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Isaac can purchase a certain item in four different ways, as shown in the table. What is the lowest price for which he can purchase this item?
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:
Process Skill: TRANSLATE - Converting the problem description into clear mathematical steps
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 = \(\$80 + \$10 = \$90\)
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 = \(\$81 + \$8.10 = \$89.10\)
Store Y is straightforward - the item costs \(\$90\) with no tax, discount, or additional fees.
Final cost = \(\$90\)
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 = \(\$99 - \$10 = \$89\)
Now let's compare all four final costs to find the lowest:
The lowest cost is Store Z at \(\$89\).
The lowest price for which Isaac can purchase this item is \(\$89.00\), which corresponds to answer choice B.
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.
2. Confusion about when rebates are applied for Store ZStudents 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.
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 \(\$89.10\).
2. Adding instead of subtracting discounts and rebatesStudents 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.
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.