Difference between revisions of "2003 AMC 10A Problems/Problem 20"
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| − | == | + | == Problem 20 == |
| − | A base-10 three digit number <math>n</math> is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of <math>n</math> are both | + | A base-10 three digit number <math>n</math> is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of <math>n</math> are both three-digit numerals? |
<math> \mathrm{(A) \ } 0.3\qquad \mathrm{(B) \ } 0.4\qquad \mathrm{(C) \ } 0.5\qquad \mathrm{(D) \ } 0.6\qquad \mathrm{(E) \ } 0.7 </math> | <math> \mathrm{(A) \ } 0.3\qquad \mathrm{(B) \ } 0.4\qquad \mathrm{(C) \ } 0.5\qquad \mathrm{(D) \ } 0.6\qquad \mathrm{(E) \ } 0.7 </math> | ||
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Thus, there are <math>608</math> base-10 three-digit numbers that are three digit numbers in base-9 and base-11. | Thus, there are <math>608</math> base-10 three-digit numbers that are three digit numbers in base-9 and base-11. | ||
| − | Therefore the desired probability is <math>\frac{608}{900}\approx 0.7 \Rightarrow E</math>. | + | Therefore the desired probability is <math>\frac{608}{900}\approx 0.7 \Rightarrow\boxed{\mathrm{(E)}\ 0.7}</math>. |
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| + | == Video Solution by OmegaLearn == | ||
| + | https://youtu.be/SCGzEOOICr4?t=596 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/YaV5oanhAlU | ||
| + | |||
| + | ~IceMatrix | ||
| + | |||
| + | ==Video Solution by WhyMath== | ||
| + | https://youtu.be/-ei6Ni-jnlc | ||
| + | |||
| + | ~savannahsolver | ||
== See Also == | == See Also == | ||
| − | + | {{AMC10 box|year=2003|ab=A|num-b=19|num-a=21}} | |
| − | |||
| − | |||
[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 23:04, 31 July 2023
Contents
Problem 20
A base-10 three digit number 
is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of are both three-digit numerals?
Solution
To be a three digit number in base-10:
Thus there are 
three-digit numbers in base-10
To be a three-digit number in base-9:
To be a three-digit number in base-11:
So, 
Thus, there are 
base-10 three-digit numbers that are three digit numbers in base-9 and base-11.
Therefore the desired probability is 
.
Video Solution by OmegaLearn
https://youtu.be/SCGzEOOICr4?t=596
~ pi_is_3.14
Video Solution
~IceMatrix
Video Solution by WhyMath
~savannahsolver
See Also
| 2003 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 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 AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
