Brenda and Sally run in opposite direction on a circular tra

Lets say respective speeds are B and S for Brenda and Sally and distance of the circular track be 2d.

1st case, they run half of the distance of the circular track before meeting since they are running in opposite direction.
so,

dB/(B+S)=100 (Brinda covered 100 m)

2nd case, they run for full circular track and Sally covered 150m.

2dS/(B+S)=150

solve and we will get 3B = 4S

put this in any of the above equation and calculate for 2d = 350.

Hi Bunuel,

Why do we assume that Brenda runs for 2*100 = 200 meters for the second part of the problem?

Does opposite direction mean running towards each other or running behind each other??
Bunuel, could you please elaborate on the concept of running half the circumference and full circumference..
I am confused

opposite direction means running towards each other ....if they run behind each other then its in same direction.

moreover since starting point is diametrically opposite point and they are running towards each other so the sum of distances travelled by both will be equal to perimeter of semicircle. ==>this is the first meeting point.

now from this point they are running opposite to each other again.
so second time when they will meet ...sum of distance travelled by both will be equal to perimeter of full circle i.e full circumference.

moreover since they have constant speed.
for covering distance equal to semi circle==>brenda contributes 100 m
now since speed is constant so for covering distance 2 times of distance equal to semicircle==>brenda will contribute 2*100 = 200 m

hope its clear

Aaahh. If Brenda accounts for 200 meters, then shouldn't Sally cover 150 meters + X meters for the entire track? Thanks!

No. Second time they rum full circumference and we are told that second time Sally runs 150 meters.

1. Since they start from diametrically opposite points and they run in opposite directions, the first time they meet they would have together run half the circular track. So when together they run half the track , Brenda covers 100 m
2. Second time they meet , together, they would have covered the full circular track.
3. Brenda would have covered 200 m since she covers 100m m in the first case.
4. It is given, Sally covers 150 m. Total length of the full circular track is 200+150=350m

Dear Brunel ,

Please explain with some diagram . I got answer but my view point is little different .

Since together Brenda and Samuel ran half the circle and in that Brenda ran 100m, When they run the complete circle Brenda will run 2*100= 200 m

Is this implied that the speed of both the girls is same? Would this method work if the speed of both the girls is different.?

I could not understand why the next time they both meet, Brenda would cover 200m. So, I took a different approach:

If t is the time they meet for the first time and d is the distance of semi circular path, then
Brenda's speed = 100/t
Sally's speed = (d-100)/t

Next they both start from same point in opposite directions and meet at some random point on the track.
So, relative speed = 100/t +(d-100)/t = d/t
total track length = 2d
Point of meeting = 2d/d/t = 2t

So, distance covered by Sally:
150 = 2t*(d-100)/t
2d = 350 m

Ref: https://www.hitbullseye.com/Quant/Circular-Motion.php

Given:
1. Brenda and Sally run in opposite direction on a circular track, starting at diametrically opposite points.
2. They first meet after Brenda has run 100 meters.
3. They next meet after Sally has run 150 meters past their first meeting point.
4. Each girl runs at a constant speed.

Asked: What is the length of the track in meters?

Let the distance covered by Sally when they first met = x m

First meeting point: Half track length covered
100 m in direction of Brenda's movement
Length of half track = (100 + x) m
Length of full track = 200 + 2x

Second meeting point: Full track length covered
Brenda moved = 200 m
Sally moved = 2x = 150 m
x = 75 m
Length of full track = 200 + 2x = 200 + 150 = 350 m


IMO C

I am trying to explain my understanding:

As Brenda and Sally started diametrically opposite and started running towards each other (because, question mentions that they run in opposite direction), 1st time when they meet, they meet at some point of the semi-perimeter of the circle.

Brenda ran 100meters.
Let us assume, Sally ran x meters.
So,the length of the semi-perimeter of the circular track is (100 + x) meters. Definitely, the full perimeter equals to (200+2x) meters.

Brenda and Sally are running in constant speeds (different but constant i.e. their speed did not change for the complete race). So, we can derive a ratio of distance covered by each, from 'speed is proportional to distance covered'. This means Sally and Brenda, each will cover the distances in the same ratio as respective speeds are constant.

so, the ratio =[the distance covered in the semi circle by Brenda] : [the distance covered in the semi circle by Sally] = 100 : x.

Now, coming to the 2nd time meeting, starting point is the point where they met 1st time.
So, to meet again, Brenda and Sally together must cover the complete circumference i.e. (200 +2x) meters.

Sally covers = (200+2x)*[x/(100+x)] = 2x which is equal to 150 as question states.

So complete track length = circumference of the circle = 200 +2x = 200 +150 = 350 meters[Option C].

Please give kudos, if the explanation helps you.

They start at diametrically opposite points and run towards each other. So they complete a semi circle together when they meet. Brenda runs 100 m in the semi circle. Rest is run by Sally.

When they meet next, they cover a full circle together. Sally has run 150 mts in the full circle. Brenda must have run 200 mts in the full circle because their relative speeds remain constant.
Hence total circumference of the circle = 150 + 200 = 350 m

Answer (C)

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