Question

If \(\frac{1}{4x} + \frac{1}{y} = \frac{1}{3} (\frac{1}{x} + \frac{1}{y})\), what is the ratio of x to y?
Note: The OCR from the scanned image is ambiguous. This solution assumes the most likely intended equation is with plus signs, as this leads to one of the answer choices.

• A. 3 to 4
• B. 2 to 3
• C. 1 to 2
  (D) 1 to 8
• D. 1 to 9
• E. 1 to 8

Hint:

When dealing with complex fractions in an equation, a good first step is often to multiply the entire equation by the least common multiple of all the denominators to eliminate the fractions and work with integers. In this case, multiplying by \(12xy\) would also lead to the correct answer.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is an algebra problem where we must manipulate an equation involving two variables to determine their ratio, \(x:y\), which is equivalent to finding the value of the fraction \(\frac{x}{y}\).
Step 2: Key Formula or Approach:
The strategy is to simplify the equation and then isolate all terms with \(x\) on one side and all terms with \(y\) on the other. The assumed equation is: \[ \frac{1}{4x} + \frac{1}{y} = \frac{1}{3} \left(\frac{1}{x} + \frac{1}{y}\right) \] Step 3: Detailed Explanation:
First, distribute the \(\frac{1}{3}\) on the right side of the equation: \[ \frac{1}{4x} + \frac{1}{y} = \frac{1}{3x} + \frac{1}{3y} \] Next, rearrange the equation to group the \(x\) terms and \(y\) terms on opposite sides. \[ \frac{1}{y} - \frac{1}{3y} = \frac{1}{3x} - \frac{1}{4x} \] Find a common denominator for each side to combine the fractions. - For the left side (common denominator is 3y): \[ \frac{3}{3y} - \frac{1}{3y} = \frac{2}{3y} \] - For the right side (common denominator is 12x): \[ \frac{4}{12x} - \frac{3}{12x} = \frac{1}{12x} \] Now, set the simplified expressions equal to each other: \[ \frac{2}{3y} = \frac{1}{12x} \] To solve for the ratio \(\frac{x}{y}\), we can cross-multiply: \[ 2 \times (12x) = 1 \times (3y) \] \[ 24x = 3y \] To get \(\frac{x}{y}\), divide both sides by \(y\) and then by 24: \[ \frac{x}{y} = \frac{3}{24} = \frac{1}{8} \] The ratio of \(x\) to \(y\) is 1 to 8. Step 4: Final Answer:
The ratio of \(x\) to \(y\) is 1 to 8.