When we’re able to create an incomplete table using two categories of information, in this case, “wood vs. plastic” and “pink vs. blue,” we can fill out any third cell horizontally or vertically if we know the other two, since our interior cells sum to each total row/column. In this case, if we know that 10% of the toys are wood, pink toys, and 50% of all toys are wood, we can fill in the blue, wood column with 40%. Similarly, if we know that wood, pink toys are 10% and total pink toys are 20%, we can fill in pink, plastic toys with the remaining 10%.

From here, we can also fill out our other total categories as follows:

Now, we have several ways to prove that blue, plastic toys make up the remaining 40% of the toys. So, if 40% of our total is equal to 80 toys, we can solve for the total number of toys as follows:

40%(T) = 80

10%(T) = 20

T = 200

The key to incomplete tables is the “total” designation for the bottom row and for the right-hand column. Here two of those cells are filled in for you, and they form your “foothold” for solving the rest. You know that 35% of all athletes were members of varsity teams, meaning that the Varsity Girls cell plus the Varsity Boys cell must equal 35%. That also tells you that, since the total has to add up to 100%, the JV Total cell must be 65% so that Varsity + JV can add to 100%. Similarly, you know that the Girls Total is 60%, meaning that the sum of Varsity Girls and JV Girls must add to 60%. And you also know that 40% of the total athletes were boys so that that column can add to 100%. Filling in what you know now, you have the bottom row and the right-most column filled in:

You’re also told that there are twice as many JV girls as varsity girls, which allows you to set up a system of equations: Varsity Girls + JV Girls = 60 JV Girls = 2(Varsity Girls) Therefore, you can conclude that the JV Girls cell is 40% and the Varsity Girls cell is 20%, which leads you another step closer to filling in the whole table:

Since each column needs to sum to its total in the bottom row, you can conclude that Varsity Boys = 15% and JV Boys = 25%. And since you’re responsible for the JV Boys number, you can now take that 25% and apply it to the total number of athletes, 480. 25% of 480 is 120, so the correct answer is “120.”

From the initial information presented to us in the stem, we can create a two way table using the categories left vs. right-handed, and take the bus vs. do not take the bus. We can fill in the table as follows: (*note - be very cautious mixing percent/fraction information with actual value information. It may be easier to leave the “30” out of the table and solve for right-handed seniors who take the bus in terms of a fraction.)

From here, if we know that half of the students who are left-handed take the bus, we can split up the  of our total that is left-handed into  and .

If  of the students take the bus, and  of the students are left-handed students who take the bus, the remaining  are right-handed students who take the bus. Since we know the value of right-handed students who take the bus is 30, and  of our total students, the total number of students must be 

*Note, we did not need to complete the entire table to answer this question, but we could! The rest of the table is as follows:

For questions where we’re exclusively given “scalable relationships” (for instance, fractions or percents) and we’re asked for a scalable relationship, we can pick a total to make the question more concrete and easier to work with! 

When we’re given fractions, we’ll want to pick a total that is divisible by all the denominators present in the question stem. So, the least common multiple of our denominators will likely be a convenient and easy-to-work-with option. In this case, we’ll want to pick a total of 60, since 60 is divisible by 4, 3, and 5. From here, we can begin to fill in the table as follows:

From here, we can fill out our remaining information from the question stem, as well as the remaining totals:

Now, we’re able to fill in our remaining cells horizontally and vertically:

So, if we’re looking for what fraction of the women are not members of the chess team, we want , in this case .

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who have a curfew or do not have a curfew and the rows will contain the students who are on the honor roll or are not on the honor roll. The first bit of information that we were given from the question was that  students had a curfew; therefore,  needs to go in the "curfew" column as the row total. Next, we were told that of those students,  were on the honor roll; therefore, we need to put  in the "curfew" column and in the "honor roll" row. Then, we were told that  students do not have a curfew but were on the honor roll, so we need to put  in the "no curfew" column and the "honor roll" row. Finally, we were told that  students do not have a curfew or were on the honor roll, so  needs to go in the "no curfew" column and "no honor roll" row. If done correctly, you should create a table similar to the following:

Our question asked how many students were not on the honor roll. We add up the numbers in the "no honor roll" row to get the total, but first we need to fill in a gap in our table, students who have a curfew but were not on the honor roll. We can take the total number of students that have a curfew, , and subtract the number of students who are on the honor roll, 

This means that  students who have a curfew, aren't on the honor roll. 

Now, we add up the numbers in the "no honor roll" row to get the total:

This means that  students were not on the honor roll.

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who have a laptop or do not have a laptop and the rows will contain the students who have a car or do not have a car. The first bit of information that we were given from the question was that  students had a laptop; therefore,  needs to go in the "laptop" column as the row total. Next, we were told that of those students,  owned a car; therefore, we need to put  in the "laptop" column and in the "car" row. Then, we were told that  students do not own a laptop, but own a car, so we need to put  in the "no laptop" column and the "car" row. Finally, we were told that  students do not have a laptop or a car, so  needs to go in the "no laptop" column and "no car" row. If done correctly, you should create a table similar to the following: 

Our question asked how many students have a car. We add up the numbers in the "car" row to get the total:

This means that  students have a car.

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who have a curfew or do not have a curfew and the rows will contain the students who are on the honor roll or are not on the honor roll. The first bit of information that we were given from the question was that  students had a curfew; therefore,  needs to go in the "curfew" column as the row total. Next, we were told that of those students,  were on the honor roll; therefore, we need to put  in the "curfew" column and in the "honor roll" row. Then, we were told that  students do not have a curfew but were on the honor roll, so we need to put  in the "no curfew" column and the "honor roll" row. Finally, we were told that  students do not have a curfew or were on the honor roll, so  needs to go in the "no curfew" column and "no honor roll" row. If done correctly, you should create a table similar to the following:

Our question asked how many students did not have a curfew. We add up the numbers in the "no curfew" column to get the total:

This means that  students do not have a curfew.

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who have a curfew or do not have a curfew and the rows will contain the students who are on the honor roll or are not on the honor roll. The first bit of information that we were given from the question was that  students had a curfew; therefore,  needs to go in the "curfew" column as the row total. Next, we were told that of those students,  were on the honor roll; therefore, we need to put  in the "curfew" column and in the "honor roll" row. Then, we were told that  students do not have a curfew but were on the honor roll, so we need to put  in the "no curfew" column and the "honor roll" row. Finally, we were told that  students do not have a curfew or were on the honor roll, so  needs to go in the "no curfew" column and "no honor roll" row. If done correctly, you should create a table similar to the following:

Our question asked how many students were on the honor roll. We add up the numbers in the "honor roll" row to get the total:

This means that  students were on the honor roll.

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who are athletes or are not athletes and the rows will contain the students who drink soda or do not drink soda. The first bit of information that we were given from the question was that  students were athletes; therefore,  needs to go in the "athlete" column as the row total. Next, we were told that of those students,  drinks soda; therefore, we need to put  in the "athlete" column and in the "drinks soda" row. Then, we were told that  students were not athletes, but drink soda, so we need to put  in the "not an athlete" column and the "drinks soda" row. Finally, we were told that  students are not athletes or soda drinkers, so  needs to go in the "not an athlete" column and "doesn't drink soda" row. If done correctly, you should create a table similar to the following:

Our question asked how many students drink soda. We add up the numbers in the "drinks soda" row to get the total:

This means that  students drink soda.

To help answer this question, we can construct a two-way table and fill in our known quantities from the question.

The columns of the table will represent the students who are athletes or are not athletes and the rows will contain the students who drink soda or do not drink soda. The first bit of information that we were given from the question was that  students were athletes; therefore,  needs to go in the "athlete" column as the row total. Next, we were told that of those students,  drinks soda; therefore, we need to put  in the "athlete" column and in the "drinks soda" row. Then, we were told that  students were not athletes, but drink soda, so we need to put  in the "not an athlete" column and the "drinks soda" row. Finally, we were told that  students are not athletes or soda drinkers, so  needs to go in the "not an athlete" column and "doesn't drink soda" row. If done correctly, you should create a table similar to the following:

Our question asked how many students don't drink soda. We add up the numbers in the "doesn't drink soda" row to get the total, but first, we need to fill in a gap in our table, students who were athletes, but don't drink soda. We can take the total number of students who are athletes, , and subtract the number of students who drink soda, 

This means that  students who are athletes, don't drink soda.  

Now, we add up the numbers in the "doesn't drink soda" row to get the total:

This means that  students don't drink soda.