AMC 10 · 2013 · #3
Grade 4 countingPick an answer.
The question literally asks for a position in a list read from the end, so Tool #11 (Work Backwards) fits directly: start at 201 and count down until 53 is reached. Tool #7 (Identify Subproblems) supplies the one reusable fact this rests on — how many whole numbers lie in a stretch from one value to another, inclusive. Tool #5 (Look for a Pattern) gives a fast cross-check, since a forward position and a backward position in the same list always add to a fixed total.
Count-inclusive rule
The forward list confirms the counting rule.
Subtracting the endpoints counts the gaps between numbers; the +1 adds back the starting number itself so both ends are included.
Subtracting the endpoints counts the gaps, so one is added back to include the starting number itself.
▸ Why?
Each number in the block is one item and each gap is a space between two of them, so items outnumber gaps by one.
▸ Why?
The count of gaps is just the difference of the endpoints, wherever on the line the block sits.
Reframe the backward count
The backward list counts from the other end.
Reading the list from the top, 53's rank is just the size of the block of numbers from the top end 201 down to 53.
2.OA.A.1Work BackwardsCompute n
The same rule gives 149, choice (D).
The single multi-digit subtraction 201 - 53 does all the work; the +1 is the same include-both-ends correction as before.
4.NBT.B.4Work BackwardsTo find a spot in a backward list, count how many numbers sit from the top end down to yours: 201 - 53 + 1 = 149, using include-both-ends counting you already know.
- Count-inclusive rule
- Reframe the backward count
- Compute n