Problem Solving

e-GMAT

James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James started from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James exactly 90 minutes after Patrick started his journey and then continued driving at the same speed for another 2 hours. By what percentage should Patrick reduce his speed so that James could catch up with Patrick in exactly 8 hours after Patrick overtook James?

A. 25%
B. 33%
C. 50%
D. 67%
E. 75%

OA D.

AAPL wrote:e-GMAT

James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James started from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James exactly 90 minutes after Patrick started his journey and then continued driving at the same speed for another 2 hours. By what percentage should Patrick reduce his speed so that James could catch up with Patrick in exactly 8 hours after Patrick overtook James?

A. 25%
B. 33%
C. 50%
D. 67%
E. 75%

OA D.

Let James' rate = 3 mph.

James started from his home and drove eastwards at a constant speed. Exactly 90 minutes after James stated from his home, his brother Patrick started from the same point and drove in the same direction as James did at a different constant speed. Patrick overtook James exactly 90 minutes after Patrick started his journey.
When James is overtaken by Patrick, James has traveled for a total of 3 hours, implying the following distance:
rt = 3*3 = 9 miles.
Patrick leaves 90 minutes after James and thus overtakes James by traveling these 9 miles in only 1.5 hours, implying the following rate for Patrick:
d/t = 9/(1.5) = 90/15 = 6 mph.

Patrick then continued driving at the same speed for another 2 hours.
When people COMPETE, we SUBTRACT THEIR RATES.
Difference between Patrick's rate and James' rate = 6-3 = 3 mph.
Implication:
Every hour after overtaking James, Patrick travels 3 miles ahead of James.
Thus:
Over the next 2 hours, Patrick travels a total of 6 miles head of James.

By what percentage should Patrick reduce his speed so that James could catch up with Patrick in exactly 8 hours after Patrick overtook James?
Since 2 hours have passed since Patrick overtook James, James must catch up to Patrick in the next 6 hours.
Implication:
For James to catch up by 6 miles over the next 6 hours, he must catch up by 1 mile every hour.
Thus:
Over the next 6 hours, James' rate must be ONE MPH GREATER than Patrick's rate, so that James catches up by 1 mile every hour.
Since James' rate = 3 mph, Patrick's rate must decrease from 6 mph to 2 mph.
Percent decrease from 6 to 2 = Difference/Larger * 100 = (6-2)/6 * 100 = 66.66%.

The correct answer is D.