We know that \(3/4\) does not equal \(4/3\). But if we say that \(3\text{ oranges}\) cost \(4\text{ dollars}\), its equivalent to saying that \(4\text{ dollars}\) buys \(3\text{ oranges}\). In other words,

\begin{align*} \frac{3\text{ oranges}}{4\text{ dollars}}=\frac{4\text{ dollars}}{3\text{ oranges}} \end{align*}This does not mean that \(3/4=4/3\) because in this case the \(3\) and the \(4\) have different dimensions, whereas in the fraction \(3/4\) both the "\(3\)" and the "\(4\)" are dimensionless. In dimensional analysis, problems are solved by treating the dimension of a number in an arithmetic way. The following examples illustrate how this works.

Suppose that oranges are on sale: \(3\text{ oranges}\) for \(4\text{ dollars}\). How many oranges, \(N\), can we buy for \(12\text{ dollars}\)?

Suppose that oranges are on sale: \(3\text{ oranges}\) for \(4\text{ dollars}\). How many oranges, \(N\), can we buy for \(12\text{ dollars}\)?

The relationship between oranges and dollars can be expressed as either

\begin{align*} \frac{3\text{ oranges}}{4\text{ dollars}}\text{, or, }\frac{4\text{ dollars}}{3\text{ oranges}} \end{align*}Since the answer we seek will have the dimension of oranges, we choose the relationship that has "oranges" in the numerator. Multiplying this by the number of dollars we have to spend gives: \begin{align*} N=\frac{3\text{ oranges}}{4\text{ dollars}}*12\text{ dollars}=9\text{ oranges} \end{align*}

Notice that dollars over dollars has been canceled out much the same as numbers would be canceled out in a numerical ratio.

If we want to know how many dollars it takes to buy \(9\text{ oranges}\), we would use the relationship that has "dollars" in the numerator:

\begin{align*} N=\frac{4\text{ dollars}}{3\text{ oranges}}*9\text{ oranges}=12\text{ dollars} \end{align*}Suppose we have a car that gets \(30\text{ miles per gallon}\), and want to know how much it costs, \(C\), to make a \(300\text{ mile}\) trip if gas costs \(3\text{ dollars per gallon}\).

Suppose we have a car that gets \(30\text{ miles per gallon}\), and want to know how much it costs, \(C\), to make a \(300\text{ mile}\) trip if gas costs \(3\text{ dollars per gallon}\).

In this case, there are two basic relationships to consider: \begin{align*} \frac{3\text{ dollars}}{1\text{ gallon}}\text{ or }\frac{1\text{ gallon}}{3\text{ dollars}}\text{; and }\frac{30\text{ miles}}{1\text{ gallon}}\text{ or }\frac{1\text{ gallon}}{30\text{ miles}} \end{align*}

Since we want to know cost in dollars, we first choose the relationship that has "dollars" in the numerator, and then multiply it by the relationship that will cancel out the undesired dimension in the denominator: \begin{align*} C=\frac{3\text{ dollars}}{1\text{ gallon}}*\frac{1\text{ gallon}}{30\text{ miles}}*300\text{ miles}=30\text{ dollars} \end{align*}

If we had \(10\text{ dollars}\) and wanted to know how far, \(F\), we could go with that amount, we would set up the problem as follows: \begin{align*} F=\frac{30\text{ miles}}{1\text{ gallon}}*\frac{1\text{ gallon}}{3\text{ dollars}}*10\text{ dollars}=100\text{ miles} \end{align*}

Given the following relationships:

\(1\text{ gallon }=4\text{ quarts}\)

\(16\text{ tablespoons }=1\text{ cup}\)

\(1\text{ quart}=4\text{ cups}\)

\(3\text{ teaspoons}=1\text{ tablespoon}\)

How many teaspoons, \(N\), are there in \(1\text{ gallon}\)?

Given the following relationships:

\(1\text{ gallon }=4\text{ quarts}\)

\(16\text{ tablespoons }=1\text{ cup}\)

\(1\text{ quart}=4\text{ cups}\)

\(3\text{ teaspoons}=1\text{ tablespoon}\)

How many teaspoons, \(N\), are there in \(1\text{ gallon}\)?

Solution: \begin{align*} N=\frac{3\text{ teaspoons}}{1\text{ tablespoon}}*\frac{16\text{ tablespoons}}{1\text{ cup}}*\frac{4\text{ cups}}{1\text{ quart}}*\frac{4\text{ quarts}}{1\text{ gallon}}=\frac{768\text{ tablespoons}}{1\text{ gallon}} \end{align*}