AMC 8 · 2007 · #13

Grade 4 counting

set-partitionlinear-equations-one-var identify-subproblemsconvert-to-algebra ↑ Prerequisites: set-partitionlinear-equations-one-var

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Problem

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 $A$ .

$\mathrm{(A)}\ 503 \qquad \mathrm{(B)}\ 1006 \qquad \mathrm{(C)}\ 1504 \qquad \mathrm{(D)}\ 1507 \qquad \mathrm{(E)}\ 1510$

Pick an answer.

(A)

503

(B)

1006

(C)

1504

(D)

1507

(E)

1510

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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 $A$?

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

#12 Draw a Venn Diagram 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 $A$).

$$\text{middle} = 1001,\;\text{left crescent} = a,\;\text{right crescent} = b$$

💡 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.

#7 Identify Subproblems 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.

$$a + 1001 = b + 1001 \;\Rightarrow\; a = b$$

💡 If two totals are equal and they share the same chunk, the leftover chunks are equal too — Grade 3 "explain patterns in arithmetic."

#7 Identify Subproblems 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.

$$a + 1001 + b = 2007 \;\Rightarrow\; 2a + 1001 = 2007$$

💡 Splitting a union into non-overlapping pieces is the standard Grade 4 multi-step word-problem move — count each region once.

#7 Identify Subproblems 4.OA.A.3 Step 4

Solve for the crescent.
Subtract $1001$ from both sides, then halve.

$$2a = 2007 - 1001 = 1006 \;\Rightarrow\; a = 503$$

💡 After removing the shared middle, the leftover $1006$ is split evenly between the two equal crescents, so each gets half.

#12 Draw a Venn Diagram 4.OA.A.3 Step 5

Read off $|A|$.
Set $A$ is the left crescent plus the middle.

$$|A| = a + 1001 = 503 + 1001 = 1504 \;\Rightarrow\; \textbf{(C)}$$

💡 From the Venn picture, $A$ is its own crescent plus everything it shares with $B$.

[1] #12 2.MD.D.10 Draw the Venn diagram. Two overlapping circles: the left circle is $A$, the righ

[2] #7 3.OA.D.9 Translate $|A| = |B|$ into the picture. Set $A$ contains the left crescent plus

[3] #7 4.OA.A.3 Write the union as the three disjoint regions and use $a = b$. The union covers

[4] #7 4.OA.A.3 Solve for the crescent. Subtract $1001$ from both sides, then halve.

[5] #12 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.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 $|A| = 503 + 1001 = 1504$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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