The sum of a number and its reciprocal is `2(1)/(20).` The number is

Options

A. `(5)/(4)" or "(4)/(5)`
B. `(4)/(3)" or "(3)/(4)`
C. `(5)/(6)" or (6)/(5)`
D. `(1)/(6)" or "6`

Explanation

To solve the problem where the sum of a number and its reciprocal is \( \frac{2}{20} \), we will follow these steps:

### Step 1: Set up the equation
Let the number be \( a \). The reciprocal of \( a \) is \( \frac{1}{a} \). According to the problem, we have:
\[ a + \frac{1}{a} = \frac{2}{20} \]

### Step 2: Simplify the right side
We can simplify \( \frac{2}{20} \) to \( \frac{1}{10} \):
\[ a + \frac{1}{a} = \frac{1}{10} \]

### Step 3: Multiply through by \( a \)
To eliminate the fraction, multiply both sides by \( a \):
\[ a^2 + 1 = \frac{1}{10} a \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ a^2 - \frac{1}{10} a + 1 = 0 \]

### Step 5: Clear the fraction
To eliminate the fraction, multiply the entire equation by 10:
\[ 10a^2 - a + 10 = 0 \]

### Step 6: Identify coefficients
In the quadratic equation \( 10a^2 - a + 10 = 0 \), the coefficients are:
- \( A = 10 \)
- \( B = -1 \)
- \( C = 10 \)

### Step 7: Use the quadratic formula
The quadratic formula is given by:
\[ a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting in the values of \( A \), \( B \), and \( C \):
\[ a = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 10 \cdot 10}}{2 \cdot 10} \]
\[ a = \frac{1 \pm \sqrt{1 - 400}}{20} \]
\[ a = \frac{1 \pm \sqrt{-399}}{20} \]

### Step 8: Analyze the discriminant
Since the discriminant \( 1 - 400 = -399 \) is negative, this indicates that there are no real solutions for \( a \). Thus, the number does not exist in the real number system.

### Conclusion
The number that satisfies the condition does not exist in the real numbers.

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