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Compute APY from compound interest: What, why, how - YouTube
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hi in this video i explain
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effective weight or annual percentage
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yield ppy and show how to compute it
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annual percentage u or apy
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is defined as the simple interest rate
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that would have yielded the same return
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or ending balance as the given
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compound interest rate
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this is what it means let's say there's
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one account with a five percent interest
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rate
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starting at this black dot here simple
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interest rate grows like a straight line
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so it grows like this and
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over one year you end up say at this
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point over here
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now that's simple interest rate and you
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know the compound interest goes
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faster than simple interest so
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if we have a second account also with a
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five percent interest rate
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but it's compounded daily instead
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then the second account would go faster
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than the first account
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the growth is in a curve an exponential
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curve like this
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and by the end of the year so at the
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same
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ending time it would end up all the way
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up there at the blue dots
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so here's the question if i want to make
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a simple interest account that can catch
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up with this compound interest account
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then it will need to be some sort of
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higher interest rate
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in order to catch up to this point right
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there
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so what kind of high interest rate well
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i need some straight line because this
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simple interest
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that's higher than the red line so i
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need this third account right here
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that has a higher interest rate to catch
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up to the compound interest
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account the interest rate for that third
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account
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the dashed green line that is
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the apy is the interest rate that would
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catch up to the given five percent
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compound interest
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right so that's the theory now let's go
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compute
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some apy
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before computing the apy let's try this
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example
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an account with unknown interest rate
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has principal one thousand dollars
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and grows to 1030.42 in one year
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if this account were a simple interest
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account what would be its rate so we're
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assuming simple interest
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so we're using the simple interest
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formula
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in this account the principal is one
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thousand dollars so p
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equals one thousand and it goes to
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one thousand thirty dollars and forty
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two cents
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so the interest amount is i is equal to
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103 0.42
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minus one thousand so it's thirty
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dollars and forty two cents
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and the term t is one year
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and the interest rate r is unknown
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so we use a simple interest formula i is
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equal to p
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or t and we plug it in we have
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30.42 is equal to 1000
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times r times one
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30.42 is equal to 1000r
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so r is equal to 30.42
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divided by 1000 is equal to 0.03042
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or 3.042 percent
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moving on let me put on another problem
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right here
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example two an account with principle of
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one thousand dollars has interest rate
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of three percent compounded monthly
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compute it balance after one year it's a
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compound interest problem so we're going
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to use the compound interest formula
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a is equal to p times 1 plus r
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over n to the power n of t
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and we're asked for a we ask for the
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ending balance
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uh we're given that p is equal to a
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thousand
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we're given that r is equal to 3 or 0.03
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n is the number of times compounding per
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year
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and this is monthly so n is equal to 12
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times a year
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and t is one year so we plug all of this
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in we get one thousand times one plus
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0.03 over 12
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to the power mt which is 12 times one
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and all that is just 12. so if you plug
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all this into your calculator and you do
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it carefully
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you get 1030 and 42 cents
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ha look at that what a coincidence
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so right now if i were to ask you what
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kind of
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simple interest what kind of simple
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interest
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would get you the same result of one
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thousand thirty dollars and 42 cents
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then all you do is you would look up to
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the previous problem and you tell me
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well it says right here 3.042
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because this would be a simple interest
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account that would achieve the same
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ending balance in one year
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that's apy you have found the apy
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by definition apy is the simple interest
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rate that would get you the same result
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as the given compound interest rate
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so what these two problems together tell
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us
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is that a three percent compounded
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monthly interest rate
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is equivalent to an apy of 3.042
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simple interest rate so we can put both
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problems into one
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and that will be an apy problem
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example three an account has interest
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rate of two percent compounded daily
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compute it's apy notice that this
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problem doesn't say
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what the principle is and the reason is
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it doesn't matter
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it only matters what the interest rate
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is and what the
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compounding period is it doesn't matter
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what the principle is
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you're gonna say wait okay so if it
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doesn't matter
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i still need it for my calculations
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right
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true but since it does not matter you
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get to pick your own
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principle let's write it down
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uh we're missing the principle but
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because it doesn't matter we got to
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choose our own
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and i usually pick p equals 1 000.
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it's nice and round and it gives me just
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about the right number of
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decimals to solve this apy problem we're
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gonna do the previous two examples in
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reverse right
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we're gonna do is the example two first
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find the final balance
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and based on that final balance we're
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gonna do like example one
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and find the simple interest rate so
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these are the steps
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step one compute the ending balance
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after one year so for this step we're
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going to need to use the compound
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interest formula
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a is equal to p times 1 plus r
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over n to the power n t
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that will get us the ending balance and
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once we have the ending balance
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we're gonna compute back if it were
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simple interest what would the simple
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interest be
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part two find the equivalent
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simple interest rate that means we're
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gonna use
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simple interest formula i is equal to p
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or t
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this is a new r it's not the same or as
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the one
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in step one it's not this two percent
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remember the graph is the interest rate
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of the new green
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dotted line
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so use i equals prt and solve
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for a new little r and that would be
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the apy so let's do this
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step one p is we're gonna pick a
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thousand dollars
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or is 0.02
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n is compounded daily so that's 365
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times a year
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and t is one year for apy calculation
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because the definition is based on one
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year
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t is gonna be one all the time then a
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is one thousand times one plus
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.02 divided by 365
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to the power n is 365
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times one and i put all of this in my
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calculator
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and i get one thousand twenty dollars
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and twenty cents all right step two i'm
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gonna find the equivalent
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simple interest rate using i equals prt
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so new unknown r
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p is still one thousand t is
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still one year and i
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is the interest part of this so the
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interest part is twenty dollars and
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twenty cents
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twenty dollars and twenty cents is equal
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to p
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rt and prt is one thousand
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times r times one so
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twenty dollars and twenty cents is equal
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to one thousand
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or and i divide by one thousand both
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sides i have
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or is equal to 20.20 divided by 1000
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which is 0.0202 which is
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2.02 percent
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that's my apy apy is 2.02 percent
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all right hope that helps thanks for
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watching
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bye
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