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Algebraic word problems | Worked example

Video transcript

- [Instructor] Alfonso earns $16 per hour for the first eight hours of his shift. For each additional hour he works beyond the first eight hours, he earns $24 per hour. If Alfonso earned $320 from a single shift, how many hours long was that shift? Pause this video, see if you can figure this out. Okay, now let's do this together. So let's remind ourselves, they want us to figure out how many hours long was Alfonso's shift? So let's just make that what we're trying to figure out. Let's say that x is equal to the number of hours of the shift. So if the shift is x hours long, how much would Alfonso make? Well, we can see that he's working more than eight hours because eight times 16, you might be able to do that in your head, but it's nowhere close to $320. So he definitely worked more than eight hours. So he's going to make $16 per hour, times the first eight hours, and then plus, then he's going to make $24 per hour, times the remaining hours. Now what's going to be the remaining hours? You might be tempted to say x, but x would be his total shift. But we're saying the remaining hours after he works the first eight hours. So the remaining hours would be x minus eight. So for example, x were 12, he's gonna make $16 an hour for eight hours, and then 12 minus eight, he's going to make $24 for the remaining four hours. And so this expression right over here, this is going to be the total amount that he's going to make for the shift, and they tell us that is $320. And that is going to be equal to $320. And so now we have this algebraic expression and we can solve for x. The number of hours of his shift. So 16 times eight, you can use a calculator for that, but you might be able to do it in your head, eight times six is 48, plus eight times 10 is 80. So that is 128, plus, now we can distribute this 24. 24 times x and then minus 24 times eight. So let's see, 20 times eight is 160. Four times eight is 32, so it's minus 192. And once again, you can use a calculator on the practice if you don't feel comfortable doing that or if you don't want to do it on paper. In fact, I encourage you to use a calculator if you feel a little bit uncomfortable. And that's going to be equal to $320. And like we've seen before, we want to isolate the xs on one side of the equation. So we can get rid of this 128 and this negative 192 by subtracting 128 and adding 192. Now of course, if we do that on the left-hand side, we have to do that on the right-hand side too. So we can, well we can add 192 and then we can subtract 128. And I did all of that in one step. I'm just adding and subtracting the same amount from both sides. So on our left-hand side we're just left with a 24x, and then our right-hand side, you could do this by hand, but I keep talking about a calculator, so let me just get a calculator out and do it. So you have 320 plus 192, minus 128, is equal to 384. And we're going to have to divide by 24. Actually, let me just do that. So this is going to be 384. Now to solve for x, you just divide both sides by 24, and we get x is equal to, I'll just get the calculator out. Again, just take that and divide by 24. We get 16, x is equal to 16 which is that choice right over there. Now there might have been another way to do this. You could've tried out values. You could said, okay, if Alfonso works eight hours, well that's too little. 10 hours, and keep trying values where for the first eight hours, you do 16 times that. 16 times eight, and then after that, you each incremental hour's $24 an hour. So it depends which strategy works better for you. But the algebraic way, we'll definitely, if you do it in a logical way it'll definitely get you to the right end.