If candy bars that regularly sell for $0.40 each are on sale at two for $0.75, what is the percent...

1. Translate the problem requirements: We need to find the percent reduction when buying two candy bars at sale price (\(\$0.75\) for two) compared to regular price (\(\$0.40\) each, so \(\$0.80\) for two)
2. Calculate the regular cost for two candy bars: Multiply the individual price by 2 to get the baseline comparison amount
3. Find the actual dollar savings: Subtract the sale price from the regular price to determine the reduction amount
4. Convert dollar savings to percentage reduction: Divide the savings by the original price and express as a percentage

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're comparing. We have candy bars that normally cost \(\$0.40\) each, but there's a special deal where you can buy two for \(\$0.75\).

To find the percent reduction, we need to compare:

• What two candy bars would cost at regular price
• What two candy bars cost at the sale price (\(\$0.75\))

The question asks: "How much less are we paying as a percentage when we buy two candy bars on sale?"

Process Skill: TRANSLATE - Converting the sale scenario into a clear comparison between regular price and sale price for the same quantity

2. Calculate the regular cost for two candy bars

At regular price, each candy bar costs \(\$0.40\).

So for two candy bars at regular price:
\(\mathrm{Cost = 2 \times \$0.40 = \$0.80}\)

This \(\$0.80\) will be our starting point - this is what we would normally pay for two candy bars.

3. Find the actual dollar savings

Now let's see how much money we actually save:

Regular price for two candy bars: \(\$0.80\)
Sale price for two candy bars: \(\$0.75\)

\(\mathrm{Dollar\ savings = \$0.80 - \$0.75 = \$0.05}\)

So we save 5 cents when buying two candy bars on sale.

4. Convert dollar savings to percentage reduction

To find the percentage reduction, we need to ask: "The 5 cents we saved represents what percent of the original \(\$0.80\)?"

\(\mathrm{Percentage\ reduction = (Amount\ saved \div Original\ price) \times 100\%}\)
\(\mathrm{Percentage\ reduction = (\$0.05 \div \$0.80) \times 100\%}\)
\(\mathrm{Percentage\ reduction = (5 \div 80) \times 100\%}\)
\(\mathrm{Percentage\ reduction = (\frac{1}{16}) \times 100\%}\)

To convert \(\frac{1}{16}\) to a percentage:
\(\mathrm{1 \div 16 = 0.0625}\)
\(\mathrm{0.0625 \times 100\% = 6.25\%}\)

We can also express \(6.25\%\) as a mixed number:
\(6.25\% = 6\frac{1}{4}\%\)

Final Answer

The percent reduction in the price of two candy bars purchased at the sale price is \(6\frac{1}{4}\%\).

Looking at our answer choices, this matches choice (B) \(6\frac{1}{4}\%\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what quantity to compare: Students often get confused about whether to find the percent reduction per candy bar or for two candy bars total. Since the question asks for "the percent reduction in the price of two such candy bars," the comparison must be between the regular cost of two bars (\(\$0.80\)) versus the sale price for two bars (\(\$0.75\)). Some students mistakenly try to find the per-unit savings first.

2. Using the wrong base for percentage calculation: Students may incorrectly use the sale price (\(\$0.75\)) as the denominator instead of the original regular price (\(\$0.80\)). Remember that percent reduction is always calculated as \(\mathrm{(amount\ saved \div original\ price) \times 100\%}\), not \(\mathrm{(amount\ saved \div new\ price) \times 100\%}\).

Errors while executing the approach

1. Arithmetic errors in fraction-to-decimal conversion: When converting \(\frac{1}{16}\) to a decimal, students may incorrectly calculate \(1 \div 16\). The correct calculation is \(1 \div 16 = 0.0625\), which equals \(6.25\%\). Common mistakes include getting 0.625 (forgetting the leading zero) or 0.00625 (adding too many decimal places).

2. Calculation errors in the percentage formula: Students may make errors when computing \(\mathrm{(\$0.05 \div \$0.80) \times 100\%}\). A common mistake is simplifying \(\frac{5}{80}\) incorrectly - the correct simplification is \(\frac{5}{80} = \frac{1}{16}\), not \(\frac{1}{8}\) or other incorrect fractions.

Errors while selecting the answer

1. Converting between decimal and mixed number formats: After correctly calculating \(6.25\%\), students may struggle to recognize this equals \(6\frac{1}{4}\%\). Some students might incorrectly convert 0.25 to \(\frac{1}{3}\) (since \(\frac{1}{3} \approx 0.33\)) or \(\frac{1}{5}\) instead of the correct \(\frac{1}{4}\). Remember that \(0.25 = \frac{25}{100} = \frac{1}{4}\).