CAT Practice : Averages, Ratios, Mixtures

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The following topics are covered in the CAT quant section from arithmetic: Averages, Ratios, Mixtures. Detailed explanatory answers, solution videos and slide decks are also provided.
  1. Averages - Maximum value

    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?

    1. 40.18 kgs
    2. 40.56 kgs
    3. 40.67 kgs
    4. 40.49 kgs
    • Correct AnswerChoice (B).
      Maximum possible average is 40.56 kgs
      Correct answer
    • Explanatory Answer
    • Averages - Maximum
    • Hard
  2. Averages - maximum possible value

    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......

    1. 99
    2. 100
    3. 87
    4. 95
    • Correct AnswerChoice (A).
      maximum possible score of the topper is 99
      Correct answer
    • Explanatory Answer
    • Averages- Maximum
    • Hard
  3. Ratios - Number Systems

    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?

    1. 99
    2. 198
    3. 330
    4. 132
    • Correct AnswerChoice (C).
      330 pairs
      Correct answer
    • Explanatory Answer
    • Two-digit numbers
    • Medium
  4. Mixtures - Fun question

    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?

    1. 51.5%
    2. 52.67%
    3. 53%
    4. Cannot be determined
    • Correct AnswerChoice (B).
      Correct answer
    • Explanatory Answer
    • Mixtures - Fun question
    • Hard
  5. Averages - Range

    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?

    1. 72
    2. 71.2
    3. 70.6
    4. 69.2
  6. Averages - Mean Median

    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?

    • Correct AnswerMedian is 4, average is 3.875. Correct answer
    • Explanatory Answer
    • Mean Median
    • Medium
  7. Mean and Range

    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?

    1. 2.8 kgs
    2. 3.2 kgs
    3. 3 kgs
    4. 4 kgs
  8. Ratios linear Equations

    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. Rs. 800
    2. Rs. 700
    3. Rs. 600
    4. Rs. 900
    • Correct AnswerChoice (A).
      Rs. 800
      Correct answer
    • Explanatory Answer
    • Fixed + variable component
    • Medium
  9. Mixtures - Number Systems

    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?

    1. 19
    2. 20
    3. 21
    4. 13
  10. Averages - Mean Median

    In a sequence of 25 terms, can 20 terms be below the average? Can 20 terms be between median and average?

  11. Averages - Sets of numbers

    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?

  12. Mixture of two mixtures

    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?

    • Correct AnswerChoice (C).
      Correct answer
    • Explanatory Answer
    • Mixing two mixtures
    • Medium
  13. Averages - Mixtures


    • Correct AnswerDifference between average of class A and class C: 8.5 Correct answer
    • Explanatory Answer
    • Mixtures and Averages
    • Hard
  14. Averages - Sets of numbers

    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?

  15. Mean and Range

    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?

    • Correct AnswerRange from 32.5 to 37.5 Correct answer
    • Explanatory Answer
    • Mean and Range
    • Medium
  16. Mixtures - Alloys

    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?

    • Correct AnswerChoice (D).
      75 kgs ≤ x ≤ 250 kgs
      Correct answer
    • Explanatory Answer
    • Mixtures - Alloys
    • Medium
  17. Mean, median, maximum

    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?

    • Correct Answer23, 25, 38, 39, 40 Correct answer
    • Explanatory Answer
    • Mean, Median, Maximum
    • Medium
  18. Change in average

    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.

    • Correct AnswerAscending order should be e, c, a, d, b Correct answer
    • Explanatory Answer
    • Mean value changes
    • Medium
  19. Mean, Median, Maximum, Minimum

    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?

    • Correct AnswerHighest and lowest possible median is 38 Correct answer
    • Explanatory Answer
    • Mean, Median, Maximum, Minimum
    • Medium
  20. Ratios of two mixtures

    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?

  21. Number of students

    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?

    1. 30
    2. 26
    3. 11
    4. 31
  22. Age of the twin children

    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?

    1. 14 years
    2. 15 years
    3. 11 years
    4. 12 years
    • Correct AnswerChoice (D).
      12 years
      Correct answer
    • Explanatory Answer
    • Age of the twin children
    • Medium
  23. Minimum number of fruits

    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 11. What is the minimum number of fruits in his shop?

    1. 240
    2. 360
    3. 120
    4. 90
    • Correct AnswerChoice (C).
      Correct answer
    • Explanatory Answer
    • Minimum number of fruits
    • Easy
  24. Number of coins

    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?

    1. 110
    2. 127
    3. 154
    4. 165
  25. Alligation

    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.

    1. Rs 12/kg
    2. Rs 14/kg
    3. Rs 16/kg
    4. Rs 6/kg
  26. Alligation

    3L of milk are drawn from a container containing 30L of milk. It is replaced by water and the process is repeated 4 times. What is the ratio of milk to water at the end?

  27. Alligation

    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?

    1. 26%
    2. 25%
    3. 24%
    4. 23%
  28. Alligation

    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?

    1. 3.33 litres
    2. 4 litres
    3. 5.67 litres
    4. 7.14 litres
  29. Alligation

    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 t 2% interest rate?

  30. Alligation

    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:

    1. 10 l
    2. 12 l
    3. 14 l
    4. 4 l
  31. Alligation

    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

    1. 12:1
    2. 11:1
    3. 1:10
    4. 1:12
  32. Alligation

    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?

    1. 3 l
    2. 2.33 l
    3. 2.67 l
    4. 4 l
  33. Alligation

    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.

    1. 0.1 l
    2. 0.909 l
    3. 0.125 l
    4. 0.111 l
  34. Alligation

    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?

    1. 1:3
    2. 17:52
    3. 151:304
    4. 17:35
  35. Alligation

    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?

    1. 40 l
    2. 30 l
    3. 50 l
    4. 36 l
  36. Alligation

    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.

    1. 72
    2. 24
    3. 48
    4. 80
  37. Alligation

    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?

    1. 747:443
    2. 787:1260
    3. 787:473
    4. 747:473
  38. Averages

    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.

    1. 80 years
    2. 60 years
    3. 30 years
    4. 10 years
  39. Averages

    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.

    1. 33.33 kmph
    2. 40 kmph
    3. 50 kmph
    4. 45 kmph
  40. Averages

    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?

    1. 9500
    2. 11,500
    3. 11,000
    4. 10,500
  41. Averages

    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 144 trees one day before the planned finish date. What was the number of trees the group was planning to cut per day?

    1. 16
    2. 32
    3. 8
    4. 64
  42. Averages

    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?

    1. 10 floors 50 rooms
    2. 20 floors 20 rooms
    3. 20 floors 25 rooms
    4. 50 floors 10 rooms
  43. Averages

    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?

    1. 44 and 65
    2. 21 and 42
    3. 34 and 55
    4. 54 and 75
  44. Averages

    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?

    1. 25 and 25
    2. 22.5 and 25
    3. 23.33 and 26
    4. 23.33 and 25
  45. Averages

    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?

    1. 20.8 and 22.4
    2. 20.6 and 22.6
    3. 20.8 and 22.8
    4. 20.8 and 21.8
  46. Averages

    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?

    1. 3
    2. 1.5
    3. T
    4. Can’t be determined
  47. Averages

    If the product of n distinct positive integers is n^n. What is the minimum value of their average if n=6?

    1. 6
    2. 10
  48. Averages

    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?

    1. 16
    2. 20
    3. 24
    4. No such M exists
  49. Averages

    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?

  50. Averages

    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?

    1. 23
    2. 25
    3. 29
    4. Can't be determined

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