- Translate the problem requirements: We need to understand what "ratio of boys to girls" means in each class, what "twice as many students in Class A as Class B" means, and how to find the combined ratio when classes merge.
- Set up concrete numbers for each class: Use the given ratios and size relationship to assign actual student counts that satisfy all conditions.
- Calculate boys and girls in each class separately: Apply the ratios to find the exact number of boys and girls in Class A and Class B.
- Combine and find the overall ratio: Add up total boys and total girls from both classes, then express as a simplified ratio.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what this problem is telling us in plain English:
- "Ratio of boys to girls in Class A is \(1:4\)" means for every 1 boy in Class A, there are 4 girls. So if we imagine 5 students total in some portion of Class A, 1 would be a boy and 4 would be girls.
- "Ratio of boys to girls in Class B is \(2:5\)" means for every 2 boys in Class B, there are 5 girls. So if we imagine 7 students total in some portion of Class B, 2 would be boys and 5 would be girls.
- "Twice as many students in Class A as in Class B" means if Class B has some number of students, Class A has exactly double that number.
- We need to find the "resulting ratio of boys to girls" when both classes are combined - this means we'll add up all the boys from both classes and all the girls from both classes, then express this as a simple ratio.
Process Skill: TRANSLATE - Converting the ratio language and relationships into clear mathematical understanding
2. Set up concrete numbers for each class
Let's pick actual numbers that satisfy all our conditions. This makes the problem much easier to work with than trying to use variables.
Since Class B has a ratio of 2 boys to 5 girls, the smallest possible size for Class B would have \(2 + 5 = 7\) students total. Let's use this: Class B has 7 students.
If Class B has 7 students, then Class A has twice as many, so Class A has \(2 \times 7 = 14\) students.
Now let's check if this works with Class A's ratio of \(1:4\):
- The ratio \(1:4\) means students come in groups of \(1 + 4 = 5\)
- Since Class A has 14 students total, we have \(14 \div 5 = 2.8\) groups
This doesn't work! We need Class A to have a number of students that's divisible by 5.
Let's try Class B with 14 students (which is \(2 \times 7\)). Then Class A would have 28 students.
- Class A: \(28 \div 5 = 5.6\) groups - still doesn't work!
Let's try Class B with 21 students (which is \(3 \times 7\)). Then Class A would have 42 students.
- Class A: \(42 \div 5 = 8.4\) groups - still doesn't work!
Let's try Class B with 35 students (which is \(5 \times 7\)). Then Class A would have 70 students.
- Class A: \(70 \div 5 = 14\) groups - this works!
So: Class B has 35 students, Class A has 70 students.
3. Calculate boys and girls in each class separately
Now let's find the exact number of boys and girls in each class:
Class A (70 students total, ratio \(1:4\)):
- Students come in groups of 5 (1 boy + 4 girls)
- We have \(70 \div 5 = 14\) such groups
- Boys in Class A: \(14 \times 1 = 14\) boys
- Girls in Class A: \(14 \times 4 = 56\) girls
- Check: \(14 + 56 = 70\) ✓
Class B (35 students total, ratio \(2:5\)):
- Students come in groups of 7 (2 boys + 5 girls)
- We have \(35 \div 7 = 5\) such groups
- Boys in Class B: \(5 \times 2 = 10\) boys
- Girls in Class B: \(5 \times 5 = 25\) girls
- Check: \(10 + 25 = 35\) ✓
4. Combine and find the overall ratio
When we combine both classes:
Total boys = Boys from Class A + Boys from Class B = \(14 + 10 = 24\) boys
Total girls = Girls from Class A + Girls from Class B = \(56 + 25 = 81\) girls
Combined ratio of boys to girls = \(24:81\)
Let's simplify this ratio by finding the greatest common divisor of 24 and 81:
- \(24 = 2^3 \times 3\)
- \(81 = 3^4\)
- GCD = 3
So \(24:81 = (24 \div 3):(81 \div 3) = 8:27\)
Final Answer
The resulting ratio of boys to girls when the two classes are combined is \(8:27\).
This matches answer choice (C) \(8:27\).
