CAT Question Bank  Arithmetic

With a leak it takes 10 hr to fill the tank
1/6  1/x = 1/10
1/x = 1/6  1/10
1/x = 4/60
1/x = 1/15
X = 15 hrs[solved by rajendra rajput]

Q6) A take 5 days more than B to do a job & 9 days more than C; A & B together can do the job in the same time as C. How many days A would take to do it?

Clearly
1/A + 1/B = 1/C
1/A + 1/(A5) = 1/(A9)
Solve this quadratic
You will get
A = 15 Days[solved by rajendra rajput]

Q7) The vessels contain water and milk in the ratio 1:2 and 2:5 are mixed in the ration is 1:4 then what will be the resulting ratio?

Quantity of water in the resultant mixture
1 * 1/3 + 4 * 2/7 = 1/3 + 8/7 = 31/21Quantity of milk in the resultant mixture
1 * 2/3 + 4 * 5/7 = 2/3 + 20/7 = 74/21Required ratio
31/21 : 74/21
= 31 : 74[solved by rajendra rajput]

Q8) Ram and Shyam start at same end of a swimming competition.The length of pool is 50m.The race is for completing 1000m. If Ram beats Shyam and meets him 17 times during the race what could be the speed of Shyam if speed of Ram is 5m/s?

Length of pool is 50m
So 100m (2x of pool)
Since they meet 17 times
So ram covers 17 * 100 = 1700 m
But race is of 1000m
When ram covers the race I.e 1000m
So shyam covers 17001000 = 700m
Ram  shyam
1000> 700
5 > ?
5 * 700/1000
= 3.5 m/s[solved by rajendra rajput]

Q9) A and B can finish a work together in 12 days, and B and C together in 16 days. If A alone works for 5 days and then B alone continues for 7 days, then remaining work is done by C in 13 days. In how many days can C alone finish the complete work?

Lcm of (12,16) = 48 units <  total work
So A+B = 48/12 = 4 units per day
B+C = 48/16 = 3 units per day
Now,
5A+7B+13C = 48 units
5 (A+B)+2 (B+C)+11C = 48
5 (4)+2 (3)+11C = 48
20+6+11C = 48
11C = 22
C = 2 units
So C does 2 units of work daily
So total time required to complete work alone is 48/2 = 24 days[solved by rajendra rajput]

Q10) Around a circular path of 360km, three men start cycling at 48, 60 and 72 km per day. After how many days will they meet?

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