## Real World Applications – Algebra I

### Topic

How much Americans Spend While Traveling to Different Countries

### Student Exploration

People love to travel. Many people save a lot of money so they can go to foreign countries and enjoy life. The US Department of Commerce likes to keep track of the amount of money that is spent in different countries each year by Americans. Mathematicians have helped the US Department of Commerce to help figure out a pattern that best represents the amount that is spent each year.

For the sake of this exercise, the formula that represents the total amount that Americans have spent every year since 2000 can be represented by the function, \begin{align*}T(x)=\frac{(3x^2+15x)}{(3x^2+12x)}\end{align*}

A formula has also been derived to represent the number of people that have left the United States to travel each year. This formula is represented by the function, \begin{align*}P(x)=\frac{(x^2-x-30)}{(x^2-7x+6)}\end{align*}, where \begin{align*}x\end{align*} also represents the number of years that have passed since 1990.

If we want to find the average amount of money spent by each American per year, we can divide the total amount of money spent by the total number of Americans that have traveled. We would have:

Since we’re dividing this rational expression, we want to first factor all of the expressions, if possible.

Now that we’ve factored both of these expressions, we notice that some of the factors can be canceled from both the numerator and from the denominator.

Once we have canceled all of the same factors, we can rewrite our expression.

When we divide rational expressions, we know that we have to multiply by the reciprocal of the second rational fraction.

Once we rewrite our new expression, we can multiply and then cancel out more factors that are the same.

Since the \begin{align*}(x + 5)\end{align*} factors are the same on the numerator and the denominator of the fraction, we can cancel them out.

What we have left is:

What does this resulting expression mean?

We were to find a way to find the average amount of money that American travelers spend each year. This is the expression that represents this relationship. We can also use this and multiply by the number of Americans that travel and verify the total amount of money that’s spent outside of the United States every year. Let’s try it.

We’re multiplying two rational expressions, so we want to make sure that before we multiply them, that all expressions are fully factored.

Now that the fraction on the right has been fully factored, we can simplify. We know that the factor \begin{align*}(x - 6)\end{align*} can be canceled.

As a matter of fact, we also know that the factor \begin{align*}(x - 1)\end{align*} is present on both the top and the bottom of the multiplication problem, so we can cancel them out too.

What we have as the result of multiplying the two rational expressions is the total amount of money that’s spent outside of the United States, which is our original expression that we started with.

### Extension Investigation

What are other similar situations that represent a similar relationship? When can we use the relationship of multiplying or dividing rational expressions?