A certain list consists of 21 different numbers. If n is in the list and n is 4 times the...

1. Translate the problem requirements: We have 21 numbers total, where one specific number n equals 4 times the average of the remaining 20 numbers. We need to find what fraction n represents of the total sum of all 21 numbers.
2. Express the constraint algebraically: Set up the relationship between n and the sum of the other 20 numbers using the given condition that \(\mathrm{n = 4 \times (average\, of\, other\, 20\, numbers)}\).
3. Connect individual relationships to the total sum: Use the relationship from step 2 to express everything in terms of the total sum of all 21 numbers.
4. Solve for the desired fraction: Algebraically manipulate to find what fraction n represents of the total sum.

Execution of Strategic Approach

1. Translate the problem requirements

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

• We have a list of 21 different numbers
• One of these numbers is called 'n'
• This number n has a special property: it equals 4 times the average of the other 20 numbers
• We want to find what fraction n represents of the total sum of all 21 numbers

Think of it this way: imagine you have 21 test scores, and one particular score is exactly 4 times the average of the other 20 scores. We want to know what portion of the total points this special score represents.

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

2. Express the constraint algebraically

Now let's put our understanding into mathematical terms:

Since n equals 4 times the average of the other 20 numbers, we can write:
\(\mathrm{n = 4 \times (average\, of\, the\, other\, 20\, numbers)}\)

Let's call the sum of the other 20 numbers 'S'. Then:

• The average of the other 20 numbers = \(\mathrm{S \div 20}\)
• So \(\mathrm{n = 4 \times (S \div 20) = 4S \div 20 = S \div 5}\)

This tells us something important: n equals one-fifth of the sum of all the other numbers!

3. Connect individual relationships to the total sum

Now we need to connect this to the total sum of all 21 numbers.

The total sum of all 21 numbers = n + S (where S is still the sum of the other 20 numbers)

We found that \(\mathrm{n = S \div 5}\), which means \(\mathrm{S = 5n}\)

So the total sum = \(\mathrm{n + S = n + 5n = 6n}\)

This is a key insight: the total sum of all 21 numbers equals 6n!

Process Skill: MANIPULATE - Using algebraic relationships to express everything in terms of one variable

4. Solve for the desired fraction

Now we can answer the question: What fraction is n of the total sum?

Fraction = \(\mathrm{n \div (total\, sum) = n \div (6n) = \frac{1}{6}}\)

Let's verify this makes sense:

• If the total sum is 6n, then n represents \(\mathrm{\frac{1}{6}}\) of that total
• The remaining 5n represents the sum of the other 20 numbers
• This confirms our earlier finding that \(\mathrm{n = (5n) \div 5 = S \div 5}\)

Final Answer

n represents \(\mathrm{\frac{1}{6}}\) of the sum of all 21 numbers.

The answer is (B) \(\mathrm{\frac{1}{6}}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "4 times the average of the other 20 numbers"

Students may incorrectly think that n equals 4 times the average of ALL 21 numbers, rather than specifically the OTHER 20 numbers (excluding n itself). This fundamental misreading changes the entire setup and leads to a completely different equation.

2. Confusing what the question is asking for

Students might get confused about whether they need to find n as a fraction of the average, or n as a fraction of the sum of the other 20 numbers, rather than understanding that the question asks for n as a fraction of the TOTAL sum of all 21 numbers.

3. Not recognizing the need to express everything in terms of one variable

Students may struggle to see that they need to relate n, the sum of the other 20 numbers (S), and the total sum all together. They might try to assign specific numerical values or get stuck because they think they need more information.

Errors while executing the approach

1. Algebraic manipulation errors when working with the constraint

When converting \(\mathrm{n = 4 \times (S \div 20)}\), students might make arithmetic errors, such as getting \(\mathrm{n = 4S \div 20 = S \div 4}\) instead of the correct \(\mathrm{n = S \div 5}\). This fraction error carries through the entire solution.

2. Incorrectly expressing the total sum

After finding that \(\mathrm{n = S \div 5}\) (so \(\mathrm{S = 5n}\)), students might incorrectly write the total sum as something other than \(\mathrm{n + S = n + 5n = 6n}\). They might forget to add n to S, or make errors in the substitution.

Errors while selecting the answer

1. Taking the reciprocal of the correct answer

Students may correctly find that the total sum equals 6n, but then incorrectly calculate the fraction as \(\mathrm{(total\, sum) \div n = 6n \div n = 6}\), or somehow arrive at \(\mathrm{\frac{6}{1}}\) instead of \(\mathrm{\frac{1}{6}}\). They flip the numerator and denominator in the final step.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for the sum of the other 20 numbers

Let's say the sum of the other 20 numbers = 20

This choice makes the average of these 20 numbers = \(\mathrm{\frac{20}{20} = 1}\), which will lead to clean calculations.

Step 2: Find the value of n

Given that \(\mathrm{n = 4 \times (average\, of\, other\, 20\, numbers)}\)

\(\mathrm{n = 4 \times 1 = 4}\)

Step 3: Calculate the total sum of all 21 numbers

Total sum = n + sum of other 20 numbers

Total sum = 4 + 20 = 24

Step 4: Find what fraction n is of the total sum

Fraction = \(\mathrm{\frac{n}{Total\, sum} = \frac{4}{24} = \frac{1}{6}}\)

Why this approach works: By choosing the sum of the other 20 numbers to be 20, we made their average equal to 1, which simplified all subsequent calculations. The relationships in the problem remain intact, and we get the same answer as the algebraic approach.

Verification: We can verify our relationships hold:

• Average of other 20 numbers = \(\mathrm{\frac{20}{20} = 1}\) ✓
• \(\mathrm{n = 4 \times 1 = 4}\) ✓
• n as fraction of total = \(\mathrm{\frac{4}{24} = \frac{1}{6}}\) ✓