If the average (arithmetic mean) of $t$ and $t+5$ is $x$ and if the average of $t$ and $t-1$ is $y$, what is the average of $x$ and $y$?

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[STEP]Setup the equation of the average of $t$ and $t+5$[/STEP]

The average of $t$ and $t+5$ is $x$:

$$\frac{t+t+5}{2}=x$$

[STEP]Setup the equation of the average of $t$ and $t-1$[/STEP]

The average of $t$ and $t-1$ is $y$:

$$\frac{t+t-1}{2}=y$$

[STEP]Calculate the average of $x$ and $y$[/STEP]

The average of $x$ and $y$ is 

\begin{eqnarray*}
\frac{x+y}{2} & = & \frac{\frac{t+t+5}{2}+\frac{t+t-1}{2}}{2}\\
& = & \frac{4t+4}{4}\\
& = & t+1
\end{eqnarray*}

[ANS]The average of $x$ and $y$ is $t+1$[/ANS]

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