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 need to find the average age of women when we know the total number of people (\(22 + 26 = 48\) people), the overall average age (35 years), and the average age of men (38 years). This is a weighted average problem where we need to work backwards from the total.
2. Calculate the total age of all people: Use the overall average and total count to find the sum of all ages at the party.
3. Calculate the total age contributed by men: Use the men's average age and their count to find how much of the total age comes from the men's group.
4. Find the total age contributed by women: Subtract the men's total age from the overall total age to isolate the women's contribution.
5. Calculate the women's average age: Divide the women's total age by the number of women to get their average age.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find in plain English:

• We have 22 men and 26 women at a party (total of 48 people)
• The average age of ALL people at the party is 35 years
• The average age of just the MEN is 38 years
• We need to find the average age of just the WOMEN

Think of this like a seesaw: the men's average (38) is above the overall average (35), so the women's average must be below 35 to balance things out. This is how weighted averages work - different groups contribute different amounts to pull the overall average in their direction.

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

2. Calculate the total age of all people

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

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

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

3. Calculate the total age contributed by men

Now let's find how much age the men contribute to that total of 1,680 years.

If 22 men have an average age of 38 years, their total contribution is:
Total age of men = \(22 \times 38 = 836\) years

So the men contribute 836 years out of the total 1,680 years.

4. Find the total age contributed by women

Since we know the total age of everyone (1,680 years) and the total age contributed by men (836 years), we can find what the women contribute:

Total age of women = Total age of all people - Total age of men
Total age of women = \(1,680 - 836 = 844\) years

The 26 women together contribute 844 years to the total.

5. Calculate the women's average age

Now we can find the average age of the women by dividing their total age by how many women there are:

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

Rounding to one decimal place: 32.5 years

Final Answer

The average age of the women is approximately 32.5 years.

Looking at our answer choices: (A) 31, (B) 31.5, (C) 32, (D) 32.5, (E) 33

Our calculated value of 32.5 exactly matches choice (D).

The answer is (D) 32.5

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what needs to be found
Students may confuse which group's average they need to calculate. The question asks for the average age of women, but some students might accidentally set up equations to solve for the men's average age or get confused about which unknown they're solving for.

Faltering Point 2: Not recognizing this as a weighted average problem
Students might try to simply average the given averages (35 and 38) without considering that the groups have different sizes (22 men vs 26 women). They might incorrectly think: "If overall average is 35 and men's average is 38, then women's average should be around 32" without properly accounting for the different group sizes.

Faltering Point 3: Setting up incorrect equations
Some students may attempt to use weighted average formulas directly but set them up incorrectly. For example, they might write: \((22 \times \text{women's age} + 26 \times 38) \div 48 = 35\), mixing up which numbers go with which groups.

Errors while executing the approach

Faltering Point 1: Arithmetic calculation errors
Students often make mistakes in basic multiplication and division. Common errors include: \(48 \times 35 = 1,680\) (might calculate as 1,680 or other wrong values), \(22 \times 38 = 836\) (might get 844 or other incorrect products), or \(844 \div 26 = 32.46\) (division errors are very common).

Faltering Point 2: Subtraction error when finding women's total age
When calculating \(1,680 - 836 = 844\), students might make simple subtraction mistakes, potentially getting 844 confused with 836, or making borrowing errors that lead to incorrect totals like 854 or 834.

Faltering Point 3: Using wrong totals in calculations
Students might carry forward an earlier calculation error. For instance, if they incorrectly calculated the total age of all people or the total age of men, they would use these wrong values in subsequent steps, leading to an incorrect final answer even if their method is sound.

Errors while selecting the answer

Faltering Point 1: Rounding errors or precision issues
Since \(844 \div 26 = 32.461...\), students need to identify that this is closest to 32.5. Some might incorrectly round to 32.4 or 32 instead of recognizing that 32.5 is the closest available option among the choices.

Faltering Point 2: Choosing based on rough estimation rather than calculation
Students might use intuitive reasoning like "the answer should be slightly below 35" and pick 33 or 32 without doing the full calculation, missing the precision needed to identify 32.5 as the correct choice.