The set S contains fractions a/b, where a and be are positive integers

Question

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

mikemcgarry · Answer

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 Denominator Here 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} imes \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 :-)

vitaliyGMAT · Answer

Dear Mike
Thank you very much for your reply and expertise!

I’m new in this forum and I haven’t taken GMAT yet. To say the truth, I feel more comfortable with calculus and differential equations rather than GMAT. But as usually happens I faced the necessity of taking this exam now. That question and previous one about composite powers I took from GMAT prep center in our country. I had no means nor gmat experience to judge their suitability. Dear Myke and other guys reading this please take my apologies.

As for the question at hand here is how I solved it.

I think the trickiest part in it is to compose right equation.

\frac{(x+1)}{(y+1)}=(1+\frac{1}{10})\frac{x}{y} 
 \frac{(x+1)}{(y+1)}=\frac{11x}{10y}

Simplifying this :

11x(y+1)=10y(x+1) 
 11xy+11x=10xy+10y 
 11x-10y+xy=0

This equation we can factorize. We just add  -110  to right and left sides:
 x(11-y)-10y-110=-110 
 x(11-y)-10(y+11)=-110 
 (x-10)(y+11)=-110 
 (10-x)(y+11)=110

Now the number theory comes into action. We need only to factorize 110
As x,y>0 we have only 3 possible pairs  1*110 ;  2*55  and  5*22  (negative numbers and  10*11  we can discard)

Plugging in we have following pairs of x and y:
 (9,99) ,  (8,44)  and  (5,11)

The question demands that x and y should have no common factors except 1, so they are co-prime. From that list only on pair is co-prime. So answer is B.
As I said, by my point of view, the most difficult part is to generate correct equation.

Dear Mike I will appreciate much any your suggestions and recommendations.
Thank you ones again!
Have a nice day
Vitaliy

RiskChamp · Answer

There is a simpler solution actually.

(a+1)/(b+1)=1.1a/b
Simplifying gives b=(1.1a)/(1-0.1a) .....(1)
We know that a and b are positive.

As a is positive, then 1-0.1a>0.
Thus, 0.1a<1

OR

a<10

Now, we know that a is prime. So, possible values of a are 2,3,5,7.

From Eqn (1),

For a = 2, we get b = 11/4, not an integer or a prime.

For a = 3, we get b = 33/7, not an integer or a prime.

For a = 5, we get b = 11, both an integer and a prime.

For a = 7, we get b = 77/3, neither an integer nor a prime.

Thus, there is only one solution.

Kinshook · Answer

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%?

S = {a/b = k : where a and b are positive integers, which don’t share any prime factor e.g. co-primes}

\frac{a+1}{b+1} = \frac{1.1a}{b} = 1.1k

(a+1)b = 1.1a(b+1)
ab + b = 1.1ab + 1.1a
.1ab + 1.1a = b
ab + 11a = 10b
a(b+11) = 10b
 a = \frac{10b}{b+11} = 10(b+11) - \frac{110}{b+11} = 10 - \frac{110}{b+11}

110 = 2*5*11 should be multiple of b+11; Since b>0; b+11>11

b+11 = {22,55,110}

Case 1: b+11 = 22; b=11; a = 10 - 110/22 = 10 -5 = 5; Fraction a/b = 5/11; a+1/b+1 = 6/12 = 1/2 = 1.1*5/11; Feasible
Case 2: b+11 = 55; b=44; a = 10 - 110/55 = 10 - 2 = 8; Fraction a/b = 8/44; Not Feasible since a & b are co-prime
Case 3: b+11 = 110; b=99; a = 10 - 110/99 ; Not feasible since a is a positive integer

IMO B

The set S contains fractions a/b, where a and be are positive integers

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The set S contains fractions a/b, where a and be are positive integers

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards