Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #9
Grade 4 counting
Pick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The picture is already a Venn diagram: three regions for A, B, C with A overlapping both B and C but B and C kept apart. Tool #12 (Draw a Venn Diagram) lets us label each region with the right count and read the total straight from the picture, without memorizing a three-set formula. Tool #7 (Identify Subproblems) splits the work into two clean Grade 4 steps: (i) add the three bed totals as if nothing were shared, then (ii) subtract the overlap plants that got counted twice. Because B and C never touch, there is no triple-overlap to add back, which keeps the bookkeeping simple.
Label the Venn regions
The picture is a Venn diagram with just five regions — A-only, B-only, C-only, A∩B, A∩C — and no B∩C or triple region.
Look at the picture before doing any arithmetic. The shape of the overlaps tells you which regions you need to track and which ones are empty.
4.OA.A.3Draw A Venn DiagramAdd the three counts
Pretend nothing is shared and add the beds: 500 + 450 + 350 = 1300, which double-counts every shared plant.
Adding the three sizes treats every plant as if it lived in only one bed. The shared plants get counted twice, so this sum is too big by exactly the amount of overlap.
4.NBT.B.4Identify SubproblemsAdd up the overlaps
The overlaps counted twice total |A∩B| + |A∩C| = 50 + 100 = 150; there is no B∩C to remove.
Each shared plant was counted twice in step 2; subtracting it once leaves it counted exactly once, which is what we want.
4.NBT.B.4Identify SubproblemsSubtract the overlaps
Subtract the overlap from the over-count: 1300 - 150 = 1150, choice (C); with no triple region there is nothing to add back.
The Venn-diagram bookkeeping (add the parts, take back the double-count) lands on choice (C).
The true number of distinct plants equals the sum of the three bed totals minus the plants shared between a pair of beds.
▸ Why?
The true total is what you get by counting each separate region of the picture once — the plants in a single bed, plus each patch two beds share.
▸ Why?
Adding the three bed totals counts each single-bed region once but each shared patch twice.
▸ Why?
A bed's total already includes the patch it shares, so that patch sits inside the totals of both beds that meet there.
▸ Why?
Subtracting each shared patch once brings its count back down from two to one, so every plant ends up counted exactly once.
▸ Why?
Taking away one copy of an amount that was added an extra time cancels that extra addition.
Add the three bed totals as if no plant were shared, then subtract the shared plants once so they aren't counted twice — 1300 - 150 = 1150 turns this AMC 8 problem into a Grade 4 Venn-diagram exercise.
- Label the Venn regions
- Add the three counts
- Add up the overlaps
- Subtract the overlaps
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