Understanding the Problem: Ratio and Percentage Increase

The problem asks us to find the new ratio of students in three classes after their initial numbers, which are in a given ratio, increase by specific percentages.

Initially, the numbers of students in the three classes are in the ratio 4 : 5 : 6.

This means if the number of students in the first class is $4x$, then the number of students in the second class is $5x$, and in the third class is $6x$, for some common factor $x$.

Calculating the Increase in Each Class

The number of students in each class increases by a certain percentage:

• Class 1: Increase by 25%
• Class 2: Increase by 20%
• Class 3: Increase by 25%

Finding the New Number of Students

To find the new number of students in each class, we add the percentage increase to the original number.

An increase of P% means the new value is the original value plus P% of the original value, which can be written as: New Value = Original Value $\times (1 + \frac{P}{100})$.

Let's calculate the new number of students for each class:

• Class 1: Original number = $4x$. Increase = 25%. New number = $4x \times (1 + \frac{25}{100}) = 4x \times (1 + 0.25) = 4x \times 1.25$. $\qquad 4 \times 1.25 = 5$. So, new number of students in Class 1 is $5x$.
• Class 2: Original number = $5x$. Increase = 20%. New number = $5x \times (1 + \frac{20}{100}) = 5x \times (1 + 0.20) = 5x \times 1.20$. $\qquad 5 \times 1.20 = 6$. So, new number of students in Class 2 is $6x$.
• Class 3: Original number = $6x$. Increase = 25%. New number = $6x \times (1 + \frac{25}{100}) = 6x \times (1 + 0.25) = 6x \times 1.25$. $\qquad 6 \times 1.25 = 7.5$. So, new number of students in Class 3 is $7.5x$.

Determining the New Ratio

The new numbers of students in the three classes are $5x$, $6x$, and $7.5x$.

The new ratio is the ratio of these new numbers:

New Ratio = $5x : 6x : 7.5x$

Simplifying the New Ratio

To simplify the ratio, we can divide each term by the common factor $x$ (assuming $x \neq 0$).

Ratio = $5 : 6 : 7.5$

To express the ratio with whole numbers, we can multiply each term by a suitable number to remove the decimal. Multiplying by 2 will remove the decimal in 7.5:

$5 \times 2 : 6 \times 2 : 7.5 \times 2$

$10 : 12 : 15$

The new ratio of the numbers of students in the three classes is 10 : 12 : 15.