Difference between revisions of "2015 AMC 12A Problems/Problem 15"
(→Solution 3) |
|||
(5 intermediate revisions by 3 users not shown) | |||
Line 5: | Line 5: | ||
<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math> | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | 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 = | + | 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 = \boxed{\textbf{(C)}\ 26}</math> |
+ | ==Solution 2== | ||
+ | Multiply the numerator and denominator of the fraction by <math>5^{22}</math> (which is the same as multiplying by 1) to give <math>\frac{5^{22} \cdot 123456789}{10^{26}}</math>. Now, instead of thinking about this as a fraction, think of it as the division calculation <math>(5^{22} \cdot 123456789) \div 10^{26}</math> . The dividend is a huge number, but we know it doesn't have any digits to the right of the decimal point. Also, the dividend is not a multiple of 10 (it's not a multiple of 2), so these 26 divisions by 10 will each shift the entire dividend one digit to the right of the decimal point. Thus, | ||
+ | <math>\boxed{\textbf{(C)}\ 26}</math> is the minimum number of digits to the right of the decimal point needed. | ||
− | == | + | ==Solution 3== |
− | + | The denominator is <math>10^4 \cdot 2^{22}</math>. Each <math>10</math> adds one digit to the right of the decimal, and each additional <math>2</math> adds another digit. The answer is <math>4 + 22 = \boxed{\textbf{(C)}\ 26}</math>. | |
+ | ==Solution 4 (for those like myself who couldn't think of something smarter to do)== | ||
+ | First, we can write the denominator as <math>2^{22}\cdot 10^4</math> and forget about the <math>10^4</math> (however, we will need to add <math>4</math> back to our final answer). Noticing that <math>2^{22}=\left(2^{11} \right)^2=2048^2,</math> we divide 123456789 by 2048 using long division. We get 60281.63525390625 as the result (though the process seems intimidating, it actually doesn't take that long, just a couple of minutes). From there, we notice that there are 11 places after the decimal point in the quotient, which means there will be another 11 after we divide this quotient by 2048 again. So, there will be <math>22</math> places after the decimal in the final quotient. We add back <math>4</math> for a total of <math>26.</math> | ||
== 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}} |
Latest revision as of 15:53, 23 June 2022
Contents
Problem
What is the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal?
Solution 1
We can rewrite the fraction as . Since the last digit of the numerator is odd, a
is added to the right if the numerator is divided by
, and this will continuously happen because
, itself, is odd. Indeed, this happens twenty-two times since we divide by
twenty-two times, so we will need
more digits. Hence, the answer is
Solution 2
Multiply the numerator and denominator of the fraction by (which is the same as multiplying by 1) to give
. Now, instead of thinking about this as a fraction, think of it as the division calculation
. The dividend is a huge number, but we know it doesn't have any digits to the right of the decimal point. Also, the dividend is not a multiple of 10 (it's not a multiple of 2), so these 26 divisions by 10 will each shift the entire dividend one digit to the right of the decimal point. Thus,
is the minimum number of digits to the right of the decimal point needed.
Solution 3
The denominator is . Each
adds one digit to the right of the decimal, and each additional
adds another digit. The answer is
.
Solution 4 (for those like myself who couldn't think of something smarter to do)
First, we can write the denominator as and forget about the
(however, we will need to add
back to our final answer). Noticing that
we divide 123456789 by 2048 using long division. We get 60281.63525390625 as the result (though the process seems intimidating, it actually doesn't take that long, just a couple of minutes). From there, we notice that there are 11 places after the decimal point in the quotient, which means there will be another 11 after we divide this quotient by 2048 again. So, there will be
places after the decimal in the final quotient. We add back
for a total of
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 |