Question

If \(x / 2\) divided by \(y / 4=10\), what percentage of \(x\) is \(x-y\) ? A. 40 B. 20 C. 80 D. 60 E. Cannot be determined

Short answer

80%

Step-by-step solution

- Simplify the Given Equation

We start with the given equation: \(\frac{x / 2}{y / 4} = 10\). Simplify this equation. \[\frac{x}{2} \times \frac{4}{y} = 10\]This simplifies to: \[\frac{2x}{y} = 10\]

- Solve for y in terms of x

Rearrange the equation \(\frac{2x}{y} = 10\) to solve for \( y \).\[ \frac{2x}{10} = y \]Simplify to get: \[ y = \frac{x}{5} \]

- Find the Expression for x - y

Use the value of \( y \) found in step 2 to express \( x - y \).\[ x - y = x - \frac{x}{5} \]Combine the terms: \[ x - \frac{x}{5} = \frac{5x}{5} - \frac{x}{5} = \frac{4x}{5} \]

- Calculate the Percentage of x

To find what percentage \( \frac{4x}{5} \) is of \( x \), we use the formula: \[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]So, \[ \text{Percentage} = \frac{\frac{4x}{5}}{x} \times 100 \]Simplify: \[ \text{Percentage} = \frac{4}{5} \times 100 = 80\text{%} \]

Key concepts

Problem Simplification

Problem simplification is a crucial step in solving GMAT questions efficiently. The original problem states: If \(\frac{x}{2}\) divided by \(\frac{y}{4}=10\), what percentage of \(\text{x}\) is \(\text{x}-y\)?
Breaking down this problem starts with simplifying the given equation. We have: \(\frac{x / 2}{y / 4} = 10\).
Simplify the fraction by multiplying both sides to remove the denominators. Transform \(\frac{x}{2} \times \frac{4}{y} = 10\) into \(\frac{2x}{y} = 10\). It seems tricky initially, but tackling complex expressions step by step unveils a simpler form, easing the journey to the solution.

Algebraic Manipulation

Algebraic manipulation is another essential skill. After deriving \(\frac{2x}{y} = 10\), rearrange to isolate \(\text{y}\).
Moving terms around gives \(\frac{2x}{10} = y\) which simplifies to \(\text{y} = \frac{x}{5}\).
This substitution makes the problem easier to handle.
Next, find the value of \(\text{x} - y\) by substituting \(\text{y} = \frac{x}{5}\) back into the expression.
We get \(\text{x} - \frac{x}{5} \) which further simplifies to \(\frac{4x}{5}\).
Combining fraction terms correctly saves effort and minimizes errors, which is crucial in algebraic manipulations.

Percentage Calculation

Finally, understanding percentage calculations can help find what percentage of \(\text{x}\) is \(\text{x} - y\).
Recall the formula: \(\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\).
Given \(\frac{4x}{5}\) as part and \(\text{x}\) as the whole, substitute into the formula: \(\text{Percentage} = \frac{\frac{4x}{5}}{x} \times 100\).
Simplifying yields \(\text{Percentage} = \frac{4}{5} \times 100 = 80\text{\text{%}}\).
These calculations show that \(\text{x} - y\) is 80% of \(\text{x}\), answering and concluding our question effectively.