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GMAT Quant
Sets and Overlap
Compute unions, neither, exactly one, and the possible range of overlap between two sets.
Mental model
Two-set problems are bookkeeping problems.
Every person belongs in one of four buckets: A only, B only, both, or neither.
Union
Union means at least one of the two sets.
A union B = A + B - Both. The overlap is subtracted because A + B counted it twice.
If A = 65%, B = 50%, Both = 25%
- Union
- 65 + 50 - 25 = 90%
- Neither
- 100 - 90 = 10%
- Exactly one
- 65 + 50 - 2*25 = 65%
Minimum overlap
Minimum overlap appears when you separate the sets as much as possible.
Both_min = max(0, A + B - 100%). If A + B exceeds 100%, the excess must overlap.
Maximum overlap
Maximum overlap is the size of the smaller set.
Both_max = min(A, B), because the smaller set cannot overlap more people than it contains.
Example: A = 72%, B = 41%
- Minimum both
- 72 + 41 - 100 = 13%
- Maximum both
- min(72,41) = 41%
- Possible range
- 13% to 41%
Recognition cues
- 01"At least one" -> union.
- 02"Neither" -> 100% - union.
- 03"Exactly one" -> remove the overlap from both sides.
- 04"At least how many in both" -> minimum overlap.
Mini application
Try this: 80% are in A and 35% are in B. What is the minimum overlap?
Think about how many people can fit outside A.
Solution
The minimum overlap is 15%.
Only 20% can be outside A, so at least 15 percentage points of B must spill into A.
Remember this
When A + B exceeds the whole, the excess must overlap.
That is the fastest way to see the minimum-overlap rule.