The ratio of boys and girls in a school is 27 : 23. If the difference between the number of boys and girls is 200, then find the number of boys.
1350
Solving Ratio Problems: Boys and Girls in School
This problem involves finding the number of boys in a school given the ratio of boys to girls and the difference between their numbers. Let's break down the steps to solve this ratio problem.
Understanding the Ratio of Boys and Girls
The ratio of boys and girls in the school is given as 27 : 23. This means that for every 27 boys, there are 23 girls. We can represent the number of boys and girls using a common multiple.
- Let the common multiple be \(x\).
- Number of boys = \(27x\)
- Number of girls = \(23x\)
Analyzing the Difference in Numbers
We are given that the difference between the number of boys and girls is 200. Since the ratio of boys (27) is greater than the ratio of girls (23), the number of boys is greater than the number of girls. The difference can be expressed as:
Number of boys - Number of girls = 200
Substituting the expressions in terms of \(x\):
\(27x - 23x = 200\)
Solving for the Common Multiple
Now, we solve the equation for \(x\):
\(27x - 23x = 200\)
\(4x = 200\)
To find \(x\), divide both sides by 4:
\(x = \frac{200}{4}\)
\(x = 50\)
The common multiple \(x\) is 50.
Calculating the Number of Boys
The number of boys is given by \(27x\). Now we substitute the value of \(x\) we found:
Number of boys = \(27 \times 50\)
Number of boys = \(1350\)
Verifying the Solution
Let's find the number of girls and check the difference:
- Number of girls = \(23x = 23 \times 50 = 1150\)
- Difference = Number of boys - Number of girls = \(1350 - 1150 = 200\)
The calculated difference (200) matches the given difference in the problem, so our calculation for the number of boys is correct.
Therefore, the number of boys in the school is 1350.
| Item | Ratio Part | Expression (with \(x\)) | Actual Number (with \(x=50\)) |
|---|---|---|---|
| Boys | 27 | \(27x\) | \(27 \times 50 = 1350\) |
| Girls | 23 | \(23x\) | \(23 \times 50 = 1150\) |
| Difference | \(27-23=4\) | \(4x\) | \(1350 - 1150 = 200\) (Given) |
Revision Table: Key Concepts in Ratio Problems
| Concept | Description | How it applies here |
|---|---|---|
| Ratio | A comparison of two or more quantities of the same kind, expressed as a:b. | The ratio of boys to girls is 27:23. |
| Parts of a Ratio | The numbers in a ratio represent parts of a whole or parts being compared. | 27 parts for boys, 23 parts for girls. |
| Common Multiple (\(x\)) | A value that, when multiplied by the ratio parts, gives the actual quantities. | Used as \(27x\) and \(23x\) to represent actual numbers of boys and girls. |
| Difference in Ratio Parts | The difference between the ratio parts corresponds to the difference in the actual quantities. | Difference in parts is \(27-23=4\), which corresponds to the difference of 200. So, \(4x=200\). |
Additional Information on Ratio and Proportion
Ratios are fundamental in mathematics and are used to compare quantities. A proportion is an equation stating that two ratios are equal. Understanding ratios helps in solving various real-world problems involving comparisons, scaling, and distribution.
- Ratio Simplification: Ratios can be simplified by dividing all parts by their greatest common divisor (GCD). The ratio 27:23 is already in its simplest form as 27 and 23 have no common factors other than 1.
- Total Quantity: If the total number of students was given, say \(T\), and the ratio is 27:23, the total parts would be \(27+23=50\). The number of boys would be \(\frac{27}{50} \times T\) and girls \(\frac{23}{50} \times T\). In this problem, we used the difference instead of the total.
- Applications: Ratios are used in various fields like cooking (recipes), map scales, financial analysis, and mixing chemicals.
Similar Questions
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Important Questions from Simple Ratios
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