Anand starts from a point P towards point Q, where PQ = 90

Question

Anand starts from a point P towards point Q, where PQ = 90 km. After 1 hour, Ram starts from P and catches up with Anand after 2 more hours. After meeting they continue to travel towards Q. On reaching Q, Ram reverses his direction and meets Anand 6 hours after the first meeting. Find Anand's speed.

(A) (45/7) kmph
(B) (60/7) kmph
(C) (40/7) kmph
(D) (30/7) kmph
(E) (65/7) kmph

Show Answer

Official Answer

B

(A) (45/7) kmph
(B) (60/7) kmph
(C) (40/7) kmph
(D) (30/7) kmph
(E) (65/7) kmph

fluke · Accepted Answer

Let the first meeting point be at a distance of x km from P, the remaining distance until Q would be (90-x) km

Anand traveled this x kms in 3 hours, making his speed x/3 kmph
Ram traveled the same x kms in 2 hours, making his speed x/2 kmph

So, in 6 hours:
Anand will cover=6x/3=2x km
Ram will cover=6x/2=3x km

And between their first meeting point and second, they both together covered a distance of 2(90-x) km.

2x+3x=2(90-x)
5x=180-2x
7x=180
x=180/7 km

Anand's speed=x/3=180/(3*7)=60/7 kmph

Ans: "B"

GyanOne · Answer

Let a be Anand's speed and r be Ram's speed.When Ram and Anand meet for the first time, distance travelled by them is the same
=> a(3) = r(2) 
When Ram and Anand meet for the second time, they have travelled twice the distance PQ, or 180 km, together
=> a(9) + r(8) = 180

Solving these equations for a, we get a = 180/21 = 60/7 km/hr 
=>Option (B)

144144 · Answer

Nice solution. I made it much more complicated...

I used - 3A=2R
8R=90+x
9A=90-x

solving giving the same result - but i guess it took me longer than to you and increased my chances to make mistakes...

viks4gmat · Answer

yup nice and easy one fluke!! kudos for the wonderfully solving the problem in 3 steps

Depaulian · Answer

Speed of Anand =a
 '" Ram=r
Till Anand and Ram meets, Anand travels for 3 hrs, and Ram trvels for 2 HRs
 When Two people meet, the distance travelled by them is equal 
=>> 3a = 2r
which gives r = 3a/2------------(i)
Now for next 6 hrs, Anand travels 6a Kms and Ram trvls 6r Kms.

Refer the pic:
 ....(3a)......||...........(6a)..................|||............... 
P|_________||______________________|||_________Q
 ....(2r)......||..............................(6r)....................

The (6r) covers the distance from (|| to Q)plus( Q to ||| )
PQ = 90
2PQ = 180
3a + 6a + 6r + 2r = 180
9a + 8r = 180
9a + 8*3a/2 = 180 -----using (i)
9a + 12a = 180
a = 180/21 = 60/7

From this

petrifiedbutstanding · Answer

Thank you, fluke and GynaOne. Really simple solutions!

sunny3011 · Answer

why the distance covered is 2(90-x) ? ram meets anand after 6hrs on first meet.
so not necessary that anand would have travelled same distance as ram did.?

bunuel help ?

knicks1288 · Answer

I solved it like this:

Let R = Ram's speed
A = Anand's speed

Since it took Ram 2 hours to cover the same distance as Anand did in 3 hours we know

2R = 3A
R = (3/2)A

We know they don't meet again for another 6 hours, at which point Anand will have been traveling for 9 hours to a distance I'll call "D". Ram will have been traveling for only 8 hours when he reaches D and the total distance he will have traveled is 90 + (90 - D) = 180 - D

So:

9A = D
8R = 180 - D or 180 - 8R = D

180 - 8R = 9A

Substitute for R

180 - 8(3/2)A = 9A
180 - 12A = 9A
180 = 21A
60/7 = A

pauc · Answer

Same question: Why is it 2(90-x)? I understand why we need the first (90-x), but why is it assumed that they meet at the same point X when they're headed back?

Bunuel · Answer

Look at the diagram below: Untitled4.png Blue dot is the second meeting point. In blue is the distance covered by Anand after the first meeting. In green is the distance covered by Ram after the first meeting. Their sum is twice 90-x. Hope it's clear.

zerosleep · Answer

Here is a quick approach- since we know the time taken by anand : time taken by ram (same distance when they meet after 3 hrs) = 3 : 2. therefore, ratio of their speed= 2 : 3. 
Let speed be 2x, 3x.
Now in three hours distance covered = 6x. remaining 90-6x covered twice in 6 hours.
=> 12x + 18x = 2(90-6x) giving 2x (remember this is the speed of anand and not just x) = 60/7

SVaidyaraman · Answer

Since Anand takes 3 hrs to travel the distance that Ram travels in 2 hrs their speed is in the ratio 2:3. We also know s1=(90-x)/9 and s2=(90+x)/8 where s1 and s2 are the speed of Anand and Ram resp. 
(90-x) /9 / (90+x)/8 = 2/3
From the above equations we get s1=60/7

Fdambro294 · Answer

Up to the 1st Meeting, the Distance that Anne and Ram will cover when Ram "CATCHES UP" and Meets Anne will be the SAME Distance = X

Let A = Anne's Speed
Let R = Ram's Speed

It takes Ram 2 Hours to cover X, while it takes Anne 1 + 2 = 3 hours to cover X

Rate * Time = Distance

R * 2 = X
A * 3 = X

3*A = X ----- (Equation I)

2nd: Over the Same Constant Distance = X that Anne and Ram cover, their Speeds will be Inversely Proportional to the Times it takes them to cover that same distance.

Ratio of Time: Anne - to - Ram = 3 : 2

Ratio of Speeds: Anne - to - Ram = 2 : 3

A / R = 2 / 3 ----- Equation II

3rd: When Ram heads to Point Q and then turns around to meet Anne for the 2nd Time, its as if they are starting at 2 Opposite Points from each other and heading towards each other.

together, the Distance they will cover is TWICE of the Remaining Distance after X ---- or (90 - X) km
It takes them together 6 hours to Meet the 2nd Time.

Distance Anne Covers + Distance Ram Covers = 2 * (90 - X)

A * (6) + R * (6) = 2 * (90 - X) ----- Equation III

Bringing all the Equations together:

Equation I: 3*A = X
Equation II: A/R = 2/3
Equation III: 6*A + 6*R = 2 * (90 - X)

Solving for Anne's Speed = A --- in Equation III:

6A + 6R = 180 - 2X

Substituting: 3A = X (Equation I) in for X
and
Substituting: R = (3/2) * A (Equation II) in for R

6A + 6 * (3/2 * A) = 180 - 2 * (3A)

6A + 9A = 180 - 6A

21A = 180

A = 180 / 21 = 60/7 k.m.p.h = Anne's Speed

Answer Choice - B

Kinshook · Answer

Given: Anand starts from a point P towards point Q, where PQ = 90 km. After 1 hour, Ram starts from P and catches up with Anand after 2 more hours. After meeting they continue to travel towards Q. On reaching Q, Ram reverses his direction and meets Anand 6 hours after the first meeting. 
Asked: Find Anand's speed.

Let Anand's speed be x kmh
In 1 hour, Anand travels = x km
In 3 hours, Anand travels = 3x km
In 2 hours, Ram travels = 3x km
Ram's speed = 1.5x kmh

In 6 hours Ram travels = 6*1.5x = 9x km
They first met from P = 3x km from P towards Q
In 6 hours Anand travels = 6x km

P---------3x km--------First meet-------------6x km---------Second meet-------1.5x km----------Q

3x + 6x + 1.5x = 10.5x = 90 km
x = 90/10.5 = 900/105 = 180/21 = 60/7 kmh

IMO B

GmatPoint · Answer

Since Ram covered the same distance in 2 hours that Anand did in 3 hours, the ratio of their speeds will be 3:2, respectively. Let the speed of Anand and Ram be 2y and 3y, respectively.

Now let us consider a point X in between P and Q, where Ram and Anand met.
Since Anand reached this point in 3 hours, thus, the distance of X from Q is 3*2y = 6y.

The distance from X and Q will be 90 - 6y.

Now, since Ram and Anand will meet again on the return journey of Ram, they both will cover twice the distance XQ in the 6 hours.

So, 2*(90 - 6y) = 6*(3y + 2y)
180 -12y = 30 y
y = 180/42

Since the speed of Anand is 2y,
Thus, speed of Anand = 2*180/42 = 60/7

Thus, the correct answer is B.

Venu01298 · Answer

Am i interpreting the question incorrectly?

The question says  "After 1 hour, Ram starts from P and catches up with Anand after 2 more hours."  . So doesn't that mean Ram takes 3 hours to cover the distance that Anand took 4 hours? (Ram starts 1 hour after and takes 2 more hours to catch up with Anand)

Bunuel · Answer

No. Anand travels for 3 hours before Ram, who has been traveling for 2 hours (1 hour less), catches up to him.

Adit_ · Answer

There are two segments:
One segment of the overtaking part, here we get relation:

3v1=2v2

Second part:

Total distance by Ram = 8v2 = 90+90-9v1 (Or its the distance of total 90 + the distance remaining from 90 after Anand travelled that distance)

Solving you get: 8v2=180-9v1

Two equations when solved you get v1 = 60/7

Answer: B

Anand starts from a point P towards point Q, where PQ = 90

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Anand starts from a point P towards point Q, where PQ = 90

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