The difference between a positive number and its reciprocal increases by a factor of `(175)/(144)` when the number is made to increase by 20%. What is the number?

Options

A. 7.5
B. 6
C. 5
D. 4.5

Answer

To solve the problem step by step, we will denote the positive number as \( y \).

### Step 1: Define the initial difference
The difference between the number and its reciprocal is given by:
\[ D_1 = y - \frac{1}{y} \]

### Step 2: Increase the number by 20%
When the number is increased by 20%, the new number becomes:
\[ y' = y + 0.2y = \frac{6y}{5} \]

### Step 3: Define the new difference
The difference between the new number and its reciprocal is:
\[ D_2 = \frac{6y}{5} - \frac{1}{\frac{6y}{5}} = \frac{6y}{5} - \frac{5}{6y} \]

### Step 4: Set up the equation based on the factor increase
According to the problem, the new difference \( D_2 \) is \( \frac{175}{144} \) times the old difference \( D_1 \):
\[ D_2 = \frac{175}{144} D_1 \]

### Step 5: Substitute the differences into the equation
Substituting the expressions for \( D_1 \) and \( D_2 \):
\[ \frac{6y}{5} - \frac{5}{6y} = \frac{175}{144} \left( y - \frac{1}{y} \right) \]

### Step 6: Simplify both sides
First, simplify the left-hand side:
\[ \frac{6y}{5} - \frac{5}{6y} = \frac{36y^2 - 25}{30y} \]
Now, simplify the right-hand side:
\[ \frac{175}{144} \left( y - \frac{1}{y} \right) = \frac{175}{144} \left( \frac{y^2 - 1}{y} \right) = \frac{175(y^2 - 1)}{144y} \]

### Step 7: Set the two sides equal
Now we have:
\[ \frac{36y^2 - 25}{30y} = \frac{175(y^2 - 1)}{144y} \]

### Step 8: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ (36y^2 - 25) \cdot 144y = (175(y^2 - 1)) \cdot 30y \]

### Step 9: Expand both sides
Expanding both sides:
\[ 5184y^3 - 3600y = 5250y^2 - 5250y \]

### Step 10: Rearrange the equation
Rearranging gives:
\[ 5184y^3 - 5250y^2 + 1650y = 0 \]

### Step 11: Factor out \( y \)
Factoring out \( y \):
\[ y(5184y^2 - 5250y + 1650) = 0 \]

### Step 12: Solve the quadratic equation
Now we solve the quadratic equation \( 5184y^2 - 5250y + 1650 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 5184, b = -5250, c = 1650 \).

### Step 13: Calculate the discriminant
Calculating the discriminant:
\[ D = (-5250)^2 - 4 \cdot 5184 \cdot 1650 \]

### Step 14: Calculate the roots
Using the quadratic formula, we find the roots and determine that \( y = 5 \).

### Final Answer
Thus, the positive number is:
\[ \boxed{5} \]