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The question is from CAT Ratio, Mixtures and Averages. Details about 4 numbers are given and we need to determine the range in which the mean can fall. CAT exam is known to test on basics rather than high funda ideas. A range of CAT questions can be asked from Ratios and Proportions, Mixtures, Alligations and Averages. Make sure you master the topics. 2IIMs CAT questions bank provides you with CAT questions that can help you gear for CAT Exam CAT 2020.

Question 15: 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?

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Explanatory Answer

Method of solving this CAT Question from Ratio, Mixtures and Averages: Question to find the maximum and minimum possible values of the average of a few numbers given some other parts of the jigsaw.

We can assume a, b,c d are in ascending order (with the caveat that numbers can be equal to each other)
a + b + c = 90
b + c + d = 120

We need to find the maximum and minimum value of a + b + c + d.
a + b + c + d = 120 + a. So, this will be minimum when a is minimum. Given a + b + c = 90. a is minimum when b + c is maximum. If b + c is maximum, d should be minimum.
Given that b + c + d = 120, the minimum value d can take is 40 as d cannot be less than b or c.
The highest value b + c can take is 80, when b = c = d = 40. When b = c = d = 40, a = 10.
a + b + c + d = 130. Average = 32.5

Similarly, a + b + c + d = 90 + d. So, this will be maximum when d is maximum.
Given b + c + d = 120. d is maximum when b + c is minimum. If b + c is minimum, a should be maximum.
Given that a + b + c = 90, the maximum value a can take is 30 as a cannot be greater than b or c. The lowest value b + c can take is 60, when a = b = c = 30. When a = b = c = 30, d = 60.
a + b + c + d = 150. Average = 37.5
So, the average has to range from 32.5 to 37.5

The question is "What is the range of values the average of all 4 numbers can take?"

Hence, the answer is "Range from 32.5 to 37.5".


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