9 months 1 year = 0.75 years 1 12 months

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1 Free Pre-Algebra Lesson 4 page 1 Lesson 4 Ierest The financial world is in large part based on loaning and borrowing money at ierest. A credit union is a good example of how this works on a small scale. Members of the credit union who save, earn ierest. The ierest comes from loaning the money saved to other members, who pay ierest for the privilege of borrowing. There are a variety of ways to calculate ierest, but they are all based on some perce of the money invested or borrowed. The money invested or borrowed is called the principal (P). The principal is the base of the perce. The perce is called the ierest rate (r). The time during which the money is invested or borrowed is conveniely called the time (t). The amou in the perce equation is the ierest (I), the extra money paid to the investor or by the borrower. Simple Ierest In Lesson 42 we calculated the simple ierest on some loans and savings. Example: Charlie had $4944 in his savings, earning 1.5% ierest per year. After a year the credit union deposited the ierest in his accou. How much was the ierest? amou = perce base ierest = rate principal = $4944 = $4.16 The equation with just the amou, perce, and base, represes one year of ierest, since the ierest rate is 1.5% per year. If the money were there only half a year, it should earn half as much ierest (although the terms of the accou may or may not allow this.) If the principal stays the same and is invested for two years, it should earn twice as much ierest. So the formula for ierest can be adjusted so that we can accou for the time the money is invested. I = Prt Ierest ($) = Principal ($) rate (% per year) time (years) Example: Olivia had $,868 in her savings, earning 1.9% ierest per year. How much ierest did she earn in 9 mohs? I = Prt Ierest ($?) = Principal ($868) rate (1.9%/year) time (9/12 years) The time is eered in terms of years, so we convert 9 mohs to years: 9 mohs 1 year = 0.5 years 1 12 mohs I = ($868)(0.019)(0.5) = $ $ The amou of ierest in 9 mohs is 9/12 of the ierest for the year.

2 Free Pre-Algebra Lesson 4 page 2 Example: Syeda saved $10,000 and invested in a -year CD earning.5% ierest per year. How much ierest did she earn over the course of the three years? I = Prt Ierest ($?) = Principal ($10,000) rate (.5% / year) time ( years) I = ($10,000)(0.05)() = $50 $1050 Syeda earned $50 ierest each year, and that comes to $1050 over the course of years. But suppose instead of receiving the ierest payme, Syeda had the ierest invested in the same accou each year. Year 1: There is $10,000 dollars invested at.5% so the ierest is $50. Syeda deposits the ierest io the accou and the new balance is $10,50. Year 2: Now there is $10,50 invested at.5%, so the ierest is $ Although the ierest rate has stayed the same, the principal has increased, so the amou of ierest earned has increased. Syeda deposits the ierest io the accou. The new balance is $10, Year : The principal is $10,12.25, the rate is.5%, so the ierest is $ She will get $4.9. By re-investing the ierest rather than taking it out of the accou, the ierest earned is greater. The total amou of ierest earned in the -year time period is not $1050, but $ ($50 + $ $4.9). The re-investme of ierest, so that ierest earns ierest, is called compound ierest. Compound Ierest When an accou earns compound ierest, the terms of the accou will include how often the ierest is added. The number of times each year that the ierest is added to the accou is called the number of compounding periods (n). Some common compounding periods are: Ierest is added. Annually once per year. n = 1 Semi-annually every 6 mohs, which is twice per year. n = 2 Quarterly every mohs, which is four times per year. n = 4 Mohly every moh, twelve times per year n = 12 Weekly every week, 52 times per year n = 52 Daily every day, but some financial institutions use a 60-day calendar. n = 65 n = 60 While we can calculate compound ierest as we did above for Syeda s accou, it would be very computation-iensive if we waed to calculate ierest compounded daily for 20 years. Instead the repetitive work is built io a formula: The variable P still stands for principal, the variable r is still the rate, and the variable t is still the time, in years. As above, n is the number of compounding periods. This formula calculates the total amou in the accou at the end of the given time period, rather than just the ierest, so the variable A stands for the amou of money in the accou, including the original principal.

3 Free Pre-Algebra Lesson 4 page Example: Calculate Syeda s accou balance after years using the compound ierest formula. She invested $10,000 at.5% ierest compounded annually. We substitute P = $10,000, r = 0.05, n = 1 (compounded annually), t =. A = $10, $ 1 % = $10,000 ( 1.05) = $11,08.18 The accou will have $11,08.18 at the end of the three years. Since the original principal was $10,000, the ierest earned is 1 11, ,000 = $1,08.18 This is the same result we found by compounding by hand in the previous example. The compound ierest formula is like other formulas you just determine which numbers should go where and plug them in. But the order of operations can make the formula tricky to compute with a calculator. Pay atteion to the order: A = $10, $ 1 % 1 Simplify inside pareheses first. You divide, then add 1. = $10, = $10, ( ) 1 Multiply n t in the expone. ( ) The expone applies to what is in pareheses. ( ) Expones come before multiplication in the order of operations. = $10, = $11,08.18 Key poi: Don t multiply the principal uil the LAST STEP. Example: When Nicole was born, her grandmother put $2,000 in an accou for her than earned 4.2% ierest compounded quarterly. Now she is 18. How much money is in the accou? A = $2, $ 4 % = $2, ( ) 2 ( ) = $2, = $4, Nicole has $4,242.2 in her accou.

4 Free Pre-Algebra Lesson 4 page 4 Combining the Ierest From Two Accous Investors sometimes divide their principal io differe accous with differe ierest rates. Usually some accous are riskier investmes with higher returns, and others are safer, but with lower ierest rates. Example: Adé invested a total of $50,000 in two accous. She put $20,000 in an accou earning 2% simple ierest per year, and the rest in an accou earning 5% simple ierest per year. What is the total ierest expected from the two accous at the end of the year? Ade started with $50,000, and put $20,000 in one accou and the rest in the other. So she must have put $50,000 $20,000 = $0,000 in the other accou. INTEREST FROM ACCOUNT 1 + INTEREST FROM ACCOUNT 2 = TOTAL INTEREST 2% of $20, % of (the rest of the $50,000) 5% of $0, $20, $0,000 $400 + $1500 = $1900 The only tricky part of this problem is to figure out how much is invested in the second accou. Otherwise it is just two simple ierest problems put together. The main reason for practicing these problems now is that they are often found in algebra texts in a slightly tougher format. If you get used to how they work in this simpler situation, it will help in your algebra course. Ierest on Loans and Credit Cards It s certainly more fun to think about earning ierest than paying ierest, but unfortunately, most of us find ourselves paying ierest more often than receiving it. Mortgages, car loans, and credit cards are common loans that allow people to have and use their house, car, clothing, furniture, etc. before they are paid for. The ierest consumers pay on loans and credit cards is usually much higher than the ierest they can earn in savings or CDs. Further, the ierest on these loans is often calculated in quite complex ways (usually not to the advaage of the consumer). Still, it s importa to get some grasp on how much you re paying in ierest so you can make an informed decision about your situation and future loans and purchases. In the lab for this lesson you can try some online calculators that are set up with the formulas actually used by financial institutions in the U.S. You can calculate mortgage paymes and ierest, and find out how much you pay over the price of the house on your mortgage. Another calculator figures credit card ierest and lets you choose a mohly payme amou over the minimum to find how long it will take to pay off your cards. You may be shocked.

5 Free Pre-Algebra Lesson 4: Ierest Worksheet Simple Ierest I = Prt I = Ierest earned, in $ P = Principal invested, in $ r = % ierest rate per year, written as a decimal t = time invested, in years Lesson 4 page 5a Name Compound Ierest Same as simple ierest, except A = amou in accou after t years n = number of compounding periods per year 1. Find the simple ierest earned if $,500 is invested for six mohs at 1.6% per year. 2. Find the simple ierest earned if $6,200 is invested for years at 1.8% per year.. Liling has $60,000 to invest in two accous. She puts $25,000 in one accou earning 2.1% simple ierest and the rest in another accou earning 4.% simple ierest. How much ierest will she receive from the two accous at the end of the year? 4. $50,000 was invested at.5% compounded quarterly for 5 years. How much was in the accou at the end of that time? 5. $80,000 was invested for 18 years at 6.2% compounded daily. How much was in the accou after 18 years?

6 Free Pre-Algebra Lesson 4 page 6 Lesson 4: Ierest Homework 4A Name 1. Alison had a basket with 140 apples to peel. She was 4/5ths done with the job. How many apples had she peeled? 2. The large roll of plastic wrap coained 200 feet. How many 2 1 / 2 foot lengths can be cut from the roll?. Find equivale fractions with a common denominator Find the sum and difference. a b Solve the equation. 8 x = Solve the equation. 5y = 1. Fill in the missing values. FRACTION DECIMAL PERCENT 8/25 19/ Round answers to the nearest teh of a perce if rounding is necessary. a. 19 of the 2 people thought the deal was a good one. What perce of the people is that? % 5% b. 8 of the 000 packages we astray. What perce of the packages we astray? 9/1000

7 Free Pre-Algebra Lesson 4 page 9. Ari told me that 45% of the 200 studes failed the first exam. How many studes failed? 10. In the other section, 4 studes failed the exam. This was 2% of the studes. How many studes are in that section? 11. Rose bought her friend a diaper bag for a shower gift. It cost $ The sales tax rate was 9.25% How much was the gift, including tax? 12. Pete left a $5 tip for his meal that cost $ What perce did he tip? 1. The hardcover edition was originally priced at $24.99, but it was offered for 44% off. How much did the book cost? 14. If you invest $1,000 at 1.2% simple ierest, how much ierest will you earn in 8 mohs? 15. $10,000 is split between two accous. $2,000 is invested at % simple ierest and the rest at % simple ierest. How much ierest did the two accous earn last year? 15. $12,250 is invested at 4.25% ierest compounded mohly. How much is in the accou at the end of eight mohs?

8 Free Pre-Algebra Lesson 4 page 8 Lesson 4: Ierest Homework 4A Answers 1. Alison had a basket with 140 apples to peel. She was 4/5ths done with the job. How many apples had she peeled? 4/5 of 140 apples is how many? = = 112 She had peeled 112 apples. 2. The large roll of plastic wrap coained 200 feet. How many 2 1 / 2 foot lengths can be cut from the roll? How many 2 1 /2 are in 200? = = = 80 You can cut 80 lengths each 2 1 /2 feet.. Find equivale fractions with a common denominator = = 5 50 = = Find the sum and difference. a. b = = = = Solve the equation. 8 x = x = 49 8 x = Solve the equation. 5y = 1 5y = 1 5y + = 1 + 5y = 20 5y / 5 = 20 / 5 y = 4. Fill in the missing values. FRACTION DECIMAL PERCENT 8/ % 19/ % 11/ % 1/ % 5/ % 5/ % 9/ % 8. Round answers to the nearest teh of a perce if rounding is necessary. a. 19 of the 2 people thought the deal was a good one. What perce of the people is that? 19/2 = about 82.6% b. 8 of the 000 packages we astray. What perce of the packages we astray? 8/000 = = 2.6%

9 Free Pre-Algebra Lesson 4 page 9 9. Ari told me that 45% of the 200 studes failed the first exam. How many studes failed? 45% of = studes failed the first exam. 11. Rose bought her friend a diaper bag for a shower gift. It cost $ The sales tax rate was 9.25% How much was the gift, including tax? $54.90 = $ to the nearest ce, $ In the other section, 4 studes failed the exam. This was 2% of the studes. How many studes are in that section? % of the studes is 6 studes 0.x = 6 0.x / 0. = 6 / 0. x = 20. about 20 studes 12. Pete left a $5 tip for his meal that cost $ What perce did he tip? $5 / $26.50 = He left about 19%. 1. The hardcover edition was originally priced at $24.99, but it was offered for 44% off. How much did the book cost? 100% 44% = 56% 0.56 $24.99 = $ It cost $ If you invest $1,000 at 1.2% simple ierest, how much ierest will you earn in 8 mohs? I = Prt I = 1, /12 I = 8 You will earn $ $10,000 is split between two accous. $2,000 is invested at % simple ierest and the rest at % simple ierest. How much ierest did the two accous earn last year? 10,000 2,000 = 8,000 % of % of 8000 is 0.0 2, ,000 = = 620 $620 ierest 15. $12,250 is invested at 4.25% ierest compounded mohly. How much is in the accou at the end of eight mohs? A = $ 12 % A = ( ) 8 ( ) = = There iss $12, in the accou.

10 Free Pre-Algebra Lesson 4 page 10 Lesson 4: Ierest Homework 4B Name 1. Carlos has to read 240 pages for his history class. He is 2/ done. How many pages has he read? 2. The large roll of aluminum foil coained 500 feet. How many 1 / 4 foot lengths can be cut from the roll?. Find equivale fractions with a common denominator. 4. Find the sum and difference. 10 a b Solve the equation x = Solve the equation. y = Fill in the missing values. FRACTION DECIMAL PERCENT 1/1000 1/ Round answers to the nearest teh of a perce if rounding is necessary. a. 6 of the 55 studes ate lunch on campus. What perce of the studes is this? % 1000% b. 40 of the 5000 park visitors lost their tickets. What perce of the visitors is this? 1/10,000

11 Free Pre-Algebra Lesson 4 page Corey had completed 80% of the 60 units required for his degree. How many units had Corey completed? 10. There were 80 calories from fat, which was 40% of the total calories. How many calories in all? 11. Hamid bought a new laptop, which cost $656. If the sales tax rate was 8.25%, how much was the sales tax? 12. Hamid left a $4 tip for his meal that cost $ What perce did he tip? 1. The CD was originally $18.95, but was on sale for 4% off. What was the sale price? 14. If you invest $,000 at 1.4% simple ierest, how much ierest will you earn in mohs? 15. $20,000 is split between two accous. $,000 is invested at 2% simple ierest and the rest at 6% simple ierest. How much ierest did the two accous earn last year? 15. $18,50 is invested at.15% ierest compounded mohly. How much is in the accou at the end of nine mohs?