M03-36

Official Solution:

Fanny and Alexander are 360 miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and 65 mph respectively, how far apart will they be exactly 1.5 hours before they meet?

A. 135 miles
B. 90 miles
C. 70 miles
D. 65 miles
E. 25 miles


Keep it simple! The question is: how far apart will they be exactly 1.5 hours before they meet? As Fanny and Alexander's combined rate is 25+65 mph then 1.5 hours before they meet they'll be (25+65)*1.5=135 miles apart.


Answer: A
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I hope this helps

D/ R/ T
F : 25t/ 25/ t
A : 65t/ 65/ t
Total : 360/ 90/ t

360=90t
t=4, so in 4 hr, they will meet up.
But the problem says 1.5 hr before they meet
So, t=2.5
360-(25*2.5)-(65*2.5)=135

Is not 135 miles how much they have traveled and therefore the distance between them should be greater? Or the question should state how far will they have traveled or how much the distance between them shortened...Please correct if wrong.

135 miles is the distance 1.5 hours prior their meeting, because in 1.5 hours they cover exactly that distance.

Check similar question to practice: two-boats-are-heading-towards-each-other-at-constant-speeds-131737.html
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question is about to find "how far apart will they be exactly 1.5 hours before they meet?". They meet in 4hours[T=360/(25+65)]. So 1.5 hours before they meet means how much they travel in 2.5 hours, Fanny traveled [25*2.5=125/2] and Alexander traveled [65*2.5=325/2], now we can subtract the distance [325/2-125/2]=100. I think they are 100 miles apart.

Please correct me if I am wrong. Thanks in Advance Bunuel

Your solution is unnecessarily long and incorrect.

Why are you subtracting distances covered? You should add them to get the distance they covered in 2.5 hours and subtract that from the initial distance:

360 - (325/2 + 125/2) = 135.
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ohh.. Yess My Bad.. We have to add the distances and then subtract from total distance. Thanks for correcting me.

How to do this using an algebraic approach and setting up a R x T = D grid?

This is not the fastest way to do this, I made this mistake too. You have to look at this in the opposite way. The question asks how far apart they are in the final 1.5 hours of their journey. This is the same thing as asking how far they will travel in 1.5 hours. The last 1.5 hours of their journey they will travel the same distance as the first 1.5 hours of their journey since the rate is constant

D = RT
So D = R1T + R2T = (R1+R2)T = 90(1.5) = 135

I don't agree with the explanation. The question is "how far apart will they be...", which meant to ask the untravelled distance left of 360 miles. I agree with the approach way, but the number of solution itself. it should be: 360 - 135 = 225 miles ( which is not stated in answer choices) .
Hope i am right, please clarify this. Thanks.

No, you are not right. 135 miles is the distance 1.5 hours prior their meeting, because in 1.5 hours they cover exactly that distance.
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I calculated this way,
Let distance at which they meet is=D miles
So setting up equation for equal time taken to meet at point D:
D/25=(360-D)/65

So D=100.
So T=4 hrs.

I did not solve further, meeting point D is 100 miles from one end. Since question asks how far apart before they meet, so the answer has to be somewhere between 100 and 360 miles. If u look at options, only option A is >100 miles.

Bunuel generis niks18 chetan2u pikolo2510 pushpitkc



Is this same as:

How much distance would each have covered after 1.5 hours ?

Since I know individual speeds of both and I have total time as 1.5 hours,
I can calculate total total distance by adding individual distance traveled by each of them.

The given initial total distance seems irrelevant in my approach too.
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Yes this approach is also correct
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adkikani , as pikolo2510 notes, your approach is correct and as you note, the given initial distance IS irrelevant to your approach. Good catch. Just one question.

You say you solved for distance by using individual distance traveled. I think you solved this way:

Fanny, distance:
\((25 mph * \frac{3}{2}hrs)=\frac{75}{2}\) miles
Alexander, distance:
(65 mph * \(\frac{3}{2}hrs)=\frac{195}{2}\) miles

Add their distances: \((\frac{75}{2}mi+\frac{195}{2}mi)=\frac{270}{2}=135\) miles

If you used that method . . . what you did is the basis for adding speeds when two people travel in opposite directions. In other words, the "combined speed" approach might make a lot of sense to you.

Use whatever method works best for you. But combining speeds saves a step, avoids some weird fractions here, and on many problems can save time.

If travelers go in opposite directions, towards OR away from one another, add (combine) their speeds.

Combined speed: (25 + 65) = 90 mph
How far, total, do they travel in 1.5 hours?
Combined distance, RT=D:
(90 mph * 1.5 hrs) = 135 miles

If combining speeds does not seem as easy or intuitive as your approach, use yours!

P.S. Bunuel , +1 for the allusion
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A bit longer, but also helps understand

GAP is 360Miles

Gap is being reduced at a speed of 65+25Mph so 90 mph

Total time

360/90 = 4 hours

so if 1.5 hours left, it means the gap has been reducing for 2.5 hours

2.5*90 = 90 + 90 +45 = 225

Total distance less distance traveled = 360 - 225= 135

I solved it in 20 seconds because I know the trick.

The distance between two doesn't matter. Since they are covering 90 miles in 1 hours they will cover 135 miles.

Imagine this. They are at the center of the line and move away from each other. So F will move 25m in 1 hour and A 65 miles. Let's make them walk 30 more minutes. F will cover 12.5 and A will cover 32.5. So that's 45 more miles. So we get 90+45=135 miles.

I hope that helps!
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I think this is a high-quality question and I agree with explanation.

I solved it as per the below method:

time taken for the entire journey= 360/(25+65)= 4 hours
4-1.5= 2.5 hours
now if they both traveled for 2.5 hours, the distance yet to be covered between them is:

360(total distance)- (2.5*(25+65)) (distance already covered)
And: 135 miles-A