MAT 142 Final Review 1. You have $2500 that you invest at 6% simple interest. What is the balance after four years? Solution: A = 2500 + 2500 · 0.06 · 4 = 3100 2. You have $7000 that you invest at 9% simple interest. What is the balance after 14 years? Solution: A = 7000 + 7000 · 0.09 · 14 = 7000 + 8820 = 15820 3. You have $4300 that you invest at 5% simple interest. How long will it take for your balance to reach $7525? Solution: 7525 = 4300 + 4300 · 0.05 · t Subtract 4300 from both sides to get 3225 = 215t =⇒ t = 15. It will take 15 years. 4. A man cuts back on his latte habit and instead makes $20 deposits each month into a savings account earning 6% interest compounded monthly. He continues these deposits for eight years. How much will the account be worth after eight years? Solution: (1 + A = 20 0.06 8·12 12 ) 0.06 12 −1 = 2456.57 5. A man makes $10 deposits each quarter into a savings account earning 8% interest compounded quarterly. He continues these deposits for 12 years. How much will the account be worth after 12 years? Solution: (1 + A = 10 0.08 12·4 4 ) 0.08 4 −1 = 793.54 6. A man borrowed $29,000 for two years under simple interest. At the end of the two years his balance due was $31,900. What annual simple interest rate did he pay? Solution: 2900 31, 900 = 29, 000 + 29, 000 · 2 · r =⇒ 2900 = 58000r =⇒ r = = 0.05 58000 He paid an annual interest rate of 5%. 7. A man borrowed $9,000 for four years under simple interest. At the end of the four years his balance due was $11,160. What annual simple interest rate did he pay? MAT 142 Final Review Solution: 2160 11, 160 = 9, 000 + 9, 000 · 4 · r =⇒ 2160 = 36000r =⇒ r = = 0.06 36000 He paid an annual interest rate of 6%. 8. Brad invests in a savings account that pays 8% interest compounded quarterly. What is the EAY for this account? Solution: APY = (1 + 0.08 4 4 ) − 1 ≈ 0.0824 = 8.24% 9. Suppose a student loan has an interest rate of 5% compounded monthly with monthly payments and the borrower has ten years to repay. If $9000 is borrowed, what are the monthly payments? Solution: 9000 · 0.05 12 d= ≈ $95.46 −120 1 − (1 + 0.05 12 ) 10. A credit card bill shows a balance due of $2500 with a monthly interest rate of 1.53%. What is the APR? What is the EAY? Solution: APR = 0.0153 · 12 ≈ 18.36% EAR = (1 + 0.0153)12 − 1 ≈ 19.97% 11. How much would you have to invest each month in an annuity earning 6% monthly to earn $50,000 at the end of 30 years? Solution: 50000 · 0.06 12 d= ≈ 49.78 30·12 − 1 (1 + 0.06 ) 12 You will need to invest approximately $49.78 per month. 12. How much would you have to invest each quarter in an annuity earning 5% quarterly to earn $5000 at the end of five years? Solution: 5000 · 0.05 4 d= ≈ 221.60 5·4 − 1 (1 + 0.05 ) 4 You will need to invest approximately $221.60 per quarter. 13. Inez wants to borrow $2100 to start a small business. She has found a bank that offers a 7% add-on loan to be repaid in monthly installments over five years. How much is the monthly payment? MAT 142 Final Review Solution: 2100 + 2100 · 0.07 · 5 2835 Her monthly payment is = ≈ $47.25 60 60 14. Suppose that you earned a bachelor’s degree and now you’re teaching high school. The school district offers teachers the opportunity to take a year off to earn a master’s degree. To achieve this goal, you deposit $2000 at the end of each year in an annuity that pays 7.5% compounded annually. a. How much will you have saved at the end of 5 years? b. Find the interest. Solution: a. $11,617 b. $1617 15. How much should you deposit at the end of each month into an IRA that pays 6.5% compounded monthly to have $2 million when you retire in 45 years? How much of the $2 million comes from interest? Solution: a. $620 b. $1, 665, 200 16. Suppose that you drive 15,000 miles per year and gas averages $3.50 per gallon. a. What will you save in annual fuel expenses by owning a hybrid car averaging 60 miles per gallon rather than an SUV averaging 15 miles per gallon? b. If you deposit your fuel savings at the end of each month into an annuity that pays 5.7% compounded monthly, how much will you have saved at the end of six years? Solution: a. $2625 b. $18, 747 17. Suppose that you are thinking about buying a car and have narrowed down your choices to two options: Option 1: The new car costs $68,000 and can be financed with a four-year loan at 7.14%. Option 2: A three-year old model of the same car costs $28,000 and can be financed with a fouryear loan at 7.92%. What is the difference in monthly payments between financing the new car and financing the used car? Solution: $950 MAT 142 Final Review 18. The price of a home is $160,000. The bank requires a 15% down payment. The buyer is offered two mortgage options: 15-year fixed at 8% or 30-year fixed at 8%. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15-year option? Solution: $125, 280 19. Below are the ages of 15 people in a room. 27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28 a. b. c. d. Find the median age. Find the mean age. Find the first and third quartile of the ages. Draw a stem plot of the ages. Solution: a. First list the numbers in increasing order to find the median age, which is 34. 620 b. x̄ = = 41.3 15 c. Q1 = 27 and Q2 = 50 Leaf Stem 2 05578 3 13467 4 4 d. 5 09 6 7 8 56 20. Below is a list of the number of dogs owned by families in a particular neighborhood: 2, 3, 1, 4, 1, 2, 0, 3 What is the standard deviation for this data? Solution: n = 8 and x̄ = 16 8 =2 xi xi − x̄ (xi − x̄)2 0 0 - 2 = -2 (−2)2 = 4 1 1 - 2 = -1 (−1)2 = 1 1 1 - 2 = -1 (−1)2 = 1 q √ 2 2-2=0 (0)2 = 0 12 =⇒ s = 8−1 = 1.71 ≈ 1.31 2 2 2-2=0 (0) = 0 3 3-2=1 (1)2 = 1 3 3-2=1 (1)2 = 1 4 4-2=2 (2)2 = 4 12 MAT 142 Final Review 21. Find the five number summary for the following set of data. 4, 7, 2, 10, 12, 3, 4, 6, 5, 8 Solution: First sort the given data to get: 2, 3, 4, 4, 5, 6, 7, 8, 10, 12 =⇒ the five-number summary (L, Q1 , M, Q3 , H) is 2, 4, 5.5, 8, 12. 22. The scores of students on a standardized test form a normal distribution with a mean of 300 and a standard deviation of 40. What are the lower- and upper-quartile scores for this test? Solution: The lower quartile is Q1 = 300 − 23 40 = 273.33 and the upper quartile is Q3 = 300 + 23 40 = 326.67. 23. The mean length of time, per week, that students at a certain school spend on their homework is 24.3 hours, with a standard deviation of 1.2 hours. Assuming the distribution of study times is normal, what percent of students spend more than 25.5 hours per week on homework? Solution: 16% of the students spend more than 25.5 hours per week. 24. The annual income of residents in a county is $42,000 with a standard deviation of $10,000. Between what two values do 95% of the incomes of county residents lie? Solution: 95% of the incomes of the county lie between 42000 − 2(10000) = 22000 and 40000 + 2(10000) = 60000. 25. The length of students’ college careers at Anytown University is known to be normally distributed, with a mean length of 5.5 years and a standard deviation of 1.75 years. What percent of students have college careers lasting between 2 and 9 years? Solution: 95% of the students have college careers lasting between 2=5.5-2(1.75) and 9=5.5+2(1.75) years. 26. A marketing company conducted a survey of college students to obtain data for an advertising campaign. They selected 1421 students randomly from campus directories of 132 colleges and universities. The 1421 students represent a. the population. b. the sample. MAT 142 Final Review Solution: 1421 students represent the sample of the survey. 27. A sample of 50 people at a local fast-food restaurant found 15 in favor of new fat-free menu items. In . this example, the sample proportion is Solution: 15 p̂ = · 100% = 30%. 50 28. A die is rolled and a coin is flipped simultaneously. The number rolled on the die and whether the coin lands heads or tails is recorded. How many outcomes are in the sample space? Solution: 6 · 2 = 12 29. We need to create serial numbers that start with one of the letters a, b, c, d, or f, followed by three non-repeating digits. How many serial numbers can be created? Solution: 5(10)(9)(8)= 3600 30. A raffle ticket costs $5. First and second prize winners will be drawn at random. The probability of winning the $100 first prize is 1/40 and the probability of winning the $25 second prize is 1/20. What is the mean winnings for one play, taking into account the $5 cost of the ticket? Solution: 1 1 µ = 95( 40 ) + 20( 20 ) − 5( 37 40 ) = −1.25 31. A fair die is rolled. If a number 3 or 4 appears, you will receive $10. If any other number appears, you will pay $5. What is the mean value of one trial of this game? Solution: µ = 10( 26 ) − 5( 46 ) = 0 32. The mean volume of a can of Super Soda is 12 oz., with a standard deviation of 0.6 oz. The volumes of all cans of Super Soda are normally distributed. What percent of cans of Super Soda contain less than 10.8 ounces? Solution: 10.8 = 12 - 2(0.6) =⇒ 2.5% of the cans contain less than 10.8 ounces. 33. In a room of 10 people, what is the probability that there are two people in this room with the same birthday? [Assume that there are 365 possible birthdays] MAT 142 Final Review Solution: The opposite (compliment) event for ”2 w/same birthday” is ” no 2 w/same birthday” 365·364·363·362·361·360·359·358·357·356 ≈ 1 − 0.8831 = 0.1169 =⇒ p(2 w/same birthday) = 1 − 365·365·365·365·365·365·365·365·365·365 =⇒ There is ≈ 12% chance that there are 2 w/same birthday. 34. A student is taking a five-question True/False test. If the student chooses answers at random, what is the probability of getting all questions correct? Solution: p(all correct) = 1 2 · 1 2 · 1 2 · 1 2 · 1 2 = 0.03125 35. In how many different ways can 6 people line up for a group photograph? Solution: 6 people can line up in 6 · 5 · 4 · 3 · 2 · 1 = 720 ways. 36. Two cards are dealt from a standard deck of cards. What is the probability that both cards are spades? Solution: p(2 spades) = 13 52 · 12 51 ≈ 0.059