The owner of an apartment purchased 1 window screen, 1 door handle, and 1 ceiling fan for her apartment. The...

1. Translate the problem requirements: We need to find the regular price of the door handle. We're told the window screen costs 80% more than the door handle, the ceiling fan costs twice the sum of the other two items, and after a 50% discount on all items, the total paid was $50.40.
2. Set up price relationships using the door handle as base: Since the door handle has the simplest price structure, use it as our foundation and express all other prices in terms of it.
3. Calculate the total regular price before discount: Add up all three item prices in terms of the door handle price to get one expression for the total.
4. Apply the discount condition to solve: Since 50% discount means they paid half the regular total, and we know they paid $50.40, we can find the door handle price.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for in everyday terms. We have three items: a window screen, a door handle, and a ceiling fan. Think of this like shopping - we know how much more expensive one item is compared to another, and we know the final discounted total.

Here's what we know:

- The window screen costs 80% more than the door handle (so if the door handle costs $10, the window screen would cost $18)
- The ceiling fan costs twice the combined price of the window screen and door handle
- All three items were bought at 50% off (half price)
- The total paid after the discount was $50.40
- We need to find the regular price of the door handle

Process Skill: TRANSLATE - Converting the percentage relationships and discount information into clear mathematical understanding

2. Set up price relationships using the door handle as base

Since the door handle has the simplest price structure, let's call its regular price 'd' dollars and express everything else in terms of 'd'.

Window screen price: The window screen costs 80% more than the door handle. This means it costs the original price PLUS 80% more. So if the door handle costs 'd', then:

- Window screen = \(\mathrm{d + 0.80d = 1.80d}\)

Ceiling fan price: The ceiling fan costs twice the sum of the other two items:

- Sum of door handle and window screen = \(\mathrm{d + 1.80d = 2.80d}\)
- Ceiling fan = \(\mathrm{2 \times (2.80d) = 5.60d}\)

3. Calculate the total regular price before discount

Now let's add up all three regular prices:

- Door handle: d
- Window screen: 1.80d
- Ceiling fan: 5.60d
- Total regular price = \(\mathrm{d + 1.80d + 5.60d = 8.40d}\)

This means the total regular price for all three items is 8.40 times the price of the door handle.

4. Apply the discount condition to solve

The key insight here is that a 50% discount means they paid exactly half of the regular total price.

If the regular total is 8.40d, then after a 50% discount:

- Amount paid = 50% of 8.40d = \(\mathrm{0.50 \times 8.40d = 4.20d}\)

We know they actually paid $50.40, so:
\(\mathrm{4.20d = \$50.40}\)

Solving for d:
\(\mathrm{d = \$50.40 \div 4.20 = \$12.00}\)

Process Skill: MANIPULATE - Using the discount relationship to create a solvable equation

Final Answer

The regular price of the door handle is $12.00.

Let's verify: If the door handle costs $12.00, then:

- Window screen: \(\mathrm{\$12.00 \times 1.80 = \$21.60}\)
- Ceiling fan: \(\mathrm{\$12.00 \times 5.60 = \$67.20}\)
- Total regular price: \(\mathrm{\$12.00 + \$21.60 + \$67.20 = \$100.80}\)
- After 50% discount: \(\mathrm{\$100.80 \div 2 = \$50.40}\) ✓

The answer is C: $12.00

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "80% greater than" as "80% of"

Students often confuse "80% greater than" with "80% of". If the door handle costs d, they might incorrectly set the window screen price as 0.80d instead of the correct 1.80d (which is d + 0.80d). This fundamental misunderstanding of percentage increase language leads to completely wrong relationships.

2. Misunderstanding what "twice the sum" means

When the problem states the ceiling fan costs "twice the sum of the regular prices of the window screen and the door handle," students might interpret this as twice the window screen price OR twice the door handle price, rather than twice their combined total. This leads to setting the ceiling fan price incorrectly.

3. Applying the discount to individual items instead of the total

Students may attempt to apply the 50% discount to each item separately and then work backwards, rather than recognizing that they should first establish the total regular price and then apply the single discount to get the equation they need to solve.

Errors while executing the approach

1. Arithmetic errors when combining coefficients

When adding up the total regular price (d + 1.80d + 5.60d), students frequently make arithmetic mistakes, possibly getting 7.40d or 8.20d instead of the correct 8.40d. These small errors cascade through the rest of the solution.

2. Incorrectly calculating the discount amount

Students might calculate 50% discount incorrectly - either by subtracting 50% from the total (getting 0.50 × regular price) when they meant to find what was paid, or by dividing by 0.50 instead of 0.50 multiplication, leading to wrong equations.

3. Division errors when solving for d

When solving 4.20d = 50.40, students often make basic division mistakes, perhaps getting $12.20 or $11.80 instead of the correct $12.00, especially if they're rushing or not being careful with decimal calculations.

Errors while selecting the answer

No likely faltering points - the final step is straightforward once the correct value of d is calculated, and the answer choices are clearly differentiated dollar amounts that match the problem's request for the door handle's regular price.

Alternate Solutions

Smart Numbers Approach

This problem can be solved effectively using smart numbers by working backwards from the final discounted amount.

Step 1: Choose a smart number for the door handle price
Since we need to find the door handle price and the answer choices are all clean dollar amounts, let's work with the given information that the total discounted price is $50.40. This means the total regular price before discount was \(\mathrm{\$50.40 \div 0.5 = \$100.80}\).

Step 2: Set up the relationship using a variable approach with smart substitution
Let's say the door handle costs $12 (testing answer choice C):
• Door handle: $12
• Window screen: \(\mathrm{\$12 \times 1.8 = \$21.60}\) (80% greater means 180% of original)
• Sum of door handle and window screen: \(\mathrm{\$12 + \$21.60 = \$33.60}\)
• Ceiling fan: \(\mathrm{2 \times \$33.60 = \$67.20}\)

Step 3: Calculate total regular price
Total regular price = \(\mathrm{\$12 + \$21.60 + \$67.20 = \$100.80}\)

Step 4: Apply the 50% discount
Discounted total = \(\mathrm{\$100.80 \times 0.5 = \$50.40}\) ✓

Step 5: Verify our smart number choice
Since our calculation with $12 for the door handle gives us exactly the required discounted total of $50.40, we can confirm that $12 is the correct regular price for the door handle.

Why this smart numbers approach works:
We logically selected $12 from the answer choices and verified it satisfies all the given conditions. This method is efficient because we can test the most likely answer choices directly rather than setting up complex algebraic equations.