PA_M8_S4_T2_Applications Involving the Extended Simple Interest
Formula Transcript
The simple interest formula itself only gives us the interest. In order
to find out what our balance is, we have to add the interest accrued to
the account balance.
We can actually extend our simple interest formula. We know that our
simple interest is given by I = Prt, and therefore the amount in the
account, I'm going to call that A, is going to be equal to the initial
principle, P, plus the interest, which is Prt; A = P + Prt. I can use the
distributive property and factor out a P from both terms, and get (1 +
rt), and so here we have A is the amount that includes the principle and
the interest; A = P(1 + rt). So now that we've combined it all into a
single equation, so we don't have to do two separate equations, we can do
it all in one.
Example 1: Let's practice it. If $4000 is invested at 6 1/4% simple
interest what will the balance be after two years?
I'm going to use my A equation,
that's the amount after two years
that I'm looking for, = P(1 + rt).
I'm going to plug things in; my
initial investment is $4,000,
that's my principle, (1 +
0.0625*2). Using my order of
operations and completing what's
in parentheses first, I get
4000*1.125. I do this
multiplication and I found out that the amount in the account after two
years is $4,500. I could have broken it up into two steps, calculated the
interest, and then added that interest on to the principle, but I would
have ended up exactly the same place.
Let's try another couple of examples…
Example 2: Here's one where the balance due on a 2-year loan is
$3,360. If the simple interest rate is 6%, what was the principle
borrowed?
Well I know that the amount due, or the balance due is going to be my A
in my A equation. A = P(1 + rt), because that's going to be the total
amount of principle and interest that is going to come due. So I'm going
to plug in $3,360 for A, is equal to P, because I don't know what the
principle is, times (1 + 0.06*2). This becomes 3360 = P(1.12).
I'm going to divide through by 1.12 on
both sides to isolate the P, and I
find out that my principle borrowed is
going to be $3000.
Example 3: Let's look at another one. The balance due on a 2-year loan is
$2200, if the principal was $2,000 what was the rate of simple interest?
Once again the $2,200 is the amount owed that includes principal and
accrued interest. The principal itself is $2,000, and I know that in this
case, this is multiplied by 1 plus the rate that I don't know times 2. I
am going to go ahead and simplify the right hand side by distributing the
2000 first to the 1, and then to the r term and I get 2000 + 4000r. I'll
subtract 2000 from both sides and I get 200 = 4000r. I isolate the r by
dividing through by 4000, and I find out that my interest rate is equal
to 1/20 which is the same thing as 0.05 or 5%, and that's my interest
rate on that 2-year note.
These are some applications where we can use the extended simple interest
equation to calculate the total amount that will be a owed on a loan or
invested in an account when I have simple interest accruing over period
of time.
Example 4: Here is one final example that I want to work through for you.
We have Karen and Sam who need a new set of living room furniture and
they're at a store that offers two choices for their payment. The first
choice is that they make no payments for 12 months, but that interest of
10.9% accrues, and this is simple interest, and they are going to have to
pay it all off at the end of 12 months.
Their second choice is a two year loan at 5 1/2%, simple interest. If the
total purchase price for the furniture is $4,500 which option should they
choose?
Let's look at this, let's look at my A = P(1 + rt). In the first choice,
my amount is going to be given by $4500(1 + 0.109*1), this gives me a
total due at the end of one year of $4990.50. In the second option we
still have my A = $4500(1 + 0.055*2) this time, and when I do all of that
math I end up with only $4,995. The best option is the one year option if
they can afford to pay it off in 12 months, but you'll notice that
there's not a huge amount of difference between the two. $4.50 may not
make a lot of sense to try and save, if you don't feel you can pay this
off in 12 months.
So it's a very similar kind of offer, but it's a good exercise in trying
to figure out how these different options that you are offered for buying
on credit will play out in the end.