A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all...

1. Translate the problem requirements: We have 22 men (average age 38) and 26 women (unknown average age) at a party. The overall average age of all 48 people is exactly 35 years. We need to find the average age of the women.
2. Calculate total age of all people: Use the overall average and total count to find the sum of all ages at the party.
3. Calculate total age of all men: Use the men's average age and count to find the sum of all men's ages.
4. Find total age of women by subtraction: Subtract men's total age from overall total age to get women's total age.
5. Calculate women's average age: Divide women's total age by the number of women to get the average.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know from the problem in simple terms:

• We have a party with 22 men and 26 women
• Total people at the party: \(22 + 26 = 48\) people
• The average age of ALL people at the party is exactly 35 years
• The average age of the men is exactly 38 years
• We need to find the average age of the women

Think of this like a balance scale - the overall average of 35 is influenced by both the men's ages and the women's ages. Since the men's average (38) is above the overall average (35), the women's average must be below 35 to balance things out.

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

2. Calculate total age of all people

If 48 people have an average age of 35 years, what's the total of all their ages combined?

Think of it this way: if everyone were exactly 35 years old, the total would be \(35 \times 48\). Since the average is 35, this gives us the actual total.

Total age of all people = Average age × Number of people
Total age of all people = \(35 \times 48 = 1,680\) years

3. Calculate total age of all men

Now let's find the combined age of just the men:

We have 22 men with an average age of 38 years.

Total age of all men = Average age of men × Number of men
Total age of all men = \(38 \times 22 = 836\) years

4. Find total age of women by subtraction

Here's the key insight: the total age of everyone equals the total age of men plus the total age of women.

Total age of all people = Total age of men + Total age of women
\(1,680 = 836 + \text{Total age of women}\)

Solving for the women's total age:
Total age of women = \(1,680 - 836 = 844\) years

5. Calculate women's average age

Finally, we can find the average age of the women:

We have 26 women with a combined age of 844 years.

Average age of women = Total age of women ÷ Number of women
Average age of women = \(844 \div 26 = 32.46...\) years

Since \(844 \div 26 = 32.46...\), this rounds to approximately 32.5 years.

6. Final Answer

The average age of the women is approximately 32.5 years.

Looking at our answer choices:

1. 31 - too low
2. 31.5 - too low
3. 32 - close but not as close as 32.5
4. 32.5 - matches our calculation exactly
5. 33 - too high

The answer is (D) 32.5.

As a quick check: our answer makes sense because the women's average (32.5) is below the overall average (35), which balances out the men's higher average (38), just as we expected from our initial reasoning.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the relationship between individual group averages and overall average
Students often struggle to recognize that this is a weighted average problem. They might try to simply average the men's average (38) and women's average (unknown) to get the overall average (35), forgetting that there are different numbers of men (22) and women (26). The correct approach requires understanding that the total age of all people equals the sum of total ages of each group.

Faltering Point 2: Attempting to set up complex equations instead of using the straightforward total-parts relationship
Some students may try to create algebraic equations with variables for the women's average age, making the problem unnecessarily complicated. While this approach can work, it's more error-prone than the simpler method of calculating totals first (total age of everyone, then total age of men, then finding total age of women by subtraction).

Errors while executing the approach

Faltering Point 1: Arithmetic errors in basic multiplication and division
Key calculations like \(35 \times 48 = 1,680\), \(38 \times 22 = 836\), and especially \(844 \div 26 = 32.46...\) are prone to computational mistakes. Students may rush through these calculations or make errors when dividing 844 by 26, potentially getting results like 32.15 or 33.2 instead of the correct 32.46.

Faltering Point 2: Incorrect subtraction when finding women's total age
When calculating \(1,680 - 836 = 844\), students might make simple subtraction errors, especially if they've made earlier multiplication mistakes. An error here would cascade through to the final average calculation, leading to an incorrect answer choice.

Errors while selecting the answer

Faltering Point 1: Misreading 'closest to' and selecting an exact but incorrect calculation
If a student made an arithmetic error and calculated something like 32.15, they might confidently select choice (C) 32 as the 'closest' value, not realizing their calculation was wrong. The question asks for the closest value, so students need to both calculate correctly AND identify which answer choice is nearest to their result.

Faltering Point 2: Rounding errors or selecting the wrong approximation
Even with the correct calculation of 32.46..., students might incorrectly round this to 32 (choice C) instead of recognizing that 32.5 (choice D) is actually closer to 32.46 than 32 is. This type of rounding judgment error is common when the calculated value falls between two answer choices.