Problem Solving

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 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%

The OA is D.

Is there a strategic approach to solve this PS question? Can anyone help? Thank you!

Hello AAPL.

This is a tricky question. I will show you how I solved it.

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

Hence we have the following diagram:

Home -------------- Distance after 90 minutes --------------- Distance after 180 minutes
James-------------- D_J1 = V_J*90 ----------------------------- D_J2=V_J*180
Patrick ------------ D_P=V_P*90

Now, since Patrick overtook James exactly 90 minutes after Patrick started his journey, we have that D_J2 = D_P, that is to say $$D_{J1}=D_P\ \ \Leftrightarrow\ \ V_J\cdot180=V_P\cdot90\ \ \Rightarrow\ \ V_P=2V_J.$$ To simplify the process, let's suppose that the velocity of James is 50 mts/min and therefore the velocity of Patrick is 100mts/min.

We also know that then Patrick continued driving at the same speed for another 2 hours. After two hours their covered distances are

Overtook--------- Distance after 2 hours = 120 minutes
Patrick ------------ 100*120 = 12000 m
James-------------- 50*120 = 6000 m

At this moment the distance between Patrick and James is 6000 meters. Now, we arrive to the question: By what percentage should Patrick reduce his speed so that James could catch up with Patrick in exactly 8 hours after Patrick overtook James?

From this 8 hours, the first two hours have passed already. So, we have to calculate the new velocity of Patrick in order that after 6 hours they have covered the same distance, that is to say:

Overtook-------- Distance after 2 hours = 120 minutes ----------- Distance after 8 hours = 120min + 360 min
James------------ 50*120 = 6000 m ---------------------------------- 6000m + 50*360 = 24000
Patrick ---------- 100*120 = 12000 m ------------------------------ 12000m + V_P*360 = 24000

This implies that the new velocity of Patrick is $$12000+V_P\cdot360=24000\ \Rightarrow\ \ V_P=33.33333\ \frac{mts}{\min}$$ Since the initial velocity of Patrick was 100 mts/min and his new velocity is 33.333 mts/min, this implies that he reduces its velocity by 100-33.333 = 66.66667 ~ 67%.

Therefore, the correct answer is the option D.

I hope it can help you.

AAPL wrote: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 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%

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.

AAPL wrote: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 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%

Alternate approach:

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.

...so that James could catch up with Patrick in exactly 8 hours after Patrick overtook James.
Since James must catch up after traveling for 8 hours at 3 mph, the following distance is implied:
rt = 3*8 = 24 miles.

Patrick then continued driving at the same speed for another 2 hours.
After overtaking James, Patrick travels for 2 hours at 6 mph, implying the following distance:
rt = 6*2 = 12 miles.

By what percentage should Patrick reduce his speed?
Since Patrick has traveled 12 miles in 2 hours -- and James travels a total of 24 miles in the final 8 hours of his journey -- Patrick msut travel 12 more miles over the next 6 hours, implying the following rate:
d/t = 12/6 = 2 mph.
Since Patrick's rate decreases from 6 mph to 2 mph, we get.
Percent decrease = Difference/Larger * 100 = (6-2)/6 * 100 = 66.66%.

The correct answer is D.