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In a certain region, the number of children who have been vaccinated against rubella is twice the number who have been vaccinated against mumps. The number who have been vaccinated against both is twice the number who have been vaccinated only against mumps. If 5,000 have been vaccinated against both, how many have been vaccinated only against rubella?
Let's break down what each phrase means in plain English:
Process Skill: TRANSLATE - Converting problem language into mathematical understanding is crucial here
Let's use simple letters to represent our groups:
Now let's convert our plain English understanding into mathematical relationships:
Relationship 1: "Rubella vaccinations are twice mumps vaccinations"
\(\mathrm{R = 2M}\)
Relationship 2: "Both vaccinations are twice the only-mumps vaccinations"
\(\mathrm{Both = 2 \times (Only\ M)}\)
Since Both = 5,000:
\(\mathrm{5{,}000 = 2 \times (Only\ M)}\)
Starting with what we know: Both = 5,000
Step 1: Find Only M
\(\mathrm{5{,}000 = 2 \times (Only\ M)}\)
\(\mathrm{Only\ M = 5{,}000 \div 2 = 2{,}500}\)
Step 2: Find total mumps vaccinations (M)
Total mumps = Only M + Both
\(\mathrm{M = 2{,}500 + 5{,}000 = 7{,}500}\)
Step 3: Find total rubella vaccinations (R)
\(\mathrm{R = 2M = 2 \times 7{,}500 = 15{,}000}\)
Now we can find what we're looking for:
Only rubella = Total rubella - Both
\(\mathrm{Only\ R = R - Both}\)
\(\mathrm{Only\ R = 15{,}000 - 5{,}000 = 10{,}000}\)
Process Skill: INFER - Understanding that "only rubella" means total rubella minus the overlap is key
The number of children vaccinated only against rubella is 10,000.
This matches answer choice C. 10,000
Quick verification:
Faltering Point 1: Misinterpreting "twice the number" relationships
Students often confuse which quantity is being doubled. When the problem states "rubella vaccinations is twice mumps vaccinations," some students might incorrectly write \(\mathrm{M = 2R}\) instead of \(\mathrm{R = 2M}\). This fundamental misunderstanding of the relationship direction will lead to completely wrong calculations throughout the solution.
Faltering Point 2: Confusing "only mumps" with "total mumps"
The statement "vaccinated against both is twice the number vaccinated only against mumps" specifically refers to the "only mumps" group (excluding those who got both vaccines). Students frequently misinterpret this as referring to the total mumps group, leading them to write: \(\mathrm{Both = 2 \times (Total\ M)}\) instead of \(\mathrm{Both = 2 \times (Only\ M)}\).
Faltering Point 3: Missing the set relationship between totals and subgroups
Students often fail to recognize that \(\mathrm{Total\ Mumps = Only\ Mumps + Both}\), and \(\mathrm{Total\ Rubella = Only\ Rubella + Both}\). Without understanding these fundamental set relationships, they cannot properly connect the given information to find the answer.
Faltering Point 1: Arithmetic errors in sequential calculations
Since this problem requires multiple sequential calculations (Only M → Total M → Total R → Only R), students often make simple arithmetic mistakes. For example, incorrectly calculating \(\mathrm{5{,}000 \div 2 = 2{,}000}\) instead of 2,500, or \(\mathrm{2 \times 7{,}500 = 14{,}000}\) instead of 15,000, which compounds through subsequent steps.
Faltering Point 2: Using wrong values in subsequent steps
Students might correctly calculate an intermediate step but then use the wrong value in the next calculation. For instance, they might correctly find Only M = 2,500 but then forget to add Both = 5,000 when calculating Total M, leading to M = 2,500 instead of M = 7,500.
Faltering Point 1: Selecting an intermediate calculation instead of the final answer
Since the problem generates several numerical values (Only M = 2,500, Total M = 7,500, Total R = 15,000, Only R = 10,000), students might mistakenly select one of the intermediate values that appears in the answer choices. For example, selecting 7,500 (Total M) or 15,000 (Total R) instead of 10,000 (Only R).