The following is an example calculation from Wikipedia's page on Percentage:
I understand everything up to the point where 3% is divided by 10%. I cannot seem to understand why these two percentages are divided, why do we divide? I understand how we get 3% of all students are female computer science majors since we understand that 60% of all students are female and 5% of those females are computer science majors. However, I do not understand the reasoning behind 10% of all students being computer science majors and having to divide the 3% of female computer science majors from that. Can someone explain why we divide and what the reasoning behind the division is as it relates to the problem? Thanks!
Remember that a percentage is just a fraction written in a different way. For example, in this case:
$F_{CS} = \mbox{Fraction of students who are computer science majors} = \frac{\mbox{Number of computer science majors}}{\mbox{Number of students}} = \frac{N_{CS}}{N_S}$
$F_{FCS} = \mbox{Fraction of students who are female computer science majors} = \frac{\mbox{Number of female computer science majors}}{\mbox{Number of students}} = \frac{N_{FCS}}{N_S}$
Notice that both of these fractions have the same denominator. As a result, when we divide one fraction by the other, the common denominator vanishes:
$\frac{F_{FCS}}{F_{CS}} = \frac{N_{FCS}}{N_S} \div \frac{N_{CS}}{N_S} = \frac{N_{FCS}}{N_S} \times \frac{N_S}{N_{CS}} = \frac{N_{FCS}}{N_{CS}} = \mbox{Fraction of computer science students who are female}$