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The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34?
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The mean of six positive integers is 15. The median is 18, and the onl...
Given:
- Mean of six positive integers = 15
- Median = 18
- Mode is less than 18
To find:
The maximum possible value of the largest of the six integers
Solution:
Let's assume the six positive integers as a, b, c, d, e, and f, where a ≤ b ≤ c ≤ d ≤ e ≤ f.
Mean of six positive integers:
Mean is the sum of all the numbers divided by the total count.
Mean = (a + b + c + d + e + f)/6
Given that the mean is 15, we have the equation:
(a + b + c + d + e + f)/6 = 15
a + b + c + d + e + f = 90 ...(Equation 1)
Median:
The median is the middle number when the numbers are arranged in ascending order.
Since the median is 18, we have three possible cases:
Case 1: If c = d = 18
Case 2: If b = c = 18
Case 3: If a = b = 18
Mode:
The mode is the number that appears most frequently in the given set of numbers.
Given that the mode is less than 18, we have two possible cases:
Case 1: a < />
Case 2: b < />
Now, let's consider all the possible cases to find the maximum value of the largest number.
Case 1: If c = d = 18
Given that c = d = 18, the equation (Equation 1) becomes:
a + b + 36 + 36 + e + f = 90
a + b + e + f = 18 ...(Equation 2)
Since the mode is less than 18, we have two sub-cases:
Sub-case 1: a < />
In this case, the maximum value of b + e + f will occur when b = e = f = 17. (b, e, and f should be as close to 18 as possible without exceeding it)
So, a + 17 + 17 + 17 = 18
a + 51 = 18
a = -33 (which is not a positive integer)
Sub-case 2: a = 18
In this case, the maximum value of b + e + f will occur when b = e = f = 17. (b, e, and f should be as close to 18 as possible without exceeding it)
So, 18 + 17 + 17 + 17 = 18
69 = 18 (which is not possible)
Hence, this case is not valid.
Case 2: If b = c = 18
Given that b = c = 18, the equation (Equation 1) becomes:
a + 36 + 36 + d + e + f = 90
a + d + e + f = 18 ...(Equation 3)
Since the mode is less than 18, we have two sub-cases:
Sub-case 1: a < />
In
- Mean of six positive integers = 15
- Median = 18
- Mode is less than 18
To find:
The maximum possible value of the largest of the six integers
Solution:
Let's assume the six positive integers as a, b, c, d, e, and f, where a ≤ b ≤ c ≤ d ≤ e ≤ f.
Mean of six positive integers:
Mean is the sum of all the numbers divided by the total count.
Mean = (a + b + c + d + e + f)/6
Given that the mean is 15, we have the equation:
(a + b + c + d + e + f)/6 = 15
a + b + c + d + e + f = 90 ...(Equation 1)
Median:
The median is the middle number when the numbers are arranged in ascending order.
Since the median is 18, we have three possible cases:
Case 1: If c = d = 18
Case 2: If b = c = 18
Case 3: If a = b = 18
Mode:
The mode is the number that appears most frequently in the given set of numbers.
Given that the mode is less than 18, we have two possible cases:
Case 1: a < />
Case 2: b < />
Now, let's consider all the possible cases to find the maximum value of the largest number.
Case 1: If c = d = 18
Given that c = d = 18, the equation (Equation 1) becomes:
a + b + 36 + 36 + e + f = 90
a + b + e + f = 18 ...(Equation 2)
Since the mode is less than 18, we have two sub-cases:
Sub-case 1: a < />
In this case, the maximum value of b + e + f will occur when b = e = f = 17. (b, e, and f should be as close to 18 as possible without exceeding it)
So, a + 17 + 17 + 17 = 18
a + 51 = 18
a = -33 (which is not a positive integer)
Sub-case 2: a = 18
In this case, the maximum value of b + e + f will occur when b = e = f = 17. (b, e, and f should be as close to 18 as possible without exceeding it)
So, 18 + 17 + 17 + 17 = 18
69 = 18 (which is not possible)
Hence, this case is not valid.
Case 2: If b = c = 18
Given that b = c = 18, the equation (Equation 1) becomes:
a + 36 + 36 + d + e + f = 90
a + d + e + f = 18 ...(Equation 3)
Since the mode is less than 18, we have two sub-cases:
Sub-case 1: a < />
In
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The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34?
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The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34?.
The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is a) 26 b) 28 c) 30 d) 32 e) 34?.
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