Two boats are heading towards each other at constant speeds

Question

Two boats are heading towards each other at constant speeds of 5 miles/hr and 20 miles/hr respectively. They begin at a distance 20 miles from each other. How far are they (in miles) one minute before they collide ?

A. 1/12
B. 5/12
C. 1/6
D. 1/3
E. 1/5

A. 1/12
B. 5/12
C. 1/6 
D. 1/3 
E. 1/5

Bunuel · Accepted Answer

The question asks: how far apart will they be 1 minute=1/60 hours before they collide? Since the combined rate of the boats is 5+20=25 mph then 1/60 hours before they collide they'll be rate*time=distance --> 25*1/60=5/12 miles apart. Answer: B. Similar question to practice: 12-easy-pieces-or-not-126366.html#p1033924 Hope it helps.

KarishmaB · Answer

Bunuel has already given you a great and most direct approach for this question. But if you did go the round about way, you could have solved it in this way:

Since their combined speed is 20 + 5 = 25 miles/hr, they cover 20 miles in  \frac{20}{25}  hrs i.e.  \frac{20}{25} * 60  mins = 48 mins

Now, in 48 mins, they cover 20 miles. So in 1 min, they must have covered 20/48 = 5/12 miles

gmihir · Answer

Thanks Bunnel for the great link - awesome questions!!

For this one (two boats heading towards each other) - i calculated the time they would collide with each other (48th min) and then "tried" calculating from there on, taking eons to solve - how can the problem be solved with this apporach ? is 25*1/60 the only way to solve such problems ?

Bunuel · Answer

It's not the only way but for sure the shortest way to solve this problem.

fameatop · Answer

Hi Bunuel, 
Your explanation is always the best. I have a slightly different Logic to solve this question. Hope you will like the logic
The question asks: How much distance will both travel in 1 minute?
Since the combined rate of the boats is 5+20=25 mph = 25/60 mile/min = 5/12 mile/min
Hence
Answer: B.

alphonsa · Answer

Sorry I don't get it

The question states that we must find out the distance 1 min from the time of collision. So if they collided at the 48th minute, question asks us to find the distance of the 2 boats from each other in the 47th minute right?

KarishmaB · Answer

Yes, there are two ways we can find the distance between the boats in the 47th minute:

One way is we find the distance traveled by the boats in 47 mins and then subtract that out of the total distance. 
Another way is to find the distance they will cover in the last 1 min. That must be the distance between them in the 47th minute. Because at the end of the 48th minute, the boats meet - i.e. there is no distance between them. Knowing their speed, we know the distance traveled by them in 1 min and hence know the distance between them 1 min before they meet. The solutions above use this approach to get the answer.

shreygupta3192 · Answer

what is the problem with my solution

after calculating the time of collision i.e 4/5=48 mins.

since we need to calculate the distance b/w them in 47 mins,can`t we do like this ..

boat 1
48 mins=4miles
47 mins=(4/48) *47

similarly for boat 2
47 mins=(16/48)*47

then subtracting the distance of boat 2 from boat 1..
(16/48)*47-(4/48) *47=answer.

kindly correct me on this.

thank you

Bunuel · Answer

It should be  20 - ((\frac{16}{48})*47 + (\frac{4}{48}) *47) .

shreygupta3192 · Answer

thank you for the valuable reply but why are we subtracting the total from 20 and not subtracting them by each other(as per my solution), as that will give us how far are they from each other..
confused.

Bunuel · Answer

In 47 minutes boat 1 covers (16/48)*47 = 47/3 miles and boat 2 covers (4/48) *47 = 47/12 miles. Together in 47 minutes they cover 47/3 + 47/12 = 235/12 miles. Therefore after 47 minutes, so 1 minute before they meet, they will be 20 - 235/12 = 5/12 miles apart. Hope it's clear. P.S. 30-second approach to deal with this problem is given here: two-boats-are-heading-towards-each-other-at-constant-speeds-131737.html#p1080945

EMPOWERgmatRichC · Answer

In these types of Combined Rate questions, you have to stay focused on what the question asks you to figure out and what the two vehicles are doing.

In this question, they two vehicles are heading TOWARDS one another, so the movement of BOTH vehicles decreases the distance between them. As such, we have to ADD their rates so that we can figure out how much the distance decreases with each passing time period (second, minute, hour, etc.).

In a different type of question, you might have one vehicle chasing another. In these situations, the vehicle "in front" is moving AWAY from the vehicle "in back." So even though the "back vehicle" might be moving faster, the "front vehicle" is still moving AWAY from it. As such, we have to SUBTRACT their rates to figure out how long the "back vehicle" takes to catch up to the "front vehicle".

GSBae · Answer

Another quick way to think about this: go backwards. Assume they are at the same point and are going in opposite directions. One at t/12 miles per hour and the other at t/2 miles per hour. After one minute, they are

\(\frac{1}{12} + \frac{1}{3} = \frac{15}{36} = \frac{5}{12}\) miles apart.

\frac{1}{12} + \frac{1}{3} = \frac{15}{36} = \frac{5}{12}  miles apart.

KarishmaB · Answer

Responding to a pm:

The question involves relative speed and in relative speed, we often talk about the combined distance covered by two objects.

Say A and B are standing 20 feet apart. They walk toward each other and now they are 10 feet apart. "They" have covered a distance of 10 feet together. We don't know how much distance which one covered but we know that together they covered 10 feet. Say they took 1 minute to cover the 10 feet together. Then their combined speed is 10 feet per min.
Perhaps A's speed is 4 feet/ min and B's speed is 6 feet /min so that in one minute, they together covered 10 feet.
or perhaps A's speed is 8 feet/ min and B's speed is 2 feet /min so that in one minute, they together covered 10 feet.
or some other such case.

Similarly, when the boats head toward each other, they are together reducing the initial distance between them of 20 miles. When they finally meet, distance between them is 0 miles so they covered 20 miles together. Since they meet each other in 48 mins, it means their combined speed is such that they covered 20 miles in 48 mins.
Does this clarify the situation?

nishthagupta · Answer

This question REALLLYY made my mind jog!
Are there more questions like this one I can refer to ?
 VeritasKarishma   Bunuel  PLEASE REPLY!!

Bunuel · Answer

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Two boats are heading towards each other at constant speeds

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