Terry had x dollars in her bank account at the beginning of a certain week. During the week, four transactions...

1. Translate the problem requirements: Terry starts with \(\mathrm{x}\) dollars, then has four transactions in order: +$50, -$70, -$40, +$80. The variable \(\mathrm{b}\) represents her account balance at ANY point during the week, and we need to find what must ALWAYS be true about \(\mathrm{b}\).
2. Calculate all possible account balances: Track Terry's account balance after each transaction to determine all possible values that \(\mathrm{b}\) could represent.
3. Identify the range of possible values for b: Determine the minimum and maximum values \(\mathrm{b}\) can take during the week.
4. Test each answer choice against the range: Check which statement must be true for ALL possible values of \(\mathrm{b}\), regardless of the starting amount \(\mathrm{x}\).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. Terry begins the week with some amount of money, which we'll call \(\mathrm{x}\) dollars. Throughout the week, she makes exactly four transactions in this specific order:

• First: She deposits $50 (adds money to her account)
• Second: She withdraws $70 (takes money out)
• Third: She withdraws $40 (takes more money out)
• Fourth: She deposits $80 (adds money back in)

The key insight here is that \(\mathrm{b}\) represents her account balance at ANY point during the week - not just at the end. This means \(\mathrm{b}\) could be her balance after the first transaction, after the second, after the third, after the fourth, or even at the very beginning.

Process Skill: TRANSLATE - Converting the problem's language about "some time during the week" into the mathematical understanding that b can take on multiple possible values.

2. Calculate all possible account balances

Now let's track Terry's account balance step by step. Since \(\mathrm{b}\) represents the balance "at some time during the week," we need to find all possible values:

Starting balance: \(\mathrm{x}\) dollars
After 1st transaction (deposit $50): \(\mathrm{x} + 50\) dollars
After 2nd transaction (withdraw $70): \(\mathrm{x} + 50 - 70 = \mathrm{x} - 20\) dollars
After 3rd transaction (withdraw $40): \(\mathrm{x} - 20 - 40 = \mathrm{x} - 60\) dollars
After 4th transaction (deposit $80): \(\mathrm{x} - 60 + 80 = \mathrm{x} + 20\) dollars

So the possible values for \(\mathrm{b}\) are: \(\mathrm{x}\), \(\mathrm{x} + 50\), \(\mathrm{x} - 20\), \(\mathrm{x} - 60\), and \(\mathrm{x} + 20\).

3. Identify the range of possible values for b

Looking at our five possible values for \(\mathrm{b}\): {\(\mathrm{x}\), \(\mathrm{x} + 50\), \(\mathrm{x} - 20\), \(\mathrm{x} - 60\), \(\mathrm{x} + 20\)}

Let's arrange these from smallest to largest:

• Smallest value: \(\mathrm{x} - 60\)
• Next: \(\mathrm{x} - 20\)
• Next: \(\mathrm{x}\)
• Next: \(\mathrm{x} + 20\)
• Largest: \(\mathrm{x} + 50\)

Therefore, \(\mathrm{b}\) ranges from a minimum of \(\mathrm{x} - 60\) to a maximum of \(\mathrm{x} + 50\).
In mathematical terms: \(\mathrm{x} - 60 \leq \mathrm{b} \leq \mathrm{x} + 50\)

Process Skill: CONSIDER ALL CASES - We must check every possible moment during the week, not just the final balance.

4. Test each answer choice against the range

Now we'll check each answer choice to see which one must ALWAYS be true for every possible value of \(\mathrm{b}\):

Choice A: \(\mathrm{b} < \mathrm{x}\)
This says \(\mathrm{b}\) is always less than \(\mathrm{x}\). But we found that \(\mathrm{b}\) can equal \(\mathrm{x} + 50\) or \(\mathrm{x} + 20\), both of which are greater than \(\mathrm{x}\). So this is false.

Choice B: \(\mathrm{b} = \mathrm{x} + 20\)
This says \(\mathrm{b}\) always equals \(\mathrm{x} + 20\). But \(\mathrm{b}\) can take on other values like \(\mathrm{x} - 60\) or \(\mathrm{x} + 50\). So this is false.

Choice C: \(\mathrm{b} \leq \mathrm{x} + 40\)
This says \(\mathrm{b}\) is always less than or equal to \(\mathrm{x} + 40\). But we found \(\mathrm{b}\) can equal \(\mathrm{x} + 50\), and \(\mathrm{x} + 50 > \mathrm{x} + 40\). So this is false.

Choice D: \(\mathrm{b} \geq \mathrm{x} - 60\)
This says \(\mathrm{b}\) is always greater than or equal to \(\mathrm{x} - 60\). Since our minimum value for \(\mathrm{b}\) is exactly \(\mathrm{x} - 60\), and all other possible values (\(\mathrm{x} - 20\), \(\mathrm{x}\), \(\mathrm{x} + 20\), \(\mathrm{x} + 50\)) are greater than \(\mathrm{x} - 60\), this statement is always true.

Choice E: \(|\mathrm{x} - \mathrm{b}| \leq 20\)
This says the distance between \(\mathrm{x}\) and \(\mathrm{b}\) is at most 20. But when \(\mathrm{b} = \mathrm{x} + 50\), we have \(|\mathrm{x} - (\mathrm{x} + 50)| = 50\), which is greater than 20. So this is false.

4. Final Answer

The correct answer is D: \(\mathrm{b} \geq \mathrm{x} - 60\)

This must be true because the smallest possible value for \(\mathrm{b}\) during the week is \(\mathrm{x} - 60\) (which occurs after the third transaction), and all other possible values are greater than or equal to this minimum.

Common Faltering Points

Errors while devising the approach

• Misinterpreting "some time during the week": Students often assume that \(\mathrm{b}\) represents only the final balance at the end of the week (after all four transactions). They miss that "some time during the week" means \(\mathrm{b}\) could be the balance after any individual transaction or even at the beginning. This leads them to calculate only \(\mathrm{x} + 20\) instead of considering all five possible values.
• Ignoring the sequential order of transactions: Students might treat the transactions as happening simultaneously or in a different order, failing to recognize that the balance changes step-by-step through the week. The order matters because it affects which intermediate balances are possible.
• Confusing "must be true" with "could be true": Students often look for statements that are true for some values of \(\mathrm{b}\), rather than statements that are true for ALL possible values of \(\mathrm{b}\). This leads them to select answer choices that work for only certain scenarios instead of universal truths.

Errors while executing the approach

• Arithmetic errors in tracking balances: When calculating the running balance after each transaction, students may make simple addition/subtraction mistakes. For example, calculating \(\mathrm{x} + 50 - 70\) as \(\mathrm{x} - 30\) instead of \(\mathrm{x} - 20\), or \(\mathrm{x} - 20 - 40\) as \(\mathrm{x} - 50\) instead of \(\mathrm{x} - 60\).
• Missing one or more possible balance values: Students might forget to include the starting balance (\(\mathrm{x}\)) as one of the possible values for \(\mathrm{b}\), or skip calculating intermediate balances, leading to an incomplete set of possible values for \(\mathrm{b}\).

Errors while selecting the answer

• Testing answer choices with only one value of b: Students might test each answer choice using only the final balance (\(\mathrm{x} + 20\)) or just one intermediate value, rather than checking whether the statement holds for ALL possible values of \(\mathrm{b}\). This can lead them to incorrectly eliminate the correct answer or select a wrong one.
• Misunderstanding absolute value inequalities: For choice E, students might incorrectly evaluate \(|\mathrm{x} - \mathrm{b}| \leq 20\) by only checking cases where \(\mathrm{b}\) is close to \(\mathrm{x}\), missing that when \(\mathrm{b} = \mathrm{x} + 50\), the absolute value becomes 50, which violates the inequality.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient starting value
Let's use \(\mathrm{x} = 100\) dollars as Terry's initial account balance. This round number will make calculations easy to follow.

Step 2: Track account balance through each transaction
Starting balance: $100
After deposit of $50: \($100 + $50 = $150\)
After withdrawal of $70: \($150 - $70 = $80\)
After withdrawal of $40: \($80 - $40 = $40\)
After deposit of $80: \($40 + $80 = $120\)

Step 3: Identify all possible values of b
Since \(\mathrm{b}\) represents the account balance "at some time during the week," \(\mathrm{b}\) could be any of these values:
\(\mathrm{b} = 100\) (initial), \(\mathrm{b} = 150\) (after first transaction), \(\mathrm{b} = 80\) (after second transaction), \(\mathrm{b} = 40\) (after third transaction), or \(\mathrm{b} = 120\) (final)

Step 4: Test each answer choice with our concrete values
A. \(\mathrm{b} < \mathrm{x}\): Not always true (\(150 > 100\), \(120 > 100\))
B. \(\mathrm{b} = \mathrm{x} + 20\): Not always true (only works for final balance: \(120 = 100 + 20\))
C. \(\mathrm{b} \leq \mathrm{x} + 40\): Not always true (\(150 > 140\))
D. \(\mathrm{b} \geq \mathrm{x} - 60\): Let's check: \(100 \geq 40\) ✓, \(150 \geq 40\) ✓, \(80 \geq 40\) ✓, \(40 \geq 40\) ✓, \(120 \geq 40\) ✓
E. \(|\mathrm{x} - \mathrm{b}| \leq 20\): Not always true (\(|100 - 150| = 50 > 20\), \(|100 - 40| = 60 > 20\))

Step 5: Verify with a different starting value
Let's try \(\mathrm{x} = 200\) to confirm our answer:
Possible \(\mathrm{b}\) values: 200, 250, 180, 140, 220
Minimum value of \(\mathrm{b} = 140\)
Does \(\mathrm{b} \geq \mathrm{x} - 60\)? Does \(\mathrm{b} \geq 200 - 60 = 140\)? Yes, all values are ≥ 140.

Conclusion: Answer choice D must be true regardless of the starting amount.