Store N gives a 50% discount on the list price of all its items and Store W gives a 60%...

1. Translate the problem requirements: We need to find what percent more or less Store W's selling price is compared to Store N's selling price for the same item, given that Store N gives 50% discount, Store W gives 60% discount, and Store W's list price is 20% higher than Store N's list price.
2. Set up concrete values to avoid fractions: Choose a simple list price for Store N that makes calculations straightforward and allows us to work with whole numbers throughout the problem.
3. Calculate actual selling prices: Apply the respective discounts to each store's list price to determine what customers actually pay at each store.
4. Compare selling prices using percentage relationship: Express Store W's selling price as a percentage of Store N's selling price to determine if it's more or less, and by what percentage.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in everyday language:
• Store N reduces all prices by 50%, so customers pay half the list price
• Store W reduces all prices by 60%, so customers pay 40% of the list price
• The same item has a list price that's 20% higher at Store W compared to Store N
• We need to find: Is Store W's actual selling price higher or lower than Store N's, and by what percentage?

Think of it this way: Store W starts with a higher sticker price but gives a bigger discount. Store N starts with a lower sticker price but gives a smaller discount. Which store ends up cheaper for the customer?

Process Skill: TRANSLATE - Converting the discount percentages and price relationships into a clear comparison framework

2. Set up concrete values to avoid fractions

To make this easy to follow, let's pick a simple number for Store N's list price. Since we're dealing with percentages like 50% and 20%, let's use $100 as Store N's list price.

Here's what we know:
• Store N's list price = $100
• Store W's list price = 20% higher than Store N = \(\mathrm{\$100 + \$20 = \$120}\)

Using $100 makes all our percentage calculations turn into simple whole numbers, which prevents arithmetic errors.

Process Skill: SIMPLIFY - Choosing convenient numbers that make percentage calculations straightforward

3. Calculate actual selling prices

Now let's figure out what customers actually pay at each store:

Store N's selling price:
Store N gives 50% discount, so customers pay 50% of the list price
Selling price = \(\mathrm{50\% \times \$100 = \$50}\)

Store W's selling price:
Store W gives 60% discount, so customers pay 40% of the list price
Selling price = \(\mathrm{40\% \times \$120 = 0.40 \times \$120 = \$48}\)

So the actual prices customers pay are:
• Store N: $50
• Store W: $48

Store W is cheaper! But by how much?

4. Compare selling prices using percentage relationship

We need to express Store W's price as a percentage of Store N's price.

Store W's price compared to Store N's price:
$48 compared to $50

As a percentage: \(\mathrm{(\$48 \div \$50) \times 100\% = 0.96 \times 100\% = 96\%}\)

This means Store W's selling price is 96% of Store N's selling price.
Since 96% is less than 100%, Store W costs less than Store N.
The difference is: \(\mathrm{100\% - 96\% = 4\%}\)

Therefore, Store W's selling price is 4% less than Store N's selling price.

4. Final Answer

Store W's selling price is 4% less than Store N's selling price.

This matches answer choice B. 4% less.

Verification: Even though Store W starts with a higher list price (+20%), their larger discount (-60% vs -50%) more than compensates, resulting in a final price that's 4% lower than Store N.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "percent discount" means
Students often confuse discount percentages with what customers actually pay. For example, when told "Store N gives a 50% discount," some students think customers pay 50% instead of understanding they pay the remaining 50% (100% - 50% = 50%). This fundamental misunderstanding derails the entire solution from the start.

2. Confusion about the comparison direction
The question asks "what percent (more or less) of the selling price in Store N is the selling price of the item in Store W?" Students often get confused about which store's price should be the base for comparison. They might mistakenly compare Store N's price to Store W's price instead of expressing Store W's price as a percentage of Store N's price.

3. Misinterpreting "20% higher list price"
When the problem states that Store W's list price is "20% higher," some students incorrectly apply this percentage to the final selling price rather than to the original list price. They fail to recognize that the 20% difference applies before any discounts are calculated.

Errors while executing the approach

1. Arithmetic errors in discount calculations
Students frequently make calculation mistakes when computing discounts. For example, calculating 60% discount on $120 might be incorrectly computed as \(\mathrm{\$120 \times 0.60 = \$72}\) (thinking this is the selling price) instead of correctly computing \(\mathrm{\$120 \times 0.40 = \$48}\) (recognizing that 60% discount means paying 40%).

2. Incorrectly setting up the percentage comparison
When comparing the two selling prices, students might set up the fraction backwards. Instead of calculating \(\mathrm{(\$48 \div \$50) \times 100\%}\), they might calculate \(\mathrm{(\$50 \div \$48) \times 100\%}\), leading to a completely different percentage and wrong conclusion about which store is more or less expensive.

Errors while selecting the answer

1. Getting the direction wrong in the final answer
Even after correctly calculating that Store W's price is 96% of Store N's price, students might conclude that Store W is "4% more" instead of "4% less." They correctly identify the 4% difference but fail to properly interpret whether this represents "more" or "less" than the base price.

Alternate Solutions

Smart Numbers Approach

This problem is ideal for the smart numbers method because we need to compare percentage relationships between stores with different discount rates and list prices.

Step 1: Choose a smart number for Store N's list price

Let's set Store N's list price = $100
This is a logical choice because:

• It makes percentage calculations straightforward
• Working with multiples of 100 keeps all calculations clean
• It allows us to easily compute both discounts and price differences

Step 2: Calculate Store W's list price

Store W's list price is 20% higher than Store N's:
Store W's list price = \(\mathrm{\$100 + (20\% \times \$100) = \$100 + \$20 = \$120}\)

Step 3: Calculate selling prices after discounts

Store N:
50% discount means customers pay 50% of list price
Store N's selling price = \(\mathrm{50\% \times \$100 = \$50}\)

Store W:
60% discount means customers pay 40% of list price
Store W's selling price = \(\mathrm{40\% \times \$120 = \$48}\)

Step 4: Compare the selling prices

We need to find what percent of Store N's selling price is Store W's selling price:
Percentage = \(\mathrm{\frac{\text{Store W's selling price}}{\text{Store N's selling price}} \times 100\%}\)
Percentage = \(\mathrm{\frac{\$48}{\$50} \times 100\% = 0.96 \times 100\% = 96\%}\)

Step 5: Determine the difference

Since Store W's selling price is 96% of Store N's selling price:
Store W's price is \(\mathrm{100\% - 96\% = 4\%}\) less than Store N's price

Answer: B. 4% less