2015 AMC 10A Problems/Problem 3

Problem

Ann made a $3$ -step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$ -step staircase?

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24$

Solution

We can see that a $1$ -step staircase requires $4$ toothpicks and a $2$ -step staircase requires $10$ toothpicks. Thus, to go from a $1$ -step to $2$ -step staircase, $6$ additional toothpicks are needed and to go from a $2$ -step to $3$ -step staircase, $8$ additional toothpicks are needed. Applying this pattern, to go from a $3$ -step to $4$ -step staircase, $10$ additional toothpicks are needed and to go from a $4$ -step to $5$ -step staircase, $12$ additional toothpicks are needed. Our answer is $10+12=\boxed{\textbf{(D)}\ 22}$

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AMC 10 Problems and Solutions

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2015 AMC 10A Problems/Problem 3

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