Question: Two friends A and B simultaneously start running around a circular track in the same direction, from the same point. A runs at 6 m/s and B runs at b m/s, where b is a positive integer. If they cross each other at exactly two points on the circular track, how many values can b take?

1. 1
2. 2
3. 3
4. 4
5. 5

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

The problem statement states that

Given:

• Two friends A and B simultaneously start running around a circular track in the same direction, from the same point.
• A run at 6 m/s and B run at b m/s, where b is a positive integer.
• They cross each other at exactly two points on the circular track.

Find out:

• The number of values of b.

If two objects move around a circle at different speeds, we need to find the difference between the (most) reduced ratios of their speeds. These differences can be used to determine the number of distinct spots on the circle the two objects will intersect.

Since b can be either greater than 6 or less than 6
Let’s first consider that b < 6. If b < 6, we can let the reduced ratio be 6/b = r/s such that r-s=2
After being reduced, we can say, r < 6.

Therefore, (r, s) can be (3,1),(5,3)
If (r, s) = (3,1), we have 6/b=3/1 this gives b = 2
If (r,s)=(5,3), we have 6/b=5/3 this gives b=18/5
Since b is a positive integer, it cannot be 18/5. Therefore, we only have one value for b < 6.

Now let us assume b > 6
Again we can let the reduced ratio be 6/b = r/s such that s - r = 2.

After being reduced, we can say r < 6.
Therefore, (r, s) can be (1, 3), (3, 5).
If (r,s)=(1,3), we have, 6/b=1/3, therefore b = 18
If (r, s)=(3,5) we have, 6/b=3/5, therefore b = 10

Since both values of b are integers, we have two integer values for b when b > 6.
Therefore there are 3 integer values for b

Approach Solution 2:
The problem statement informs that

Given:

• Two friends A and B simultaneously start running around a circular track in the same direction, from the same point.
• A run at 6 m/s and B run at b m/s, where b is a positive integer.
• They cross each other at exactly two points on the circular track.

Find out:

• The number of values of b.

We know that the speed of A =6 m/s
We know that the speed of B = b m/s
Therefore, the ratio of A : B = 6 : b = m:n (after eradicating the common factor)

The number of distinct points at which they will meet in a circular track can be resolved by finding out the reduced ratios of their speeds (m: n).
m-n=2 or n-m=2

Case1: If m > n and m-n = 2
Then, 6 > b, the possible value b = 2
6/2 = 3/1 and 3 - 1 = 2

Case2: If m < n and n-m = 2
If 6 < b the possible value b =10 or 18
6/10=3/5 and 5-3=2
6/18=1/3 and 3-1=2

Therefore b takes 3 different possible values.

Approach Solution 3:
The problem statement informs that:

Given:

• Two friends A and B simultaneously start running around a circular track in the same direction, from the same point.
• A run at 6 m/s and B run at b m/s, where b is a positive integer.
• They cross each other at exactly two points on the circular track.

Find out:

• The number of values of b.

Let the total length be L

The relative speed of A with regard to B = b - 6
Therefore, the time they first meet= L/(b-6)
Time A takes to complete one full lap of the track = L/6
Time A takes to complete one full lap of the track=L/b
Therefore the time they meet at the starting point for the first time= L/HCF(b,6)

Number of times they meet on starting point = time taken to meet at the starting point/time taken for meeting the first time = (b-6)/HCF(b,6)
This is equal to 2 as per the question.

Therefore,
(b-6)/HCF(b,6) = 2
Since less than 30, the values of b that satisfy the above equation are 2,10,18.

Hence b will take 3 values

“Two friends A and B simultaneously start running around a circular track”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT Problem Solving questions enable the candidates to enhance their knowledge of mathematics in order to crack numerical problems. GMAT Quant practice papers assist the candidates to go through several sorts of questions that will enable them to score higher marks in the GMAT exam.

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