2 hours total from 1pm to 3 pm.
FinalFall2019Solutions Math 123 Final Exam Name: Fall Semester 2019 Student Number: Total marks: 95. Read all the questions carefully. Show your work. Use correct notation. Clearly indicate your final answer. Give answers correct to 2 decimal places. Good luck! 1. (2 marks) 7 students are meeting each other for the first time and shaking hands. How many handshakes will there be? 2. (2 marks) As a promotion at a local garden show, you can choose one packet of seeds, one flower pot, and one garden ornament for $10. There are 3 different types of seeds, 2 different flower pots, and 3 different garden ornaments. How many different combinations are there? 3. (4 marks) There are 15 apple trees in an orchard. Some trees have 4 apples growing on them and some trees have 3 apples growing on them. If there are 52 apples in total, how many trees have 4 apples? 4. (4 marks) One elephant weighs the same as two rhinos plus one giraffe. Also, one elephant weights the same as seven giraffes. How many giraffes weigh the same as one rhino? /12 Score: Formula for number of handshakes between people: For 7 students, we have so: n n(n − 1) 2 n = 7 7(7 − 1) 2 = 7 × 6 2 = 42 2 = 21 Answer: 21 Let S = seeds, F = flower pot, O = ornament Answer: 18O2 S1 F1 F2 O2O1 O3 O1 O3 S2 F1 F2 O2O1 O3 O1 O3 S3 F1 F2 O2O1 O3 O1 O3 = 6 + 6 + 6 = 18 OR: 3 × 2 × 3 = 18 Let = trees with 4 apples, let = trees with 3 apples (Equation for total trees) (Equation for total apples) x y x + y = 15 → y = 15 − x 4x + 3y = 52 4x + 3(15 − x) = 52 4x + 45 − 3x = 52 x + 45 = 52 x = 52 − 45 = 7 OR: Draw 15 trees with 3 apples on them, then add one extra apple to each tree until you count 52 apples in total 2 Rhinos = 6 Giraffes so 1 Rhino = 3 Giraffes = 1 Elephant 1 Elephant Let = elephants, = Rhinos, = Giraffes Therefore: E R G E = 2R + G E = 7G 2R + G = 7G 2R = 7G − G = 6G R = 6G 2 = 3G Answer: 3 Answer: 7 SOLUTIONS 5. (3 marks) Write as a Babylonian numeral. 6. (3 marks) Write as a Mayan numeral. 7. (2 marks) Write as a Roman numeral. 8. (1 mark) If today is Thursday, what day will it be in 23 days time? 9. (2 marks) Find if 10. (2 marks) Write (Base 5) in Hindu-Arabic. 11. In a video game world, Mario is buying supplies at the market. There are flowers, mushrooms, and stars for sale. Mario uses a binary (Base 2) numeration system. Answer the following questions. Do all your calculations in Base 2 (without converting to Base 10). (a) (2 marks) Mario spends $ on mushrooms and $ on flowers. How much in total has he spent so far? Give your answer as a Base 2 numeral. (b) (2 marks) He notices that the stars are on sale today for only $ each. If he has a total of $ in his wallet, does he have enough money left to buy some stars? If so, how many stars can he buy? 8787 5454 1942 x 6 + x ≡ 4 mod 8 4445 10112 11112 12 110112 /17 4445 = 4 × 52 + 4 × 51 + 4 × 50 = 4 × 25 + 4 × 5 + 4 × 1 = 100 + 20 + 4 = 124 2 3600 8787 - 7200 1587 26 60 1587 - 1560 27 8787 = 2 × 3600 + 26 × 60 + 27 × 1 15 360 5454 - 5400 54 2 20 54 - 40 14 5454 = 15 × 360 + 2 × 20 + 14 × 1 1942 = 1000 + 900 + 40 + 2 = 1000 + (1000 − 100) + (50 − 10) + 2 0 1 2 3 4 5 6 7 where is an integer take 6 + x ≡ 4 + 8y y 6 + x = 4 + 8(1) y = 1 6 + x = 12 x = 12 − 6 = 6 OR: Try different values for starting at 0, then 1, 2 etc. Count on an 8 hour clock each time, from 0 up to 6+ . When , we are counting to 6+6 = 12, and on an 8 hour clock this lands us on 4, which is what we want!. x x x = 6 In 21 days it will be Thursday again, so in 23 days it will be Saturday + 10112 11112 110102 Mario has spent dollars so far 110102 - 110112 110102 12 He only has dollars left so he can buy one star! 12 M CM XL II 12. (2 marks) Find . 13. A design for a Christmas decoration is shown. The design is made out of triangles and a regular convex polygon (thick lines). Answer the following questions: (a) (2 marks) Find the size of the angle . (b) (3 marks) If the area of the shaded region is 75 cm2, find the perimeter of the regular convex polygon. 14. (5 marks) The shape shown below is made entirely out of semicircles. The radius of the largest semicircles is 5m as shown. Using the other measurement given, find the area of the shaded region. 15. (1 mark) What percentage of 60 is 7? 16. (1 mark) Write 256% as a decimal. 17. (1 mark) 8% of which number is 6? x x /15 15cm x 5m2m 4x − 5 2x + 5 3x We add up the interior angles of the triangle. They must be equal to 180∘ 2x + 5 + 4x − 5 + 3x = 180 9x = 180 x = 180 9 = 20 (n − 2) × 180 n = (5 − 2) × 180 5 = 3 × 180 5 = 540 5 = 108∘ The shaded region is a triangle with area 75 cm2 and height 15 cm. The base of the triangle is one fifth of the perimeter of the regular convex polygon. so: A = 1 2 b × h 75 = 1 2 b × 15 b = 2 × 75 15 = 10 Perimeter = cm 5 × 10 = 50 1 2 3 Area of semicircle 1: Area of semicircle 2: Area of semicircle 3: Area of shaded region 1 2 π × r21 = 1 2 π × 52 = 1 2 π × 25 ≈ 39.27 1 2 π × r22 = 1 2 π × (72 ) 2 = 1 2 π × 49 4 ≈ 19.24 1 2 π × r23 = 1 2 π × (32 ) 2 = 1 2 π × 9 4 ≈ 3.53 ≈ 39.27 − 19.24 + 3.53 ≈ 23.56 m2 p × 60 = 7 p = 7 60 = 0.116667 = 11.67 % 256 % = 2.56 0.08 × B = 6 B = 6 0.08 = 75 18. (2 marks) A travel agency is offering a discount of 12% on a vacation package to Mexico. If you pay $2200 for the vacation package, what was the price before the discount? 19. (4 marks) Assume that the annual interest rate on your credit card is 13% and your unpaid balance at the beginning of last month was $764. Since then, you purchased a new iPhone for $1300, a winter coat for $140 and movie tickets for $34. You also returned a pair of shoes for $100 and returned a book for $35. What is your unpaid balance this month? What will be your finance charge next month? 20. (4 marks) Meghan is investing money to save up for year long trip around the world. How much money must she place in an investment now, that pays 8% annual interest compounded two times per year, if she wants to have have $45,000 in 3 years time? 21. You are taking out an add-on loan to do renovations on your new condo in downtown Vancouver. You agree to pay off the loan with 24 monthly payments of $700. The annual interest rate applied is 2.5%. (a) (3 marks) What is your principal? (b) (1 mark) How much total interest will you pay? /14 You paid 100% - 12% = 88% of the price before the discount. Let the price before the discount = so: x 0.88 x = 2200 x = 2200 0.88 = 2500 Unpaid balance from last month: Finance charge this