Answer:
1/6
Step-by-step explanation:
The three numbers, a, b, c, satisfy the equations ...
- a +b +c = 1
- a -b -c = 0
- a +b -5c = 0
Subtracting the last equation from the first gives ...
(a +b +c) -(a +b -5c) = (1) -(0)
6c = 1
c = 1/6
Now, we know the sum of the first two numbers is 5/6 (five times the last), and the difference of the first two numbers is 1/6 (equal to the last). Then the least of the first two numbers is ...
(5/6 -1/6)/2 = 1/3
and the third number, 1/6, is shown to be the smallest.
The smallest of these numbers is 1/6.
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In order, the numbers are 1/2, 1/3, 1/6.
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<em>Comment on finding the second-smallest number</em>
For ...
- a +b = 5/6 . . . . . sum of the two numbers
- a -b = 1/6 . . . . . . difference of the two numbers
We can find b by subtracting the second equation from the first:
(a +b) -(a -b) = (5/6) -(1/6)
2b = 4/6 = 2/3 . . . . . simplify
b = 1/3 . . . . . . . . . . . . divide by 2
Please note that half the difference of the sum and difference is the generic solution to finding the smallest in a "sum and difference" problem.
