Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #24
Grade 4 logicPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Cryptarithms are pure place-value puzzles. Tool #3 (Write an Equation) turns each column of the addition and subtraction into a small equation in the unknown digits (with carry/borrow bits). Tool #7 (Identify Subproblems) tells us to work column by column — start with the ones column, which forces B, then move to the tens column, which forces A, C, and finally D — instead of trying to guess all four digits at once.
Read the addition ones column
Ones column of the addition: B + A ends in A, so B = 0 — and no carry into the tens column.
Adding A to something and getting back A in the ones place is exactly the Grade 4 "add multi-digit whole numbers using place value" idea — the only digit that vanishes under addition is 0.
4.NBT.B.4Eliminate PossibilitiesRead the subtraction ones column
Subtraction ones column: with B = 0, the 0 - A needs a borrow, so 10 - A = A and A = 5.
Borrowing in column subtraction is the standard Grade 4 algorithm; the borrow turns the impossible "0 - A = A" into the solvable "10 - A = A".
The digit A must equal 5.
▸ Why?
The ones column of the subtraction, B - A, has to end in the digit A, and since B = 0 that column reads 0 - A; taking A away from nothing forces a borrow of one ten, so the column really computes 10 - A.
▸ Why?
A subtraction can be worked one column at a time because each place keeps its own count, so the ones digit of the answer depends only on the ones digits, apart from a borrow.
▸ Why?
You cannot remove A ones from just 0 ones, so you unbundle one ten from the tens place into ten ones — one ten is exactly ten ones — which makes the ones column 10 - A.
▸ Why?
That borrowed column 10 - A is required to equal A, which means A and another A together rebuild the whole ten, so A is exactly half of ten.
▸ Why?
Moving the subtracted A to the other side — addition undoes subtraction — turns 10 - A = A into A + A = 10.
▸ Why?
The part taken away and the part left over, both equal to A, must add back with no overlap to the whole ten they came from, so the two equal parts split ten evenly.
Use the subtraction tens column
Subtraction tens column: the borrow took 1 from the tens, so (A - 1) - C = 0 and C = 4.
Tool #7: once the ones column is solved, the tens column becomes its own small subproblem with one unknown.
4.NBT.B.4Identify SubproblemsUse the addition tens column
Addition tens column: no carry from the ones, so D = A + C = 5 + 4 = 9.
With A and C pinned down, the tens column of the addition reads off D directly.
4.NBT.B.4Eliminate PossibilitiesThis AMC 8 problem only needs the Grade 4 standard column-by-column add/subtract algorithm — once you read each column carefully, the digits fall out one at a time.
- Read the addition ones column
- Read the subtraction ones column
- Use the subtraction tens column
- Use the addition tens column
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