## Real World Applications – Algebra I

### Topic

Are cab rides worth it?

### Student Exploration

In the “Linear Systems with Addition or Subtraction” concept, you solved a problem regarding cab fares. This activity is an extension of that problem. We’re going to look at different cab fares in San Francisco and we’ll use Algebra to find cab fares and determine if it’s worth it. As an extension, you can also look at cab fares for your favorite city or the city closest to you and you can determine for yourself if taking a cab is worth it.

Let’s look at two different routes Denise wants to go. She used www.taxifarefinder.com to find out how much it’ll cost her to get from the Four Seasons Hotel to Fisherman’s Wharf. After typing in the required information into the website, we figured that it’ll cost her approximately $14.89 to take a cab in medium traffic. Looking at the “Approximate Breakdown,” it tells us that the initial fare is $3.50, the additional metered fare is $11.39, and the 20% tip is $2.98. She also used this website to find out that it costs $26.44 to get from the Hostel in Downtown SF to the Golden Gate Bridge. Again, the initial fare is $3.50 and the additional metered fare is $22.94. It’s nice that we’re given all of this information, isn’t it? But let’s use Algebra to find out the specifics of these routes.

Similar to the problem in the exercises, we’re going to use a system of equations to calculate the pick-up fee and the per-mile fee. Let \begin{align*}x\end{align*} represent the pick-up fee, and let \begin{align*}y\end{align*} represent the per-mile fee. Our equation for our first route would be \begin{align*}14.89 = x + 2.8y\end{align*}, because 14.89 is our total cab fare and the whole trip was 2.8 miles. Our equation for our second route would be \begin{align*}26.44 = x + 5.5y\end{align*}, because $26.44 is our total cab fare and the whole trip was 5.5 miles.

Let’s stack these equations on top of each other and multiply one of the equations by -1 so that one of the variables will get eliminated.

\begin{align*}14.89 &= x + 2.8y\\ 26.44 &= x + 5.5y\end{align*}

Multiply one of the equations by -1.

\begin{align*}26.44 &= x + 5.5y\\ -14.89 &= -x - 2.8y\\ 11.55 &= 2.7y\\ 4.27 &= y\end{align*}

Now, let’s substitute this value in one of our original equations to solve for \begin{align*}x\end{align*}.

\begin{align*}14.89 &= x + 2.8(4.27)\\ 14.89 &= x + 11.97\\ 2.91 &= x\end{align*}

Using the Elimination Method to solve this system of linear equations, we found that the pick-up fee is $2.91 and the per-mile fee is $4.27. But, the website claims that the pick-up fee is $3.50 We could have also easily calculated the per-mile fee by dividing the total “additional metered fare” by the number of miles of the trip. Why don’t the numbers match up?

There are a lot of factors that contribute to the total cost of a cab ride. If you look at the website, the approximated numbers for the cab fare vary, depending on traffic. You can see that a cab ride is cheaper if there is no traffic, and more expensive if there is heavy traffic. Why do you think this is?

What are some other factors that contribute to a cab ride? If you want to take a cab during rush hour, do you think it would be worth it? Why or why not?

### Extension Investigation

- Go to the website www.taxifarefinder.com and search for your favorite city or a city close to where you live. Enter two different routes and have the website calculate two different cab fares for the two different distances.
- Use a system of equations to represent each trip, and calculate the pick-up fee and the per-mile fee.
- How close are your calculations with the website’s calculations? What are some factors that contribute to the differences in your calculations with the website’s calculations?