CAT Question Bank  Arithmetic

The time each takes to complete a round of circular field,
360/48 = 7.5
360/60 = 6
360/72 = 5 days.
They'll meet at the time equal to the LCM of (7.5,6,2)
which is 30 days.[solved by rajendra rajput]

Q11) Read the information below and answer the questions that follow: Harsh and his son Rakesh go for running in a park. The track is circular with a circumference of 600m. They run with a constant speeds of 7m/s and 5m/s respectively. They start from the same point but run in opposite directions till they meet each other at the starting point. Then they start running in the same direction till they meet again at the starting point.
For what duration do they run?
a) 5 minutes
b) 10 minutes
c) 15 minutes
d) 20 minutesWhat is the distance covered by Rakesh during the entire session?
a) 4 km
b) 5 km
c) 6 km
d) 8 km

When they run in the opposite direction:
Relative speed = 12 m/s.
So they meet every 600/12 = 50 seconds.
When they meet for the first time, harsh would have run 350m and Rakesh 250m.
LCM of 350 and 600 = 4200
LCM of 250 and 600 = 3000
So they’ll meet at the starting point after covering a total distance of 7200m
Time taken = 4200/7 = 600 sec.
When they run in the same direction:
Relative speed = 2 m/s.
So they meet every 600/2 = 300 seconds.
When they meet for the first time, harsh would have run 2100m and Rakesh 1500m.
LCM of 2100 and 600 = 4200
LCM of 1500 and 600 = 3000
So they’ll meet at the starting point for the second time after covering a total distance of 7200m.
Time taken = 4200/7 = 600 sec.
Total duration = 1200 sec = 20 min.Since both of them ran for 20 minutes, distance covered by Rakesh
= 5x20x60 = 6000m = 6km
[solved by ADURO team]

Q12) Two shooters who are 510m apart shoot a bullet each whose speeds are in the ratio 8:15. The bullets collide at right angle after 3 seconds and are reflected by 90 degrees with respect to their original direction. The speed of the first bullet remains the same while the speed of the second bullet is reduced by 60%. How far are the bullets from each other five seconds after their collision?
a) 400 m
b) 500 m
c) 600 m
d) 700 m

Let the original speeds of the bullets be 8x and 15x respectively.
Distance covered by them in 3 seconds is 24x and 45x.
Since they are shot in perpendicular direction,
51x = 510
x = 10
Hence their speeds before collision were 80m/s and 150m/s.
After collision, their speeds are 80m/s and 60m/s.
Distance travelled by them in 5 seconds after collision = 400m and 300m respectively.
Since they are perpendicular after collision, distance between them after 5 seconds[solved by ADURO team]

Q13) A boy is walking along the direction of 2 parallel railway tracks. on one of these tracks, trains are going on 1 direction at equal intervals. on the other track, trains are going in the opp direction at the same equal intervals. the speed of every train is same . In one direction, a train crosses the boy every 20 mins. and in the opp direction the train passes the boy every 30 mins. If the boy stands still beside the tracks, at intervals of how many minutes will two consecutive trains going in the same direction cross him?

When boy & train are travelling in same direction  relative speed = t  b; for opposite direction = t + b; when boy is stand stil, train will move at = t ; so here tb, t, t+b are in AP => time must be in HP ==> t = 2 * 20 * 30/50 = 24 min

Q14) If 60ml of a drink containing ‘a’ % alcohol is mixed with 240ml of another drink containing ‘b’ % alcohol to give a resultant drink containing 45% alcohol. If a > b, and it is known that b is an integer, then how many integer values can ‘a’ assume?
a) 12
b) 14
c) 13
d) 18

t is given that a > b, and the resulting mixture contains 45% of alcohol.
Combining the two we get a > 45 > b,
We are here combining 60 ml of drink containing ‘a’% alcohol, and 240 ml of drink containing ‘b’% of alcohol.
Thus by using principle of Alligations, we get
60 x (a – 45) = 240 x (45 – b)
or (a – 45) = 4(45 – b)
But since b is an integer, (45 – b) is also an integer
In that case (a – 45) has to be a multiple of 4 such that a > 45
‘a’ can take the following values {45, 49, 53, … , 93, 97}
we have to discard a = 45 as a > b
Thus we get a total of 14  1 = 13 different integer values of ‘a’[solved by ADURO team]

Q15) Akbar and Shiva are princes of two states. One day both of them decided to give a surprise visit to each other without informing the other. Coincidentally they started from their respective capitals at the same time towards each other on the common road between their capitals. Akbar’s horse is a slow starter, for the first hour it gallops at 10 km/h but its speed increases by 5km every hour. On the other hand Shiva’s horse starts at 35 km/h but its speed retards by 5km per hour. They meet each other 5 hours from the start. What is the distance between their capitals?
a)180 km
b) 200 km
c) 225 km
d) 270 km

Distance covered by Akbar’s horse = 10+15+20+25+30 = 100 km
Distance covered by Shiva’s horse = 35+30+25+20+15 = 125 km
Thus, the distance between their capitals = 100+125 = 225 km.[Solved by ADURO team]

Q16) A train A starts from town A for town B at 6 am. Another train B leaves town C, which is 600km from town A in the direction of town B, at 7 am. The two trains meet each other at 11 am at a place which is exactly between towns A and C. thereafter they continue to their destination. If train A reaches B 3.5 hours after train B reaches A, find the distance between the towns A and B?
a) 750 km
b) 720 km
c) 675 km
d) 650 km

5 hrs and A travels 300 km so A has a speed of 60 kmph.
4 hrs and B travels 300km so B would take another 4 hrs to reach A => A would take another 7.5 hrs to reach B.
So, net distance = 300km + 7.5 * 60 = 750km

Q17) A tank has two filling taps, each of which can fill the tank in 160 minutes. The tank also has three emptying pipes connected to it, fixed at 1/4th, 1/2 and 3/4th of the height of the tank respectively. The efficiency of a filling tap, is twice that of an emptying pipe, and the taps and pipes are working simultaneously, from the start. How much time will it take for an empty tank to be filled?
a) 350/3 minutes
b) 400/3 minutes
c) 150 minutes
d) 500/3 minutes

The bottom 1/4th portion of the tank will be subject to only the filling taps, hence time required to fill the bottom 1/4th portion of the tank is 1/4 * 160/2 = 20 mins.
The next 1/4th portion of the tank will be subject to both the filling taps and one of the emptying pipes, hence time required to fill the bottom 1/4th portion of the tank is 1/4 * 160/(21/2) = 80/3 mins.
Similarly the next 1/4th portion of the tank will be filled in 1/4 * 160/(21) = 40 mins.
The upper 1/4th portion of the tank will be filled in 1/4 *160/(23/2) = 80 mins.
Hence total time required to fill the tank = (20 + 80/3 + 40 + 80) = 500/3 minutes.[solved by ADURO team]

Q18) Bikram, Chetan, Ehsan, Gautam, Kapil and Milkha were the top six runners of SK College. Their coach Sunil had to select top 4 for intercollege relay race. Since all of them were not available at the same time, he conducted the race with them in pairs. In the first race, which was of 300m, he defeated Gautam by 30m and Kapil by 40m. In a race of 400m he defeted Milkha by 25m and Ehsan by 40m. In the third race of 500m, he finished ahead of Chetan and Bikram by 50m and 65m respectively. Assuming all run at the same speed in every race, which of these should be chosen for the relay team?
a) Gautam, Milkha, Chetan, Kapil
b) Gautam, Milkha, Chetan, Ehsan
c) Chetan, Milkha, Ehsan, Bikram
d) Gautam, Milkha, Ehsan, Kapil

Q19) Suresh can work for three hours non–stop but then needs to rest for half an hour. His wife can work for two hours but rests for 15 min after that, while his son can work for 1 hour before resting for half an hour. If a work takes 50 man–hours to get completed, then approximately how long will it take for the three to complete the same? Assume all of them all equally skilled in their work.
a) 15
b) 17
c) 20
d) 24

Q20) A big machine takes 20 hrs to fill a large production order and a small machine 30 hrs.. What is the min no. of such small machines that should be operated with 2 big machines such that a large production order would be completed in 5 hours?

Q21) In a examination, students had to find out the average of nine 3digit natural numbers. Shweta by mistake copied down one of the numbers in her answer sheet, in the reverse order of the digits, and proceeded to calculate the average. She got an answer which was 22 more than that of the actual average. How many distinct values are possible for the 3digit number that Shweta erroneously copied down?
a) 7
b) 9
c) 63
d) 70

Q22) Rajat leaves goes to his office by bike driving at 60 km/h. Yesterday was his friend’s birthday and he had bought a gift for him. When he was 5km from his house, he realized he had forgot the gift in his house. He increased his speed by 10km/h for the rest of his journey. It took him 5 minutes at his house to go in the elevator, search for the gift and return to his bike. If he reached his office 10 minutes late than the usual time, find the distance between his house and his office?
a) 35 km
b) 30 km
c) 25 km
d) 18 km