SchachtN
2020-11-06
Answered

Sandy is upgrading her Internet service. Fast Internet charges $15 for installation and $56.45 per month. Quick Internet has free installation but charges $58.95 per month. Complete the equation that can be used to find the number of months after which the Internet service would cost the same. Use the variable x to represent the number of months of Internet service purchased.

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Dora

Answered 2020-11-07
Author has **98** answers

If xx represents the number of months and Fast Internet charges $56.45 per month, then the monthly cost for Fast Internet is 56.45x dollars. Since Fast Internet also charges a $15 installation fee, then the total cost for Fast Internet is $56.45x+15$

Since Quick Internet charged $58.95 per month, then the monthly cost for Quick Internet is 58.95x dollars. Since Quick Internet does not charge an installation fee, then the monthly cost and total cost are the same for Quick Internet. The total cost for Quick Internet is then 58.95x.

If the total costs of the two companies are the same, then$56.45x+15=58.95x$

Since Quick Internet charged $58.95 per month, then the monthly cost for Quick Internet is 58.95x dollars. Since Quick Internet does not charge an installation fee, then the monthly cost and total cost are the same for Quick Internet. The total cost for Quick Internet is then 58.95x.

If the total costs of the two companies are the same, then

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Adding fractions of Groups of People

I understand the rules of adding fractions perfectly well. I know how to find common denominators, and understand why adding fractions without common denominators doesn't make sense.

But, today someone asked me about adding $\frac{5}{6}$ and $\frac{21}{28}$. They were wondering why dividing the two factions and taking the average ($\frac{0.8333+0.75}{2}=0.7917$) was giving them a different value than adding them together, and then dividing ($\frac{5}{6}+\frac{21}{28}=\frac{26}{34}=0.7647$).

My initial reaction was the same as yours: I was taken aback by someone adding fractions in this way, and I gave a quick refresher of why this doesn't make sense, and how to properly add fractions.

I thought I had helped them fix their problem, until they gave more more context: The $\frac{5}{6}$ was five people in a group of six who were observed washing their hands after a certain activity. The $\frac{21}{28}$ was twenty-one out of twenty-eight people who were observed washing their hands after a certain activity. The goal was to find the total number of people who had washed their hands, as a fraction of the total number of observed people. So, $\frac{26}{34}$ is actually correct in this case.

But, I'm still having trouble reconciling this with what I know about fractions. At least, what I think I know. Is there a term for these sorts of fractions? This isn't like having five slices of a six-slice pizza combined with twenty-one slices of a twenty-eight-slice pizza. This is like having one oven holding six pizzas, five of which have mushrooms, and another (huge) oven holding twenty-eight pizzas, twenty-one of which have mushrooms. What fraction of the pizzas have mushrooms? Is it $\frac{5}{6}+\frac{21}{28}$ ? I don't think so.

Is there some terminology I'm forgetting here? Why am I getting so tripped up by this?

I understand the rules of adding fractions perfectly well. I know how to find common denominators, and understand why adding fractions without common denominators doesn't make sense.

But, today someone asked me about adding $\frac{5}{6}$ and $\frac{21}{28}$. They were wondering why dividing the two factions and taking the average ($\frac{0.8333+0.75}{2}=0.7917$) was giving them a different value than adding them together, and then dividing ($\frac{5}{6}+\frac{21}{28}=\frac{26}{34}=0.7647$).

My initial reaction was the same as yours: I was taken aback by someone adding fractions in this way, and I gave a quick refresher of why this doesn't make sense, and how to properly add fractions.

I thought I had helped them fix their problem, until they gave more more context: The $\frac{5}{6}$ was five people in a group of six who were observed washing their hands after a certain activity. The $\frac{21}{28}$ was twenty-one out of twenty-eight people who were observed washing their hands after a certain activity. The goal was to find the total number of people who had washed their hands, as a fraction of the total number of observed people. So, $\frac{26}{34}$ is actually correct in this case.

But, I'm still having trouble reconciling this with what I know about fractions. At least, what I think I know. Is there a term for these sorts of fractions? This isn't like having five slices of a six-slice pizza combined with twenty-one slices of a twenty-eight-slice pizza. This is like having one oven holding six pizzas, five of which have mushrooms, and another (huge) oven holding twenty-eight pizzas, twenty-one of which have mushrooms. What fraction of the pizzas have mushrooms? Is it $\frac{5}{6}+\frac{21}{28}$ ? I don't think so.

Is there some terminology I'm forgetting here? Why am I getting so tripped up by this?