Problem Solving

Brent@GMATPrepNow wrote:Rhonda runs at an average speed of 12 kilometers per hour, and she bicycles at an average speed of 30 kilometers per hour. When she bicycles to work, her travel time is 2.25 minutes less than when she runs to work. The distance to work, in kilometers, is

A) 27/40
B) 3/4
C) 7/8
D) 11/12
E) 6/5

Source: GMAT Prep Now
Difficulty level: 700

Rhonda's running speed = 12km/hour or 12km/60minutes or 1km/5minutes.
Biking speed = 30km/hour = 30km/60minutes = 1km/2minutes

If it takes her t minutes to run to work, it would take her t - 2.25 minutes, or t - 9/4 to bike.

Running she covers (1/5)t kilometers. Biking she covers (1/2)(t-9/4) kilometers. So (1/5)t = (1/2)(t-9/4)
(1/5)t = (1/2)t - 9/8
(1/5)t - (1/2)t = - 9/8
(-3/10)t = -9/8
t = 90/24 = 15/4

If she covers (1/5)t running, she'll cover (1/5)* (15/4) = 15/20 = 3/4 km. The answer is B

Brent@GMATPrepNow wrote:Rhonda runs at an average speed of 12 kilometers per hour, and she bicycles at an average speed of 30 kilometers per hour. When she bicycles to work, her travel time is 2.25 minutes less than when she runs to work. The distance to work, in kilometers, is

A) 27/40
B) 3/4
C) 7/8
D) 11/12
E) 6/5

We can PLUG IN THE ANSWERS, which represent the distance.
2.25 minutes = 9/4 minutes = (9/4)/60 hours = 3/80 hours.
When the correct answer choice is plugged in, the difference between the two times will be 3/80 hours.

B: 3/4
At a speed of 12 kph, the time to bike 3/4 kilometers = d/r = (3/4)/12 = 1/16 hours.
At a speed of 30 kph, the time to walk 3/4 kilometers = d/r = (3/4)/30 = 1/40 hours.
Time difference = 1/16 - 1/40 = 5/80 - 2/80 = 3/80 hours.
Success!

The correct answer is B.

Let's use D = RT.

Distance, running:
D = distance
R = 12 km/hr = 1/5 km/min
T = m

Distance, biking:
D = distance
R = 30 km/hr = 1/2 km/min
T = m - 2.25

Since the two distances are the same, D = D and RT = RT, giving us

(1/5)m = (1/2)(m - 2.25)

(1/5)m = (1/2)m - 1.125

1.125 = (3/10)m

m = 3.75

From there, D = (1/5)m = (1/5) * 3.75 = 0.75, or B.

Another approach that's quicker to solve but harder to conceive in the first place:

Suppose Rhonda is cloned, and the two Rhondas head to work, one running, one biking. (What could be less glamorous than cloning yourself only to head to work! But hey ...) As before, we'll put their rates in km/min, so the runner = 1/5 km/min and the biker = 1/2 km/min.

Let's say it takes m minutes for biking Rhonda to get to work. After m minutes, running Rhonda has traveled (1/5)m and biking Rhonda has traveled (1/2)m.

That means the entire distance = (1/2)m. It will take running Rhonda 2.25 more minutes to get to work, and her remaining distance = (1/2)m - (1/5)m. So we've got

2.25 = (1/2)m - (1/5)m

3.75 = m

and from there the solution is the same, (1/5) * 3.75.