Let A be a positive real number. If B is 90% of A, C is 110% of B, D is...

1. Translate the problem requirements: We need to find the ratio \(\mathrm{E/A}\) by following a chain of percentage changes: \(\mathrm{A} \rightarrow \mathrm{B}\) (\(90\%\) of \(\mathrm{A}\)) \(\rightarrow \mathrm{C}\) (\(110\%\) of \(\mathrm{B}\)) \(\rightarrow \mathrm{D}\) (\(80\%\) of \(\mathrm{C}\)) \(\rightarrow \mathrm{E}\) (\(120\%\) of \(\mathrm{D}\)). Each percentage needs to be converted to its decimal multiplier form.
2. Convert percentages to multipliers: Transform each percentage change into its corresponding decimal multiplier to set up the multiplication chain.
3. Chain the multipliers: Multiply all the decimal factors together to get the overall ratio \(\mathrm{E/A}\) in decimal form.
4. Match to answer choice format: Convert the decimal result into the specific algebraic form presented in the answer choices by recognizing the pattern of \((1 + \text{fraction})(1 + \text{fraction})\) or \((1 - \text{fraction})(1 - \text{fraction})\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what's happening step by step. We have a positive number \(\mathrm{A}\), and we're applying a series of percentage changes to create new values \(\mathrm{B}\), \(\mathrm{C}\), \(\mathrm{D}\), and \(\mathrm{E}\). Think of this like a chain reaction where each step affects the next.

Here's what we know in plain English:

• \(\mathrm{B}\) is \(90\%\) of \(\mathrm{A}\) (\(\mathrm{B}\) gets \(90\%\) of \(\mathrm{A}\)'s value)
• \(\mathrm{C}\) is \(110\%\) of \(\mathrm{B}\) (\(\mathrm{C}\) gets \(110\%\) of \(\mathrm{B}\)'s value)
• \(\mathrm{D}\) is \(80\%\) of \(\mathrm{C}\) (\(\mathrm{D}\) gets \(80\%\) of \(\mathrm{C}\)'s value)
• \(\mathrm{E}\) is \(120\%\) of \(\mathrm{D}\) (\(\mathrm{E}\) gets \(120\%\) of \(\mathrm{D}\)'s value)

Our goal is to find the ratio \(\mathrm{E/A}\) - essentially, how does \(\mathrm{E}\) compare to our original value \(\mathrm{A}\)?

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

2. Convert percentages to multipliers

Now let's convert each percentage into its decimal multiplier form. When we say "\(90\%\) of something," we multiply by \(0.9\). When we say "\(110\%\) of something," we multiply by \(1.1\).

Here are our conversions:

• \(\mathrm{B} = 90\%\) of \(\mathrm{A}\) means \(\mathrm{B} = 0.9 \times \mathrm{A}\)
• \(\mathrm{C} = 110\%\) of \(\mathrm{B}\) means \(\mathrm{C} = 1.1 \times \mathrm{B}\)
• \(\mathrm{D} = 80\%\) of \(\mathrm{C}\) means \(\mathrm{D} = 0.8 \times \mathrm{C}\)
• \(\mathrm{E} = 120\%\) of \(\mathrm{D}\) means \(\mathrm{E} = 1.2 \times \mathrm{D}\)

This gives us a clear multiplication chain that we can follow from \(\mathrm{A}\) all the way to \(\mathrm{E}\).

3. Chain the multipliers

Now we follow the chain by substituting each equation into the next one. Think of this like following a recipe where each ingredient depends on the previous step.

Starting from the end and working backwards:

• \(\mathrm{E} = 1.2 \times \mathrm{D}\)
• But \(\mathrm{D} = 0.8 \times \mathrm{C}\), so \(\mathrm{E} = 1.2 \times (0.8 \times \mathrm{C}) = 1.2 \times 0.8 \times \mathrm{C}\)
• But \(\mathrm{C} = 1.1 \times \mathrm{B}\), so \(\mathrm{E} = 1.2 \times 0.8 \times (1.1 \times \mathrm{B}) = 1.2 \times 0.8 \times 1.1 \times \mathrm{B}\)
• But \(\mathrm{B} = 0.9 \times \mathrm{A}\), so \(\mathrm{E} = 1.2 \times 0.8 \times 1.1 \times (0.9 \times \mathrm{A}) = 1.2 \times 0.8 \times 1.1 \times 0.9 \times \mathrm{A}\)

Therefore: \(\mathrm{E/A} = 1.2 \times 0.8 \times 1.1 \times 0.9\)

Let's calculate this step by step:

• \(1.2 \times 0.8 = 0.96\)
• \(1.1 \times 0.9 = 0.99\)
• \(0.96 \times 0.99 = 0.9504\)

So \(\mathrm{E/A} = 0.9504\)

4. Match to answer choice format

Now we need to express \(0.9504\) in the form given in the answer choices. Let's look for patterns like \((1 + \text{fraction})(1 + \text{fraction})\) or \((1 - \text{fraction})(1 - \text{fraction})\).

Notice that:

• \(0.9 = 1 - 0.1 = 1 - 1/10\)
• \(1.1 = 1 + 0.1 = 1 + 1/10\)
• \(0.8 = 1 - 0.2 = 1 - 1/5\)
• \(1.2 = 1 + 0.2 = 1 + 1/5\)

So our expression \(\mathrm{E/A} = 0.9 \times 1.1 \times 0.8 \times 1.2\) can be rewritten as:
\(\mathrm{E/A} = (1 - 1/10) \times (1 + 1/10) \times (1 - 1/5) \times (1 + 1/5)\)

Using the algebraic identity \((1-x)(1+x) = 1 - x^2\):

• \((1 - 1/10)(1 + 1/10) = 1 - (1/10)^2 = 1 - 1/100\)
• \((1 - 1/5)(1 + 1/5) = 1 - (1/5)^2 = 1 - 1/25\)

Therefore: \(\mathrm{E/A} = (1 - 1/100)(1 - 1/25)\)

Process Skill: MANIPULATE - Recognizing the algebraic pattern to match answer choices

Final Answer

The answer is (B) \((1 - 1/100)(1 - 1/25)\)

We can verify: \((1 - 1/100)(1 - 1/25) = (0.99)(0.96) = 0.9504\), which matches our calculated ratio \(\mathrm{E/A}\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the sequential nature of percentage calculations: Students might think each percentage is applied to the original value \(\mathrm{A}\) instead of understanding that \(\mathrm{B}\) depends on \(\mathrm{A}\), \(\mathrm{C}\) depends on \(\mathrm{B}\), \(\mathrm{D}\) depends on \(\mathrm{C}\), and \(\mathrm{E}\) depends on \(\mathrm{D}\). This leads to incorrect setup like \(\mathrm{C} = 1.1 \times \mathrm{A}\) instead of \(\mathrm{C} = 1.1 \times \mathrm{B}\).
2. Confusion between percentage notation and decimal multipliers: Students may struggle to correctly convert percentages to decimal form, especially getting confused between "\(90\%\) of" (multiply by \(0.9\)) versus "\(90\%\) increase" (multiply by \(1.9\)), or incorrectly converting \(110\%\) as \(0.11\) instead of \(1.1\).

Errors while executing the approach

1. Arithmetic errors in decimal multiplication: When calculating \(1.2 \times 0.8 \times 1.1 \times 0.9\), students commonly make computational mistakes, especially in steps like \(0.96 \times 0.99 = 0.9504\), often getting values like \(0.95\) or \(0.954\).
2. Incorrect substitution order: Students may substitute the variables in the wrong sequence or skip steps, leading to expressions like \(\mathrm{E} = 1.2 \times 1.1 \times 0.9 \times 0.8 \times \mathrm{A}\) instead of properly chaining \(\mathrm{E} = 1.2 \times \mathrm{D}\), where \(\mathrm{D} = 0.8 \times \mathrm{C}\), etc.
3. Fraction conversion errors: When converting decimals to fractions (\(0.9 = 1 - 1/10\), \(1.2 = 1 + 1/5\)), students often make mistakes like writing \(0.8 = 1 - 1/4\) instead of \(1 - 1/5\), or \(1.1 = 1 + 1/100\) instead of \(1 + 1/10\).

Errors while selecting the answer

1. Misapplying the difference of squares identity: Students may incorrectly apply \((1-x)(1+x) = 1-x^2\) by mixing up which terms to pair together, such as pairing \((1-1/10)\) with \((1-1/5)\) instead of pairing \((1-1/10)\) with \((1+1/10)\), leading them to select answer choice (D) instead of (B).
2. Sign errors in final expression: After correctly calculating the numerical value, students might incorrectly match it to \((1+1/100)(1+1/25)\) thinking that since some original percentages were increases (\(110\%\), \(120\%\)), the final answer should have plus signs, leading them to choose answer choice (A).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for A

Let \(\mathrm{A} = 100\) (chosen because it makes percentage calculations straightforward and intuitive)

Step 2: Calculate each subsequent value using the given percentages

• \(\mathrm{B} = 90\%\) of \(\mathrm{A} = 0.90 \times 100 = 90\)

• \(\mathrm{C} = 110\%\) of \(\mathrm{B} = 1.10 \times 90 = 99\)

• \(\mathrm{D} = 80\%\) of \(\mathrm{C} = 0.80 \times 99 = 79.2\)

• \(\mathrm{E} = 120\%\) of \(\mathrm{D} = 1.20 \times 79.2 = 95.04\)

Step 3: Calculate the ratio E/A

\(\mathrm{E/A} = 95.04/100 = 0.9504\)

Step 4: Convert to fractional form to match answer choices

\(0.9504 = 9504/10000 = 0.99 \times 0.96\)

Notice that:

• \(0.99 = 99/100 = (100-1)/100 = 1 - 1/100\)

• \(0.96 = 96/100 = 24/25 = (25-1)/25 = 1 - 1/25\)

Step 5: Match with answer choices

Therefore: \(\mathrm{E/A} = (1 - 1/100)(1 - 1/25)\)

This matches answer choice (B).