A certain state charges no sales tax on food, but charges a sales tax of 2 percent on clothing and...
GMAT Word Problems : (WP) Questions
A certain state charges no sales tax on food, but charges a sales tax of 2 percent on clothing and a sales tax of 6 percent on all other items. In this state Sarah purchased items with a total price, before tax,of $210, which included $45 for food items and $90 for clothing items. The total sales tax on the items Sarah purchased was what percent of the total price of $210 ?
- Translate the problem requirements: Break down Sarah's \(\$210\) purchase into three categories (food: \(\$45\) with \(0\%\) tax, clothing: \(\$90\) with \(2\%\) tax, other items: remaining amount with \(6\%\) tax), calculate total sales tax paid, then find what percent this tax represents of the original \(\$210\).
- Identify the missing category amount: Calculate how much Sarah spent on "other items" by subtracting known amounts from the total.
- Calculate tax for each applicable category: Apply the respective tax rates to clothing and other items (food has no tax).
- Find total tax as percentage of original purchase: Sum all taxes and express as a percentage of the \(\$210\) pre-tax total.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what Sarah bought and understand the tax situation clearly.
Sarah spent a total of \(\$210\) before taxes on three types of items:
- Food items: \(\$45\) (no sales tax - \(0\%\))
- Clothing items: \(\$90\) (sales tax of \(2\%\))
- Other items: We need to figure this out (sales tax of \(6\%\))
Our goal is to find the total sales tax as a percentage of the original \(\$210\) purchase.
Process Skill: TRANSLATE - Converting the problem's tax rules and purchase breakdown into a clear mathematical framework
2. Identify the missing category amount
To find how much Sarah spent on "other items," we subtract what we know from the total.
Total spent before tax: \(\$210\)
Food items: \(\$45\)
Clothing items: \(\$90\)
Amount spent on food and clothing: \(\$45 + \$90 = \$135\)
Therefore, amount spent on other items = \(\$210 - \$135 = \$75\)
Now we have the complete breakdown:
- Food: \(\$45\) (\(0\%\) tax)
- Clothing: \(\$90\) (\(2\%\) tax)
- Other items: \(\$75\) (\(6\%\) tax)
3. Calculate tax for each applicable category
Now let's calculate the actual tax amount for each category:
Food items tax: \(\$45 \times 0\% = \$45 \times 0 = \$0\)
Clothing items tax: \(\$90 \times 2\% = \$90 \times 0.02 = \$1.80\)
Other items tax: \(\$75 \times 6\% = \$75 \times 0.06 = \$4.50\)
Total sales tax paid = \(\$0 + \$1.80 + \$4.50 = \$6.30\)
4. Find total tax as percentage of original purchase
Now we need to express the total tax of \(\$6.30\) as a percentage of the original \(\$210\) purchase.
Percentage = (Total tax ÷ Original purchase amount) × \(100\%\)
Percentage = \((\$6.30 ÷ \$210) \times 100\%\)
Percentage = \(0.03 \times 100\% = 3.0\%\)
Final Answer
The total sales tax on Sarah's purchases was \(3.0\%\) of the total price of \(\$210\).
This matches answer choice B: \(3.0\%\)
Common Faltering Points
Errors while devising the approach
Faltering Point 1: Misunderstanding the tax structure and applying taxes incorrectly to categories. Students might apply the \(2\%\) clothing tax to all items or the \(6\%\) "other items" tax to clothing, not carefully reading that there are THREE distinct categories with different tax rates (\(0\%\), \(2\%\), and \(6\%\)).
Faltering Point 2: Confusion about what constitutes "other items." Students might assume that only food and clothing exist, missing that there's a third category of purchases that aren't food or clothing, which gets taxed at \(6\%\).
Faltering Point 3: Misinterpreting what the question is asking for. Students might calculate the total tax amount in dollars but forget that the question asks for this tax as a percentage of the original \(\$210\) purchase price.
Errors while executing the approach
Faltering Point 1: Arithmetic errors when calculating the amount spent on "other items." Students might incorrectly subtract \(\$45 + \$90\) from \(\$210\), getting the wrong base amount for the \(6\%\) tax calculation.
Faltering Point 2: Percentage calculation errors when computing individual taxes. For example, calculating \(2\%\) of \(\$90\) as \(\$18\) instead of \(\$1.80\), or \(6\%\) of \(\$75\) as \(\$45\) instead of \(\$4.50\) by forgetting to convert percentages to decimals.
Faltering Point 3: Error in the final percentage conversion when dividing \(\$6.30\) by \(\$210\). Students might make decimal errors or forget to multiply by \(100\) to convert to percentage form.
Errors while selecting the answer
Faltering Point 1: Selecting the wrong answer choice due to rounding errors or decimal placement mistakes. For example, if they calculated \(0.3\%\) instead of \(3.0\%\), they might look for \(0.3\%\) among the choices rather than recognizing their calculation error.