Lesson 1: How Your Money Changes Appreciation & Depreciation

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1 : How Your Money Changes Appreciation & Depreciation Learning Target I can solve Appreciation and Depreciation word problems I can calculate simple and compound interests In your own words write answer the following question: 1. In your own words describe appreciation and depreciation (in terms of value in dollars) What does it mean appreciates in terms of money? There are many things in the real world that grow faster or decrease slower as they get smaller. We can use exponential functions to model certain situations. Some exponential functions show growth whole others show Decrease or decay. Discuss: Why do banks pay you to provide their services? Example 1 Quick Write Kyra has been babysitting since 6 th grade. She has saved $1000 and wants to open an account at the bank so that she will earn interest on her savings. Simple Bank pays simple interest at a rate of 10%. How much money will Kyra have after 1 year? After 2 years, if she does not add money to her account? Is there a formula we can create to find this amount? After 5 years?

2 Example 2 Jack has $500 to invest. The bank offers an interest rate of 6% compounded annually. How much money will Jack have after 1 year? 2 years? 5 years? Year Amount of money Interest Total amount Simple Interest Interest is calculated once per year on the original amount borrowed or invested. The interest does not become part of the amount borrowed or owed (the principal). I = Prt Compound Interest Interest is calculated once per period on the current amount borrowed or invested. Each period, the interest becomes a part of the principal. A = P(1 + r) t Example 3: A rare coin appreciates at a rate of 5.2% a year. If the initial value of the coin is $500, fill in the table to model the coin value 3 years. t How much will the coin be worth after 8 years When a quantity grows by a fixed percentage at regular intervals, the pattern can be represented by a function. In the word problems this refers to appreciation or compounded interest f(t) f(t) = a ( ) a = initial amount r = Depreciation rate (converted into a decimal) t = number of time intervals that have passed Write the formula that will model the value of the coin after t years

3 Example 3: Ryan bought a new computer for $2,100. The value of the computer decreases by 50% each year. t f(t) What will be the value of the computer after year 4? When a quantity depreciates by a fixed percent at regular intervals, the pattern can be represented by a function. In the word problems this refers to depreciations. f(t) = a ( ) a = initial amount r = Depreciation rate (converted into a decimal) t = number of time intervals that have passed Write a formula that models the value of the computer each year KEY IDEA When a quantity grows or decreases by a fixed percent at regular intervals, the pattern can be represented by the function: Exponential Functions -Growth f(t) = a(1 ) t Exponential Functions Decay f(t) = a(1 ) t

4 : How Your Money Changes Appreciation & Depreciation Problem Set 1. On average a new car depreciates in value approximately 18% per year. Your parents purchased a new car in 2013 and paid $26,000. a. Write an equation to model the value of the car b. Then use your equation to approximate the value of your parent s car in Mike deposits $350 in an account that pays 2% interest compounded annually. If there are no additional deposits or withdrawals, how much money is in the account after 6 years? 3. Daniel s Print Shop purchased a new printer for $35,000. Each year it depreciates (loses value) at a rate of 5%. What will its approximate value be at the end of the fourth year? (1) $33, (2) $30, (3) $28, (4) $27, The value, y, of a $15,000 investment over x years is represented by the equation y = 15000(1.2) x 3. What is the profit (interest) on a 6-year investment? (1) $6,600 (2) $21,600 (3) $10,799 (4) $25, Mr. Smith invested $2,500 in a savings account that earns 3% interest compounded annually. He made no additional deposits or withdrawals. Which expression can be used to determine the number of dollars in this account at the end of 4 years? (1) 2500( ) 4 (2) 2500( ) 4 (3) 2500( ) 3 (4) 2500( ) 3

5 6. Kathy plans to purchase a car that depreciates (loses value) at a rate of 14% per year. The initial cost of the car is $21,000. Which equation represents the value, v, of the car after 3 years? 1) 2) 3) 4) 7. A car depreciates (loses value) at a rate of 4.5% annually. Greg purchased a car for $12,500. Which equation can be used to determine the value of the car, V, after 5 years? 1) 2) 3) 4) 8. The value of a car purchased for $20,000 decreases at a rate of 12% per year. What will be the value of the car after 3 years? 1) $12, ) $13, ) $17, ) $28, The Booster Club raised $30,000 for a sports fund. No more money will be placed into the fund. Each year the fund will decrease by 5%. Determine the amount of money, to the nearest cent, that will be left in the sports fund after 4 years. 10. Adrianne invested $2000 in an account at a 3.5% interest rate compounded annually. She made no deposits or withdrawals on the account for 4 years. Determine, to the nearest dollar, the balance in the account after the 4 years.

6 11. $250 is invested at a bank that pays 7% simple interest. Calculate the amount of money in the account after 1 year; 3 years; 7 years; 20 years. Work space: 1 year: 3 years: 7 years: 20 years: 12. $325 is borrowed from a bank that charges 4% interest compounded annually. If you decide to default (not make any payments), how much would you owe the bank after 1 year; 3 years; 7 years; 20 years? Work space: 1 year: 3 years: 7 years: 20 years: 13. A bank is advertising that new customers can open a savings account with a 3 3 % interest rate 4 compounded annually. Robert invests $5,000 in an account at this rate. If he makes no additional deposits or withdrawals on his account, find the amount of money he will have, to the nearest cent, after three years. 14. Challenge Suppose the population of a certain endangered species is 15,000. Scientists believe that each year 2/3 of the population will remain in comparison to the previous year. Determine the population in four years. Write a formula to justify your answer