PRE-ALGEBRA

Lesson 7-7 Warm-Up

PRE-ALGEBRA Simple and Compound Interest (7-7) principal: the amount of money that is invested (put in to earn more) or borrowed (called a “loan”) Interest: the amount of money you pay the bank to borrow their money (loan) or the bank pays you to borrow your money (when you loan money to the bank, it’s called and investment) interest rate: the percentage paid for borrowing or loaning money simple interest: interest paid only on the principal (doesn’t include interest already earned on the principal) Rule: Simple Interest Formula: I = prt where I is the interest, p is the principal amount, r is the interest rate per year, and t is the time in years Example: If you deposit $500 into a savings account at a simple rate of 4% per year, how much would you have in the account after 6 years. I = prt Use the simple interest formula (p = 500, r = 4% = 0.04, t = 6) I = 500 · 0.04 · 6Substitute (replace) I = 20 · 6 or $120Simplify Total = Principal + Interest = $500 + $120 = $620 What is “principal”? What is “interest rate”? What is “simple interest”? How do you calculate “simple interest”?

PRE-ALGEBRA Simple and Compound Interest (7-7) Example: If you deposit $500 into a savings account at a simple rate of 4% per year, how much interest would you have earned after 3 months. I = prtUse the simple interest formula I = 500 · 0.04 · 0.25 Substitute: p = 500, r = 4% = 0.04, t = 3/12 = ¼ = 0.25 (write t as a fraction of a year) I = 20 · 0.25 = $5Simplify You would earn $5 in interest after 3 months.

PRE-ALGEBRA Suppose you deposit $1,000 in a savings account that earns 6% simple interest per year. a. Find the interest earned in two years. Find the total of principal plus interest. The account will earn $120 in two years. The total of principal plus interest will be $1,120. I = prtUse the simple interest formula. I = 1, Replace p with 1,000, r with 0.06, and t with 2. I = 120Simplify. total = 1, = 1,120Find the total. Simple and Compound Interest LESSON 7-7 Additional Examples

PRE-ALGEBRA (continued) b. Find the interest earned in six months. Find the total of principal plus interest. The account will earn $30 in six months. The total of principal plus interest will be $1,030. I = prtUse the simple-interest formula. I = 1, Replace p with 1,000, r with 0.06, and t with 0.5. I = 30Simplify. Total = 1, = 1,030Find the total. Write the months as part of a year. t = = = Simple and Compound Interest LESSON 7-7 Additional Examples

PRE-ALGEBRA Simple and Compound Interest (7-7) compound interest: interest paid on both the principal and on interest the principal has earned (called the balance) Rule: Compound Interest Formula: B = p (1 + r) n where B is the final balance, p is the principal amount, r is the interest rate for each interest period, and n is the number of interest periods Example: If you deposit $400 into a savings account at a rate of 5% compounded annually (once a year), what is the balance (how much would you have in the account) after 4 years. Method 1: Use the compound interest formula: B =p (1 + r) n B =p (1 + r) n Compound Interest Formula B = 400 ( ) 4 Substitute (replace): p = 400, r = 5% = 0.05, n = 4 B = 400 (1.05) 4 Simplify = 400 (1.05 · 1.05 · 1.05 · 1.05) = 400 ( )  $ Your balance after 4 years would be $ What is “compound interest”? How do you calculate compound interest?

PRE-ALGEBRA Simple and Compound Interest (7-7) Method 2: Use a Table Your balance after 4 years would be $ Principal at Beginning of Year InterestBalance Year 1: $400.00$ · 0.05 = $20.00$420 + $20 = $420 Year 2: $420.00$ · 0.05 = $21.00$420 + $21 = $441 Year 3: $441.00$ · 0.05 = $22.05$441 + $22.05 = $ Year 4: $463.05$ · 0.05 = $ » $23.15 $ $23.15 = $486.20

PRE-ALGEBRA Simple and Compound Interest (7-7) Example: If you compound the $400 from the above problem semiannually (every “half of a year, or twice a year) instead of annually, would it make a difference? B =p (1 + r) n Compound Interest Formula B = 400 ( ) 8 Substitute: p = 400, r = 5% ÷ 2 = 2.5% = (the interest for one year is divided into two interest payments for that year), n= 4 · 2 = 8 (there will be 8 interest payments in 4 years) B = 400 (1.025) 8 Simplify = 400 (1.025        1.025) = 400 (1.2184)  $ The balance after 4 years compounded semiannually is $ That’s about $1.16 more than if the same amount was compounded annually! This shows that more frequent compounding creates more interest on interest, so the balance gets bigger faster. Maybe, that’s why credit card companies compound your interest as often as possible. Does compounding more or less frequently make a difference?

PRE-ALGEBRA Year 5 : $ = = Year 6 : $ = Year 7 : $ = Year 8 : $ = You deposit $400 in an account that earns 5% interest compounded annually (once per year). The balance after the first four years is $ What is the balance in your account after another 4 years, a total of 8 years? Round to the nearest cent. After the next four years, for a total of 8 years, the balance is $ Interest Balance Principal at Beginning of Year Simple and Compound Interest LESSON 7-7 Additional Examples

PRE-ALGEBRA Find the balance on a deposit of $2,500 that earns 3% interest compounded semiannually for 4 years. The interest rate r for compounding semiannually is 0.03 ÷ 2, or The number of payment periods n is 4 years  2 interest periods per year, or 8. The balance is $2, B = p(1 + r) n Use the compound interest formula. B = 2,500( ) 8 Replace p with 2,500, r with 0.015, and n with 8. Round to the nearest cent.B 2, Simple and Compound Interest LESSON 7-7 Additional Examples

PRE-ALGEBRA Find the simple interest and the balance. 1.$1,200 at 5.5% for 2 years 2.$2,500 at 8% for 6 months 3.Find the balance on a deposit of $1,200, earning 9.5% interest compounded semiannually for 10 years. $3, $132; $1,332$100; $2,600 Simple and Compound Interest LESSON 7-7 Lesson Quiz