pre-algebra. lesson 7-7 warm-up pre-algebra simple and compound interest (7-7) principal: the amount...

PRE-ALGEBRA

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 · 6 Substitute (replace)I = 20 · 6 or $120 Simplify

Total = Principal + Interest = $500 + $120 = $620

What is “principal”?

What is “principal”?

What is “interest

rate”?

What is “simple interest”?

How do you calculate “simple interest”?

PRE-ALGEBRA


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 = prt Use the simple interest formulaI = 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 = $5 Simplify

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 = prt Use the simple interest formula.

I = 1,000 • 0.06 • 2 Replace p with 1,000, r with 0.06, and t with 2.

I = 120 Simplify.

total = 1,000 + 120 = 1,120 Find the total.

Simple and Compound InterestLESSON 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 = prt Use the simple-interest formula.

I = 1,000 • 0.06 • 0.5 Replace p with 1,000, r with 0.06, and t with 0.5.

I = 30 Simplify.

Total = 1,000 + 30 = 1,030 Find the total.

Write the months as part of a year.t = = = 0.512

612


Additional Examples

PRE-ALGEBRA


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 FormulaB = 400 (1 + 0.05)4 Substitute (replace):

p = 400, r = 5% = 0.05, n = 4B = 400 (1.05)4 Simplify= 400 (1.05 · 1.05 · 1.05 · 1.05)= 400 (1.21550625) $486.20

Your balance after 4 years would be $486.20.

What is “compound interest”?

How do you calculate compound interest?

PRE-ALGEBRA


Method 2: Use a Table

Your balance after 4 years would be $486.20.

Principal at Beginning of Year

Interest Balance

Year 1: $400.00 $400.00 · 0.05 = $20.00 $420 + $20 = $420

Year 2: $420.00 $420.00 · 0.05 = $21.00 $420 + $21 = $441

Year 3: $441.00 $441.00 · 0.05 = $22.05 $441 + $22.05 = $463.05

Year 4: $463.05 $463.05 · 0.05 = $23.152 » $23.15

$463.05 + $23.15 = $486.20

PRE-ALGEBRA


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 FormulaB = 400 (1 + 0.025)8 Substitute: p = 400, r = 5% ÷ 2 = 2.5% =

0.025 (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 1.025 1.025 1.025 1.025 1.025 1.025)= 400 (1.2184) $487.36

The balance after 4 years compounded semiannually is $487.36. 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 : $486.20 486.20 • 0.05 = 24.31 486.20 + 24.31 = 510.51

Year 6 : $510.51 510.51 • 0.05 25.53 510.51 + 25.53 = 536.04

Year 7 : $536.04 536.04 • 0.05 26.80 536.04 + 26.80 = 562.84

Year 8 : $562.84 562.84 • 0.05 28.14 562.84 + 28.14 = 590.98

You deposit $400 in an account that earns 5% interest compounded annually (once per year). The balance after the first four years is $486.20. 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 $590.98.

Interest BalancePrincipal at

Beginning of Year


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 0.015.

The number of payment periods n is 4 years 2 interest periods per year, or 8.

The balance is $2,816.23.

B = p(1 + r)n Use the compound interest formula.

B = 2,500(1 + 0.015)8 Replace p with 2,500, r with 0.015, and n with 8.

Round to the nearest cent.B 2,816.23


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. $3,035.72

$132; $1,332 $100; $2,600


Lesson Quiz

pre-algebra. lesson 7-7 warm-up pre-algebra simple and compound interest (7-7) principal: the amount...

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