AMC 8 · 2011 · #8

Easy mode Grade 3

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Problem

Imagine two bags of chips.

Bag A has three chips, labeled $1$ , $3$ , and $5$ .

Bag B also has three chips, labeled $2$ , $4$ , and $6$ .

Pick one chip from Bag A and one chip from Bag B, then add the two numbers. How many different sums are possible?

$\textbf{(A) }4 \qquad\textbf{(B) }5 \qquad\textbf{(C) }6 \qquad\textbf{(D) }7 \qquad\textbf{(E) }9$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Bag A holds three chips marked $1$, $3$, $5$ and Bag B holds three chips marked $2$, $4$, $6$. One chip is drawn from each bag and the two numbers are added. How many different sums are possible?

Givens: Bag A $= \{1, 3, 5\}$ (all odd); Bag B $= \{2, 4, 6\}$ (all even); One chip is drawn from each bag and the values are added; Answer choices: (A) $4$, (B) $5$, (C) $6$, (D) $7$, (E) $9$

Unknowns: The number of different possible values of the sum $a + b$, where $a \in \{1,3,5\}$ and $b \in \{2,4,6\}$

Understand

Givens: Bag A $= \{1, 3, 5\}$ (all odd); Bag B $= \{2, 4, 6\}$ (all even); One chip is drawn from each bag and the values are added; Answer choices: (A) $4$, (B) $5$, (C) $6$, (D) $7$, (E) $9$

Plan

Primary tool: #11 Make a Table

Secondary: #1 Find a Pattern

There are only $3 \times 3 = 9$ ways to pick one chip from each bag, so Tool #11 (Make a Table) is the cleanest way to lay every sum out in a $3 \times 3$ addition grid — nothing can be missed. Tool #1 (Find a Pattern) then explains the structure of the answer: odd $+$ even is always odd, and the sums are evenly spaced from the smallest ($1+2=3$) to the largest ($5+6=11$), so the distinct sums must be $3, 5, 7, 9, 11$.

Execute — Answer: B

#11 Make a Table 3.OA.A.1 Step 1

Set up a $3 \times 3$ addition table with Bag A values down the side and Bag B values across the top.
Each cell holds the sum $a + b$ for that row and column.

$$\begin{array}{c|ccc} + & 2 & 4 & 6 \\ \hline 1 & 3 & 5 & 7 \\ 3 & 5 & 7 & 9 \\ 5 & 7 & 9 & 11 \end{array}$$

💡 An addition table is the Grade 3 way to organize "each from this set combined with each from that set" — every possible pair gets exactly one cell.

#1 Find a Pattern 3.OA.D.9 Step 2

Read the nine sums off the table: $3, 5, 7, 5, 7, 9, 7, 9, 11$.
Several values repeat — for example, $1+4 = 3+2 = 5$ and $1+6 = 3+4 = 5+2 = 7$.
The problem asks for different sums, so collapse the duplicates.

$$\{3, 5, 7, 5, 7, 9, 7, 9, 11\} \;\longrightarrow\; \{3, 5, 7, 9, 11\}$$

💡 Spotting that sums repeat in a regular pattern — and that the set of different sums forms a Grade 3 arithmetic pattern — is the Tool #1 move.

#11 Make a Table 3.OA.A.1 Step 3

Count the distinct sums in $\{3, 5, 7, 9, 11\}$.
There are five of them, so there are $5$ different possible values of the sum.

$$|\{3, 5, 7, 9, 11\}| = 5 \;\Rightarrow\; \textbf{(B)}$$

💡 Counting the distinct entries that appear in the addition table is the same equal-groups counting move from Grade 3 multiplication.

[1] #11 3.OA.A.1 Set up a $3 \times 3$ addition table with Bag A values down the side and Bag B v

[2] #1 3.OA.D.9 Read the nine sums off the table: $3, 5, 7, 5, 7, 9, 7, 9, 11$. Several values r

[3] #11 3.OA.A.1 Count the distinct sums in $\{3, 5, 7, 9, 11\}$. There are five of them, so ther

Review

Reasonableness: Every chip in Bag A is odd and every chip in Bag B is even, so every sum is odd. The smallest sum is $1+2=3$ and the largest is $5+6=11$. The odd numbers from $3$ to $11$ are $3, 5, 7, 9, 11$ — exactly $5$ values. Each of these is reachable (for instance $3=1+2$, $5=1+4$, $7=1+6$, $9=3+6$, $11=5+6$), so the count of $5$ is tight: no gaps and no extras. Answer (B) is consistent.

Alternative: Tool #1 (Find a Pattern) directly: the smallest sum is $1+2=3$, the largest is $5+6=11$, and every sum is odd + even $=$ odd. The odd numbers from $3$ to $11$ form the arithmetic sequence $3, 5, 7, 9, 11$, and a quick check confirms each is actually attainable. That gives $5$ different sums without writing out the full table.

CCSS standards used (min grade 3)

3.OA.A.1 Interpret products of whole numbers and the equal-groups model of multiplication (Recognizing that the $3 \times 3 = 9$ entries of the addition table list every possible (Bag A, Bag B) pair exactly once, and counting the $5$ distinct sums that appear in the table.)
3.OA.D.9 Identify arithmetic patterns and explain them using properties of operations (Observing the pattern that odd $+$ even is always odd and that the distinct sums $3, 5, 7, 9, 11$ form a regular arithmetic sequence, which justifies collapsing the duplicates.)

⭐ This AMC 8 problem only needs Grade 3 skills — make an addition table, spot the pattern, and count — that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 3)

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