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# If a and b are positive numbers such that a + b =2, which of the foll

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Math Expert
Joined: 02 Sep 2009
Posts: 56307
If a and b are positive numbers such that a + b =2, which of the foll  [#permalink]

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28 Mar 2018, 02:56
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Difficulty:

55% (hard)

Question Stats:

58% (02:20) correct 42% (02:13) wrong based on 87 sessions

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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

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Posts: 1073
If a and b are positive numbers such that a + b =2, which of the foll  [#permalink]

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28 Mar 2018, 03:18
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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

We'll use the minimal and maximal values to find the range of possible answers.
This is a Logical approach.

Since a+b=2 and both are positive, then both are between 0 and 2.
The phrase 50a-100b is maximal when a is as large as possible and b as small as possible.
That is, a is almost 2 and b is almost 0. In this case, 50a-100b is just under 50*2-100*0 = 100.
That is, we can only get numbers smaller than 100: I and II are false.
Since one of our answer choices must be correct, we can mark (C).
(Note that it is unnecessary to solve fully, but if you like see that a = 5/3 and b = 1/3 give the required result)

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Re: If a and b are positive numbers such that a + b =2, which of the foll  [#permalink]

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29 Mar 2018, 17:37
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.

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Re: If a and b are positive numbers such that a + b =2, which of the foll   [#permalink] 29 Mar 2018, 17:37
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