In a certain region, the number of children who have been vaccinated against rubella is twice the number who have...

1. Translate the problem requirements: Convert the word problem into mathematical relationships. "Twice the number" means multiply by 2. "Vaccinated against both" refers to the overlap between two vaccination groups. "Vaccinated only against mumps" means those who got mumps vaccine but not rubella vaccine.
2. Define variables for each vaccination group: Set up variables for total rubella vaccinations, total mumps vaccinations, both vaccinations, and only mumps vaccinations to organize the given relationships.
3. Build equations from the stated relationships: Convert each sentence in the problem into a mathematical equation using the defined variables, focusing on the "twice" relationships and the "both" versus "only" distinctions.
4. Solve systematically using the known value: Work backwards from the given fact that 5,000 have been vaccinated against both, using this to find the other quantities step by step.
5. Calculate the final answer using set relationships: Use the relationship that "only rubella" equals "total rubella" minus "both rubella and mumps" to find the target value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what each phrase means in plain English:

• "Number of children vaccinated against rubella is twice the number vaccinated against mumps" - If 100 kids got mumps vaccine, then 200 kids got rubella vaccine
• "Vaccinated against both is twice the number vaccinated only against mumps" - If 50 kids got only mumps (and not rubella), then 100 kids got both vaccines
• "5,000 have been vaccinated against both" - This is our concrete starting point
• "How many vaccinated only against rubella?" - We want kids who got rubella vaccine but NOT mumps vaccine

Process Skill: TRANSLATE - Converting problem language into mathematical understanding is crucial here

2. Define variables for each vaccination group

Let's use simple letters to represent our groups:

• R = total number vaccinated against rubella
• M = total number vaccinated against mumps
• Both = number vaccinated against both (we know this is 5,000)
• Only M = number vaccinated only against mumps
• Only R = number vaccinated only against rubella (this is what we want to find!)

3. Build equations from the stated relationships

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:

\(\mathrm{5{,}000 = 2 \times (Only\ M)}\)

4. Solve systematically using the known value

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)

\(\mathrm{R = 2M = 2 \times 7{,}500 = 15{,}000}\)

5. Calculate the final answer using set relationships

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

Final Answer

The number of children vaccinated only against rubella is 10,000.

This matches answer choice C. 10,000

Quick verification:

• Only mumps: 2,500
• Both: 5,000 (given)
• Only rubella: 10,000
• Total mumps: 2,500 + 5,000 = 7,500
• Total rubella: 10,000 + 5,000 = 15,000
• Check: 15,000 = 2 × 7,500 ✓
• Check: 5,000 = 2 × 2,500 ✓

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the approach

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.

Errors while selecting the answer

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).