Math 20 Math 25 Math Tips davidvs.net

# Math Lecture NotesPercent Homework

## Small Questions

### Mortgages

1. The Wahl family has an annual income of \$45,000. They want to spend 20% of that income on a mortgage with a twenty-year loan. How much are they prepared to pay each month?

2. Continuing with the Wahl family, if mortgage interest rates are at 6% then how large a loan can they afford?

3. Continuing with the Wahl family, how much will be paid total over the twenty years? How much of that is interest?

4. Does doubling the income double the size of the numbers? Next consider the Moneybag family that has an annual income of \$90,000, with everything else the same as the first problem. How large a loan can they afford? How much will be paid total over the twenty years? How much of that is interest?

5. Does halving the number of years either halve or double the size of the numbers? Next consider Mr. Short, who wants a ten-year loan, with everything else the same as the first problem. How large a loan can he afford? How much will be paid total over the ten years? How much of that is interest?

6. Brenda can afford to spend \$900 per month on mortgage payments. Currently mortgage rates are 8% per year. How big a twenty-year mortgage can she afford? How big a thirty-year mortgage can she afford?

7. When the length of the term (years) of a loan is made longer, does the monthly payment increase or decrease? Does the total amount paid increase or decrease?

8. When the length of the term (years) of a loan is made longer, does the affordable mortgage size increase or decrease?

1. The Wahl family wants to spend each year \$45,000 × 0.2 = \$9,000 per year. This becomes \$9,000 ÷ 12 = \$750 per month.

2. The Amortization Table value for 6% and 20 years is \$7.16. Then \$750 ÷ \$7.16 ≈ 104.750. That means a \$104,750 loan.

3. The monthly payment of \$750 happens twelve times per year for twenty years. \$750 × 12 × 20 = \$180,000 paid total. Of that amount, all the money is either loan or interest. So there is \$180,000 − \$104,750 = \$75,250 interest total.

Often mortgage loans are rounded to the nearest thousand dollars. If you did that, then the answers to problem 2 and 3 would be a \$105,000 loan and \$75,000 interest total. In real life, the bank will use a more detailed Amortization Table and the actual monthly payment will be specified to dollars and cents. Our estimate of an affordable house size is only an estimate.

4. Yes, doubling the amount you pay per month makes all the other numbers double. The Moneybag family pays \$1,500 per month for a \$209,500 loan. Over the twenty years they pay \$360,000 total, of which \$150,500 is interest.

Again, you could round the loan size to the nearest thousand and those answers change to a \$210,000 loan and total interest paid of \$150,000.

5. Mr. Short still has a monthly payment of \$750. Halving the years to ten years changes the Amortization Table value to \$11.10. (This is not double the \$7.16 from before, so the mortage size will not be halved.) The new mortage size is \$750 ÷ \$11.10 ≈ 68, so a \$68,000 loan. The monthly payment is unchanged, so \$750 happening twelve times per year but only for ten years. \$750 × 12 × 10 = \$90,000 paid total (half what was paid before, of course). There is \$90,000 − \$68,000 = \$22,000 interest total (much less than half the interest!).

6. The Amortization Table value for 8% and 20 years is \$8.36, which results in a \$107,700 loan. The Amortization Table value for 8% and 30 years is \$7.34, which results in a \$122,600 loan. (You could round these estimates differently.)

7. When the length of the term (years) of a loan is made longer, the monthly payment decreases but the total amount paid increases. Each payment is smaller, but they eventually add up to more.

8. When the length of the term (years) of a loan is made longer, the affordable mortgage size increases.

LCC Math 25 Packet Homework

Also do some percent review in the packet on pages:
• PF-1 to PF-5
• MS-28 to MS-29

And more mortgage problems in the packet on pages:
• PF-31 to PF-36

### Yield Percent

9. Imagine that you are preparing a large meal for a non-profit fundraiser. Your task is to make the provided meal for 60 people. Find how much of each ingredient you need to purchase, keeping in mind the yield percent for produce. When using eggs you will need to round fractional amounts to the nearest whole number. (Note: spices and directions are omitted for the sake of simplicity.)

10. Use this list of prices to find the total price for the recipes after scaling all three recipes to 60 servings. (Water is considered free.)

 brown rice: \$0.76 per pound carrots: \$0.55 per pound cauliflower: \$1.01 per pound heavy cream: \$2.87 per quart dry milk: \$4.30 per pound honey: \$3.31 per pound large eggs: \$2.75 per dozen milk: \$2.65 per gallon onions: \$0.22 per pound spinach: \$2.35 per pound vegetable broth: \$2.28 per quart whole wheat flour: \$0.57 per pound

9 and 10. Lots of numbers! Answers are on a Google spreadsheet. You can also save your own copy of that spreadsheet and try fiddling with the prices or other values to instantly see how the total food cost changes.

### Three Kinds of Probability

11. When rolling two dice, what is the probability of sum being seven?

12. When rolling two dice, what is the probability of sum being ten?

13. When rolling two dice, what are the odds of sum being seven?

14. When rolling two dice, what are the odds of sum being ten?

15. When rolling two dice, what is the expected value?

16. At a carnival, a booth offers a dice game in which you roll two dice and find the sum. If the sum is even the booth operator will pay you \$1. If the sum is odd you will pay him \$1. Is this a fair game?

17. Your little brother thinks that ten is a very big number. He wants to play a dice game about the number ten. He proposes a game where you each start with a pile of candies, and he finds the sum of two dice several times. Whenever the sum is less than ten, he gives you one candy. Whenever the sum is ten or greater, you give him more than one candy—but he is not sure how many is fair. Help your brother finish inventing his game by using an expected value table to find how many candies must you give him when he "wins" so that the game has an expected value of zero.

18. Anna is about to take her calculus final exam. Her grades so far in the class are listed below. The final exam has 30 problems, each worth one point. How many does she need to get correct to earn an overall grade of 80% in the class?

19. You are going to be the instructor of a Biology class and need to write the syllabus. You want attendance to be worth 10%, the final exam to be worth 30%, and the rest divided between nine lab reports and two midterms with each midterm worth as much as three lab reports. How much is each lab report worth? How much is each midterm worth?

11. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven. So the probability is 636, which we should reduce to 16 or change to about 16.7%.

12. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of seven. So the probability is 336, which we should reduce to 112 or change to about 8.3%.

13. Looking at the green boxes on the chart, we see that six out of thirty-six possibilities have a sum of seven, and thirty do not. So the odds are 6 to 30, which could be reduced to 1 to 5.

14. Looking at the pink boxes on the chart, we see that three out of thirty-six possibilities have a sum of seven, and thirty-three do not. So the odds are 3 to 30, which could be reduced to 1 to 10.

15. This answer requires making a table, as below. The answer is not surprising. Most people already know that the "average value" when rolling two dice is seven. The expected value table confirms that common knowledge is precise instead of rounded: the expected value is indeed seven exactly, not slightly more or less. (The original Google spreadsheet is here.)

16. Yes, the game is fair. The expected value table has two rows, and the sum of the products is zero. Overall you will gain and lose the same amounts, and end up at zero.

ResultValueProbabilityProduct
even+10.5+1 × 0.5 = +0.5
odd−10.5−1 × 0.5 = −0.5

17. We need to make an expected value table. We will do it from your little brother's point of view, so giving you a candy is negative but receiving candy is positive.

ResultValueProbabilityProduct
less than ten−130363036
ten or more+y636(6 × y)36

We want the game to be fair, which means the sum of the products should be zero. Therefore the numerators of the two products should match (but one is negative and one is positive). So 30 = 6 × y. Each time your brother wins you should give him 5 candies.

18. For the top four rows, multiply across to get products of 13.5, 16, 16.4, and 16.8. That toal is 62.7. Before the final, Anna has a 62.7% overall grade in the class. To earn an 80% she only needs the bottom row to have a product of 13.3. So we solve y × 0.25 = 13.3 to find an answer of 53.2 as her required final exam percentage. The final exam has thirty problems. So we ask, "What is 53.2% of 30?". Solve this to get 16 correct problems needed.

19. Be careful, this is not an expected value problem! All we are trying to do is have the graded values add up to 100%. The nine lab reports and two midterms need to sum to 100% − 10% − 30% = 60%. Since each midterm should be worth three lab reports, that means we have a total of 9 + 3 + 3 = 15 lab report equivalent things to together make up that 60%. So each is worth 60% ÷ 15 = 4%. In other words, lab reports are worth 4%. Midterms are worth three times that, so midterms are worth 12%.

LCC Math 25 Packet Homework

Also do some probability problems in the packet on pages:
• R-6 to R-12

### Restaurant Entrée Pricing Methods

20. A restaurant meal that serves four has \$32 food cost, \$60 labor cost, and \$15 other cost. Find the price per plate using to the food cost percentage method with a 30% scale factor.

21. Using those same costs, find the price per plate according to the desired profit method with a 10% desired profit?

22. A restaurant meal that serves six has \$50 food cost, \$70 labor cost, and \$25 other cost. Find the price per plate using to the food cost percentage method with a 30% scale factor, and then with the desired profit method with a 10% desired profit.

20. cost per plate = food cost ÷ scale factor ÷ servings = \$32 ÷ 0.3 ÷ 4 = \$26.67

21. cost per plate = (food cost + labor cost + other costs) × scale factor ÷ servings = (\$32 + \$60 + \$15) × 1.1 ÷ 4 = \$29.43

22. For the food cost percentage method, the cost per plate = food cost ÷ scale factor ÷ servings = \$50 ÷ 0.3 ÷ 6 = \$27.78 For the desired profit method, the cost per plate = (food cost + labor cost + other costs) × scale factor ÷ servings = (\$50 + \$70 + \$25) × 1.1 ÷ 6 = \$26.58

### Retail Markup and Discount

23. Grace works at a store that uses a 40% markup on wholesale cost. She orders an item for \$90. What will the markup be in dollars?

24. Grafton works at a store that uses a 30% markup on wholesale cost. He orders an item for \$200. For how much should he price the item?

25. Gavin sees that an item whose selling price is \$240 has a 40% markup on selling price. What is the dollar amount of this markup?

26. Georgina works at a vitamin store that uses a 75% markup on selling price. She needs to stock a certain bottle of vitamins for no more than \$3.50. How much can she allow a supplier to charge her store for this bottle of vitamins?

27. Galina has a clock that cost her \$62.50. She wants to sell it online for \$102.50, for a profit of \$40. What is the markup rate if measured as a markup on wholesale cost? What is the markup rate if measured as a markup on selling price?

28. Ginger works at a sporting goods store, and knows that a certain kind of skis will only sell if it is priced \$109.95 or less. Currently the price is \$120. What percent discount is needed?

29. Grant buys a jacket. What a great deal! It normally cost \$275, but there was a store-wide 20% off sale and he was also able to use coupon to reduce that sale price another 15%. How much did he pay for the jacket?

30. Geoffrey works at a candy store. He knows from past years' experience that after Valentine's Day he needs to reduce the prices of the special \$30 chocolate boxes down to \$18 to clear out that inventory. He uses a store-wide sale of 10%, hoping that will attract customers. He also distributes a coupon that further discounts the sale price of those expensive chocolate boxes. What percent discount is needed on the coupon?

23. For a markup on wholesale cost, markup in dollars = wholesale cost × markup rate = \$90 × 0.4 = \$36.

24. For a markup on wholesale cost, retail selling price = wholesale cost × (1 + markup rate) = \$200 × 1.3 = \$260.

25. For a markup on selling price, markup in dollars = retail selling price × markup rate = \$240 × 0.4 = \$96.

26. For a markup on selling price, wholesale cost = retail selling price × (1 − markup rate) = \$3.50 × (1 − 0.75) = \$3.50 × 0.25 = \$0.87.

27. The markup rate if measured as a increase from the cost is \$40 ÷ \$62.50 = 0.64 = 64%. The markup rate if measured as a decrease from the selling price is \$40 ÷ \$102.50 = 0.39 = 39%.

28. The change is \$120 − \$109.95 = \$10.05. So the discount rate is \$10.05 ÷ \$120 = 0.08375 ≈ 8%.

29. The final price is \$275 × (1 − 0.2) × (1 &miuns; 0.15) = \$187.

30. The discount chain equation is \$30 × (1 − 0.1) × r = \$18. Combine the first two numbers to get \$27 × r = \$18. Divide both sides by 27 to get r = 67%. Since that is what should remains after the coupon decreases the price, the discount should be 100% − 67% = 33%.

### Interest

#### Simple Interest

31. Calculate the simple interest on \$500 invested at 14% annually for three years and three months.

32. A business buys a copy machine for \$300 by borrowing that \$300 with a nine month loan of 15% simple interest. What is the total cost (the copy machine plus the loan's interest)?

33. Someone borrows \$5,000 for 120 days at 9.25% annual simple interest. How much interest must she pay?

34. How much was borrowed at 17% annual simple interest for six months if the interest was \$85?

35. Louie borrows \$1,500 for three months. At the end of that time he pays back the \$1,500 and also \$30 interest. What was the simple interest rate? (Hint: you may rearrange the order of three numbers being multiplied without changing the result.)

36. Dale buys a painting for \$1,000 and its value increases to \$1,030. What is the percent appreciation? Chip buys a share of a mutual fund for \$100 and its value increases to \$130. What is the percent appreciation? Why did the two items have different percent appreciation if they both increased by \$30?

31. Three years and three months is 3.25 years. Simple Interest = Principal × annual rate × time in years = P × r × t = \$500 × 0.14 × 3.25 = \$226.80

32. Nine months is 0.75 years. Simple Interest = P × r × t = \$300 × 0.15 × 0.75 = \$33.75. Add back the principal to get a total cost of \$300 + \$33.75 = 333.75.

We could have used the one plus trick.

Total = P × (1 + r) × t = \$300 × 1.15 × 0.75 = \$333.75.

33. Simple Interest = P × r × t = \$5,000 × 0.0925 × (120 ÷ 365) = \$152.05

34. Now we are solving for a missing P.

I   = P × r × t

\$85  = P × 0.17 × 0.5

\$85  = P × 0.085

\$1,000 = P

35. Now we are solving for a missing r. Three months is 0.25 years.

I = P × r × t

\$30 = \$1,500 × r × 0.25

\$30 = r × \$1,500 × 0.25

\$30 = r × \$375

0.08 = r

8% = r

36. This is not a simple interest problem! There is no time measurement. We are only comparing two examples of percent change.

Dale's percent change = change ÷ original = \$30 ÷ \$1,000 = 0.03 = 3%

Chip's percent change = change ÷ original = \$30 ÷ \$100 = 0.3 = 30%

The same total change is ten times more significant when the original amount is only one-tenth as big.

LCC Math 25 Packet Homework

Also do some simple interest problems in the packet on pages:
• PF-45
• PF-50
• PF-52 to PF-56

#### Compound Interest

37. If Ms. Caruso invests \$2,000 today at 8% annual simple interest how much will she have after 30 years?

38. If Ms. Caruso invests \$2,000 today at 8% annual interest, compounded annually, how much will she have after 30 years?

39. If Dr. Julius invests \$2,000 today at 8% annual interest, compounded quarterly, how much will he have after 30 years?

40. If Dr. Julius invests \$2,000 today at 8% annual interest, compounded monthly, how much will he have after 30 years?

41. Blue Bank offers five-year certificates of deposit that pay 6.5% compounded quarterly. Red Bank offers five-year certificates of deposit that pay 7% compounded semiannually. Which is a better deal? If Rosa Klebb has \$30,000 to spend on a certificate of deposit, how much would each option be worth for her at the end of five years?

42. Mr. Largo invests \$1,000 for seven years at 6% annual interest compounded monthly. Additionally, at the beginning of the fifth year she deposits an additional \$2,000 into the account. What is the investment worth at the end of all seven years?

43. In seven years you want to have \$20,000 to buy a vacation as a retirement present for yourself. What is the present value of this gift if the investment you make today will earn 11% annual interest compounded semiannually?

44. If you gave a 20-year-old sibling a \$100 wedding present and they invested it at 11% annual compound interest, how much would it be worth when they are 60 years old?

37. Simple Interest = Principal × annual rate × time in years = P × r × t = \$2,000 × 0.08 × 30 = \$4,800. Then add back the prinicipal: \$2,000 + \$4,800 = \$6,800

We could have used the one plus trick.

Total = P × (1 + r) × t = \$2,000 × 1.08 × 30 = \$6,800.

38. For compound interest, Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$2,000 × 1.0830 = \$20,125.31

If Ms. Caruso switches from simple interest to compound interest it would make a huge difference!

39. Now the interest rate per payout is 0.08 ÷ 4 = 0.02, and the number of payout is 30 × 4 = 120. Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$2,000 × 1.02120 = \$21,530.33

40. Now the interest rate per payout is 0.08 ÷ 12 = 0.006666... To avoid rounding in the middle we should keep that division happening in the formula. The number of payout is 30 × 12 = 360. Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$2,000 × (1 + 0.08 ÷ 12)360 = \$21,871.46

More frequent compounding will earn more money for Dr. Julias, but not by a huge amount.

41. For Blue Bank we do Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$30,000 × 1.0162520 = \$41,412.59. For Red Bank we do Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$30,000 × 1.03510 = \$42,317.96. Red bank is a better deal. The income increase more from a higher percentage rate than from more frequent compounding.

The result of the previous problem is not always true. Over a long time span, more frequent compounding can be more significant than a half-percent rate increase. You could use a spreadsheet to check how long a time span is needed for Blue Bank to be a better deal.

42. At the end of four years we have Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$1,000 × 1.00548 = \$1,270.49. Then it is the beginning of the fifth year, and the \$2,000 deposin brings the current account balance up to \$3,270.49. The account grows for years five, six, and seven (three more years). So the Very Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$3,270.49 × 1.00536 = \$3,913.73.

43. Now we are solving for the principal.

Final Amount = Principal × (1 + interest rate per payout)number of payouts

\$20,000 = P × 1.117

\$20,000 = P × 2.0761

\$9,633 ≈ P

Notice how 1.117 ≈ 2. Any principal multiplied by 1.117 would double (with slight rounding). Since the stock market averages an 11% rate of increase, we get a rule of thumb that its value doubles every seven years.

44. Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$100 × 1.1140 = \$6,500.09

#### Charge Options (More Compound Interest)

45. Fendrick wants to buy a new dining room set for \$900. He is considering four methods of payment. After looking at his budget as well as his actual expenses for the past few months, he thinks he can save \$80 per month towards this purchase. He has four options, described in detail on the table below. Which option is best for him? (a) He could pay cash up front if he waits almost a year, but he would prefer not to wait that long. (b) He could use the furniture store's normal installment plan that has a 15% annual interest rate. (c) He could use the furniture store's special zero down plan with an 18% annual interest rate. (d) He could open a new credit card with a 22% annual rate.

45. Lots of numbers! Answers are on a Google spreadsheet. You can also save your own copy of that spreadsheet and try fiddling with the starting balance, monthly interest rates, or other values to instantly see how the total interest changes.

LCC Math 25 Packet Homework

Also do some compound interest problems in the packet on pages:
• PF-46 to PF-49
• PF-51
• PF-57 to PF-59

Also do some charge options problems in the packet on pages:
• PF-42 to PF-44

#### Annual Effective Interest

46. If you invest \$1 for a year, with compound interest of 6% happening monthly, what is that annual effective interest rate?

46. If you invest \$1 for a year, with compound interest of 6% happening weekly, what is that annual effective interest rate?

45. Because of the monthly compounding we have an interest rate per payout of 0.06 ÷ 12 = 0.005. We have 12 payouts during the year. So the Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1 = 1.00512 − 1 = 0.06167 ≈ 6.17%.

The compounding raises the effective interest to slightly more than 6%. Not hugely more, but it would become significant over time!

46. Because of the weekly compounding we have an interest rate per payout of 0.06 ÷ 52 = 0.00115385. We have 52 payouts during the year. So the Annual Effective Interest Rate = (1 + interest rate per payout)number of payouts − 1 = 1.0011538552 − 1 = 0.06179 ≈ 6.18%.

More frequent compounding does raise the effective interest even more, but only barely. It would take a long time for that change to become significant. That is why banks and credit cards stick with monthly interest instead of competing with each other by also offering options with weekly compounding.

#### Sum of Annuity Due

47. Someone saves \$1,200 each year for retirement. If this is invested with 8% annual interest over 45 years, how much will it grow to be worth?

48. Someone saves \$1,800 each year for retirement. If this is invested with 8% compound interest over 30 years, how much will it grow to be worth?

49. According to research by Vanguard, the average 25-year-old American has saved \$3,865 for retirement. If this single deposit is invested with 9% compound interest for 40 years, how much will it grow to be worth?

50. That same research says the average American who retires at age 65 has saved about \$176,000 for retirement. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to that amount, over 40 years and at 9% annual interest.

47. Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate = [ \$1,200 × 1.0846 − \$1,200 × 1.08] ÷ 0.08 ≈ \$500,911

48. Final Amount = ( \$1,800 × 1.0831 − \$1,800 × 1.08 ) ÷ 0.08 ≈ \$220,223

Compared to the previous problem, these deposits are 50% bigger than before. But the other problem had a time frame that was another 50% longer than this problem. Which helped more?

49. This problem has a single deposit, so we are back to the compound interet formula. Final Amount = Principal × (1 + interest rate per payout)number of payouts = \$3,865 × 1.0940\$121,397

That is a nice amount of money, but a middle-class retirement costs a lot more, especially for someone who pays for their own health insurance.

50. Final Amount = ( \$478 × 1.0941 − \$478 × 1.09 ) ÷ 0.09 ≈ \$176,044.

For this problem we had to guess-and-check with different principal amounts. The answer is about \$478. That is not very much to save each year towards retirement, especially later in life after your career has made some progress!

LCC Math 25 Packet Homework

Also do some sum of annuity due problems in the packet on pages:
• PF-65 to PF-73
• PF-60 to PF-64

### Changes in Likelihood

51. The medicine trastuzumab, which fights breast cancer in women who already have breast cancer, was popularized because of a certain study. In the control group of 1,700 women, 34 died. In the group treated with trastuzumab, 23 of 1,643 women died. What percentage of the women in the control group died? What percentage of the women in the treated group died?

52. Continuing the previous problem, what was the absolute change (subtraction) in risk? What was the relative change (percent change) in risk?

53. If you were in charge of publicity for this drug, what type of claim could you truthfully make about the medicine?

54. Trastuzumab also has some dangerous side effects. Most notably, 40% of the women who take it develop flu-like symptoms, 7% develop mild heart problems, and 5% suffer a stroke or severe heart failure. If you were trying to discredit trastuzumab (perhaps concerned about the side effects and trying to convince a family member with breast cancer not to take the medicine) what type of claim could you truthfully make about the medicine?

51. For the control, group, 34 ÷ 1,700 = 0.02 = 2%. For the treated group, 23 ÷ 1,643 = 0.0139 ≈ 1.4%.

52. The absolute percent change is 2% − 1.4% = 0.6%. The relative percent change = change ÷ original = 0.6% ÷ 2% = 0.3 = 30%.

53. You could say, "People that take this drug are 30% less likely to die from breast cancer."

54. You could say, "People that take this drug only have their cancer risk reduced by less than 1%, but have a 5% chance to suffer side effects of stroke or severe heart failure."

## Big Questions

The "big questions" in each homework assignment represent real-life situations. Sometimes they require researching additional facts not included in the problem or the Math 25 website.

B8. Oregon Senate Bill 1105 limits payday loan interest rates to 36% or less. Before that bill was passed, payday loans in Oregon often had interest rates of 120%. Nationally, payday loans can have interest rates as high as 7,000%. The average payday loan is a two-week advance on \$350. How much interest would be owed with an annual simple interest rate of 36%? 120%? 7,000%?

B9. As a birthday present, Janice receives a \$50 gift certificate to her favorite restaurant. She invites two friends to join her for dinner there. She expect great service and plans to pay a 15% tip. Restaurant sales tax where she lives is 10%. What is the most she can pay for the food if she wants the gift certificate to also cover everyone's tip and tax?

B10. Bradley has heard that it is wise to spend no more than 25% of your income on your mortgage. He earns \$39,600 per year. He wants a thirty-year home loan. Mortgage interest rates are at 5%. How large a loan can he afford? How much will he pay total over the thirty years? How much of that is interest? Where is one Lane County neighborhood that Zillow suggests is appropriate for Bradley?

B11. The historical return for the stock market is about 11%. If the bank decided not to do business with Bradley, but instead invested the loan amount in the stock market, how much would the bank have after thirty years of 11% annual compound interest each year? Why would a bank choose to offer a mortgage to Bradley considering that stock market gain is so much smaller?

B12. Geoffrey Crayon starts a new credit card that charges 22% annual interest per year to keep his bookkeeping simple when buying a \$3,000 computer. (He will use the card for nothing else.) The credit card charges him one-twelfth of its annual interest rate each month. Geoffrey pays \$400 per month until the balance is paid off. Finish the table below to find his total interest in dollars. Also express the total interest amount as a percentage of the computer's cost. Is there a way to solve answer these questions without using a table?

B13. Cindy, Clara, and Chloe are three sisters. Each has plans to save for retirement, but their plans are somewhat different. Despite their different lives, by the time each was 25 years old she had the ability to set aside \$3,000 per year for retirement. All three use a retirement account that earns 8% annual compound interest. How much will each sister have saved when she retires at age 65?

B14. Cliff is Clara's husband. He stops smoking, and decides to devote the money he used to spend on cigarettes to retirement. He used to smoke 1 pack per day, at \$5.70 per pack. How much money per year was Cliff spending on cigarettes? If he instead puts his "cigarette money" annually into a retirement account that earns 9% annual compound interest, how much will extra will he have for retirement after thirty years?

B15. The general rule for retirement savings is to, at age 65, have saved \$20 for each dollar in income your retirement expenses will exceed your retirement income (from Social Security, pensions, etc.). Typical retirement expenses are greater than retirement income by about \$20,000 per year—meaning typical Americans should save \$400,000 for retirement. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to that amount, over 40 years and at 8% annual interest.

B16. Another general guideline is to save 10% to 15% of your income each year. Make it personal, and estimates about how your future income will change in a spreadsheet to predict your retirement savings this way. How does your answer compare to the \$400,000 guideline?

B17. How does the long-term financial cost of infant care compare to college? The "big three" child care centers in Eugene (Vivian Olum at UO, Early Learning Childrens Community at LCC, and Child Development Center at EWEB) cost roughly \$13,000 per year. (Oregon is an unusually expensive state for child care.) Use the Sum of Annuity Due Formula for two years (when the infant is age 1 and 2), and then the Compound Interest Formula for sixteen years (until the child is 18). If the family insted saved the infant care money, what annual interest rate would be needed for the savings to cover the child's college tuition?