Ahab ran southeast at 10 km per hour. An hour later, Eli ran southeast

Question

Ahab ran southeast at 10 km per hour. An hour later, Eli ran southeast, along the same route as Ahab, at 15 kilometers per hour. If Eli arrived at his destination 30 minutes before Ahab arrived at the same destination, how long, in hours, was Ahab's journey?

A) 3.0
B) 3.5
C) 4.0
D) 4.5
E) 5.0

A) 3.0
B) 3.5
C) 4.0
D) 4.5
E) 5.0

hemilshihora · Accepted Answer

Dear Experts  KarishmaB   Bunuel   GMATNinja 
Please check if my method is correct

As we know Distance = Speed x Distance 
Let x be the time taken by Ahab and D is the total distance

Equation for Ahab : 
10*x = D

Equation for Eli : 
15*(x-1.5) = D
here 1.5 = 1  (Eli started 1 hour late)  + 0.5  (Eli reached 0.5 hours early)

So  10*x = 15(x-1.5) , which gives   x = 4.5 hours

KarishmaB · Answer

You can do this using the concept of races or relative speed.

Method 1: Relative Speed

When Ahab ran for an hour, he covered 10 km. Eli completed the distance ahead of Ahab by half an hour (Ahab would have needed half an hour to complete the distance which means the distance was 5 km since his speed is 10 kmph)

So Eli ran a distance of 10 + 5 = 15 km extra at an extra speed of 15 - 10 = 5 kmph. So Eli took 15/5 = 3 hrs to complete this distance. 
Then Ahab took 1+ 3 + 0.5 = 4.5 hrs to complete this distance.

Method 2:Races

Make a races diagram:

E ----(10 km)------ A--------------------(d km)-------------------A ------(5 km)----E

In the same time, Eli covered 10 + d + 5 km while Ahab covered only d km.
Since their speeds are in the ratio 3:2, the distance covered will be in the ratio 3:2 i.e. d+15 : d
Hence, the difference of 1 is actually 15 so d must be 30 km.

Then total distance is 10 + 30 + 5 = 45 km for which Ahab took 45/10 = 4.5 hrs

Answer (D)

GianKR · Answer

Ahab ran at 10 km/hr for one hour, he therefore run a distance of 10km before Eli started running along the same route.

Eli ran 10km + a certain distance "D" for a time "t" at a rate of 15 km/hr.
Ahab ran the same distance "D" for a time t+1/2hr at a rate of 10 km/hr.

We therefore obtain the following two equations:
15t=D+10
And 10t+5=D

Using both these equations, we get 15t-10 = 10t+5
Hence, t=3 (which is the time Eli spent running)

Ahab ran t + 1/2hr +1hr (in the beginning before Eli starts running). He therefore ran 3 hours + 1/2 hour + 1 hour= 4,5 hours

Correct answer is D

KarishmaB Bunuel I know this problem can be solved using the rate at which the distance between the two people changes, but I just don't know how to solve using that method.
This question felt difficult for me as I tend to struggle understanding how to approach questions that involve two bodies moving at the same time in the same direction or in opposite directions

Can you please provide an easy method to solve these type of questions? And similar questions for practice?

Thank you so much for your usual support!

Can you please provide an easy method to solve these type of questions? And similar questions for practice? Thank you so much for your usual support!

Archit3110 · Answer

Distance ran by Ahab is 
10 km in 1st hour
relative speed is 5 kmph
30 mins Ahab would have ran 5km more
total distance of Ahab is 15 km
Eli took 15/5 ; 3 hours
time of Ahab 1+ 0.5+3 ; 4.5 hours
option D

GianKR · Answer

Hello  KarishmaB

Thank you for taking the time. Much gratitude!

When you say Eli took 3 hours to complete this distance, I understand you mean the entire journey's distance and not just only the extra 15 km he had to run. Can you please elaborate on why the 3 hours correspond to the time he took to cover up the entire journey?

I believe I don't have a good understanding of the Relative Speed method to solve this type of questions. I know relative speed method is dependant on the determination of the relative speed, the gap distance and the gap time, and I struggle to define  the gap distance and the gap time  each time. What do they actually mean? Are there any questions I can ask myself that can help me define those each time I am faced with this type of questions?

kanikaa9 · Answer

Eli's speed - 15 kmph, Eli's Time = T . So  D = S x T = 15T 
A's speed - 10 kmph, A's time = T + 1 + 0.5 (A started 1 hr before Eli and ended 0.5 or 30 mins later than Eli). So  D = S X T = 10 T + 15

The distance travelled by both of them is same. So equating the two-

15T = 10T+ 15
5T = 15
T=3

We need to find A's time - T + 1.5 = 3 + 1.5 = 4.5

djangobackend · Answer

so relative speeds = 15-10 = 5 km/hr and it took Eli 2 hrs to cover the gap of 10km. 
Now, we can see that in 30mins Ahab covered 10 * 30/60 = 5 km and this is what Eli also covered, so in total Eli ran 10+5 = 15km. Now, IF only Eli ran then to cover 15km it would take him 15/15 = 1hr BUT we are given both are in motion so in order to cover 15km, we need to use relative speeds i.e., Eli is faster than Ahab by 5 km/hr so in 1 hour distance b/w then is reduced by 5km so to reduce 15km it would take 3 hours. 
OR 
we can also think that to cover gap of 10km Eli took 2 hours and then he ran 5km more as we derived above (using distance covered by Ahab), so using relative speeds to cover a gap of 10km eli takes 2 hours, so to make another gap of 5km (ahead this side) he will take 1 hour. so we can again say that total Eli ran for 2+1 = 3 hours. 
 
So overall Eli ran 3 hours, and we are given Ahab was running 1 hour prior and took 30 mins more so he overall took 1 + 3 + 0.5 = 4.5 hours

Rishm · Answer

First 1 hour - Ahab ran 10km/hr and Eli - 0, 2nd hour - 20/15, 3rd 30,30 (Both tied), 4th hour - Eli - 45, Ahab 40, now Eli reached 30 mins before Ahab considering here 45 as the stop point, he will get there in 30 mins where Eli will stop at 45. Hence 4.5 hrs. This is a logical step, I can't do it mathematically at all.

Ahab ran southeast at 10 km per hour. An hour later, Eli ran southeast

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Ahab ran southeast at 10 km per hour. An hour later, Eli ran southeast

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