Systems of linear inequalities word problems — Basic example

- Karunesh is a gym owner who wants to offer a full schedule of yoga and circuit training classes. Yoga classes are each 1 1/2 hours long, while circuit training classes are only an hour. And he wants at least 25 hours of classes. At least 25 hours of classes on the schedule each week. All of his instructors are paid $35 per class. So it doesn't matter whether it's an hour and 1/2 yoga class or a one hour circuit training class. But Karunesh doesn't want to spend more than $1,000. Does not want to spend more than $1,000 per week on salaries. Which of the following falls within Karunesh's guidelines for the weekly schedule? So we can just go through these choices or we can say okay which of these meet the at least 25 hours of classes but then don't break the bank, don't cost more than $1,000? So this first one right over here is three yoga classes at an hour and a 1/2 each. Well this is going to be 4.5 hours here, plus 19 hours. Each circuit training class is an hour. So plus 19 hours is going to get you to 23 and 1/2 hours. So that doesn't get us to at least 25 hours of classes. So we can rule that out. I see 10 yoga classes and 12 circuit training classes. 10 yoga classes at an hour and a 1/2 each just that's gonna be 15 hours. Plus the 12 circuit training classes. That does get us to 27 hours. So it checks off the minimum number of hours that he wants to get. Now let's make sure it doesn't break the bank. So the classes are $35 per class regardless of whether it's an hour and a 1/2 yoga class or it's a one hour circuit training class. So right over here, we're gonna have a total of 10 plus 12. So this is gonna be 22 classes. 22 classes times $35 per class. And what is this going to be? I can already tell it's gonna be less than 1,000 and if you're under time pressure, say hey this is less than 1,000. But if you want to verify that this right over here is less than 1,000, two times 35 is 70. And then two times... Sorry. And then 20 times 35 is 700. 700. So that's going to be 770. I know I did this in a somewhat nontraditional way. I could have done it like this. 35 times 22. So two times 35 is 70. 20 times 35 is 700. You add them together you get 770. But the important point is this is less than $1,000. So this is going to meet our constraints. We have enough hours but we haven't broken the bank. Now in these two, it looks like we have plenty of hours here if I just took yoga classes enough meet our 25 hours 'cause 20 yoga classes is gonna be 30 hours of classes and 25 is gonna be even more. But I would suspect that these are gonna break the bank. And if you look at this is gonna be 32 classes at $35 per class. Yup, that's gonna be over $1,000. This one is gonna be 31 classes. And so this is probably gonna be over $1,000 as well.