Difference between revisions of "2021 AMC 12A Problems/Problem 5"
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==Problem== | ==Problem== | ||
| − | + | When a student multiplied the number <math>66</math> by the repeating decimal <cmath>\underline{1}.\underline{a}\underline{b}\underline{a}\underline{b}...=\underline{1}.\overline{\underline{a}\underline{b}},</cmath> where <math>a</math> and <math>b</math> are digits, he did not notice the notation and just multiplied <math>66</math> times <math>\underline{1}.\underline{a}\underline{b}</math>. Later he found that his answer is <math>0.5</math> less than the correct answer. What is the <math>2</math>-digit number <math>\underline{a}\underline{b}?</math> | |
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| + | <math>\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75</math> | ||
==Solution== | ==Solution== | ||
Revision as of 14:00, 11 February 2021
Problem
When a student multiplied the number 
by the repeating decimal ![\[\underline{1}.\underline{a}\underline{b}\underline{a}\underline{b}...=\underline{1}.\overline{\underline{a}\underline{b}},\]](http://latex.artofproblemsolving.com/6/0/a/60a48ebc868080ee1035b06da5cf73954173c68c.png)
where 
and 
are digits, he did not notice the notation and just multiplied times 
. Later he found that his answer is 
less than the correct answer. What is the 
-digit number 
Solution
It is known that 
and 
. Let 
. We have that 
. Solving gives that 
so 
. ~aop2014
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=MUHja8TpKGw&t=359s
See also
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 4 |
Followed by Problem 6 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
