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A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?
Let's break down what we know from the problem in simple terms:
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
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
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
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
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.
The average age of the women is approximately 32.5 years.
Looking at our answer choices:
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.
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).
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.
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.