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

The table denoting speed distance and time for Ronaldo and Messi is shown below -



Assume at time 't', the distance left to cover by Ronaldo is half the distance left to cover by Messi.

Distance covered by Ronaldo in 't' seconds = \(\frac{xt}{8}\)

Distance left to be covered by Ronaldo = \(x -\frac{xt}{8}\) =\(\frac{x(8-t)}{8}\)

Distance left to be covered by Messi = \(x - \frac{xt}{10}\) =\(\frac{x(10-t)}{10}\)

Given -

\(\frac{x(8-t)}{8}\) = \(\frac{1}{2}\)*\(\frac{x(10-t)}{10}\)

Upon simplifying

t = 20/3

IMO D

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

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)



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)