College Ratio Calculation: New Male-Female Numbers

This solution explains how to find the new ratio of males to females in a college after specific percentage increases.

Initial Ratio Setup

The initial ratio of males to females is given as $27:28$.

Let the initial number of males be $27x$ and the initial number of females be $28x$.

Calculating Percentage Increases

Male Increase: The number of males increases by 20%. $ \text{Increase} = 20\% \times 27x = \frac{20}{100} \times 27x = 5.4x $

Female Increase: The number of females increases by 30%. $ \text{Increase} = 30\% \times 28x = \frac{30}{100} \times 28x = 8.4x $

New Population Figures

New Males: Add the increase to the initial number. $ \text{New Males} = 27x + 5.4x = 32.4x $

New Females: Add the increase to the initial number. $ \text{New Females} = 28x + 8.4x = 36.4x $

Final Ratio Simplification

Calculate the new ratio of males to females:

$ \text{New Ratio} = \frac{\text{New Males}}{\text{New Females}} = \frac{32.4x}{36.4x} = \frac{32.4}{36.4} $

Simplify the fraction by removing decimals and dividing by the GCD (4):

$ \frac{324}{364} = \frac{324 \div 4}{364 \div 4} = \frac{81}{91} $

The new ratio of males to females is $81:91$.