Difference between revisions of "2003 AMC 10B Problems/Problem 10"
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==Problem== | ==Problem== | ||
| − | Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of | + | Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? |
<math>\textbf{(A) } \frac{26}{10} \qquad\textbf{(B) } \frac{26^2}{10^2} \qquad\textbf{(C) } \frac{26^2}{10} \qquad\textbf{(D) } \frac{26^3}{10^3} \qquad\textbf{(E) } \frac{26^3}{10^2} </math> | <math>\textbf{(A) } \frac{26}{10} \qquad\textbf{(B) } \frac{26^2}{10^2} \qquad\textbf{(C) } \frac{26^2}{10} \qquad\textbf{(D) } \frac{26^3}{10^3} \qquad\textbf{(E) } \frac{26^3}{10^2} </math> | ||
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There are <math>26</math> letters and <math>10</math> digits. There were <math>26 \cdot 10^4</math> old license plates. There are <math>26^3 \cdot 10^3</math> new license plates. The number of license plates increased by | There are <math>26</math> letters and <math>10</math> digits. There were <math>26 \cdot 10^4</math> old license plates. There are <math>26^3 \cdot 10^3</math> new license plates. The number of license plates increased by | ||
| − | <cmath>\frac{26^3 \cdot 10^3}{26 \cdot 10^4} = \boxed{\ | + | <cmath>\frac{26^3 \cdot 10^3}{26 \cdot 10^4} = \boxed{\textbf{(C) \ } \frac{26^2}{10}}</cmath> |
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| + | Note: Nebraska is not actually the home of the AMC, the AMC resides in Washington D.C., where its headquarters is located. | ||
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| + | ~Andrew_Lu, Mismatchedcubing | ||
==See Also== | ==See Also== | ||
| − | |||
{{AMC10 box|year=2003|ab=B|num-b=9|num-a=11}} | {{AMC10 box|year=2003|ab=B|num-b=9|num-a=11}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 10:33, 1 August 2024
Problem
Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased?
Solution
There are 
letters and 
digits. There were 
old license plates. There are 
new license plates. The number of license plates increased by
![\[\frac{26^3 \cdot 10^3}{26 \cdot 10^4} = \boxed{\textbf{(C) \ } \frac{26^2}{10}}\]](http://latex.artofproblemsolving.com/b/c/e/bce4220f1b7d53e103e486a9453b3f8368d28bf8.png)
Note: Nebraska is not actually the home of the AMC, the AMC resides in Washington D.C., where its headquarters is located.
~Andrew_Lu, Mismatchedcubing
See Also
| 2003 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
