## Real World Applications – Algebra I

### Topic

Cost of Living – Bills!

### Student Exploration

A teacher was interviewed about her living expenses. We’re going to apply our understanding of adding, subtracting, and multiplying rational numbers to figure out the percentages of her bills in relation to her total cost of living.

The teacher gave us the following information:

Bill | How much paid per month |
---|---|

Rent | $925.00 |

Credit Card | $300.00 |

Student Loan | $392.53 |

Internet | $19.95 |

Cell Phone | $97.04 |

Water | $20.05 |

PG&E | $20.71 |

First, what does this table tell you?

We have to figure out the total that this teacher’s paying in bills to understand the fraction of the bill that’s going toward each of the costs.

\begin{align*}\text{Total Paid to Bills} = 925 + 300 + 392.53 + 19.95 + 97.04 + 20.05 + 20.71 = \$1,775.28\end{align*}

Let’s look at how much of this total is dedicated to rent. We can divide \begin{align*}\frac{925}{1775.28}\end{align*} and get 0.521. We can also multiply this decimal by 100 to represent that 52% of the total cost is dedicated to rent. Try doing this same strategy to figure out the percentages of the rest of the bills in comparison to the total cost. Which would you predict represents the smallest percentage of the total bill? How can we see that without doing all of the calculations?

Let’s look at this teacher’s cost of living in a different way. We’re going to represent this as a big expression to represent the fraction of each cost.

\begin{align*}\frac{925}{1775.28}+ \frac{300}{1775.28}+ \frac{392.53}{1775.28}+ \frac{19.95}{1775.28}+ \frac{97.04}{1775.28}+ \frac{20.05}{1775.28}+ \frac{20.71}{1775.28}\end{align*}

If we were to simplify this expression, what do you think the total would be?

To add fractions, remember to make sure that the denominators are the same and then add the numerators. In this case, since all of the denominators are the same, we add the numerators. Then we can simplify the fraction, and it equals 1! What do you think this means?

In reality, for a teacher, some of these bills are a little excessive and some are a little high. What if we were to take away some of the bills that aren’t as important? How could we do that?

Having internet or a cell phone are important, but not essential. Let’s see what would happen when we try to subtract the internet and cell phone bills from the total and see what we’d get.

\begin{align*}1775.28-(97.04 + 19.95)\end{align*}

To simplify this, we want to do the operation inside the parentheses first, and then subtract this from the total.

\begin{align*}1775.28-(116.99)\end{align*}

Our new total bill is $1,658.29

There are two different ways that we can look at what we have left, as a percentage or fraction of the old total bill.

The first strategy is to just take the new total and divide it by the old total and multiply the result by 100 to find the percentage. We would have \begin{align*}\left(\frac{1658.29}{1775.28}\right)\times 100= 93.4 \%\end{align*} left. This means that cell phone and internet only accounted for less than 7% of the total bill.

The second strategy is to use our big expression from above.

\begin{align*}1- \left(\frac{97.04}{1775.28}+ \frac{19.95}{1775.28}\right)\end{align*}

In this case, 1 represents 100% of the bills together, and we’re subtracting each portion of the bill – in this case, the cell phone and the internet. Let’s simplify.

\begin{align*}&1-\left(\frac{97.04}{1775.28}+ \frac{19.95}{1775.28}\right)\\ & \qquad 1- \left(\frac{116.99}{1775.28}\right)\\ &\quad \frac{1775.28}{1775.28}- \frac{116.99}{1775.28}\\ &\qquad \quad \frac{1658.29}{1775.28}\\ &\qquad \quad 0.933937\end{align*}

Notice how we got similar answers?

What similarities do you notice between the two strategies that we used above? What differences do you notice?

### Extension Investigation

Try looking into the costs of living for an adult that’s around you. Use your knowledge of operations on rational numbers to find the total cost of all of the person’s bills, and find the different percentages of each bill to the total.