Competition · AMC preparation · step 4 of 4
AMC 8 · 2019 · #11
Grade 2 countinglogicPick an answer.
AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Draw the two circles
Draw two overlapping circles — Math and Foreign — splitting the 93 students into Math-only, Both, and Foreign-only; no outside region.
Sorting students into named groups and counting each group is the Kindergarten "classify and count" idea.
K.MD.B.3Draw A Venn DiagramTake out the language students
A Math-only student is exactly one NOT in the foreign-language circle; since nobody sits outside both circles, Math-only = 93 - 54.
"How many are left after removing the foreign-language group?" is the Grade 1 subtraction-as-unknown-addend idea.
The number of eighth graders who take only a math class equals the total number of students minus the number who take a foreign language class.
▸ Why?
The whole class of 93 splits with no gaps or overlaps into two parts — students who take a foreign language and students who do not — so removing the foreign-language part from the total leaves exactly the count of students who take no foreign language.
▸ Why?
Because every student takes at least one class, a student who takes no foreign language must be taking only math, so the 'no foreign language' group and the 'only math' group are the very same students and therefore have the same count.
Carry out the subtraction
Subtracting with regrouping, 93 - 54 = 39 eighth graders take only a math class — choice (D).
Subtracting two two-digit numbers with regrouping is the Grade 2 fluency standard.
2.NBT.B.5Change Focus Count The ComplementThis AMC 8 problem only needs Grade 2 two-digit subtraction you already know — once a Venn diagram shows you which numbers to subtract!
- Draw the two circles
- Take out the language students
- Carry out the subtraction
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2025 #7 Gr 2Similar levelSame sub-type: Two-Set Inclusion-Exclusion: Solve for Missing RegionSame techniqueComplementary Counting
On the most recent exam on Prof. Xochi's class, 5 students earned a score of at least 95%, 13 students earned a score of…
- AMC 10 2004B #2 Gr 3HarderSame sub-type: Two-Set Inclusion-Exclusion: Solve for Missing RegionSame techniqueComplementary Counting
How many two-digit positive integers have at least one 7 as a digit?
- AMC 8 1999 #9 Gr 4Harder
Same sub-type: Two-Set Inclusion-Exclusion: Solve for Missing RegionSame techniqueComplementary CountingThree flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B s…