Understanding the School Student Ratio Problem

This problem involves understanding and calculating ratios related to the number of boys and girls in a school before and after new admissions. We are given the initial number of students, the initial ratio of boys to girls, the number of new students admitted, and the final ratio of boys to girls. Our goal is to find out how many of the newly admitted students were boys.

Initial Number of Students and Ratio

Initially, the school has 630 students. The ratio of boys to girls is 3 : 2.

The total parts in the initial ratio are \(3 + 2 = 5\).

To find the number of boys and girls initially, we can divide the total number of students by the total ratio parts:

Value of one ratio part \( = \frac{\text{Total Students}}{\text{Total Ratio Parts}} = \frac{630}{5} = 126 \)

• Initial number of boys \( = \text{Ratio of boys} \times \text{Value of one part} = 3 \times 126 = 378 \)
• Initial number of girls \( = \text{Ratio of girls} \times \text{Value of one part} = 2 \times 126 = 252 \)

Let's verify: Initial boys + Initial girls \( = 378 + 252 = 630 \). This matches the given total number of students.

New Admissions and Final Ratio

90 new students are admitted to the school. The total number of students after admission is \(630 + 90 = 720\).

After the admission, the ratio of boys to girls changes to 7 : 5.

The total parts in the final ratio are \(7 + 5 = 12\).

Now, let's find the number of boys and girls after the admission using the new total number of students and the final ratio:

Value of one ratio part (final) \( = \frac{\text{New Total Students}}{\text{Total Final Ratio Parts}} = \frac{720}{12} = 60 \)

• Final number of boys \( = \text{Final ratio of boys} \times \text{Value of one part} = 7 \times 60 = 420 \)
• Final number of girls \( = \text{Final ratio of girls} \times \text{Value of one part} = 5 \times 60 = 300 \)

Let's verify: Final boys + Final girls \( = 420 + 300 = 720 \). This matches the new total number of students.

Calculating the Number of Newly Admitted Boys

The increase in the number of boys is due to the newly admitted boys. Similarly, the increase in the number of girls is due to the newly admitted girls.

• Newly admitted boys \( = \text{Final number of boys} - \text{Initial number of boys} = 420 - 378 \)
• Newly admitted girls \( = \text{Final number of girls} - \text{Initial number of girls} = 300 - 252 \)

Calculating the number of newly admitted boys:

\( \text{Newly admitted boys} = 420 - 378 = 42 \)

Calculating the number of newly admitted girls:

\( \text{Newly admitted girls} = 300 - 252 = 48 \)

Let's verify if the total number of newly admitted students is 90:

\( \text{Newly admitted boys} + \text{Newly admitted girls} = 42 + 48 = 90 \)

This matches the total number of new students admitted.

The number of newly admitted boys is 42.

Conclusion

Based on the initial and final ratios and total student numbers, the number of newly admitted boys is 42.

Revision Table: School Ratio Problem

Additional Information: Ratio and Proportion

Ratios are fundamental in mathematics for comparing quantities. When a ratio is given as \(a : b\), it means that for every \(a\) units of the first quantity, there are \(b\) units of the second quantity. The total number of parts in the ratio is \(a + b\).

Proportion is an equation that states that two ratios are equivalent. For example, \( \frac{a}{b} = \frac{c}{d} \) is a proportion.

In problems like this, where the total quantity changes and the ratio changes, we first use the initial total and ratio to find the initial quantities. Then, we use the new total and new ratio to find the final quantities. The difference between the final and initial quantities gives the change, which in this case, represents the newly admitted students.

It is important to correctly identify the total quantity that corresponds to the given ratio. Initially, the ratio 3:2 corresponds to 630 students. Finally, the ratio 7:5 corresponds to 720 students.