Difference between revisions of "2015 AMC 10A Problems/Problem 3"
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| − | + | ==Problem== | |
| + | Ann made a <math>3</math>-step staircase using <math>18</math> toothpicks as shown in the figure. How many toothpicks does she need to add to complete a <math>5</math>-step staircase? | ||
| + | |||
| + | <math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}}\ 22\qquad\textbf{(E)}\ 24</math> | ||
| + | |||
| + | ==Solution== | ||
| + | We can see that a <math>1</math>-step staircase requires <math>4</math> toothpicks and a <math>2</math>-step staircase requires <math>10</math> toothpicks. Thus, to go from a <math>1</math>-step to <math>2</math>-step staircase, <math>6</math> additional toothpicks are needed and to go from a <math>2</math>-step to <math>3</math>-step staircase, <math>8</math> additional toothpicks are needed. Applying this pattern, to go from a <math>3</math>-step to <math>4</math>-step staircase, <math>10</math> additional toothpicks are needed and to go from a <math>4</math>-step to <math>5</math>-step staircase, <math>12</math> additional toothpicks are needed. Our answer is <math>10+12=\boxed{\textbf{(D)}\ 22}</math>. | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2015|ab=A|num-b=2|num-a=4}} | ||
| + | {{MAA Notice}} | ||
Revision as of 16:42, 4 February 2015
Problem
Ann made a 
-step staircase using 
toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 
-step staircase?
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}}\ 22\qquad\textbf{(E)}\ 24$ (Error compiling LaTeX. Unknown error_msg)
Solution
We can see that a 
-step staircase requires 
toothpicks and a 
-step staircase requires 
toothpicks. Thus, to go from a -step to -step staircase, 
additional toothpicks are needed and to go from a -step to -step staircase, 
additional toothpicks are needed. Applying this pattern, to go from a -step to -step staircase, additional toothpicks are needed and to go from a -step to -step staircase, 
additional toothpicks are needed. Our answer is 
.
See Also
| 2015 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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. 
