In a class of 49 students, the ratio of girls to boys is 4 ∶ 3. If 4 girls leave the class, the ratio of girls to boys would be
8 ∶ 7
Solving the Class Ratio Problem
This problem involves understanding ratios and how they change when the number of individuals in one group changes. We start with a total number of students and an initial ratio of girls to boys. Then, a certain number of girls leave, and we need to find the new ratio.
Understanding the Initial Class Composition
The total number of students in the class is 49. The initial ratio of girls to boys is given as 4 ∶ 3. This means that for every 4 parts of girls, there are 3 parts of boys. The total number of parts in the ratio is the sum of the parts for girls and boys.
- Total parts in the ratio = 4 (girls) + 3 (boys) = 7 parts
Since these 7 parts represent the total number of students (49), we can find the number of students per part.
- Number of students per part = \( \frac{\text{Total students}}{\text{Total ratio parts}} = \frac{49}{7} = 7 \) students/part
Calculating the Initial Number of Girls and Boys
Now that we know how many students are in each part, we can calculate the initial number of girls and boys using the initial ratio.
- Initial number of girls = Number of girls parts × Students per part = \( 4 \times 7 = 28 \) girls
- Initial number of boys = Number of boys parts × Students per part = \( 3 \times 7 = 21 \) boys
We can check this by adding the number of girls and boys: \( 28 + 21 = 49 \), which matches the total number of students.
Change in Class Composition
The problem states that 4 girls leave the class. The number of boys remains unchanged.
- Number of girls leaving = 4
- New number of girls = Initial number of girls - Number of girls leaving = \( 28 - 4 = 24 \) girls
- New number of boys = Initial number of boys = 21 boys
Calculating the New Ratio of Girls to Boys
The new ratio of girls to boys is the new number of girls compared to the number of boys.
- New ratio of girls to boys = New number of girls ∶ Number of boys = 24 ∶ 21
Simplifying the New Ratio
Ratios are usually expressed in their simplest form. To simplify the ratio 24 ∶ 21, we need to find the greatest common divisor (GCD) of 24 and 21 and divide both numbers by it.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 21: 1, 3, 7, 21
- The greatest common divisor (GCD) of 24 and 21 is 3.
Now, divide both parts of the ratio by the GCD:
- New simplified ratio = \( \frac{24}{3} : \frac{21}{3} = 8 : 7 \)
So, the new ratio of girls to boys after 4 girls leave is 8 ∶ 7.
Summary of Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Find total ratio parts | \( 4 + 3 = 7 \) |
| 2 | Find students per part | \( 49 \div 7 = 7 \) |
| 3 | Calculate initial girls | \( 4 \times 7 = 28 \) |
| 4 | Calculate initial boys | \( 3 \times 7 = 21 \) |
| 5 | Calculate new number of girls | \( 28 - 4 = 24 \) |
| 6 | Form the new ratio | \( 24 : 21 \) |
| 7 | Simplify the new ratio | \( 24 \div 3 : 21 \div 3 = 8 : 7 \) |
Revision Table: Understanding Ratios and Proportions
| Concept | Explanation | Example |
|---|---|---|
| Ratio | A comparison of two or more quantities of the same kind, usually expressed as a:b or a/b. | Ratio of girls to boys is 4:3 |
| Total Parts | The sum of the individual parts in a ratio when comparing quantities that make up a whole. | For a ratio 4:3, total parts are \( 4+3=7 \). |
| Simplifying Ratios | Dividing all parts of a ratio by their greatest common divisor (GCD) to express it in its simplest form. | Ratio 24:21 simplifies to 8:7 by dividing by 3. |
Additional Information: Ratio Problems in Context
Ratio problems often involve finding the actual quantities represented by a ratio, understanding how these quantities change, and then calculating the new ratio. It's important to correctly identify which quantity is changing and by how much. Always simplify the final ratio to its lowest terms unless otherwise specified.
Understanding ratios is fundamental in many areas, including mixing ingredients, scaling recipes, interpreting maps, and analyzing data in various fields. Practice with different scenarios, such as adding individuals to a group, removing individuals, or changing both quantities, will help solidify the concept.
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