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GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in

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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 05:06
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Bunuel wrote:
Ronaldo and Messi race each other in an x-meter race. They start simultaneously from the same point. If Ronaldo, running at a content rate, covers the distance in 8 seconds, while Messi, running at a content rate, covers the distance in 10 seconds, in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi ?

A. 3 seconds
B. 10/3 seconds
C. 5 seconds
D. 20/3 seconds
E. Cannot be determined


 


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Let's say that Ronaldo covers a ground of 'x-a' and Messi covers a ground of 'x-2a' somewhere within this race. We have to find out the time when this happens
Ronaldo's speed = x/8
Messi's speed = x/10

Now we have the time taken to be constant hence
(x-a)/(x/8) = (x-2a)/(x/10)
=> 8x(x-a) = 10x(x-2a)
=> 8x - 8a = 10x - 20a
=> 12a = 2x
=> 6a = x

Hence the total distance is 6a, therefore Ronaldo's speed is 6a/8

Time taken for him to travel 'x-a' = 6a - a = 5a is
5a/(6a/8) = 20/3

We can verify this with Messi's time as 'x-2a'/(x/10)
=> (6a - 2a)/(x/10)
=> 4a/(6a/10) = 20/3

IMHO Option D
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 06:16
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Bunuel wrote:
Ronaldo and Messi race each other in an x-meter race. They start simultaneously from the same point. If Ronaldo, running at a content rate, covers the distance in 8 seconds, while Messi, running at a content rate, covers the distance in 10 seconds, in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi ?

A. 3 seconds
B. 10/3 seconds
C. 5 seconds
D. 20/3 seconds
E. Cannot be determined

 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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Distance = x
Time (Ronaldo) = 8s
Time (Messi) = 10s

Therefore:
S(R) = x/8
S(M) = x/10

Best way is the check answer choices and let us start halfway down at C so that based on if we are near or far, we can decide to go up or down because answer choices are in ascending order

C) 5s

Distance covered by Ronaldo = 5x/8
Distance covered by Messi = 5x/10 = x/2

Distance to cover (R) = 3x/8
Distance to cover (M) = x/2

1/2 of x/2 is = x/4 (0.25) and 3x/8 is (0.375): We are close but since 1/2 of Messi is less than Ronaldo, we should be searching for answers with time more than 5 seconds, so let us try option D

D) 20/3 s

Distance covered by Ronaldo = 20x/24 = 5x/6
Distance covered by Messi = 20x/30 = 2x/3

Distance to cover (R) = x/6
Distance to cover (M) = x/3

1/2 of x/3 = x/6 WHICH IS distance to cover for Ronaldo

Answer - D
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 06:48
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Bunuel wrote:
Ronaldo and Messi race each other in an x-meter race. They start simultaneously from the same point. If Ronaldo, running at a content rate, covers the distance in 8 seconds, while Messi, running at a content rate, covers the distance in 10 seconds, in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi ?

A. 3 seconds
B. 10/3 seconds
C. 5 seconds
D. 20/3 seconds
E. Cannot be determined


 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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An interesting question. We will solve it using ratios as we usually do. In races, we love to make diagrams.
Time taken R : M = 8 : 10
Speed R : M = 10 : 8 = 5 : 4 (speed is inverse of time taken)

Attachment:
Screenshot 2022-07-19 at 7.13.57 PM.png
Screenshot 2022-07-19 at 7.13.57 PM.png [ 16.64 KiB | Viewed 1016 times ]


So at any given time, if they start together, the ratio of their distances covered will be 5:4 (the ratio of their speeds). The difference between distances covered by them will be 1 unit on the ratio scale. Since the leftover distance of M should be twice the leftover distance of R, 1 unit of distance should be remaining for R (as we can see in the diagram)

So R would have covered 5/6th of the total distance in this time.
To cover complete distance of x, R takes 8 secs.
To cover 5/6th the distance, R will take 8 * 5/6 = 20/3 secs

Answer (D)
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 07:03
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Bunuel wrote:
Ronaldo and Messi race each other in an x-meter race. They start simultaneously from the same point. If Ronaldo, running at a content rate, covers the distance in 8 seconds, while Messi, running at a content rate, covers the distance in 10 seconds, in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi ?

A. 3 seconds
B. 10/3 seconds
C. 5 seconds
D. 20/3 seconds
E. Cannot be determined


 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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WhatsApp Image 2022-07-19 at 7.28.07 PM (1).jpeg [ 64.17 KiB | Viewed 1021 times ]

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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 07:06
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OA = D

based on the given information,

speed of Ronaldo = x/8 , speed of Messi = x/10

now, let's say after t seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi

now, distance covered in t seconds by Ronaldo = \(t*(\frac{x}{8})\)
so distance left to cover by Ronaldo = \(x - \frac{t*x}{8}\)

now, distance covered in t seconds by Messi= \(t*(\frac{x}{10})\)
so distance left to cover by Messi= \(x - \frac{t*x}{10}\)

and, the distance left to cover by Ronaldo is half the distance left to cover by Messi
\(x - \frac{t*x}{8} =\frac{1}{2 }*(x - \frac{t*x}{10})\)

now, divide by x on both sides,
\(1- \frac{t}{8} =\frac{1}{2 }*(1- \frac{t}{10})\)

by solving this equation , t = 20/3
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   19 Jul 2022, 07:41
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Ronaldo and Messi race each other in an x-meter race. They start simultaneously from the same point. If Ronaldo, running at a content rate, covers the distance in 8 seconds, while Messi, running at a content rate, covers the distance in 10 seconds, in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi ?

A. 3 seconds
B. 10/3 seconds
C. 5 seconds
D. 20/3 seconds
E. Cannot be determined

Let Ronaldo's speed = R
Messi's speed = M

(Given) 8*R = 10*M
R/M = 10/8 = 5/4
so, R = 5k & M = 4k

Hence, Total distance = 8*R = 10*M = 40k

Let the time be T seconds (when the distance left to cover by Ronaldo is half the distance left to cover by Messi )

Distance left to cover (for Ronaldo after T seconds) = 40k - 5kT
Distance left to cover (for Messi after T seconds) = 40k - 4kT

(Given) (40k - 5kT) / (40k - 4kT) = 1/2
(40 - 5T) / (40 - 4T) = 1/2
80 - 10T = 40 -4T
6T = 40
T = 20/3 seconds

(D) is the CORRECT option
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink] New post   18 Dec 2023, 22:42
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Re: GMAT Club World Cup 2022 (DAY 6): Ronaldo and Messi race each other in [#permalink]
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