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Math Expert V
Joined: 02 Sep 2009
Posts: 64949
Two friends A and B simultaneously start running around a circular tra  [#permalink]

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15 00:00

Difficulty:   95% (hard)

Question Stats: 23% (01:56) correct 77% (01:57) wrong based on 111 sessions

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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?

A. 1
B. 2
C. 3
D. 4
E. 5

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GMATWhiz Representative G
Joined: 07 May 2019
Posts: 802
Location: India
Two friends A and B simultaneously start running around a circular tra  [#permalink]

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Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

Are You Up For the Challenge: 700 Level Questions

Solution:

• Speed of A = 6 m/s
• Speed of b = b m/s
o Ratio of A and B = 6: b = m : n [After eliminating the common factors]
NOTE:- Number of distinct points at which they (the bodies in circular motion) will meet can be determined by finding out the reduced ratios of their speeds (m: n).
• m -n =2 or n-m =2
• Case I: If m >n and m-n = 2
o Then, 6 > b, the possible value of b = 2.
 $$\frac{6}{2}$$ = $$\frac{3}{1}$$, and $$3-1 = 2$$
• Case II: If m <n and n-m = 2
o Then, 6 < b, the possible value of b = 10 or18.
 $$\frac{6}{18} = \frac{1}{3}$$, and $$3-1 = 2$$
$$\frac{6}{10} = \frac{3}{5}$$, and $$5-3 = 2$$
• There are 3 different possible values of b.
Hence, the correct answer is Option C.
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Senior Manager  P
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Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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Hi chetan2u , any easier way to do this ?
Senior Manager  S
Joined: 18 Dec 2017
Posts: 290
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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2
If two runners moves in same direction with simplified ratio of speed a:b where a>b then they meet a-b times in the circular track.
And if the runners moves in opposite direction then they meet a+b times on a circular track.
Now,
6:b can be 3:1, 1:3, 3:5
So b can be 2, 18 and 10 respectively.
Don't take 2:4 because it will reduced to 1:2 so only three values of b are possible.

Posted from my mobile device
Intern  B
Joined: 30 Apr 2020
Posts: 2
Concentration: Finance, Marketing
WE: Engineering (Energy and Utilities)
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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1
let say time taken by both A and B to meet is same
so Ta=Tb
and no of turns taken by B = n

Ta=Tb
2(pi)r/6=n2(pi)r/b

n=6/b
nb=6

so in order to do 2 meeting n=2
so, 2*b=6
b=3

Target Test Prep Representative V
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Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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1
2
Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

Solution:

If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.

Since b can be either greater than 6 or less than 6, let’s first assume that b < 6. If b < 6, we can let the reduced ratio be 6/b = r/s such r - s = 2. Notice that after being reduced, r < 6. So (r, s) can be (3, 1), (5, 3).

If (r, s) = (3, 1), we have 6/b = 3/1 and this gives us b = 2.

If (r, s) = (5, 3), we have 6/b = 5/3 and this gives us b = 18/5.

However, since b is a positive integer, b can’t be 18/5. So we only have one value for b when b < 6. Next, let’s assume now b > 6. Again we can let the reduced ratio be 6/b = r/s such s - r = 2. Notice that after being reduced, r < 6. So (r, s) can be (1, 3), (3, 5).

If (r, s) = (1, 3), we have 6/b = 1/3 and this gives us b = 18.

If (r, s) = (3, 5), we have 6/b = 3/5 and this gives us b = 10.

Since both values of b are integers, we have 2 integer values for b when b > 6. Therefore, altogether, there are 3 integer values for b.

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# Scott Woodbury-Stewart

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Intern  B
Joined: 11 Jun 2020
Posts: 1
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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ScottTargetTestPrep
If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.

Sorry if it too elementary, can I check if the above statement is applicable irrespective of their direction i.e. same or opposite. If not, how does it change?
Intern  B
Joined: 05 May 2020
Posts: 4
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

The number of meeting points of two runner in same direction on a circular track = a-b

The number of meeting points of two runner, running in opposite direction on a circular track = a+b

la-bl = 2 where a = 6

so b = 4 or 8

IMO B
Intern  B
Joined: 30 Sep 2019
Posts: 37
Two friends A and B simultaneously start running around a circular tra  [#permalink]

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ScottTargetTestPrep wrote:
Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

Solution:

If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.

Since b can be either greater than 6 or less than 6, let’s first assume that b < 6. If b < 6, we can let the reduced ratio be 6/b = r/s such r - s = 2. Notice that after being reduced, r < 6. So (r, s) can be (3, 1), (5, 3).

If (r, s) = (3, 1), we have 6/b = 3/1 and this gives us b = 2.

If (r, s) = (5, 3), we have 6/b = 5/3 and this gives us b = 18/5.

However, since b is a positive integer, b can’t be 18/5. So we only have one value for b when b < 6. Next, let’s assume now b > 6. Again we can let the reduced ratio be 6/b = r/s such s - r = 2. Notice that after being reduced, r < 6. So (r, s) can be (1, 3), (3, 5).

If (r, s) = (1, 3), we have 6/b = 1/3 and this gives us b = 18.

If (r, s) = (3, 5), we have 6/b = 3/5 and this gives us b = 10.

Since both values of b are integers, we have 2 integer values for b when b > 6. Therefore, altogether, there are 3 integer values for b.

Hi Scott,
Why can't (r,s) take the values (4,2) and (2,4) when r>s and s>r respectively?
Is it that the values of r must be factors of 6? That would eliminate (4,2) as an option but (2,4) is still viable.
Intern  B
Joined: 10 Oct 2017
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GMAT 1: 610 Q41 V35
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

Are You Up For the Challenge: 700 Level Questions

Bunuel what if the both start together with same speed which is 6. then B reduces his speed <6 then again increases>6 , then again decreases less than 6.

In this case b can have 4 values. . or does crossing also mean that if both are at same pace and one takes the lead from there.
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Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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ShreyasJavahar wrote:
ScottTargetTestPrep wrote:
Bunuel wrote:
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?

A. 1
B. 2
C. 3
D. 4
E. 5

Solution:

If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.

Since b can be either greater than 6 or less than 6, let’s first assume that b < 6. If b < 6, we can let the reduced ratio be 6/b = r/s such r - s = 2. Notice that after being reduced, r < 6. So (r, s) can be (3, 1), (5, 3).

If (r, s) = (3, 1), we have 6/b = 3/1 and this gives us b = 2.

If (r, s) = (5, 3), we have 6/b = 5/3 and this gives us b = 18/5.

However, since b is a positive integer, b can’t be 18/5. So we only have one value for b when b < 6. Next, let’s assume now b > 6. Again we can let the reduced ratio be 6/b = r/s such s - r = 2. Notice that after being reduced, r < 6. So (r, s) can be (1, 3), (3, 5).

If (r, s) = (1, 3), we have 6/b = 1/3 and this gives us b = 18.

If (r, s) = (3, 5), we have 6/b = 3/5 and this gives us b = 10.

Since both values of b are integers, we have 2 integer values for b when b > 6. Therefore, altogether, there are 3 integer values for b.

Hi Scott,
Why can't (r,s) take the values (4,2) and (2,4) when r>s and s>r respectively?
Is it that the values of r must be factors of 6? That would eliminate (4,2) as an option but (2,4) is still viable.

Response:

According to my solution, (r, s) must satisfy two things: 1) the difference between them must equal 2, and 2) r/s must be in the most reduced terms (meaning r and s must be relatively prime). Your examples of (4, 2) and (2, 4) both meet the first requirement; however, neither 4/2 nor 2/4 is in the most reduced terms. We can reduce 4/2 to 2/1 and 2/4 to 1/2, but in that case the difference is not 2 anymore. That’s why (r, s) can’t take the values (4, 2) or (2, 4).
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# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

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Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
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Location: United States (CA)
Re: Two friends A and B simultaneously start running around a circular tra  [#permalink]

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vardaanaro wrote:
ScottTargetTestPrep
If two objects are moving around a circle at different speeds, the number of distinct points on the circle at which they will meet can be determined by the difference between the (most) reduced ratios of their speeds.

Sorry if it too elementary, can I check if the above statement is applicable irrespective of their direction i.e. same or opposite. If not, how does it change?

Response:

The statement depends on whether the two objects move in the same direction or in opposite directions. If r/s is the most reduced ratio of their speeds, then they will meet in r - s points if they move in the same direction and r + s points if they move in opposite directions.
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# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews Re: Two friends A and B simultaneously start running around a circular tra   [#permalink] 29 Jun 2020, 07:53

# Two friends A and B simultaneously start running around a circular tra  