Two-way tables

A two-way frequency table shows how data splits across two categories at the same time.
One category runs down the rows and the other across the columns, and each inner cell counts the people or things that fall in both.

Reading a single cell answers "how many are in both groups."
In a table of students by year and commute, the cell in the "Freshman" row and "Bus" column might read $18$, meaning $18$ students are freshmen who take the bus.
Every inner cell works this way.

The totals summarize one category at a time.
The total at the end of the "Freshman" row counts all freshmen regardless of commute, and the total at the foot of the "Bus" column counts all bus riders regardless of year.
The bottom-right corner is the grand total of everyone.

Most questions are answered by picking the right cell or total and, if a fraction is asked, choosing the correct denominator.
"What fraction of freshmen take the bus?" divides the freshman-bus cell by the freshman total, not by the grand total.
Matching the denominator to the group the question asks about is the key step.

A table shows students by year and commute: $18$ freshmen take the bus, $12$ freshmen take a car, giving $30$ freshmen in all. What fraction of freshmen take the bus?
(A) $\frac{18}{30}$
(B) $\frac{18}{60}$
The question is about freshmen, so the denominator is the freshman total, $30$, not the grand total.
So (A), $\frac{18}{30} = \frac{3}{5}$, is correct.

In the same table, $27$ students take the bus in total and there are $60$ students overall. What fraction of all students take the bus?
(A) $\frac{27}{60}$
(B) $\frac{27}{30}$
The question asks about all students, so the denominator is the grand total, $60$.
So (A), $\frac{27}{60}$, is correct.

A table splits $60$ people into $27$ bus riders and the rest car riders, with $30$ freshmen and $30$ seniors. If $21$ seniors take a car, how many seniors take the bus?
The senior row must total $30$, and $21$ of them take a car.
So the seniors who take the bus are $30 - 21 = 9$.