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If \(\mathrm{m} > 0\) and \(\mathrm{x}\) is \(\mathrm{m}\) percent of \(\mathrm{y}\), then, in terms of \(\mathrm{m}\), \(\mathrm{y}\) is what percent of \(\mathrm{x}\)?
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
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:
This basic relationship will be our foundation for finding what percent y is of x.
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
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:
Therefore, y is \(\mathrm{(10000/m)}\) percent of x.
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.
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.
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.
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.
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.
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.
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:
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.