If xyz ne 0 and w/x = x/y = y/z, then w/z must be equal to which of the following?

1. Translate the problem requirements: We have four variables w, x, y, z (all non-zero) where three ratios are equal: \(\mathrm{w/x = x/y = y/z}\). We need to find what \(\mathrm{w/z}\) equals in terms of the given answer choices.
2. Establish the chain relationship: Use the equal ratios to express each variable in terms of a common ratio, creating a multiplicative chain from w to z.
3. Apply the chain rule for ratios: Recognize that when ratios are equal in a chain, the overall ratio equals the common ratio raised to the power of the number of steps in the chain.
4. Match with answer choices: Verify our result matches one of the given options by expressing it in the required form.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in plain English. We're told that four numbers w, x, y, and z are all non-zero, and we have three ratios that are all equal to each other:

• The ratio of w to x
• The ratio of x to y
• The ratio of y to z

Think of this like a chain of proportional relationships. If I told you that "A is to B as B is to C as C is to D," you'd have a similar chain where each step maintains the same proportional relationship.

Mathematically, we can write this as: \(\mathrm{w/x = x/y = y/z}\)

Our goal is to find what \(\mathrm{w/z}\) equals, and we need to express it in terms of the answer choices provided.

Process Skill: TRANSLATE - Converting the ratio relationships into a clear mathematical framework

2. Establish the chain relationship

Let's call this common ratio 'r'. This means:

• \(\mathrm{w/x = r}\), so \(\mathrm{w = rx}\)
• \(\mathrm{x/y = r}\), so \(\mathrm{x = ry}\)
• \(\mathrm{y/z = r}\), so \(\mathrm{y = rz}\)

Now here's the key insight: we can create a chain by substituting backwards. Think of it like following a path from w all the way to z.

Starting with \(\mathrm{w = rx}\), but we know \(\mathrm{x = ry}\), so:

\(\mathrm{w = r(ry) = r^2y}\)

But we also know \(\mathrm{y = rz}\), so:

\(\mathrm{w = r^2(rz) = r^3z}\)

This tells us that \(\mathrm{w = r^3z}\), which means \(\mathrm{w/z = r^3}\).

Process Skill: INFER - Recognizing that substitution creates a multiplicative chain effect

3. Apply the chain rule for ratios

We've discovered that \(\mathrm{w/z = r^3}\), where r is our common ratio. But remember, \(\mathrm{r = w/x}\).

So we can substitute: \(\mathrm{w/z = (w/x)^3}\)

This makes intuitive sense! When you have a chain of equal ratios, the overall ratio from start to finish is the common ratio raised to the power of the number of steps. We took 3 steps in our chain (w→x, x→y, y→z), so we get the third power.

Let's verify this with a concrete example. Say \(\mathrm{r = 2}\):

• If \(\mathrm{z = 1}\), then \(\mathrm{y = 2(1) = 2}\)
• If \(\mathrm{y = 2}\), then \(\mathrm{x = 2(2) = 4}\)
• If \(\mathrm{x = 4}\), then \(\mathrm{w = 2(4) = 8}\)

So \(\mathrm{w/z = 8/1 = 8}\), and \(\mathrm{(w/x)^3 = (8/4)^3 = 2^3 = 8}\). ✓

4. Match with answer choices

We found that \(\mathrm{w/z = (w/x)^3}\).

Looking at our answer choices:

• A: \(\mathrm{w/x}\)
• B: \(\mathrm{w/x^3}\)
• C: \(\mathrm{w^2/x^2}\)
• D: \(\mathrm{w^2/x^3}\)
• E: \(\mathrm{w^3/x^3}\)

We need \(\mathrm{(w/x)^3}\), which can be written as \(\mathrm{w^3/x^3}\).

This matches choice E exactly.

Process Skill: MANIPULATE - Recognizing that \(\mathrm{(w/x)^3 = w^3/x^3}\)

Final Answer

The answer is E: \(\mathrm{w^3/x^3}\)

This result shows the beautiful pattern in equal ratio chains: when you have n equal ratios in sequence, the overall ratio from first to last equals the common ratio raised to the nth power.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the chain relationship: Students often fail to recognize that the three equal ratios \(\mathrm{w/x = x/y = y/z}\) form a connected chain. Instead, they may treat these as three separate, unrelated equations. This leads them to attempt algebraic manipulation without seeing the substitution pattern that allows us to express w in terms of z.

2. Overlooking the constraint xyz ≠ 0: Students might not fully appreciate why this constraint is given. This condition ensures that all variables are non-zero, which is crucial for the ratios to be well-defined and for the algebraic manipulations to be valid. Without recognizing this, students may worry unnecessarily about division by zero or fail to use the relationships confidently.

3. Attempting direct algebraic manipulation instead of using substitution: Some students may try to solve this by cross-multiplying or setting up a complex system of equations rather than recognizing that simple substitution through the chain (\(\mathrm{w = rx, x = ry, y = rz}\)) provides the most direct path to the solution.

Errors while executing the approach

1. Making substitution errors in the chain: When working through \(\mathrm{w = rx}\), then \(\mathrm{x = ry}\), students often make mistakes in the substitution process. For example, they might write \(\mathrm{w = rx + ry}\) instead of \(\mathrm{w = r(ry) = r^2y}\), or they may lose track of the exponents when continuing the chain to get \(\mathrm{w = r^3z}\).

2. Forgetting to count the number of steps correctly: Students may incorrectly count how many ratios are in the chain. They might think there are only 2 steps instead of 3, leading them to conclude that \(\mathrm{w/z = r^2}\) instead of \(\mathrm{r^3}\). This happens when they don't carefully track each ratio step: w→x, x→y, y→z.

3. Arithmetic errors with exponents: When calculating \(\mathrm{(w/x)^3}\), students may make basic exponent errors, such as writing it as \(\mathrm{w^3/x}\) instead of \(\mathrm{w^3/x^3}\), or confusing it with \(\mathrm{3w/3x}\). These fundamental algebraic mistakes can lead to selecting an incorrect answer choice.

Errors while selecting the answer

1. Confusing \(\mathrm{(w/x)^3}\) with other similar expressions: After correctly finding that \(\mathrm{w/z = (w/x)^3}\), students may incorrectly match this with answer choices. They might select \(\mathrm{w^2/x^2}\) thinking it represents \(\mathrm{(w/x)^2}\), or choose \(\mathrm{w/x^3}\) by incorrectly distributing the exponent. The key is recognizing that \(\mathrm{(w/x)^3 = w^3/x^3}\).

2. Second-guessing the result: Students who arrive at the correct mathematical result may doubt themselves because the answer \(\mathrm{w^3/x^3}\) looks more complex than the original ratios. They might revert to a simpler-looking choice like \(\mathrm{w/x}\) (choice A), not trusting that the chain relationship genuinely requires cubing the common ratio.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for the common ratio

Since we have \(\mathrm{w/x = x/y = y/z}\), let's call this common ratio k. To work with concrete numbers, let's choose \(\mathrm{k = 2}\) (a simple value that makes calculations straightforward).

So: \(\mathrm{w/x = 2, x/y = 2, y/z = 2}\)

Step 2: Assign specific values starting from z

Let's set \(\mathrm{z = 1}\) (starting with the simplest positive value).

From \(\mathrm{y/z = 2: y = 2z = 2(1) = 2}\)

From \(\mathrm{x/y = 2: x = 2y = 2(2) = 4}\)

From \(\mathrm{w/x = 2: w = 2x = 2(4) = 8}\)

Step 3: Calculate w/z using our concrete values

\(\mathrm{w/z = 8/1 = 8}\)

Step 4: Express this result in terms of our common ratio and original variables

We had \(\mathrm{k = w/x = 2}\), so \(\mathrm{w/x = 2}\)

We found \(\mathrm{w/z = 8 = 2^3 = (w/x)^3}\)

Step 5: Verify this pattern holds generally

Let's try \(\mathrm{k = 3}\) to confirm:

If \(\mathrm{z = 1}\), then \(\mathrm{y = 3, x = 9, w = 27}\)

So \(\mathrm{w/z = 27/1 = 27 = 3^3 = (w/x)^3}\)

Conclusion: \(\mathrm{w/z = (w/x)^3 = w^3/x^3}\)

This matches answer choice E.