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CCSS.Math:

you put $3,800 in a savings account the bank will provide 1.8 percent interest on the money into account every year another way of saying that the money in the savings account will grow by 1.8% per year write an expression that describes how much money will be in the account in 15 years so let's just think about this a little bit let's just think about the starting amount so on the start we're just going to put three thousand eight hundred dollars we could view that as year zero year actually let me write it that way so the start is the same thing as year zero and we're going to start with three thousand eight hundred dollars now let's think about year one how much money will we have after one year well we would have the original amount that we put three thousand eight hundred dollars and then we're going to get the amount that we get an interest and they say that the bank will provide 1.8 percent interest on the money in the account so it'll be plus 1.8 percent times 3838 our $3,800 or our $3,800 and we could also write this as a decimal this is equal to three thousand eight hundred plus and I'll just write I'll switch the order of multiplication here plus three thousand eight hundred times zero point zero one eight one point eight percent is the same thing as eighteen thousand or one point eight hundredths depending on how you want to pronounce it and so here you might say well it's kind of an interesting potential simplification mathematically here I could factor a three thousand eight hundred out of each of these terms I have a three thousand eight hundred here I have a three thousand eight hundred here so why don't I factor it out essentially undistribute it so this is going to be three thousand eight hundred times when you factor it out here you get a one plus when you factor it out here you get zero point zero one eight and so I could just rewrite this as three thousand eight hundred times one point zero one eight so this is a interesting time to pause we're not at the full answer yet how much we have in 15 years but we have an interesting expression for how much we have after one year notice that if the money is growing by 1.8% or another way it was growing by 0.018 that's equivalent to multiplying the amount that we started the year with by 1 plus the amount that it's growing by or one point zero one eight and once again why does this make intuitive sense because at the end of the year you're going to have the original amount that you put that's what that one really represents and then plus you're going to have the amount that you grew by so you multiply both the sum here times the original amount you put and that time what you'll have at the end of year one what about year two so year two well we know what we're going to start with in year two we're going to start with whatever we finished year one with so we're going to start with 3,800 times one point zero one eight but then it's going to grow by 1.8% or grow by zero point zero one eight we already said if you're going to grow by that amount that's equivalent at to multiplying it by one point zero one eight well this is the same thing as three thousand eight hundred times one point zero one eight to the second power I think you see where this is going every time we grow by 1.8% we're going to multiply by one point zero one eight and if we're thinking about fifteen years in the future we're going to do that 15 times so one year in the future your exponent here is essentially one two years your exponent is two so year fifteen I can just cut to the chase here so year fifteen well that's just going to be we're going to have the original amount that we invested and we are going to grow one point zero one eight fifteen times so we're going to multiply by this amount fifteen times to get the final amount and even and one of the fun things this is actually called compound growth where every year you grow on top of the amount that you had before you'll see if you type this into a calculator but even though 1.8% per year does not seem like a lot over 15 years it actually would amount to a reasonable amount but this is the expression we they're not asking us to calculate it they just want us to know an expression that describes how much money will be in the account in 15 years