A camera lens filter kit containing 5 filters sells for $57.50. If the filters are purchased individually, 2 of them...

1. Translate the problem requirements: We need to find what percent we save by buying a kit ($57.50) versus buying 5 filters individually, then express this savings as a percentage of the individual purchase total
2. Calculate the total cost of individual filters: Add up the individual prices of all 5 filters to establish our baseline comparison cost
3. Determine the actual savings amount: Find the difference between individual total and kit price to quantify the dollar savings
4. Convert savings to percentage: Express the dollar savings as a percentage of the individual purchase total using the basic percentage formula

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking us to do in plain English. We have a camera filter kit that costs $57.50, and we want to compare this to buying the same 5 filters individually. The question asks: "How much do we save by buying the kit, expressed as a percentage of what we would pay if we bought each filter separately?"

To answer this, we need to:

• Find the total cost if we buy all filters individually
• Calculate how much money we save by buying the kit instead
• Express that savings as a percentage of the individual purchase total

Process Skill: TRANSLATE - Converting the problem language into clear mathematical steps

2. Calculate the total cost of individual filters

Now let's add up what it would cost to buy all 5 filters separately:

• 2 filters at $10.45 each = \(2 \times \$10.45 = \$20.90\)
• 2 filters at $12.05 each = \(2 \times \$12.05 = \$24.10\)
• 1 filter at $17.50 each = \(1 \times \$17.50 = \$17.50\)

Total individual cost = \(\$20.90 + \$24.10 + \$17.50 = \$62.50\)

So if we bought all the filters individually, we would pay $62.50 total.

3. Determine the actual savings amount

Now we can calculate our dollar savings by comparing the two options:

• Individual purchase total: $62.50
• Kit price: $57.50
• Savings = \(\$62.50 - \$57.50 = \$5.00\)

By buying the kit instead of individual filters, we save exactly $5.00.

4. Convert savings to percentage

Finally, we need to express our $5.00 savings as a percentage of the total individual cost ($62.50).

In plain English: "$5.00 is what percent of $62.50?"

To find this percentage:

• We take our savings ($5.00) and divide by the individual total ($62.50)
• Then multiply by 100 to get the percentage

Percentage = \((\$5.00 \div \$62.50) \times 100\)
Percentage = \((0.08) \times 100 = 8\%\)

Final Answer

The amount saved by purchasing the kit is 8% of the total price of the 5 filters purchased individually.

The answer is (B) 8%.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what the percentage should be calculated against

Students often get confused about the denominator for the percentage calculation. The question asks for savings "as a percent of the total price of the 5 filters purchased individually." Some students might mistakenly calculate the percentage using the kit price ($57.50) as the denominator instead of the individual purchase total ($62.50). This fundamental misunderstanding of the question's requirement would lead to an incorrect setup from the start.

2. Confusion about what constitutes "savings"

While most students correctly identify that savings = individual total - kit price, some might get the subtraction backwards, especially if they're working quickly. They might calculate kit price - individual total, which would give them a negative value and signal something is wrong, but this directional error in the approach setup can waste valuable time.

Errors while executing the approach

1. Arithmetic errors in calculating individual filter costs

This problem involves multiple multiplication and addition steps that provide several opportunities for calculation mistakes. Common errors include:

• Incorrectly calculating \(2 \times \$10.45 = \$20.90\) or \(2 \times \$12.05 = \$24.10\)
• Making addition errors when summing \(\$20.90 + \$24.10 + \$17.50 = \$62.50\)

These arithmetic mistakes in the foundation calculation will cascade through the entire solution.

2. Decimal errors in the percentage calculation

When calculating \((\$5.00 \div \$62.50) \times 100\), students might make decimal placement errors. The division \(\$5.00 \div \$62.50 = 0.08\), and some students might incorrectly get 0.8 or 0.008, leading to final answers of 80% or 0.8% respectively. This type of decimal error is particularly common when working under time pressure.

Errors while selecting the answer

1. Selecting a nearby answer choice due to minor calculation errors

If students made small arithmetic errors earlier in their calculations, they might arrive at a percentage close to but not exactly 8%. For example, if they miscalculated the individual total as $61.50 instead of $62.50, they would get approximately 8.1%, which might lead them to select (C) 8.5% as the "closest" answer rather than recognizing their calculation error and checking their work.