Cost vs career comparison
Narrator says 'subtract 70,000 from both sides' when he means 70,000T, and calls the 360,000 constant 'opportunity cost' although it combines 80,000 tuition and the 280,000 from 70,000 x 4. The final answer (T > 10.59, year 11) is correct.
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The video compares working right away at $36,000 a year with spending 4 years and $80,000 on a program, then earning $70,000 a year. It writes total earnings as 36,000T and 70,000(T - 4) - 80,000, sets up an inequality and solves it to find T > about 10.59, so option two pulls ahead by year 11.
You'll be able to
- write expressions for total earnings over T years for two career options.
- set up and solve an inequality to find when one option overtakes another.
- remember to flip the inequality sign when dividing by a negative number.
Key words
- Variable
- A letter, like T, that stands for a number that can change, such as the number of years.
- Inequality
- A statement that one expression is greater than or less than another.
- Opportunity cost
- What you give up by choosing one option, such as wages not earned while in school.
Transcript
- [Instructor] We are told a person is comparing two options. Option one: start working immediately, earning $36,000 per year. Option two: attend a four-year program costing $20,000 per year, then earn $70,000 per year. After how many years will the second option earn more money overall?
Pause this video, have a go at that before we do this together. Okay, let's just set some variables here. Well, we wanna figure out how many years are gonna go by for the second option to do better. So let's just say, let T is equal to number of years that has gone by.
Number of years that has gone by. Now, in option one, what's the total amount of money that they're going to make? Total is going to be equal to 36,000 times T. Pretty straightforward. What about option two? Well, the total, and maybe, well, I'll write it this way, the total after the four years, we could set up two separate lines or two separate algebraic equations for the first four years and then after that, but that makes it complicated.
We could just say that the total, especially after graduation, it's going to be that you spent four years at $20,000 a year. So that's negative $80,000. So you have to spend negative $80,000. You kind of start in the hole, but then you make $70,000 per year afterwards.
Now, I'm not just gonna multiply 70,000 times T, because I'm working four fewer years at $70,000. So it's gonna be times T minus four. And so what we really care about is at what point this total right here is going to be greater than this total. And so we can set up an equation here.
So I could say, let me rewrite all of this. Maybe I'll distribute this 70,000 a little bit. So this would be, 70,000T minus, 70,000 times four minus 280,000, minus 280,000 minus 80,000, minus 80,000 is going to be greater than 36,000. Is gonna be 36,000 times T.
Or another way of saying this is that we get 70,000T minus 360,000 needs to be greater than 36,000, 360,00T. And this one is actually an interesting way of looking at it because in option two, when you attend the four-year program, you'll eventually make $70,000 a year, but you lost the opportunity cost of working for four years and you had to pay the tuition.
So this 360,000 here is really that opportunity cost, while in this situation, you're just making $36,000 per year right from the get-go. But now, let's just solve for T over here. And so we could say, let's see, we could subtract 70,000 from both sides. So if we get negative 360,000 is greater than, what's this, negative 34,000T.
And then if we divide both sides by negative 34,000, that would swap the inequality. Or I could even swap these two. So this would be T would be greater than, instead of less than, it's gonna be, or instead of, I'm swapping the sides, if you get what I'm doing.
T is gonna be greater than 360 over 34. I could write 360,000 over 34,000, but this is gonna be the same thing. And so another way of saying it, T is going to be greater than, let's say, that's the same thing as 180 over 17. So that's going to be a little bit, let's see, maybe about 10.58-ish.
So let's see, this is going to be, so T is going to be greater than approximately, give or take, 10.5. Let me get a calculator out just to make sure. So 180 divided by 17 is indeed 10., let's say, if I round, 10.59. T is approximately greater than 10.59 years.
So by year 11, option two would've passed up option one. And we're done.
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