Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #3
Grade 6 arithmeticPick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Both endpoints are real numbers near small whole numbers, so the question reduces to: which whole numbers fit between them? Tool #2 (Make a Systematic List) is the cleanest path — convert each endpoint to a decimal, find the smallest whole number above the lower bound and the largest whole number below the upper bound, then list everything in between and count. No algebra is needed once the endpoints are pinned down.
Convert the lower bound
Read as 5 ÷ 3: the lower bound is about 1.667, just past 1.
Grade 5 "fraction as division": 5/3 means 5 split into 3 equal parts, which is just over 1.6.
5.NF.B.3Make A Systematic ListConvert the upper bound
Double π ≈ 3.14159: the upper bound 2π is about 6.283, just past 6.
Grade 5 decimal multiplication: doubling 3.14 lands a little past 6.28, between 6 and 7.
5.NBT.B.7Make A Systematic ListCount the whole numbers between
Between 1.667 and 6.283 sit the integers 2, 3, 4, 5, 6 — five values, so (D).
Grade 6 ordering on the number line: walk from left to right, keep each integer that lies inside both endpoints, then count.
The whole numbers that fit strictly between 5/3 and 2π form an unbroken run from 2 at the low end to 6 at the high end.
▸ Why?
The low end 5/3 lands between 1 and 2, so 2 is the smallest whole number that clears it while 1 and 0 fall short.
▸ Why?
5/3 is 5 shared into 3 equal parts, and since 3 fits into 5 once (3) but not twice (6), the value sits between 1 and 2.
▸ Why?
The high end 2π lands between 6 and 7, so 6 is the largest whole number still under it while 7 spills over.
▸ Why?
2π is two whole copies of π, and with π ≈ 3.14 that is 3.14 + 3.14 = 6.28 — past 6 but short of 7.
▸ Why?
Any whole number caught between 2 and 6 automatically clears the low end and stays under the high end, so no value inside the run is skipped.
▸ Why?
A number above 2 is above 5/3 because 2 itself already beats 5/3, and a number below 6 is below 2π because 6 itself already sits under 2π — each comparison passes down the chain.
Turn the messy endpoints and 2π into decimals, then list every whole number that fits between 1.67 and 6.28 — the five values 2, 3, 4, 5, 6 give answer (D).
- Convert the lower bound
- Convert the upper bound
- Count the whole numbers between
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