10 kmh, which means the average OVERALL speed is 10 kmh
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

[GMATGuruNY]

A cyclist rode up and down a mountain along the same route each way. What was his average speed for the full trip?

(1) His uphill speed averaged 22 kilometers per hour, and his downhill speed averaged 48 kilometers per hour.

(2) The route is 16 kilometers each way.

RULE:
If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS THAT ARE RELATIVELY CLOSE -- 20mph and 40mph, 30mph and 45mph -- then the AVERAGE SPEED for the entire trip will be equal to A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.

Statement 1:
Since the average of 22 and 48 = (22+48)/2 = 35, the average speed for the full trip is a little less than 35kph.
SUFFICIENT.

Statement 2:
Since the time and speed in each direction are unknown, INSUFFICIENT.

The correct answer is A.

[MartyMurray]

A cyclist rode up and down a mountain along the same route each way. What was his average speed for the full trip?

(1) His uphill speed averaged 22 kilometers per hour, and his downhill speed averaged 48 kilometers per hour.

(2) The route is 16 kilometers each way.

Source: VeritasPrep

Statement 1:

When you travel the same distance 2 times at 2 particular rates, the average rate for the total trip is the same for routes of any distance.

In other words, if you travel 10 kilometers at 22 kph, and then return 10 kilometers at 48 kph, your average rate will be the same as it would be were you to travel 20 kilometers at 22 kph and return 20 kilometers at 48 kph.

So if you know the rates for the two directions of a round trip you can always calculate the average rate of the round trip without any additional information.

If why this is the case it not clear to you, see Brent's work above or try calculating average rates using various distances for the round trip.

Sufficient.

Statement 2:

Given just the distance and no other information, one cannot calculate rates or average rates.

Insufficient.

The correct answer is A.

[Brent@GMATPrepNow]

RULE:
If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS, the AVERAGE SPEED for the entire trip will be equal to A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.

This is a great rule, but just in case someone wants to use it to estimate the answer to a Problem Solving question, we might want to eliminate "A LITTLE" to get:

If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS, the AVERAGE SPEED for the entire trip will be LESS THAN THE AVERAGE OF THE TWO SPEEDS.

For example, let's say the distance UP the hill is 1 inch (and the distance DOWN is also 1 inch).
If the cyclist's uphill speed is 1 inch per year, and the cyclist's downhill speed is 1000 miles per hour, then the ENTIRE 2-inch trip will take about 1 year. So the average speed is 2 inches/year

If we find the average of the two speeds (1 inch per year AND 1000 miles per hour), we get about 500 miles per hour.

As we can see, 2 inches per year is A LOT less than 500 mph.

Cheers,
Brent

[GMATGuruNY]

RULE:
If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS, the AVERAGE SPEED for the entire trip will be equal to A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.

This is a great rule, but just in case someone wants to use it to estimate the answer to a Problem Solving question, we might want to eliminate "A LITTLE" to get:

If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS, the AVERAGE SPEED for the entire trip will be LESS THAN THE AVERAGE OF THE TWO SPEEDS.

For example, let's say the distance UP the hill is 1 inch (and the distance DOWN is also 1 inch).
If the cyclist's uphill speed is 1 inch per year, and the cyclist's downhill speed is 1000 miles per hour, then the ENTIRE 2-inch trip will take about 1 year. So the average speed is 2 inches/year

If we find the average of the two speeds (1 inch per year AND 1000 miles per hour), we get about 500 miles per hour.

As we can see, 2 inches per year is A LOT less than 500 mph.

Cheers,
Brent

True:
If the two speeds are very far apart, then the average speed for the whole trip will be much closer to the slower of the two speeds.
That said:
Official problems with the same distance traveled at two different speeds typically involve speeds that are relatively close, with the result that the average speed for the whole trip is not much less than the average of the two speeds.

[Brent@GMATPrepNow]

True:
If the two speeds are very far apart, then the average speed for the whole trip will be much closer to the slower of the two speeds.
That said:
Official problems with the same distance traveled at two different speeds typically involve speeds that are relatively close, with the result that the average speed for the whole trip is not much less than the average of the two speeds.

Agreed!

However, in my paranoid mind, I envision GMAC staff reviewing these posts and looking for ways to creat questions that contradict any statements we make

Cheers,
Brent

[GMATGuruNY]

True:
If the two speeds are very far apart, then the average speed for the whole trip will be much closer to the slower of the two speeds.
That said:
Official problems with the same distance traveled at two different speeds typically involve speeds that are relatively close, with the result that the average speed for the whole trip is not much less than the average of the two speeds.

Agreed!

However, in my paranoid mind, I envision GMAC staff reviewing these posts and looking for ways to creat questions that contradict any statements we make

Cheers,
Brent

So you can sleep at night, I've amended my post as follows:

RULE:
If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS THAT ARE RELATIVELY CLOSE -- 20mph and 40mph, 30mph and 45mph -- then the AVERAGE SPEED for the entire trip will be equal to A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.

[Brent@GMATPrepNow]

So you can sleep at night, I've amended my post as follows:

RULE:
If the SAME DISTANCE is traveled at TWO DIFFERENT SPEEDS THAT ARE RELATIVELY CLOSE -- 20mph and 40mph, 30mph and 45mph -- then the AVERAGE SPEED for the entire trip will be equal to A LITTLE LESS THAN THE AVERAGE OF THE TWO SPEEDS.

That (and some warm milk) will go a long way.
Of course, it doesn't mean that they aren't watching our every move!!