In 1985 a company sold a brand of shoes to retailers for a fixed price per pair. In 1986 the...

1. Translate the problem requirements: We need to understand that revenue = quantity × price, and we're given percentage changes in both quantity (decreased by 20%) and price (increased by 20%) from 1985 to 1986, along with the 1986 revenue of $3.0 million. We need to find the 1985 revenue.
2. Set up the relationship between 1985 and 1986 revenues: Express the 1986 revenue in terms of the 1985 revenue using the percentage changes in quantity and price.
3. Calculate the net effect of percentage changes: Determine how the combined 20% decrease in quantity and 20% increase in price affects total revenue from 1985 to 1986.
4. Work backwards from 1986 revenue: Use the relationship established to calculate the 1985 revenue from the known 1986 revenue of $3.0 million.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in simple terms:

• In 1985, the company sold some number of shoe pairs at a certain price per pair
• In 1986, they sold 20% fewer pairs (so only 80% of the original quantity)
• But in 1986, the price per pair was 20% higher (so 120% of the original price)
• The total money they made in 1986 was $3.0 million
• We need to find how much money they made in 1985

The key insight is that Revenue = Quantity × Price. When both quantity and price change, we need to see how their combined effect changes the total revenue.

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

2. Set up the relationship between 1985 and 1986 revenues

Let's think about this step by step using concrete terms:

If we call the 1985 revenue "R", then:

• 1985 revenue = (original quantity) × (original price) = R
• 1986 revenue = (80% of original quantity) × (120% of original price)

In everyday language: "1986 revenue equals 0.8 times the original quantity, multiplied by 1.2 times the original price"

So: 1986 revenue = 0.8 × (original quantity) × 1.2 × (original price)
Which gives us: 1986 revenue = 0.8 × 1.2 × [(original quantity) × (original price)]
Therefore: 1986 revenue = 0.8 × 1.2 × (1985 revenue)

3. Calculate the net effect of percentage changes

Now let's figure out what 0.8 × 1.2 equals:

0.8 × 1.2 = 0.96

This tells us something important: even though the price went up by 20%, the overall revenue in 1986 was actually 96% of what it was in 1985. This makes sense because the 20% decrease in quantity had a bigger impact than the 20% increase in price.

So we can write: 1986 revenue = 0.96 × (1985 revenue)

Process Skill: INFER - Recognizing that the net effect is less than 100% despite the price increase

4. Work backwards from 1986 revenue

Now we can solve for the 1985 revenue:

We know that 1986 revenue = $3.0 million
And we established that 1986 revenue = 0.96 × (1985 revenue)

So: $3.0 million = 0.96 × (1985 revenue)

To find 1985 revenue, we divide both sides by 0.96:
1985 revenue = $3.0 million ÷ 0.96

Let's calculate this:
$3.0 ÷ 0.96 = $3.0 × (1 ÷ 0.96) = $3.0 × (100 ÷ 96) = $3.0 × (25 ÷ 24)

$3.0 × 25 ÷ 24 = $75.0 ÷ 24 = $3.125 million

So the 1985 revenue was approximately $3.1 million.

4. Final Answer

The 1985 revenue was approximately $3.1 million.

Let's verify this makes sense: If 1985 revenue was $3.1 million, then 1986 revenue should be $3.1 × 0.96 = $2.976 million, which rounds to $3.0 million ✓

Looking at our answer choices, $3.1 million corresponds to choice (D).

Answer: (D) $3.1 million

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the direction of percentage changes
Students often confuse which year had which change. They might think that 1985 had the 20% decrease in quantity and 20% increase in price, rather than correctly identifying that these changes occurred in 1986. This leads them to set up the relationship backwards, calculating 1986 revenue as 1985 revenue divided by 0.96 instead of multiplied by 0.96.

2. Incorrectly applying percentage changes
When the problem states "decreased by 20 percent," students might use 0.2 instead of 0.8 in their calculations. Similarly, for "increased by 20 percent," they might use 0.2 instead of 1.2. This fundamental misunderstanding of how to convert percentage changes to multipliers derails the entire solution.

3. Setting up additive instead of multiplicative relationships
Some students might think that a 20% decrease followed by a 20% increase results in no net change (0% total change), failing to recognize that percentage changes multiply rather than cancel out. They might incorrectly conclude that 1985 and 1986 revenues are equal.

Errors while executing the approach

1. Arithmetic errors in multiplication and division
Students often make computational mistakes when calculating 0.8 × 1.2 = 0.96, or when dividing 3.0 ÷ 0.96. They might get 0.84 instead of 0.96 for the multiplier, or make errors in the long division, leading to incorrect final answers.

2. Incorrect fraction conversions
When converting 3.0 ÷ 0.96 to 3.0 × (25/24), students might make errors in the fraction manipulation. They could incorrectly write it as 3.0 × (24/25) or make arithmetic mistakes when calculating 75 ÷ 24, arriving at values like 2.9 or 3.6 instead of 3.125.

Errors while selecting the answer

1. Choosing the calculated 1986 revenue instead of 1985 revenue
After working through the problem, some students lose track of what the question is asking for. Since they see $3.0 million prominently in their work (the given 1986 revenue), they might select choice (C) $3.0 million instead of recognizing that this was given information, not the answer.

2. Rounding errors or selecting non-approximated values
Students who calculate 3.125 million might look for exactly this value among the choices rather than recognizing that 3.125 is approximately 3.1. They might incorrectly select a choice that seems numerically closer, like $3.0 million, instead of properly rounding to $3.1 million.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient smart numbers for 1985

Let's assign concrete values to the 1985 situation that will make our calculations clean:

• 1985 quantity = 100 pairs (round number for easy percentage calculations)
• 1985 price per pair = $25 (chosen so that percentage changes yield whole numbers)
• 1985 revenue = 100 × $25 = $2,500

Step 2: Calculate 1986 values using the given percentage changes

• 1986 quantity = 100 - (20% of 100) = 100 - 20 = 80 pairs
• 1986 price per pair = $25 + (20% of $25) = $25 + $5 = $30
• 1986 revenue = 80 × $30 = $2,400

Step 3: Find the scaling factor

We know that 1986 revenue is actually $3.0 million, but our smart numbers gave us $2,400.
Scaling factor = $3,000,000 ÷ $2,400 = 1,250

Step 4: Scale up to find the actual 1985 revenue

Actual 1985 revenue = $2,500 × 1,250 = $3,125,000 = $3.125 million

Step 5: Match to closest answer choice

$3.125 million is closest to $3.1 million, so the answer is (D) $3.1 million.

Why this smart numbers approach works: By choosing convenient concrete numbers that satisfy all the given relationships, we can work with simple arithmetic throughout the problem while maintaining all the proportional relationships described in the original scenario.