2014 AMC 10A Problems/Problem 10

Problem

Five positive consecutive integers starting with $a$ have average $b$ . What is the average of $5$ consecutive integers that start with $b$ ?

$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$ (Error compiling LaTeX. Unknown error_msg)

Solution 1

Let $a=1$ . Our list is $\{1,2,3,4,5\}$ with an average of $15\div 5=3$ . Our next set starting with $3$ is $\{3,4,5,6,7\}$ . Our average is $25\div 5=5$ .

Therefore, we notice that $5=1+4$ which means that the answer is $\boxed{\textbf{(B)}\ a+4}$ .

Solution 2

We are given that

\begin{aligned}\frac{a+a+1+a+2+a+3+a+4}5 & =b\rightarrow \\ 
b & =a+2\end{aligned} (Error compiling LaTeX. Unknown error_msg)

We are asked to find the average of the 5 consecutive integers starting from $b$ in terms of $a$ . By substitution, this is $\frac{a+2+a+3+a+4+a+5+a+6}5=a+4$

Thus, the answer is $\boxed{\textbf{(B)}\ a+4}$

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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All AMC 10 Problems and Solutions

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2014 AMC 10A Problems/Problem 10

Contents

Problem

Solution 1

Solution 2

See Also