SOLUTION: The mean of three numbers is 5/9. The difference between the largest and smallest number is 1/2, which happens to be one of the numbers. What is the smallest number? Express your a

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Question 203413: The mean of three numbers is 5/9. The difference between the largest and smallest number is 1/2, which happens to be one of the numbers. What is the smallest number? Express your answer as a common fraction.
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
Let the three numbers be x, y, z in such a way that x < y < z.

Mean of the numbers:
%28x%2By%2Bz%29%2F3=5%2F9 i.e. x+%2B+y+%2B+z+=+5%2F3 ______ (1)

The difference between the largest and smallest number:
z+-+x+=+1%2F2 __________ (2)

One of the numbers is 1%2F2.
(A) If the largest number is 1%2F2 then the smallest number would be 0 and hence the sum of three numbers will be less than 5%2F3.
(B) If the smallest number is 1%2F2 then the largest number would be 1 and hence the middle number would be 5%2F3+-+3%2F2+=+1%2F6 which is less than the smallest number 1%2F2.
Since options (A) and (B) results in contradiction with the given conditions, both of them are false. So the middle number is 1%2F2.
To solve for the numbers:
Adding both sides of equations (1) and (2) we have
y+%2B+2z+=+13%2F6
But, y = middle number = 1%2F2.
So, 1%2F2+%2B+2z+=+13%2F6 i.e. z+=+5%2F6

Thus, x+=+z+-+1%2F2+=+5%2F6+-+1%2F2+=+1%2F3

Hence, the numbers are 1%2F3, 1%2F2 and 5%2F6.