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Difference between revisions of "2003 AMC 10A Problems/Problem 20"

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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>
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== Solution ==
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The smallest base-11 number that has 3 digits in base-10 is <math>100_{11}</math> which is <math>121_{10}</math>.
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The largest number in base-9 that has 3 digits in base-10 is <math>8\cdot9^2+8\cdot9^1+8\cdot9^0=888_{9}=728_{10}</math>
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The smallest number in base-9 that has 3 digits in base-10 is <math>1\cdot9^2+2\cdot9^1+1\cdot9^0=121_{9}=100_{10}</math>
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Hence, all numbers that will have 3 digits in base-9, 10, and 11 will be between <math>728_{10} and </math>121_{10}, thus the total amount of numbers that will have 3 digits in base-9, 10, and 11 is <math>728-121+1=608</math>
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There are 900 possible 3 digit numbers in base 10.
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Hence, the answer is <math>\frac{608}{900}\approx .675 \approx \boxed{0.7}</math>
  
 
== See Also ==
 
== See Also ==

Revision as of 15:31, 14 June 2019

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

The smallest base-11 number that has 3 digits in base-10 is $100_{11}$ which is $121_{10}$.

The largest number in base-9 that has 3 digits in base-10 is $8\cdot9^2+8\cdot9^1+8\cdot9^0=888_{9}=728_{10}$

The smallest number in base-9 that has 3 digits in base-10 is $1\cdot9^2+2\cdot9^1+1\cdot9^0=121_{9}=100_{10}$

Hence, all numbers that will have 3 digits in base-9, 10, and 11 will be between $728_{10} and$121_{10}, thus the total amount of numbers that will have 3 digits in base-9, 10, and 11 is $728-121+1=608$

There are 900 possible 3 digit numbers in base 10.

Hence, the answer is $\frac{608}{900}\approx .675 \approx \boxed{0.7}$

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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