Use the Totals to Fill the Blanks
A two-way table summarizes counts for combinations of categories from two categorical variables. In Reading a Two-Way Table: Rows, Columns, and Totals, you learned to read the inside cells, row totals, column totals, and grand total. Now use those totals to work backward: if some cell counts are missing, totals can show how many individuals must be in the blank cells.
The basic idea is subtraction. If a row total is known and all but one of the counts in that row are known, subtract the known counts from the row total to find the missing count. The same reasoning works down a column. When a table has several blanks, fill in the ones that can be found first, then use the newly completed cells to find others.
A table can give you row totals, column totals, both, or only some of them. A grand total can be found by adding the row totals or by adding the column totals; those two sums should agree. A cell count represents a number of individuals, so a completed count should be a nonnegative whole number. If your arithmetic produces a negative number or a fraction, recheck the entries and the totals.
A Step-by-Step Strategy
Before calculating, scan the entire table. Note which entries are given, which are missing, and which totals are available. Do not start by guessing a blank or by subtracting numbers from the grand total without checking what those numbers represent. Instead, use a row or column where a total and all but one cell are known. That gives a direct calculation.
Identify what each row and column represents, and note the available row totals, column totals, and grand total.
Subtract its known cell counts from its total. Record the result in the blank cell.
After filling a cell, check whether its row or column now has only one blank. Repeat the subtraction method.
Add across each completed row and down each completed column. Confirm that the row totals and column totals agree with the grand total.
It is helpful to write each subtraction before entering a value in the table. For example, if a row total is 70 and the other cells in that row are 26 and 19, show \(70-26-19=25\). This makes the work easy to check and helps prevent a common mistake: subtracting a count that is not part of the row or column you are using.
Sometimes there is not enough information to find every blank uniquely. For instance, a row with two missing cells and a known total tells you only their combined count. You need another usable total or a given cell to separate the two counts. Do not invent information to fill a blank. State that the table does not determine a unique answer unless its totals and given cells provide enough constraints.
Worked Examples: Completing Counts from Totals
Worked Example: One Missing Count in Each Row
A fictional school survey records whether students bring a reusable water bottle and which bus route they take. Complete the two missing counts.
| Bus route | Brings a bottle | Does not bring a bottle | Total |
|---|---|---|---|
| Route A | 31 | ? | 52 |
| Route B | 40 | ? | 68 |
| Total | 71 | 49 | 120 |
Start with Route A. Its row total is 52, and 31 students in that row bring a bottle. Therefore, the count who do not bring one is
For Route B, subtract the 40 students who bring a bottle from the row total of 68:
The completed table is:
| Bus route | Brings a bottle | Does not bring a bottle | Total |
|---|---|---|---|
| Route A | 31 | 21 | 52 |
| Route B | 40 | 28 | 68 |
| Total | 71 | 49 | 120 |
Check the columns: \(31+40=71\) students bring a bottle, and \(21+28=49\) do not. The grand total is \(52+68=120\), which also agrees with \(71+49=120\).
Conclusion: In this fictional survey, 21 students on Route A and 28 students on Route B do not bring a reusable bottle. The row totals, column totals, and grand total all agree.
Worked Example: Work Across Rows, Then Check Columns
A fictional community poll records how respondents contacted a local service desk and whether their issue was resolved. Find the missing cell counts and complete the totals.
| Contact method | Issue resolved | Issue not resolved | Total |
|---|---|---|---|
| Online form | 46 | ? | 80 |
| Phone | ? | 35 | 65 |
| In person | 42 | ? | 75 |
| Total | ? | ? | 220 |
Each row has one blank, so begin with the row totals. For online forms, \(80-46=34\) issues were not resolved. For phone contacts, \(65-35=30\) issues were resolved. For in-person contacts, \(75-42=33\) issues were not resolved.
Now add down each outcome column. The resolved total is \(46+30+42=118\). The not-resolved total is \(34+35+33=102\). The completed table is:
| Contact method | Issue resolved | Issue not resolved | Total |
|---|---|---|---|
| Online form | 46 | 34 | 80 |
| Phone | 30 | 35 | 65 |
| In person | 42 | 33 | 75 |
| Total | 118 | 102 | 220 |
Verify the results in both directions. The row totals are \(46+34=80\), \(30+35=65\), and \(42+33=75\). The column totals are 118 and 102, which add to 220. The row totals also add to \(80+65+75=220\), matching the stated grand total.
Conclusion: The missing counts are 34 online-form issues not resolved, 30 phone issues resolved, and 33 in-person issues not resolved. The completed margins are 118 resolved and 102 not resolved, for 220 respondents.
Worked Example: Let Column Totals Unlock Several Blanks
A fictional recreation survey records the preferred activity for people in three age groups. Some rows have two blanks, so use the column totals as well as the row totals to fill the table.
| Age group | Walking | Swimming | Cycling | Total |
|---|---|---|---|---|
| Under 18 | 20 | ? | 15 | 60 |
| 18–39 | ? | 10 | ? | 50 |
| 40 and older | 15 | ? | ? | 40 |
| Total | 55 | 45 | 50 | 150 |
The under-18 row has one blank, so its swimming count is \(60-20-15=25\). The walking column now has one blank: the 18–39 walking count is \(55-20-15=20\). Next, the swimming column has one blank, so the 40-and-older swimming count is \(45-25-10=10\).
The 40-and-older row now has one blank: its cycling count is \(40-15-10=15\). The 18–39 row’s cycling count is then \(50-20-10=20\). The completed table is:
| Age group | Walking | Swimming | Cycling | Total |
|---|---|---|---|---|
| Under 18 | 20 | 25 | 15 | 60 |
| 18–39 | 20 | 10 | 20 | 50 |
| 40 and older | 15 | 10 | 15 | 40 |
| Total | 55 | 45 | 50 | 150 |
Check each column: walking gives \(20+20+15=55\), swimming gives \(25+10+10=45\), and cycling gives \(15+20+15=50\). The row totals add to \(60+50+40=150\), and the column totals add to \(55+45+50=150\). All the margins match.
Conclusion: The completed table has 25 under-18 respondents who prefer swimming, 20 respondents ages 18–39 who prefer walking, 10 respondents ages 40 and older who prefer swimming, and the remaining cycling counts of 20 and 15 in the two older groups. The row and column totals confirm the results.
When the Table Does Not Give Enough Information
A total constrains the sum of cells in its row or column; it does not always determine each cell separately. Suppose a row total is 50 and two cells are blank. The information tells you that the two counts add to 50, but possible pairs include 20 and 30 or 35 and 15. Unless another given count or margin distinguishes between them, there is no unique way to fill both cells.
A useful check is to ask whether you can find a row or column with just one unknown cell. If you can, calculate it and then look again for a newly solvable row or column. If no such row or column exists, the information may not determine every blank. Avoid treating a reasonable-looking guess as a calculated count.
Also distinguish a missing cell from a missing total. If a total is not printed but all the counts needed for it are shown, add those counts. For instance, once a column is complete, its column total is the sum of its cells. Then compare the resulting total with the grand total or other margins, if they are provided.
Common Mistakes and AP Exam Tips
- Using the wrong total. A row calculation must use that row’s total; a column calculation must use that column’s total. Identify the relevant margin before subtracting.
- Forgetting more than one known cell. If a row has a total of 60 and two known counts of 18 and 17, the blank is \(60-18-17=25\), not \(60-18=42\).
- Mixing up rows and columns. Check that every count you subtract belongs to the same row or column as the missing cell. A nearby number in another row is not part of the calculation.
- Stopping as soon as the blanks are filled. Re-add every row and column. A table can contain a subtraction error even if every blank has a number.
- Ignoring a mismatch with the grand total. Row totals and column totals must each add to the same grand total. If they do not, revisit the calculations or check whether a given entry was copied incorrectly.
- Forcing a unique answer when information is insufficient. A total for two blanks gives their sum, not each count individually. Use another margin or given cell, or explain that the counts cannot be determined uniquely.
- Reporting a count that cannot represent individuals. Counts must be nonnegative whole numbers. A negative or fractional result signals an arithmetic or data inconsistency that should be checked.
A clear written solution shows which margin you used, the subtraction that finds each blank, and a check of the completed totals. For example: “The Route A count who do not bring a bottle is \(52-31=21\), using the Route A row total. The completed row and column totals agree with the grand total of 120.” That statement identifies both the arithmetic and what the result means in context.
Check Your Understanding
Use the given counts and totals to complete each table or explain what information is missing.
- A row total is 74. Two counts in that row are 28 and 19. Find the third count and show the subtraction.
- A column total is 90. Its known cell counts are 22, 31, and one blank. Find the missing count.
- A row total is 40, but two cells in the row are blank and no other relevant counts or totals are given. What can you determine about the two cells, and why can’t you find each count separately?
- Row totals are 32 and 48. Column totals are 25 and 55. What grand total should both sets of margins produce?
- A completed table’s row totals add to 125, but its column totals add to 123. What should you do before reporting the table?