A positive number x is decreased by 60% and the result is increased by 80%. If y is the value...

1. Translate the problem requirements: We start with positive number \(\mathrm{x}\), apply two operations (decrease by \(60\%\), then increase by \(80\%\)) to get \(\mathrm{y}\), and need to express \(\mathrm{x}\) in terms of \(\mathrm{y}\)
2. Track the sequential transformations: Follow \(\mathrm{x}\) through each percentage operation to establish the relationship between starting value \(\mathrm{x}\) and final value \(\mathrm{y}\)
3. Set up the algebraic relationship: Express \(\mathrm{y}\) in terms of \(\mathrm{x}\) using the results from the transformations
4. Solve for x in terms of y: Rearrange the equation to isolate \(\mathrm{x}\) and match with answer choices

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what's happening in plain English: We start with some positive number \(\mathrm{x}\). We're going to do two things to it in order - first decrease it by \(60\%\), then increase the result by \(80\%\). After these two operations, we get a final value called \(\mathrm{y}\). Our job is to figure out how to express the original number \(\mathrm{x}\) in terms of this final value \(\mathrm{y}\).

Think of it like this: if someone tells you "I started with a number, decreased it by \(60\%\), then increased that result by \(80\%\), and ended up with 72", we want to be able to work backwards to find what number they started with.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical understanding of what transformations occur

2. Track the sequential transformations

Let's follow our number \(\mathrm{x}\) through each step:

Step 1: Decrease x by 60%
If we decrease something by \(60\%\), we're keeping \(40\%\) of it (since \(100\% - 60\% = 40\%\)).
So after the first operation: \(\mathrm{x}\) becomes \(0.40\mathrm{x}\)

Step 2: Increase the result by 80%
Now we take that \(0.40\mathrm{x}\) and increase it by \(80\%\). When we increase something by \(80\%\), we're making it \(180\%\) of what it was (since \(100\% + 80\% = 180\%\)).
So after the second operation: \(0.40\mathrm{x}\) becomes \(1.80 \times (0.40\mathrm{x}) = 0.72\mathrm{x}\)

Therefore, after both operations: \(\mathrm{y} = 0.72\mathrm{x}\)

3. Set up the algebraic relationship

From our tracking above, we found that:
\(\mathrm{y} = 0.72\mathrm{x}\)

Let's convert \(0.72\) to a fraction to make our work cleaner:
\(0.72 = \frac{72}{100} = \frac{18}{25}\)

So our relationship is:
\(\mathrm{y} = \frac{18}{25}\mathrm{x}\)

4. Solve for x in terms of y

Now we need to rearrange our equation to get \(\mathrm{x}\) by itself:

Starting with: \(\mathrm{y} = \frac{18}{25}\mathrm{x}\)

To isolate \(\mathrm{x}\), we multiply both sides by \(\frac{25}{18}\):
\(\mathrm{x} = \mathrm{y} \times \frac{25}{18}\)
\(\mathrm{x} = \frac{25}{18}\mathrm{y}\)

Process Skill: MANIPULATE - Rearranging the equation to solve for the desired variable

4. Final Answer

We found that \(\mathrm{x} = \frac{25}{18}\mathrm{y}\), which matches answer choice D: \(\frac{25}{18}\mathrm{y}\).

Let's verify this makes sense: if we start with \(\mathrm{x} = \frac{25}{18}\mathrm{y}\) and apply our operations:

• After \(60\%\) decrease: \(\frac{25}{18}\mathrm{y} \times 0.40 = \frac{25}{18}\mathrm{y} \times \frac{2}{5} = \frac{10}{18}\mathrm{y}\)
• After \(80\%\) increase: \(\frac{10}{18}\mathrm{y} \times 1.80 = \frac{10}{18}\mathrm{y} \times \frac{9}{5} = \frac{90}{90}\mathrm{y} = \mathrm{y}\) ✓

Answer: D. \(\frac{25}{18}\mathrm{y}\)

Common Faltering Points

Errors while devising the approach

1. Misunderstanding percentage decrease/increase operations
Students often confuse what "decrease by \(60\%\)" means. They might think it means multiply by \(0.60\) instead of keeping \(40\%\) (multiply by \(0.40\)). Similarly, for "increase by \(80\%\)", they might multiply by \(0.80\) instead of \(1.80\). This fundamental misunderstanding of percentage operations will lead to an incorrect relationship between \(\mathrm{x}\) and \(\mathrm{y}\).

2. Confusion about the direction of the problem
The question asks for "\(\mathrm{x}\) in terms of \(\mathrm{y}\)" but students might set up their work to find "\(\mathrm{y}\) in terms of \(\mathrm{x}\)" and then forget to solve for \(\mathrm{x}\). They might stop at \(\mathrm{y} = 0.72\mathrm{x}\) and think they're done, missing that they need to rearrange to get \(\mathrm{x} = \frac{25}{18}\mathrm{y}\).

Errors while executing the approach

1. Arithmetic errors when converting decimals to fractions
Students might incorrectly convert \(0.72\) to a fraction. Common mistakes include writing \(0.72\) as \(\frac{72}{10}\) instead of \(\frac{72}{100}\), or making errors when simplifying \(\frac{72}{100}\) to \(\frac{18}{25}\). These fraction conversion errors will lead to the wrong final answer.

2. Algebraic manipulation errors when solving for x
When rearranging \(\mathrm{y} = \frac{18}{25}\mathrm{x}\) to solve for \(\mathrm{x}\), students might incorrectly divide by the fraction instead of multiplying by its reciprocal, or they might flip the fraction incorrectly. For example, they might write \(\mathrm{x} = \frac{18}{25}\mathrm{y}\) instead of \(\mathrm{x} = \frac{25}{18}\mathrm{y}\).

Errors while selecting the answer

1. Selecting the reciprocal of the correct answer
Students who correctly find that \(\mathrm{y} = \frac{18}{25}\mathrm{x}\) might mistakenly select answer choice A (\(\frac{18}{25}\mathrm{y}\)) instead of the correct answer D (\(\frac{25}{18}\mathrm{y}\)). They forget that they need to solve for \(\mathrm{x}\), not express the relationship they initially found.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient starting value
Let \(\mathrm{x} = 100\) (chosen because percentage calculations with 100 are straightforward)

Step 2: Apply the first operation - decrease by 60%
After decreasing by \(60\%\): \(100 - (60\% \text{ of } 100) = 100 - 60 = 40\)

Step 3: Apply the second operation - increase by 80%
After increasing by \(80\%\): \(40 + (80\% \text{ of } 40) = 40 + 32 = 72\)
So \(\mathrm{y} = 72\)

Step 4: Test the answer choices
We need to find which expression gives us \(\mathrm{x} = 100\) when \(\mathrm{y} = 72\):

1. \(\frac{18}{25} \times 72 = \frac{18 \times 72}{25} = \frac{1296}{25} = 51.84\) ✗
2. \(\frac{5}{60} \times 72 = \frac{5 \times 72}{60} = \frac{360}{60} = 6\) ✗
3. \(\frac{6}{5} \times 72 = \frac{6 \times 72}{5} = \frac{432}{5} = 86.4\) ✗
4. \(\frac{25}{18} \times 72 = \frac{25 \times 72}{18} = 25 \times 4 = 100\) ✓
5. \(\frac{25}{12} \times 72 = \frac{25 \times 72}{12} = 25 \times 6 = 150\) ✗

Step 5: Verify the result
Only choice D gives us our original value \(\mathrm{x} = 100\), confirming that \(\mathrm{x} = \frac{25}{18} \times \mathrm{y}\)