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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Scott Woodbury-Stewart

Founder and CEO

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