Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #1
Grade 4 arithmeticPick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The numbers look scary (six digits!), but because we only need the last digit, we can replace each big number with just its units digit. That turns the whole problem into single-digit arithmetic — a much easier related problem (Tool #9). The pattern of "every number ends in 2" (Tool #5) makes the shortcut leap out, and since this is multiple-choice, Tool #3 lets us verify by elimination at the end.
Look only at the units digits
Every number here — 222,222 down to 2 — ends in the digit 2, so keep only each ones digit and drop the rest.
Reading a multi-digit number and naming its units digit is exactly the place-value skill from Grade 4.
The ones digit of the whole expression is fixed by the ones digits of the six numbers alone.
▸ Why?
Each number splits into a chunk of whole tens plus its ones digit — everything above the ones place is a bundle of tens.
▸ Why?
Those whole-tens chunks carry a 0 in the ones place, so adding or subtracting them never disturbs the ones digit of the answer.
▸ Why?
Bundling ten of a unit up into the next place leaves the ones place empty, so every multiple of ten ends in 0.
▸ Why?
A part that holds 0 in the ones place adds nothing there, so it leaves that digit unchanged.
▸ Why?
That leaves only the ones digits to combine, and we may gather them together first because reordering and regrouping the terms does not change the total.
▸ Why?
The order in which we add numbers does not change their total.
▸ Why?
How we group the numbers being added does not change their total.
Add the five units digits
Add the five subtracted ones digits: 2+2+2+2+2 = 10, so together they end in 0.
Adding five small numbers within 20 is fluent mental math from Grade 2.
2.OA.B.2Look For A PatternSubtract in the ones column
In the ones column the whole thing acts like 2 - 0 = 2; the leading 2 stays and no borrow occurs.
Subtracting multi-digit numbers column-by-column (Grade 4) lets us isolate the ones column safely.
4.NBT.B.4Solve An Easier Related ProblemMatch against the choices
Match against the choices: our ones digit 2 is choice (B); 10 isn't a single digit, and 0, 4, 8 can't come from 2 - 0.
Identifying the ones place of a number to match it to a multiple-choice option uses place value from Grade 2.
2.NBT.A.1Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 multi-digit place-value sense you already know!
- Look only at the units digits
- Add the five units digits
- Subtract in the ones column
- Match against the choices
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