Sets A and B, shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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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 $A$?
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 $A$?
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
#12 Draw a Venn Diagram 2.MD.D.10Step 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 $A$).
💡 From the Venn picture, $A$ is its own crescent plus everything it shares with $B$.
[1]
#12 2.MD.D.10Draw the Venn diagram. Two overlapping circles: the left circle is $A$, the righ
[2]
#7 3.OA.D.9Translate $|A| = |B|$ into the picture. Set $A$ contains the left crescent plus
[3]
#7 4.OA.A.3Write the union as the three disjoint regions and use $a = b$. The union covers
[4]
#7 4.OA.A.3Solve for the crescent. Subtract $1001$ from both sides, then halve.
[5]
#12 4.OA.A.3Read 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.10 Draw 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.9 Identify 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.3 Solve 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 $|A| = 503 + 1001 = 1504$.
