AMC 10 · 2004 · #10
Grade 4 arithmeticA grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains 100 cans, how many rows does it contain?
Pick an answer.
AMC 10 2004 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: Cans are stacked in rows. The top row has $1$ can, and every row below holds $2$ more cans than the row directly above it. The whole stack uses $100$ cans. Find how many rows the stack has.
Givens: The top row has $1$ can.; Each row below has exactly $2$ more cans than the row above it.; The stack contains $100$ cans in total.; Answer choices: (A) $5$, (B) $8$, (C) $9$, (D) $10$, (E) $11$.
Unknowns: The number of rows in the stack.
Understand
Restated: Cans are stacked in rows. The top row has $1$ can, and every row below holds $2$ more cans than the row directly above it. The whole stack uses $100$ cans. Find how many rows the stack has.
Givens: The top row has $1$ can.; Each row below has exactly $2$ more cans than the row above it.; The stack contains $100$ cans in total.; Answer choices: (A) $5$, (B) $8$, (C) $9$, (D) $10$, (E) $11$.
Plan
Primary tool: #5 Look for a Pattern
Secondary: #1 Draw a Diagram, #6 Guess and Check
Because each row grows by the same step of $2$, the row sizes are the odd numbers $1, 3, 5, 7, \ldots$, and the running totals hide a clean pattern — the signature trigger for Tool #5 (Look for a Pattern). Adding the rows one at a time gives $1, 4, 9, 16, \ldots$, which are perfect squares, so the total after $n$ rows is $n\times n$. Tool #1 (Draw a Diagram) shows why: each new odd row wraps an L-shape around a square and completes the next larger square. Tool #6 (Guess and Check) then just asks which square equals $100$.
Execute — Answer: D
4.OA.C.5 Step 1 Read the rows as odd numbers
- Start at the top with $1$ can and add $2$ cans for each row going down.
- That gives $1$, then $1+2=3$, then $3+2=5$, then $5+2=7$, and so on.
- The number of cans in the rows, from top to bottom, is $1, 3, 5, 7, 9, \ldots$ — the odd numbers in order.
💡 Stepping up by $2$ every time walks you straight through the odd numbers.
3.OA.D.9 Step 2 Add row by row and watch the totals
- Add the rows one at a time and keep a running total.
- One row: $1$.
- Two rows: $1+3=4$.
- Three rows: $4+5=9$.
- Four rows: $9+7=16$.
- Five rows: $16+9=25$.
- The totals are $1, 4, 9, 16, 25$, which are exactly $1\times 1,\ 2\times 2,\ 3\times 3,\ 4\times 4,\ 5\times 5$.
- So the total after $n$ rows is $n\times n$.
💡 Each new odd row is an L-shape that hugs the current square and turns it into the next bigger square.
3.OA.C.7 Step 3 Match the total to 100
- The stack must total $100$ cans, and the total after $n$ rows is $n\times n$.
- So find the number that multiplied by itself gives $100$.
- Since $10\times 10=100$, the stack needs $n=10$ rows.
- The answer is (D).
💡 Reaching a total of $100$ means reaching the square $10\times 10$, which happens at exactly $10$ rows.
4.OA.C.5 Start at the top with $1$ can and add $2$ cans for each row going down. That giv 3.OA.D.9 Add the rows one at a time and keep a running total. One row: $1$. Two rows: $1+ 3.OA.C.7 The stack must total $100$ cans, and the total after $n$ rows is $n\times n$. So Review
Reasonableness: Check the neighbors: $9$ rows give $9\times 9=81$ cans and $11$ rows give $11\times 11=121$ cans, so only $10$ rows land exactly on $100$. The wrong choices match squares that miss: $5$ rows make $25$ and $8$ rows make $64$, both far short. Landing precisely on the offered choice $10$ is a strong sign the count is right.
Alternative: Use the arithmetic-series shortcut instead of spotting squares. With $n$ rows, the bottom row holds $2n-1$ cans, and the sum of the rows is $\tfrac{n}{2}\times(\text{first}+\text{last})=\tfrac{n}{2}\times(1+(2n-1))=\tfrac{n}{2}\times 2n=n\times n$. Setting $n\times n=100$ gives $n=10$ again. Or simply keep adding odd cans until the running total hits $100$: $1,4,9,16,25,36,49,64,81,100$ — that is $10$ additions, so $10$ rows.
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern that follows a given rule (Generating the row sizes $1, 3, 5, 7, \ldots$ from the rule 'start at 1 and add 2 each row.')3.OA.D.9Identify arithmetic patterns and explain them using properties of operations (Noticing that the running totals $1, 4, 9, 16, 25$ are the perfect squares, so the total after $n$ rows is $n\times n$.)3.OA.C.7Fluently multiply and divide within 100 (Recognizing that $10\times 10=100$ to find the number of rows.)
⭐ Adding up the odd numbers $1+3+5+\cdots$ always gives a perfect square, so a stack that grows by $2$ cans per row reaches $100=10\times 10$ after exactly $10$ rows.
⭐ Adding up the odd numbers $1+3+5+\cdots$ always gives a perfect square, so a stack that grows by $2$ cans per row reaches $100=10\times 10$ after exactly $10$ rows.
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