# Applying Direct Variation

Direct variation can be used to model real-world situations.

Many real-world situations can be modeled by direct variation. In geometry, the perimeter $P$ of a square is proportional to the side length $s$ of the square by the formula $P=4s$. The formula $d=rt$ describes how distance $d$ is proportional to time $t$ by the constant of proportionality $r$, which represents a constant rate of travel. When shopping for several same-price items, the total cost $C$ is proportional to the number of items $n$ by the price $p$ of each item, or $C = pn$.

Sales tax is another example of direct variation. The amount of sales tax on an item is proportional to the price. If the sales tax rate is $r$, then the amount of sales tax on an item with price $x$ is $rx$. The total cost $y$ of the item with tax is:
\begin{aligned}y&=x + rx\\y&=(1+r)x\end{aligned}
Another application of direct variation is converting units, such as distance, volume, or currency. The constant of variation is the conversion factor.
Step-By-Step Example
Applying Direct Variation to Sales Tax
Graph and then determine the exact total cost of an item with a price of 5 after a sales tax of 10%. Step 1 The total cost $y$ of an item with price $x$ after a sales tax rate of $r$ is represented by the sales tax equation: $y=(1 + r)x$ For a 10% sales tax, the value of $r$ is 0.10, which can also be written as 0.1. Substitute 0.1 for $r$. \begin{aligned}y &= (1 + 0.1)x\\y &= 1.1x\end{aligned} Step 2 Determine two points on the line of the equation. The graph of a direct variation passes through the point $(0, 0)$ and $(1, k)$, where $k$ is the constant of proportionality. In this situation, $k$ is 1.1. \begin{aligned} y&=kx\\y&=1.1x\end{aligned} So, the line of the equation goes through the points $(0, 0)$ and $(1, 1.1)$. Solution Plot the points and determine the slope, or $k$, of the equation. The graph of the equation goes through the origin with a slope of 1.1. From the graph, when the price of the item $x$ is5, the total cost $y$ after tax is between $5 and$6. Substitute 5 for $x$ in the equation to identify the exact total cost.
\begin{aligned}y &= 1.1x\\ &=1.1(5)\\&=5.5\end{aligned}
The exact total cost is \$5.50.
Step-By-Step Example
Applying Direct Variation to Currency
Currency in U.S. dollars varies directly with the value in other currencies. On a given day, 8 euros are exchanged for 9.25 U.S. dollars. How many euros would be exchanged for 50 U.S. dollars?
Step 1
Write the direct variation equation. Let $y$ represent the number of euros and $x$ represent the number of U.S. dollars.
$y=kx$
Step 2
Substitute the given values of $x$ and $y$ into the equation, and solve for $k$.
\begin{aligned}8&=k(9.25)\\ \frac{8}{9.25}&=k\\ 0.865&=k\end{aligned}
This means that the exchange rate is about 0.865 euro per U.S. dollar.
Step 3
Substitute the value of $k$ into the direct variation equation.
$y=0.865x$
Step 4
The number of U.S. dollars is given, which is represented by $x$. Substitute 50 for the value of $x$ into the equation.
\begin{aligned}y&=0.865(50)\\y&=43.25\end{aligned}
Solution
At the given exchange rate, 43.25 euros would be exchanged for 50 U.S. dollars.