1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 4.8 Applications: Profit, Simple Interest, and Compound Interest

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand percent of profit. o Calculate simple interest. o Calculate compound interest.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Percent of Profit Profit and Percent of Profit Profit:The difference between selling price and cost. Profit = Selling price – Cost Profit = $100 – $80 = $20 Percent of Profit:There are two types; both are ratios with profit in the numerator.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Profit and Percent of Profit (cont.) 1.Percent of profit based on cost: (Cost is the denominator.) Profit and Percent of Profit

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Profit and Percent of Profit (cont.) 2.Percent of profit based on selling price: (Selling price is the denominator.) Profit and Percent of Profit

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A retail store markets calculators that cost the store $45 each and are sold to customers for $60 each. a.What is the profit on each calculator? Solution To find the profit, The profit is $15 per calculator. Example 1: Calculating Percent of Profit

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Calculating Profit and Percent of Profit (cont.) For b. and c., use a ratio and then change the fraction to a percent to find each percent of profit. b.What is the percent of profit based on cost? Solution For profit based on cost, remember that cost is in the denominator.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Calculating Profit and Percent of Profit (cont.) c.What is the percent of profit based on selling price? Solution For profit based on selling price, remember that selling price is in the denominator.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Formula for Calculating Simple Interest I  P ∙ r ∙ t, where I =interest (earned or paid) P =principal (the amount invested or borrowed) r =rate of interest (stated as an annual rate) in decimal number or fraction form t =time (one year or fraction of a year) Simple Interest

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Simple Interest Formula for Calculating Simple Interest (cont.) Note:For calculation purposes, we will use 360 days in one year and 30 days in a month. However, many lending institutions now (with the advent of computers) base their calculations on 365 days per year and pay or charge interest on a daily basis.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Calculating Simple Interest You want to borrow $2000 from your bank for one year. If the interest rate is 5.5%, how much interest would you pay? Solution Use the formula for simple interest: with You will pay $110 in interest.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Sylvia borrowed $2400 at 5% interest for 90 days (3 months). How much interest did she have to pay? Solution Sylvia had to pay ________ in interest. Completion Example 3: Calculating Simple Interest

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 3: Calculating Simple Interest (cont.) If you know the values of any three of the variables in the formula I = P ∙ r ∙ t, you can find the unknown value by substituting into the formula and solving for the unknown. This procedure is illustrated in Examples 4 and 5.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Calculating Principal using Simple Interest What principal would you need to invest at a rate of 6% to earn $450 in 6 months? Solution Here the principal P is unknown. We do know

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Calculating Principal using Simple Interest (cont.) Substituting and solving for P, we have You would need to invest $15,000 to earn $450 in 6 months at a rate of 6%.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Calculating Time using Simple Interest Stuart wants to borrow $1500 from his father and is willing to pay $15 in interest. His father told Stuart that he would want interest at 4%. How long can Stuart keep the money? Solution In this case, the unknown is time t. We do know

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Calculating Time using Simple Interest (cont.) Substituting and solving for t gives Stuart can keep the money for (or 3 months).

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Calculate Compound Interest Step 1:Use the formula I = P  r  t, to calculate simple interest. Let where n is the number of periods per year for compounding. For example: for compounding annually, n = 1 and Compound Interest

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Calculate Compound Interest (cont.) for compounding semiannually, n = 2 and for compounding quarterly, n = 4 and for compounding bi-monthly, n = 6 and for compounding monthly, n = 12 and Compound Interest

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Calculate Compound Interest (cont.) for compounding daily, n = 360 and Step 2:Add this interest to the principal to create a new value for the principal. Step 3:Repeat steps 1 and 2 however many times the interest is to be compounded. Compound Interest

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Calculating Compound Interest If a savings account of $1200 is compounded annually (once a year) at 5%, how much interest will be earned in three years? Solution The account is compounded annually, so n = 1 and Use the formula for simple interest, I = P ⋅ r ⋅ t, with r = 5% = 0.05 and t = 1. The principal will change each year.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Calculating Compound Interest (cont.) a.First year: the principal is P = $1200 interest for the first year. b.Second year: the new principal is P = $1200 + $60 = $1260 interest for the second year. c.Third year: the new principal is P = $1260 + $63 = $1323 interest for the third year.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Calculating Compound Interest (cont.) The total interest earned in three years will be (Note that, because the principal is larger each year, the interest earned increases each year.)

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Calculating Compound Interest If an account of $5000 is compounded monthly (12 times a year) at 6%, what will be the balance in the account at the end of four months? Solution The account is compounded monthly, so n = 12 and Use the formula for simple interest, I = P ⋅ r ⋅ t, with r = 6% = 0.06 and The principal will change each month.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Calculating Compound Interest (cont.) a.First month: the principal is P = $5000. interest for the first month b.Second month: the new principal is P = $5000 + $25 = $5025.00. interest for the second month

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Calculating Compound Interest (cont.) c.Third month: the new principal is P = $5025 + $25.13 = $5050.13. interest for the third month d.Fourth month: the new principal is P = $5050.13 + $25.25 = $5075.38. interest for the fourth month

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Calculating Compound Interest (cont.) The total interest earned in four months will be The balance in the account will be $5000.00 + $100.76 = $5100.76.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Notes Loans in savings accounts, house payments, and credit card debts are based on compound interest, compounded daily over periods of years. The calculations, including monthly earnings or payments, are generally performed with computers. These calculations are related to the compound interest formula Compound Interest

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 1.A company manufactures and sells plastic boxes that cost $21 each to produce, and that sell for $28 each. a.How much profit does the company make on each box? b.What is the percent of profit based on cost? c.What is the percent of profit based on selling price? 2.If you were to borrow $1000 at 5% for nine months, how much interest would you pay? Practice Problems

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) 3.What interest rate would you be paying if you borrowed $1000 for 6 months and paid $60 in interest? 4.You deposit $1500 at 4% to be compounded semiannually. How much interest will you earn in 3 years? 5.A principal of $2500 is deposited at 6% to be compounded monthly. How much will the account be worth in 6 months?

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.a.$7b.c. 25% 2.$37.50 3.12% 4.$189.24 5.$2575.94