- Translate the problem requirements: We have 42 people total, each buying one bouquet. 28 bouquets contain red carnations (could be red-only or red-and-white mix), 24 bouquets contain white carnations (could be white-only or red-and-white mix). We need to find how many bouquets contain red carnations only.
- Identify the overlapping groups: Recognize this as a set problem where some bouquets contain both red and white carnations, creating an overlap between the red group and white group.
- Apply the total-parts relationship: Use the principle that \(\mathrm{Total} = \mathrm{Red-only} + \mathrm{White-only} + \mathrm{Both\ colors}\), and that \(\mathrm{Red\ total} = \mathrm{Red-only} + \mathrm{Both\ colors}\).
- Solve using the overlap logic: Calculate the overlap (both colors) first, then subtract from the red total to find red-only bouquets.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in everyday language:
Imagine 42 people walking into a flower shop, and each person buys exactly one bouquet. Now, these bouquets come in three types:
- Red carnations only
- White carnations only
- Mixed bouquets (both red and white carnations)
The problem tells us:
- Total people = 42
- 28 bouquets contained red carnations (this includes both red-only and mixed bouquets)
- 24 bouquets contained white carnations (this includes both white-only and mixed bouquets)
We need to find: How many bouquets contained red carnations only?
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
2. Identify the overlapping groups
This is a classic overlapping sets problem. Think of it like this:
Imagine drawing two circles:
- Circle 1: All bouquets with red carnations (28 bouquets)
- Circle 2: All bouquets with white carnations (24 bouquets)
These circles overlap because some bouquets contain BOTH red and white carnations. The overlapping area represents the mixed bouquets.
Since we have 42 people total, but \(28 + 24 = 52\), we're counting some bouquets twice. This tells us there must be an overlap!
Process Skill: VISUALIZE - Recognizing the overlapping structure helps us see the solution path
3. Apply the total-parts relationship
Let's use plain English logic:
\(\mathrm{Total\ bouquets} = \mathrm{Red-only} + \mathrm{White-only} + \mathrm{Mixed\ bouquets}\)
\(42 = \mathrm{Red-only} + \mathrm{White-only} + \mathrm{Mixed}\)
Also, we know:
- All red bouquets = Red-only + Mixed = 28
- All white bouquets = White-only + Mixed = 24
If we add all red bouquets + all white bouquets:
\(28 + 24 = 52\)
But we only have 42 people! The extra 10 \((52 - 42 = 10)\) represents the mixed bouquets that we counted twice.
So: Mixed bouquets = 10
4. Solve using the overlap logic
Now we can find red-only bouquets:
Since All red bouquets = Red-only + Mixed
\(28 = \mathrm{Red-only} + 10\)
\(\mathrm{Red-only} = 28 - 10 = 18\)
Let's verify our answer:
- Red-only: 18 bouquets
- White-only: \(24 - 10 = 14\) bouquets
- Mixed: 10 bouquets
- Total: \(18 + 14 + 10 = 42\) ✓
Final Answer
The number of bouquets containing red carnations only is 18.
This matches answer choice C.
Common Faltering Points
Errors while devising the approach
- Misinterpreting "contained red carnations" as "red carnations only": Students often confuse the statement "28 contained red carnations" to mean 28 bouquets had ONLY red carnations, when it actually means 28 bouquets had red carnations (including mixed bouquets). This fundamental misreading leads to treating the problem as simple subtraction rather than recognizing it as an overlapping sets problem.
- Not recognizing this as an overlapping sets problem: Students may try to solve this using simple arithmetic \((42 - 28 = 14, \text{etc.})\) without understanding that some bouquets contain both colors and are counted in both the "red" and "white" categories. This leads to incorrect approaches that don't account for the overlap.
- Setting up incorrect variable relationships: Students might define variables incorrectly, such as letting R = 28 and W = 24 represent red-only and white-only bouquets respectively, rather than understanding these represent total bouquets containing each color.
Errors while executing the approach
- Arithmetic errors in calculating the overlap: Even when students understand the overlapping sets concept, they may make calculation errors when finding the overlap: \(28 + 24 - 42 = 10\). Common mistakes include getting \(28 + 24 = 54\) instead of 52, or subtracting incorrectly.
- Incorrect application of the overlap formula: Students might calculate the overlap correctly (10) but then subtract it from the wrong quantity, such as calculating \(42 - 10 = 32\) instead of \(28 - 10 = 18\) for red-only bouquets.
Errors while selecting the answer
- Selecting the overlap value instead of red-only: After correctly calculating that 10 bouquets contain both colors, students might mistakenly select answer choice B (10) thinking this answers the question, rather than using this to find red-only bouquets (18).
- Confusing red-only with white-only: Students might correctly calculate both red-only (18) and white-only (14) but then select the wrong value, especially if they lose track of what the question is specifically asking for.
