Sensim Math Original · sm-4

SM Original Grade 4 arithmeticnumber-theory

Inspired by AMC 8 2024 #1

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

To celebrate a "lucky-three" promotion, a bank teller closes the day with a ledger in which every entry is written using only the digit 3. The day's net balance is

33{,}333 + 3{,}333 + 333 + 33 - 3.

What is the units digit of this net balance?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: The bank teller records four credits ($33{,}333$, $3{,}333$, $333$, $33$) and one debit ($3$). We need only the units (ones) digit of the net balance $33{,}333 + 3{,}333 + 333 + 33 - 3$, not the full dollar amount.

Givens: Four credit entries: $33{,}333$, $3{,}333$, $333$, $33$; One debit entry: $3$; Every entry is built entirely from the digit 3; Every entry ends in the digit 3 (units digit is 3 for every term); Mixed sign pattern: four additions and one subtraction; Five answer choices: (A) 0, (B) 3, (C) 5, (D) 7, (E) 9

Unknowns: The units digit of $33{,}333 + 3{,}333 + 333 + 33 - 3$

Understand

Plan

Primary tool: #9 Solve an Easier Related Problem

Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities

A five-digit credit plus three more credits and one debit looks like a heavy column-subtraction problem, but the only column that affects our answer is the ones column. Replacing each multi-digit entry with just its units digit collapses the whole ledger to single-digit arithmetic (Tool #9). The shared 'every entry ends in 3' fingerprint is the pattern (Tool #5) that makes the reduction clean, and Tool #3 lets us cross-check against the choices at the end.

Execute — Answer: E

#9 Solve an Easier Related Problem 4.NBT.A.2 Step 1

Every ledger entry — $33{,}333$, $3{,}333$, $333$, $33$, and $3$ — ends in the digit 3.
The units digit of a sum or difference depends only on the ones column of each term, so we may ignore the tens, hundreds, thousands, and ten-thousands columns entirely.
This is the easier-related-problem move: five multi-digit numbers shrink to five copies of the digit 3.

$$33{,}333 \to 3,\ 3{,}333 \to 3,\ 333 \to 3,\ 33 \to 3,\ 3 \to 3$$

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

#5 Look for a Pattern 2.OA.B.2 Step 2

Add the four credit units digits and then subtract the one debit units digit, just as the original signs prescribe: $3 + 3 + 3 + 3 - 3$.
Compute left to right: $3 + 3 = 6$, $6 + 3 = 9$, $9 + 3 = 12$, $12 - 3 = 9$.
So the ones column of the net balance is 9.

$$3 + 3 + 3 + 3 - 3 = 9$$

💡 Adding and subtracting four small numbers under 20 is Grade 2 mental arithmetic — the repeated 3s make the pattern obvious.

#9 Solve an Easier Related Problem 4.NBT.B.4 Step 3

Confirm that no carry or borrow from the tens column muddies the ones place.
In the actual sum the four credits' ones columns add to $12$, which puts a 2 in the ones place and carries a 1 — but then the debit subtracts a 3 from a tens column that has plenty to spare, so no borrow reaches back into the ones column.
Either way the ones digit of the final balance is the value we just computed, namely 9.

$$\text{ones column of net balance} = (3+3+3+3-3) \bmod 10 = 9$$

💡 Grade 4 multi-digit add-subtract reasoning lets us check that no carry/borrow leaks across columns once the signs are settled.

#3 Eliminate Possibilities 2.NBT.A.1 Step 4

Match against the five offered digits.
The candidates are 0, 3, 5, 7, 9.
Our ones digit is 9, which is exactly choice (E).
The other digits — 0, 3, 5, 7 — never arise from $3+3+3+3-3$ in the ones column, so they are ruled out.

$$\text{Units digit} = 9 \;\Rightarrow\; \textbf{(E)}$$

💡 Reading off the ones place to pick a multiple-choice option is Grade 2 place-value work.

[1] #9 4.NBT.A.2 Every ledger entry — $33{,}333$, $3{,}333$, $333$, $33$, and $3$ — ends in the d

[2] #5 2.OA.B.2 Add the four credit units digits and then subtract the one debit units digit, ju

[3] #9 4.NBT.B.4 Confirm that no carry or borrow from the tens column muddies the ones place. In

[4] #3 2.NBT.A.1 Match against the five offered digits. The candidates are 0, 3, 5, 7, 9. Our one

Review

Reasonableness: Sanity check by computing the full balance: $33{,}333 + 3{,}333 = 36{,}666$; $36{,}666 + 333 = 36{,}999$; $36{,}999 + 33 = 37{,}032$; $37{,}032 - 3 = 37{,}029$. The ones digit of $37{,}029$ is indeed 9, so the units-digit shortcut agrees with the full ledger arithmetic and the answer (E) 9 is correct.

Alternative: An equivalent route is Tool #13 (Convert to Algebra) using modular arithmetic mod 10. Each entry is $\equiv 3 \pmod{10}$, so the net balance is $3 + 3 + 3 + 3 - 3 \equiv 9 \pmod{10}$. Same digit, but for a young solver the easier-problem move reads more naturally than naming the modulus.

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 net balance to match it against the offered answer choices.)
2.OA.B.2 Fluently add and subtract within 20 using mental strategies (Combining the five units-digit terms $3 + 3 + 3 + 3 - 3 = 9$ in one's head.)
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Recognizing that each ledger entry — up to $33{,}333$ — has units digit 3.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Checking that no carry or borrow from the tens column reaches into the ones column once the four credits and one debit are combined.)

⭐ This AMC 8 problem only needs Grade 4 multi-digit place-value sense you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

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