Bunuel wrote:
John and Amanda stand at opposite ends of a straight road and start running towards each other at the same moment. Their rates are randomly selected in advance so that John runs at a constant rate of 3, 4, 5, or 6 miles per hour and Amanda runs at a constant rate of 4, 5, 6, or 7 miles per hour. What is the probability that John has traveled farther than Amanda by the time they meet?
(A) 3/16
(B) 5/16
(C) 3/8
(D) 1/2
(E) 13/16
John can run at 4 different speeds {3, 4, 5, 6}
and Amenda can run at 4 different speeds {4, 5, 6, 7}
Hence, Total Combination of the speeds maintained by the two runners can be given by = 4 x 4 = 16
Now Let's make FAVORABLE and UNFAVORABLE cases
Case 1: If John Runs are speed of 3 miles per hourAmenda can run at speeds of {4, 5, 6, 7}
in every case Amenda runs at greater speed than the speed of John hence will cover more distance than John in each case
i.e. Favorable case = 0
i.e. Unfavorable case = 4
Case 2: If John Runs are speed of 4 miles per hourIf Amenda Runs at 4, The distance covered by them both will be same due to same speed of each runner
If Amenda Runs at 5, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
If Amenda Runs at 6, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
If Amenda Runs at 7, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
i.e. Favorable case = 0
i.e. Unfavorable case = 4
Case 3: If John Runs are speed of 5 miles per hourIf Amenda Runs at 4, The distance covered by John will be more than the distance covered by Amenda till they meet due to higher speeed of John than Amenda's speed
If Amenda Runs at 5, The distance covered by them both will be same due to same speed of each runner
If Amenda Runs at 6, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
If Amenda Runs at 7, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
i.e. Favorable case = 1
i.e. Unfavorable case = 3
Case 4: If John Runs are speed of 6 miles per hourIf Amenda Runs at 4, The distance covered by John will be more than the distance covered by Amenda till they meet due to higher speeed of John than Amenda's speed
If Amenda Runs at 5, The distance covered by John will be more than the distance covered by Amenda till they meet due to higher speeed of John than Amenda's speed
If Amenda Runs at 6, The distance covered by them both will be same due to same speed of each runner
If Amenda Runs at 7, The distance covered by Amenda will be more than the distance covered by John till they meet due to higher speeed of Amenda than john's speed
i.e. Favorable case = 2
i.e. Unfavorable case = 2
Total Favorable cases when John covers more distance than distance covered by Amenda = 0+0+1+2 = 3
Total Cases = 16
Probability = Favorable cases / Total cases
i.e. Required Probability = 3/16Answer: Option