Problem Solving

For the question below, I assume the formula is Distance = Rate X Time, however the solution doesn't seem to be using it... can any experts explain why:

After driving to a riverfront parking lot, Bob plans to
run south along the river, turn around, and return to
the parking lot, running north along the same path.
After running 3.25 miles south, he decides to run for
only 50 minutes more. If Bob runs at a constant rate
of 8 minutes per mile, how many miles farther south
can he run and still be able to return to the parking lot
in 50 minutes?
(A) 1.5
(B) 2.25
(C) 3.0
(D) 3.25
(E) 4.75
SOLUTION

After running 3.25 miles south, Bob has been
running for (3.25)(8) = 26minutes. Thus, if t is the number of additional minutes
that Bob can run south before turning around,
then the number of minutes that Bob will run
north, after turning around, will be t + 26. Since
Bob will be running a total of 50 minutes after
the initial 26 minutes of running, it follows that
t + (t + 26) = 50, or t = 12. Therefore, Bob can run
south an additional 12 / 8 = 1.5 miles before turning around.
The correct answer is A.


My first concern is the calculation of (3.25)(8) = 26minutes. This is essentially saying Distance x Rate = time. That does not match Distance = Rate x Time??

My second conceern is the calculation of 12 / 8 = 1.5 miles. This is essentially saying Time / Rate = Distance. However, this does not match the Distance = Rate x time forumula?

Is there another formula? Or is my formula incorrect?? I'm confused

LiquidFireAK wrote:For the question below, I assume the formula is Distance = Rate X Time, however the solution doesn't seem to be using it... can any experts explain why:

After driving to a riverfront parking lot, Bob plans to
run south along the river, turn around, and return to
the parking lot, running north along the same path.
After running 3.25 miles south, he decides to run for
only 50 minutes more. If Bob runs at a constant rate
of 8 minutes per mile, how many miles farther south
can he run and still be able to return to the parking lot
in 50 minutes?
(A) 1.5
(B) 2.25
(C) 3.0
(D) 3.25
(E) 4.75

In the solution below, the units in red CANCEL OUT.

Time to travel 3.25 miles at a rate of 8 minutes per mile = 3.25 miles * (8 minutes)/(1 mile) = (3.25)(8) minutes = 26 minutes.

Since Bob travels for 50 more minutes -- for a total of 76 minutes -- the total distance traveled = 76 minutes * (1 mile/8 minutes) = 76/8 miles = 19/2 miles.

Since half the total distance is traveled in each direction, the total distance traveled south = (19/2) * (1/2) = 19/4 miles.

Since 3.25 miles south have already been traveled, the additional distance traveled south = 19/4 - 3.25 = 19/4 - 13/4 = 6/4 = 1.5 miles.

The correct answer is A.

To use r*t = d -- where r is in terms of MILES PER MINUTE -- we could proceed as follows:

d = 3.25 miles.
r = 1 mile per 8 minutes = 1/8 mile per minute.
Since t = d/r, we get:
t = (3.25)/(1/8) = (3.25)(8) = 26 minutes.

Since Bob travels for an additional 50 minutes -- for a total of 76 minutes -- at a rate of 1/8 mile per minute, we get:
Total distance = r*t = 1/8 * 76 = 19/2 miles.

From here, we could proceed as in my solution above.

Speed of Bod = 1/8 mile per min

S = D/T

Time needed to travel 3.25 miles = 3.25/1/8 ==> 8 * 3.25 = 26

Total Distance in another 50 minutes = 1/8 * 50 = 6.25

So distance per side = (6.25 - 3.25)/2 = 1.5

Answer [spoiler]{A}[/spoiler]

LiquidFireAK wrote:
My first concern is the calculation of (3.25)(8) = 26minutes. This is essentially saying Distance x Rate = time. That does not match Distance = Rate x Time??

8 minutes = 1 mile

1 minute = 1/8 mile

So, Speed of Bod = 1/8 mile per min

S = D/T

Time needed to travel 3.25 miles = 3.25/1/8 ==> 8 * 3.25 = 26

LiquidFireAK wrote: My second conceern is the calculation of 12 / 8 = 1.5 miles. This is essentially saying Time / Rate = Distance. However, this does not match the Distance = Rate x time forumula?

Is there another formula? Or is my formula incorrect?? I'm confused

8 minutes = 1 mile

1 minute = 1/8 mile

So, Speed of Bod = 1/8 mile per min

t = 12

S = D/T

S * T = D

1/8 * 12 = D

D = 1.5

I hope both of your confusions are gone now

GMATGuruNY wrote:

LiquidFireAK wrote:For the question below, I assume the formula is Distance = Rate X Time, however the solution doesn't seem to be using it... can any experts explain why:

After driving to a riverfront parking lot, Bob plans to
run south along the river, turn around, and return to
the parking lot, running north along the same path.
After running 3.25 miles south, he decides to run for
only 50 minutes more. If Bob runs at a constant rate
of 8 minutes per mile, how many miles farther south
can he run and still be able to return to the parking lot
in 50 minutes?
(A) 1.5
(B) 2.25
(C) 3.0
(D) 3.25
(E) 4.75

In the solution below, the units in red CANCEL OUT.

Time to travel 3.25 miles at a rate of 8 minutes per mile = 3.25 miles * (8 minutes)/(1 mile) = (3.25)(8) minutes = 26 minutes.

Since Bob travels for 50 more minutes -- for a total of 76 minutes -- the total distance traveled = 76 minutes * (1 mile/8 minutes) = 76/8 miles = 19/2 miles.

Since half the total distance is traveled in each direction, the total distance traveled south = (19/2) * (1/2) = 19/4 miles.

Since 3.25 miles south have already been traveled, the additional distance traveled south = 19/4 - 3.25 = 19/4 - 13/4 = 6/4 = 1.5 miles.

The correct answer is A.

To use r*t = d -- where r is in terms of MILES PER MINUTE -- we could proceed as follows:

d = 3.25 miles.
r = 1 mile per 8 minutes = 1/8 mile per minute.
Since t = d/r, we get:
t = (3.25)/(1/8) = (3.25)(8) = 26 minutes.

Since Bob travels for an additional 50 minutes -- for a total of 76 minutes -- at a rate of 1/8 mile per minute, we get:
Total distance = r*t = 1/8 * 76 = 19/2 miles.

From here, we could proceed as in my solution above.

If the inverted fractions are confusing, try this:

The rate is given as 8 minutes per mile instead of miles per minute (to catch people out), so it needs to be inverted to speed = miles per minute

So Speed = 1/8 = 0.125

Now you can use your formula, distance = speed x time, etc...

(Yes normally rate is the other way round, but they were clever to use the word rate instead of speed. Speed is never the other way round.)

I hope this helps, if not too late!