Two-third of a positive number and `16/216` of its reciprocal are equal. Find the positive number.

Options

A. a. `9/25`
B. b. `14/4`
C. c. `4 /12`
D. d. `144/25`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and derive the solution systematically.

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

### Step 2: Set Up the Equation
According to the problem, two-thirds of the positive number is equal to \( \frac{16}{216} \) of its reciprocal. The reciprocal of \( x \) is \( \frac{1}{x} \). Therefore, we can write the equation as:
\[ \frac{2}{3} x = \frac{16}{216} \cdot \frac{1}{x} \]

### Step 3: Simplify the Right Side
First, simplify \( \frac{16}{216} \):
\[ \frac{16}{216} = \frac{16 \div 8}{216 \div 8} = \frac{2}{27} \]
Now the equation becomes:
\[ \frac{2}{3} x = \frac{2}{27} \cdot \frac{1}{x} \]

### Step 4: Cross-Multiply
To eliminate the fractions, we can cross-multiply:
\[ \frac{2}{3} x \cdot x = \frac{2}{27} \]
This simplifies to:
\[ \frac{2}{3} x^2 = \frac{2}{27} \]

### Step 5: Eliminate the Coefficient
Next, we can multiply both sides by \( 3 \) to eliminate the fraction:
\[ 2 x^2 = \frac{2 \cdot 3}{27} \]
This simplifies to:
\[ 2 x^2 = \frac{6}{27} \]

### Step 6: Further Simplify
Now simplify \( \frac{6}{27} \):
\[ \frac{6}{27} = \frac{6 \div 3}{27 \div 3} = \frac{2}{9} \]
So we have:
\[ 2 x^2 = \frac{2}{9} \]

### Step 7: Divide Both Sides by 2
Now, divide both sides by 2:
\[ x^2 = \frac{2}{9} \div 2 = \frac{2}{9} \cdot \frac{1}{2} = \frac{1}{9} \]

### Step 8: Take the Square Root
Finally, take the square root of both sides:
\[ x = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

### Step 9: Conclusion
Thus, the positive number is:
\[ \boxed{\frac{1}{3}} \]