We are back with another set. Standard rules. Solve the next five questions in five minutes. Come back and check the solutions.
Q.1 In a 4000 m race around a circular stadium having a circumference of 1000 m, the fastest runner and the slowest runner reach the same point at the end of the fifth minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race. (CAT 2003)
(1) 20 minutes (2) 15 minutes (3) 10 minutes (4) 5 minutes
Q.2 There are 6 boxes numbered 1, 2, 3, 4, 5, and 6. Each box is to be filled with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is (CAT 2003)
(1) 5 (2) 21 (3) 33 (4) 60
Q.3 Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is (CAT 2005)
(1) 4 (2) 5 (3) 6 (4) 7
Q.4 A student finds the sum of 1, 2, 3 … as his patience run out. He found the sum as 575. When the teacher declared the result wrong, the student realized that he had missed a number. What was the number that the student missed? (CAT 2002)
(1) 16 (2) 18 (3) 14 (4) 20
Q.5 Let D be a recurring decimal of the form, D = 0.a1a2a1a2a1a2…., where digits a1 and a2 lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D? (CAT 2000)
(1) 18 (2) 108 (3) 198 (4) 288
Solutions:
Q.1 In a 4000 m race around a circular stadium having a circumference of 1000 m, the fastest runner and the slowest runner reach the same point at the end of the fifth minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race. (CAT 2003)
Solution:
Lets say there are only two people running this race. One wins and the other loses. Lets say the speed of the fast runner is 2 and slow runner is 1. Which means that he will take double the time to run the same distance. So the distance covered by slower person in 5 minutes is covered by faster one in 2.5 minutes. For the faster person to meet the slower person he has to take one circumference more. Hence, in 2.5 minutes the faster person covers one more round = 1000m. Hence, he will take 10 minutes to complete the race. Answer = 10 minutes
Q.2 There are 6 boxes numbered 1, 2, 3, 4, 5, and 6. Each box is to be filled with a red or a green ball in such a way that at least 1 box contains a green ball and the boxes containing green balls are consecutively numbered. The total number of ways in which this can be done is (CAT 2003)
Solution:
If we put 1 green ball, there are 6 possibilities. If we put two, and they must be consecutive, we have 5. If we put three, we have 4. Hence, 6 + 5 + 4 + 3 + 2 + 1 = 21
Alternate approach: If there is only one box, 1 case will have a green ball.
If there are two boxes (multiple green to be considered consecutive) = 3 cases will have green balls matching condition
If there are three boxes (multiple green to be considered consecutive) = 6 cases will have green balls matching condition
Extending this logic, if there are six boxes, we will have 21 cases matching condition. Answer = 21
Or this can be looked at in the following way:
6C1 + 5C1 + 4C1 + 3C1 + 2C1 + 1C1 = 21
Q.3 Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is (CAT 2005)
Solution:
First of all, find the maximum possible area that small tiles can take. Small tile area is 21. Large tile area is 143. The maximum possible is 143 / 21 = 6.something.
Alternate approach: One edge is 130 which can accommodate 70 cm of one tile and 30-30 cm of two other tiles. So one edge will have three and the other will have three. In total, 6.
Q.4 A student finds the sum of 1, 2, 3 … as his patience run out. He found the sum as 575. When the teacher declared the result wrong, the student realized that he had missed a number. What was the number that the student missed? (CAT 2002)
Solution:
Sum of n natural numbers is given as n (n+1)/2. By trial and error*, we reach sum of first 34 natural numbers as 595. Which means he missed the number 595 – 20 to get 575. Answer is 20
*It is not trial and error. When you have some of n natural numbers given which is either correct or incorrect or exact, multiply it by 2. So 595*2 = 1190. Find the closest square to 1190. 33^2 = 1089, 34^2 = 1156, 35^2 = 1225. This will give us the number where we should focus (34 in this case)
Q.5 Let D be a recurring decimal of the form, D = 0.a1a2a1a2a1a2…., where digits a1 and a2 lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D? (CAT 2000)
Solution:
As a1 and a2 lie between 0 and 9, we can take any two random numbers and substitute. Say 3 and 7. D can be then written as 0.373737… As two numbers are recurring, we will multiply with 100 and subtract to get rid of them.
100D = 37.373737… and D = 0.373737… → 99D = 37. Hence, D = 37/99
Then, the only task left is to find a multiple of 99 in the options. Answer = 198
P.S. – 75 days to CAT articles series is highly recommended for serious CAT aspirants. Read the entire series here.
