At a certain supplier, a machine of type A costs $20{,}000 and a machine of type B costs $50{,}000. Each...

1. Translate the problem requirements: Clarify what "finance charges = \(40\%\) of the remainder" means and identify that we need to find the difference between total costs of 2 Type A machines versus 1 Type B machine
2. Calculate the total cost structure for each machine type: Determine the down payment (\(20\%\) of cost) and remainder for each machine, then apply the \(40\%\) finance charge to find total costs
3. Scale up for the comparison scenario: Calculate total cost for 2 Type A machines and compare with 1 Type B machine
4. Find the cost difference: Subtract to determine how much less 2 Type A machines cost compared to 1 Type B machine

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in plain English. We have two types of machines with different costs, and a specific payment structure that we need to understand.

Machine costs:

• Type A machine: \(\$20,000\)
• Type B machine: \(\$50,000\)

Now, let's understand the payment structure. When you buy a machine, you don't pay the full amount upfront. Instead:

• You make a down payment of \(20\%\) of the machine's cost
• The remaining \(80\%\) of the cost has finance charges added to it
• The finance charges equal \(40\%\) of that remaining amount

The question asks: How much less would 2 Type A machines cost compared to 1 Type B machine?

Process Skill: TRANSLATE - Converting the financing terms into a clear mathematical structure

2. Calculate the total cost structure for each machine type

Let's work through the actual dollar amounts for each machine type, step by step.

For Type A machine (\(\$20,000\)):

• Down payment = \(20\%\) of \(\$20,000 = \$4,000\)
• Remainder after down payment = \(\$20,000 - \$4,000 = \$16,000\)
• Finance charges = \(40\%\) of the remainder = \(40\%\) of \(\$16,000 = \$6,400\)
• Total cost = Down payment + Remainder + Finance charges
• Total cost = \(\$4,000 + \$16,000 + \$6,400 = \$26,400\)

For Type B machine (\(\$50,000\)):

• Down payment = \(20\%\) of \(\$50,000 = \$10,000\)
• Remainder after down payment = \(\$50,000 - \$10,000 = \$40,000\)
• Finance charges = \(40\%\) of the remainder = \(40\%\) of \(\$40,000 = \$16,000\)
• Total cost = Down payment + Remainder + Finance charges
• Total cost = \(\$10,000 + \$40,000 + \$16,000 = \$66,000\)

3. Scale up for the comparison scenario

Now we need to find the costs for the specific scenario mentioned in the question.

Cost of 2 Type A machines:
We calculated that 1 Type A machine costs \(\$26,400\) total
So 2 Type A machines cost: \(2 \times \$26,400 = \$52,800\)

Cost of 1 Type B machine:
We calculated that 1 Type B machine costs \(\$66,000\) total

4. Find the cost difference

The question asks how much less 2 Type A machines would cost compared to 1 Type B machine.

This means we need to find: Cost of 1 Type B machine - Cost of 2 Type A machines

Difference = \(\$66,000 - \$52,800 = \$13,200\)

Therefore, 2 Type A machines cost \(\$13,200\) less than 1 Type B machine.

Final Answer

The answer is E. \(\$13,200\)

This matches our calculation exactly. Two Type A machines cost \(\$13,200\) less than one Type B machine when accounting for the down payment and finance charge structure.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what finance charges are applied to: Students often think finance charges apply to the entire machine cost (\(\$20,000\) or \(\$50,000\)) rather than only to the remainder after the down payment. This leads to calculating \(40\%\) of the full cost instead of \(40\%\) of the remaining \(80\%\).

2. Confusion about the comparison being asked: Students may set up the wrong comparison, such as comparing 1 Type A to 1 Type B machine, or comparing 2 Type A to 2 Type B machines, instead of the specific comparison requested: 2 Type A machines versus 1 Type B machine.

3. Misinterpreting "total cost": Some students calculate only the finance charges or only the down payment, forgetting that the total cost includes the down payment + remainder + finance charges. They might think they only need to pay the down payment plus finance charges, excluding the original remainder.

Errors while executing the approach

1. Arithmetic errors in percentage calculations: Students commonly make errors when calculating \(20\%\) of machine costs for down payments or \(40\%\) of remainders for finance charges. For example, calculating \(40\%\) of \(\$16,000\) as \(\$6,000\) instead of \(\$6,400\).

2. Forgetting to double the Type A machine cost: Students correctly calculate the total cost for one Type A machine (\(\$26,400\)) but forget to multiply by 2 when comparing to Type B, leading them to use \(\$26,400\) instead of \(\$52,800\) in their final calculation.

3. Adding components incorrectly: When calculating total cost, students may add the components incorrectly, such as adding down payment + finance charges but forgetting to include the remainder amount, or double-counting the remainder by adding original cost + remainder + finance charges.

Errors while selecting the answer

1. Calculating the difference in the wrong direction: Students may subtract in the wrong order, calculating (Cost of 2 Type A machines - Cost of 1 Type B machine) instead of (Cost of 1 Type B machine - Cost of 2 Type A machines), giving them \(-\$13,200\) and then selecting a positive value that seems close but is incorrect.

2. Selecting the closest "reasonable" answer: If students make arithmetic errors in their calculations and arrive at a value like \(\$12,400\) or \(\$11,800\), they might select the closest answer choice (\(\$12,000\) or \(\$12,800\)) rather than recognizing their calculation error and rechecking their work.