CAT Questions from Ratios and Proportions, Mixtures, Alligations and Averages are a part of CAT Arithmetic in the CAT Exam. They are regularly tested and have been constantly appearing in Quantitative Aptitude Section of CAT Exam. The concepts from Ratios and Proportions, Mixtures and Averages are particularly interesting as they are relatable and can be applied in real life scenarios. CAT exam does not only check for formulaic knowledge in this idea, but also for strong fundamentals and application of the concepts involved. One can usually expect 5~6 questions from these ideas in the CAT exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.
Consider a class of 40 students whose average weight is 40 kgs. m new students join this class whose average weight is n kgs. If it is known that m + n = 50, what is the maximum possible average weight of the class now?
The average score in an examination of 10 students of a class is 60. If the scores of the top five students are not considered, the average score of the remaining students falls by 5. The pass mark was 40 and the maximum mark was 100. It is also known that none of the students failed. If each of the top five scorers had distinct integral scores, the maximum possible score of the topper is......
The ratio of a two-digit natural number to a number formed by reversing its digits is 4 : 7. Which of the following is the sum of all the numbers of all such pairs?
Three friends A, B and C play a game in a pub. The rules are simple. Whenever there is a contest between any two of them, the one who has a higher percentage alcohol should pour 200 ml of his wine into the one having lower percentage alcohol. The game starts as a contest between A and B, then B and C and then C and A. Post this, the game continues in the same cycle on and on. If a player has emptied all his alcohol, then the remaining two play the game with the same rules. If two players have the alcohol of the same percentage level, the younger one pours 200 ml of his alcohol into the elder one’s glass. All three of them start the game with 600 ml of wine. A’s wine has 60% alcohol, B’s has 48% alcohol and C’s has 50% alcohol. They take 3 minutes to play one round of this game. D, a fourth friend leaves the pub immediately after the game begins, returns after an hour and drinks wine from the person who has the highest alcohol percentage. What is the concentration of the alcohol that D had?
5 Scores in a classroom are broken into 5 different ranges, 51-60, 61-70, 71-80, 81-90 and 91-100. The number of students who have scored in each range is given below.
51 to 60 - 3 students, 61 to 70 - 8 students, 71 to 80 - 7 students, 81 to 90 - 4 students, 91 to 100 - 3 students.
Furthermore, we know that the number of students who scored 76 or more is atleast one more than those who scored below 75. What is the minimum possible average overall of this class?
The median of n distinct numbers is greater than the average, does this mean that there are more terms above the average than below it?
In a class of 5 students, average weight of the 4 lightest students is 40 kgs, Average weight of the 4 heaviest students is 45 kgs. What is the difference between the the maximum and minimum possible average weight overall?
Janta Airline has a free luggage allowance for its passengers. If any passenger carries excess luggage, it is charged at a constant rate per kg. The total luggage charge paid by Ravind Jekriwal and Pranas Shubhan is Rs. 1100. If both Ravind and Pranas had carried luggage twice the weight than they actually did, their luggage charges would have been Rs. 2000 and Rs. 1000 respectively. What was the charge levied on Ravind’s luggage?
1 unit of x% alcohol is mixed with 3 units of y% alcohol to give 60% alcohol. If x > y, how many integer values can x take?
In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?
From 10 numbers, a, b, c,...j, all sets of 4 numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the 10 numbers?
Class A has boys to girls in the ratio 2 : 3, Class B has girls to boys in the ratio 5 : 3. If the number of students in Class A is at least twice as many as the number of students in Class B, what is the minimum percentage of boys when both classes are considered together?
One needs to recognize that averages and mixtures are effectively two sides of the same coin. This question is designed to drive home that point.
Natural numbers 1 to 25 (both inclusive) are split into 5 groups of 5 numbers each. The medians of these 5 groups are A, B, C, D and E. If the average of these medians is m, what are the smallest and the largest values m can take?
Consider 4 numbers a, b, c and d. Ram figures that the smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40. What is the range of values the average of all 4 numbers can take?
What percent of students took English and Math but not Science?
100 kgs of an alloy of tin and lead in the ratio 1:3 is mixed with x kgs of an alloy of tin and lead in the ratio 3:2. If the overall alloy should contain between 40% and 50% tin, what is the range of values x can take?
The average of 5 distinct positive integers if 33. What are the maximum and minimum possible values of the median of the 5 numbers if the average of the three largest numbers within this set is 39?
Consider 5 distinct positive numbers a, b, c, d, and e. The average of these numbers is k. If we remove b from this set, the average drops to m (m is less than k). Average of c, b, d and e is K. We also know that c is less than d and e is less than k. The difference between c and b is equal to the difference between e and d. Average of a, b, c and e is greater than m. Write down a, d, c, d and e in ascending order.
Average of 6 distinct positive integers is 33. The median of the three largest numbers is 43. What is the difference between the highest and lowest possible median of the 6 numbers?
In class A, the ratio of boys to girls is 2 : 3. In class B the ratio of boys to girls is 4 : 5. If the ratio of boys to girls in both classes put together is 3 : 4, what is the ratio of number of girls in class A to number of girls in class B?
A certain number of badges were distributed among a class of students. The student who got 1/6th of the total number of badges actually got 5 times the average number of badges the others got! How many students were there in the class?
The average age of a couple was 24 years. After their 1st and 2nd children (twins) were born, the average age of the family became 13.5 years. The average age of the family just after 3rd child was born was 13.2 years. The average age of the family after 4th child was born was 16 years. The current average age of the family is 19 years. What is the current age of the twin children?
A fruit seller has oranges, apples and guavas in the ratio 2:5:8. The number of apples is more than the number of oranges by a number that is a multiple of both 6 and 8. What is the minimum number of fruits in his shop?
A, B and C have a few coins with them. 7 times the number of coins that A has is equal to 5 times the number of coins B has while 6 times the number of coins B has is equal to 11 times the number of coins C has. What is the minimum number of coins with A, B and C put together?
6 kg of Rs 8/kg wheat is mixed with 3 kg of another type of wheat to get a mixture costing Rs 10/kg. Find the price of the costlier wheat.
3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 2 times. What is the ratio of milk to water at the end?
40% of a club’s revenue comes from people of 25 years of age while 60% of its revenue comes from people of 35 years of age. If the club raises its fee by 20% for its 25 years old members and 30% for 35 years old members, what is the percentage increase in overall revenue of the club?
A mixture of 100 litres of spirit and alcohol contains 25% alcohol. How much more alcohol should be added to the mixture to increase the percentage of alcohol to 30% in the new mixture?
Ram borrows Rs 4000 on simple interest from Shyam for a period of 4 years. He borrows a portion of amount at 2% interest and the remaining at 5%. If the interest Shyam earns is Rs 480, How much money did Ram borrow at 2% interest rate?
A milkman purchases milk at Rs 20/litre and mixes 4 litres of water in it. By selling the resultant mixture at the rate of Rs 20/litre, he earns a profit of 40%. The amount of mixture he had with him to sell was:
What is the ratio in which water should be mixed with a coke concentrate costing Rs 15/litre to make a profit of 30% by selling the resultant drink at Rs 18/litre?
A mixture of 40 litres of milk and water, contains 20% of water. How much water must be added to the above mixture to make the water 25% of the resultant mixture?
A man buys juice at Rs 10/litre and dilutes it with water. He sells the mixtures at the cost price and thus gains 11.11%. Find the quantity of water mixed by him in every litre of juice.
Two tanks of similar volume are full of a mixture of oil and water. In the first, the ratio of oil and water is 5:8 and in the second, it is 7:19. If both these tanks are poured in a larger tank, what would be the resultant ratio of oil and water?
A vessel is full of a mixture of methanol and ethanol in which there is 20% ethanol. 10 litres of mixture are drawn off and filled with methanol. If the ethanol is now 15%, what is the capacity of the vessel?
In Kaziranga national park, the residents are either Hippopotamus or Peacocks. When the heads are counted, it comes out to be 96 and when the legs are counted it is 336 in number. Find the number of peacocks in the park.
What would be the ratio of milk and water in a final mixture formed by mixing milk and water that are present in three vessels of capacity 1l, 2l, and 3l respectively and in the ratios 5:1, 3:2 and 4:3 respectively?
A group of 20 people has the oldest person with 90 years of age. The average of the group is reduced by 4, if the oldest person is reduced by someone new, Find the age of the new person.
Ram travels half of his journey by train at 80 kmph, half of the remaining with bus at 40 kmph and the rest with cycle at 20 kmph. Find his average speed during the entire journey.
Sambhunath had a great job in India but he went abroad to earn more money. He realized he had to make at least USD 6000/month in order to justify his foreign trip. He recorded an average of USD 5,500/ month for the first 11 months. What should be his earning on the last month in order for his foreign visit to make sense?
A group of people decided to cut 128 trees in a certain number of days. For the first 4 days, they were able to achieve their planned per day target. However, for the remaining days, the group was able to cut 4 more trees daily than planned. In this way, the group had cut 164 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?
There are 500 rooms in a multi-floored hotel. However, due to a change in rule, the hotel has to decrease the number of floors by 5. However, the management is able to put 5 more rooms in each floor. Over all, the number of rooms in the hotel decreases by 10%. Find the number of floors and the number of rooms/floor the hotel originally had?
For a global sports event, Nike has to make 810 pair of shoes while Adidas has to make 900 pair of shoes, in the same period of time. Nike could complete the order 3 days before the scheduled time while Adidas completed the order 3 days before Nike. How many pair of shoes did each make per day if Adidas made 21 more shoes per day than Nike?
In olympics, a game has 2 groups A and B having participants from 20 and 25 countries respectively having an average score of 20 and 25 respectively. Also,
A: Highest score: 25 Lowest score: 15
B: Highest score: 32 Lowest score: 24
What can be the minimum and maximum value of B’s average if 5 teams are transferred from A to B?
In the previous question, what can be the minimum and maximum value of A’s average if 5 teams are transferred from B to A?
If the average of 9 consecutive number is T. How much will the average increase by if the next 3 consecutive numbers are also added?
If the product of n distinct positive integers is nn. What is the minimum value of their average if n = 6?
If M is a positive integer such that average of 31, 33, M, 36. 37 lies b/w 40 and 43 (both inclusive). What is the number of possible values of M?
Average of ‘n’ number is t. One of the numbers ‘s’ is replaced by ‘z’ and the new average becomes u. What is the relation b/w n, t, s, u and z?
Arun has 13 boxes of chocolates with him, with an average of 17 chocolates per box. If each box has at least 11 chocolates and no two boxes have equal number of chocolates, then what can be the maximum possible number of chocolates in any box?
A Container contains X Litres of Milk. A thief stole 50 Litres of Milk and replaced it with the same quantity of water. He repeated the same process further two times. And thus Milk in the container is only X-122 litres. Then what is the quantity of water in the final mixture?
Given ratio are a : b = 2 : 3, b : c = 5 : 2, c : d = 1 : 4. Find a : b : c.
If there are Rs 495 in a bag in denominations of one-rupee, 50 paisa and 25 paisa coins, which are in the ratio 1 : 8 : 16. How many 50 paisa coins are there in the bag?
Raj invested Rs 76000 in a business. After few months Monty joined him and invests Rs 57000. At the end of the year, both of them share the profits at the ratio of 2:1. After how many months Monty joined Raj?
Three friends divided some apples in the ratio 3 : 5 : 7 among themselves. After consuming 16 apples they found that the remaining number of apples with them was equal to largest number of apples received by one of them at the beginning. Total number of apples these friends initially had was
The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is
An alloy P has copper and zinc in the proportion of 5: 2 (by weight), while another alloy Q has the same metals in the proportion of 3: 4 (by weight). If these two alloys are mixed in the proportion of a : b (by weight), a new alloy R is formed, which has equal contents of copper and zinc. Then, the proportion of copper and zinc in the alloy S, formed by mixing the two alloys P and Q in the proportion of b : a (by weight) is
The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.
Dick is thrice as old as Tom and Harry is twice as old as Dick. If Dick's age is 1 year less than the average age of all three, then Harry's age, in years, is
A batsman played n + 2 innings and got out on all occasions. His average score in these n + 2 innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38 runs in each of the first n innings. In these n innings, his average score was 30 runs and lowest score was x runs. The smallest possible value of x is
Two alcohol solutions, A and B, are mixed in the proportion 1:3 by volume. The volume of the mixture is then doubled by adding solution A such that the resulting mixture has 72% alcohol. If solution A has 60% alcohol, then the percentage of alcohol in solution B is
The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is
In a group of 10 students, the mean of the lowest 9 scores is 42 while the mean of the highest 9 scores is 47. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by
A sum of money is split among Amal, Sunil and Mita so that the ratio of the shares of Amal and Sunil is 3:2, while the ratio of the shares of Sunil and Mita is 4:5. If the difference between the largest and the smallest of these three shares is Rs 400, then Sunil’s share, in rupees, is
Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is
An alloy is prepared by mixing metals A, B, C in the proportion 3 : 4 : 7 by volume. Weights of the same volume of metals A, B, C are in the ratio 5 : 2 : 6. In 130 kg of the alloy, the weight, in kg, of the metal C is
A solution, of volume 40 litres, has dye and water in the proportion 2 : 3. Water is added to the solution to change this proportion to 2 : 5. If one-fourths of this diluted solution is taken out, how many litres of dye must be added to the remaining solution to bring the proportion back to 2 : 3?
The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?
In an examination, Rama's score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11:10:3. Then Anjali's score exceeded Rama's score by
The salaries of Ramesh, Ganesh and Rajesh were in the ratio 6:5:7 in 2010, and in the ratio 3:4:3 in 2015. If Ramesh’s salary increased by 25% during 2010-2015, then the percentage increase in Rajesh’s salary during this period is closest to
The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt. Each of three vessels A, B, C contains 500 ml of salt solution of strengths 10%, 22%, and 32%, respectively. Now, 100 ml of the solution in vessel A is transferred to vessel B. Then, 100 ml of the solution in vessel B is transferred to vessel C. Finally, 100 ml of the solution in vessel C is transferred to vessel A. The strength, in percentage, of the resulting solution in vessel A is
Amala, Bina, and Gouri invest money in the ratio 3 : 4 : 5 in fixed deposits having respective annual interest rates in the ratio 6 : 5 : 4. What is their total interest income (in Rs) after a year, if Bina's interest income exceeds Amala's by Rs 250?
A chemist mixes two liquids 1 and 2. One litre of liquid 1 weighs 1 kg and one litre of liquid 2 weighs 800 gm. If half litre of the mixture weighs 480 gm, then the percentage of liquid 1 in the mixture, in terms of volume, is
Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is
There are two drums, each containing a mixture of paints A and B. In drum 1, A and B are in the ratio 18 : 7. The mixtures from drums 1 and 2 are mixed in the ratio 3 : 4 and in this final mixture, A and B are in the ratio 13 : 7. In drum 2, then A and B were in the ratio
A 20% ethanol solution is mixed with another ethanol solution, say, S of unknown concentration in the proportion 1:3 by volume. This mixture is then mixed with an equal volume of 20% ethanol solution. If the resultant mixture is a 31.25% ethanol solution, then the unknown concentration of S is
A jar contains a mixture of 175 ml water and 700 ml alcohol. Gopal takes out 10% of the mixture and substitutes it by water of the same amount. The process is repeated once again. The percentage of water in the mixture is now
The scores of Amal and Bimal in an examination are in the ratio 11 : 14. After an appeal, their scores increase by the same amount and their new scores are in the ratio 47 : 56. The ratio of Bimal’s new score to that of his original score is
The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7. The ratio of the strength of D to that of A is
A trader sells 10 litres of a mixture of paints A and B, where the amount of B in the mixture does not exceed that of A. The cost of paint A per litre is Rs. 8 more than that of paint B. If the trader sells the entire mixture for Rs. 264 and makes a profit of 10%, then the highest possible cost of paint B, in Rs. per litre, is
A CAT aspirant appears for a certain number of tests. His average score increases by 1 if the first 10 tests are not considered, and decreases by 1 if the last 10 tests are not considered. If his average scores for the first 10 and the last 10 tests are 20 and 30, respectively, then the total number of tests taken by him is [TITA]
Two types of tea, A and B, are mixed and then sold at Rs. 40 per kg. The profit is 10% if A and B are mixed in the ratio 3 : 2, and 5% if this ratio is 2 : 3. The cost prices, per kg, of A and B are in the ratio
A wholesaler bought walnuts and peanuts, the price of walnut per kg being thrice that of peanut per kg. He then sold 8 kg of peanuts at a profit of 10% and 16 kg of walnuts at a profit of 20% to a shopkeeper. However, the shopkeeper lost 5 kg of walnuts and 3 kg of peanuts in transit. He then mixed the remaining nuts and sold the mixture at Rs. 166 per kg, thus making an overall profit of 25%. At what price, in Rs. per kg, did the wholesaler buy the walnuts?
Raju and Lalitha originally had marbles in the ratio 4 : 9. Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became 5 : 6. What fraction of her original number of marbles was given by Lalitha to Raju?
In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years?
Bottle 1 contains a mixture of milk and water in 7 : 2 ratio and Bottle 2 contains a mixture of milk and water in 9 : 4 ratio. In what ratio of volumes should the liquids in Bottle 1 and Bottle 2 be combined to obtain a mixture of milk and water in 3 : 1 ratio?
The average height of 22 toddlers increases by 2 inches when two of them leave this group. If the average height of these two toddlers is one-third the average height of the original 22, then the average height, in inches, of the remaining 20 toddlers is
If a, b, c are three positive integers such that a and b are in the ratio 3 : 4 while b and c are in the ratio 2 : 1, then which one of the following is a possible value of (a + b + c)?
Consider three mixtures - the first having water and liquid A in the ratio 1 : 2, the second having water and liquid B in the ratio 1 : 3, and the third having water and liquid C in the ratio 1 : 4. These three mixtures of A, B, and C, respectively, are further mixed in the proportion 4 : 3 : 2. Then the resulting mixture has
An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group? (TITA)
Suppose, C1, C2, C3, C4, and C5 are five companies. The profits made by C1, C2, and C3 are in the ratio 9 : 10 : 8 while the profits made by C2, C4, and C5 are in the ratio 18 : 19 : 20. If C5 has made a profit of Rs 19 crore more than C1, then the total profit (in Rs) made by all five companies is:
A stall sells popcorn and chips in packets of three sizes: large, super, and jumbo. The numbers of large, super, and jumbo packets in its stock are in the ratio 7 : 17 : 16 for popcorn and 6 : 15 : 14 for chips. If the total number of popcorn packets in its stock is the same as that of chips packets, then the numbers of jumbo popcorn packets and jumbo chips packets are in the ratio:
A class consists of 20 boys and 30 girls. In the mid-semester examination, the average score of the girls was 5 higher than that of the boys. In the final exam, however, the average score of the girls dropped by 3 while the average score of the entire class increased by 2. The increase in the average score of the boys is:
CAT® (Common Admission Test) is a registered trademark of the Indian Institutes of Management. This website is not endorsed or approved by IIMs.
2IIM Online CAT Coaching
A Fermat Education Initiative,
58/16, Indira Gandhi Street,
Kaveri Rangan Nagar, Saligramam, Chennai 600 093