Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #13
Grade 4 algebranumber-theoryPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question "how many values of k work?" begs for a Tool #2 systematic list — pick an ordering for b (the smaller of the two numbers) and list every b that produces a valid positive k. To set up the list cleanly we first use Tool #7 (Subproblems): combine the two facts (a = 2b + k and a + b = 28) into a single arithmetic relationship 3b + k = 28, so k = 28 - 3b. Then Tool #6 (Guess and Check) lets us walk b = 1, 2, 3, … and stop the moment k would drop below 1. This avoids reaching for Tool #13 (algebra) when ordinary subtraction and a list do the job.
Turn the sentences into equations
"a is k more than twice b" means a = 2b + k; substitute into a + b = 28 to get 3b + k = 28.
"Twice another" is a multiplicative comparison — a Grade 4 word-problem move — and we just add the second number to both sides of the count.
4.OA.A.2Identify SubproblemsExpress the extra by itself
Solve for k: k = 28 - 3b. Now ask which positive integers b keep 28 - 3b positive.
Turning a two-condition problem into a single subtraction is the Tool #7 "break it down" move — pure Grade 4 multi-step arithmetic.
4.OA.A.3Identify SubproblemsFind the largest smaller number
k ≥ 1 needs 28 - 3b ≥ 1, so 3b ≤ 27 and b ≤ 9; b runs over the whole numbers 1 to 9.
Asking "how big can b be before k runs out?" is Tool #6 (Guess and Check) on the boundary; the test reduces to 27 ÷ 3 = 9, a Grade 4 fact.
4.OA.A.3Guess And CheckList every valid pair
List (b, k) for b = 1 to 9: k = 25, 22, 19, …, 1, and a = 28 - b stays positive throughout.
Generating the table by the rule "subtract 3 from k every time b goes up by 1" is exactly the Grade 4 pattern-from-a-rule standard.
4.OA.C.5Make A Systematic ListCount the distinct values
Count the distinct k values 25, 22, 19, 16, 13, 10, 7, 4, 1 — that is 9 different numbers, choice (D).
Counting the entries of a finished list is a Grade 3 two-step word-problem skill — no fancier tools needed.
Let b be the smaller of the two numbers and let the blank be k. How many positive integers can fill the blank comes down to two facts: each allowed b gives exactly one k with no repeats, and b can be any whole number in the unbroken run from 1 up to 9.
▸ Why?
Matching each allowed b against its k, the rule k = 28 - 3b sends each b to exactly one k, and down the list those values 25, 22, 19, … step down without ever repeating, so the b-list and the k-list pair off one for one with nothing left over.
▸ Why?
Since b is a positive whole number it is at least 1, and it can be no larger than 9, because a bigger b would drag k = 28 - 3b below 1, which a positive k forbids.
▸ Why?
The relation k = 28 - 3b holds because the total 28 splits with no overlap into b plus the bigger number, and the bigger number is itself 2b + k, so 28 = b + (2b + k) = 3b + k.
▸ Why?
So the largest allowed b is 9: keeping k ≥ 1 forces 3b ≤ 27, and undoing the times-three by dividing 27 by 3 gives 9.
This AMC 8 problem only needs Grade 4 "twice as many" thinking and a careful list you already know how to make!
- Turn the sentences into equations
- Express the extra by itself
- Find the largest smaller number
- List every valid pair
- Count the distinct values
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