Solve each problem. See Example 4. Cody sells some property for $...

Question

Solve each problem. See Example 4. Cody sells some property for $240,000. The money will be paid off in two ways: a short-term note at 2% interest and a long-term note at 2.5%. Find the amount of each note if the total annual interest paid is $5500.

Accepted Answer

Hey, everyone in this problem, Sam takes a loan of 572,000 from two banks. If the interest rate of one bank is six And that of another bank is 7%,, we're asked to find out the amount borrowed at each rate. We're told that the total interest paid annually is $37,040. We're given four answer choices. Option A, The amount at the 6% rate is 300,000 and the amount at the 7% rate is 272,000. Option B the amount at the 6% rate is $272,000 and the amount at the 7% rate is $300,000. Option C the amount at the 6% rate is $200,000 and the amount at the 7% rate is $372,000. In option D, the amount at the 6% rate is $372,000 and the amount at the 7% rate is $200,000. Now, when we have a word problem like this, the first thing we wanna do is lay out this problem and make it very clear what all of our variables mean, sort out what we're looking for. Ok. That way if you get stuck and you need to come back and look at your problem, you can look through your work and understand what you're doing and whoever is reading your solution will be able to follow it easily too. So let's start by saying that X is going to be the amount borrowed From the 6% interest rate bank. And why is going to be the amount borrowed From the 7% interest rate bank? So what we're looking to find is X and Y now the interest we pay at 6%. OK. It's going to be 0.06. Ok. And that's on the value X and X is the amount borrowed from the 6% bank. So we're gonna have X Multiplied by 6%,, Which is equal to 0.06 X we can do the same thing for the interest At the 7% bank. OK. We're borrowing an amount why Multiplied by the interest of 7%. So we're gonna be paying 0.07 Y in interest at that bank. So if we're talking about our total interest, the total interest we pay on these loans, It's gonna be the interest of the, at 6% plus the interest at 7%. So we have 0.06 x Plus 0.07 Y. And we're told that that total annual interest is $37,040. And so the sum of these two is equal to 37, and $40. Now we're gonna call this equation two. OK. What about equation one? If we have an equation two, we should have an equation one, an equation one is gonna relate to X and Y we know we borrow a total of $572,000. And so X plus Y, the amount we borrow from one bank, plus the amount we borrow from the other bank should be equal to $572,000. And that is gonna be equation one. And let me add the dollar sign to our $37,000. And uh so sorry, $37,040 in equation two. So now we have two equations, equation one and equation two with two unknowns X and Y. So we've gone from a work problem down to two equations with two unknowns solved for the variables. We know how to do that, right. So we're gonna start, we're gonna take equation one, X plus Y is equal to 572,000. We're gonna isolate one of the variables in order to use substitution method. So let's isolate X. We can write this as X is equal to $572, -6. OK. We're gonna call this one, equation one star. It has the exact same information as equation one. It's just been rearranged. So that's gonna be one star And we are gonna substitute that into equation two. And let me, All right. So equation two tells us that 0. multiplied by X. Ok. Well, we know that X is going to be 572,000 minus Y And then we have plus 0.07, y is equal to $37,040. If we expand on the left hand side, it is 0.06 has to apply to both terms. We get 34,000 320. Mine is 0.06, Y Less 0.07. Y is equal to 34. Whoops. Not 34, We can move this 34,320 to the right hand side in order to isolate why. And when we do that, it's going to subtract if we simplify the Y term negative 0.6, Y plus 0.7, Y gives us 0.1 Y And this is equal to 2720. And when we subtract that 34,320, We can divide to isolate y and we get that Y is equal to $272,000. Remember why is the amount borrowed? From the 7% bank. And this is why we write down our variables at the start. So once we're done solving, we're not confused about what, why means we've written it down, we know that why is the amount borrowed from the bank with a 7% interest rate? So we have our value of Y and now we need to find the corresponding value of X. Now we can substitute this back into any equation, either one or two that we'd like Equation one is gonna be simpler to solve for because the coefficient on the X is just one. So let's substitute back into equation one. So we're gonna substitute into equation one and we had X plus Y is equal to $572,000. Hm That means that X plus $272,000 Is equal to $572,000. We can subtract that 272,000 to isolate X. And we get that X is equal to $300,000. All right. So we have our x value of $300,000. We have our y value of $272,000. Let's go back up. Look at our answer choices. Compare what we found. We found that the amount That Sam Borrows from the 6% bank is $300,000 and the amount they borrow from the 7% interest rate bank is $272,000, which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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Solve each problem. See Example 4. Cody sells some property for $240,000. The money will be paid off in two ways: a short-term note at 2% interest and a long-term note at 2.5%. Find the amount of each note if the total annual interest paid is $5500.