AMC 8 · 2000 · #3
Grade 6 arithmeticProblem
How many whole numbers lie in the interval between and ?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Count the whole numbers $n$ that lie strictly between $\tfrac{5}{3}$ and $2\pi$.
Givens: Lower bound: $\tfrac{5}{3}$; Upper bound: $2\pi$; $\pi \approx 3.14159$; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) infinitely many
Unknowns: The number of whole numbers in the open interval $\left(\tfrac{5}{3},\, 2\pi\right)$
Understand
Restated: Count the whole numbers $n$ that lie strictly between $\tfrac{5}{3}$ and $2\pi$.
Givens: Lower bound: $\tfrac{5}{3}$; Upper bound: $2\pi$; $\pi \approx 3.14159$; Answer choices: (A) $2$, (B) $3$, (C) $4$, (D) $5$, (E) infinitely many
Plan
Primary tool: #2 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.
Execute — Answer: D
5.NF.B.3 Step 1 - Convert the lower bound to a decimal.
- Reading $\tfrac{5}{3}$ as $5 \div 3$ gives a value between $1$ and $2$.
💡 Grade 5 "fraction as division": $\tfrac{5}{3}$ means $5$ split into $3$ equal parts, which is just over $1.6$.
5.NBT.B.7 Step 2 - Convert the upper bound to a decimal.
- Using $\pi \approx 3.14159$, double it.
💡 Grade 5 decimal multiplication: doubling $3.14$ lands a little past $6.28$, between $6$ and $7$.
6.NS.C.7 Step 3 - List the whole numbers strictly between $1.667$ and $6.283$.
- The smallest whole number above $1.667$ is $2$; the largest below $6.283$ is $6$.
- Every integer in between also fits.
💡 Grade 6 ordering on the number line: walk from left to right, keep each integer that lies inside both endpoints, then count.
5.NF.B.3 Convert the lower bound to a decimal. Reading $\tfrac{5}{3}$ as $5 \div 3$ gives 5.NBT.B.7 Convert the upper bound to a decimal. Using $\pi \approx 3.14159$, double it. 6.NS.C.7 List the whole numbers strictly between $1.667$ and $6.283$. The smallest whole Review
Reasonableness: Sketch a number line: mark $\tfrac{5}{3} \approx 1.67$ (just past $1$) and $2\pi \approx 6.28$ (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 $6.28 - 1.67 \approx 4.6$, which comfortably holds $5$ integers, matching answer (D).
Alternative: Tool #1 (Draw a Diagram): plot $\tfrac{5}{3}$ and $2\pi$ on a number line and shade the segment between them. The integer tick marks inside the shaded region are $2, 3, 4, 5, 6$ — five of them. The picture rules out (E) at a glance because the interval is finite, and it shows that $1$ and $7$ fall outside the shading.
CCSS standards used (min grade 6)
5.NF.B.3Interpret a fraction as division of the numerator by the denominator (Reading $\tfrac{5}{3}$ as $5 \div 3 \approx 1.667$ so the lower bound can be compared to integers.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Doubling $\pi \approx 3.14159$ to get $2\pi \approx 6.28318$ for the upper bound.)6.NS.C.7Understand ordering and absolute value of rational numbers (Comparing each candidate whole number against the two decimal endpoints, listing the ones that fit, and counting them.)
⭐ Turn the messy endpoints $\tfrac{5}{3}$ and $2\pi$ 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).
⭐ Turn the messy endpoints $\tfrac{5}{3}$ and $2\pi$ 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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