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**Introducing the Mathematics of Finance**

Simple Interest Introducing the Mathematics of Finance

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**Notes about Simple Interest**

For a short-term loan (perhaps a loan to a friend), simple interest may be used. For simple interest, the interest is determined just once, at the end of the loan. Later, we’ll study compound interest where the interest is applied at regular periods.

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3 months of interest Suppose you make a loan of $2000 for which you charge 8% annually in simple interest. If the loan is repaid at the end of 3 months, how much interest should you be paid?

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3 months of interest If the rate is 8%, the interest is 8% of $2000 per year. A total of ($2000)(0.08) = $160 interest for one year. But since the loan was for just 3 months or 3/12 of a year, you are paid this fraction of the yearly interest. The interest paid is ($2000)(0.08)(3/12) = $40.00

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**Simple Interest Formula**

For a loan of P dollars at a simple interest rate r, the interest I is given by where t is the duration (in years) of the loan. The total amount A due at the end of the loan is dollars borrowed + interest charged

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18 month investment Suppose a short-term investment offers 10% simple interest. If you invest $4000 in such an account, what is the value of the investment 18 months later? The interest earned is Prt = ($4000)(0.10)(18/12) = $600. And so, the total value of the investment is A = P + Prt = $ = $4600.

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Investing today Again, suppose a short-term investment offers 10% simple interest. How much should be invested in the account today in order for the account to have a future value of $7500 at the end of 18 months? Here, we’re asked to find the initial amount P. And so, we note the “present value” of the account.

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**Present Value To have a future value of $7500 at the end of 18 months,**

is the amount we need to invest today. Note the parentheses!

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**Now and Later Initial amount or principal, P**

Accumulated total value, A “present value” “future value” $ $ 7500. 0 years after 1.5 years

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**Examples Suppose a short-term investment offers 6.5% simple interest.**

If you invest $1200 in such an account, what is the value of the investment 14 months later? How much needs to be invested today in order for the account to have a value of $1500 at the end of 14 months?

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**Find the simple interest rate**

You borrow $1400 and agree to pay simple interest. At the end of 18 months, you write a check for $1589 to payoff the loan. What was the rate of simple interest? Set up the equation A = P + Prt , then solve for r.

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Time to double? How long does it take for an investment to double in value if it earns 15% simple interest? Since the value has doubled, we know A = 2P.

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