If m > 0 and x is m percent of y, then, in terms of m, y is what percent...

1. Translate the problem requirements: We need to understand that "x is m percent of y" means \(\mathrm{x = (m/100) × y}\), and we want to find what percent y is of x in terms of m
2. Set up the basic relationship: Express the given condition "x is m percent of y" as an algebraic equation
3. Solve for the target relationship: Rearrange the equation to find y in terms of x, then convert this to percentage form
4. Express as percentage: Convert the ratio y/x into percentage terms and simplify to match the answer choices

Execution of Strategic Approach

1. Translate the problem requirements

Let's start with what the problem is telling us in plain English. When we say "x is m percent of y," we're describing a relationship between three quantities: x, y, and m.

Think about it this way: if you have $100 and someone takes 25% of it, they would have $25. In this example, the $25 is 25 percent of the $100.

So when the problem says "x is m percent of y," it means x is the result when we take m percent of y. In mathematical terms, this translates to: \(\mathrm{x = (m/100) × y}\)

The question asks: "y is what percent of x?" This is asking us to flip the relationship around - instead of finding what percent y gives us when we take m percent of it, we want to know what percent x gives us when we take some unknown percent of it to get y.

Process Skill: TRANSLATE - Converting the percentage language into a clear mathematical relationship

2. Set up the basic relationship

Now let's write down what we know algebraically. From our translation above:

\(\mathrm{x = (m/100) × y}\)

This equation captures the relationship "x is m percent of y." Let's verify this makes sense with a concrete example:

• If y = 200 and m = 50, then \(\mathrm{x = (50/100) × 200 = 100}\)
• Indeed, 100 is 50% of 200 ✓

This basic relationship will be our foundation for finding what percent y is of x.

3. Solve for the target relationship

We want to find what percent y is of x. Let's call this unknown percentage "P".

If y is P percent of x, then: \(\mathrm{y = (P/100) × x}\)

But we already have a relationship between x and y from step 2: \(\mathrm{x = (m/100) × y}\)

Let's solve for y in terms of x by rearranging our original equation:

Starting with: \(\mathrm{x = (m/100) × y}\)
Multiply both sides by 100: \(\mathrm{100x = m × y}\)
Divide both sides by m: \(\mathrm{(100x)/m = y}\)

So: \(\mathrm{y = (100/m) × x}\)

Comparing this with our percentage form \(\mathrm{y = (P/100) × x}\), we can see that:

\(\mathrm{(P/100) = (100/m)}\)

Process Skill: MANIPULATE - Rearranging equations to isolate the desired relationship

4. Express as percentage

Now we solve for P by working with our equation \(\mathrm{(P/100) = (100/m)}\):

Multiply both sides by 100:
\(\mathrm{P = 100 × (100/m)}\)
\(\mathrm{P = 10000/m}\)

Let's verify this with our earlier example:

• We had y = 200, m = 50, and x = 100
• Our formula says \(\mathrm{P = 10000/50 = 200}\)
• So y should be 200% of x
• Check: 200% of 100 = 200 ✓

Therefore, y is \(\mathrm{(10000/m)}\) percent of x.

Final Answer

The answer is E. 10000/m

This result makes intuitive sense: if x is a small percentage of y (small m), then y must be a large percentage of x. The relationship \(\mathrm{10000/m}\) captures this inverse relationship perfectly - as m gets smaller, \(\mathrm{10000/m}\) gets larger, which matches our expectation.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the direction of the percentage relationship

Students often confuse which variable is the "part" and which is the "whole" when translating percentage statements. When the problem says "x is m percent of y," students might incorrectly write \(\mathrm{y = (m/100) × x}\) instead of \(\mathrm{x = (m/100) × y}\). This fundamental misunderstanding of the percentage relationship will lead to an incorrect setup that produces the wrong final answer.

2. Incorrectly setting up the reverse percentage question

When asked "y is what percent of x," students may struggle to properly set up the equation \(\mathrm{y = (P/100) × x}\) where P is the unknown percentage. Some students might write \(\mathrm{x = (P/100) × y}\) instead, essentially asking the same question they already answered rather than finding the reverse relationship.

Errors while executing the approach

1. Algebraic manipulation errors when isolating y

When rearranging \(\mathrm{x = (m/100) × y}\) to solve for y, students commonly make mistakes in the algebraic steps. For example, they might incorrectly get \(\mathrm{y = (m/100) × x}\) instead of \(\mathrm{y = (100/m) × x}\). This happens when they don't properly handle the fraction manipulation or division steps.

2. Errors in comparing coefficient forms

When students have \(\mathrm{y = (100/m) × x}\) and need to compare it with \(\mathrm{y = (P/100) × x}\), they might incorrectly conclude that \(\mathrm{P = 100/m}\) instead of recognizing that \(\mathrm{(P/100) = (100/m)}\), which leads to \(\mathrm{P = 10000/m}\). This coefficient comparison step is where many algebraic errors occur.

Errors while selecting the answer

1. Selecting the intermediate result instead of the final percentage

Students might arrive at the relationship \(\mathrm{y = (100/m) × x}\) and mistakenly think that \(\mathrm{100/m}\) is the percentage, selecting answer choice (C) \(\mathrm{1/m}\) or missing that they need to account for the percentage conversion factor of 100. They fail to complete the final step of recognizing that if \(\mathrm{y = (100/m) × x}\), then y is \(\mathrm{(10000/m)}\) percent of x.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose smart values for the variables

Since we need "x is m percent of y," let's choose values that make the percentage calculation clean and easy to work with.

Let's choose: m = 25 and y = 100

This means x is 25% of 100, so x = 25

Step 2: Verify our relationship

We have: x = 25, y = 100, m = 25

Check: Is x equal to m percent of y?

\(\mathrm{25\% \text{ of } 100 = (25/100) × 100 = 25}\) ✓

Step 3: Find what percent y is of x

We need to find what percent 100 is of 25.

\(\mathrm{Percent = (y/x) × 100 = (100/25) × 100 = 4 × 100 = 400\%}\)

Step 4: Express in terms of m

We found that y is 400% of x when m = 25.

Let's check which answer choice gives us 400 when m = 25:

• A. \(\mathrm{100m = 100(25) = 2,500}\) ✗
• B. \(\mathrm{1/(100m) = 1/(100×25) = 1/2,500}\) ✗
• C. \(\mathrm{1/m = 1/25 = 0.04}\) ✗
• D. \(\mathrm{10/m = 10/25 = 0.4}\) ✗
• E. \(\mathrm{10,000/m = 10,000/25 = 400}\) ✓

Step 5: Verify with another set of smart numbers

Let's try m = 50 and y = 200:

Then \(\mathrm{x = 50\% \text{ of } 200 = 100}\)

What percent is y of x? \(\mathrm{(200/100) × 100 = 200\%}\)

Check answer choice E: \(\mathrm{10,000/m = 10,000/50 = 200\%}\) ✓

Therefore, the answer is E.