Compound Interest Formula Explained, Investment, Monthly & Continuously, Word Problems, Algebra



in this video we're going to work on some word problems using the compound interest formula there's two equations that you need to know a is equal to P times 1 plus R divided by n raised to the N times T now in this formula P is basically the principal that's how much you would deposit in a savings account or a checking account on something like that a represents the future of value of that amount in the account after it's been credited for interests over a period of time so you can think of P as the present value how much you put in in the present and a is like how much it will be worth 10 or 20 years later r is the annual interest ring so let's say if the annual interest rate is 8% you need to plug in point zero eight for our you need to convert eight percent into a decimal to do that you would divide by 100 n is how many times you receive interest in a given year so let's say if it's compounded monthly that's 12 months in a year so n would be 12 for muffle now what about for weekly what is the value of n there's 52 weeks in years so n will be 52 daily and it's 365 quarterly n is 4 semi-annually and is 2 and annually and is 1 and T is basically the time in years now there's one more equation that you need and it's this one a is equal to P times e raised to the RT a and P are basically the same as the last equation P is the principle that's the amount that you put in to an account a is the future value after some time maybe 10 or 20 years later he is basically the inverse of the natural log function so if you have your calculator to find e you may have to type in shift from natural log or second Ln something like that R is the annual interest rate as a decimal and T is the time in years so when do we use this equation compared to the other equation now if you hear the key word compounded continuously use this equation if it's compounded times something else let's say monthly daily weekly quarterly you would use the other equation but the only time you would use this equation if the problem says it's compounded continuously so let's work on some problems susan puts 20,000 in a savings account paying 8% annual interest compounded monthly at this rate how much money will be in the account after 40 years so it's not compounding continuously therefore we need to use this equation so P is the principle she puts in twenty thousand in the account R is the annual interest rate which is eight percent if we divide that by 100 that's point zero eight and she receives that total eight percent annual interest in twelve months so basically that eight percent is divided into twelve so her account is credited with interest every month so we're going to divide this point zero eight by n which is twelve and we want to find out how much money will be in the account after forty years T is forty point zero eight divided by 12 plus one is basically one point zero zero six repeating twelve times 40 is 480 so if you type this in a calculator you should get a value of around four hundred eighty five thousand four hundred sixty seven dollars and seventy nine cents so that's how much money will be in the account after forty years so as you can see it pays to save early here's another problem you can pause the video and work on it john wants to have two million for e time in forty five years he invests in a mutual fund pain an average of nine point five percent each year compounded quarterly how much should he deposit into his mutual fund so we need to use this equation a equals P one plus R divided by n raised to the NT so we know the future value he wants to have two million in his account so he needs to decide how much he should put in now to get to that level so we're looking for paying the problem R is the annual interest rate nine point five divided by 100 is 0.09 five and it's compounded quarterly that is four times a year so four times a year his account is credited with interest so we're going to divide it by four and then it's raised to the NT or four times forty five T is the time in years so you can type it in exactly the way you see it let's find out what this value is equal to first point zero nine five divided by four is point zero two three seventy five and let's add one to it so that's one point zero two three seven five and four times 45 is 180 so this is going to be P times sixty eight point three seven six one five two so to solve for P one needs to divide both sides by this number so P is 2 million divided by sixty eight point three seven six one five two so that's going to be about twenty nine thousand two hundred forty nine and ninety six cents so if he invests about 29,000 let's round it to two fifteen if he invest that much and if he finds an account pain and an interest rate of 9.5% compounded quarterly then in forty five years he should have two million in his retirement so if he starts invested in the famous twenties by his mid or upper 60s he can have that much in savings so as you can see due to the effect of compound interest it pays to save early Sara wishes to turn her ten thousand investment into a hundred thousand and twenty years how much interest does she need to receive compounded annually to reach her goal so in this problem we need to solve for R so let's use this equation again so a is the value in twenty years she wants a hundred thousand P is her initial deposit the principal which is ten thousand R is the annual interest rate which we're looking for and n is one since its compounded annually which means that she receives interest once per year and T is 20 so the first thing we should do in order to solve for R is divide both sides by ten thousand a hundred thousand divided by ten thousand is ten so basically she wants to multiply her investment by a factor of ten so now what can we do to solve our how can we get rid of this exponent in order to open the parentheses we need to turn the 20 into a 1 to do that raise both sides to the reciprocal of 20 or run over 20 20 times 1 over 20 is 1 so we have is 10 raised to the 1 over 20 equals 1 plus our so to solve for our we need to subtract both sides by one so it's 10 raised to the 1 over 20 minus 1 10 raised to the 1 over 20 is about one point one to two and subtracted by 1 this is equal to point one to two now to turn into a percentage multiplied by one hundred percent so R is 12 point two percent so if she wants to multiply her investment by a factor of 10 she needs an account that is paying 12 point two annual interest if he could find that than in 20 years she's going to multiply her investment by a factor of 10 so if she invests a hundred thousand and twenty years is going to be a million if she invests two hundred thousand 20 years is going to be 2 million Mary invests 50 thousand dollars into an index annuity that's averaging 8.4 percent per year compounded semi-annually at this ring how many years will it take for her account to reach 1 million so let's write the equation a is equal to P times 1 plus R divided by n raised to the NT so her goal is to reach a million that's d that's the a value so to speak her investment the principle is 50,000 the interest rate is 8 point 4 percent which is point zero eight four and it's compounded semi-annually which means she receives interest twice a year so n is two so what we need to do is solve for T so first let's divide both sides by 50,000 so what's the 1 million / 50,000 that's equal to 20 so she wants to multiply her investment by 20 point zero eight four divided by two plus one is one point zero four two in order to solve 14 we need to use logarithms so let's take the log of both sides so on the Left we're going to have a log 20 and on the right we're going to have a log one point zero 4 2 raised to the 2t a property of logs allows us to take the exponent and move it to the front so therefore we now have is log 20 is equal to 2t times log 1.0 for tune now to get T by itself let's divide by 2 log 1.04 to both sides so therefore T is equal to I'm going to take it one step at a time log 20 is about one point three zero one zero three log one point zero four two times two is point zero three five seven three five four if you divide these two numbers you should get thirty six point four years if I typed it in correctly mistakes do happen but this is how long it's going to take her to multiply her investment by a factor of twenty so in thirty six point four years if she can find an account that is averaging eight point four percent per year in interest she could turn this fifty thousand investment to a million Juliette invests a hundred thousand in an account paying seven point two percent interest compounded continuously how much money will be in her account after 30 years now anytime you see this key expression compounding continuously this is the equation that you need so we're looking for the future value of her account 30 years from now so we're solvent for a we have her principle investment it's 100,000 and the interest that she's receiving is 7.2 percent or point zero seven two as a decimal and her account will be active for 30 years point zero seven two times 30 is basically two point sixteen and E which is the inverse of the natural log function e raise to two point one six is about like eight point six seven one one three seven six something times 100,000 so her investment is going to be worth eight hundred sixty seven thousand one hundred thirteen dollars and seventy seven cents mark wants to have 1.5 million in 50 years how much should he invest now in an account paying 12% interest compounded continuously so here is our key expression which means we need to use this equation again so we have the future value the value in 50 years so that's a and this problem we're looking for ping we need to know how much he should deposit into his account in order to reach this goal R is 12% or 0.12 and the time is 50 years so first let's multiply point 12 times 15 and that's equal to 6 now to get P by itself let's divide both sides by e to the 6 so P is going to be 1.5 million divided by e to the 6 and basically this is equal to three thousand seven hundred and eighteen dollars and thirteen cents which seems very very small but the reason why this small amount turns into this large amount is because of the time 50 years is a long time that's one and two the interest rate is much higher than the interest of the other problems which were like seven 8% a 12% interest rate compounded continuously will greatly increase this account value over a long period of time as you can see a small investment was greatly multiplied over 50 years John invest five million in an account paying eleven percent interest compounded continuously how long will it take for his investment to turn into two million so let's try this problem so we have the same formula a is equal to pert and we have the future value of two million and his deposit of five thousand R is point 1 1 but this problem will looking for T so let's begin by dividing both sides by 5000 so if you wish to do this in your head you can get rid of three zeros so you have 2000 divided by 5 mm is basically 20 times 120 divided by 5 is 4 4 times 100 is 400 so if you take 2 million and divided by 5,000 it will give you 400 so we have 400 is equal to e raised to the point 11 T now instead of using log we're going to use natural log the reason being is the natural log of E is equal to 1 so natural log 400 is equal to the natural log e raised to the point xi e so whenever you have a variable in the exponent you can use the log function or the natural log function but when you're dealing with E is easier to deal with or use the natural log function so what should we do now once you get to this part take the exponent and move it to the front so we have the natural log of 400 and that's equal to point 11 T times the natural log of e now the natural log of e as you mentioned is 1 so that's just going to disappear so our last step is simply to divide both sides by point 11 natural log of 400 is about 5.99 146 if we divide that by point 11 this is going to be 54 point four seven so that's how long he needs to invest if he wants to have 2 million now 5,000 is a small investment but if he invests early that's the key he can take the advantage of the effective compound or compound interest his money will grow to 2 million if he does it in if he invest early let's say 54 years early so as you can see whenever you invest early and you can use time to help multiply your investment now granted this effect will be greatly increased if you can find account our savings account or checking account that's paying a very high interest rate typically in mutual funds and indexed annuities is probably the best place where you can get such high interest rates but that is it for this video thanks for watching and have a great day

Constructing an Amortization Schedule 141-37



this video is provided as supplementary material for courses taught at Howard Community College and in this video I'm going to show how to construct an amortization schedule so imagine you've got a situation that's like this you've taken out a loan for $15,000 it's at four-and-a-half percent compounded monthly and you've got monthly payments for three years and we have to make an amortization table for that situation okay so two steps we have to do before we can start filling in the table the first thing is we're going to use the present value formula and we'll figure out what the monthly payment amount should be so I've already done that and given this this situation the monthly payment would be 446 dollars and 20 cents the other thing we have to do is figure out what the interest rate for each payment period is so what we're going to do is take the angular annual interest rate which is four and a half percent we'll turn that into a decimal that would be 0.055 and then we want to divide that by the number of compounding periods every year and since we're making monthly payments we would divide that by twelve so if we do that calculation we get point zero zero three seven five that's the monthly interest rate so I'm going to put that in here now in reality in a normal table we wouldn't fill in these two amounts I wouldn't put the four hundred and forty six thousand twenty cents up here at the top where it says payment and I wouldn't feel any interest rate but I want to be able to refer to that so I'm just going to leave it in there for now okay now we can start filling in the table at zero months there are no payments you've just taken out the loan and so there's no interest on that and there's no balance reduction but we have to fill in the unpaid balance the unpaid balance in this first line is the original amount of the loan so we're going to start out with $15,000 as unpaid balance now at the end of the first month we have to make a payment the payment is four hundred and forty six dollars and twenty cents it's always going to be that amount I want to figure out how much of that payment was interest so what I have to do is multiply the interest rate this point zero three seventy five times the unpaid balance when I do that multiplication I get fifty six dollars and twenty five cents so in other words out of the payment fifty-six twenty five cents was interest the rest of it was paying off the principal so what we're going to do is find the unpaid balance reduction all that is is what happens when you take the payment and subtract the interest so we'll take the payment $446 and 20 cents subtract the interest which for this month was fifty six dollars and twenty five cents and we get three hundred eighty nine dollars and ninety five cents this is the unpaid balance reduction which means it's the amount we're reducing the unpaid balance by so we'll take this amount and subtract it from the unpaid balance which is fifteen thousand dollars and what I'm going to get is fourteen thousand six hundred and ten dollars and five cents and now we're going to begin the process again for the second month we'll fill in a payment the payment is always going to be four hundred forty six dollars and twenty cents and then we want to find out how much of that was interest so once again we'll take that monthly interest rate and multiply it by the new unpaid balance in other words we're multiplying the interest rate by whatever the bottom line is in the unpaid balance column which is now fourteen thousand six hundred ten dollars and five cents when I do that multiplication I get fifty four dollars and seventy nine cents that's the amount of interest I paid I want to find out how much of the principal I paid which is what happens when I take the payment and subtract the interest when I do that what I get is three hundred ninety one dollars and forty one cents that's the amount I'm reducing the unpaid balance by so I'm always going to be taking the bottom line in this unpaid balance reduction column and subtracting it from the bottom line in the unpaid balance column or in other words subtracting three hundred ninety one dollars and forty one cents from fourteen thousand six hundred ten dollars and five cents cents that means the new unpaid balance is fourteen thousand two hundred eighteen dollars and 64 cents okay I'm going to do one more line of this once again now we're at the end of the third month the payment is four hundred forty six dollars and 20 cents I want to find out how much of that was interest so when I get to the interest column what I'm always going to do is multiply the interest rate for that period in other words in this case is the monthly interest rate times the bottom line in the unpaid balance column when I do that multiplication I get fifty three dollars and 32 cents when I get to the unpaid balance reduction column what I'm always going to do is take the payment and subtract the bottom line in the interest column so when I do that I get three hundred ninety two dollars and eighty-eight cents and then when I'm ready for the unpaid balance column what I do is I take the unpaid balance reduction the bottom line in that column and subtract it from the bottom line in the unpaid balance column to give me the new unpaid balance so in this case we're going to get thirteen thousand eight hundred twenty-five dollars and seventy six cents and then I would begin again and before I finished I would get rid of these notes that I made for myself okay now I've got a nicer version of this this is the same numbers we did plus a couple more months I just want to make some comments about this what you should notice is that your interest the numbers in your interest column are getting smaller and smaller as you get further and further down in the schedule you should also notice that the numbers in the unpaid balance reduction column are getting bigger and bigger if you wanted to check things as you went along you could multi could add the number in the interest column and the number next to it in the unpaid balance reduction column and you should always get the payment amount when you look at the unpaid balance column that number should also be getting smaller and smaller now when you get to the very end of this process you would expect that there would be a zero in the unpaid balance column in other words there should be no more balance left to pay probably you're going to end up with either a few cents as a positive number or a few cents as a negative number the reason for that is this in a problem like this where we've got 36 payments and you're rounding numbers off each time you work to a payment it's very likely that you're going to end up end up off by one or two cents either way okay so if you're off by a couple of cents don't worry about it doesn't mean you made a mistake it just means is what we call a rounding error okay that's about it take care I'll see you next time

Geometry: 1-3 Measuring & Constructing Angles



geometry and constructing angles all right it sticks in one batch three men constructing angles all right the world ruler in this section is pretty much just about angles okay then one of the basic things in geometry example exhibit this right here that's an angle okay made up by two rays from meet at a youth vertex that point right there it's called a vertex all right now in an angle some of the basic things you need to know inside that angle call the interior outside everything outside of it all called the there's stuff thanks to you exterior is very good all right like I said before the point where they meet is called the town virgin all right good deal next thing you know about angles it's called measure mostly I've heard the word measure before it's talking about how big or small some is right here how far it's open it's called the in a world thanks for being in small measure comes into play all right professor all right now we measure vegetables in degrees okay a degree most time you think about what temperature oil guess what it's by this like it was 45 degrees that's the degree symbol okay that's how open or why these things are now all the way around would be how many degrees my circle is under 360 360 is all the way around so pretty much every degree is one out of that 360 okay well we have an angle and went all the way around be 360 there were times I equals there's a cute which means that it is less than 90 all right it's less than 90 degrees they begin in there listen cute is less tremendously right right angles you'll see these a lot and do a lot of stuff with them right angle means it is exactly 90 degrees in cynically 90 degrees all right now we usually mark it by putting that little box there because guess what I'm a corner piece of paper that's 90 degree angle boxes perfect squares are oh nine degree angles so we put a little square there for 90 degrees all right and then obtuse is take a log s greater stop it please greater than 90 degrees all right so you got a cute write that up too lovely now a straight angle is the last one you won't see it a lot called straight usually just called the line but that's what it is exactly all the way across so therefore it's 180 degrees okay so you got a cute which is less than I mean the way love you'll remember that is it's a cute blue angle okay you think of a cute little puppy alright Sam not at all but that's the way some people memory okay smaller right we just crack away it's perfect okay it's 90 degrees I'm too sis bigger and if it's a straight line it's always 180 degrees that all these angles one solution do the work good deal yeah last thing we talked about the words congruent and bisect yesterday congruent means it's the exact same which means easily so two angles are congruent you can just set them equal to each other all right so current means equal bisector means it cuts it in half so if you have an angle please stop all right if you have an angle and let's say the angle is 60 degrees all right you have an angle bisector if it box of X and it cuts it in the middle okay therefore 60 will become half so it be learning and urgent very good all right heart regime problems they never succeed in this math game find out on the next episode of Dragon Ball Z new label angels that is how you do it all right say we want to talk about this thing right here ABC okay we just right angle a b c the important thing when you label angles you put the little angle front of it put all three points that are involved the important thing is the middle point it has to be the vertex it has to be where you know it changes directions see we're changing directions at the B that means it has to be in the middle okay that pretty much labels how bangle is examples how do you do them all right pretty much if they just give it to you in words you got to draw it out they said L is in the interior of angle of jkm all right so first control jkm okay he has to be the vertex because it's in the middle and we know L is an interior that means it's somewhere inside this okay so we'll just draw help all right now they wants to find JK m 2j k m so we're trying to find the whole thing it's a measure of JK l is 42 degrees please don't say that every time you say next one is l km is 28 degrees now obviously if we're trying to find the whole thing and they gave us some number one part and another part you can put it together and find the whole thing's all you do is you had it 42 plus 28 now equal 70 degrees you know the whole thing is 70 degrees and you do the same thing say they gave us the whole thing and apart and we want to find the other part we were just subtracted like we do the whole thing with 70 and we knew that part was 42 we wanted to find this right here we would just do 72-42 and we would get 20 degrees good job

Constructing a Dilation 128-3.12



this video is provided as supplementary material for courses taught at Howard Community College and in this video I'm going to demonstrate how to construct a dilation of an image so we're going to start with the image of this triangle triangle ABC I'm going to dilate it by a scale factor of 2 and I'll use point P as the center of dilation so the first step is going to be to draw array starting at point P and going through point a so use a straightedge for that I'll draw that ready I'll continue quite a ways then I'm going to take my compass and set it so it's the same distance as line segment PA and with a point of the compass over to point a and draw an arc along the line so with that arc intersects the line is twice as far from point P as point a was so I've dilated that line by a scale factor of 2 so that's going to be that point there where the arc intersects the line is going to be point a prime of my dilated image now we'll repeat that process two more times for each of the other points on the original image so I'll draw array starting at Pig and going through point B I'll set a compass equal to the distance for line segment PB move the point of the compass over to point B draw an arc where the arc intersects the line is now going to be point B Prime and one more time this time the Rays going to start at line say at Point P and go through Point C take the compass set it for the length of line segment PC and repeat that length with an arc that's going to get me point C Prime and now for my new image all I have to do is connect points a prime B Prime and C Prime so there's line segment a prime C prime B prime B prime and B prime C Prime so triangle a prime B prime C prime is a dilated image of triangle ABC I used to scale factor of 2 and the center of dilation was at Point P that's it take care I'll see you next time

Constructing a Parallel Line Through a Given Point 128-2.21



this video is provided as supplementary material for courses taught at Howard Community College and in this video I'm going to demonstrate how to construct a parallel line through a given point using a compass and a straightedge there are a number of methods for doing this here's the one I'm going to use I've drawn a line which I'm calling line ail and above the line I've got point P what I want to do is draw a line through point P that is parallel to L now here's how I'm going to do it I'm going to place a point on line L I'll call that point a and you can put this point just about anywhere then I'm going to draw a line through points a and pig now that line that I've just drawn creates an angle at a my next step is going to be to copy that angle over to point P so I'll do this by placing my compass point on point a and drawing an arc that goes through those two lines and then I'll copy that I place in the compas point on pig and drug another arc and now I'm going to copy the angle at a by setting my compass equal to the distance between where the arc touches each of those two lines now I copy that distance onto my second arc and now I can create a parallel line by connecting point P where the place where these two arcs intersect now I've got a line which is parallel to my first line here's how I know they're parallel I've got two lines and transversal that's the second line that I drew the corresponding angles angle a and angle P are congruent if that's the case those two lines have to be parallel lines so now I know that I've drawn a line parallel to line L to point P okay that's it take care I'll see you next time