(a) A man has $$ 4.55$ in change composed entirely of dimes and quarters. What are the maximum and minimum number of coins that he can have? Is it possible for the ...

Solve the coin word problems.
Connor has a collection of dimes and quarters with a total value of $6.30. The number of dimes is 14 more than the number of quarters. How many of each coin does he have?

Okay, so you have is your 0.5 x plus 0.1 line, peoples 2.55 Andi, I'll just substitute this in. So instead of x o. C. Three y minus nine plus 0.1. Why he goes to 0.55 This is your 0.15 Why minus is your 0.45 plus 0.1 y equals 2.55 is your 0.25? Why equals three? And finally I have why equals three divided by 20.25 which gives me 12. And why is the number of times so I have 12 times on dhe exes. Three times Y minus nine. I have three times total minus nine. So this is 36 minus in time, and this gives me 27 Nichols.

So, firstly, in the question, it says there are five more dimes and quarters, so I'm gonna have why equals Explosive Fi? And also it says nine more nickels and quarters. The number of nickels is nine more than the quarters, and the next they conform my equation. Joe quantify explosive 0.1 by plus, Still going to I said you called 2.75 second substitute These into the main equation toe have just one unknown. So, uh, X plus five. Plus, you're a bunch of by X plus nine equals 3.75 So this is gonna be through one X plus your one stop one show five x plus sarah 0.45 equals 3.75 So I added up all the ex terms, I'm gonna have 0.25 point 35 0.4 x and 0.5 on the left side. Subtract that on the right side to get 3.25 and so x is gonna equal are sorry. I made a mistake here. This is actually going to be 3.75 minus 145 plus 0.5 is your 0.95 and now ex is gonna be point being divided by you're going for just seven. You got 7/4. Why is exposed by which is seven plus 5 to 12? Uh, dimes that is exposed. Nine. So seven plus nine, which is are nine, which is 16 Nichols.

That each thing is a penny and the value of a penny is 0.1 0.1 And so we got 0.1 for the pennies as the value plus when you have a dime a dime is 10 cents of 10 times time and then 1/4 is 25 cents a 250.25 Q is equal to $3 and 69 cents a 3.69 So we're gonna be plugging into this eventually. But we gotta take apart some information. So here's what they tell us that the number of pennies is equal to three more than the number of dimes. 4/4 were told that their number of quarters is equal to two times the amount of times there are. So this is a situation where we can plug in and replace three plus D for P and two D for Q. And then we will have an equation with one letter one variable so that we can actually solve for the number of dimes first and foremost. But then our final goal is to also be ableto plugged back in and state what Thea not of pennies are and what the amount of quarters are. So we're gonna do some substitution. We're gonna dio 0.1 times three plus d plus 0.10 the chain there because we're focusing on the fact that all are variable letters will be the letter D plus 0.25 times to D is equal to the 3 69 So let's clean this up. The first thing we're gonna need to do is distribute year, so you're gonna get 0.3 plus 0.1 d plus 0.10 d waas point 50 p is equals e 69. So ready to combine like terms. So there's three life terms that can be combined. 50 cents plus 10 cents plus one cent is 61 cents at 61 beat plus point only, visible to $3.69. So to solve for D, for the number of dying's against attract point of re both sets and the threes were gonna cancel out. You're gonna be left with my sits one b is equal $3 says when we divide both sides by 61 cents, our number of dimes ends up being six of their six times and Now, our task is to take a bee's number of dimes. Andy plot back in up here to figure out the number of penny. So if there are six penny or sick of sorry, six times that, I'm gonna do three plus six. That's gonna give me the number of pennies. Just nine pennies. So that's an answer. And then, lastly, to figure out the number of quarters que is equal to two times six the number of dying, which is gonna be 12 12 quarters, nine Chinese and

So Given that there are 74 coins in total, all of which are either quarters dimes for nickels Represented by Q. D. & N. respectfully. The total value of all points is equal to $8.85. And there are four more 4th and Nickels and Dimes. And the question is asking how many of each coin does Matthew have? So how many quarters dimes and nickels? So to begin solving this, we can turn this into a system of equations. The first equation comes from the fact that there are 74 total coins and this means that the total number of quarters, Dimes and Nickels is equal to 74. The second equation comes from the fact that the total value of all the coins is go to the $8.85. This tells us that the value of each coin times the quantity of that point for all three coins is equal to $8.85. So 0.25 times Q plus 0.1 times D plus 0.5 times end He called to 8.85. And the last equation comes from are the fact that there are four more quarters and nickels and dimes and this can be written as Q plus n is equal to D plus four. And voila, we have our three equations for a system of equations. The next step in solving this would be to use matrices to solve our system. So we need to turn this, our system of equations into a coefficient matrix, a variable matrix and a result matrix. So let us rewrite the equation so we can more clearly see the coefficient our first equation is Q plus the plus an Is equal to 74. And since there's no numbers in front of our variables, all the coefficients are one. And let us add that to the first row of the matrix. Let's make that look a little nicer our second equation and I'll wait just a moment to put the coefficients in. Our coefficients are 0.25 Plus 0.1, g plus 0.05. So our coefficients are 0.25, 0.1 And 0.05. And our final equation is Q U plus n is equal to D plus four. But we need to move all of the variables to one side of the equal sign. So we have we can rewrite this as q minus t plus at Is equal to four. It's our coefficients would be one -1 and positive mm. There we go for a variable matrix. We just include all three of our variables here and they have to be in the same order as our equations except from going across its down. So since we start with q q is our top variable and then D is in the middle and it's at the bottom just like our equations. And then finally for our result matrix we just include are the three results we got from the equations or the three values that don't have variables attached to them. So this would be 74 8.5. And for so now the last step is solving for the variable matrix. And we were able to do this because if we have if we have two matrices and we're multiplying them by each other, let's say we have one matrix named a, another matrix named X. And that's equal to another product, matrix named B. Uh We can rewrite this in terms of X. If we multiply each side by the inverse of A. So the matrix X is equal to the inverse of matrix A times B. And the same thing is going on up here where our X matrix or the matrix we're trying to solve for is our variable matrix and this is that and our coefficient matrix is the one we have to manipulate. And we multiply that matrix by our results results matrix. So to do this, let's first rewrite are matrices in terms of X or are variable matrix. Mhm 0.10.05. And we're taking the inverse of us to move it to the side of the equation of the equal sign. And we're multiplying this by 74 8.5. And for. And to solve this, we're going to be using a graphing calculator. So let me put it up real quick and we can enter all the digits into the matrix by using the matrix function on our calculator. I've gone in heaven, entered in all the values already to save some time. But here's what it would look like. The 1st 1-3 by three and here are all of the coefficients. And the second matrix is a three x 1. And it just has our result values in here. So 74, and four. The last step is we need to multiply the inverse of A times B. So a. To the -1. Power times matrix B. Our our results matrix And this gives us the Matrix 1735 and two. So our variable matrix Q D N is equal to the matrix 17 35 22. And this tells us That the total number of corridors is equal to 17. The total number of dives is equal to 35 And the total number of Nickels is equal to 22.