Question

Of the 84 parents who attended a meeting at a school, 35 volunteered to supervise children during the school picnic and 11 volunteered both to supervise children during the picnic and to bring refreshments to the picnic. If the number of parents who volunteered to bring refreshments was 1.5 times the number of parents who neither volunteered to supervise children during the picnic nor volunteered to bring refreshments, how many of the parents volunteered to bring refreshments?

• A. 25
• B. 36
• C. 38
• D. 42
• E. 45

Hint:

For problems with two overlapping sets, it's often easiest to break the total down into the four distinct regions: Only Group A, Only Group B, Both, and Neither. Set up an equation where the sum of these four regions equals the total.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a set theory problem that can be solved using a Venn diagram or the principle of inclusion-exclusion. We are given information about a total group and various overlapping subgroups.
Step 2: Key Formula or Approach:
Let S be the set of parents who volunteered to supervise, and R be the set of parents who volunteered to bring refreshments.
The formula for two sets is: \[ \text{Total} = |S| + |R| - |S \cap R| + |\text{Neither}| \] Alternatively, a more intuitive formula is: \[ \text{Total} = |\text{Only S}| + |\text{Only R}| + |\text{Both}| + |\text{Neither}| \] We are given:

Total = 84
\(|S|\) = 35
\(|S \cap R|\) (Both) = 11
\(|R| = 1.5 \times |\text{Neither}|\)
Step 3: Detailed Explanation:
Let's use the second formula. First, find the number of parents who only supervised. \[ |\text{Only S}| = |S| - |S \cap R| = 35 - 11 = 24 \] Let \(N = |\text{Neither}|\). Then the total number who brought refreshments is \(|R| = 1.5N\). The number of parents who only brought refreshments is: \[ |\text{Only R}| = |R| - |S \cap R| = 1.5N - 11 \] Now, plug all the parts into the total formula: \[ \text{Total} = |\text{Only S}| + |\text{Only R}| + |\text{Both}| + |\text{Neither}| \] \[ 84 = 24 + (1.5N - 11) + 11 + N \] Simplify the equation: \[ 84 = 24 + 1.5N + N \] \[ 84 = 24 + 2.5N \] Subtract 24 from both sides: \[ 60 = 2.5N \] Solve for N: \[ N = \frac{60}{2.5} = \frac{600}{25} = 24 \] The question asks for the number of parents who volunteered to bring refreshments, which is \(|R|\). \[ |R| = 1.5 \times N = 1.5 \times 24 = 36 \] Step 4: Final Answer:
36 parents volunteered to bring refreshments.