Internet: $80, with no sales tax, plus $10 shipping fee Store X: $90, a 10% discount, plus 10% tax on...

1. Translate the problem requirements: We need to calculate the final cost for each of the four purchasing options and identify which gives the lowest total price Isaac will pay.
2. Calculate Internet option cost: Apply the shipping fee to the base price with no tax.
3. Calculate Store X cost: Apply the discount first, then calculate tax on the discounted amount.
4. Calculate Store Y cost: Simply use the listed price since there are no additional costs.
5. Calculate Store Z cost: Apply tax to the base price, then subtract the rebate from the total.
6. Compare all options: Identify the minimum cost among all four calculated totals.

Execution of Strategic Approach

1. Translate the problem requirements

We need to find out which of the four purchasing options gives Isaac the lowest final cost. Each option has different combinations of discounts, taxes, shipping fees, and rebates that we need to calculate step by step.

Let me clearly identify what each option involves:

• Internet: Base price \(\$80\), no tax, but add \(\$10\) for shipping
• Store X: Base price \(\$90\), get \(10\%\) discount first, then pay \(10\%\) tax on the reduced amount
• Store Y: Base price \(\$90\), no additional costs
• Store Z: Base price \(\$90\), pay \(10\%\) tax on full amount, then get \(\$10\) rebate back

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

2. Calculate Internet option cost

For the Internet option, we start with \(\$80\) and simply add the \(\$10\) shipping fee since there's no sales tax.

Final cost = Base price + Shipping fee
Final cost = \(\$80 + \$10 = \$90\)

3. Calculate Store X cost

For Store X, we need to apply the discount first, then calculate tax on the discounted amount.

Step 1: Apply \(10\%\) discount to \(\$90\)
Discount amount = \(10\% \text{ of } \$90 = 0.10 \times \$90 = \$9\)
Price after discount = \(\$90 - \$9 = \$81\)

Step 2: Apply \(10\%\) tax to the discounted price
Tax amount = \(10\% \text{ of } \$81 = 0.10 \times \$81 = \$8.10\)
Final cost = \(\$81 + \$8.10 = \$89.10\)

4. Calculate Store Y cost

Store Y is straightforward - the item costs \(\$90\) with no tax, discount, or additional fees.

Final cost = \(\$90\)

5. Calculate Store Z cost

For Store Z, we apply tax to the full price first, then subtract the rebate.

Step 1: Apply \(10\%\) tax to \(\$90\)
Tax amount = \(10\% \text{ of } \$90 = 0.10 \times \$90 = \$9\)
Price after tax = \(\$90 + \$9 = \$99\)

Step 2: Apply \(\$10\) rebate
Final cost = \(\$99 - \$10 = \$89\)

6. Compare all options

Now let's compare all four final costs to find the lowest:

• Internet: \(\$90\)
• Store X: \(\$89.10\)
• Store Y: \(\$90\)
• Store Z: \(\$89\)

The lowest cost is Store Z at \(\$89\).

Final Answer

The lowest price for which Isaac can purchase this item is \(\$89.00\), which corresponds to answer choice B.

Common Faltering Points

Errors while devising the approach

Students may incorrectly calculate the tax on the original \(\$90\) price instead of on the discounted price. The problem clearly states "\(10\%\) tax on discounted price," but students might apply tax first, then discount, or calculate tax on the full amount.

Students might subtract the \(\$10\) rebate before calculating the tax instead of after. The problem states "\(\$10\) rebate after tax," but students may misinterpret this sequence and apply rebate to the base price first.

Errors while executing the approach

Students commonly make calculation mistakes when computing \(10\%\) of \(\$81\) for Store X's tax, potentially getting \(\$8.01\) instead of \(\$8.10\), leading to a final answer of \(\$89.01\) instead of \(\$89.10\).

Students may accidentally add the \(\$9\) discount to Store X's price or add the \(\$10\) rebate to Store Z's price instead of subtracting these amounts, leading to inflated final costs.

Errors while selecting the answer

If students made arithmetic mistakes in earlier steps (like getting \(\$89.10\) for Store Z instead of \(\$89.00\)), they might incorrectly select answer choice C (\(\$89.10\)) instead of the correct answer B (\(\$89.00\)), thinking Store X gives the lowest price.