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:
x + 3
Step-by-step explanation:
Because he bought an unknown number of soccer balls that is where we use the variable, x. And we know that he bought 3 baseballs, and we add them both to get the total number of things bought. Hope this helped and you could give brainliest :)
Answer:
<em>18x² - 84x + 96</em>
Step-by-step explanation:
(5 × 2 - 6x + 2) × (4 × 2 - 3x)
5 × 2 (4 × 2 - 3x) - 6x(4 × 2 - 3x) + 2(4 × 2 - 3x)
80 - 30x - 6x(4 × 2 - 3x) + 2(4 × 2 - 3x)
80 - 30x - 48x + 18x² - 6x
96 - 84x + 18x²
18x² - 84x + 96
Hope this helps! :)
