At a certain corporation, the planning committee has 23 members and the finance committee has 20 members. If a total...

1. Translate the problem requirements: We have two committees with some overlap. "Members of only one committee" means people who are in exactly one committee but not both. We need to find how many people are in both committees.
2. Set up the relationship between total members and overlaps: Use the principle that total unique people = people in planning only + people in finance only + people in both committees.
3. Apply the given constraint about single-committee members: Since 21 people are in only one committee, we can work backwards to find the overlap.
4. Calculate the number in both committees: Use the relationship between total committee memberships and actual people to solve for the overlap.

Execution of Strategic Approach

1. Translate the problem requirements

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

• The planning committee has 23 members
• The finance committee has 20 members
• Some people might be on both committees (the overlap we need to find)
• Exactly 21 people are members of "only one" committee, meaning they're on planning OR finance, but NOT both

What we're looking for: How many people are on BOTH committees?

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

2. Set up the relationship between total members and overlaps

Let's think about this step by step using plain English:

Imagine we're counting people in three groups:

• Group A: People ONLY on planning committee
• Group B: People ONLY on finance committee
• Group C: People on BOTH committees

The problem tells us that \(\mathrm{Group\,A + Group\,B = 21}\) people (those in "only one" committee).

Now, when we count the total committee memberships:

• Planning committee (23) = \(\mathrm{Group\,A + Group\,C}\)
• Finance committee (20) = \(\mathrm{Group\,B + Group\,C}\)

So the total memberships = \(\mathrm{23 + 20 = 43}\)
But this counts Group C people twice (once for each committee they're on).

Process Skill: VISUALIZE - Setting up the relationship between overlapping groups

3. Apply the given constraint about single-committee members

We know that exactly 21 people are in only one committee.
This means: \(\mathrm{Group\,A + Group\,B = 21}\)

Since the total number of actual people = \(\mathrm{Group\,A + Group\,B + Group\,C}\)
We get: \(\mathrm{Total\,actual\,people = 21 + Group\,C}\)

Alternatively, we can think about it this way:
\(\mathrm{Total\,actual\,people = Total\,memberships - People\,counted\,twice}\)
\(\mathrm{Total\,actual\,people = 43 - Group\,C}\)

Setting these equal: \(\mathrm{21 + Group\,C = 43 - Group\,C}\)

4. Calculate the number in both committees

Now we solve our equation from plain English reasoning:

\(\mathrm{21 + Group\,C = 43 - Group\,C}\)
\(\mathrm{21 + Group\,C + Group\,C = 43}\)
\(\mathrm{21 + 2 \times Group\,C = 43}\)
\(\mathrm{2 \times Group\,C = 43 - 21}\)
\(\mathrm{2 \times Group\,C = 22}\)
\(\mathrm{Group\,C = 11}\)

Let's verify this makes sense:

• People on both committees: 11
• People only on planning: \(\mathrm{23 - 11 = 12}\)
• People only on finance: \(\mathrm{20 - 11 = 9}\)
• People on exactly one committee: \(\mathrm{12 + 9 = 21}\) ✓

This matches our given constraint!

Final Answer

The number of people who are members of both committees is 11.

This corresponds to answer choice B. 11.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "only one committee": Students often confuse "only one committee" with "at least one committee." The phrase "only one" means exactly one committee (not both), but students might set up equations thinking it means anyone who is on any committee.

2. Incorrect overlap visualization: Students may struggle to properly identify the three distinct groups: (people only on planning), (people only on finance), and (people on both). They might incorrectly think there are just two groups or fail to recognize that the "both" group gets counted twice in the individual committee totals.

3. Misunderstanding the constraint: Students might interpret "21 people are members of only one committee" as meaning there are 21 total people across both committees, rather than understanding this refers specifically to people who are NOT on both committees.

Errors while executing the approach

1. Arithmetic errors in equation setup: When setting up the equation \(\mathrm{21 + C = 43 - C}\), students may make sign errors or incorrectly combine terms, such as writing \(\mathrm{21 - C = 43 + C}\) or failing to properly isolate the variable.

2. Double-counting confusion: Students might incorrectly calculate the total memberships \(\mathrm{(23 + 20 = 43)}\) or fail to understand why people in both committees get counted twice, leading to errors in the fundamental equation.

3. Verification calculation errors: When checking their answer, students may incorrectly calculate the number of people in only planning \(\mathrm{(23 - 11 = 12)}\) or only finance \(\mathrm{(20 - 11 = 9)}\), or make errors when adding these to verify they sum to 21.

Errors while selecting the answer

No likely faltering points - once students correctly solve for the overlap value of 11, the answer choice B is straightforward to select.