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The mean of four numbers is 37. The mean of the smallest three of them is 34. If the range of the data is 15, what is the mean of the largest three?
  • a)
    41
  • b)
    38
  • c)
    40
  • d)
    39
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The mean of four numbers is 37. The mean of the smallest three of them...
Calculation:
Let the numbers be x1, x2, x3, x4.
The mean of four numbers x1, x2, x3, x4 = 37
The sum of four numbers x1, x2, x3, x4 = 37 × 4 = 148.
The mean of the smallest three numbers x1, x2, x3 = 34
The sum of the smallest three numbers x1, x2, x3 = 34 × 3 = 102.
∴ The value of the largest number x4 = 148 – 102 = 46.
The Range (Difference between largest and smallest value) x4 – x1 = 15.
∴ Smallest number x1 = 46 – 15 = 31.
Now,
The sum of x2, x3 = Total sum – (sum of smallest and largest number).
⇒ 148 – (46 + 31)
⇒ 148 – 77
⇒ 71
Now,
The mean of the Largest three numbers x2, x3, x4 = (71 + 46)/3 = 117/3 = 39
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The mean of four numbers is 37. The mean of the smallest three of them...
Given information:
- The mean of four numbers is 37.
- The mean of the smallest three of them is 34.
- The range of the data is 15.

To find: The mean of the largest three numbers.

Let's consider the four numbers as a, b, c, and d, where a ≤ b ≤ c ≤ d.

Finding the Mean of the Four Numbers:
The mean of the four numbers is given as 37. The mean is the sum of all the numbers divided by the total number of values. Therefore, we can write the equation as:
(a + b + c + d) / 4 = 37

Finding the Mean of the Smallest Three Numbers:
The mean of the smallest three numbers is given as 34. Since a, b, and c are the smallest three numbers, we can write the equation as:
(a + b + c) / 3 = 34

Finding the Range:
The range is the difference between the largest and smallest numbers. Here, the range is given as 15. Therefore, we can write the equation as:
d - a = 15

Solving the Equations:
We have three equations with three unknowns (a, b, c). We can solve them simultaneously to find the values of a, b, c, and d.

From equation 2, we can express a in terms of b and c:
a = 102 - b - c

Substituting a in equation 3, we have:
(102 - b - c) + b + 15 = 15
102 - c = 15
c = 87

Substituting the value of c in equation 2, we have:
a + b + 87 = 102
a + b = 15

Substituting the values of a and c in equation 1, we have:
(15 + b + 87 + d) / 4 = 37
(102 + b + d) / 4 = 37
102 + b + d = 148
b + d = 46

Solving the equations b + d = 46 and a + b = 15, we find:
a = 0, b = 15, c = 87, d = 31

Therefore, the numbers are 0, 15, 87, and 31.

Mean of the Largest Three Numbers:
To find the mean of the largest three numbers (b, c, d), we can calculate the sum of these numbers and divide it by 3:
Mean = (15 + 87 + 31) / 3
Mean = 133 / 3
Mean ≈ 44.33

Therefore, the mean of the largest three numbers is approximately 44.33, which is closest to option D) 39.
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The mean of four numbers is 37. The mean of the smallest three of them is 34. If the range of the data is 15, what is the mean of the largest three?a)41b)38c)40d)39Correct answer is option 'D'. Can you explain this answer?
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