Competition · AMC preparation · step 4 of 4
AMC 8 · 2011 · #21
Grade 4 number-theoryPick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Three clean conditions slice through a finite list — perfect setup for Tool #3 (Eliminate Possibilities). Tool #2 (Systematic List) builds the short candidate set from the "off by one" clue: scan the guess list for pairs that differ by exactly 2, and the age sits squarely in the middle of each such pair. Then the "at least half too low" condition kills the candidates that are too small, and the "prime" condition finishes off whatever remains. No algebra, just three filters.
Turn the clue into a bound
Half of 10 is 5, so the five smallest guesses (24, 28, 30, 32, 36) must all be too low — Norb's age is greater than 36.
Computing half of 10 and counting the first five values is Grade 3 multiplication/division reasoning.
3.OA.A.3Eliminate PossibilitiesList the off-by-one ages
Both x-1 and x+1 must be guesses, so scan for pairs differing by 2: (28,30), (30,32), (36,38), (47,49) give candidates 29, 31, 37, 48.
Listing every "x-1 and x+1 both in the list" candidate turns the clue into a finite checklist — the core move of Tool #2.
Norb's age must be one more than the smaller number in a pair of guesses that are exactly 2 apart.
▸ Why?
The clue says one guess lands one below his age and another lands one above it, so the distance from the lower guess up to the higher guess is one step plus one step, a gap of 2.
▸ Why?
That gap of 2 is just the below-part and the above-part laid end to end with no overlap and nothing left out, so the two single steps add up to the whole gap.
▸ Why?
The lower guess is his age with one taken away, so putting that one back gives his age exactly, which is why the age is pinned right between the pair.
▸ Why?
Taking one away made the lower guess, and adding one back reverses that removal to return to the very number you started from.
Combine the two clues
Apply the 'more than 36' filter to the candidates: 29 and 31 drop out, leaving 37 and 48.
Comparing multi-digit whole numbers to a threshold is Grade 4 place-value reasoning.
4.NBT.A.2Eliminate PossibilitiesTest which one is prime
37 is prime, but 48 = 2 × 24 is composite, so only 37 survives the prime clue.
Identifying a number as prime or composite within 100 is exactly the Grade 4 factor-pair standard.
4.OA.B.4Eliminate PossibilitiesRead off the age
Norb is 37 years old, which is choice (C).
After three filters leave one candidate, that candidate is forced — the cleanest finish for Tool #3.
4.OA.A.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 prime-or-composite checking and a couple of "is it bigger than 36?" comparisons you already know!
- Turn the clue into a bound
- List the off-by-one ages
- Combine the two clues
- Test which one is prime
- Read off the age
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