Difference between revisions of "2015 AMC 12A Problems/Problem 15"
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==Solution 1== | ==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== | ==Solution 2== |
Revision as of 14:47, 18 August 2017
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.
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 |