Quantitative Aptitude
RACES AND GAMES MCQs
Races And Games Of Skill
Let the speeds of P, Q and R be denoted by p, q and r respectively. We are given that in a 2 km race, P can give Q 200 m and R 560 m. This means that while P covers the entire 2 km distance, Q covers only 1800 m (i.e., 200 m less than P), and R covers only 1440 m (i.e., 560 m less than P). We can use this information to write the following equations:
- Speed of P / Speed of Q = 2000 / 1800
- Speed of P / Speed of R = 2000 / 1440
- Speed of P = (4/5) speed of Q
- Speed of P = (5/3) speed of R
- d - x = 1800 (since P gives Q a head start of 200 m)
- d / q = (d - x) / r
- (5/3) r / (4/5) q = (5/3) (d - x) / r
- q / r = (4/5) (d - x) / d
- q / r = (4/5) (1800) / (d)
- d = 4500 meters
- 2000 - 4500 = -2500 meters
- |d - x| = 4500 - x
- q / r = (4/5) (4500 - x) / 4500
- (5/3) r / q = (4/5) (4500 - x) / 4500
- 25r / 9q = (4500 - x) / 5625
- 25r / 9q = (4/5) - (4x/22500)
- 25r / 9q + (4x/22500) = 4/5
- 25r / 9q + 4x = 9000
- 4x = 9000 - (25r / 9q)
- x = (22500/9) - (25r/36q)
- x = 400
Let us assume that the speed of A is x and that of B is y.
Then, according to the given information, x = 1.5y.
The time taken by A and B to cover a distance of d meters will be given by:
Time taken by A = d/x
Time taken by B = d/y
Now, since A and B have to reach the winning post at the same time, we have:
d/x = d/y
On solving, we get:
x = y
Substituting the value of x in the equation x = 1.5y, we get:
1.5y = y
y = 0
This is not possible as the speed of B cannot be zero.
Therefore, the given statement is not possible.
Now, let us assume that A has a start of 50 m.
Then, the total distance of the race will be given by:
Total distance = Distance covered by A + Distance covered by B + 50 m
Let us assume that this total distance is d.
Then, the time taken by A and B to cover this distance will be given by:
Time taken by A = (d - 50)/x
Time taken by B = d/y
Now, since A and B have to reach the winning post at the same time, we have:
(d - 50)/x = d/y
On solving, we get:
d = 150 m
Therefore, the winning post should be 150 m away in order for A and B to reach at the same time.
Hence, the correct answer is Option D 150 m.
If you think the solution is wrong then please provide your own solution below in the comments section .
Let P,Q and R be the three runners.
• Let P's speed be x m/sec
• Let Q's speed be y m/sec
• Let R's speed be z m/sec
• According to the question, P wins by 1 minute over Q and by 375 m over R.
• Also, Q wins by 30 seconds over R.
• We can write the following equations using the above information:
• 1 minute = 60 seconds
• P's distance covered = Q's distance covered + 60 seconds * x m/sec
• P's distance covered = R's distance covered + 375 m
• Q's distance covered = R's distance covered + 30 seconds * y m/sec
• From the first equation, we get:
• 60 seconds * x m/sec = 60 seconds * y m/sec + 375 m
• x m/sec = y m/sec + 375/60 m/sec
• From the second equation, we get:
• x m/sec = z m/sec + 375/60 m/sec
• z m/sec = x m/sec - 375/60 m/sec
• Substituting the value of x m/sec from the first equation, we get:
• z m/sec = y m/sec + 375/60 m/sec - 375/60 m/sec
• z m/sec = y m/sec
• We can calculate the individual speeds of the runners using the above equations.
• x m/sec = y m/sec + 375/60 m/sec
• x m/sec = z m/sec + 375/60 m/sec
• y m/sec = z m/sec
• x m/sec = 2y m/sec + 375/60 m/sec
• x = 2y + 375/60
• y = x - 375/60
• z = x - 375/60
• From the above equations, we can calculate the individual speeds of the runners.
• x = 2(x - 375/60) + 375/60
• x = 3x - 375/30
• 4x = 375/30
• x = 375/120 m/sec
• y = x - 375/60 = 375/120 - 375/60 = 375/240 m/sec
• z = x - 375/60 = 375/120 - 375/60 = 375/240 m/sec
• We can calculate the time taken by each runner to run a kilometre by using the formula:
• Time taken = Distance/Speed
• Time taken by P = 1000/ x = 1000/ 375/120 = 150 seconds
• Time taken by Q = 1000/ y = 1000/ 375/240 = 210 seconds
• Time taken by R = 1000/ z = 1000/ 375/240 = 240 seconds
Hence, the time taken by each runner to run a kilometre is 150 seconds, 210 seconds and 240 seconds respectively.
Thus, the correct answer is option A. 150 sec, 210 sec, 240 sec.
If you think the solution is wrong then please provide your own solution below in the comments section .
Let the speed of Ram and Shyam be x m/s and y m/s respectively.
Since their speeds are in the ratio 5 : 4, we have x/y = 5/4
Now, for the given race, the total distance to be covered is 4 km = 4000 m.
Time taken by Ram to cover 4000 m = 4000/x = 4000/5x
Time taken by Shyam to cover 4000 m = 4000/y = 4000/4y
Since the time taken by Ram is less than that of Shyam, Ram will complete the race before Shyam.
Therefore, the number of times the winner (Ram) passes the other (Shyam) can be calculated as follows:
Number of times the winner (Ram) passes the other (Shyam) = (Distance covered by the winner)/(Distance covered by the other)
Distance covered by Ram in 4000/5x = 4000/4y
Distance covered by Ram in 4000/4x = 4000/5y
Therefore, the number of times the winner (Ram) passes the other (Shyam) = 4000/4x / 4000/5y
Number of times the winner (Ram) passes the other (Shyam) = 4x/5y = 5/4
Since the ratio of the speed of Ram and Shyam is 5 : 4, the number of times the winner (Ram) passes the other (Shyam) is 3.
Hence, the correct answer is Option B. Ram passes Shyam thrice.
If you think the solution is wrong then please provide your own solution below in the comments section .
Let's assume that P can run 1 km in x minutes, which means P's speed is 1/x km per minute. Using the given information, we can deduce that Q can run 1 km in (x + 0.5) minutes (as P runs half a minute faster than Q). Therefore, Q's speed is 1/(x+0.5) km per minute.
Now, we are given that in a 1 km race, Q gets a start of 100 m, which means Q only needs to run 900 m to finish the race. However, despite the head start, Q loses the race by 100 m, which means P finishes the race in 800 m (i.e., P covers 1 km - 100 m).
Let's use the formula distance = speed x time to calculate the time taken by P and Q to run a kilometre:
- P's time to run 1 km = time taken to run 800 m + head start of 200 m = (800/1/x) + (200/1/(x+0.5)) = (800x+400)/(2x+1) minutes
- Q's time to run 1 km = time taken to run 900 m = (900/1/(x+0.5)) = (1800x+900)/2(x+0.5) minutes
(800x+400)/(2x+1) = (1800x+900)/2(x+0.5) + (1/2)Simplifying this equation, we get:4x^2 - 3x - 1 = 0Solving this quadratic equation, we get:x = 1 or x = -1/4
Since x cannot be negative, we choose x = 1, which means P takes 1 minute to run 1 km. Using this value, we can calculate Q's time to run 1 km:
Q's time to run 1 km = (1800 + 900)/2.5 = 2.5 minutes
Therefore, the correct answer is option B, i.e., P takes 1 minute and Q takes 2 minutes 30 seconds to run a kilometre.