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Difference between revisions of "2015 AMC 12A Problems/Problem 15"

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==Solution==
 
==Solution==
 
We can rewrite the fraction as <math>\frac{123456789}{2^{22} \cdot 10^4} = \frac{12345.6789}{2^{22}}</math>. Since the last digit of the numerator is odd, a <math>5</math> is added to the right if the numerator is divided by <math>2</math>, and this will continuously happen because <math>5</math>, itself, is odd. Indeed, this happens twenty-two times since we divide by <math>2</math> twenty-two times, so we will need <math>22</math> more digits. Hence, the answer is <math>4 + 22 = 26 \textbf{ (C)}</math>.
 
We can rewrite the fraction as <math>\frac{123456789}{2^{22} \cdot 10^4} = \frac{12345.6789}{2^{22}}</math>. Since the last digit of the numerator is odd, a <math>5</math> is added to the right if the numerator is divided by <math>2</math>, and this will continuously happen because <math>5</math>, itself, is odd. Indeed, this happens twenty-two times since we divide by <math>2</math> twenty-two times, so we will need <math>22</math> more digits. Hence, the answer is <math>4 + 22 = 26 \textbf{ (C)}</math>.
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==Alternate Solution==
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Note that <math>123456789</math> is not a multiple of <math>2</math> or <math>5</math>, and therefore shares no factors with the original denominator. Multiple the numerator and denominator of the fraction by <math>5^{22}</math> to give <math>\frac{5^{22} \cdot 123456789}{10^{26}}</math>. This fraction will require <math>26</math> divisions by ten to write as a decimal, and since the original fraction is less than <math>1</math> all of the digits will be to the right of the decimal point. Answer: <math>\textbf{ (C)}</math>
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== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2015|ab=A|num-b=14|num-a=16}}
 
{{AMC12 box|year=2015|ab=A|num-b=14|num-a=16}}

Revision as of 12:17, 5 February 2015

Problem

What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^4}$ as a decimal?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}}\ 30\qquad\textbf{(E)}\ 104$ (Error compiling LaTeX. Unknown error_msg)

Solution

We can rewrite the fraction as $\frac{123456789}{2^{22} \cdot 10^4} = \frac{12345.6789}{2^{22}}$. Since the last digit of the numerator is odd, a $5$ is added to the right if the numerator is divided by $2$, and this will continuously happen because $5$, itself, is odd. Indeed, this happens twenty-two times since we divide by $2$ twenty-two times, so we will need $22$ more digits. Hence, the answer is $4 + 22 = 26 \textbf{ (C)}$.


Alternate Solution

Note that $123456789$ is not a multiple of $2$ or $5$, and therefore shares no factors with the original denominator. Multiple the numerator and denominator of the fraction by $5^{22}$ to give $\frac{5^{22} \cdot 123456789}{10^{26}}$. This fraction will require $26$ divisions by ten to write as a decimal, and since the original fraction is less than $1$ all of the digits will be to the right of the decimal point. Answer: $\textbf{ (C)}$


See Also

2015 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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
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