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2003 AMC 10A Problems/Problem 20: Difference between revisions

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<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>
== Solution ==
To be a three digit number in base-10: 
<math>10^{2} \leq n \leq 10^{3}-1</math>
<math>100 \leq n \leq 999</math>
Thus there are <math>900</math> three-digit numbers in base-10
To be a three-digit number in base-9: 
<math>9^{2} \leq n \leq 9^{3}-1</math>
<math>81 \leq n \leq 728</math>
To be a three-digit number in base-11: 
<math>11^{2} \leq n \leq 11^{3}-1</math>
<math>121 \leq n \leq 1330</math>
So, <math>121 \leq n \leq 728</math>
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\boxed{\mathrm{(E)}\ 0.7}</math>.
== 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 ==

Latest revision as of 23:04, 31 July 2023

Problem 20

A base-10 three digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?

$\mathrm{(A) \ } 0.3\qquad \mathrm{(B) \ } 0.4\qquad \mathrm{(C) \ } 0.5\qquad \mathrm{(D) \ } 0.6\qquad \mathrm{(E) \ } 0.7$

Solution

To be a three digit number in base-10:

$10^{2} \leq n \leq 10^{3}-1$

$100 \leq n \leq 999$

Thus there are $900$ three-digit numbers in base-10

To be a three-digit number in base-9:

$9^{2} \leq n \leq 9^{3}-1$

$81 \leq n \leq 728$

To be a three-digit number in base-11:

$11^{2} \leq n \leq 11^{3}-1$

$121 \leq n \leq 1330$

So, $121 \leq n \leq 728$

Thus, there are $608$ base-10 three-digit numbers that are three digit numbers in base-9 and base-11.

Therefore the desired probability is $\frac{608}{900}\approx 0.7 \Rightarrow\boxed{\mathrm{(E)}\ 0.7}$.

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

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

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