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Problem 56

Your roommate, John, offered to buy household supplies for you and your other roommate. You live

near the border of three states, each of which has a different sales tax. The total amount of money spent

was $\$ 100.75$ . Your supplies were bought with 5$\%$ tax, John's with 8$\%$ tax, and your third roommates with 9$\%$ sales tax. The total amount of money spent without taxes is $\$ 93.50 .$ If your supplies before tax were $\$ 1$ more than half of what your third roommate's supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.

Answer

There were 150 cats, 130 dogs and 70 rabbits.

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## Discussion

## Video Transcript

all right, Question 56. We have three people on the roommates and they're buying them some supplies. They lived next to three different states. Is that they different? Each person pays different amounts of sales stats. Um, the total amount that they spend, um, altogether with taxes, $100.75 before taxes, the supplies were $93.50 and the first person we call Capital X or Apple X spends one more than the third person. O Z Um Z person s o. Let's actually define variables. Let's have X equal. The amount spent, the amount that capital expense before tax. Why will be the amount X Why spends you want tax and then see is the amount Z spends before tax. All right, so we know that X plus y Z equals 93 50. We also know that together all three people spent $7.25 in tax, so that means that 105 times X plus zero 08 times why plus 0.9 I see equal 7 25 $7.25. Right. Then our 30 question we need to have free is this last one here. Where? Expense? It all more than half of what c stands. So, um Okay, So that means that X Mount basemen iss a dollar. More than half of what Z spends. So agency spends at once. Okay, so we need to solve this system right here. All right, so, um, various ways we could start this. Um, Let's go ahead and substitute this third equation into the other two to get us get us rolling here. All right, so by the first equation gone, more than half of disease. Sorry. Okay, so here we go. I got, um what, half z? 1/2 z plus why No one has Z plus one. There's my ex. Plus why equals 93 and I know that 105 times 1/2 Z plus one plus 10 What? Those 0.9 z Asta e equals 7 25 case. Not back to equations and two variables. Um, I should simplify those. Okay, So the top equation been a to track one from both sides. So it was 92. 50. I'm gonna get keep my wise. There's, like, one y no life terms with that and then 1/2 Z plus one z is rehab easy. And my words were said to a desperate We'll see what goes on the next one. Now, with this equation here, I'm really tempted right from the start right now is to multiply this through by 100 and, um, see if I must fight through by 100. Then this becomes by this becomes eight. This becomes nine, and this becomes 720. Okay, so I'm go ahead and do that. We don't have to have those decimals floating around with it. Um, so I get, uh, this is five. Have Z. Hey, close five waas. Eat what? Plus nine z. It was 7 25 So, um, subtract the five from both sides of this 7 20 Um, You see, I've got a wise and I have out. Let me see. Nine Z. That's 18 over too. So they tell you 23 over two. Easy. All right. So, um, let's multiply the top equation by eight. All right, Um, by this equation by eight, the wise the same. It's like it. Why? Plus eight times this. So wars off 12 z equals C eight times 92 50 ISS 740. Okay, so let me see. This is 24 24 over to minus 23. Over two is 1/2. So when I subtract these so eight schools, why might say why? 0 24 halves, minus 23 houses. 1/2 z. Well, when that A room here. 1/2 Z. Two people 20. It's that easy equals 40. That that Now I need to find out. What? Why is he too? Okay, Um, why then has been a equal that anything that sofa. Why already? I don't. So let's go ahead and find that's next. So x using this equation, I'm gonna get X equals half of Z plus one. So X equals 21. Right? So so far. Expend 21 zee spent 40. We know when we add up all three of them, we get 93 50. So that means that my why is going to be equal to we're here. Sorry about the organization, but it's starting a new page right now. So why's unequal 93 50 rg 3 50 minus with the other two did so minus 61 which is 32 50. So Why spend 30 two? All right, so there's our three amounts ex spent 21 That's the smallest and wife spent 30 to 50 s. And next on the Z is 40. Um, if you want to go back, they wanted to find out how much they actually spend. Um, that's before taxes with taxes. Um, you would multiply this time's 1.5 now the how much? The middle bird 1.8 And that will be what they spent with tax Extreme will buy that one by 1.55% tax and then where this one you multiply that by 1.9 on the helmets z paid because that wasn't a 9%.

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