AMC 8 · 2007 · #13
Easy mode Grade 4Problem
Picture two overlapping circles, like a Venn diagram. One circle is set A, the other circle is set B.
Sets A and B have the same number of items inside.
If you put all items from A and B together (counting shared items only once), you get 2007 items in total.
The items that sit inside both circles at the same time — there are 1001 of those.
How many items are in set A?
Pick an answer.
AMC 8 2007 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.
Toolkit + CCSS Solution
Understand
Restated: Two sets $A$ and $B$ have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. How many elements are in $A$?
Givens: $|A \cup B| = 2007$; $|A \cap B| = 1001$; $|A| = |B|$; Answer choices: (A) $503$, (B) $1006$, (C) $1504$, (D) $1507$, (E) $1510$
Unknowns: The number of elements in set $A$, i.e. $|A|$
Understand
Restated: Two sets $A$ and $B$ have the same number of elements. Their union has $2007$ elements and their intersection has $1001$ elements. How many elements are in $A$?
Givens: $|A \cup B| = 2007$; $|A \cap B| = 1001$; $|A| = |B|$; Answer choices: (A) $503$, (B) $1006$, (C) $1504$, (D) $1507$, (E) $1510$
Plan
Primary tool: #12 Draw a Venn Diagram
Secondary: #7 Identify Subproblems
The problem hands us a Venn diagram and asks about union and intersection sizes — Tool #12 (Draw a Venn Diagram) is the literal fit. Sketch two overlapping circles, label the middle (intersection) $1001$, and let the two crescents be the "only-$A$" and "only-$B$" regions. Since $|A| = |B|$, those crescents have equal size. Tool #7 (Identify Subproblems) then splits the union of $2007$ into three pieces — both crescents and the middle — and the symmetry gives each crescent in one short calculation. Add one crescent back to the middle to get $|A|$.
Execute — Answer: C
2.MD.D.10 Step 1 - Draw the Venn diagram.
- Two overlapping circles: the left circle is $A$, the right circle is $B$, the middle lens is $A \cap B$.
- Label the middle with $1001$.
- Call the left-only crescent $a$ (the elements in $A$ but not in $B$) and the right-only crescent $b$ (the elements in $B$ but not in $A$).
💡 Sorting the union into three disjoint regions is the Grade 2 "picture graph" idea — a visual way to break a set into non-overlapping parts.
3.OA.D.9 Step 2 - Translate $|A| = |B|$ into the picture.
- Set $A$ contains the left crescent plus the middle, and set $B$ contains the right crescent plus the middle.
- Both have the same middle, so the crescents must match.
💡 If two totals are equal and they share the same chunk, the leftover chunks are equal too — Grade 3 "explain patterns in arithmetic."
4.OA.A.3 Step 3 - Write the union as the three disjoint regions and use $a = b$.
- The union covers the left crescent, the middle, and the right crescent exactly once each.
💡 Splitting a union into non-overlapping pieces is the standard Grade 4 multi-step word-problem move — count each region once.
4.OA.A.3 Step 4 - Solve for the crescent.
- Subtract $1001$ from both sides, then halve.
💡 After removing the shared middle, the leftover $1006$ is split evenly between the two equal crescents, so each gets half.
4.OA.A.3 Step 5 - Read off $|A|$.
- Set $A$ is the left crescent plus the middle.
💡 From the Venn picture, $A$ is its own crescent plus everything it shares with $B$.
2.MD.D.10 Draw the Venn diagram. Two overlapping circles: the left circle is $A$, the righ 3.OA.D.9 Translate $|A| = |B|$ into the picture. Set $A$ contains the left crescent plus 4.OA.A.3 Write the union as the three disjoint regions and use $a = b$. The union covers 4.OA.A.3 Solve for the crescent. Subtract $1001$ from both sides, then halve. 4.OA.A.3 Read off $|A|$. Set $A$ is the left crescent plus the middle. Review
Reasonableness: Plug back into the Venn diagram. Left crescent $503$, middle $1001$, right crescent $503$. Then $|A| = 503 + 1001 = 1504$ and $|B| = 503 + 1001 = 1504$, so $|A| = |B|$ as required. The union is $503 + 1001 + 503 = 2007$, matching the given. The intersection is $1001$, also given. Magnitude check: $|A|$ must be between $1001$ (the intersection) and $2007$ (the union), and $1504$ sits squarely in that range. Choices (A) $503$ and (B) $1006$ are too small to contain the $1001$ intersection.
Alternative: Tool #13 (Convert to Algebra) gives the inclusion-exclusion formula directly: $|A \cup B| = |A| + |B| - |A \cap B|$. With $|A| = |B| = x$, this becomes $2007 = 2x - 1001$, so $2x = 3008$ and $x = 1504$. Same answer, but the Venn picture makes the algebra unnecessary — the three-region split is the formula.
CCSS standards used (min grade 4)
2.MD.D.10Draw a picture graph and a bar graph to represent a data set (Drawing the two-circle Venn diagram and labeling the three disjoint regions (left-only, middle, right-only) to organize the count.)3.OA.D.9Identify arithmetic patterns and explain them using properties of operations (Using $|A| = |B|$ together with the shared middle of $1001$ to conclude the two crescents must be equal in size.)4.OA.A.3Solve multistep word problems using the four operations (Splitting the union $2007$ into three non-overlapping parts, solving $2a + 1001 = 2007$ for the crescent size, and adding it back to the middle to get $|A| = 1504$.)
⭐ Drawing the Venn diagram turns this AMC 8 problem into a Grade 4 split-and-add: the union $2007$ is the middle $1001$ plus two equal crescents, so each crescent is $503$ and $|A| = 503 + 1001 = 1504$.
⭐ Drawing the Venn diagram turns this AMC 8 problem into a Grade 4 split-and-add: the union $2007$ is the middle $1001$ plus two equal crescents, so each crescent is $503$ and $|A| = 503 + 1001 = 1504$.
More like this
Same archetype — closest grade level first.
