- Define the total allowance as \(A\).
- Calculate the remainder after Saturday's spending as \(\frac{1}{2}A\).
- Calculate the expenditure on Sunday as \(\frac{1}{3}\) of \(\frac{1}{2}A\), which is \(\frac{1}{6}A\).
- The correct remaining amount is \(\frac{1}{2}A - \frac{1}{6}A = \frac{1}{3}A\), which differs from \(\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\).
- Final Answer: \(\boxed{\text{No, because she spent one third of her remaining half on Sunday. This is modeled by } \frac{1}{3} \cdot \frac{1}{2}. \text{ So she had } \frac{1}{2} - (\frac{1}{3} \cdot \frac{1}{2}) \text{ remaining after Sunday.}}\)
Explanation
- Analyzing the Spending Process
Let the total allowance be represented by the value \(A\).
On Saturday, Kara spends \(\frac{1}{2}\) of her allowance:
Amount spent Saturday = \(\frac{1}{2}A\)
Amount remaining after Saturday = \(A - \frac{1}{2}A = \frac{1}{2}A\)
On Sunday, Kara spends \(\frac{1}{3}\) of the remainder (\(\frac{1}{2}A\)):
Amount spent Sunday = \(\frac{1}{3} \times (\text{Amount remaining after Saturday}) = \frac{1}{3} \times \frac{1}{2}A = \frac{1}{6}A\)
The amount remaining after Sunday = (Amount remaining after Saturday) - (Amount spent Sunday) = \(\frac{1}{2}A - \frac{1}{6}A = \frac{3}{6}A - \frac{1}{6}A = \frac{2}{6}A = \frac{1}{3}A\)
- Evaluating the Proposed Model
We examine the expression \(\frac{1}{2} - \frac{1}{3}\):
\(\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}\).
Comparing this to the remaining amount calculated in the previous step (\(\frac{1}{3}A\)), we see that \(\frac{1}{6}A
eq \frac{1}{3}A\).
Furthermore, the mathematical structure of the problem involves the product of fractions, as she spends a fraction of the remaining amount, not a flat subtraction from the initial fraction. Specifically, the amount remaining is \(A - (\frac{1}{2}A + \frac{1}{6}A) = A - \frac{2}{3}A = \frac{1}{3}A\). The term \(\frac{1}{3} \cdot \frac{1}{2}\) correctly represents the portion spent on Sunday, and subtracting this from the initial remainder correctly models the total reduction. Thus, the second option is the correct one.
- Conclusion
Based on our analysis, the situation cannot be modeled as \(\frac{1}{2} - \frac{1}{3}\) because the second expenditure is a fraction of the remainder, not a simple subtraction of fractions from the whole. The correct model for the amount remaining is \(\frac{1}{2} - (\frac{1}{3} \cdot \frac{1}{2}) = \frac{1}{3}\). Therefore, the second option is correct: No, because she spent one third of her remaining half on Sunday. This is modeled by \(\frac{1}{3} \cdot \frac{1}{2}\). So she had \(\frac{1}{2} - (\frac{1}{3} \cdot \frac{1}{2})\) remaining after Sunday.
Examples
Understanding how to handle sequential deductions is vital in everyday life, such as when calculating discounts on sale items. For example, if a store offers a 50% discount on an item and then an additional 33.3% (a third) off the sale price, you don't simply add the percentages or subtract them from the original price. By understanding that each successive discount applies to the new, reduced price, you can accurately determine the final cost and ensure you are getting the correct deal.