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Two buses P and Q start from cities A and B towards each other. They m

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New post 20 Oct 2017, 07:37
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Two buses P and Q start from cities A and B towards each other. They meet at a distance of 22 km from City B, cross each other to reach city B and A and turn back. During the second time they meet at a distance of 10 km from City A. Find the distance between cities A and B.

A. 24 km
B. 56 km
C. 60 km
D. 72 km
E. 84 km
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Oct 2017, 07:45, edited 1 time in total.
Renamed the topic, edited the question, moved to PS forum and added the OA.

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afa13 wrote:
Two buses P and Q start from cities A and B towards each other. They meet at a distance of 22 km from City B, cross each other to reach city B and A and turn back. During the second time they meet at a distance of 10 km from City A. Find the distance between cities A and B.

A. 24 km
B. 56 km
C. 60 km
D. 72 km
E. 84 km


let's concentrate on Bus Q that starts from city B..

when they meet first time they both combined have traveled entire distance say x, out of which Q travels 22..
when they meet second time, a total of 2 time the entire distance over and above the initial has been covered, so 3x..

If Q covers 22 in x, it will cover 3*22 = 66 in 3x..
But they meet 10 kms from A, so Q has traveled x+10...
x+10=66...x=56

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Two buses P and Q start from cities A and B towards each other. They m [#permalink]

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New post 20 Oct 2017, 08:21
chetan2u wrote:
If Q covers 22 in x, it will cover 3*22 = 66 in 3x..


Can you please explain this part a little more?

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New post 20 Oct 2017, 08:38
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saswata4s wrote:
chetan2u wrote:
If Q covers 22 in x, it will cover 3*22 = 66 in 3x..


Can you please explain this part a little more?



Q starts from B and P starts from A and meet 22 kms from B
so when they meet each other, they would have covered a combine of x and B covers 22..

Now when they meet second time..
Q reaches A from B and turns back, so covers x and some distance
In the same time, P covers x distance from A to B and turns back..
Now when they meet they have again covered the distance x..
so TOTAL 3x..
Q covers 22 when they cover x distance, so another 22 when they have covered another x that is x+x
another 22, when they have travelled another x or 3x.. so total 22+22+22=66
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Re: Two buses P and Q start from cities A and B towards each other. They m [#permalink]

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New post 20 Oct 2017, 08:54
chetan2u wrote:
saswata4s wrote:
chetan2u wrote:
If Q covers 22 in x, it will cover 3*22 = 66 in 3x..


Can you please explain this part a little more?



Q starts from B and P starts from A and meet 22 kms from B
so when they meet each other, they would have covered a combine of x and B covers 22..

Now when they meet second time..
Q reaches A from B and turns back, so covers x and some distance
In the same time, P covers x distance from A to B and turns back..
Now when they meet they have again covered the distance x..
so TOTAL 3x..
Q covers 22 when they cover x distance, so another 22 when they have covered another x that is x+x
another 22, when they have travelled another x or 3x.. so total 22+22+22=66


Thanks. Its clear now.. :-)

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Re: Two buses P and Q start from cities A and B towards each other. They m [#permalink]

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New post 20 Oct 2017, 09:00
I have figured out the algebraic approach:

Let us say that train P leaves A at speed a and covers distance x.
Q leaves B at the same time P leaves A at at constant velocity b and meet at 22 from B (let us call this point R.
We can get these equations: x/a=22/b => a/b=x/22.
After P leaves R it will cover 22 and the distance from B to miles away from A. Thus, it will cover 22+(x+22)-10=x+34
After Q leaves R it will cover x then reach A then the 10 miles to meet with P. Therefore, it will cover x+10.
we can set up these equations: (x+34)/p=(x+10)/q => x+34=(p/q)(x+10) => x+34=(x/22)(x+10) leading to the following quadratic equation:
x^2-12x-(22)(34)=0
(x-34)(x+22)=0
since distance is greater than zero then x=34 and total distance is 34+22=56.

Is there a shorter way?

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New post 20 Oct 2017, 09:19
afa13 wrote:
Two buses P and Q start from cities A and B towards each other. They meet at a distance of 22 km from City B, cross each other to reach city B and A and turn back. During the second time they meet at a distance of 10 km from City A. Find the distance between cities A and B.

A. 24 km
B. 56 km
C. 60 km
D. 72 km
E. 84 km


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Re: Two buses P and Q start from cities A and B towards each other. They m [#permalink]

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New post 20 Oct 2017, 10:28
Seems to me that chetan2u has the shortest method. You have to think about it differently, but I like how quick you can get an answer. I solved a little longer method.

We know that d=r*t. We can assign Rp as rate for P and Rq as rate for Q. We also know that meet at first at one time, call it T1, and then meet again at another time, call it T2.

At the first meeting point 22=Rq*T1 and d-22=Rp*T1.
I solved each for T1 and then substituted for T1 to get 22/Rq = (d-22)/Rp....I

At the second meeting point we have d+10=Rq*T2 and 2d-10=Rp8T2
Here I solved for T2 and substituted for T2 to get (d+10)/Rq = (2d-10)/Rp......II

Since Rq and Rp are constant, their ratio also has to be constant, so I solved I and II for Rp/Rq to get the following:

Rp/Rq = (d-22)/22 using I
Rp/Rq = (2d-10)/(d+10) using II

I first thought to make these two equal and solve, but the equation looked ugly and I already spent too much time, so I noticed that the first equation is divisible by 22, so the second equation must also be divisible by a multiple of 22. I then asked what value of d in the answer choices will make (d+10) a multiple of 22 and 56 was the only option (choice B)

With all that, chetan2u's method is much quicker and works out the same.

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Re: Two buses P and Q start from cities A and B towards each other. They m   [#permalink] 20 Oct 2017, 10:28
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