Walk Away (2)

[ahistegt]

Tom and Linda stand at point A. Linda begins to walk in a straight line away from Tom at a constant rate of 2 miles per hour. One hour later, Tom begins to jog in a straight line in the exact opposite direction at a constant rate of 6 miles per hour. If both Tom and Linda travel indefinitely, what is the positive difference, in minutes, between the amount of time it takes Tom to cover the exact distance that Linda has covered and the amount of time it takes Tom to cover twice the distance that Linda has covered?

A 60
B 72
C 84
D 90
E 120

I don't understand why OE, below, uses "T" for both Tom and Linda. I'd thought it should be "T" and "T1+1".
I'd appreciate quick response; G-day is Tuesday :)

***********************************************************
OE:
The formula to calculate distance is Distance = (Rate)(Time). So at any given moment Tom's distance (let's call it DT) can be expressed as DT = 6T. So, at any given moment, Linda's distance (let's call it DL) can be expressed as DL = 2(T + 1) (remember, Linda's time is one hour more than Tom's). The question asks us to find the positive difference between the amount of time it takes Tom to cover Linda's distance and the time it takes him to cover twice her distance. Let's find each time separately first.

When Tom has covered Linda's distance, the following equation will hold: 6T = (2(T + 1)). We can solve for T:
6T = (2(T + 1))
6T = 2T + 2
4T = 2
T = 1/2

So it will take Tom 1/2 hour, or 30 minutes, to cover Linda's distance. When Tom has covered twice Linda's distance, the following equation will hold: 6T = 2(2(T + 1)). We can solve for T:
6T = 2(2(T + 1))
6T = 2(2T + 2)
6T = 4T + 4
2T = 4
T = 2

So it will take Tom 2 hours, or 120 minutes, to cover twice Linda's distance.
We need to find the positive difference between these times: 120 - 30 = 90.

The correct answer is D.

[JonathanSchneider]

The solution is laid out this way because we want to first calculate the distance that Linda has traveled on her own. This = 2 miles. From there, we can say that they are traveling for the same amount of time, so that we can use "t" for both. We just need to add +2 to Linda's distance.

[craig]

Why do you take
6T = (2(T + 1)) for Linda. Linda depart one hour before Tom

instead
2T = (6(T - 1)) for Tom. Tom depart one hour after Linda

[RonPurewal]

Why do you take
6T = (2(T + 1)) for Linda. Linda depart one hour before Tom

instead
2T = (6(T - 1)) for Tom. Tom depart one hour after Linda

that will also work.

if you use that instead:

first situation:
2t = 6(t - 1)
2t = 6t - 6
6 = 4t
3/2 = t
(notice this is the same as above: the two times are t = 3/2 and (t - 1) = 1/2. in the above, they were t = 1/2 and (t + 1) = 3/2.)

second situation:
2(2t) = 6(t - 1)
4t = 6t - 6
6 = 2t
3 = t
(notice this is the same as above: the two times are t = 3 and (t - 1) = 2. in the above, they were t = 2 and (t + 1) = 3.)

[elaine1920]

I was trying to use another way to approach but got stuck in the middle. Could any expert please shed some lights on my approach? thank you.

If Linda has walked one hour ahead of Tom, then Linda has travelled 2 miles already ( 2 mile/hour * 1 hour) . So in this case, in order for Tom to cover the same distance Linda traveled, Tom needs to make up the difference for the 2 miles. Tom travelled 4 miiles faster than Linda, so he use 1/2 hours to walked the distance that Linda walked, which is 2miles / 4 miles per hour. So in this step, I got the first number for 30 mins.

But when come to calculate the second part, which is the amount of time for Tom to cover twice the distance that Linda travelled, I don't know how to continue this way. Please give me some pointers. thanks!

[Ben Ku]

If Linda has walked one hour ahead of Tom, then Linda has travelled 2 miles already ( 2 mile/hour * 1 hour) . So in this case, in order for Tom to cover the same distance Linda traveled, Tom needs to make up the difference for the 2 miles. Tom travelled 4 miiles faster than Linda, so he use 1/2 hours to walked the distance that Linda walked, which is 2miles / 4 miles per hour. So in this step, I got the first number for 30 mins.

Hi Elaine: I'm not sure that this approach is correct. There are two portions that need to be considered: the portion where Linda walked by herself, and the portion where they both traveled away from each other. Here, the 2 miles you used is ONLY the distance Linda walked by herself, while the 4miles per hour is the difference in rate when they traveled together. Even though the numerical answer is correct, the process is not correct. That is also why I think you are having trouble setting up the second part.

[karchopra]

Unfortunately, I still dont completely get the answer to this question.

For the first part we get T = 1/2

As Tom travels at 6 miles/hr. Distance = 3 miles. This is the same distance covered by Linda.

Now the second part asks us "amount of time it takes Tom to cover twice the distance that Linda has covered"

This informs us to double the distance that Linda has covered = 3 * 2 = 6 miles.

Now the new time is 6 miles/6 miles/hr = 1 hr!

Thus the difference should be 30 minutes!

[tim]

Try again. By the time Tom has covered 6 miles, Linda has covered more than just 3. You need to adjust both of their distances upward as you run the clock forward..

[Eddie Gutia]

I approached this question in a completely different way. I picked a smart number for the distance, say 18 miles and solved for the time. Of course, this approach makes little/no sense.

My question: how do I resist picking smart numbers on question types like these?

Thanks a lot.

[RonPurewal]

I approached this question in a completely different way. I picked a smart number for the distance, say 18 miles and solved for the time. Of course, this approach makes little/no sense.

My question: how do I resist picking smart numbers on question types like these?

Thanks a lot.

the point of "smart numbers" is that you can specify, essentially at random, quantities that are not determined in the problem itself.

so, you need to think carefully about what you know and what you don't know.

in this problem, the two people are walking away from each other at specified rates. so, any specific condition (e.g., "xxx distance is twice yyyy distance") is going to occur after an equally specific amount of time, and after the two people have covered an equally specific distance.

[RonPurewal]

more generally—
remember that there are 2 secrets to success on this thing:
1/ do stuff.
2/ don't NOT do stuff.

if you're not sure whether you can pick a "smart number", then, in the spirit of #1 here, you should just try it and see what happens.
if it turns out that you can't actually pick smart numbers, that's something you'll discover quickly enough in the process of trying to do so.

[RonPurewal]

^^ the point of the post above, in other words, is that you're going to have to think about whether smart numbers will apply to any given situation.

there are some students who would rather memorize 1000 templates than trust themselves to think through these issues—but those students aren't going to do very well on this exam, which is specifically designed to thwart learning-by-rote methods.

sure, there are some signals that you can memorize. for instance, if the answer choices are full of variables, or if you see "in terms of..." in the question, then the chance that you'll be able to pick smart numbers is pretty much 100%.
however, there are plenty of other opportunities that don't fall into these two neat categories. as always, you'll have to think about which strategies are well suited to the particular problem at hand.

[BK87]

Why do you take
6T = (2(T + 1)) for Linda. Linda depart one hour before Tom

instead
2T = (6(T - 1)) for Tom. Tom depart one hour after Linda

Why isn't it 2T = 6(T+1)? If Tom is departing one hour after Linda, why don't you add the exra hour to his time?

[RonPurewal]

If Tom is departing one hour after Linda, why don't you add the exra hour to his time?

"t" is not a clock time; it's the total time spent traveling by the designated person.

if tom leaves an hour later, then, from that point onward, linda has been in transit for an hour longer than he has.
so, if "t" represents tom's time in transit, then linda's time in transit is "t + 1".
alternatively, if "t" represents linda's time in transit, then tom's time in transit is "t – 1".

[deem432]

I tried to answer this question by making an RTD chart and quickly discovered it was possible, but definitely not the best approach. I keep finding myself trying to make charts when there are better approaches to solving a problem. Are there signs I should be looking for that indicate using an RTD chart is not the best option? Although I understand this problem and can now easily solve it, I am not confident I would be able to solve similar problems. I have a serious problem with memorizing solutions instead of learning them.