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minimum value of (a+b)?

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by uptowngirl92 » Sat Sep 26, 2009 6:30 pm
If a & b are real, and ab + 2(a+b) = 21, what is the minimum value of (a+b)?
a) 2
b) 3
c) 4
d) 5
e) 6


How to approach?I do not even kno how to proceed:(
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Re: minimum value of (a+b)?

by Brent@GMATPrepNow » Sat Sep 26, 2009 9:01 pm
uptowngirl92 wrote:If a & b are real, and ab + 2(a+b) = 21, what is the minimum value of (a+b)?
a) 2
b) 3
c) 4
d) 5
e) 6
How to approach?I do not even kno how to proceed:(
I don't think this question is valid.
For example, one solution to the equation is a=-12 and b=-4.5
Here, a+b=-16.5, which is less than all of the answer choices.

Notice, that we can take the equation ab+2(a+b)=21 and solve for a+b to get a+b=(21-ab)/2

From here, we can make 21-ab infinitely small (e.g., -1,000,000,000) by making ab very very big. Subsequently, a+b can become infinitely small. So, I don't believe there is a lower bound for the value of a+b
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by uptowngirl92 » Sat Sep 26, 2009 9:37 pm
But the OA is E :roll:
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Re: minimum value of (a+b)?

by mohitsharda » Sat Sep 26, 2009 10:00 pm
Brent Hanneson wrote:
uptowngirl92 wrote:If a & b are real, and ab + 2(a+b) = 21, what is the minimum value of (a+b)?
a) 2
b) 3
c) 4
d) 5
e) 6
How to approach?I do not even kno how to proceed:(
I don't think this question is valid.
For example, one solution to the equation is a=-12 and b=-4.5
Here, a+b=-16.5, which is less than all of the answer choices.

Notice, that we can take the equation ab+2(a+b)=21 and solve for a+b to get a+b=(21-ab)/2

From here, we can make 21-ab infinitely small (e.g., -1,000,000,000) by making ab very very big. Subsequently, a+b can become infinitely small. So, I don't believe there is a lower bound for the value of a+b

I agree that the question is not valid if a and b are taken as real numbers.
If we take them to be integers, then i think we will get the answer as 6.
Taking that, I approached the question as follows:

ab + 2(a+b) =21
since the sum is odd => one of the numbers is odd, and the other even.

2(a+b) will always be even => ab is odd

Now, looking at the options we have I made the following table
(a+b) ab
6 9
5 11
4 13
3 15
2 17

Clearly, for a+b = 6 & ab = 9, we get a=b=3 as a solution.
In other cases we cant get an ordered pair (a,b) to satisfy the constraints.

If we do get such a question, the process of elimination could be an option.
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Re: minimum value of (a+b)?

by Brent@GMATPrepNow » Sun Sep 27, 2009 6:52 am
mohitsharda wrote: I agree that the question is not valid if a and b are taken as real numbers.
If we take them to be integers, then i think we will get the answer as 6.
Taking that, I approached the question as follows:

ab + 2(a+b) =21
since the sum is odd => one of the numbers is odd, and the other even.

2(a+b) will always be even => ab is odd

Now, looking at the options we have I made the following table
(a+b) ab
6 9
5 11
4 13
3 15
2 17

Clearly, for a+b = 6 & ab = 9, we get a=b=3 as a solution.
In other cases we cant get an ordered pair (a,b) to satisfy the constraints.

If we do get such a question, the process of elimination could be an option.
Even if we say that a and b must be integers, the question is not valid.
Given the equation ab + 2(a+b)=21, we could have the solution a= -7 and b= -7, in which case a+b= -14, which is less than the "official" answer of 6.

Another solution is a= -27 and b= -3, in which case the sum is -30

What is the source of this question?
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