A certain company's yearly revenue was 20% less in 2005 than in 2004 and 10% greater in 2006 than in...

1. Translate the problem requirements: We need to understand what happens to revenue each year: 2005 is 20% less than 2004, 2006 is 10% greater than 2005, and 2007 equals 2004. We want the percent increase from 2006 to 2007.
2. Set up a concrete baseline: Use a simple number for 2004 revenue to make calculations straightforward and track the actual dollar amounts year by year.
3. Calculate revenue for each relevant year: Work through 2005 and 2006 revenues using the given percentage changes to find the exact values we need.
4. Apply percent increase formula: Calculate the percent increase from 2006 to 2007 using the standard percent change formula and match to the closest answer choice.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening to this company's revenue year by year in plain English:

• 2004: This is our starting point
• 2005: Revenue dropped by 20% compared to 2004
• 2006: Revenue increased by 10% compared to 2005
• 2007: Revenue returned to exactly the same level as 2004

Our goal is to find: How much did revenue increase from 2006 to 2007?

Process Skill: TRANSLATE - Converting the percentage changes into a clear sequence of events

2. Set up a concrete baseline

To make our calculations simple, let's say the company's revenue in 2004 was \(\$100\). This is a smart choice because working with percentages becomes much easier when we start with 100.

Why \(\$100\)? Because:
• 20% of \(\$100 = \$20\) (easy to calculate)
• 10% of any amount will be simple to find
• We can focus on the logic rather than getting bogged down in arithmetic

So our baseline: 2004 Revenue = \(\$100\)

3. Calculate revenue for each relevant year

Now let's track the revenue year by year:

2005 Revenue:
• 2005 was 20% less than 2004
• 20% less means we keep 80% of the original
• \(2005 = 80\% \text{ of } \$100 = \$80\)

2006 Revenue:
• 2006 was 10% greater than 2005
• 10% greater means we add 10% to what we had
• \(10\% \text{ of } \$80 = \$8\)
• \(2006 = \$80 + \$8 = \$88\)

2007 Revenue:
• 2007 equals 2004 revenue
• \(2007 = \$100\)

So we have: \(2006 = \$88\) and \(2007 = \$100\)

4. Apply percent increase formula

Now we can find the percent increase from 2006 to 2007:

The increase in dollars: \(\$100 - \$88 = \$12\)

\(\text{Percent increase} = \frac{\text{Amount of increase}}{\text{Starting amount}} \times 100\%\)
\(\text{Percent increase} = \frac{\$12}{\$88} \times 100\%\)

Let's calculate this step by step:
• \(\frac{\$12}{\$88} = \frac{12}{88}\)
• Simplify: \(\frac{12}{88} = \frac{3}{22}\)
• Convert to decimal: \(3 \div 22 = 0.1364...\)
• Convert to percentage: \(0.1364 \times 100\% = 13.64\%\)

Looking at our answer choices, \(13.64\%\) is closest to 14%.

Final Answer

The percent increase in the company's yearly revenue from 2006 to 2007 is approximately 14%.

This matches answer choice (D) 14%.

Verification: Our calculation of \(13.64\%\) rounds to \(14\%\), which confirms our answer is correct.

Common Faltering Points

Errors while devising the approach

• Misunderstanding the reference point for percentage calculations: Students often confuse which year serves as the base for each percentage change. For example, they might think "10% greater in 2006 than in 2005" means 10% greater than 2004, not realizing that 2005 is the reference point for the 2006 calculation.
• Incorrectly interpreting "20% less" as "80% decrease": Students may misread "20% less than 2004" and think the revenue dropped to 20% of the original instead of understanding it means the revenue is 80% of the original \((100\% - 20\% = 80\%)\).
• Setting up the wrong equation for the final calculation: Students might set up the percent change formula backwards, using 2007 as the denominator instead of 2006, since we want the percent increase FROM 2006 TO 2007, meaning 2006 should be the base (denominator).

Errors while executing the approach

• Arithmetic errors when calculating 10% of the 2005 revenue: When finding 2006 revenue (10% greater than 2005's \(\$80\)), students might incorrectly calculate 10% of \(\$80\) as \(\$10\) instead of \(\$8\), leading to 2006 revenue of \(\$90\) instead of \(\$88\).
• Fraction simplification and decimal conversion mistakes: When calculating \(\frac{12}{88}\), students might make errors in simplifying to \(\frac{3}{22}\) or converting \(3 \div 22\) to a decimal, potentially getting \(0.136\) instead of \(0.1364\), which affects the final percentage.
• Forgetting to multiply by 100 when converting to percentage: Students might calculate \(12 \div 88 = 0.1364\) but forget the final step of multiplying by 100 to get the percentage, leading them to think the answer is \(0.14\%\) instead of \(14\%\).

Errors while selecting the answer

• Choosing the exact calculated value instead of the closest approximation: Students might calculate \(13.64\%\) correctly but then look for an answer choice of exactly \(13\%\) or \(13.6\%\), not recognizing that \(14\%\) is the closest option among the given choices.
• Second-guessing and selecting a "round number" like 10% or 15%: After getting \(13.64\%\), students might doubt their calculation because it seems like an "odd" result and instead pick \(10\%\) or \(15\%\) thinking these are more likely to be correct since they're "rounder" numbers.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for 2004 revenue

Let's set 2004 revenue = \(\$100\). This makes percentage calculations very straightforward since we can work directly with the percentage values as dollar amounts.

Step 2: Calculate 2005 revenue

2005 revenue was 20% less than 2004:
\(2005 \text{ revenue} = \$100 - (20\% \text{ of } \$100) = \$100 - \$20 = \$80\)

Step 3: Calculate 2006 revenue

2006 revenue was 10% greater than 2005:
\(2006 \text{ revenue} = \$80 + (10\% \text{ of } \$80) = \$80 + \$8 = \$88\)

Step 4: Identify 2007 revenue

2007 revenue equals 2004 revenue:
\(2007 \text{ revenue} = \$100\)

Step 5: Calculate percent increase from 2006 to 2007

\(\text{Percent increase} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\)
\(\text{Percent increase} = \frac{\$100 - \$88}{\$88} \times 100\%\)
\(\text{Percent increase} = \frac{\$12}{\$88} \times 100\%\)
\(\text{Percent increase} = \frac{12}{88} \times 100\% = \frac{3}{22} \times 100\% \approx 0.136 \times 100\% \approx 13.6\%\)

Step 6: Match to closest answer choice

\(13.6\%\) is closest to \(14\%\), so the answer is (D) \(14\%\).

Why this smart number works: Choosing \(\$100\) as the base makes all percentage calculations immediate and intuitive, eliminating the need for complex algebraic manipulation while maintaining full accuracy.