SSC IMD SA 2017 – Question 38

A fraction is greater than three times its reciprocal by 23/6. What is the fraction?

• 9/2
• 2/9
• 9/4
• 4/9

Detailed Explanation:

We are given that a fraction exceeds three times its reciprocal by $23/6$.

Step-by-step Solution:

1. Let the fraction be $x$. Its reciprocal is $\frac{1}{x}$.
2. Three times its reciprocal is $\frac{3}{x}$.
3. According to the problem statement:$$x – \frac{3}{x} = \frac{23}{6}$$
4. Multiply the entire equation by $6x$ to clear fractions:$$6x^2 – 18 = 23x$$$$6x^2 – 23x – 18 = 0$$
5. Factorize the quadratic equation:
   Find two numbers whose product is $6 \times (-18) = -108$ and sum is $-23$. The numbers are $-27$ and $+4$:$$6x^2 – 27x + 4x – 18 = 0$$$$3x(2x – 9) + 2(2x – 9) = 0$$$$(2x – 9)(3x + 2) = 0$$
6. Solving for $x$:$$2x – 9 = 0 \implies x = \frac{9}{2}$$$$3x + 2 = 0 \implies x = -\frac{2}{3}$$
7. Among the given positive options, $x = \frac{9}{2}$.
8. Verification:
   $$\frac{9}{2} – 3\left(\frac{2}{9}\right) = \frac{9}{2} – \frac{2}{3} = \frac{27 – 4}{6} = \frac{23}{6}\quad \text{(Verified)}$$

Therefore, the fraction is 9/2. The correct option is a).