The sum of x numbers is what percent greater than the average (arithmetic mean) of the x numbers?

1. Translate the problem requirements: We need to find by what percent the sum of \(\mathrm{x}\) numbers exceeds their arithmetic mean (average)
2. Express sum and average in terms of each other: Use the fundamental relationship that sum = average × number of values
3. Calculate the difference between sum and average: Find how much greater the sum is compared to the average in absolute terms
4. Convert the difference to percentage terms: Apply the percent increase formula to express the difference as a percentage of the average

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked. We have \(\mathrm{x}\) numbers, and we want to know: by what percent is their sum greater than their average?

Think of it this way: if you have 3 test scores of 80, 90, and 100, their sum is 270 and their average is 90. The question asks: how much bigger is 270 compared to 90, expressed as a percentage?

In general terms, we need to find by what percent the sum of \(\mathrm{x}\) numbers exceeds their arithmetic mean (average).

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical comparison

2. Express sum and average in terms of each other

Let's use the fundamental relationship between sum and average that we all know from everyday math.

If we call the average of our \(\mathrm{x}\) numbers 'A', then:

- The sum of all \(\mathrm{x}\) numbers = \(\mathrm{A \times x}\)

This makes perfect sense - if 5 students each scored an average of 80 points, the total points scored by all students is \(\mathrm{80 \times 5 = 400}\).

So we have:

- Sum = \(\mathrm{A \times x}\)

- Average = \(\mathrm{A}\)

3. Calculate the difference between sum and average

Now let's find how much greater the sum is compared to the average in absolute terms.

Difference = Sum - Average
Difference = \(\mathrm{(A \times x) - A}\)
Difference = \(\mathrm{A \times (x - 1)}\)

This tells us that the sum exceeds the average by \(\mathrm{A \times (x - 1)}\) units.

For example, with our 3 test scores (sum = 270, average = 90):
Difference = \(\mathrm{270 - 90 = 180}\)
Using our formula: \(\mathrm{A \times (x - 1) = 90 \times (3 - 1) = 90 \times 2 = 180}\) ✓

4. Convert the difference to percentage terms

To find by what percent the sum is greater than the average, we use the percent increase formula:

Percent increase = \(\mathrm{(Difference ÷ Original\ value) \times 100\%}\)

Here:

- Difference = \(\mathrm{A \times (x - 1)}\)

- Original value (what we're comparing to) = Average = \(\mathrm{A}\)

So: Percent increase = \(\mathrm{[A \times (x - 1) ÷ A] \times 100\%}\)

The A's cancel out: Percent increase = \(\mathrm{(x - 1) \times 100\%}\)

This simplifies to: \(\mathrm{100(x - 1)\%}\)

Process Skill: SIMPLIFY - Recognizing that terms cancel to give us a clean final expression

4. Final Answer

The sum of \(\mathrm{x}\) numbers is \(\mathrm{100(x - 1)\%}\) greater than their average.

This matches answer choice E: \(\mathrm{100(x - 1)\%}\)

Let's verify with our example: 3 test scores with sum = 270, average = 90
Using our formula: \(\mathrm{100(3 - 1)\% = 100(2)\% = 200\%}\)
Direct calculation: \(\mathrm{\frac{270 - 90}{90} \times 100\% = \frac{180}{90} \times 100\% = 200\%}\) ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "percent greater than" as "percent of"

Students often confuse "what percent greater than" with "what percent of." They might try to find what percent the sum is OF the average (which would be \(\mathrm{\frac{Sum}{Average} \times 100\%}\)), rather than BY what percent the sum is GREATER THAN the average. This leads to the wrong formula setup from the beginning.

2. Confusion about what to compare

Some students might misread the question and think they need to compare individual numbers to the average, or compare the average to the sum (backwards), rather than clearly identifying that we need to find how much bigger the sum is compared to the average.

3. Not establishing the fundamental sum-average relationship

Students might struggle to express both the sum and average in comparable terms. They may not recognize that if the average is A, then the sum must be \(\mathrm{A \times x}\), which is the key relationship needed to solve this problem algebraically.

Errors while executing the approach

1. Algebraic manipulation errors when canceling terms

When calculating the percent increase as \(\mathrm{[A \times (x-1) ÷ A] \times 100\%}\), students might make errors in canceling the A terms, or forget to distribute the 100 correctly, leading to expressions like \(\mathrm{\frac{(x-1)}{100}}\) instead of \(\mathrm{100(x-1)}\).

2. Sign errors in the difference calculation

Students might incorrectly calculate the difference as Average - Sum instead of Sum - Average, leading to a negative result or the wrong expression \(\mathrm{A(1-x)}\) instead of \(\mathrm{A(x-1)}\).

Errors while selecting the answer

1. Choosing the decimal form instead of percentage form

Students might arrive at the correct mathematical result \(\mathrm{(x-1)}\) but select answer choice C: \(\mathrm{(x-1)\%}\), forgetting that they need to multiply by 100 to convert from decimal to percentage form. The correct answer should be \(\mathrm{100(x-1)\%}\).

2. Misreading similar-looking answer choices

Answer choices like \(\mathrm{\frac{(x-1)}{100}\%}\) and \(\mathrm{100(x-1)\%}\) can look similar at first glance. Students might select choice B thinking they have the right form, when actually their calculation gives \(\mathrm{100(x-1)\%}\), not \(\mathrm{\frac{(x-1)}{100}}\).

Alternate Solutions

Smart Numbers Approach

We can solve this problem by choosing specific values for the \(\mathrm{x}\) numbers and their sum, then calculating the percent difference directly.

Step 1: Choose convenient smart numbers
Let's choose \(\mathrm{x = 3}\) numbers with values that give us a clean average:
Numbers: 10, 20, 30
Sum = \(\mathrm{10 + 20 + 30 = 60}\)
Average = \(\mathrm{60 ÷ 3 = 20}\)

Step 2: Calculate the percent increase
We need to find by what percent the sum (60) is greater than the average (20).
Difference = \(\mathrm{Sum - Average = 60 - 20 = 40}\)
Percent increase = \(\mathrm{\frac{Difference}{Average} \times 100\% = \frac{40}{20} \times 100\% = 200\%}\)

Step 3: Verify with the answer choices
With \(\mathrm{x = 3}\):
• Choice E: \(\mathrm{100(x - 1) = 100(3 - 1) = 100(2) = 200\%}\) ✓

Step 4: Test with different smart numbers
Let's verify with \(\mathrm{x = 2}\) numbers: 5, 15
Sum = 20, Average = 10
Percent increase = \(\mathrm{\frac{20 - 10}{10} \times 100\% = 100\%}\)
Choice E: \(\mathrm{100(2 - 1) = 100\%}\) ✓

The smart numbers approach confirms that the answer is Choice E: \(\mathrm{100(x - 1)\%}\)