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Hey,vipulgoyal wrote:1,1,18,18,19,32 could be the feasible solution but in any case max value could be 32
If median is 18, middle numbers doesnt necessarily be 18,18 or 17,19. It could be any two numbers whose sum is 36.fcabanski wrote:Identify the problem type. It's a mean, median and mode problem, and all the numbers are positive integers. What do you know about all of this?
First they are all positive integers. None are 0, because it's neither positive nor negative.
mean = sum/(number of numbers) which in this case is 15 = sum/6 so sum = 90
median - is the middle of the numbers. When there's an even number of numbers the median is the average of the middle two numbers. In this case that means the middle numbers are either both 18 (x, y, 18, 18, w, z), or they are 17 and 19 (x, y, 17, 19, w, z).
mode - the number repeated the most, and a group of numbers can have multiple modes. The mode is less than 18, which means the middle numbers have to be 17 and 19. If the two middle numbers were 18, then 18 would be a mode because there are only two numbers that can be below 18 (x and y). If those numbers are the same, then both 18 and x,y would be modes. 18 can't be a mode because it isn't less than 18. It wouldn't matter to the average because 18+18 is the same as 17+19.
It does matter to w and z, however. Since the only mode is less than 18, 19 can't repeat. So w has to be at least 20.
To maximize the largest of the numbers, minimize all the other numbers.
x,y,17,19,w,z
x and y can be 1, so the mode is 1 (less than 18). That's minimizing x and y.
1,1,17,19,w,z
20 is the lowest possibility for w. If w were 19, then 19 would be a mode. 19 is not less than 18.
1,1,17,19,20,z
Now find z, which will be the largest possibility since all the other numbers are as small as they can be.
1+1+17+19+20+z (the sum) = 90
58+z=90
z = 90-58=32
Let's turn this into a system of equations, and say that our set is a, b, c, d, e, and f. Without loss of generality, we'll say that c and d are the two "middle numbers". (In other words, a ≥ b ≥ c ≥ d ≥ e ≥ f.)vipulgoyal wrote: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
Ans 32[spoiler][/spoiler]
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