Last year Manfred received 26 paychecks. Each of his first 6 paychecks was $750; each of his remaining paychecks was...

1. Translate the problem requirements: Manfred received 26 total paychecks - the first 6 were $750 each, and the remaining 20 were $30 more than $750 (so $780 each). We need to find the average amount across all 26 paychecks.
2. Calculate total earnings from each paycheck group: Find the total money from the first 6 paychecks and the total from the remaining 20 paychecks separately.
3. Sum total annual earnings: Add the earnings from both groups to get Manfred's total income for the year.
4. Calculate the average paycheck amount: Divide total annual earnings by 26 paychecks to find the arithmetic mean.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know about Manfred's paychecks in plain English:

• Manfred received a total of 26 paychecks last year
• His first 6 paychecks were each $750
• His remaining paychecks were each $30 more than his first 6 paychecks

So his remaining paychecks were each: \(\$750 + \$30 = \$780\)

How many remaining paychecks did he have? Since he had 26 total paychecks and 6 were the smaller amount, he had: \(26 - 6 = 20\) paychecks at the higher amount.

We need to find the average (arithmetic mean) of all 26 paychecks.

Process Skill: TRANSLATE - Converting the problem's everyday language into clear mathematical understanding

2. Calculate total earnings from each paycheck group

Now let's calculate how much money Manfred earned from each group of paychecks separately:

From his first 6 paychecks:
Each paycheck was $750, and he had 6 of them
Total from first 6 paychecks = \(6 \times \$750 = \$4,500\)

From his remaining 20 paychecks:
Each paycheck was $780, and he had 20 of them
Total from remaining 20 paychecks = \(20 \times \$780 = \$15,600\)

This approach of separating the calculation makes the arithmetic much simpler and helps us avoid errors.

3. Sum total annual earnings

To find Manfred's total earnings for the year, we add up the money from both groups of paychecks:

Total annual earnings = Money from first 6 paychecks + Money from remaining 20 paychecks
Total annual earnings = \(\$4,500 + \$15,600 = \$20,100\)

So Manfred earned $20,100 total last year from his 26 paychecks.

4. Calculate the average paycheck amount

The average (arithmetic mean) paycheck amount is simply the total money earned divided by the number of paychecks:

Average paycheck = Total annual earnings ÷ Number of paychecks
Average paycheck = \(\$20,100 ÷ 26\)

Let's calculate this division:
\(\$20,100 ÷ 26 = \$773.076...\)

Since the problem asks for the answer "to the nearest dollar," we round $773.076... to $773.

Final Answer

The average amount of Manfred's paychecks for the year was $773.

Looking at our answer choices, this matches choice (D) $773.

Verification: We can quickly check this makes sense - the average should be between $750 (the lower paycheck amount) and $780 (the higher paycheck amount), and closer to $780 since there were more paychecks at the higher amount (20 vs 6). Our answer of $773 fits this expectation perfectly.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the paycheck structure: Students may incorrectly assume that ALL paychecks were $30 more than the first 6, rather than understanding that only the REMAINING paychecks (after the first 6) were $30 more. This leads to calculating 26 paychecks at $780 instead of separating them into two groups.

2. Confusion about the number of remaining paychecks: Students might forget to subtract the first 6 paychecks from the total 26, incorrectly thinking there are 26 remaining paychecks instead of 20 remaining paychecks (26 - 6 = 20).

Errors while executing the approach

1. Arithmetic errors in multiplication: Students may make calculation mistakes when computing 6 × $750 = $4,500 or 20 × $780 = $15,600, leading to an incorrect total annual earnings.

2. Division errors: When dividing $20,100 by 26, students might make computational errors or use incorrect long division, arriving at the wrong average before rounding.

3. Incorrect addition of totals: Students may add $4,500 + $15,600 incorrectly, getting a wrong total annual earnings amount which then leads to an incorrect average.

Errors while selecting the answer

1. Rounding errors: Students might round $773.076... incorrectly, either not rounding to the nearest dollar as requested, or rounding up to $774 instead of down to $773, leading them to select a different answer choice.