Bunuel wrote:
If a and b are positive numbers such that a + b =2, which of the following could be a value of 50a - 100b ?
I. 150
II. 100
III. 50
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
Let’s test each Roman number
I. 150
a + b = 2
and
50a - 100b = 150
a - 2b = 3
Subtracting our two equations, we have:
3b = -1
b = -1/3
Since b is negative, I is not true.
II. 100
a + b = 2
and
50a - 100b = 100
a - 2b = 2
Subtracting our two equations, we have:
3b = 0
b = 0
Since b is not positive, II is not true.
III. 50
a + b = 2
and
50a - 100b = 50
a - 2b = 1
Subtracting our two equations, we have:
3b = 1
b = 1/3
So a = 2 - 1/3 = 5/3. Since both a and b are positive, III is true.
(note: once we saw that I and II were not true, we could have inferred that III must be the correct answer).
Alternate Solution:
We can express a in terms of b as a = 2 - b. Then, the given expression becomes
50a - 100b = 50(2 - b) - 100b = 100 - 150b
We see that for any positive value of b, 100 - 150b is less than 100. Since Roman numeral III is the only choice which is less than 100, that is the only possible value of 50a - 100b.
Answer: C
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