- Translate the problem requirements: We have a class split into two equal-sized sections A and B. We know the girl-to-boy ratio for the entire class (\(3:5\)) and for section A (\(5:7\)). We need to find the girl-to-boy ratio for section B.
- Set up the ratio framework using concrete numbers: Convert the given ratios into actual student counts by choosing convenient numbers that make the equal section sizes work cleanly.
- Apply the equal sections constraint: Use the fact that sections A and B have the same total number of students to determine the student distribution in section B.
- Calculate section B's ratio: Subtract section A's girls and boys from the total to find section B's composition, then express as a simplified ratio.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in simple terms:
- We have a class that's split into two sections: A and B
- Both sections have exactly the same number of students
- For the entire class, the ratio of girls to boys is \(3:5\)
- For section A alone, the ratio of girls to boys is \(5:7\)
- We need to find the ratio of girls to boys in section B
Think of it like this: if you have two equal-sized groups that when combined give you a certain mix of girls and boys, and you know the mix in one group, you can figure out what the mix must be in the other group.
Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships
2. Set up the ratio framework using concrete numbers
Instead of working with variables, let's use actual numbers to make this concrete and easy to follow.
Since the whole class has a girl-to-boy ratio of \(3:5\), let's say there are:
- \(3\mathrm{k}\) girls and \(5\mathrm{k}\) boys in the entire class
- Total students = \(3\mathrm{k} + 5\mathrm{k} = 8\mathrm{k}\) students
Since sections A and B are equal in size:
- Each section has \(8\mathrm{k} \div 2 = 4\mathrm{k}\) students
For section A, the girl-to-boy ratio is \(5:7\), which means:
- Out of every \(5 + 7 = 12\) students in section A, 5 are girls and 7 are boys
- Since section A has \(4\mathrm{k}\) students total, we need \(4\mathrm{k} = 12\mathrm{m}\) for some number m
- This means \(\mathrm{k} = 3\mathrm{m}\)
Let's use \(\mathrm{k} = 3\) to keep our numbers simple, so \(\mathrm{m} = 1\):
- Total class: 24 students (9 girls, 15 boys)
- Each section: 12 students
- Section A: 5 girls, 7 boys (total = 12) ✓
3. Apply the equal sections constraint
Now we can use the fact that the total girls and boys must add up correctly:
For the entire class:
- Total girls = 9
- Total boys = 15
We know section A has:
- Girls in A = 5
- Boys in A = 7
So section B must have:
- Girls in B = Total girls - Girls in A = \(9 - 5 = 4\) girls
- Boys in B = Total boys - Boys in A = \(15 - 7 = 8\) boys
- Total in B = \(4 + 8 = 12\) students ✓
This confirms our sections are equal as required.
Process Skill: APPLY CONSTRAINTS - Using the equal section sizes to determine section B's composition
4. Calculate section B's ratio
Section B has 4 girls and 8 boys.
To find the ratio of girls to boys in section B:
- Girls : Boys = \(4 : 8\)
- Simplify by dividing both by 4: \(4\div4 : 8\div4 = 1 : 2\)
So the ratio of girls to boys in section B is \(1:2\).
Final Answer
The ratio of girls to boys in section B is \(1:2\), which corresponds to answer choice D.
Let's verify: If section A has ratio \(5:7\) and section B has ratio \(1:2\), with equal section sizes of 12 students each:
- Section A: 5 girls, 7 boys
- Section B: 4 girls, 8 boys (since \(1:2\) ratio with 12 total means \(4:8\))
- Total class: 9 girls, 15 boys → ratio \(9:15 = 3:5\) ✓
The answer is D: \(1:2\).
Common Faltering Points
Errors while devising the approach
- Misinterpreting the equal sections constraint: Students often overlook that sections A and B have exactly the same number of students. They might try to solve by assuming different section sizes, which leads to incorrect setup and impossible equations.
- Confusion about ratio interpretation: Students may misunderstand what 'ratio of girls to boys is \(3:5\)' means, thinking it represents fractions or percentages rather than parts in a ratio. This leads to incorrect total student calculations.
- Attempting to solve with pure algebra: Students might immediately jump into setting up complex algebraic equations with multiple variables instead of using the concrete number approach, making the problem unnecessarily complicated and error-prone.
Errors while executing the approach
- Incorrect constraint application: When determining that \(4\mathrm{k} = 12\mathrm{m}\) (since section A needs to accommodate the \(5:7\) ratio), students often make errors in finding compatible values for k and m, or forget this constraint entirely when choosing their working numbers.
- Arithmetic errors in subtraction: When calculating section B's composition by subtracting section A's numbers from the total (like \(9-5=4\) girls, \(15-7=8\) boys), students frequently make simple arithmetic mistakes, especially when working with larger numbers.
- Failing to verify equal section sizes: Students often forget to check that their calculated section B has the same total number of students as section A, missing a crucial verification step that would catch their computational errors.
Errors while selecting the answer
- Reversing the ratio order: After correctly calculating 4 girls and 8 boys in section B, students might express this as 'boys to girls' (\(2:1\)) instead of 'girls to boys' (\(1:2\)), selecting answer choice B instead of the correct answer D.
- Using unsimplified ratios: Students might arrive at the ratio \(4:8\) for girls to boys but fail to simplify it to \(1:2\), looking for \(4:8\) among the answer choices and getting confused when it's not there.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose smart numbers based on the given ratios
For the whole class with girls to boys ratio of \(3:5\), let's use convenient numbers:
- Girls in whole class = \(3\mathrm{k}\)
- Boys in whole class = \(5\mathrm{k}\)
- Total students = \(8\mathrm{k}\)
Since sections A and B are equal, each section has \(4\mathrm{k}\) students.
Step 2: Apply the constraint for section A
Section A has girls to boys ratio of \(5:7\), so if section A has \(4\mathrm{k}\) total students:
- Girls in section A = \(5\mathrm{m}\)
- Boys in section A = \(7\mathrm{m}\)
- Total in section A = \(12\mathrm{m} = 4\mathrm{k}\)
- Therefore: \(\mathrm{m} = \mathrm{k}/3\)
Step 3: Choose a specific value to make calculations clean
Let \(\mathrm{k} = 12\) (chosen because it's divisible by 3, making \(\mathrm{m} = 4\))
- Total students = 96
- Each section = 48 students
- Whole class: 36 girls, 60 boys
- Section A: 20 girls, 28 boys
Step 4: Calculate section B composition
- Girls in section B = \(36 - 20 = 16\)
- Boys in section B = \(60 - 28 = 32\)
- Ratio of girls to boys in section B = \(16:32 = 1:2\)
The smart numbers approach works here because we can logically choose \(\mathrm{k} = 12\) to ensure all calculations yield whole numbers, making the arithmetic clean while preserving the ratio relationships.
