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Diana bought a stereo for \(\$530\), which was the retail price plus a \(6\%\) sales tax. How much money could she have saved if she had bought the stereo at the same retail price in a neighboring state where she would have paid a sales tax of \(5\%\)?
Let's start by understanding what happened and what we need to find.
Diana went shopping and bought a stereo. The total amount she paid was $530. This $530 includes two parts: the original retail price of the stereo plus a 6% sales tax on top of that retail price.
Now we're asking: what if Diana had bought the exact same stereo (same retail price) in a neighboring state where the sales tax is only 5% instead of 6%? How much money would she have saved?
So we need to find the difference between what she actually paid (with 6% tax) and what she would have paid (with 5% tax on the same retail price).
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical understanding of what we're comparing
We know that Diana paid $530 total, and this represents the retail price plus 6% tax.
Let's think about this step by step. If the retail price is some amount, and we add 6% tax to it, we get $530.
In everyday terms: if something costs $100 and we add 6% tax, we pay $100 + $6 = $106. This means we're paying 106% of the original price.
So if the retail price is R dollars, then:
\(\mathrm{R + 6\% \text{ of } R = \$530}\)
\(\mathrm{R + 0.06R = \$530}\)
\(\mathrm{1.06R = \$530}\)
To find R: \(\mathrm{R = \$530 ÷ 1.06 = \$500}\)
Let's verify: \(\mathrm{\$500 + (6\% \text{ of } \$500) = \$500 + \$30 = \$530}\) ✓
Now we know the retail price is $500. Let's calculate what Diana would have paid in the neighboring state with 5% sales tax.
Using the same retail price of $500:
\(\mathrm{\text{Total cost with 5\% tax} = \$500 + (5\% \text{ of } \$500)}\)
\(\mathrm{\text{Total cost with 5\% tax} = \$500 + \$25 = \$525}\)
Alternatively, we can think of this as paying 105% of the retail price:
\(\mathrm{\text{Total cost} = 1.05 × \$500 = \$525}\)
Now we can find the difference between what she actually paid and what she would have paid:
\(\mathrm{\text{Savings} = \text{What she actually paid} - \text{What she would have paid}}\)
\(\mathrm{\text{Savings} = \$530 - \$525 = \$5}\)
This makes intuitive sense: the difference in tax rates is \(\mathrm{6\% - 5\% = 1\%}\), and 1% of the $500 retail price is $5.
Diana could have saved $5.00 if she had bought the stereo in the neighboring state with 5% sales tax instead of 6% sales tax.
The answer is (D) $5.00.
Faltering Point 1: Misunderstanding what the $530 represents
Students may think that $530 is just the retail price, forgetting that it already includes the 6% sales tax. This leads them to calculate 6% of $530 instead of working backwards to find the original retail price first.
Faltering Point 2: Confusion about what "savings" means in this context
Students might think they need to find how much Diana could save compared to not paying any tax at all, rather than understanding that the question asks for the difference between paying 6% tax versus 5% tax on the same item.
Faltering Point 1: Incorrect calculation when finding the retail price
When solving \(\mathrm{1.06R = \$530}\), students may incorrectly multiply instead of divide, calculating \(\mathrm{R = \$530 × 1.06 = \$561.80}\) instead of \(\mathrm{R = \$530 ÷ 1.06 = \$500}\).
Faltering Point 2: Using the wrong base amount for tax calculations
After incorrectly finding the retail price, students may compound their error by calculating the 5% tax on their incorrect retail price amount, leading to a completely wrong final answer.
Faltering Point 1: Calculation verification oversight
Students may arrive at the correct numerical answer of $5 but fail to double-check their work using the logical shortcut (1% of retail price), missing the opportunity to catch any computational errors they may have made along the way.