Problem 188 - 1,001 GED Practice Questions For Dummies

Question

If \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{w} \cdot p=1\), which of the following represents the value of \(p\) ? (A) 1 (B) \(\frac{x}{w}\) (C) \(\frac{w}{x}\) (D) \(\frac{1}{p}\)

Step-by-Step Solution

Answer

The value of \(p\) is \(\frac{w}{x}\) (Option C).

Step 1: Write the equation

Write the given equation: \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{w} \cdot p=1\).

Step 2: Simplify the equation

Simplify the equation by cancelling the common terms in the numerator and denominator: \((x \cdot y \cdot z \cdot p) / (y \cdot z \cdot w) = 1\).

Step 3: Solve for p

Now, isolate p in the equation to find its value: \(p = \frac{y \cdot z \cdot w}{x \cdot y \cdot z}\).

Step 4: Cancel common terms

Simplify the expression further by cancelling the common terms y and z: \(p = \frac{w}{x}\).

Step 5: Check the options

Now, compare the value of p that we found to the given answer options. The correct option is (C) \(\frac{w}{x}\).

Key Concepts

Fractions

Fractions can often be seen as intimidating, but they're just a way of representing parts of a whole. A fraction consists of a numerator and a denominator. The numerator is the top number, while the denominator is the bottom one. When multiplying fractions, as in the example, you multiply the numerators together and the denominators together.
For this exercise, multiplying \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{w}\) exemplifies how simplifying can make a task easier. Notice that certain terms can cancel each other out. This is because when a term appears in both the numerator and the denominator, they effectively divide to 1.
This cancellation is crucial in fraction multiplication and simplification. When solved correctly, it allows us to see what is truly affecting the value in a series of multiplications. Becoming comfortable with multiplying and canceling terms in fractions is essential in algebra.
Remember:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify by canceling any identical terms.

Simplifying Equations

Simplifying equations is about making them easier to solve. This involves combining like terms, using arithmetic operations, or canceling terms when necessary. Simplification helps you to see the core components of an equation, making it simpler to isolate the variable you are solving for.
In the given exercise, simplification occurs when cancelling the common terms in the fraction \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{w}\). Terms like \(y\) and \(z\) appear in both the numerator and the denominator. This allows them to be cancelled out, reducing the expression to \(\frac{xz}{yw} \cdot p = 1\).
Once the equation is simplified, your job is far easier, as fewer operations are needed to get to the solution. This approach not only saves time but also reduces the likelihood of errors. Make it a habit to simplify where possible in math to see the solution path more clearly.
Tips for simplifying:

• Look for common terms to cancel.
• Combine like terms as much as possible.
• Re-check each step to ensure accuracy.

Solving for a Variable

Solving for a variable means isolating the variable you want to know more about, often resulting in a solution for that variable. The goal is to get the variable on one side of the equation and everything else on the other. This often involves reversing operations or unraveling steps taken in setting up the equation.
In our example, once the equation is simplified, solving for \(p\) involves getting \(p\) alone. With the expression \(\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{w} \cdot p = 1\), you simplify to find that common terms cancel out, reducing it further. Then, we isolate \(p\) by dividing both sides of the equation by \(\frac{x}{w}\), resulting in \(p = \frac{w}{x}\).
Isolating variables is a fundamental skill in algebra, letting you find unknown values with known quantities. Always be methodical:

• Try to get the variable alone on one side of the equation.
• Use inverse operations, like division to undo multiplication.
• Double-check your manipulation to prevent mistakes.

Understanding how to solve for variables lays the groundwork for more complex algebraic problem-solving techniques.