After an appliance store decreased the selling price of a certain hot-water heater, the store's gross profit on the heater...

1. Translate the problem requirements: We need to understand that gross profit is the difference between selling price and cost. The profit decreased from \(20\%\) of cost to \(5\%\) of cost when the selling price was reduced, while cost remained constant. We need to find what percent the new selling price is of the old selling price.
2. Set up the profit relationships using cost as the base: Express both the old and new selling prices in terms of the cost, using the given profit percentages.
3. Calculate the ratio of new to old selling price: Compare the new selling price directly to the old selling price to find the percentage relationship.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. The appliance store has a hot-water heater that they buy at a certain cost and then sell at a higher price to make a profit.

Originally, their gross profit was \(20\%\) of what they paid for the heater. Then they reduced the selling price, so now their gross profit is only \(5\%\) of what they paid for the heater. The cost they pay for the heater hasn't changed - only the selling price went down.

We need to figure out: the new selling price is what percentage of the old selling price?

To make this concrete, let's say the cost of the heater is \(\$100\). This makes our math much easier to follow.

Process Skill: TRANSLATE - Converting the problem language about profit percentages into a clear mathematical relationship

2. Set up the profit relationships using cost as the base

Now let's work out the actual selling prices using our \(\$100\) cost example.

Original situation:
- Cost = \(\$100\)
- Profit = \(20\%\) of cost = \(20\% \times \$100 = \$20\)
- Since Selling Price = Cost + Profit
- Original Selling Price = \(\$100 + \$20 = \$120\)

New situation (after price decrease):
- Cost = \(\$100\) (same as before)
- Profit = \(5\%\) of cost = \(5\% \times \$100 = \$5\)
- New Selling Price = \(\$100 + \$5 = \$105\)

This approach works because gross profit is always the difference between what you sell something for and what it cost you. When the problem tells us the profit as a percentage of cost, we can directly calculate the selling price.

3. Calculate the ratio of new to old selling price

Now we can directly compare the two selling prices:

- New Selling Price = \(\$105\)
- Original Selling Price = \(\$120\)

To find what percent the new price is of the old price:

Percentage = \((\text{New Selling Price} ÷ \text{Original Selling Price}) \times 100\%\)
Percentage = \((\$105 ÷ \$120) \times 100\%\)
Percentage = \(\frac{105}{120} \times 100\%\)

Let's simplify this fraction:
\(\frac{105}{120} = \frac{21}{24} = \frac{7}{8} = 0.875\)

So: \(0.875 \times 100\% = 87.5\%\)

Process Skill: SIMPLIFY - Using a concrete cost value (\(\$100\)) makes the arithmetic straightforward and avoids complex algebraic manipulation

4. Final Answer

The new selling price is \(87.5\%\) of the old selling price.

Looking at our answer choices, this matches choice B: \(87.5\%\).

We can verify this makes sense: if the selling price dropped to \(87.5\%\) of its original value, that's a \(12.5\%\) decrease in selling price, which would significantly reduce profit from \(20\%\) to \(5\%\) of cost - exactly what the problem described.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "gross profit" means

Students often confuse gross profit with markup or selling price. When the problem states "gross profit on the heater decreased from \(20\) percent of the cost," some students might think this means the selling price is \(20\%\) of cost, rather than understanding that gross profit = selling price - cost. This leads to setting up completely wrong equations.

2. Confusing the base for percentage calculations

The problem clearly states profits as "percent of the cost," but students sometimes misinterpret this as "percent of the selling price." This confusion about whether cost or selling price is the base completely changes the setup and leads to incorrect profit calculations.

3. Setting up overly complex algebraic equations

Many students immediately jump into using variables like C for cost without realizing they can use a simple concrete number. They create unnecessarily complex algebraic expressions when a straightforward numerical example (like cost = \(\$100\)) would make the problem much clearer and less error-prone.

Errors while executing the approach

1. Arithmetic errors when calculating selling prices

Even with the correct setup, students often make basic calculation mistakes. For example, when calculating "cost + \(20\%\) of cost," they might incorrectly compute it as just "\(20\%\) of cost" instead of "\(120\%\) of cost," or make errors in percentage-to-decimal conversions.

2. Incorrect fraction simplification

When computing \(\frac{105}{120}\), students frequently make algebraic errors in simplification. Common mistakes include incorrectly reducing to fractions like \(\frac{21}{25}\) instead of \(\frac{21}{24}\), or making arithmetic errors when converting \(\frac{7}{8}\) to its decimal form (\(0.875\)).

Errors while selecting the answer

1. Selecting the percentage decrease instead of the percentage ratio

After correctly calculating that the new price is \(87.5\%\) of the old price, some students get confused and select the complement (\(12.5\%\)) thinking the question asks "by what percent did the price decrease?" rather than "what percent of the old price is the new price?"

2. Forgetting to convert decimal to percentage form

Students who correctly calculate \(0.875\) as their final ratio sometimes forget the final step of converting to percentage form and look for \(0.875\) among the answer choices, or make errors in the conversion process (like reporting \(8.75\%\) instead of \(87.5\%\)).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for cost
Since we're dealing with percentages of cost, let's use cost = \(\$100\). This makes percentage calculations very clean since any percentage of \(100\) equals the percentage number itself.

Step 2: Calculate the old selling price
Original profit = \(20\%\) of cost = \(20\%\) of \(\$100 = \$20\)
Old selling price = Cost + Profit = \(\$100 + \$20 = \$120\)

Step 3: Calculate the new selling price
New profit = \(5\%\) of cost = \(5\%\) of \(\$100 = \$5\)
New selling price = Cost + Profit = \(\$100 + \$5 = \$105\)

Step 4: Find what percent the new selling price is of the old selling price
Percentage = \((\text{New selling price} / \text{Old selling price}) \times 100\%\)
Percentage = \((\$105 / \$120) \times 100\%\)
Percentage = \(\frac{105}{120} \times 100\%\)
Percentage = \(\frac{7}{8} \times 100\% = 87.5\%\)

Answer: B. \(87.5\%\)

Why this smart number works: Using cost = \(\$100\) eliminates the need for variables and makes all percentage calculations immediate. The choice is logical because the problem is fundamentally about ratios and percentages, so the actual dollar amount doesn't matter - only the relationships between the values.