Two-third of a positive number and `(25)/(216)` of its reciprocal are equal. The number is

Options

A. `(25)/(144)`
B. `(5)/(12)`
C. `(144)/(25)`
D. `(12)/(5)`

Answer

To solve the problem, we need to find a positive number \( x \) such that two-thirds of this number is equal to \( \frac{25}{216} \) of its reciprocal. Let's break this down step by step.

### Step 1: Define the variable
Let the positive number be \( x \).

### Step 2: Express two-thirds of the number
Two-thirds of the number \( x \) can be expressed as:
\[ \frac{2}{3} x \]

### Step 3: Express \( \frac{25}{216} \) of its reciprocal
The reciprocal of \( x \) is \( \frac{1}{x} \). Therefore, \( \frac{25}{216} \) of its reciprocal can be expressed as:
\[ \frac{25}{216} \cdot \frac{1}{x} = \frac{25}{216x} \]

### Step 4: Set up the equation
According to the problem, these two expressions are equal:
\[ \frac{2}{3} x = \frac{25}{216x} \]

### Step 5: Cross-multiply to eliminate the fractions
To eliminate the fractions, we can cross-multiply:
\[ 2 \cdot 216x = 3 \cdot 25 \]
This simplifies to:
\[ 432x = 75 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by 432:
\[ x = \frac{75}{432} \]

### Step 7: Simplify the fraction
To simplify \( \frac{75}{432} \), we can find the greatest common divisor (GCD) of 75 and 432. The GCD is 3, so we divide both the numerator and denominator by 3:
\[ x = \frac{75 \div 3}{432 \div 3} = \frac{25}{144} \]

### Step 8: Check if it can be expressed as a square
We can express \( \frac{25}{144} \) as:
\[ x = \left(\frac{5}{12}\right)^2 \]

### Step 9: Take the square root
Taking the square root gives us:
\[ x = \frac{5}{12} \]

Since we are looking for a positive number, we discard the negative root.

### Final Answer
Thus, the positive number is:
\[ \boxed{\frac{5}{12}} \]