Okay, let’s move away from “Divisibility and Remainders” on the GMAT, at least temporarily, and let’s focus our attention on another topic. If you have read some of my previous posts, I guess you know that I like to solve questions “logically.” I like to avoid making equations. Instead, I try to make myself “figure it out.”

A few days back, I came across a Time-Speed-Distance problem which was a perfect example of how you could “figure stuff out” without dealing with any equations. Actually, you can do that with a majority of GMAT questions (and save yourself loads of time!) but what was special about this question was that a couple of instructors of one of our competitor (I am not at liberty to disclose exactly who this competitor is!) had told their students that it is not possible to solve it logically. That got me thinking that perhaps, logical thinking is not as widely utilized as we at Veritas would like to believe. (I think our love for logical thinking is also apparent in the way we teach Sentence Correction!) Anyway, I thought of sharing the question and its logical solution with you! Here goes…

Now, before you look at the solution, try and solve it on your own. Do not use a pen/pencil/marker/any other writing instrument and don’t try to make equations in your head. Just try and reason out the solution! It will be a great intellectual exercise and if you are able to get the answer by using your mind alone, let me know for a virtual pat on the back!

Two friends, Tanaya and Stephen were standing together. Tanaya begins to walk in a straight line away from Stephen at a constant rate of 3 miles per hour. One hour later, Stephen begins to run in a straight line in the exact opposite direction at a constant rate of 9 miles per hour. If both Tanaya and Stephen continue to travel, what is the positive difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?

(A) 60 mins

(B) 72 mins

(C) 90 mins

(D) 100 mins

(E) 120 mins

Solution:

Following is what goes on in my mind when I read the question. I will quote sentences from the question to explain you the thought process.

“Two friends, Tanaya and Stephen were standing together.”

When I read the above, I think, “Ok, so two people, Tanaya and Stephen, are standing at the same point at a particular time, say 12 noon.”

“Tanaya begins to walk in a straight line away from Stephen at a constant rate of 3 miles/hour.”

Now I think, “Stephen is still standing where they were standing together but Tanaya starts walking away from Stephen at a speed of 3 miles/hour. Women!”

“One hour later, Stephen begins to run in a straight line in the exact opposite direction at a constant rate of 9 miles per hour.”

Stephen was shell shocked! After a whole hour (i.e. at 1:00 pm), when Tanaya was actually 3 miles away from him, he regains control of his legs and starts running in the opposite direction at 9 miles/hr, three times the speed at which Tanaya was walking away (vengeance!).

“What is the positive difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and …”

So, I need to find the time it takes Stephen to cover the exact distance that Tanaya has covered. To cover the same distance as Tanaya, Stephen needs to cover the distance that Tanaya is covering now plus he has to cover the extra 3 miles that Tanaya has already covered. The thing going for him is that he is running much faster than Tanaya. Out of his speed of 9 miles/hr, 3 miles/hr get used to cover what Tanaya is covering right now (since Tanaya’s speed is 3 miles/hr). So he uses the extra 6 miles/hr to cover the 3 miles that Tanaya has already covered! That means it will take him half an hour (3 miles/6 miles/hr) to cover the extra distance. In half an hour, i.e. at 1:30 pm, Stephen would have covered the same distance as Tanaya. At 1:30, Tanaya must be 4.5 miles (= 3 miles/hr*1.5 hrs) away from the starting point and Stephen would also be 4.5 miles away from the starting point (since he has covered the same distance as Tanaya).

“…and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?”

I also need to find the time it takes Stephen to cover twice the distance that Tanaya has covered. So, out of his speed of 9 miles/hr, 6 miles/hr will be used to cover twice of what Tanaya will now cover at a speed of 3 miles/hr. The leftover 3 miles/hr (= 9 miles/hr – 6 miles/hr) of his speed will be used to cover an extra 4.5 miles. If you are wondering why he has to cover an extra 4.5 mile, recall that at 1:30 pm, both of them are at a distance of 4.5 miles from the starting point. Stephen should have been at a distance of 9 miles from the starting point if he wants to cover double the distance that Tanaya covers from the starting point.

Hence, now he needs to cover an extra 4.5 miles. At a speed of 3 miles/hr (which we obtained above using 9 miles/hr – 6 miles/hr), it will take him 1.5 hrs to cover the extra 4.5 miles ( = 4.5 miles/3 miles per hour). Therefore, at 3:00 pm he would have covered twice the distance that Tanaya would have covered since 12 noon.

The time difference between 1:30 pm and 3:00 pm is 1.5 hrs (90 mins). This is the required time difference.

Answer (C)

How long do you think it takes to think all this?

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, and regularly participates in content development projects such as this blog!*

## Quarter Wit, Quarter Wisdom: An Intellectual Exercise in TSD

*Posted on June 1, 2011, filed in: GMAT, Quarter Wit, Quarter Wisdom*

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Hi, Karishma

thank you for your valuable posts.

my question is if we are asked to find the time difference to cover the distances, why do we need to find the distances and time spent to cover that distances separately?

I mean we are asked to find “twice the distance that Tanaya has covered” minus “the distance that Tanaya has covered”=”the distance that Tanaya has covered”,yes?

So,to find the time needed by Stephen to cover Tanaya’s distance is sufficient.

we know that Stephen has 1 hour gap (it is stated in the q.stem that Tanaya went for a walk 1 hour before)

we also know the rate difference of these two =6 m/h (9m/h-3m/h). 3 miles/6 miles/h=1/2 hour

total=1+1/2=1.50 hours

please correct me , if I am wrong. I cant say I am strong enough in such kind of problems, so ur correction is welcome.

Actually, I am a little doubtful about the logic used here. We need to find the time in which Stephen covers twice the distance covered by Tanaya. The point is that in that time, Tanaya needn’t have covered twice the distance she did initially (when Stephen covered same distance as her). It works out over here due to the given numbers. Let’s change the numbers a little:

Say, all is same except Stephen’s speed is 12 mph. So he uses 3 mph to equal Tanaya’s speed and uses the 9 mph to cover the extra 3 miles. He takes 1/3 hr i.e. 20 mins. So it is 1:20 pm now.

In what time does he cover twice of Tanaya’s distance? Starting from the beginning again, he uses 6 mph of his speed to cover double of Tanaya’s distance and he uses the rest of the 6 mph to cover 2*3 miles. He takes 1 hr to do it. So it must be 2:00 pm now.

The time difference is 40 mins (answer)

Find out the answer using your method. I think you get 1 hr 20 mins.

Does it make sense or did I miss something?

dear karishma,

didn’t take pen or pencil !

but after reading LalaB and ur explaination to it, i doubted my own solution (though i got the anwer correct) !!

according to ur quest -

tanya covers 3 in 1st hr + 3in 2nd hr ( when stephen start running and cover 9) +3 in 3rd hr( when stephen covers 18)

now, amount of time to cover twice of tanya’s distance – tanya’s distance

= 18 miles in 3hr (as tanya has 9miles in 3 hr)

– 9 (tanya’s distance – which stephen covers in 2 hr) = 1 hr.

is it so? what is going wrong? please help..

I see you found the time taken by Stephen to cover twice the distance covered by Tanaya (total 3 hrs in which Stephen covered 18 miles and Tanaya covered 9 miles but Stephen ran for only 2 hrs). It’s certainly fine. But I don’t see where you calculated the amount of time taken by Stephen to cover the exact same distance as covered by Tanaya (total 1.5 hrs in which Tanaya covers 4.5 miles and Stephen covers 4.5 miles too but he ran for only 1/2 an hour)

Diff in the time = 1.5 hrs

Still, I would say, for these type of probs the equation method works just fine for me.

Simple:

Case 1:

Distance covered by both are same.

9 x t1 = 3 x (60 + t1) => t1 = 30

Case 2:

Distance By Stephen = 2 x Distance by Tanya

9 x t2 = 2 x 3 x (60 + t2) => t2 = 120

Difference: t2 – t1 = 90

How much, time does it take ?

During the test, the psychological pressure makes it challenging to think so much than to solve 2 simple equations.

I think, one must develop a where-to-use-what strategy, i.e. whether to go for logical way or simple equation way.

Let me point out a few things:

1. In the actual exam, you should use the method that comes first to your mind, whatever it is. The method that will come first to your mind will be the one you will be most comfortable with. I am obviously not suggesting that everyone should use my methods.

2. There will always be an algebraic solution. You don’t NEED logical methods because you cannot solve the question otherwise. You use logical solutions only because it is a choice you make.

3. For me, solving logically is more intuitive. It’s like looking at the big picture and thinking to reason out the answer instead of following a mathematical process. Most GMAT questions, when solved logically, take less than a minute. Therefore, I like to share my thoughts. I absolutely agree that these methods are not for everyone. It takes time and effort to get used to them.

This question is certainly tricky and that is why it is ‘an intellectual exercise’.

There is no debate on which method is better/faster. Use whatever works for you. You just have more options now.

I happened to bump into this old post and found the discussion pretty intriguing. I gave this query some thought and have a slightly different take on it – in fact I can think of two viewpoints. So here goes : Since the time factor is a constant here, we can simply rephrase the question as the time taken by Stephen to catch up Tanya when her speed is 6m/hr – time taken by her with her original speed(3m/hr). So addressing both speeds of Stephen(9 and 12 miles), in the former it takes 2.0-0.5 and in the later,its 1hr – 20 min’s.

Secondly, a slightly longer approach would be to use the knowledge of ratio’s of the 2 speeds. Assuming Stephen’s speed to be 9m/hr, we know Speed(Tanya):Speed(Stephen) = 1:3. We just do not know which multiplying factor to use,meaning, we do not know the multiplying factor to use to make the ratio of distance 1:2. So once we figure out the time taken for them to achieve 1:1 (total)distance, we see the ratio we need is 1.5:4.5(for the distance 4.5). From here on the problem simply boils down to figuring out how many multiples of this ratio we need to achieve the overall 1:2 distance, that is, 4.5:4.5, 6:9, 7.5:13.5 and finally 9:18 which is our answer. If Stephen’s speed is 12m/hr, the ratio’s would proceed as 4:4,5:8,6:12(the answer).

the wording of the problem sounds wearied.

problem says:

“difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?”

It gives impression that we need to take absolute time taken by Stephen.

time taken by stephen to cover the distance that is covered by tanaya= .5 hr

time taken by stephen to cover twice the distance covered by Tanaya= 1.5 hr

So difference is 60 min.

How can i identify that time difference is not absolute but from the starting point at which tanaya has started?

It is an intellectual exercise in TSD. Besides, a convoluted question stem is not uncommon in GMAT.

The question is quite clear in my opinion.

difference between A and B

A = amount of time it takes Stephen to cover the exact distance that Tanaya has covered = 30 min (Stephen starts are 1:00 pm and covers same distance as Tanaya by 1:30 pm)

B = amount of time it takes Stephen to cover twice the distance that Tanaya has covered = 2 hrs (Stephen starts are 1:00 pm and covers twice the distance as Tanaya by 3:00 pm)

B – A = 90 mins

Thanks for the article, Karishma… Another method added to the arsenal :)

I agree, it always helps to know different options that are available…

Thanks Karishma, for the nice articles. Your way of teaching is exceptional, and the questions, you pose inside the articles, really make the reader involved and think through.

Hey Deepak and Sri,

Thanks. We are glad you liked our post.

Best,

Karishma

Hi karishma,

here is how I tried to figure it out,

Tanya is walking at 3 miles per hour i.e. 1 miles in every 20 mins

Stephen aka steve is going at 9 miles per hour so after an hour it would take him around 20 mins to catch up the 3 miles.

Now during these 20 mins tanya would have gone another 1 mile, steve is going 9 miles per hour i.e. 1 miles approx per 7 mins. So steve would take around 26 mins to catch up with tanya.

He has to now cover twice the distance that she has covered till now i.e. 8 mins. It takes him approx 7 mins for 1 mile so for 8 miles he will take around 56 mins.

The total time = 26 + 56 = 82. The nearest answer is C

This is the way I figured it out, hope my logic is correct. Please let me know if it is. Or if there is any ambiguity then kindly point it out to me.

It’s a little painful when you don’t consider relative speed. So Steve is trying to cover the distance between him and Tanya. You find the distance between them and find Steve’s speed and use it. The problem is that Tanya is moved forward in that time. So you make adjustments for the extra distance. Then you use Steve’s speed to cover that distance. But Tanya is still on the move so she has moved further ahead. Now you need to make another adjustment. The adjustments required would be smaller and smaller but continuous. So try to use relative speed i.e. Steve is moving at a speed of only 6 mph relative to Tanya. The rest of his 3 mph speed is used to cover the same distance that Tanya is covering right now while Steve is trying to catch up.