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Difference between revisions of "2000 AMC 8 Problems/Problem 3"

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==Problem==
  
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'''(D)'''
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How many whole numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi</math>?
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The smallest whole number in the interval is <math>2</math> because 5/3 is more than 1 but less than 2. The largest whole number in the interval is 6 because <math>2\pi</math> is more than 6 but less than 7. There are five whole numbers in the interval. They are 2, 3, 4, 5, and 6.
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<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ \text{infinitely many}</math>
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==Solution 1 ==
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The smallest whole number in the interval is <math>2</math> because <math>5/3</math> is more than <math>1</math> but less than <math>2</math>. The largest whole number in the interval is <math>6</math> because <math>2\pi</math> is more than <math>6</math> but less than <math>7</math>. There are five whole numbers in the interval. They are <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, and <math>6</math>, so the answer is <math>\boxed{\text{(D)}\ 5}</math>.
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==Solution 2==
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We can approximate <math>2\pi</math> to <math>6</math>. Now we approximate <math>5/3</math> to <math>2</math>. Now we list the integers between <math>2</math> and <math>6</math> including <math>2</math> and <math>6</math>:
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<cmath>2,3,4,5,6</cmath>
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Hence, the answer is <math>\boxed{D}</math>
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==See Also==
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{{AMC8 box|year=2000|num-b=2|num-a=4}}
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{{MAA Notice}}

Latest revision as of 08:37, 24 July 2024

Problem

How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$?

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ \text{infinitely many}$

Solution 1

The smallest whole number in the interval is $2$ because $5/3$ is more than $1$ but less than $2$. The largest whole number in the interval is $6$ because $2\pi$ is more than $6$ but less than $7$. There are five whole numbers in the interval. They are $2$, $3$, $4$, $5$, and $6$, so the answer is $\boxed{\text{(D)}\ 5}$.

Solution 2

We can approximate $2\pi$ to $6$. Now we approximate $5/3$ to $2$. Now we list the integers between $2$ and $6$ including $2$ and $6$: \[2,3,4,5,6\] Hence, the answer is $\boxed{D}$

See Also

2000 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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