The Greek soldier Pheidippides runs the 40 miles from Marathon to Athens

BTGModeratorVI wrote: ↑

Mon Jul 13, 2020 7:44 am

The Greek soldier Pheidippides runs the 40 miles from Marathon to Athens at a constant speed. Halfway through the run he injures his foot, and continues to run at half his previous speed. If the second half takes him 10 hours longer than the first half, how many hours did it take Pheidippides to run the second half?
A. 2
B. 10
C. 15
D. 20
E. 40

Answer: D
Source: Economist GMAT

Several ways to answer this. Here's one:

Given: the second half takes him 10 hours longer than the first half

Write this information as a WORD EQUATION:
(Time to complete first 20 miles) + 10 hours = (Time to complete second 20 miles)

let x = speed traveled for first 20 miles
So, x/2 = speed traveled for second 20 miles

Time = distance/speed so, we take our WORD EQUATION and replace values to get:
20/x + 10 = 20/(x/2)
Simplify to get: 20/x + 10 = 40/x
Multiply both sides by x to get: 20 + 10x = 40
Solve to get x = 2
In other words, the speed traveled for first 20 miles = 2 miles per hour

How many hours did it take Pheidippides to run the second half?
Time to complete second 20 miles = 40/x = 40/2 = 20 = D

Cheers,
Brent

BTGModeratorVI wrote: ↑

Mon Jul 13, 2020 7:44 am

The Greek soldier Pheidippides runs the 40 miles from Marathon to Athens at a constant speed. Halfway through the run he injures his foot, and continues to run at half his previous speed. If the second half takes him 10 hours longer than the first half, how many hours did it take Pheidippides to run the second half?
A. 2
B. 10
C. 15
D. 20
E. 40

Answer: D
Source: Economist GMAT

This is also a great candidate for CHECKING THE ANSWER choices.

How many hours did it take Pheidippides to run the second half?

A) 2
Impossible, since the travel time is 10 hours longer for the second half, which means the travel time for the first half = 2 - 10 = -8 hours (?!?)
ELIMINATE A

B) 10
Impossible, since the travel time is 10 hours longer for the second half, which means the travel time for the first half = 10 - 10 = 0 hours (?!?)
ELIMINATE B

C) 15
So, the travel time for the first half = 15 - 10 = 5 hours
This means speed for first half = (20 miles)/(5 hours) = 4 mph
Speed for second half = (20 miles)/(15 hours) = 4/3 mph
The question says that the travel speed for the second leg is HALF the travel speed for the first leg, HOWEVER 4/3mph is NOT half of 4mph
ELIMINATE C

D) 20
So, the travel time for the first half = 20 - 10 = 10 hours
Hmm, I like this already. The first leg takes HALF the time to complete. So, he must have been traveling twice as fast during the first leg. Everything checks out, so D is the correct answer.

To be safe, let's confirm the numbers.
Speed for first half = (20 miles)/(10 hours) = 2 mph
Speed for second half = (20 miles)/(20 hours) = 1 mph
Perfect, the travel speed for the second leg is HALF the travel speed for the first leg, AND 1mph IS half of 4mph
Great!
Answer: D

BTGModeratorVI wrote: ↑

Mon Jul 13, 2020 7:44 am

The Greek soldier Pheidippides runs the 40 miles from Marathon to Athens at a constant speed. Halfway through the run he injures his foot, and continues to run at half his previous speed. If the second half takes him 10 hours longer than the first half, how many hours did it take Pheidippides to run the second half?
A. 2
B. 10
C. 15
D. 20
E. 40

Answer: D

Solution:

We can let r = the original speed (i.e., the speed for the first half of the run) and t = the time needed to finish the first half. We can create the equations:

First half: rt = 20

Second half: ½r(t + 10) = 20

For the first equation, we have t = 20/r. Substituting this into the second equation, we have:

½r(20/r + 10) = 20

r(20/r + 10) = 40

20 + 10r = 40

10r = 20

r = 2

Thus, it takes 20/2 = 10 hours to run the first half and 10 + 10 = 20 hours to run the second half.

Answer: D