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.
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 DiagramPretend 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 SubproblemsThe 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 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).
4.OA.A.3Draw A Venn DiagramAdd 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.
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