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Study Guide Solutions – Exam 1 Exam #1 on Chapters 1-4 is now through March 6th in ETS testing lab 1A-1) The argument given or described involves some kind of fallacy. Identify the fallacy You should brush your teeth every day because brushing your teeth is very important Since this only restates the proposition this is Circular Reasoning 1B-1) Determine whether the statement is a proposition. 0. 2 = .0 2 Do you like this color? The equation 0.2=.02 is always false so it is a proposition “Do you like this color?” makes no statement (it is a question) so it is not a proposition 1B-2) Write the negation of the proposition. Susie lives in a green house. Everyone is asleep. Susie does not live in a green house. There is somebody who is not asleep. 1B-3) Make a truth table for the given statement. The letters p, q, r, s represent propositions. q∧~r∧s q r ~r s q ~ r q ~ r s T T F T F F T T F F F F T F T T T T T F T F T F F T F T F F F T F F F F F F T T F F F F T F F F 1B-4) Write the converse, inverse, ad contrapositive of the proposition. If you received a refund of over $1000, then you cannot make a claim. If you received a refund of over $1000, then you cannot make a claim. Converse: If you cannot make a claim, then you received a refund of over $1000. Inverse: If you did not receive a refund of over $1000, then you can make a claim. Contrapositive: If you can make a claim, then you did not receive a refund of over $1000. 1B-5) Two statements are listed in which p, q, and r represent propositions. Are the two statements logically equivalent? ~(p∧q); ~ p∨q p q p q ~ p q ~ p ~ p q T T T F F T T F F T F F F T F T T T F F F T T T 1C-1) Solve the problem using Venn Diagrams. The following Venn diagram describes the types of cookies in a bakery. Use it to determine how many chocolate chip cookies do not also have walnuts. 13 1D-1) Decide whether the argument is inductive or deductive. All U.S. Presidents have come from the contiguous 48 states. No person from Alaska can be President. This is inductive (based on selected observations) For any positive number p, |-p| = p. Therefore, |-23| = 23 This is deductive (a specific case of know properties) 1D-2) Evaluate the validity of the chain of conditionals. Premise: If the moon is made of cheese, then what goes up must come down. Premise: If what goes up must come down, then most Americans like apple pie. Conclusion: If the moon is made of cheese, then most Americans like apple pie. This is valid (p implies r AND r implies q, THEREFORE p implies q) 2A-1) Carry out the indicated unit conversion. Round your answer, if appropriate. Convert a distance of 42 feet into yards. 1yd 42 ft 14yd 3 ft Convert a weight of 13 pounds into ounces; there are 16 ounces in a pound. 16oz 13lb 208oz 1lb A container holds 6 gallons of water. How many fluid ounces is that? 128oz 6gal 768oz 1gal 2B-1) Convert the measurement to the units specified. Round your answer to the nearest tenth. 31 liters to gallons (1gal = 3.785 L) 1gal 31l 8.19gal 8.2gal 3.785l 35 pounds to grams (1 pound = 453 grams) 453g 35lb 15855g 15.8kg 1lb 2B-2) Convert the temperature, as indicated. Round your answer, if appropriate. 60°F, into Celsius (C=5(F-32)/9) C = 5(60-32)/9 = 15.6°C 20°C, into Fahrenheit (F=9C/5+32) F = 9(20)/5+32 = 68°F 2C-1) Solve the problem. A traffic counter consists of a thin black tube stretched across a street or highway and connected to a "brain box" at the side of the road. The device registers one "count" each time a set of wheels (that is, wheels on a single axle) rolls over the tube. A normal automobile (two axles) registers two counts, and a light truck (three axles) registers three counts. Suppose that, during a one-hour period, a particular counter registers 38 counts on a residential street on which only two-axle vehicles (cars) and three-axle vehicles (light trucks) are allowed. How many cars and light trucks passed over the traffic counter? Find all the possible solutions to the problem. 2C-1) Solve the problem. This amounts to solving the equation 2x+3y=38 with nonegative integers x and y. Starting with x=19 and y=0, we can use the slope of this line (m=- 2/3 or m=2/(-3)) and generate the other points x=16, y=2 x=13, y=4 x=10, y=6 x=7, y=8 x=4, y=10 x=1, y=12 There are seven in all. 3A-1) Write the percentage as a fraction or decimal. Write 13.5% as a decimal. To write a percent as a decimal we divide by 100 13.5 0.135 100 3A-2) Write as a percent. Write 0.829 as a percent To write a decimal as a percent we multiply by 100 0.829x100 = 82.9% 3A-3) Find Percentage $973 is % of $1676 To find percent of, we use the formula Part P Whole 1676 is the whole 973 is the part 973 P 0.58 58% 1676 3A-4) Find percentage part. An outlet store had monthly sales of $119,400 and spent 4% of it on health insurance. How much was spent on health insurance? To find percent part, we use the formula Part Whole Percent 119400 is the whole 4% = 0.04 is the percent Part 119400 0.04 $4776 3A-5) Find percentage mark-up/down. A store manager paid $57 for an item and set the selling price at $82.65. What was the percent markup? To find percent mark-up, we use the formula New Old Percent Old 82.65 is the new 57 is the old 82.65 57 Percent 0.45 45% 57 3A-6) Find mark-up/down value. The regular selling price of an item is $ 182. For a special year-end sale the price is at a markdown of 20%. Find the discount sale price. To find marked-down price, we use the formula New Old Old Percent 20%=0.20 is the percent 182 is the old New 182 182 0.20 $145.60 3B-1) Write the number in standard notation. a) 5.8x106 b) 7. 83 x 10-4 If the exponent is positive, we move the decimal point to the right. So in 5.8x106 we move the decimal point in 5.8, 6 places to the right. 5.800000 becomes 5,800,000 If the exponent is negative, we move the decimal point to the left. So in 7.83x10-4 we move the decimal point in 7.83, 4 places to the left. 00007.83 becomes 0.000783 3B-2) Write the number in scientific notation. a) 3,400,000 b) 0.0000561 To get a mantissa between 1 and 10, we have to move the decimal point to immediately after the first non-zero number. If we move to the left, the exponent is positive, if we move to the right, the exponent is negative. 3400000.0 becomes 3.4 after moving the decimal point 6 places to the left so 3400000= 3.4 106 0.0000561 becomes 5.61 after moving the decimal point 5 places to the right so 0.0000561= 5.61105 3B-3) Use scientific notation to perform the following operations. Leave your answer in scientific notation form. a) (4x103)( 9x104) When multiplying numbers in scientific notation, the resulting mantissa is the product of the factors’ mantissas. The exponent is the sum of the factors’ exponents. (4 10 3 )( 9 10 4 ) =(4 9) 10 3+4 =36 10 7 If the resulting mantissa is greater than 10, we must divide it by 10 and add 1 to the exponent. 36 10 7 =3.6 108 3B-3) Use scientific notation to perform the following operations. Leave your answer in scientific notation form. 2.8 10 8 b) 7.0 10 1 When dividing numbers in scientific notation, we divide the mantissas. The exponent is the difference of the factors’ exponents. 2.8 10 8 2.8 10 8(1) 0.4 109 7.0 10 1 7.0 If the resulting mantissa is less than 1, we must multiply it by 10 and subtract 1 from the exponent. 0.4 109 4.0 108 4A-1) Use the compound interest formula for compounding more than once a year to determine the accumulated balance after the stated period. a) $ 900 deposit at an APR of 12% with quarterly compounding for 5 years The value of an investment into which P dollars is invested for t units of time at an interest rate of r per unit time and compounded n times per unit time is nt r A P 1 P 1 r / n (n t) n P=900, r=12%=0.12, n=4, and t=5 45 0.12 A 900 1 900 1 0.12 / 4 (4 5) 4 =1625.50 4A-1) Use the compound interest formula for compounding more than once a year to determine the accumulated balance after the stated period. b) $ 5500 deposit at an APR of 5.5% with monthly compounding for 8 years The value of an investment into which P dollars is invested for t units of time at an interest rate of r per unit time and compounded n times per unit time is nt r A P 1 P 1 r / n (n t) n P=5500, r=5.5%=0.055, n=12, and t=8 128 0.055 A 5500 1 5500 1 0.055 /12 (12 8) 12 = 8531.31 4A-2) Find the annual percentage yield (APY). A bank offers an APR of 3.6% compounded monthly. The annual percentage yield of an account with an annual interest rate of r and compounded n times per year is n r APY 1 1 (1 r / n) n 1 n r=3.6%=0.036 and n=12 12 0.036 APY 1 1 (1 0.036 /12) 12 1 12 = 0.0366 = 3.66% 4B-1) Calculate the balance under the given assumptions. Find the savings plan balance after 27 months with an APR of 11% and monthly payments of $306. If PMT dollars are added to an investment account, n times per year, the annual interest rate is r and the payments are made for t years, the value of the account would be r nt 1 1 n A PMT PMT ((1 r / n) (n t) 1) / (r / n) r n A=306, r=11%=0.11, n=12, and t=2.25 0.11 12 2.25 1 1 12 A 306 306((1 0.11 /12) (12 2.25) 1) / (0.11 /12) 0.11 12 =9326.08 4B-2) Calculate the payment for the given situation. You intend to create a college fund for your baby. If you can get an APR of 7.0% and want the fund to have a value of $154,826 after 18 years, how much should you deposit monthly? The payment required for a savings plan to have A dollars if it is deposited n times per year for t years at an annual interest rate of r is r PMT A n =A r/n/((1+r/n)^(n t)-1) nt r 1 n 1 A=154826, r=7%=0.07, n=12, and t=18 0.07 PMT 154826 12 =154826 0.07/12/((1+0.07/12)^(12 18)-1) 1218 0.07 1 1 12 =359.46 4B-3) Compute the total and annual returns on the described investment. Five years after buying 200 shares of XYZ stock for $ 30 per share, you sell the stock for $10,000. The total return formula is TR = (SalePrice - BuyingPrice) / BuyingPrice SalePrice=10000 BuyingPrice=200x30=6000 TR = (10000 - 6000) / 6000 = .6667 = 66.67% The annual return formula is AR TR+1 1 1/t TR=.667 t=5 AR .667+1 1 .1076 10.76% 1/5 4B-4) Calculate current yield of a bond. Calculate the current yield for a $1000 Treasury bond with a coupon rate of 2.9% that has a market value of $750. The interest that any bond pays is given by the formula Interest = FaceValue x CouponRate FaceValue=1000 CouponRate=0.029 Interest = 1000 x 0.029 = $29 Interest The Current Yield is given by the formula CurrentYield MarketValue MarketValue=750 29 CurrentYield 0.0387 3.87% 750 4C-1) Find loan payment Calculate the monthly payment for a student loan of $ 67,519 at a fixed APR of 8% for 17 years. A loan payment is given by the formula r P n PMT nt P r / n / (1- (1 r / n) (-n t)) r 1 1 n P=67519 r=0.08 n=12 t=17 PMT 67519 0.08 /12 / (1- (1 0.08 /12) (-12 17)) $606.49 4C-2) Compute the monthly payment for 30 vs. 15-year mortgage. Assume that the loans are fixed rate and that closing costs are the same in both cases. You need a $ 104,473 loan. Option 1: a 30 year-loan at an APR of 8% Option 2: a 15-year loan at 7% A loan payment is given by the formula r P n PMT nt P r / n / (1- (1 r / n) (-n t)) r 1 1 n P=104473 r=0.08 n=12 t=30 PMT 104473 0.08 /12 / (1- (1 0.08 /12) (-12 30)) $766.59 P=104473 r=0.07 n=12 t=15 PMT 104473 0.07 /12 / (1- (1 0.07 /12) (-12 15)) $939.03

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