vitaliyGMAT wrote:
The set \(S\) contains fractions \(\frac{a}{b}\), where \(a\) and \(b\) are positive integers, which don’t share any prime factor. How many of these fractions have the following property: when both numerator and denominator are increase by 1, the value of fraction is increased by 10%?
A) 0
B) 1
C) 2
D) 3
E) 4
Dear
vitaliyGMAT,
I'm happy to respond.
My friend, I don't know where you found this question, but this is NOT a GMAT question. This is an advanced-math brain-teaser, puzzle type question, far more difficult and time-consuming than what the GMAT would ask.
For background, see:
GMAT Shortcut: Adding to the Numerator and DenominatorHere was my approach for finding an answer. The fraction with 1 added to the numerator & denominator is 10% larger than the original. A 10% increase is equivalent to multiplying by the multiplier 1.1 or 11/10. Thus
\(\frac{a+1}{b+1} = \frac{11}{10} \times \frac{a}{b}\)Now, cross multiply
10ab + 10 b = 11ab + 11a
10 b = 1ab + 11a Divide by b.
\(10 = a + 11\frac{a}{b}\)
\(\frac{10-a}{11} = \frac{a}{b}\)From here, it's obvious that b has to have a factor of 11 in it, so I just said: let
b = 11. Then (
10 - a) = a, so
a = 5, and we get the fraction
\(\frac{a}{b} = \frac{5}{11}\)When we add 1 to the numerator and denominator, we get
6/12 = 1/2, which is 10% greater than 5/11.
From here, I didn't see a good way to prove it, but it's my intuition that each number less than one would increase by a different percentage when we add 1 to the numerator & denominator, so it seems as though this answer would be the only answer. If there's a more elegant proof of this aspect, I would love to see it.
Once again, this is several levels beyond what the GMAT would ask. Don't be fooled by the fact that someone wrote this in the form of a GMAT question: this would have no place on the GMAT.
Does all this make sense?
Mike