Sensim Math Original · sm-3
SM Original Grade 4 arithmeticnumber-theoryProblem
A city clerk keeps a "lucky sevens" license-plate ledger. The day begins with blank plates on the shelf. During the morning the clerk hands out three batches: first plates to the registry office, then plates to a dealership, and finally plates to a walk-in customer. What is the units digit of the number of plates still on the shelf at the end of the day?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Starting from $7{,}777$ 'lucky sevens' license plates on a shelf, the clerk gives out $777$, then $77$, then $7$ more plates. We must report the units (ones) digit of the leftover count, $7{,}777 - 777 - 77 - 7$, not the full number.
Givens: Opening shelf count: $7{,}777$ plates; Three successive outflows: $777$, then $77$, then $7$; Every quantity in the expression is built entirely from the digit 7; Every quantity ends in the digit 7 (units digit is 7); Five answer choices: (A) 0, (B) 3, (C) 4, (D) 6, (E) 8
Unknowns: The units digit of $7{,}777 - 777 - 77 - 7$
Understand
Restated: Starting from $7{,}777$ 'lucky sevens' license plates on a shelf, the clerk gives out $777$, then $77$, then $7$ more plates. We must report the units (ones) digit of the leftover count, $7{,}777 - 777 - 77 - 7$, not the full number.
Givens: Opening shelf count: $7{,}777$ plates; Three successive outflows: $777$, then $77$, then $7$; Every quantity in the expression is built entirely from the digit 7; Every quantity ends in the digit 7 (units digit is 7); Five answer choices: (A) 0, (B) 3, (C) 4, (D) 6, (E) 8
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities
Four numbers with up to four digits look intimidating, but only the ones column matters. Replacing each plate count by its units digit collapses the whole ledger to single-digit arithmetic — a much easier related problem (Tool #9). The repeated 7s expose a clean pattern (Tool #5), and on a multiple-choice problem Tool #3 lets us cross-check the survivor at the end.
Execute — Answer: D
4.NBT.A.2 Step 1 - Every quantity in the ledger — $7{,}777$, $777$, $77$, and $7$ — ends in the digit 7.
- Because the units digit of a sum or difference is fixed by the ones column alone, we can throw away the thousands, hundreds, and tens columns.
- This is the easier-related-problem move: four big numbers shrink to four single digits.
💡 Reading off the rightmost digit of a multi-digit number is the Grade 4 place-value move.
2.OA.B.2 Step 2 - Adding the three outflow units digits gives $7 + 7 + 7 = 21$.
- So the three subtracted plate counts together end in the digit 1 (the ones place of 21).
- The thousands and hundreds carries from the actual sum don't reach the ones column, so we can ignore them.
💡 Adding three small numbers under 30 mentally is the Grade 2 fluency.
4.NBT.B.4 Step 3 - Now look only at the ones column of $7{,}777 - (\text{number ending in } 1)$.
- We need to compute $7 - 1$ in the ones place.
- Since $7 > 1$, no borrow comes in from the tens column, and the ones digit of the answer is simply $7 - 1 = 6$.
💡 Subtracting multi-digit numbers column by column (Grade 4) lets us safely isolate the ones place when no borrow is needed.
2.NBT.A.1 Step 4 - Match against the five offered digits.
- The candidates are 0, 3, 4, 6, 8.
- Our ones digit is 6, which is exactly choice (D).
- The other choices — 0, 3, 4, 8 — never arise from $7 - 1$ in the ones column under these inflow/outflow signs, so they are ruled out.
💡 Reading the ones place of a number to pick the matching choice is direct Grade 2 place-value work.
4.NBT.A.2 Every quantity in the ledger — $7{,}777$, $777$, $77$, and $7$ — ends in the dig 2.OA.B.2 Adding the three outflow units digits gives $7 + 7 + 7 = 21$. So the three subtr 4.NBT.B.4 Now look only at the ones column of $7{,}777 - (\text{number ending in } 1)$. We 2.NBT.A.1 Match against the five offered digits. The candidates are 0, 3, 4, 6, 8. Our one Review
Reasonableness: Let's confirm by computing the full leftover count: $777 + 77 + 7 = 861$, and $7{,}777 - 861 = 6{,}916$. The ones digit of $6{,}916$ is indeed 6, so the shortcut matches the full subtraction and the answer (D) 6 is correct.
Alternative: An equivalent route is Tool #13 (Convert to Algebra) via modular arithmetic mod 10: every term is $\equiv 7 \pmod{10}$, so the expression is $7 - 7 - 7 - 7 = -14 \equiv 6 \pmod{10}$. Same digit 6, but for a young solver Tool #9 reads more naturally.
CCSS standards used (min grade 4)
2.NBT.A.1Understand that the three digits of a three-digit number represent hundreds, tens, and ones (Reading the ones place of the surviving count to match against the answer choices.)2.OA.B.2Fluently add and subtract within 20 using mental strategies (Adding the three outflow units digits $7 + 7 + 7 = 21$ in one's head.)4.NBT.A.2Read and write multi-digit whole numbers and compare using symbols (Recognizing that each plate count (up to $7{,}777$) has units digit 7.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Reasoning column-by-column about the multi-digit subtraction so the ones digit can be read directly.)
⭐ This AMC 8 problem only needs Grade 4 multi-digit place-value sense you already know!
⭐ This AMC 8 problem only needs Grade 4 multi-digit place-value sense you already know!