AMC 8 · 2000 · #3

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

interval-arithmeticfraction-decimal-conversionestimation bound-inequality-then-enumeratesystematic-enumeration ↑ Prerequisites: fraction-decimal-conversionmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Count the whole numbers n that lie strictly between

\frac{5}{3}

and 2π.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

infinitely many

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

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.

1STEP 1

Read $\frac{5}{3}$ as 5 ÷ 3: the lower bound is about 1.667, just past 1.

\frac{5}{3}

= 5 ÷ 3 ≈ 1.667

2STEP 2

Double π ≈ 3.14159: the upper bound 2π is about 6.283, just past 6.

2π ≈ 2 × 3.14159 = 6.28318

3STEP 3

Between 1.667 and 6.283 sit the integers 2, 3, 4, 5, 6 — five values, so (D).

{2, 3, 4, 5, 6} → 5 values → (D)

Answer

Sketch a number line: mark

\frac{5}{3}

≈ 1.67 (just past 1) and 2π ≈ 6.28 (just past 6). The integers 2, 3, 4, 5, 6 all sit inside; 1 is to the left of 1.67, and 7 is to the right of 6.28, so both are excluded. The width of the interval is about 6.28 - 1.67 ≈ 4.6, which comfortably holds 5 integers, matching answer (D).

💡Key takeaway

Turn the messy endpoints $\frac{5}{3}$ 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).

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