AMC 10 · 2015 · #3
Grade 4 algebraPick an answer.
We are not told whether 28 is the number repeated twice or the number repeated three times, so there are only two possible setups. Tool #3 (Eliminate Possibilities) tests each setup and throws out the one that cannot produce a whole number, leaving exactly one valid answer. Tool #4 (Introduce a Variable) names the unknown integer x so we can track what each case forces it to be. The decisive clue is that the other integer must come out whole — only one of the two cases survives that test.
Name the unknown, list the two cases
There are two possible cases.
There are only two ways to share the five slots between 28 and x, so checking both covers everything.
4.OA.A.3Introduce A VariableTest Case A: 28 three times
The first gives a clean whole 8.
If three of the numbers are pinned down, the leftover total has to be split evenly between the last two.
If three of the numbers are pinned down, the leftover total has to be split evenly between the last two.
▸ Why?
The whole total is its five numbers added together, so removing three leaves exactly the other two.
▸ Why?
The two remaining numbers are equal, so each is that leftover total shared over two.
Test Case B, eliminate it, read the answer
The second fails to give a whole number, choice (A).
If splitting the leftover into equal whole pieces is impossible, that case can't happen.
4.NBT.B.6Eliminate PossibilitiesWhen you don't know which slot a number fills, try every option and keep only the one that comes out as a whole number.
- Name the unknown, list the two cases
- Test Case A: 28 three times
- Test Case B, eliminate it, read the answer