AMC 10 · 2011 · #6
Grade 4 arithmeticPick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks for the smallest possible union, which is a min/max question — exactly what Tool #14, the Extreme Principle, is for. Instead of trying every arrangement, jump straight to the extreme: shared elements are the only way to shrink a union, so push the overlap as large as it can go. A Venn diagram (Tool #12) makes that extreme easy to see — slide B's circle completely inside A's circle so nothing pokes out. Then only counting is left.
Aim for maximum overlap
The union counts each element once, so shared elements are the only brake on the count — make the overlap as big as possible.
Overlap is the only discount on a union, so grab the biggest discount you can.
Overlap is the only discount on a union, so the biggest overlap gives the smallest union.
▸ Why?
Adding two group sizes counts the shared part twice, so the union is that sum minus the overlap.
▸ Why?
The overlap can grow until the smaller group sits wholly inside the larger one, and no further.
Slide B inside A
B has only 15 elements, so the overlap tops out at 15: draw B's circle wholly inside A's and B adds nothing new.
If the small circle sits wholly inside the big circle, the picture of the union is just the big circle.
4.OA.A.3Draw A Venn DiagramCount the union
With B tucked inside A the union is all of A, and the formula agrees: 20 + 15 - 15 = 20, choice (C).
When the smaller set adds nothing new, the union is exactly the bigger set's size.
2.NBT.B.5Extreme PrincipleA union is smallest when the sets overlap the most, so the smallest possible union is just the size of the bigger set.
- Aim for maximum overlap
- Slide B inside A
- Count the union