Simple Interest Formula

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Louise Miller
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1 Accelerated Precalculus 5.7 (Financial Models) 5.8 (Exponential Growth and Decay) Notes Interest is money paid for the use of money. The total amount borrowed (whether by an individual from a bank in the form of a loan or by a bank from an individual in the form of a savings account) is called the principal. The rate of interest, expressed as a percent, is the amount charged for the use of the principal for a given period of time, usually on a yearly (that is, per annum) basis. Simple Interest Formula If a principal of P dollars is borrowed for a period of t years at a per annum interest rate r, expressed as a decimal, the interest I charged is I = Prt Interest charged according to this formula is called simple interest. In problems involving interest, the term payment period is defined as follows: Annually: Once per year Semiannually: Twice per year Quarterly: Four times per year Monthly: 12 times per year Daily: 365 times per year (Actually, most banks use a 360-day year because the calculations used to be much simpler that way when they had to do them by hand.) When the interest due at the end of a payment period is added to the principal so that the interest computed at the end of the next payment period is based on this new principal amount (old principal + interest), the interest is said to have been compounded. Compound interest is interest paid on the principal and on previously earned interest.

2 Example 1 A credit union pays interest of 2% per annum compounded quarterly on a certain savings plan. If $1000 is deposited in such a plan and the interest is left to accumulate, how much is in the account after 1 year? The pattern of calculations performed in Example 1 leads to a general formula for compound interest. Compound Interest Formula The amount A after t years due to a principal P invested at an annual interest rate r, expressed as a decimal, compounded n times per year is A = P (1 + r n ) nt The amount A is typically referred to as the future value of the account, while P is called the present value. Example 2 Investing $1000 at an annual rate of 10% compounded annually, semiannually, quarterly, monthly, and daily will yield what amounts after one year?

3 Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is A = Pe rt Example 3 Find the amount A that results from investing a principal P of $1000 at an annual rate r of 10% compounded continuously for a time t of 1 year. Example 4 What annual rate of interest compounded annually is needed in order to double an investment in 5 years?

4 Example 5 (a) How long will it take for an investment to double in value if it earns 5% compounded continuously? (b) How long will it take to triple at this rate?

5 Many natural phenomena have been found to follow the law that an amount A varies with time t according to the function A(t) = A 0 e kt Here A 0 is the original amount (t = 0) and k 0 is a constant. If k > 0, the above equation states that the amount A is increasing over time; if k < 0, the amount A is decreasing over time. In either case, when an amount A varies over time according to the above equation, it is said to follow the exponential law or the law of uninhibited growth (k > 0) or decay (k < 0). Example 6 A colony of bacteria that grows according to the law of uninhibited growth is modeled by the function N(t) = 100e 0.045t, where N is measured in grams and t is measured in days. (a) (b) (c) (d) (e) (f) Determine the initial amount of bacteria. What is the growth rate of the bacteria? Graph the function using a graphing utility. What is the population after 5 days? How long will it take for the population to reach 140 grams? What is the doubling time for the population?

6 Example 7 A colony of bacteria increases according to the law of uninhibited growth. (a) If N is the number of cells and t is the time in hours, express N as a function of t. (b) If the number of bacteria doubles in 3 hours, find the function that gives the number of cells in the culture. (c) How long will it take for the size of the colony to triple? (d) How long will it take for the population to double a second time (i.e., increase four times)?

7 Example 8 Traces of burned wood along with ancient stone tools in an archeological dig in Chile were found to contain approximately 1.67% of the original amount of carbon-14. If the half-life of carbon-14 is 5730 years, approximately when was the tree cut and burned? Assignment: 5.7 (p ) #7, 9, 15, 17, 17, 29, 31, 39, 44, 50, 57 (read the explanation at the bottom of page 332), & (p ) #1, 3, 5, 7, 9, 11, 17, 19, 21

4.7 Compound Interest

4.7 Compound Interest 4.7 Compound Interest Objective: Determine the future value of a lump sum of money. 1 Simple Interest Formula: InterestI = Prt Principal interest rate time in years 2 A credit union