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

Let the total distance, i.e., x = 80 mtrs

Therefore, Ronaldo's speed is 10 m/s
Messi's speed is 8 m/s

Just directly use the options.

In 20/3 seconds, Ronaldo covers 66.67 mtrs - 13.33 mtrs to go
Messi covers 53.33 mtrs - 26.67 mtrs to go, which is double of Ronaldo's.

Hence, option D is the answer.

From the stem of the question, both Ronaldo and Messi cover x meters in 8 and 10 secs respectively.
Let constant speed of Ronaldo Ro = x/8 m/sec
constant speed of Messi Me = x/10 m/sec

Let the time they travelled be t secs. After t secs, Ronaldo has y meters left to cover and Messi has 2y meters left (from stem)

time = distance/speed

For Ronaldo, t = (x-y)/ (x/8)
For Messi, t = (x-2y)/(x/10)

equating t and solving, we get x=6y

Plugging back x = 6y in time for Ronaldo or Messi , we get t = 20/3 seconds

Hence choice D is the correct answer

distance of race is x meter
speed of Ronaldo x/8 and speed of Messi is x/10
use answer option to solve
in how many seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi
let time be 20/3 seconds
so distance covered by Messi
20/3 *x / 10 ; 2x/3 ; distance left to cover is x/3 and half of it is x/6

distance covered by Ronaldo in 20/3 seconds
20/3 * x/ 8 ; 5x/6 distance left to cover is x/6
this is sufficient
OPTION D is correct

Given ; race distance = x meters, time taken by Ronaldo = 8 sec, and time taken by Messi = 10 sec.

Hence, Speed of Ronaldo = x/8 m/sec and Speed of Messi = x/10 m/sec.

We have to find the time in which the distance left to cover by Ronaldo is half the distance left to cover by Messi.
So, we know that both Ronaldo and Messi have travelled for the same interval of time up untill the the distance left to cover by Ronaldo is half the distance left to cover by Messi.

Suppose this time interval is t sec.

Now distance travelled by Ronaldo in t sec = xt/8 mts
and distance travelled by Messi in t sec = xt/10 mts

Distance left to cover by Ronaldo = x - xt/8
Distance left to cover by Messi = x - xt/10

As per the equation -

=> x - xt/8 = (x - xt/10)/2
=> 2x - xt/4 = x - xt/10
=> x = xt/2 (1/2 - 1/5)
=> x = 3xt/20

Now cancel x from both sides. (NOTE : x cannot be 0. They cant race for 0 meters.)

t = 20/3 ANSWER. OPTION D

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

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

at 4 sec - Ronaldo left .5 distance; Messi left .6
at 6 sec - Ronaldo left .25 distance; Messi left .4 ; nearly double

answer D

IMO Option D is the answer.
I chose the total distance as 2400m.
Ronaldo speed is 300 meters per second.
Messi speed is 240 meters per second.
At 20/3 seconds, Ronaldo ran 2000m. It remains 400m to the end of the finish line.
At 20/3 seconds, Messi ran 1600m. It remains 800m to the end of the finish line.
400m is half of 800m.
Therefore IMO, option D is correct.

Ans - D) 20/3

Steps to solution is attached

Posted from my mobile device

Ronaldo covers x meters in 8 seconds. Ronaldo’s speed = x/8 m/s
Messi covers x meters in 10 seconds. Messi’s speed = x/10 m/s

Say in t seconds the distance left to cover by Ronaldo is half the distance left to cover by Messi. So in this t seconds, the distance covered by them:
Ronaldo covers = (x*t)/8 meters
Messi covers = (x*t)/10 meters

As per the question we have relation between the remaining distances,
=> (x – xt/8) = (1/2) * (x – xt/10)
Solving for t we get, t = 20/3

Hence answer choice (D)

the correct answer is D,
distance =x,
Ronaldo rate =x/8
Messi rate = x/10
distance remaining ronaldo = half that of messi
(x-(x/8)*t)=1/2(x-(x/10)*t) by simplification of x we get t=20/3

Speed of Ronaldo = x/8
Speed of Messi = x/10

Let t be the time at which distance left to be covered by Messi is twice that to be covered by Ronaldo

Equation for the condition

2(x-x/8t)=(x-x/10t)

Therefore
t= 20/3

We know that Distance=Speed*Time
Let's say total distance x=80 meters
It is given that Ronaldo covers this distance in 8 seconds, so his speed would be 80/8=10 meter/second
Similarly, it is given that Messi covers this distance in 10 seconds, so his speed would be 80/10=8 meter/second
Now, we can use the options to our advantage to solve this question.

Let's start with option D:
In 20/3 seconds, distance covered by Ronaldo=10*20/3=200/3 meters
Therefore, distance remaining=80-200/3=40/3 meters ---(i)

In 20/3 seconds, distance covered by Messi=8*20/3=160/3 meters
Therefore, distance remaining=80-160/3=80/3 meters
Half of this distance=40/3 meters ---(ii)

Since, (i) and (ii) above are equal, answer is D.

Let total distance be =80
then speed of ronaldo =10m/sec
Speed of Messi = 8m/sec

At t=20/3 sec
Distance covered by R =200/3
Distance left =40/3

Distance covered by M = 160/3
Distance left =80/3

OA:D

Refer to the image

speed of ronaldo = x/8; speed of messi = x/10

assume after y second distance left to covered by ronaldo be 1/2 of distance left to cover by messi

(x - (xy/8)) = 1/2 (x-(xy/10))
dividing both sides by x we have
(8-y)/8 = (10-y)/20
(8-y)5 = 2(10-y)
40-5y = 20 -2 y
20 = 3y
y = 20/3
hence option D

Imo D

Lets say
The distance left to cover for Ronaldo = d/2
The distance left to cover for Messi = d
Speed of Messy = M
Speed of Ronaldo = R
Time to find = t
The distance covered by Ronaldo = y
Total Distance = y + d/2

M = (y+ d/2)/10 ---(1)
Also M = (y+ d/2 - d)/ t --- (2)

R= (y + d/2) / 8 ----(3)
R = y / t ---- (4)

Equate time from (2) and (4)
M/R = (y - d/2)/ y ---- (5)

From (1) and (3)
M/R = 4/5 ----------(6)
From (5) and (6)
y= 5d/2 ------(7)

Using eq (3) and (4) and putting the value of y from equation 7
8y/(y + d/2 ) = t
4*5d/3d = t
t = 20/3

Rate of ronaldo :
x--> 8 sec
d--> 8d/x, where d is the distance left to finish the race for ronaldo
Rate of messi
x-->10 sec
2d-->20d/x (distance remaining for messi is twice that of ronaldo)

now the distance covered by ronaldo
x-->8 sec
x-d--> 8(x-d)/x (*)

the distance covered by messi
x-->10 sec
x-2d--> 10(x-2d)/x

we are going to calculate the timing one left distance is twice the other, so the time covered by both racers is the same then :
8(x-d)=10(x-2d) => x=6d (**)
using (*) in (**), we'llfind that the timing is 20/3 sec
Answer is D