Solve each problem. See Examples 5 and 9. A sparkling-water distr...

Question

Solve each problem. See Examples 5 and 9. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for $6.00 per gallon. She wishes to mix three grades of water selling for $9.00, $3.00, and $4.50 per gallon, respectively. She must use twice as much of the $4.50 water as of the $3.00 water. How many gallons of each should she use?

Accepted Answer

Hello everybody. Everything. All right. Today, today we're going to be looking at this question that states the manufacturer of cement plans to produce 500 kg of cement and sell it for $5 per kilogram. She intends to mix three grades of cement with different prices. $10.08 dollars and $4 per kilogram. She must use three times more of the $4 cement than the $8 cement. What amount of each grade should she mix? Now, the answers provided are a zero k zero kg of $ cement. 125 kg of $8 cement and 375 kg of $4 cement. B 125 kg of $10 cement, zero kg of $8 cement and 375 kg of $ cement. C 375 kg of $10 cement. 125 kg of $8 cement and zero kg of $ cement and d 375 kg of $ cement. zero kg of $8 cement and 100 25 kg of $4 cement. Now for a question like this. I first want to focus on all the information they gave me and see if I can turn that information into an equation and I can. So I'm going to assign a variable to each value that I want. And, and have, for example, I'll write that X is equal to amount oh $10 cement. Y will be the amount, the amount of eight daughter cement and Z will be the amount of your daughter cement. So now that everything has a variable assigned to it, I can write an equation with it. So we know that we're trying to make 500 kg of cement. So that means that when we add up all amounts. So when we have X plus Y plus Z, that will equal 500 because if we add up all the amounts of the cements that should equal a total of 500 kg, which is what we want. Now, we also have that we must use three times more of the $4 cement than the $8 cement. That can be written as Z which is the amount of the $4 cement is equal to three times Y which is the amount of $8 cement. So we will be, that is translating that we are using three times more of the $ cement. So now I can write an equation for the price and the total money and be because they told us $10.08 dollars and $4. I can write that 10 X because they told us $ is for the, uh, we're assigning X as the amount of $10 cement. So we're having 10 X because $10 multiplied by that amount will tell me how, how much money that cement will have. So I have 10 X plus eight Y once again because we have $8 multiplied by the amount of the $8 cement. And that'll be plus four Z because we have $ multiplied by the amount and that'll be equal to five, multiplied by 500. And this is because we're trying to make 500. So that's where our 500 comes from and we're sending it for $5 per kilogram. So that's where the five comes from and we multiply them by, by the four, the 500 kg, which is the amount. So now that we have that equation, we can plug in R Z into the four Z. So we'll have 10 X plus eight Y or we'll plug it into our first, we'll plug it into our X plus Y plus Z equation. So we'll have X plus Y plus three, Y equals which means that we also have 10 X plus eight Y plus four, multiplied by three Y. So what we did here is we multiplied or we plugged in our Z equals three Y into each Z that we had So the X plus Y plus Z equation we have now X plus Y plus three Y because our Z is three Y and our other equation, we have 10 plus 10, X plus eight Y plus four multiplying three Ys since we have four multiplying Z and that second equation is equal to 2500 since we distribute the five multiplying the 500. And now let's simplify our X plus Y plus three Y equation. So we'll have that X plus. Well, actually let's just simplify both equations. So we'll have X plus four Y equals 500. That'll be our first equation. And our second equation will be 10 X plus 20 Y equals 2000 500. And that 20 Y simply comes from distributing. So if we distribute four multiplied by a three Y, that's 12 Y and 12 Y plus eight Y is 20 Y. So now we have two equations with the same variables and to kind of solve a variable here, we can use a method called elimination where basically we try to multiply one of the equations by a number that once that multiplication is done, if we add both equations up, one of the variables will be eliminated. And I'll explain a little bit better now. So I see that for equation one, if I multiply equation one by negative 10, so we'll have equation one will be negative 10 X or I'll, I'll write it broadly first So an equation will, will have negative 10, multiplying X plus four Y equals 500. So we're multiplying the whole thing by negative 10, including the answer that will give us that equation one will now be negative 10 X minus 40 Y equals negative 5000. So now I can plug, plug in equation. So I'll let's write equation one and under it, we'll write equation two which was 10 X plus 20 Y equals 2500. And then we can draw a line under that. Now imagine that you're adding up both of the equations. So I write them one on top of the other because that's usually how we write addition, the numbers one on top of the other with the line under and the addition sign. So if we write it like this, we can visualize that better. So now let's add them up. So we have negative 10 X plus 10 X that will cancel out. And that is where the elimination comes in. We took out the variable X, then we have negative 40 Y plus 20 Y that will be negative Y. So that'll be negative 20 Y equals and then we had negative 500 or negative 5000 plus 2500 that will be negative 2500. And if we multiply both sides by negative one, we'll have 20 Y equals 2500. And then we divide both sides by 20 which will give us Y equals 125. And that will be our first answer for this question. Why being the amount of $8 cement? So we have that, we will have 100 and 25 kg of the $8 cement. So now we can start doing some plugging in. So we had that, we had that X plus four, Y equals 500. That was our equation number one. So if we plug in the Y, now we'll have that X plus four multiplied by 25 equals 500 which means X plus 500 equals because four multiplied by 100 and 25 is 500. So that means that X will be equal to zero and that will be our second answer. Which means that we will have zero kg of the $10 cement. We won't have any amount for the $10 cement. And finally, we had that Z is equal to three Y. That was one of our earlier equations. And if we plug in the Y, we'll have Z is equal to three, multiplied by 100 and 25 which is Z equals 375. And that will be our last answer here, which means we'll have 375 kg of the $4 cement. And all these amounts match up with answer choice. A which is zero kg of $10 cement. 100 and 25 kg of $8 cement and 375 kg of the $4 cement. So I hope that was helpful. And until next time.

Key Concepts

Systems of Equations

Variable Representation

Proportional Relationships

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