# Venn Diagram Problems — Free Timed Practice | Quant Daily > Practise venn diagram problems free and timed: union, neither, and three-group overlaps solved by inclusion-exclusion. Worked solutions every time, no sign-up. Source: https://quantdaily.app/drill-types/venn-diagram-problems --- Sets ## Venn diagram problems — subtract the overlap once Venn diagram problems are solved by inclusion-exclusion. For two groups, the union is the first total plus the second minus the overlap, because everyone in both was counted twice. The number in neither group is the overall total minus that union, and three-group questions extend the same alternating pattern. Nothing here needs to be drawn. The whole topic is one principle applied carefully: a person counted in two lists contributes twice to the naive sum, so the overlap comes off exactly once. Questions then ask for the union, for neither, or for the count in exactly one group, and those are three different numbers from the same figures. **Try it against the clock.** Sets & Venn runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading. [Start a timed Sets & Venn drill →](https://quantdaily.app/?drill=quant_sets&src=seo-venn-diagram-problems) ### How to solve venn diagram problems by inclusion-exclusion - Add the groups and subtract the overlap With 25 in the first group, 18 in the second and 9 in both, the union is 25 + 18 − 9 = 34. The overlap comes off once, not twice. - Find 'neither' from the overall total Subtract the union from the total surveyed. With 40 people and a union of 34, exactly 6 are in neither group. - Extend the alternating pattern to three groups Add the three totals, subtract each pairwise overlap, then add back the triple overlap, which has otherwise been removed once too often. - Check what 'only' means The number in the first group only is its total minus the overlap. That is a different figure from the group total, and both are offered. ### Which formula answers which question | Asked for | Formula | | --- | --- | | In either group | A + B − both | | In neither group | Total − (A + B − both) | | In A only | A − both | | In exactly one | A + B − 2 × both | | Three groups | A + B + C − pairs + triple | ### 3 sample venn diagram problems, with worked solutions Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers. #### Question 1 A survey of 31 people found that 14 are in group A, 16 are in group B, and 5 are in both groups. How many people are in at least one of the two groups? - 6 - 30 - 14 - 25 - 16 - |A ∪ B| = |A| + |B| − |A ∩ B| = 14 + 16 − 5 = 25. - The 6 people in neither group are not asked. - Answer: 25. #### Question 2 A survey of 22 people found that 13 are in group A, 14 are in group B, and 7 are in both groups. How many people are in at least one of the two groups? - 20 - 27 - 13 - 14 - 2 - |A ∪ B| = |A| + |B| − |A ∩ B| = 13 + 14 − 7 = 20. - The 2 people in neither group are not asked. - Answer: 20. #### Question 3 A survey of 34 people found that 17 are in group A, 16 are in group B, and 5 are in both groups. How many people are in neither group? - 5 - 33 - 17 - 6 - 28 - |A ∪ B| = |A| + |B| − |A ∩ B| = 17 + 16 − 5 = 28. - Neither = 34 − 28 = 6. - Answer: 6. ### Common mistakes this drill is built from The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks. #### Adding the group totals without subtracting That double-counts everyone in both groups, inflating the union by exactly the size of the overlap. #### Subtracting the overlap twice Taking it off from both group totals removes the shared members entirely. They belong in the union once. #### Answering the union when 'neither' was asked They are complementary numbers from the same data and both are printed. The union is not the final step for a 'neither' question. ### Why your answer came out wrong The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here. #### 25 like tea, 18 like coffee, 9 like both — do 43 like at least one? The answer is 34. Adding 25 and 18 counts the 9 who like both twice, so the overlap comes off once: 25 + 18 − 9 = 34. The uncorrected 43 is always among the options because it is what the natural first instinct produces. #### If 34 are in at least one group out of 40, how many like only tea? That is a different question. The number in neither is 40 − 34 = 6, while the number who like only tea is 25 − 9 = 16. Union, neither and 'only' are three separate figures from the same data, and mixing them up is the main risk here. ### FAQ #### Why do I subtract the overlap when finding the union? Because adding the two group totals counts everyone in both groups twice — once in each total. Subtracting the overlap once removes the duplicate while keeping those people in the union, which is where they belong. #### How do I find how many are in neither group? Work out the union first, then subtract it from the overall total. If 40 people were surveyed and 34 are in at least one group, then 6 are in neither. Skipping the union step and subtracting the group totals directly gives a number that is too small. #### How does inclusion-exclusion work with three groups? Add the three totals, subtract each of the three pairwise overlaps, then add the triple overlap back. The triple overlap is added three times and subtracted three times by the earlier steps, so it needs restoring once to be counted properly. ### Practise the next topic - [All 51 drill types](https://quantdaily.app/drill-types) — the shared daily cycles 36 of 51, with every published time limit. - [Probability questions — count the favourable](https://quantdaily.app/drill-types/probability) — probability questions, timed. - [Permutation and combination questions — does the order matter?](https://quantdaily.app/drill-types/permutation-combination) — permutation and combination questions, timed. - [Speed tricks](https://quantdaily.app/drill-types) — the seven mental-math shortcuts that make a 15-second answer possible. - [How to improve calculation speed](https://quantdaily.app/calculation-speed) — the six-week arc these drills fit into. Quant Daily is free, needs no account, and is not affiliated with ETS or GMAC. GRE/GMAT-style practice only.