CAT Arithmetic Questions & Solutions

A sample of real CAT Arithmetic past-year questions, each with a worked solution and the correct answer highlighted. AthenaPrep has 354 Arithmetic questions in all — sign up free to practise them timed.

1. Q1.CAT 2025

   Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours. Her speeds in these four parts follow an arithmetic progression, and the corresponding time taken to cover these four parts follow another arithmetic progression. If she travels at a speed of 960 meters per minute for 30 minutes to cover the first part, then the distance, in meters, she travels in the fourth part is

   • 76800

   • 112000

   • 96000

   • 86400

   Show solution

   According to question, Shruti travels a distance of 224 km in four parts for a total travel time of 3 hours.

   Given, in the first part time required is 30 minutes.

   Also, the time taken to cover these four parts follow arithmetic progression.

   So, let us say the times be 30 minutes, (30+d) minutes, (30+2d) minutes and (30+3d) minutes

   So,  $30+\left(30+d\right)+\left(30+2d\right)+\left(30+3d\right)=180$

   or,  $6d=180-120$

   or,  $6d=60$

   or,  $d=\dfrac{60}{6}=10$ minutes.

   So, the time required in these four parts are 30 minutes, 40 minutes, 50 minutes and 60 minutes respectively.

   Now, in the first part, speed is 960 metres per minute.

   Speed in the four parts is also in arithmetic progression.

   Let's say the speeds be $(960+x)$ , $(960+2x)$ , $(960+3x)$ metres per minute

   So, total distance covered =  $960\times\ 30+\left(960+x\right)\times\ 40+\left(960+2x\right)\times\ 50+\left(960+3x\right)\times\ 60$ metres 

   So,  $960\times\ 30+\left(960+x\right)\times\ 40+\left(960+2x\right)\times\ 50+\left(960+3x\right)\times\ 60=224000$

   or,  $960\left(30+40+50+60\right)+40x+\left(2x\right)\left(50\right)+\left(3x\right)\left(60\right)=224000$

   or,  $172800+320x=224000$

   or,  $320x=224000-172800=51200$

   or,  $x=\dfrac{51200}{320}=160$

   So, the speed in the fourth part is  $960+3x=960+3\times\ 160=1440$ metres per minute

   Time in the fourth part is 60 minutes

   So, distance covered in the fourth part =  $1440\times\ 60=86400$ metres

   So, option D is the correct answer.

2. Q2.CAT 2024

   A bus starts at 9 am and follows a fixed route every day. One day, it traveled at a constant speed of 60 km per hour and reached its destination 3.5 hours later than its scheduled arrival time. Next day, it traveled two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at 40 km per hour to reach just on time. The scheduled arrival time of the bus is

   • 7 : 30 pm

   • 7 : 00 pm

   • 9 : 00 pm

   • 10 : 30 pm

   Show solution

   Let's take the scheduled time taken by the bus to be t
   From the first statement (bus travelling at 60 kmph), we can get the total distance travelled by bus to 60(t+3.5)

   The second scenario gives us that the bus covered two-thirds of the distance in one-third of the time, meaning that the remaining one-third distance was covered in two-thirds of the time, giving us the relation  $\frac{1}{3}st$ covered in  $\frac{2}{3}t$ giving the speed to be  $\frac{s}{2}$ which is given as 40 km/h, thereby giving the usual speed of the bus to be 80 km/hr

   Now the first relation we get 60(t+3.5)=80t
   Giving us t=10.5 hours

   Thus, the bus usually takes 10.5 hours on its journey. 
   Staring at 9:00, it will complete the journey at 7:30 pm

   Therefore, Option A is the correct answer. 

3. Q3.CAT 2023

   There are three persons A, B and C in a room. If a person D joins the room, the average weight of the persons in the room reduces by x kg. Instead of D, if person E joins the room, the average weight of the persons in the room increases by 2x kg. If the weight of E is 12 kg more than that of D, then the value of x is

   • 2

   • 0.5

   • 1

   • 1.5

   Show solution

   Let us assume that A, B, C, D, and E weights are a, b, c, d, and e.

   1st condition

   $\frac{\left(a+b+c\right)}{3}-\frac{\left(a+b+c+d\right)}{4}=x$

   2nd condition

   $\frac{\left(a+b+c+e\right)}{4}-\frac{\left(a+b+c\right)}{3}=2x$

   Adding both the equations, we get:

   $\frac{\left(e-d\right)}{4}=3x$

   => $\frac{\left(e-d\right)}{4}=3x$ => e - d = 12x

   Given that 12x = 12 => x = 1.

4. Q4.CAT 2022

   A glass contains 500 cc of milk and a cup contains 500 cc of water. From the glass, 150 cc of milk is transferred to the cup and mixed thoroughly. Next, 150 cc of this mixture is transferred from the cup to the glass. Now, the amount of water in the glass and the amount of milk in the cup are in the ratio

   • 1 : 1

   • 10 : 13

   • 3 : 10

   • 10 : 3

   Show solution

   Initially: a glass 500cc milk and a cup 500cc water

   Step 1: 150 cc of milk is transferred to the cup from glass

   After step 1: Glass - 350 cc milk, Cup - 150 cc milk and 500 cc water

   Step 2: 150 cc of this mixture is transferred from the cup to the glass

   After step 2: 

   Glass - 350 cc milk + 150 cc mixture with milk:water ratio 3:10

   Cup - 500 cc mixture with milk:water ratio 3:10

   water in glass : milk in cup = $\frac{10}{13}\times150\ :\ \frac{3}{13}\times500=1:1$

   The answer is option A.

5. Q5.CAT 2021

   A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

   Answer: 34

   Show solution

   Let price of smallest cup be 2x and medium be 5x and large be y 
   Now by condition  1 
   we get  $2x\ \times\ \ 5x\ \times\ y\ =800$
   we get  $x^2y\ =80$     (1)
   Now as per second condition ;
   $\left(2x+6\right)\times\ \left(5x+6\right)\ y\ =3200$        (2)
   Now dividing (2) and (1)
   we get $\frac{\left(\left(2x+6\right)\times\ \left(5x+6\right)\right)}{x^2}=40$
   we get $10x^2+42x+36\ =\ 40x^2$
   we get $\ 30x^2-42x-36=0$
   $5x^2-7x-6=0$
   we get x=2
   So 2x=4 and 5x=10
   Now substituting in (1) we get y =20
   Now therefore sum = 4+10+20 =34
    

6. Q6.CAT 2020

   Two circular tracks T1 and T2 of radii 100 m and 20 m, respectively touch at a point A. Starting from A at the same time, Ram and Rahim are walking on track T1 and track T2 at speeds 15 km/hr and 5 km/hr respectively. The number of full rounds that Ram will make before he meets Rahim again for the first time is

   • 5

   • 3

   • 2

   • 4

   Show solution

   

   To complete one round Ram takes 100m/15kmph and Rahim takes 20m/5kmph
   

   They meet for the first time after L.C.M of (100m/15kmph , 20m/5kmph) = 100m/5kmph=20m/kmph.

   Distance traveled by Ram =20m/kmph * 15kmph =300m.

   So, he must have ran 300/100=3 rounds.

   Note:

   CAT gave both 2 and 3 as correct answers because of the word 'before'. 

7. Q7.CAT 2018

   Point P lies between points A and B such that the length of BP is thrice that of AP. Car 1 starts from A and moves towards B. Simultaneously, car 2 starts from B and moves towards A. Car 2 reaches P one hour after car 1 reaches P. If the speed of car 2 is half that of car 1, then the time, in minutes, taken by car 1 in reaching P from A is

   Answer: 12

   Show solution

   Let the distance between A and B be 4x. 
   Length of BP is thrice the length of AP.
   => AP = x and BP = 3x
   
   Let the speed of car 1 be s and the speed of car 2 be 0.5s. 
   Car 2 reaches P one hour (60 minutes) after Car 1 reaches P.
   
   => x/s + 60 = 3x/0.5s
   x/s + 60 = 6x/s
   5x/s = 60
   x/s = 12
   
   Time taken by car 1 in reaching P from A = x/s = 12 minutes.
   Therefore, 12 is the correct answer.  
   

8. Q8.CAT 2017

   Ravi invests 50% of his monthly savings in fixed deposits. Thirty percent of the rest of his savings is invested in stocks and the rest goes into Ravi's savings bank account. If the total amount deposited by him in the bank (for savings account and fixed deposits) is Rs 59500, then Ravi's total monthly savings (in Rs) is

   Answer: 70000

   Show solution

   Let his total savings be 100x.

   He invests 50x in fixed deposits. 30% of 50x, which is 15x is invested in stocks and 35x goes to savings bank.

   It is given 85x = 59500

   x = 700

   Hence, 100x = 70000

9. Passage

   Directions for the following two questions: Shabnam is considering three alternatives to invest her surplus cash for a week. She wishes to guarantee maximum returns on her investment. She has three options, each of which can be utilized fully or partially in conjunction with others.

   Option A: Invest in a public sector bank. It promises a return of +0.10%.

   Option B: Invest in mutual funds of ABC Ltd. A rise in the stock market will result in a return of +5%, while a fall will entail a return of – 3%.

   Option C: Invest in mutual funds of CBA Ltd. A rise in the stock market will result in a return of – 2.5%, while a fall will entail a return of + 2%.

   Q9.CAT 2007

   What strategy will maximize the guaranteed return to Shabnam?

   • 100% in option A

   • 36% in option B and 64% in option C

   • 64% in option B and 36% in option C

   • 1/3 in each of the three options

   • 30% in option A, 32% in option B and 38% in option C

   Show solution

   Let a, b and c be the percentages of amount invested in options A, B and C respectively => a + b + c = 100

   Return attained if there is a rise in the stock market => 0.001a + 0.05b - 0.025c

   Return attained if there is a fall in the stock market => 0.001a - 0.03b + 0.02c

   Maximum guaranteed return is attained when both are equal because it is indifferent to rise and fall in the market.

   0.001a + 0.05b - 0.025c = 0.001a - 0.03b + 0.02c

   => 0.08b = 0.045c => 16b = 9c

   Let's put the values for a, b and c that satisfy the above equation.

   b = 9, c = 16, a = 75 => return = 0.125

   b = 18, c = 32, a = 50 => return = 0.15

   b = 27, c = 48, a = 25 => return = 0.175

   b = 36, c = 64, a = 0 => return = 0.2

   Hence, the maximum guaranteed return is 0.2% and it is attained when 36% is invested in option B and 64% is invested in option C.

10. Q10.CAT 2001

    Three classes X, Y and Z take an algebra test.

    The average score in class X is 83.

    The average score in class Y is 76.

    The average score in class Z is 85.

    The average score of all students in classes X and Y together is 79.

    The average score of all students in classes Y and Z together is 81.

    What is the average for all the three classes?

    • 81

    • 81.5

    • 82

    • 84.5

    Show solution

    Let x , y and z be no. of students in class X, Y ,Z respectively.

    From 1st condition we have

    83*x+76*y = 79*x+79*y which give 4x = 3y.

    Next we have 76*y + 85*z = 81(y+z) which give 4z = 5y .

    Now overall average of all the classes can be given as $\frac{83x+76y+85z}{x+y+z}$

    Substitute the relations in above equation we get, 

    $\frac{83x+76y+85z}{x+y+z}$  = (83*3/4 + 76 + 85*5/4)/(3/4 + 1 + 5/4) = 978/12 = 81.5

11. Q11.CAT 1998

    A yearly payment to the servant is Rs. 90 plus one turban. The servant leaves the job after 9 months and receives Rs. 65 and a turban. Then find the price of the turban.

    • Rs. 10

    • Rs. 15

    • Rs. 7.50

    • Cannot be determined

    Show solution

    Let's say price of turban is x.

    So total price for 12 months will be = $90+x$
    
    total price for 9 months = $\frac{(90+x) \times 9}{12} = (65+x)$
    
    By solving above equation, we will get value of x= 10.

12. Q12.CAT 1996

    Instead of a metre scale, a cloth merchant uses a faulty 120 cm scale while buying, but uses a faulty 80 cm scale while selling the same cloth. If he offers a discount of 20%, what is his overall profit percentage?

    • 20%

    • 25%

    • 40%

    • 15%

    Show solution

    Let's say the cost of the cloth is x rs per metre. Because of the faulty meter, he is paying x for 120 cms when buying. 

     

    So cost of 100 cms = 100x/120.

    He is selling 80 cms for x, so selling price of 100cms of cloth is 100x/80. 

    discount = 20% 

    so the effective selling price is .8*100x/80= x

    profit = SP-CP= x - 100x/120 = x/6 

    Profit % = x/6 divided by 100x/120 = 20%
     
     

CAT Arithmetic previous year questions with solutions — AthenaPrep