AMC 8 · 2019 · #11

Grade 2 countinglogic
Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region
set-partitionsystematic-enumerationlinear-equations-one-var caseworkcomplementary-counting ↑ Prerequisites: set-partition
📏 Short solution 💡 2 insights
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Problem
Lincoln Middle School's eighth grade has 93 students. Every student takes at least one of two classes — math or a foreign language — and some take both. 70 students are in math and 54 are in foreign language. How many take a math class but NOT a foreign language class?

Pick an answer.

(A)
16
(B)
53
(C)
31
(D)
39
(E)
70

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Venn Diagram

The word "only" plus two overlapping categories (math and foreign language) is the textbook trigger for Tool #12 (Venn Diagram). Drawing two overlapping circles and labeling the three regions — Math-only, Both, Foreign-only — makes the structure of the problem visible. From the diagram, Tool #16 (Complement) gives a shortcut: because every student is in at least one circle, anyone NOT in the Foreign-language circle must be in the Math-only region. So Math-only = Total - Foreign-language takers, no need to find the overlap first.

1STEP 1

Draw two overlapping circles — Math and Foreign — splitting the 93 students into Math-only, Both, and Foreign-only; no outside region.

Math-only + Both + Foreign-only = 93
2STEP 2

A Math-only student is exactly one NOT in the foreign-language circle; since nobody sits outside both circles, Math-only = 93 - 54.

Math-only = 93 - 54
3STEP 3

Subtracting with regrouping, 93 - 54 = 39 eighth graders take only a math class — choice (D).

93 - 54 = 39 → (D)
Answer
39
Check that all three Venn regions add to 93. Math-only = 39. Foreign-only = 93 - 70 = 23 (students not in the math circle). Both = 70 - 39 = 31 (math takers minus math-only). Sum: 39 + 23 + 31 = 93. Every region is non-negative and the total matches, so the answer is internally consistent. Also 39 is reasonable: it's between the wrong-direction traps 31 (the overlap) and 70 (all of math).
💡Key takeaway

This AMC 8 problem only needs Grade 2 two-digit subtraction you already know — once a Venn diagram shows you which numbers to subtract!

More like this

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