2005 AMC 8 Problems/Problem 24

Problem

A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Solution

It is easier to work backwards from $200$ , and have keys that display [-1] and [x0.5]. Use the second key when the number is even, and the first key when the number is odd until you get one. We get:

$200, 100, 50, 25, 24, 12, 6, 3, 2, 1$

This took $\boxed{\textbf{(B)}\ 9}$ keystrokes.

2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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
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2005 AMC 8 Problems/Problem 24

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