Sensim Math Original · sm-3

SM Original Grade 4 arithmeticnumber-theory

Inspired by AMC 8 2024 #1

Archetype Units-Digit Tracking └ Sub-archetype Digitwise Additive Reduction

units-digit-trackingplace-valuemulti-digit-arithmetic units-digit-trackingmodular-arithmetic-mod-10 ↑ Prerequisites: multi-digit-arithmeticplace-value

📏 Medium solution 💡 3 insights

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Problem

A city clerk keeps a "lucky sevens" license-plate ledger. The day begins with $7{,}777$ blank plates on the shelf. During the morning the clerk hands out three batches: first $777$ plates to the registry office, then $77$ plates to a dealership, and finally $7$ 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?

7{,}777 - 777 - 77 - 7

Pick an answer.

(A)

(B)

(C)

(D)

(E)

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

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

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

#9 Solve an Easier Related Problem 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.

$$7{,}777 \to 7,\ 777 \to 7,\ 77 \to 7,\ 7 \to 7$$

💡 Reading off the rightmost digit of a multi-digit number is the Grade 4 place-value move.

#5 Look for a Pattern 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.

$$7 + 7 + 7 = 21 \;\Rightarrow\; \text{units digit of outflow total} = 1$$

💡 Adding three small numbers under 30 mentally is the Grade 2 fluency.

#9 Solve an Easier Related Problem 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 $7 - 1 = 6$.

$$\underbrace{7{,}777}_{\text{ends in }7} - \underbrace{(777 + 77 + 7)}_{\text{ends in }1} \;\Rightarrow\; 7 - 1 = 6$$

💡 Subtracting multi-digit numbers column by column (Grade 4) lets us safely isolate the ones place when no borrow is needed.

#3 Eliminate Possibilities 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.

$$\text{Units digit} = 6 \;\Rightarrow\; \textbf{(D)}$$

💡 Reading the ones place of a number to pick the matching choice is direct Grade 2 place-value work.

[1] #9 4.NBT.A.2 Every quantity in the ledger — $7{,}777$, $777$, $77$, and $7$ — ends in the dig

[2] #5 2.OA.B.2 Adding the three outflow units digits gives $7 + 7 + 7 = 21$. So the three subtr

[3] #9 4.NBT.B.4 Now look only at the ones column of $7{,}777 - (\text{number ending in } 1)$. We

[4] #3 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.1 Understand 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.2 Fluently 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.2 Read 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.4 Fluently 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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