In a college union, there are 48 students. The ratio of the number of boys to the number of girls is 5 : 3. The number of girls to be added in the union, so that the number of boys to girls in 6 : 5 is
7
Understanding the Ratio Problem
This problem involves working with ratios to determine how changing one part of a group affects the overall ratio. We start with a known total number of students and an initial ratio of boys to girls. We then need to find out how many girls must be added to achieve a new, desired ratio while keeping the number of boys constant.
Initial State of the College Union
We are given that the total number of students in the college union is 48. The initial ratio of the number of boys to the number of girls is 5 : 3.
To find the actual number of boys and girls, we need to consider the total number of parts in the ratio. The total parts are the sum of the parts for boys and girls.
Total ratio parts = 5 (boys) + 3 (girls) = 8 parts
Now, we can find the value of one ratio part by dividing the total number of students by the total ratio parts:
Value of one part = \(\frac{\text{Total students}}{\text{Total ratio parts}} = \frac{48}{8} = 6\)
Using the value of one part, we can calculate the initial number of boys and girls:
- Initial number of boys = Ratio part for boys \(\times\) Value of one part = \(5 \times 6 = 30\) boys
- Initial number of girls = Ratio part for girls \(\times\) Value of one part = \(3 \times 6 = 18\) girls
Let's verify: \(30 \text{ boys} + 18 \text{ girls} = 48 \text{ students}\), which matches the given total.
| Category | Ratio Part | Number of Students (Ratio Part \(\times\) 6) |
|---|---|---|
| Boys | 5 | 30 |
| Girls | 3 | 18 |
| Total | 8 | 48 |
Achieving the New Ratio of Boys to Girls
The problem states that we need to add a certain number of girls to the union so that the new ratio of boys to girls becomes 6 : 5. No boys are added.
Let 'x' be the number of girls to be added.
- New number of boys = Initial number of boys = 30
- New number of girls = Initial number of girls + Number of girls added = \(18 + x\)
The new ratio of boys to girls is given as 6 : 5. We can write this as an equation:
\(\frac{\text{New number of boys}}{\text{New number of girls}} = \frac{6}{5}\)
Substituting the values:
\(\frac{30}{18 + x} = \frac{6}{5}\)
Solving for the Number of Girls to Add
To find the value of 'x', we can cross-multiply the equation:
\(30 \times 5 = 6 \times (18 + x)\)
\(150 = 6 \times 18 + 6 \times x\)
\(150 = 108 + 6x\)
Now, isolate the term with 'x' by subtracting 108 from both sides:
\(150 - 108 = 6x\)
\(42 = 6x\)
Finally, solve for 'x' by dividing both sides by 6:
\(x = \frac{42}{6}\)
\(x = 7\)
So, 7 girls need to be added to the college union.
Verification
If 7 girls are added, the new number of girls will be \(18 + 7 = 25\).
The number of boys remains 30.
The new ratio of boys to girls is \(30 : 25\).
We can simplify this ratio by dividing both numbers by their greatest common divisor, which is 5:
\(30 \div 5 = 6\)
\(25 \div 5 = 5\)
The new ratio is \(6 : 5\), which matches the target ratio given in the problem. Thus, our calculation is correct.
| Initial | Change | New | |
|---|---|---|---|
| Boys | 30 | +0 | 30 |
| Girls | 18 | +7 | 25 |
| Total | 48 | +7 | 55 |
| Ratio (Boys : Girls) | 30 : 18 (5:3) | 30 : 25 (6:5) |
The number of girls to be added is 7.
Revision Table: College Union Ratios
| Concept | Description | Formula/Method |
|---|---|---|
| Ratio | Comparison of two or more quantities of the same kind. | \(a:b\) or \(\frac{a}{b}\) |
| Finding quantities from total and ratio | Divide total by sum of ratio parts to find value of one part; multiply value of one part by each ratio part. | Value per part = \(\frac{\text{Total}}{\text{Sum of ratio parts}}\) Quantity = Ratio part \(\times\) Value per part |
| Changing Ratios | If one quantity changes, adjust it and form a new ratio with the unchanged quantity. Set up an equation if the target ratio is known. | \(\frac{\text{Quantity 1}}{\text{Quantity 2 (after change)}} = \frac{\text{New Ratio Part 1}}{\text{New Ratio Part 2}}\) |
Additional Information: Working with Ratios
Ratios are a fundamental concept in mathematics used to express proportional relationships. Here are some key points about working with ratios:
- Simplifying Ratios: Ratios can be simplified by dividing all parts by their greatest common divisor, similar to simplifying fractions. For example, \(10 : 15\) simplifies to \(2 : 3\) by dividing by 5.
- Equivalent Ratios: Multiplying or dividing all parts of a ratio by the same non-zero number results in an equivalent ratio. For example, \(2:3\) is equivalent to \(4:6\) or \(6:9\).
- Comparing Ratios: To compare ratios, it's often helpful to write them as fractions or decimals, or find a common denominator or common term.
- Ratios and Fractions: A ratio \(a:b\) can be represented as a fraction \(\frac{a}{b}\). The total amount can be represented as \(\frac{a}{a+b}\) for the first quantity and \(\frac{b}{a+b}\) for the second quantity.
- Applications: Ratios are used widely in various fields, including scaling recipes, mixing solutions, map scales, financial analysis, and many areas of science and engineering.
Understanding how to manipulate ratios and solve problems involving changes to quantities is crucial for many mathematical applications.
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