## Juneau Average Temperature

If the average daily temperature in Juneau, Alaska, during the first $20$ days of January was $30.9^\circ\text{F}$, what must the daily average be during the remainder of the month if Juneau’s January average is to match the historical average of $25.7^\circ\text{F}$?
Source: NCTM Mathematics Teacher 2008

SOLUTION
Let $a_1,a_2,a_3,\cdots,a_{31}$ represent the respective daily temperatures of day $1,2,3,\cdots,31$ of January in Juneau, Alaska.
By definition of average we want
$\dfrac{a_1+a_2+a_3+\cdots+a_{20}+a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{31}=25.7^\circ$
Multiply by $31$
$a_1+a_2+a_3+\cdots+a_{20}+a_{21}+a_{22}+a_{23}+\cdots+a_{31}=796.7^\circ$
Divide by $20$
$\dfrac{a_1+a_2+a_3+\cdots+a_{20}}{20}+\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=\dfrac{796.7^\circ}{20}$
Substitute the value of the average daily temperature during the first twenty days
$30.9^\circ+\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=39.835^\circ$
Simplify
$\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=8.935^\circ$
Multiply by $20$ and divide by $11$
$\dfrac{20}{11}\times \dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{20}=8.935^\circ\times \dfrac{20}{11}$
$\dfrac{a_{21}+a_{22}+a_{23}+\cdots+a_{31}}{11}=16.2^\circ$
The average daily temperature for the remaining $11$ days of January should be $16.2^\circ$ if the Juneau’s January average is to match the historical average of $25.7^\circ$.

Answer: $16.2^\circ\text{F}$