Of the 84 parents who attended a meeting at a school, 35 volunteered to supervise children during the school picnic...

1. Translate the problem requirements: We have 84 total parents with overlapping volunteer commitments. 35 supervise children, 11 do both activities, and the number bringing refreshments equals 1.5 times those doing neither activity. We need to find how many bring refreshments.
2. Set up the group relationships using variables: Define variables for each group (refreshments-only, supervision-only, both, neither) and establish how they relate to the given totals.
3. Create the key constraint equation: Use the relationship that refreshments volunteers = 1.5 × neither group, combined with the fact that all groups must sum to 84 total parents.
4. Solve systematically for the refreshments group: Substitute known values and the constraint relationship to find the number of parents who volunteered to bring refreshments.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

• We have 84 parents total at this meeting
• 35 parents volunteered to supervise children during the picnic
• 11 parents volunteered for BOTH supervising children AND bringing refreshments
• There's a special relationship: the number bringing refreshments = 1.5 × the number doing neither activity
• We need to find how many parents volunteered to bring refreshments

Think of this like organizing volunteers for a community event. Some people do one job, some do both jobs, some do the other job, and some don't volunteer at all. We need to figure out how these groups fit together.

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

2. Set up the group relationships using variables

Let's organize the parents into four distinct groups. Imagine drawing two overlapping circles - one for "supervise" and one for "refreshments":

• Parents who ONLY supervise children (but don't bring refreshments)
• Parents who do BOTH activities (supervise AND bring refreshments) = 11
• Parents who ONLY bring refreshments (but don't supervise)
• Parents who do NEITHER activity

Let's call the number who bring refreshments "R" (this is what we're looking for).
Let's call the number who do neither activity "N".

Now we can figure out each group:

• BOTH activities = 11 (given)
• ONLY supervise = \(35 - 11 = 24\) (since 35 total supervise, minus the 11 who do both)
• ONLY refreshments = \(\mathrm{R} - 11\) (since R total bring refreshments, minus the 11 who do both)
• NEITHER = N

3. Create the key constraint equation

We have two important relationships:

First, all parents must add up to 84:
(ONLY supervise) + (BOTH) + (ONLY refreshments) + (NEITHER) = 84
\(24 + 11 + (\mathrm{R} - 11) + \mathrm{N} = 84\)
\(24 + \mathrm{R} + \mathrm{N} = 84\)
Therefore: \(\mathrm{R} + \mathrm{N} = 60\)

Second, we're told that refreshments volunteers = 1.5 times those doing neither:
\(\mathrm{R} = 1.5\mathrm{N}\)

Now we have two simple equations:

• \(\mathrm{R} + \mathrm{N} = 60\)
• \(\mathrm{R} = 1.5\mathrm{N}\)

Process Skill: APPLY CONSTRAINTS - Using all given conditions to create solvable equations

4. Solve systematically for the refreshments group

Let's substitute the second equation into the first:

Since \(\mathrm{R} = 1.5\mathrm{N}\), we can replace R in the equation \(\mathrm{R} + \mathrm{N} = 60\):
\(1.5\mathrm{N} + \mathrm{N} = 60\)
\(2.5\mathrm{N} = 60\)
\(\mathrm{N} = 60 ÷ 2.5 = 24\)

Now we can find R:
\(\mathrm{R} = 1.5\mathrm{N} = 1.5 × 24 = 36\)

Let's verify this makes sense:

• ONLY supervise: 24 parents
• BOTH activities: 11 parents
• ONLY refreshments: \(36 - 11 = 25\) parents
• NEITHER: 24 parents
• Total: \(24 + 11 + 25 + 24 = 84\) ✓

And checking our constraint: \(\mathrm{R} = 1.5\mathrm{N}\) means \(36 = 1.5 × 24 = 36\) ✓

Final Answer

The number of parents who volunteered to bring refreshments is 36.

This matches answer choice B. 36.

Common Faltering Points

Errors while devising the approach

Students often confuse which group is 1.5 times the other. The problem states "the number of parents who volunteered to bring refreshments was 1.5 times the number of parents who neither volunteered," but students might incorrectly set up the equation as \(\mathrm{N} = 1.5\mathrm{R}\) instead of \(\mathrm{R} = 1.5\mathrm{N}\). This reversal completely changes the problem setup and leads to incorrect answers.

When setting up the problem, students frequently forget that the 11 parents who do BOTH activities are already counted in the 35 who supervise children. They might incorrectly add all numbers together (\(35 + 11 + \mathrm{R} + \mathrm{N} = 84\)) instead of recognizing that the "ONLY supervise" group is \(35 - 11 = 24\). This double-counting error completely throws off the constraint equations.

Students sometimes confuse whether R represents "total parents bringing refreshments" or "parents ONLY bringing refreshments." The problem asks for the total number bringing refreshments, but if students think R represents only those bringing refreshments (excluding the 11 who do both), they'll set up incorrect equations and arrive at wrong answers.

Errors while executing the approach

When dividing \(60 ÷ 2.5\), students often make calculation mistakes. They might incorrectly calculate \(60 ÷ 2.5\) as 25 instead of 24, or struggle with the decimal division and get confused about the process. This leads to \(\mathrm{N} = 25\), which then gives \(\mathrm{R} = 37.5\), creating further confusion.

When substituting \(\mathrm{R} = 1.5\mathrm{N}\) into \(\mathrm{R} + \mathrm{N} = 60\), students might make algebraic mistakes such as writing \(1.5\mathrm{N} + \mathrm{N} = 2\mathrm{N}\) instead of \(2.5\mathrm{N}\), or incorrectly distribute when combining like terms. These errors lead to wrong values for both N and R.

Errors while selecting the answer

After correctly calculating \(\mathrm{N} = 24\) and \(\mathrm{R} = 36\), students might accidentally select 24 (the number doing neither activity) instead of 36 (the number bringing refreshments). Since the question asks specifically for "how many parents volunteered to bring refreshments," selecting the value of N instead of R is a common final-step error, especially if students lose track of what each variable represents.