This is the first post in the CAT 2016 Sprint Preparation Series – Arithmetic. We have posted 20 questions from previous year CAT papers, forums, mock tests, and other entrances that are on par with the level of difficulty you can expect in CAT 2016. We will be posting the solutions and traps/things to look at while solving similar questions so that you are avoid making silly mistakes during the test.

You can go through the entire series by clicking on this link: CAT 2016 Sprint Preparation Series by Learningroots.

# Quant | Set 2 | Arithmetic

## CAT 2016 Sprint Preparation Series

1.

Which of the following statement(s) is/are true?

I. 5^401 > 5.05^400

II. 2^840 > 3^560

III. 2^840 < 5^360

a. Only I

b. Only III

c. Both I and III

d. None of these

2. Four men start simultaneously at uniform speeds from a point ‘X’ for the point ‘Y’. Though they move along the same route, they arrive at ‘Y’ at equal intervals of time. If the speed of the fastest man is 70 km/hr and that of the slowest man is 30 km/hr, then the speeds of the other two men are

a. 39 km/hr, 51 km/hr

b. 33 km/hr, 68 km/hr

c. 36.2 km/hr, 44.8 km/hr

d. 37 km/hr, 48.5 km/hr

3. A truck traveling at 70 mph uses 30% more diesel to travel a certain distance than it does when it travels at the speed of 50 mph. If the truck can travel 19.5 kilometers on a liter of diesel at 50 mph, how far can the truck travel on 10 liters of diesel at a speed of 70 mph.

4. 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.

a. 20 minutes

b. 15 minutes

c. 10 minutes

d. 5 minutes

5. Students of two different sections of a school appeared for a test. Average score of the students of the 1st section is 65 and the average scores of boys and girls of the 1st section are 68 and 64 respectively. For the 2nd section: the average score of the section is 71; and the average score of the boys and girls are also 80 and 65 respectively. It number of girls in 2nd section is half the number of boys in the 1st section then the average score of the students of the two sections together is

a. 67.7

b. 66.6

c. 69.2

d. 70.1

e. 66

6. A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?

a. 2 =< x =< 6

b. 5 =< x =< 8

c. 9 =< x =< 12

d. 11 =< x =< 14

e. 13 =< x =< 18

7. 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?

a. 16

b. 18

c. 14

d. 20

8. A car is being driven, in a straight line and at a uniform speed, towards the base of a vertical tower. The top of the tower is observed from the car and, in the process, it takes 10 min for the angle of elevation to change from 45° to 60°. After how much more time will this car reach the base of the tower?

a. 5 (√3 +1)

b. 6 (√3 + 2)

c. 7 (√3 −1)

d. 8 (√3 − 2)

9. If log3 = 0.47712, log2 = 0.30103, determine how many zeros there are between the decimal point and the first significant digit in (2/3)^100?

10. A woman goes to a mall to withdraw some money from an ATM. She sees that there is a long queue for the same and so, she dejectedly starts walking down a downward-moving escalator and steps down 10 steps to reach the bottom. Just as she reaches the bottom of the escalator, someone announces that another ATM has been replenished with cash on the floor above. She runs back up the downward moving escalator at a speed five times that which she walked down. She covers 25 steps in reaching the top. How many steps are visible on the escalator when it is switched off?

11. Clark Kent and Bruce Wayne run in opposite directions from a point P on a circle with different but constant speeds. Clark Kent runs in the clockwise direction. They meet for the first time at a distance of 900 m in the clockwise direction from P and for the second time at a distance of 800 m in the anticlockwise direction from P. If Bruce Wayne is yet to complete 1 round, then what is the circumference of the circle?

12. The ratio of expenditure and savings is 3:2. If the income increases by 15% and the savings increases by 6%, then by how much percent should his expenditure increase?

13. Two shots are fired from a cannon at an interval of 35 seconds. A soldier who is approaching enemy territory by traveling underwater hears the two shots at a gap of 30 seconds. What is the speed of the sound in water (in m/s) if the soldier is traveling at a speed of 50 m/s towards the source?

14. There are two tanks T1 and T2. Two pipes P1 and P2 are used to fill these tanks. When operating alone, P1 takes 12 hours more to fill T1 than to fill T2, but P2 takes 24 hours more to fill T1 than to fill T2. When operating together how many hours more, will they take to fill T1 than to fill T2?

15. When I left home for office on a particular day my father’s watch showed 7:00 PM, and when I reached my office my watch showed 7:50 PM. Next day in the morning when my father’s watch showed 6:00 AM, my watch showed 7:30 AM. If my father’s watch runs at normal speed and my watch gains 5 minutes per hour, then the time taken by me to reach my office from home, on that day, was closest to

a. 15 minutes

b. 13 minutes 38 seconds

c. 13 minutes 50 seconds

d. 14 minutes 50 seconds

16. Let D be a recurring decimal of the form, D = 0.a1 a2 a1 a2 a1 a2 …., 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?

a. 18

b. 108

c. 198

d. 288

17. A milkman mixes 20 L of water with 80 L of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he has sold. What is the current proportion of water to milk?

a. 2:3

b. 1:2

c. 1:3

d. 3:4

18. Arun, Barun and Kiranmala start from the same place and travel in the same direction at speeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barun and Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmala start?

19. What is the largest sum of rupees which can never be paid using infinite number of coins of denominations Rs. 5, Rs. 7 and Rs. 11?

a. 13

b. 28

c. 48

d. 8

e. More than 48

20. Two cars race around a circular track at constant speeds starting from the same point. If they travel in opposite directions, then they meet every 35 seconds. If they travel in the same direction, then they meet every 3 minutes and 30 seconds. How long would the faster car take to complete one round of the circular track?

a. 96 s

b. 84 s

c. 60 s

d. 72 s

e. Data Insufficient

Solutions:

1. The second and the third ones are easy. We have to simply balance the power and we are done:

II. 2^840 = (2^3)^280 or 8^280 and 3^560 = (3^2)^280 = 9^280. So, it is false.

III. 2^840 = (2^7)^120 = 128^120 and 5^360 = (5^3)^120 = 125^120. So, it is false.

For the first one, binomial expansion helps

(5.05)^400 = 5^400 * (1.01)^400

Now, 1.01^400 is nothing but

400c0 * 1^400 + 400c1 * 1^399 * 0.01^1 + …

1 + 4 + …

So, slightly greater than 5.

So, the entire expression will be slightly greater than 5^401 and so, it is false as well.

PS: Remember that if you go for a smaller number and check say 5^3 vs. 5.05^2, we will get the wrong answer. So, be careful while generalizing in these situations.

2. Let the total distance be 210 km. So, the fastest person takes 3 hours and the slowest person takes 7 hours. As the two men in between are equally spaced, they would take 3 + 4/3 = 13/3 hours and 13/3 + 4/3 = 17/3 hours respectively.

So, their speeds are 210/(13/3) and 210/(17/3) respectively. So, option d is correct.

3. The question is confusing primarily because it deals with a lot of units (mph, %, kilometers per liter, and finally distance). The best thing to do here is to look at the units only and ignore the rest. For e.g. If you multiply the volume in liters with the efficiency in kilometers per liter, you will get the total number of kilometers traveled.

Also, it helps to be suspicious of the information and numbers given to you in any question. In this case, you can see that the question talks in terms of 50 and 70 only. So, we can manage this question without using any formula whatsoever.

50 mph – 1 liter – 19.5 km

70 mph – 1.3 liter – 19.5 km

This is the obvious part. Same distance at 30% more fuel.

So, 1.3 liter for 19.5 km means that 1 liter gives 15 km. So, 10 liters give 150 km. Not at all difficult. Again, be suspicious of a random number like 19.5. It is bound to get canceled out at some point in time.

4. Again, a lot of randomness in the question and very few numbers may lead to the illusion of the question being a bit difficult. It isn’t if you adhere to your 3 basic things from circular motion.

Let the speed of the fastest runner be 2x m/min. The speed of the slowest runner is x m/min.

The time after which two people traveling along the same direction at speeds a and b on a circular track of circumference C is given by C/|a-b|.

In this case, we know that

1000/|2x-x|=5

x = 200 m/min

Fastest runner runs at a speed of 400 m/min.

To cover 4000 m, he will take 10 mins.

5. Simple alligation will tell us that the ratio of boys to girls in the 1^{st} section is 1:3 and that in the 2^{nd} section is 2:3. Let there be 4x students in the 1^{st} section and 5y students in the 2^{nd} section.

3y = x/2 or x = 6y.

Average will be

(65*4x + 71*5y)/(4x + 5y)

1915/29 = approximately 66.

6. There are at least a couple of ways of solving this. The commonest one is from top to bottom. Let’s look at the other one – from bottom to top. If the third customer buys half of the remaining amount and still has half a kg left, the amount of rice remaining before the third customer split it was 1 kg. Similarly, before the second customer bought the extra half kg, there was 1.5 kg of rice left. So, before the second customer made the purchase, there was 3 kg of rice left. And similarly, we can say that the amount of rice at the start was 7 kg.

7. Sum of first n natural numbers is given by

n(n+1)/2

As a number was missed, the result which the student got is less than the total that we get using the formula.

n(n+1)/2 > 575

n(n+1) > 1150

Now, we know that n(n+1) will be greater than n^2 and so, we find the smallest perfect square greater than 1150. This is given by 34^2 = 1156 and so, n will be equal to 34.

This means that the student added the first 34 natural numbers which amounted to 595. As the difference is 20, that is the number that was missed.

8. A bit of visualization would help here. The only length that is being constant is the height of the tower. Let this be x. If the angle of elevation was 45 degrees, the distance from the tower was also x.

After some time, the angle of elevation is 60 degrees and so, using the 30-60-90 theorem, we can say that the distance between the base of the tower and the car is x/√3. So, distance covered is x – x/√3. This is covered in 10 mins and so, the remaining distance i.e. x/√3 will be covered in what time is the question. As speed is constant in both the cases, we get:

(x – x/√3)/10 = (x/√3)/t

x = 10/(√3 – 1)

Multiplying and dividing by the conjugate, we get

x = 5(√3 + 1)

9. Again, you have to simply find the form of log (2/3)^{100}

Approximately equal to 100 {log 2 – log 3} = 100 * (-0.17609) = -17.609 which is equal to .391 and so, number of zeros post the decimal point would be |-18+1| = 17.

10. A tad tricky and lengthy if you ask some. In reality, it is pretty straightforward.

Let the speed of the woman be s and that of the escalator be e.

In the first case, the woman takes 10 steps. Time taken to cover these 10 steps is 10/s. So we get the total steps to be:

Total steps = 10 + 10e/s

In the second case, her speed becomes 5s. Similar to what we did in the first case, we get:

Total steps = 25 – 25e/5s

Understand that we subtract the steps covered by the escalator simply because the two entities are moving in the opposite direction.

Equating the two, we get

10 + 10e/s = 25 – 5e/s

15e/s = 15

or, e = s

Substituting it in the first equation, we get:

Total steps = 20.

11.

The diagram shows us how the motion would be happening. We know that A – B – C = 900 m and A – C – B = 800 m. Let’s say that AC = x m.

So, we know that, the ratio of the speeds of Bruce Wayne and Clark Kent is a ratio of the distance covered:

Sc/Sb = 900/x = (1000 + 2x)/800

Solving this (or by simple observation), you will get x = 400 and the circumference = 1300 m.

12. There was a question in CAT 2015 on incomes, expenditures and savings. Generally these questions are pretty easy and although you might need to assign 2 or may be 3 variables, you will get through easily. This is an easy one. Let expenditure be 3x and savings be 2x. Total income is 5x. New income will be 5.75x and new savings will be 2.12x. So, new expenditure will be 3.63x or will increase by 21%.

13. The distance between the two shots is nothing but the distance traveled by sound in 35 seconds. Let the speed of sound be s. So, the distance between the two shots is 35s.

Now, when the soldier hears the first shot, the distance between him and the second shot is 35s. This distance is covered at a relative speed of (50 + s) as they are moving in the opposite direction. Time taken for this motion to complete is 30 sec. So, we get a simple equation:

35s/(50 + s) = 30

Or, s = 300 m/sec.

14. Let the capacity of the two tanks be a and b respectively. Let P1 fill x units per hour and P2 fill y units per hour. So,

a/x – b/x = 12

a/y – b/y = 24

So, 12x = 24y or x = 2y.

While operating together,

a/(x + y) – b/(x + y) = ?

(a – b)/(x + y) = 24y/3y = 8 hours.

15. Again, an interesting question that looks easy but isn’t quite. As is the case with questions on time zones, we have to take a fixed frame of reference and see how it goes. In this case, the fixed frame of reference is the father’s watch. Let the son’s watch show some time say t when the father’s watch showed 7 pm. So, 11 hours have passed since then (as the father’s watch shows 6 am). In these 11 hours, the son’s watch should have gone ahead by 55 mins. So, the son’s watch should have showed 7.35 pm when he left home. But the son’s watch shows 7.50 pm when he reaches. Now, the time by which the son’s watch has advanced is 15 mins. So, actual time taken will be given by the simple comparison:

60 mins in father’s watch = 65 mins in son’s watch

x mins in father’s watch = 15 mins in son’s watch

So, x = 13.84 minutes or 13 mins 50 seconds approximately.

16. D = 0.a1 a2 a1 a2…

100D = a1a2 + 0.a1 a2 a1 a2…

a1a2 = 99D

As a1 and a2 are integers, D multiplied by 99 will definitely give us an integer and so, 198 fits.

The question is extremely typical and though there could be a lot of variations, the thing you have to look for is the repetition leading to the original D.

17. Extremely straightforward. You can gobble this up during the test. The only thing I want to suggest here is to avoid making equations and think along these lines: If 1/4^{th} of the mixture is removed, it means that 1/4^{th} of the total water and 1/4^{th} of the total milk are removed. So, 20 L of milk is removed. So, 60 L of milk is left. As the basic volume remains the same (100 L), we know that there is 40 L water and 60 L milk. Easy peasy!

18. The thing to remember here is that, relative motion will take place only when __both__ the entities are in motion. So, by the time Barun starts moving, Arun would have covered 60 km. This 60 km will be covered at a relative speed of 10 km per hour. So, Barun would be traveling for 6 hours. Ergo, Barun would have covered 240 km. To cover 240 km, Kiranmala would take 4 hours and so, Kiranmala would start 4 hours after Arun has started.

19. While there is a concept called Chicken McNugget theorem, even a routine jotting down of case will do here:

5a + 7b + 11c should not give us a solution.

From observation, we can see that 1, 2, 3, 4, 6, 8, 9 are definitely not possible. We can make 11 and 12 but 13 isn’t happening. Post this, we can write down cases and check.

14 = 5(0) + 7(2) + 11(0)

15 = 5(3) + 7(0) + 11(0)

16 = 5(1) + 7(0) + 11(1)

17 = 5(2) + 7(0) + 11(1)

18 = 5(0) + 7(1) + 11(1)

Do we need to check anymore? No. Because, after 18, we will essentially need to add a 5 to the corresponding previous cases and we will get the answer. In short, if you get n consecutive combinations possible where n is the smallest denomination, you are done.

20. Again, the basic formula for circular motion that we used in another question.

C/|a-b| will be the time after which, two objects moving with speeds a and b in the same direction will meet along a circular track of circumference C and the time corresponding to objects moving in the opposite direction will be C/|a+b|

So,

C/|a-b|=210

C/|a+b|=35

So, 6(a – b) = (a + b) or, 5a = 7b.

We have to find C/a

C/(2a/7) = 210

C/a = 60 seconds.

Hope you liked today’s set. Will be back tomorrow with a fresh set of 20 questions.

You can go through the entire series by clicking on this link: CAT 2016 Sprint Preparation Series by Learningroots.