The ratio of the number of boys and girls in a school is 3 : 2. If 20% of the boys and 25% of the girls are scholarship holders, the percentage of the school students who are not scholarship holders is:
78
Understanding the School Student Ratio and Scholarship Data
The question provides us with the ratio of boys to girls in a school and the percentage of scholarship holders within each group (boys and girls). We need to find the overall percentage of students in the school who are not scholarship holders.
Given Information:
- Ratio of boys : girls = 3 : 2
- Percentage of boys who are scholarship holders = 20%
- Percentage of girls who are scholarship holders = 25%
Goal:
Find the percentage of total school students who are not scholarship holders.
Step-by-Step Calculation
Let's assume a total number of students in the school to make the calculation easier. Since the ratio of boys to girls is 3:2, the total number of parts in the ratio is \(3 + 2 = 5\). A convenient number of total students would be a multiple of 5. Let's assume the total number of students is 100 (as percentages are based on 100).
Based on the ratio 3:2, the number of boys and girls can be calculated as follows:
- Total parts in ratio = 3 + 2 = 5
- Number of boys = \(\frac{3}{5} \times \text{Total Students}\)
- Number of girls = \(\frac{2}{5} \times \text{Total Students}\)
Assuming Total Students = 100:
- Number of boys = \(\frac{3}{5} \times 100 = 3 \times 20 = 60\)
- Number of girls = \(\frac{2}{5} \times 100 = 2 \times 20 = 40\)
- Check: Total students = 60 boys + 40 girls = 100 students.
Now, let's calculate the number of scholarship holders in each group:
- Scholarship holders among boys = 20% of the number of boys
- Scholarship holders among girls = 25% of the number of girls
Calculations:
- Number of scholarship boys = 20% of 60 = \(\frac{20}{100} \times 60 = 0.20 \times 60 = 12\)
- Number of scholarship girls = 25% of 40 = \(\frac{25}{100} \times 40 = 0.25 \times 40 = 10\)
The total number of scholarship holders in the school is the sum of scholarship boys and scholarship girls:
- Total scholarship holders = Number of scholarship boys + Number of scholarship girls
- Total scholarship holders = 12 + 10 = 22
The total number of students is 100. The number of students who are not scholarship holders is:
- Students not scholarship holders = Total Students - Total scholarship holders
- Students not scholarship holders = 100 - 22 = 78
To find the percentage of students who are not scholarship holders, we compare the number of students not scholarship holders to the total number of students:
- Percentage of students not scholarship holders = \(\frac{\text{Number of students not scholarship holders}}{\text{Total Students}} \times 100\%\)
- Percentage of students not scholarship holders = \(\frac{78}{100} \times 100\% = 78\%\)
Alternatively, we could calculate the percentage of students *not* holding scholarships in each group first:
- Percentage of boys not scholarship holders = 100% - 20% = 80%
- Percentage of girls not scholarship holders = 100% - 25% = 75%
Using the assumed numbers (60 boys, 40 girls):
- Number of boys not scholarship holders = 80% of 60 = \(\frac{80}{100} \times 60 = 0.80 \times 60 = 48\)
- Number of girls not scholarship holders = 75% of 40 = \(\frac{75}{100} \times 40 = 0.75 \times 40 = 30\)
Total students not scholarship holders = 48 (boys) + 30 (girls) = 78.
Percentage of students not scholarship holders = \(\frac{78}{100} \times 100\% = 78\%\).
Both methods yield the same result.
Thus, 78% of the school students are not scholarship holders.
Revision Table: Key Information
| Category | Ratio/Percentage | Assumed Number (out of 100) | Scholarship Holders (Assumed Numbers) | Not Scholarship Holders (Assumed Numbers) |
|---|---|---|---|---|
| Boys | 3 parts (Ratio 3:2) | 60 | 20% of 60 = 12 | 80% of 60 = 48 |
| Girls | 2 parts (Ratio 3:2) | 40 | 25% of 40 = 10 | 75% of 40 = 30 |
| Total School | 5 parts | 100 | 12 + 10 = 22 | 48 + 30 = 78 |
From the table, the total number of students not holding scholarships is 78 out of 100, which is 78%.
Additional Information: Ratio and Percentage Concepts
This problem combines the concepts of ratios and percentages. Understanding how to work with these is fundamental in many quantitative problems.
- Ratio: A ratio compares two or more quantities. A ratio of 3:2 means that for every 3 units of the first quantity (boys), there are 2 units of the second quantity (girls). The total number of units is the sum of the ratio parts (3+2=5).
- Percentage: A percentage is a way of expressing a number as a fraction of 100. 'Per cent' means 'out of one hundred'. For example, 20% means 20 out of 100, or \(\frac{20}{100}\).
- Calculating Percentage of a Quantity: To find a percentage of a number, convert the percentage to a decimal or fraction and multiply it by the number. For example, 20% of 60 is \(0.20 \times 60\) or \(\frac{20}{100} \times 60\).
- Calculating Overall Percentage from Subgroups: When you have subgroups with different percentages (like boys and girls with different scholarship percentages), you cannot simply average the percentages. You must calculate the actual numbers for each subgroup and then sum them up to find the total for the whole group.
Problems like this are common in tests and help assess your ability to combine different mathematical concepts to solve a practical scenario.
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