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Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed1/3; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 1/10. Find the fraction.
- a)3/8
- b)4/15
- c)5/24
- d)None
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Letus consider a fraction whose denominator is smaller than the square...
Let the numerator be x and the denominator be y.
Given, y = x^2 - 1
Adding 2 to both numerator and denominator, we get (x+2)/(y+2) > 1/3
Substituting y = x^2 - 1, we get (x+2)/(x^2+1) > 1/3
Multiplying both sides by 3(x^2+1), we get 3x + 6 > x^2 + 1
Rearranging, we get x^2 - 3x + 5 < />
This quadratic has no real roots, which means there are no values of x that satisfy this inequality. Therefore, the given fraction cannot exceed 1/3 after adding 2 to both numerator and denominator.
Now, subtracting 3 from both numerator and denominator, we get (x-3)/(y-3) < />
Substituting y = x^2 - 1, we get (x-3)/(x^2-4) < />
Multiplying both sides by 10(x^2-4), we get 10x - 30 < x^2="" -="" />
Rearranging, we get x^2 - 10x + 26 > 0
This quadratic has no real roots, which means there are no values of x that satisfy this inequality. Therefore, the given fraction cannot be smaller than 1/10 after subtracting 3 from both numerator and denominator.
Therefore, there is no fraction that satisfies both conditions, and the answer is option D (None).
Given, y = x^2 - 1
Adding 2 to both numerator and denominator, we get (x+2)/(y+2) > 1/3
Substituting y = x^2 - 1, we get (x+2)/(x^2+1) > 1/3
Multiplying both sides by 3(x^2+1), we get 3x + 6 > x^2 + 1
Rearranging, we get x^2 - 3x + 5 < />
This quadratic has no real roots, which means there are no values of x that satisfy this inequality. Therefore, the given fraction cannot exceed 1/3 after adding 2 to both numerator and denominator.
Now, subtracting 3 from both numerator and denominator, we get (x-3)/(y-3) < />
Substituting y = x^2 - 1, we get (x-3)/(x^2-4) < />
Multiplying both sides by 10(x^2-4), we get 10x - 30 < x^2="" -="" />
Rearranging, we get x^2 - 10x + 26 > 0
This quadratic has no real roots, which means there are no values of x that satisfy this inequality. Therefore, the given fraction cannot be smaller than 1/10 after subtracting 3 from both numerator and denominator.
Therefore, there is no fraction that satisfies both conditions, and the answer is option D (None).
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Community Answer
Letus consider a fraction whose denominator is smaller than the square...
According to the question the given fraction is N2−1N
⟹ Adding 2 to Numerator and denominator gives N2+1N+2>31
⟹N2−3N−5<0
⟹Now subtracting 3 gives N2−2N−1<101
⟹N2−10N+8>0
There is no such Integer value of N that can satisfy the conditions
So, the correct answer is D
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