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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?
  • a)
    32.5 to 37.5
  • b)
    22.5 to 27.5
  • c)
    33.5 to 37.5
  • d)
    23.5 to 27.5
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Consider 4 numbers a, b, c and d. Ram figures that the smallest avera...
Solution:

Given, smallest average of some three of these four numbers is 30 and the largest average of some three of these 4 is 40.

Let the four numbers be a, b, c, and d.

Smallest average of some three numbers is 30

This means that the sum of the three smallest numbers divided by 3 is equal to or greater than 30.

a + b + c >= 90

Similarly, the largest average of some three numbers is 40

This means that the sum of the three largest numbers divided by 3 is equal to or less than 40.

b + c + d <=>

Average of all 4 numbers can be found out as:

(a + b + c + d)/4

We need to find the range of values that this expression can take.

Let's assume that b and c are the middle two numbers. Then, we can write:

a + b + c >= 90 (1)

b + c + d <= 120="">

Adding (1) and (2), we get:

a + 2b + 2c + d >= 210

(a + b + c + d) + b + c >= 210

(a + b + c + d) >= 210 - (b + c)

(a + b + c + d) >= 210 - (a + d)

(a + b + c + d)/4 >= (210 - (a + d))/4

(a + b + c + d)/4 >= 52.5 - (a + d)/4

We know that (a + d)/2 <= (a="" +="" b="" +="" c="" +="" d)/4=""><= (a="" +="">

So, we have:

(a + d)/2 <= (a="" +="" b="" +="" c="" +="" d)/4=""><= (a="" +="">

(a + d) <= 2(a="" +="" b="" +="" c="" +="" d)/4=""><= (a="" +="">

(a + d) <= (a="" +="" b="" +="" c="" +="" d)/2=""><= (a="" +="">

2(a + d) <= a="" +="" b="" +="" c="" +="" d=""><= 2(a="" +="">

2a + 2d <= a="" +="" b="" +="" c="" +="" d=""><= 2a="" +="">

a + d <= (a="" +="" b="" +="" c="" +="" d)/4=""><= (a="" +="">

Therefore, the range of values that the average of all 4 numbers can take is from (a + d)/2 to (a + b + c + d)/4, which is from 27.5 to 32.5.

Hence, the correct option is (a) 32.5 to 37.5.
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Community Answer
Consider 4 numbers a, b, c and d. Ram figures that the smallest avera...
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.
Hence, the correct option is (a).
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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?a)32.5 to 37.5b)22.5 to 27.5c)33.5 to 37.5d)23.5 to 27.5Correct answer is option 'A'. Can you explain this answer?
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