Understanding the Ratio Problem

The problem asks us to find the new ratio of girls and boys in a school after their numbers increase by certain percentages. We are given the initial ratio and the percentage increase for each group.

Initial Ratio and Representation

The initial ratio of the number of girls to the number of boys in the school is given as 8 : 7.

We can represent the initial number of girls and boys using a common variable. Let the common multiplier be \(x\).

• Initial number of girls = \(8x\)
• Initial number of boys = \(7x\)

Calculating the Increase in Numbers

The number of girls increases by 10%.

Increase in girls = \(10\%\) of initial girls

Increase in girls = \(\frac{10}{100} \times 8x = \frac{1}{10} \times 8x = \frac{8x}{10}\)

The number of boys increases by 20%.

Increase in boys = \(20\%\) of initial boys

Increase in boys = \(\frac{20}{100} \times 7x = \frac{1}{5} \times 7x = \frac{7x}{5}\)

To make calculations easier, we can express the increase in boys with the same denominator as girls:

Increase in boys = \(\frac{7x}{5} = \frac{7x \times 2}{5 \times 2} = \frac{14x}{10}\)

Calculating the New Numbers of Girls and Boys

The new number of girls is the initial number plus the increase.

New number of girls = Initial girls + Increase in girls

New number of girls = \(8x + \frac{8x}{10} = \frac{80x}{10} + \frac{8x}{10} = \frac{80x + 8x}{10} = \frac{88x}{10}\)

The new number of boys is the initial number plus the increase.

New number of boys = Initial boys + Increase in boys

New number of boys = \(7x + \frac{14x}{10} = \frac{70x}{10} + \frac{14x}{10} = \frac{70x + 14x}{10} = \frac{84x}{10}\)

Finding the New Ratio

The new ratio of girls to boys is the new number of girls divided by the new number of boys.

New Ratio = New number of girls : New number of boys

New Ratio = \(\frac{88x}{10} : \frac{84x}{10}\)

We can multiply both sides of the ratio by 10 to remove the denominator:

New Ratio = \(88x : 84x\)

Since \(x\) is a common factor (assuming \(x \neq 0\), as the number of students cannot be zero), we can cancel \(x\) from both sides:

New Ratio = \(88 : 84\)

Simplifying the New Ratio

To simplify the ratio \(88 : 84\), we need to find the greatest common divisor (GCD) of 88 and 84 and divide both numbers by it.

• Factors of 88: 1, 2, 4, 8, 11, 22, 44, 88
• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The greatest common divisor of 88 and 84 is 4.

Divide both parts of the ratio by 4:

\(88 \div 4 = 22\)

\(84 \div 4 = 21\)

The simplified new ratio is 22 : 21.

Summary of Calculations

New Ratio = \(\frac{88x}{10} : \frac{84x}{10} = 88 : 84\)

Simplifying \(88 : 84\) by dividing by GCD(88, 84) = 4 gives \(22 : 21\).

Thus, the new ratio of girls and boys in the school will be 22 : 21.

Ratio and Percentage Increase Revision

Ratio problems often involve understanding how quantities relate to each other. A ratio like \(a:b\) means that for every \(a\) units of one quantity, there are \(b\) units of another. Percentage increase is calculated on the original amount, and the increased amount is added to the original to find the new total.

Additional Information on Ratios and Percentages

When dealing with ratios and percentage changes, it's crucial to apply the percentage change to the specific quantity it relates to (girls or boys in this case) based on their initial proportion. The common multiplier (\(x\)) used in this problem helps maintain the original ratio while allowing for calculation of absolute or proportional increases. Simplifying the final ratio is a standard step to present it in its most concise form.

• Ratio: A comparison of two or more quantities.
• Percentage Increase: The amount of increase shown as a percentage of the original amount. Formula: \(\frac{\text{Increase}}{\text{Original Amount}} \times 100\%\).
• New Amount = Original Amount + Increase Amount.