Answer:
Step-by-step explanation:
2005 AMC 8 Problems/Problem 20
Problem
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 24$
Solution
Alice moves $5k$ steps and Bob moves $9k$ steps, where $k$ is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, $14k$, is a multiple of $12$. Since this number must be a multiple of $12$, as stated in the previous sentence, $14$ has a factor $2$, $k$ must have a factor of $6$. The smallest number of turns that is a multiple of $6$ is $\boxed{\textbf{(A)}\ 6}$.
See Also
2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 Followed by
Problem 21
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All AJHSME/AMC 8 Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
Answer:
The base of the window is 7.5 feet long.
Step-by-step explanation:
The area of a parallelogram can be found with the formula A = bh
Plug in the known values to get 30 = 4b
30 = 4b
30/4 = b
7.5 = b
Answer:
-46
Step-by-step explanation:
We need to find the common difference of this arithmetic sequence:
So, the common difference d = -4. We can now write the equation:
So, the 15th term is -46.
Hope this helps!
Answer:
(x+4)(x-3)
Step-by-step explanation:
I'm italian. I used a formula that we Italians call "somma e prodotto". Tell me if it's correct.
Answer:
9π, or 28.26
Step-by-step explanation:
The Arc of the semicircle is the circumference of half a circle.
The radius of the semicircle is 9, and the circumference of a circle is 2πr, so it is 2π9 = 18π. since this is a semicircle, it is half of that, so 9π, or 28.26, since π is 3.14 here
