How to Solve Venn Diagram and Survey Problems on the SSAT Upper Level
Survey problems and questions about group overlaps on the SSAT Upper Level always ask one of three questions: how many items are in both groups, how many are in either group, or how many are in neither group. Each question type has its own formula, and the key word at the end of the problem — “both,” “either,” or “neither” — tells you exactly which formula to use.
The short answer: SSAT survey problems come in three types, identified by a single keyword. “Both” problems use \( A + B \ – \ \text{Total} \). “Either” problems use \( A + B \ – \ \text{Both} \). “Neither” problems use \( \text{Total} \ – \ (A + B \ – \ \text{Both}) \). Read the last sentence of the problem first, identify the keyword, select the formula, and plug in. These problems are among the most reliably solvable on the test once the method is clear.
The setup of a survey problem looks complicated — a large group of people, two overlapping categories, and several numbers to track. Students often try to draw a Venn diagram and fill it in from scratch, which works but takes time and leaves room for arithmetic errors.
The shortcut is simpler: you don’t need the diagram. You need the keyword and the formula. The diagram is useful for understanding why the formulas work, but on test day, recognizing the keyword and plugging in is faster and just as accurate.
The Three Formulas
Before the worked examples, here they are in one place for easy reference:
Question Type
Keyword
Formula
Intersection
“both”
\( \text{Both} = A + B \ – \ \text{Total} \)
Union
“either”
\( \text{Either} = A + B \ – \ \text{Both} \)
Neither
“neither” or “not either”
\( \text{Neither} = \text{Total} \ – \ (A + B \ – \ \text{Both}) \)
Note: “Both” questions will only appear when every person or item in the survey belongs to at least one of the two groups. If some people belong to neither group, the problem will ask for “either” or “neither” instead.
These formulas are also listed in the formulas to memorize post — if you haven’t bookmarked that one yet, it’s a useful companion reference.
Type 1: “Both” Problems
Example:
In a survey of 70 software engineers, each reported owning a cat, a dog, or both. If 50 engineers own a cat and 30 own a dog, how many own both?
Step 1: Identify the keyword.
The question asks “how many own both.” This is a “both” problem.
Step 2: Apply the formula.
\( \text{Both} = A + B \ – \ \text{Total} \)
\( \text{Both} = 50 + 30 \ – \ 70 = 10 \)
10 engineers own both a cat and a dog.
Notice the setup: “each reported owning a cat, a dog, or both” tells us everyone belongs to at least one group — which is the condition required for the “both” formula.
Type 2: “Either” Problems
“Either” problems give you the overlap (both) and ask for the total who belong to at least one group.
Example:
In a survey of 145 travelers, 58 have traveled to Thailand, 35 have traveled to Vietnam, and 28 have been to both. How many have traveled to either Thailand or Vietnam?
Step 1: Identify the keyword.
“Either Thailand or Vietnam” — this is an “either” problem.
Step 2: Apply the formula.
\( \text{Either} = A + B \ – \ \text{Both} \)
\( \text{Either} = 58 + 35 \ – \ 28 = 65 \)
65 travelers have been to either Thailand or Vietnam.
Type 3: “Neither” Problems — The Fully Worked Example
“Neither” problems are the trickiest of the three types, because they involve one more step. The good news is that the formula is just a short extension of the “either” formula.
Worked Sample Problem:
Miss Vy surveys 198 people about drinks they enjoy. She finds that 86 people enjoy coffee, 42 people enjoy tea, and 31 people enjoy both. How many people surveyed enjoy neither coffee nor tea?
(A) 97
(B) 98
(C) 99
(D) 100
(E) 101
Step 1: Identify the keyword.
“Neither coffee nor tea” — this is a “neither” problem.
Step 2: Apply the formula.
\( \text{Neither} = \text{Total} \ – \ (A + B \ – \ \text{Both}) \)
Notice that the answer choices are clustered tightly — 97 through 101. This is a deliberate SSAT trap. Students who forget to subtract the “both” count from the parentheses, or who add instead of subtract inside the bracket, land on one of the wrong answers. The formula handles this correctly as long as you follow it exactly.
Set Intersection Problems: A Related Type
Survey problems deal with people. A related question type deals with number sets and asks for the intersection — the items that appear in both sets.
Example:
Define Set A as the first eight positive even integers and Set B as the first six positive multiples of three. What is the intersection of Set A and Set B?
Step 1: Write out both sets.
Set A: {2, 4, 6, 8, 10, 12, 14, 16}
Set B: {3, 6, 9, 12, 15, 18}
Step 2: Circle numbers that appear in both.
6 and 12 appear in both sets.
The intersection is {6, 12}.
These problems don’t use the survey formulas — they use direct comparison. The method is the same whether the sets are listed explicitly or defined by a rule: write them out, then find the overlap. Trying to do this in your head leads to missed items.
Reading the Problem: The One-Step Shortcut
The single most useful habit for survey problems is to read the last sentence first. The question is always at the end, and the keyword is always in the question. Once you know whether you’re solving for “both,” “either,” or “neither,” the rest of the problem is just filling in values.
This is the same identification-first approach that works for sequence problems and probability problems — on the SSAT, knowing what you’re solving for before you start calculating is almost always faster than diving straight into the numbers.
If survey problems are an area where your student keeps second-guessing themselves, book a free 60-minute trial session — a few worked examples together is usually enough to make the keyword-to-formula habit click.
Frequently Asked Questions: Venn Diagram and Survey Problems on the SSAT Upper Level
How do you solve survey problems on the SSAT Upper Level?
Read the last sentence of the problem first to identify the keyword. “Both” means use \( A + B \ – \ \text{Total} \). “Either” means use \( A + B \ – \ \text{Both} \). “Neither” or “not either” means use \( \text{Total} \ – \ (A + B \ – \ \text{Both}) \). Plug in the numbers from the problem and calculate. Drawing a Venn diagram is not required — the keyword-and-formula method is faster and just as accurate.
What is the formula for “both” in an SSAT survey problem?
When a survey problem states that every person belongs to at least one group and asks how many belong to both, the formula is: Both = A + B − Total. For example, if 50 people own a cat, 30 own a dog, and all 70 surveyed own at least one, then Both = 50 + 30 − 70 = 10.
What is the difference between “either” and “neither” on SSAT survey problems?
“Either” asks for the total who belong to at least one of the two groups: Either = A + B − Both. “Neither” asks for the total who belong to neither group: Neither = Total − (A + B − Both). The “neither” formula is just one step further — subtract the “either” count from the total. Both formulas require knowing the number who belong to both groups, which may be given directly or calculated first.
Do SSAT survey problems always use Venn diagrams?
No. While the underlying concept involves overlapping sets, which Venn diagrams illustrate, SSAT survey problems are solved algebraically using the three formulas above, not by filling in a diagram. Understanding what the Venn diagram represents helps students see why the formulas work, but on test day the keyword-to-formula method is faster. Set intersection problems, a related type, require listing both sets and finding the shared items rather than using the survey formulas.
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