# What does the relationship between dividing percentages signify in this problem?

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!

• We are told at the beginning that $10\%$ of all students are computer science majors. Jan 3, 2019 at 3:37
• Yes I just want to know why we divide and what it signifies. Jan 3, 2019 at 3:38

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:

$$\frac{F_{FCS}}{F_{CS}} = \frac{N_{FCS}}{N_S} \div \frac{N_{CS}}{N_S} = \frac{N_{FCS}}{N_S} \times \frac{N_S}{N_{CS}} = \frac{N_{FCS}}{N_{CS}} = \mbox{Fraction of computer science students who are female}$$

Suppose there are $$S$$ students in all. Then we know that there are $$.1S$$ computer science majors and $$.03S$$ female computer science majors. The fraction of computer science majors who are fmeale is $${.03S\over .1S}={.03\over.1}=30\%$$

• Ok I think I figured it out, the reason we divide is because the 10% is serving as the total right? We are looking for the percentage of computer science majors that are female, we get the percentage (3%) of female computer science majors from the total of female students and to find what percentage they occupy in regards to all the computer science majors (10%) we divide. The portion (3%) is divided by the total (10%). I know this seems really rudimentary and I just embarrassingly figured out the 10% is serving as the total. Is this explanation correct? Jan 3, 2019 at 3:46