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

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Total eighth graders: $93$ (every student is in math, foreign language, or both); Students taking a math class: $70$; Students taking a foreign language class: $54$; Answer choices: (A) $16$, (B) $53$, (C) $31$, (D) $39$, (E) $70$

Unknowns: The number of students who take math but NOT a foreign language (the "math only" group)

Understand

Plan

Primary tool: #12 Draw a Venn Diagram

Secondary: #16 Change Focus / Count the Complement

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.

Execute — Answer: D

#12 Draw a Venn Diagram K.MD.B.3 Step 1

Draw two overlapping circles, one for Math ($M$) and one for Foreign language ($F$).
The diagram splits the $93$ students into three regions: Math-only (left), Both (middle), and Foreign-only (right).
There is no "outside" region because every student takes at least one class.

$$\text{Math-only} + \text{Both} + \text{Foreign-only} = 93$$

💡 Sorting students into named groups and counting each group is the Kindergarten "classify and count" idea.

#16 Change Focus / Count the Complement 1.OA.B.4 Step 2

Use the complement insight.
A student in the Math-only region is exactly a student who is NOT in the Foreign-language circle.
Since the Foreign-language circle covers all $54$ foreign-language students and nobody sits outside both circles, the Math-only count is just the total minus the Foreign-language count.

$$\text{Math-only} = 93 - 54$$

💡 "How many are left after removing the foreign-language group?" is the Grade 1 subtraction-as-unknown-addend idea.

#16 Change Focus / Count the Complement 2.NBT.B.5 Step 3

Carry out the subtraction.
$93 - 54$ needs regrouping (you can't take $4$ from $3$), giving $39$.
So $39$ eighth graders take only a math class, matching choice (D).

$$93 - 54 = 39 \;\Rightarrow\; \textbf{(D)}$$

💡 Subtracting two two-digit numbers with regrouping is the Grade 2 fluency standard.

[1] #12 K.MD.B.3 Draw two overlapping circles, one for Math ($M$) and one for Foreign language ($

[2] #16 1.OA.B.4 Use the complement insight. A student in the Math-only region is exactly a stude

[3] #16 2.NBT.B.5 Carry out the subtraction. $93 - 54$ needs regrouping (you can't take $4$ from $

Review

Reasonableness: 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).

Alternative: Tool #13 (Convert to Algebra) with inclusion-exclusion: $|M \cup F| = |M| + |F| - |M \cap F|$ gives $93 = 70 + 54 - |M \cap F|$, so $|M \cap F| = 31$. Then Math-only $= |M| - |M \cap F| = 70 - 31 = 39$. Same answer, but it requires two subtractions instead of one — the Venn + complement path is faster.

CCSS standards used (min grade 2)

K.MD.B.3 Classify objects into given categories and count the numbers in each (Sorting the $93$ students into the three Venn regions (Math-only, Both, Foreign-only) so each student is counted exactly once.)
1.OA.B.4 Understand subtraction as an unknown-addend problem (Recognizing that "Math-only" is what is left after the Foreign-language group is removed from the total — a take-away (unknown-addend) setup.)
2.NBT.B.5 Fluently add and subtract within 100 (Computing $93 - 54 = 39$, a two-digit subtraction with regrouping.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 2)

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