Minnesota had 35.5 inches of precipitation (rain and snow) in 2022, setting a new record. It is difficult to predict the weather.
One report estimated that precipitation would increase an average of 0.2 inches per year. If that report is accurate then in 2032 (after 10 years) we would expect total precipitation to be around
\begin{equation*}
35.5 + 10 \times 0.2 = 37.5 \text{ inches}
\end{equation*}
We can use the letter \(Y\) to represent the number of years in the future (since 2022).
After \(Y\) years, we would expect total precipitation (in inches) to be around
\begin{equation*}
35.5 + Y \times 0.2
\end{equation*}
This expression generalizes our earlier calculation, replacing the 10 years by the \(Y\) years. This expression is often written as
\begin{equation*}
35.5 + 0.2Y
\end{equation*}
Notice that we are writing the number (0.2) before the letter (\(Y\)). Also, we are not writing the multiplication symbol (\(\times\)) at all! It is something you just need to know: that \(0.2Y\) really means \(0.2 \times Y\text{.}\) That shorthand is an example of algebraic notation.
Another report estimated that precipitation would increase about 0.5% per year. Notice that percentage is less than 1%. It is a bit more complicated to figure out what to expect in 2032 (after 10 years), but following some examples we saw in the last section, it turns out that we can calculate the predicted precipitation to be
\begin{equation*}
35.5 \times 1.005 \wedge 10 = 37.3154\ldots \approx 37.3 \text{ inches}
\end{equation*}
After \(Y\) years, we would expect total precipitation (in inches) to be around
\begin{equation*}
35.5 \times 1.005 \wedge Y
\end{equation*}
This expression generalizes our earlier calculation, replacing the 10 years by the \(Y\) years. This expression is often written as
\begin{equation*}
35.5\ast1.005^Y
\end{equation*}
Notice that we are writing the exponent as a superscript (meaning raised up and a little smaller). Also notice that we have replaced the usual multiplication symbol \(\times\) with an alternative symbol \(\ast\text{.}\) That’s because \(\times\) looks like the letter \(X\) which is sometimes used in algebra. This shorthand is another example of algebraic notation. The expression can also be written without the \(\ast\) symbol as
\begin{equation*}
35.5(1.005^Y)
\end{equation*}
where now we need to know that the number before the parentheses is multiplied.
For a little more practice with algebraic notation, recall that Piadina’s flatbread cost $17.49 and is usually cut into 5 slides. At that rate it cost
\begin{equation*}
17.49 \div 5 = \$3.50
\end{equation*}
per slice. If there were \(S\) slices, then we can replace the 5 slices by \(S\) slices and calculate that the flatbread cost
\begin{equation*}
17.49 \div S
\end{equation*}
in dollars per slice. In algebraic notation, we write the division as a fraction instead
\begin{equation*}
\frac{17.49}{S}
\end{equation*}
One more piece of terminology. When we have an expression, like \(\frac{17.49}{S}\text{,}\) and we want to know the value when \(S\) is a particular value, we say that we are evaluating the expression. Suppose we want to know the price per slice if Piadina’s cut their flatbread into 4 slices instead. That means we want to evaluate the expression \(\frac{17.49}{S}\) when \(S = 4\) slices.
The first step in the evaluation process is to replace the letter by its value and we write that value in parentheses. Thus we would have
\begin{equation*}
\frac{17.49}{(4)}
\end{equation*}
The second step is to replace the algebraic notation with the arithmetic notation, which is usually what we enter into the calculator:
\begin{equation*}
17.49 \div (4)
\end{equation*}
It turns out that we do not need the parentheses around the 4 so we can simply calculate
\begin{equation*}
17.49 \div 4 = 4.3725 \approx \$4.37 \text{ per slice}
\end{equation*}
