A certain telephone company offers two plans, A and B. Under plan A, the company charges a total of $0.60...

1. Translate the problem requirements: We need to find the call duration where Plan A's total cost equals Plan B's total cost. Plan A charges \(\$0.60\) for first 7 minutes plus \(\$0.06\) per additional minute. Plan B charges \(\$0.08\) per minute for the entire call.
2. Set up the cost equations: Express the total cost for each plan in terms of call duration, treating calls longer than 7 minutes separately since Plan A has different rates.
3. Identify the relevant scenario: Check if the answer is for calls under or over 7 minutes by testing the boundary case, then focus on the appropriate scenario.
4. Solve for equal costs: Set the two cost expressions equal and solve for the call duration where both plans charge the same amount.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what each plan charges in plain English:

Plan A works like a taxi fare - there's a base charge plus extra for additional time. You pay \(\$0.60\) no matter what for the first 7 minutes of your call. If your call goes longer than 7 minutes, you pay an additional \(\$0.06\) for every minute beyond those first 7 minutes.

Plan B is simpler - it's like paying by the minute for everything. You pay \(\$0.08\) for every single minute you talk, from the very first minute.

We need to find the exact call length where both plans cost exactly the same amount.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical understanding before setting up equations

2. Set up the cost equations

Now let's express what each plan costs for a call of 't' minutes.

For Plan B, this is straightforward: Cost = \(\$0.08 \times \mathrm{t}\)

For Plan A, we need to consider two scenarios:

- If the call is 7 minutes or less: Cost = \(\$0.60\)
- If the call is longer than 7 minutes: Cost = \(\$0.60 + \$0.06 \times \text{(extra minutes beyond 7)}\)

For calls longer than 7 minutes, if the total call is 't' minutes, then the extra minutes are \(\left(\mathrm{t} - 7\right)\).
So for calls over 7 minutes: Plan A cost = \(\$0.60 + \$0.06 \times \left(\mathrm{t} - 7\right)\)

3. Identify the relevant scenario

Since we're looking for when the costs are equal, let's check if this happens for short calls (7 minutes or less) or longer calls.

For a 7-minute call:

- Plan A: \(\$0.60\) (the base charge)
- Plan B: \(\$0.08 \times 7 = \$0.56\)

Plan A costs more at 7 minutes. Since Plan B's rate (\(\$0.08\)/minute) is higher than Plan A's additional rate (\(\$0.06\)/minute), Plan B will eventually cost more for longer calls. So the equal point must be somewhere beyond 7 minutes.

Process Skill: CONSIDER ALL CASES - Determining which scenario applies before solving

4. Solve for equal costs

Since the answer must be for calls longer than 7 minutes, we set the costs equal:

Plan A cost = Plan B cost
\(\$0.60 + \$0.06 \times \left(\mathrm{t} - 7\right) = \$0.08 \times \mathrm{t}\)

Let's solve this step by step:
\(\$0.60 + \$0.06\mathrm{t} - \$0.06 \times 7 = \$0.08\mathrm{t}\)
\(\$0.60 + \$0.06\mathrm{t} - \$0.42 = \$0.08\mathrm{t}\)
\(\$0.18 + \$0.06\mathrm{t} = \$0.08\mathrm{t}\)
\(\$0.18 = \$0.08\mathrm{t} - \$0.06\mathrm{t}\)
\(\$0.18 = \$0.02\mathrm{t}\)
\(\mathrm{t} = \$0.18 \div \$0.02 = 9\)

Let's verify: For a 9-minute call:

- Plan A: \(\$0.60 + \$0.06 \times (9-7) = \$0.60 + \$0.06 \times 2 = \$0.60 + \$0.12 = \$0.72\)
- Plan B: \(\$0.08 \times 9 = \$0.72\) ✓

Final Answer

The duration of a call for which both plans charge the same amount is 9 minutes.

The answer is B.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding Plan A's pricing structure

Students often misinterpret "\(\$0.60\) for the first 7 minutes" as meaning you pay \(\$0.60\) per minute for each of the first 7 minutes, rather than understanding it as a flat fee of \(\$0.60\) total for up to 7 minutes of talk time. This leads to setting up completely wrong equations.

2. Failing to recognize the piecewise nature of Plan A

Students may not realize that Plan A has two different pricing scenarios: one formula for calls ≤ 7 minutes and another for calls > 7 minutes. They might try to use a single equation for all call durations, missing the "thereafter" condition in the problem.

3. Setting up equations without checking which scenario applies

Even when students understand there are two cases, they might immediately assume the answer falls in the "≤ 7 minutes" case and only set up the equation Plan B = \(\$0.60\), without checking whether this assumption makes sense given the different rates.

Errors while executing the approach

1. Algebraic manipulation errors when distributing

When expanding \(\$0.06 \times \left(\mathrm{t} - 7\right)\), students commonly make the error: \(\$0.06\mathrm{t} - \$0.06 \times 7 = \$0.06\mathrm{t} - \$0.42\), but some students incorrectly calculate this as \(\$0.06\mathrm{t} - \$0.07\) or forget to subtract the \(\$0.42\) entirely.

2. Sign errors when rearranging the equation

In the step where \(\$0.18 = \$0.08\mathrm{t} - \$0.06\mathrm{t}\), students often make sign mistakes, such as writing \(\$0.18 = \$0.06\mathrm{t} - \$0.08\mathrm{t} = -\$0.02\mathrm{t}\), leading to a negative answer that should signal an error.

3. Decimal division errors

When calculating \(\$0.18 \div \$0.02 = 9\), students frequently make arithmetic mistakes with decimal division, getting answers like 0.9, 90, or 3.6 instead of the correct answer of 9.

Errors while selecting the answer

1. Forgetting to verify the solution

Students may arrive at \(\mathrm{t} = 9\) but fail to substitute back into both original cost formulas to confirm that Plan A and Plan B indeed give the same cost. This verification step would catch any computational errors made during solving.

No likely faltering points in answer selection beyond verification - the question asks for the duration in minutes, and if calculated correctly, 9 minutes directly corresponds to choice B.