Marge received a certain amount of money as a gift. She spent half of the amount in the first week....

1. Translate the problem requirements: Marge starts with some gift amount, spends half in week 1, then spends half of what remains each subsequent week for 12 more weeks. We need to find the first week where her total spending exceeds \(\frac{19}{20}\) of the original gift.
2. Set up the spending pattern: Express what fraction of the original gift Marge has remaining after each week, starting with a convenient variable.
3. Track cumulative spending: Calculate what fraction of the original gift has been spent by the end of each week until we exceed \(\frac{19}{20}\).
4. Compare against the threshold: Identify the first week where cumulative spending surpasses \(\frac{19}{20}\) of the original amount.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. Marge gets some money as a gift - let's call this amount G dollars to make it concrete.

In the first week, she spends half of G, which is \(\mathrm{G/2}\).
In the second week, she spends half of what's left, which is half of \(\mathrm{G/2} = \mathrm{G/4}\).
In the third week, she spends half of what's left after that, which is \(\mathrm{G/8}\).
And this pattern continues...

We need to find the first week where the total amount she has spent exceeds \(\frac{19}{20}\) of her original gift. Since \(\frac{19}{20} = 0.95\), we're looking for when she has spent more than 95% of her gift.

Process Skill: TRANSLATE - Converting the weekly spending pattern into a mathematical sequence

2. Set up the spending pattern

Let's think about what Marge has remaining after each week, since this will help us see the pattern clearly.

If she starts with amount G:

• After week 1: She has \(\mathrm{G/2}\) remaining (spent \(\mathrm{G/2}\))
• After week 2: She has \(\mathrm{G/4}\) remaining (spent \(\mathrm{G/2} + \mathrm{G/4}\))
• After week 3: She has \(\mathrm{G/8}\) remaining (spent \(\mathrm{G/2} + \mathrm{G/4} + \mathrm{G/8}\))
• After week 4: She has \(\mathrm{G/16}\) remaining
• After week 5: She has \(\mathrm{G/32}\) remaining

I can see the pattern now! After week n, she has \(\frac{\mathrm{G}}{2^n}\) remaining.

This means after week n, she has spent: \(\mathrm{G} - \frac{\mathrm{G}}{2^n} = \mathrm{G}(1 - \frac{1}{2^n})\)

3. Track cumulative spending

Now let's calculate what fraction of the original gift has been spent after each week:

After week 1: Spent = \(\mathrm{G}(1 - \frac{1}{2}) = \mathrm{G}(\frac{1}{2}) = \frac{1}{2}\) of the gift
After week 2: Spent = \(\mathrm{G}(1 - \frac{1}{4}) = \mathrm{G}(\frac{3}{4}) = \frac{3}{4}\) of the gift
After week 3: Spent = \(\mathrm{G}(1 - \frac{1}{8}) = \mathrm{G}(\frac{7}{8}) = \frac{7}{8}\) of the gift
After week 4: Spent = \(\mathrm{G}(1 - \frac{1}{16}) = \mathrm{G}(\frac{15}{16}) = \frac{15}{16}\) of the gift
After week 5: Spent = \(\mathrm{G}(1 - \frac{1}{32}) = \mathrm{G}(\frac{31}{32}) = \frac{31}{32}\) of the gift

Process Skill: VISUALIZE - Tracking the pattern of remaining amounts to see cumulative spending

4. Compare against the threshold

We need to find when the cumulative spending first exceeds \(\frac{19}{20}\) of the gift.

Let's convert our fractions to compare with \(\frac{19}{20}\):

• \(\frac{19}{20} = 0.95\)
• After week 4: \(\frac{15}{16} = 0.9375\) (less than \(\frac{19}{20}\))
• After week 5: \(\frac{31}{32} = 0.96875\) (greater than \(\frac{19}{20}\))

Let's double-check this calculation:
\(\frac{19}{20} = \frac{19}{20}\)
\(\frac{31}{32} = \frac{31}{32}\)

To compare these fractions, let's find a common denominator:
\(\frac{19}{20} = \frac{19 \times 32}{20 \times 32} = \frac{608}{640}\)
\(\frac{31}{32} = \frac{31 \times 20}{32 \times 20} = \frac{620}{640}\)

Since \(\frac{620}{640} > \frac{608}{640}\), we confirm that \(\frac{31}{32} > \frac{19}{20}\).

Therefore, the 5th week is the first week in which Marge's total spending exceeds \(\frac{19}{20}\) of her gift.

Final Answer

The answer is B. 5th week.

After the 5th week, Marge will have spent \(\frac{31}{32}\) of her gift, which is approximately 96.9% - greater than the required 95% (\(\frac{19}{20}\)) threshold.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting what "spent" means in the context of the threshold. Students may incorrectly think they need to find when the amount spent in a single week exceeds \(\frac{19}{20}\), rather than when the cumulative total amount spent across all weeks exceeds \(\frac{19}{20}\) of the original gift.

Faltering Point 2: Confusion about the spending pattern. Students might think Marge spends half of the original amount G each week (\(\mathrm{G/2}\) every week), rather than understanding she spends half of whatever remains from the previous week, creating the sequence \(\mathrm{G/2}, \mathrm{G/4}, \mathrm{G/8}\), etc.

Errors while executing the approach

Faltering Point 1: Arithmetic errors when calculating the fraction of money remaining. Students may incorrectly calculate \(1 - \frac{1}{2^n}\), particularly getting confused between what's remaining versus what's been spent. For example, after week 4, they might calculate \(\frac{1}{16}\) as the amount spent rather than the amount remaining.

Faltering Point 2: Errors in fraction comparison. When comparing \(\frac{15}{16}\) vs \(\frac{19}{20}\) or \(\frac{31}{32}\) vs \(\frac{19}{20}\), students may make mistakes finding common denominators or converting to decimals, leading them to incorrectly conclude which fraction is larger.

Errors while selecting the answer

Faltering Point 1: Off-by-one errors in identifying the "first week." Students might correctly calculate that after 5 weeks the cumulative spending exceeds \(\frac{19}{20}\), but then incorrectly conclude this means the answer is the 4th week or 6th week, misunderstanding whether the question asks for the week number when the threshold is crossed.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient starting amount

Let's say Marge received \$1,000 as her gift. This is a convenient round number that will make our calculations clean and easy to track.

Step 2: Calculate spending and remaining amounts week by week

We'll track how much Marge spends each week and how much she has left:

• Week 1: Spends half of \$1,000 = \$500. Remaining: \$500
• Week 2: Spends half of \$500 = \$250. Remaining: \$250
• Week 3: Spends half of \$250 = \$125. Remaining: \$125
• Week 4: Spends half of \$125 = \$62.50. Remaining: \$62.50
• Week 5: Spends half of \$62.50 = \$31.25. Remaining: \$31.25

Step 3: Calculate cumulative spending after each week

• After Week 1: Total spent = \$500
• After Week 2: Total spent = \$500 + \$250 = \$750
• After Week 3: Total spent = \$750 + \$125 = \$875
• After Week 4: Total spent = \$875 + \$62.50 = \$937.50
• After Week 5: Total spent = \$937.50 + \$31.25 = \$968.75

Step 4: Determine the threshold amount

We need to find when total spending exceeds \(\frac{19}{20}\) of the original gift:

\(\frac{19}{20} \times \$1,000 = 0.95 \times \$1,000 = \$950\)

Step 5: Compare cumulative spending to threshold

• After Week 4: \$937.50 < \$950 (not yet greater than \(\frac{19}{20}\))
• After Week 5: \$968.75 > \$950 (exceeds \(\frac{19}{20}\) for the first time)

Conclusion: The 5th week is the first week in which Marge's total spending exceeds \(\frac{19}{20}\) of her original gift.

Why this smart number works: Using \$1,000 makes the fraction \(\frac{19}{20}\) easy to calculate (\$950), and the pattern of halving remains clear throughout. Any starting amount would yield the same result since we're dealing with proportional relationships.