For any fraction \(x\), non integer decimal part = \(x - 1\) (example non-integer decimal part of 1.2 = 1.2 - 1 = 0.2)

The reciprocal part of \( x\)'s non-integer decimal part equals \(x + 1\)

=> Reciprocal of \(( x - 1 )\) = \(x + 1\)

=> \(\frac{1}{(x - 1)}\) = x + 1

cross multiply, we get,

=> 1 = ( \(x\) - 1 ) ( \(x\) + 1 )

Using ( a - b ) * ( a + b ) = \(a^2\) - \(b^2\)

=> 1 = \(x^2\) - \(1^2\) = \(x^2\) - 1

=> \( x^2 \)= 1 + 1 = 2

=> \(x\) = +√2 or \(x\) = -√2

But \(x\) > 0

=> \(x\) = √2

Ans: C

the explanation won't hold for x>2

Did you want to say x > 2 or x > 0? and give us the reason why it not hold (whatever u said)..............?

In such difficult question, we should ask ourselves WHY a numeric value is given other sector.
Lets play with calculator,
√2=1.4142.......
Then 1.4142.......-1=0.4142.......
then M+
Then 1/MR=2.4142...... which is Exactly √2+1.
Everything matches. So, X=√2.
c

You are purporting that x can have only 1 possible value, which is true for 1<x<2 (I'm going to assume the phrasing excludes 0<x<1) but as soon as you go above that range, x can have multiple values, which are the positive solutions for:

\(\frac{1}{(x - n)}\) = x + 1

where n={2,3,4,5...}

So, the answer should be D as there is no information allowing us to tell whether x is √2 or some other value.

The given answer is wrong.

Assume \(x = N+f\), where \(N\) is a non-negative integer (because \(x\) is positive) and \(f\) is the fractional part ranging between \(0\) and \(1\). According to the question, reciprocal of non-integral part of \(x\) is \(x+1\), which according to how we have defined \(x\) can be written as \(1/f = x+1 = (N+1) + f\)

Multiply both sides by \(f\) to get \(1=(N+1)f+f^2\).

This is a simple quadratic equation whose roots can be found by using the quadratic formula.

The positive root will be \((-(N+1)+\sqrt{(N+1)^2+4})/2\) and this will always lie between \(0\) and \(1\). Thus, we have a value of \(f\) independent of \(N\), which implies value of \(x\) is not fixed.

Hence, D is the correct option. eg - consider the numbers 2.30277563773199464655961063373525 or 0.61803398874989484820458683436564 (this is the golden ratio, btw).

PS:

An easier method is to realize that the function \(1/t\) when \(t\) is between \(0\) and \(1\) has the range from \(1\) to infinity. Thus, we can always create the desired number \(x\) because the number \(x+1\) will itself be in the range of the function \(1/t\)

Thank you. Fixed.

Regards

If the condition given was 1<x<2, then what would have been the answer? Was it going to be C?

the fourth root of 40=2.51..
the calc. shows 2.445..

Exactly. So how can this be solved algebraically?

In the previous replies it is solved algebraically such as this one https://gre.myprepclub.com/forum/the-re ... tml#p46710

Carcass the above solution seems easy and time saving to me compared to others , is this solution a good approach to solving the given question

ANY approach is good.

Even if I guess all the questions during the exam and I nail them correctly, even though I do not know 2+2=4.

GMAT, GRE, LSAT do not give you a bonus or a premium if you use the best approach, the most clean one, the most effective, or you guess all the questions.

The only thing that counts is IF you pick it correct. If yes you are great. regardless the best approach, or the most effective.

Your goal is to end the exam on time with the greatest number of questions correct.

I hope now is clear