Difference between revisions of "2007 AMC 8 Problems/Problem 25"
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− | + | ==Problem== | |
− | On the dart board shown in the | + | On the dart board shown in the figure below, the outer circle has radius <math>6</math> and the inner circle has radius <math>3</math>. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
− | + | <center><asy> | |
+ | draw(circle((0,0),6)); | ||
+ | draw(circle((0,0),3)); | ||
+ | draw((0,0)--(0,6)); | ||
+ | draw((0,0)--(rotate(120)*(0,6))); | ||
+ | draw((0,0)--(rotate(-120)*(0,6))); | ||
+ | label('1',(rotate(60)*(0,3/2))); | ||
+ | label('2',(rotate(-60)*(0,3/2))); | ||
+ | label('2',(0,-3/2)); | ||
+ | label('2',(rotate(60)*(0,9/2))); | ||
+ | label('1',(rotate(-60)*(0,9/2))); | ||
+ | label('1',(0,-9/2)); | ||
+ | </asy></center> | ||
<math>\mathrm{(A)} \frac{17}{36} \qquad \mathrm{(B)} \frac{35}{72} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{37}{72} \qquad \mathrm{(E)} \frac{19}{36}</math> | <math>\mathrm{(A)} \frac{17}{36} \qquad \mathrm{(B)} \frac{35}{72} \qquad \mathrm{(C)} \frac{1}{2} \qquad \mathrm{(D)} \frac{37}{72} \qquad \mathrm{(E)} \frac{19}{36}</math> | ||
− | + | ==Solution== | |
To get an odd sum, we must add an even number and an odd number. So we have a little casework to do. Before we do that, we also have to figure out some relative areas. You could either find the absolute areas (as is done below for completeness), or apply some proportional reasoning and symmetry to realize the relative areas (which is really all that matters). | To get an odd sum, we must add an even number and an odd number. So we have a little casework to do. Before we do that, we also have to figure out some relative areas. You could either find the absolute areas (as is done below for completeness), or apply some proportional reasoning and symmetry to realize the relative areas (which is really all that matters). | ||
− | + | The area of this entire smaller circle is <math>9\pi</math>. The area of each smaller section then must be <math>\frac{9\pi}{3}</math> or <math>3\pi</math>. The larger sections trisect a "ring" which is the difference of two circles, one with radius <math>3</math>, the other radius <math>6</math>. So, the area of the ring (''annulus'') is <math>36\pi - 9\pi</math> or <math>27\pi</math>. The area of each larger section must be <math>\frac{27\pi}{3}</math> or <math>9\pi</math>. Note that the area of the whole circle is <math>36\pi</math>. | |
One smaller section and two larger sections contain an odd number (that is, 1). So the probability of throwing an odd number is <math>3\pi + (2 \cdot 9\pi) = 21\pi</math>. Since the area of the whole circle is <math>36\pi</math>, the probability of getting an odd is <math>\frac{21\pi}{36\pi} = \frac{21}{36} = \frac{7}{12}</math>. | One smaller section and two larger sections contain an odd number (that is, 1). So the probability of throwing an odd number is <math>3\pi + (2 \cdot 9\pi) = 21\pi</math>. Since the area of the whole circle is <math>36\pi</math>, the probability of getting an odd is <math>\frac{21\pi}{36\pi} = \frac{21}{36} = \frac{7}{12}</math>. | ||
− | Since the remaining sections contain even numbers (that is, 2), the probability of throwing an | + | Since the remaining sections contain even numbers (that is, 2), the probability of throwing an even is the complement, or <math>1 - \frac{7}{12} = \frac{5}{12}</math>. |
Line 20: | Line 32: | ||
− | + | Case 1: Odd, then even. | |
Multiply the probabilities to get <math>\frac{7}{12} \cdot \frac{5}{12} = \frac{35}{144}</math>. | Multiply the probabilities to get <math>\frac{7}{12} \cdot \frac{5}{12} = \frac{35}{144}</math>. | ||
− | + | Case 2: Even, then odd. | |
Multiply the probabilities to get <math>\frac{5}{12} \cdot \frac{7}{12} = \frac{35}{144}</math>. Notice that this is the same. | Multiply the probabilities to get <math>\frac{5}{12} \cdot \frac{7}{12} = \frac{35}{144}</math>. Notice that this is the same. | ||
− | Thus, the total probability of an odd sum is <math>\frac{35}{144} \cdot \frac{2}{1} | + | Thus, the total probability of an odd sum is <math>\frac{35}{144} \cdot \frac{2}{1} = \boxed{\textbf{(B)} = \frac{35}{72}}</math>. |
+ | |||
+ | ==Video Solution 1== | ||
+ | https://youtu.be/7kLWbuoNvo8 | ||
+ | |||
+ | == Video Solution 2 == | ||
+ | https://youtu.be/tKsYSBdeVuw?t=8 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2007|num-b=24| after=Last problem}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:02, 8 July 2021
Problem
On the dart board shown in the figure below, the outer circle has radius and the inner circle has radius . Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?
Solution
To get an odd sum, we must add an even number and an odd number. So we have a little casework to do. Before we do that, we also have to figure out some relative areas. You could either find the absolute areas (as is done below for completeness), or apply some proportional reasoning and symmetry to realize the relative areas (which is really all that matters).
The area of this entire smaller circle is . The area of each smaller section then must be or . The larger sections trisect a "ring" which is the difference of two circles, one with radius , the other radius . So, the area of the ring (annulus) is or . The area of each larger section must be or . Note that the area of the whole circle is .
One smaller section and two larger sections contain an odd number (that is, 1). So the probability of throwing an odd number is . Since the area of the whole circle is , the probability of getting an odd is .
Since the remaining sections contain even numbers (that is, 2), the probability of throwing an even is the complement, or .
Now, the two cases: You could either get an odd then an even, or an even then an odd.
Case 1: Odd, then even.
Multiply the probabilities to get .
Case 2: Even, then odd.
Multiply the probabilities to get . Notice that this is the same.
Thus, the total probability of an odd sum is .
Video Solution 1
Video Solution 2
https://youtu.be/tKsYSBdeVuw?t=8
~ pi_is_3.14
See Also
2007 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last problem | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.