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# If a and b, are positive, what is the value of a + b ?

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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
soloyolodolo wrote:
u1983 wrote:
(1) $$(3^a) (3^b)$$=81 - Sufficient- as $$(3^a) (3^b)$$=$$3^{a+b}$$=$$3^4$$ ==>a+b=4
(2) $$(3^a) (5^b)$$=225 - Not Sufficient - remember the cond is only positive not positive integer.
One way to think of the proof is :
225= 3.3.5.5 = 5.45
Case 1: $$3^2$$ * $$5^2$$ ==>a=2 & b=2.Hence a+b=4
Case 2:one component is 5 so b=1
Now 45 could be $$3^a$$ ==>a=log45/log3 = 3.465 ( $$3^{3.465}$$ almost=45). Hence a+b=4.465

Hence Ans is A.

GMAT will not give false information between the two facts so if

1.) A+B = 4, then

2.) A+B = 4 as well. and in your explanation A+B = 4.465.

Wouldn't the correct solution be D?

2.) A = 2, B = 2?

Hello soloyolodolo .... my sincere apologies. I really could not follow your point. It would be great if you can elaborate a little more.
Thanks in advance.
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
u1983 wrote:
soloyolodolo wrote:
u1983 wrote:
(1) $$(3^a) (3^b)$$=81 - Sufficient- as $$(3^a) (3^b)$$=$$3^{a+b}$$=$$3^4$$ ==>a+b=4
(2) $$(3^a) (5^b)$$=225 - Not Sufficient - remember the cond is only positive not positive integer.
One way to think of the proof is :
225= 3.3.5.5 = 5.45
Case 1: $$3^2$$ * $$5^2$$ ==>a=2 & b=2.Hence a+b=4
Case 2:one component is 5 so b=1
Now 45 could be $$3^a$$ ==>a=log45/log3 = 3.465 ( $$3^{3.465}$$ almost=45). Hence a+b=4.465

Hence Ans is A.

GMAT will not give false information between the two facts so if

1.) A+B = 4, then

2.) A+B = 4 as well. and in your explanation A+B = 4.465.

Wouldn't the correct solution be D?

2.) A = 2, B = 2?

Hello soloyolodolo .... my sincere apologies. I really could not follow your point. It would be great if you can elaborate a little more.
Thanks in advance.

Gmat will never contradict.

In your explanation.

1.) A+B = 4

2.) A+B = 4.465

Does that make sense?

I agree with your explanation of the answer, however based on GMAT rules. I think answer would be D?

Maybe someone else can help. I know there are numerous people in the forums who are much more intelligent and math wizards.

Posted from my mobile device
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
soloyolodolo wrote:

I agree with your explanation of the answer, however based on GMAT rules. I think answer would be D?

Maybe someone else can help. I know there are numerous people in the forums who are much more intelligent and math wizards.

Till the time someone else pitches in, inline is the Official Explanation

Statement (1) gives you that $$(3^a)(3^b)=81$$
Remember that to find the product of two exponential expressions with a common base, you simply add their exponents.
That means that $$(3^a)(3^b)=81$$ becomes 3^(a+b)=81

The right side of the equation can then be rewritten as $$3^4$$ giving you the equation 3^(a+b) = $$3^4$$
so a+b=4

You can therefore conclude that statement (1) is sufficient when taken on its own.

Statement (2) gives that $$(3^a)(5^b)=225$$
If you factor 225, it becomes $$(3^2)(5^2)=(3^a)(5^b)$$, At this point, it is tempting to say that a+b=4
But remember -- you haven’t been told even whether a and b are integers.
It is possible to say that a=4 and that b is equal to an irrational number (log5 $$\frac{225}{81}$$, if you do the math).

Because the equation could yield multiple values of a and b

You must conclude that statement (2) is insufficient and choose answer choice (A).
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
soloyolodolo I agree with you that the two statements would not contradict each other on GMAT's official questions. I also went with D.
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
Bunuel Please I need help with this question. I got D but OA is A. Thanks in advance.
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Re: If a and b are positive, what is the value of a+b [#permalink]
Expert Reply
If a and b, are positive, what is the value of a + b ?

Notice that we are not told that a and b are integers.

(1) $$(3^a) (3^b)=81$$;

$$3^{a+b}=3^4$$;

$$a + b = 4$$.

Sufficient.

(2) $$(3^a) (5^b)=225$$.

Now, if were told that a and b are positive integers, then yes, from $$(3^a) (5^b)=3^2*5^2$$, it would follow that a = 2 and b = 2. But we are not given that, thus it's possible that a is say 1 and b is some irrational number (satisfying 5^b = 225 --> b = ~3.3652...). Not sufficient.

Answer: A.

Similar questions to practice:
https://gmatclub.com/forum/if-3-a-4-b-c- ... 06047.html
https://gmatclub.com/forum/if-x-2y-3-200 ... 92486.html
https://gmatclub.com/forum/if-x-3y-4-5- ... 76981.html

Hope it helps.
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
If we consider any value of a and b other than a=2 and b=2 then 3^a*5^b will never become perfect 225 and only via approximation we need to conclude the value as 225 (a or b will be irrational number). So whether GMAT may test on such definite vs approximate results in real test? Bunuel chetan2u
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
Expert Reply
RupamPaul13 wrote:
If we consider any value of a and b other than a=2 and b=2 then 3^a*5^b will never become perfect 225 and only via approximation we need to conclude the value as 225 (a or b will be irrational number). So whether GMAT may test on such definite vs approximate results in real test? Bunuel chetan2u

You are wrong.

5^b = 225 has a solution for b, b = log(225)/log(5) (which is irrational number). For that value, 5^b is EXACTLY 225.
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
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Re: If a and b, are positive, what is the value of a + b ? [#permalink]
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