Solving the Student Ratio Problem

This problem involves understanding and manipulating ratios to find the new ratio of successful to unsuccessful students after a change in the number of students who passed.

Initial Situation Analysis

We are given the total number of examinees and the initial ratio of successful to unsuccessful students.

• Total examinees = 132
• Initial ratio of successful to unsuccessful students = 9 : 2

The ratio 9 : 2 means that for every 9 successful students, there are 2 unsuccessful students. The total number of parts in the ratio is the sum of the parts for successful and unsuccessful students.

Total ratio parts = \(9 + 2 = 11\) parts

These 11 parts represent the total number of examinees (132). We can find the value of one ratio part.

Value of one ratio part = \(\frac{\text{Total examinees}}{\text{Total ratio parts}} = \frac{132}{11} = 12\) students per part

Now, we can calculate the initial number of successful and unsuccessful students:

• Initial number of successful students = \(9 \times \text{Value of one part} = 9 \times 12 = 108\) students
• Initial number of unsuccessful students = \(2 \times \text{Value of one part} = 2 \times 12 = 24\) students

Let's verify the total: \(108 + 24 = 132\), which matches the total number of examinees.

Calculating New Numbers After Students Pass

The problem states that 4 more students passed. This means that 4 students who were initially unsuccessful are now successful. Assuming the total group of 132 remains the same, these 4 students transferred from the unsuccessful category to the successful category.

• New number of successful students = Initial successful students + 4 = \(108 + 4 = 112\) students
• New number of unsuccessful students = Initial unsuccessful students - 4 = \(24 - 4 = 20\) students

Let's verify the new total: \(112 + 20 = 132\), which is consistent with the original total number of examinees.

Determining the New Ratio

The new ratio of successful to unsuccessful students is the new number of successful students compared to the new number of unsuccessful students.

New ratio (successful : unsuccessful) = \(112 : 20\)

To simplify this ratio, we need to find the greatest common divisor (GCD) of 112 and 20 and divide both numbers by it.

Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112

Factors of 20: 1, 2, 4, 5, 10, 20

The greatest common divisor (GCD) of 112 and 20 is 4.

Divide both parts of the ratio by 4:

• Successful part: \(\frac{112}{4} = 28\)
• Unsuccessful part: \(\frac{20}{4} = 5\)

The new simplified ratio of successful to unsuccessful students is \(28 : 5\).

Conclusion

After 4 more students passed, the number of successful students is 112 and the number of unsuccessful students is 20. The new ratio of successful to unsuccessful students is 112 : 20, which simplifies to 28 : 5.

Revision Table: Student Ratio Problem

Additional Information: Understanding Ratios

Ratios are fundamental in mathematics for comparing quantities. They can represent parts of a whole or comparisons between different quantities.

• Parts to Parts Ratio: Compares one part of a whole to another part, like the ratio of successful to unsuccessful students (9:2).
• Parts to Whole Ratio: Compares a part to the total whole, like the ratio of successful students to the total examinees (\(108:132\) or \(9:11\)) or unsuccessful students to the total examinees (\(24:132\) or \(2:11\)).

When a change occurs, it's important to identify how it affects the individual quantities before calculating the new ratio. In this problem, the change (4 more students passing) directly affected both the number of successful and unsuccessful students because the total number of examinees remained constant.

Ratios can be written in different forms: using a colon (\(9:2\)), as a fraction (\(\frac{9}{2}\)), or with the word "to" (9 to 2). The form \(a:b\) is common when comparing two quantities.