Chapter Review Problems

Transcription

1 Chapter Review Problems Unit 3.1 Mathematical symbols and expressions 1. The symbol means approximately equal to. (T or F) False. means is not equal to 2. Show three different ways to indicate 7 times (12) 3. Show two different ways to indicate 13 divided by If m 4, can m be a. 13? (yes or no) Yes b. 2? (yes or no) No c. 4? (yes or no) Yes 5. Write using exponents What is the value of 3 5? 3 5 = = 243 For Problems 7 15, simplify = = (8 + 3) 2 = (11) 2 = = = [2 + 3(4-2) - 5] + 3 = 5[2 + 3(2) - 5] + 3 = 5[ ] + 3 = 5[3] + 3 = = (3 + 2) + (2-4) - (7-5) - (2-8) = 3 + (+5) + (-2) - (+2) - (-6) = = (4)(3) = = (2)(-3)(2)(4)(-3)(-2) = -288 odd number of negatives, so answer is negative = 6 For Problems 16 20, simplify each expression using the distributive rule and/or combining terms (m + n) = 2m +2n 17. 5(3a + 2b) = 15a +10b 18. 2y - (y - 2) = 2y - y + 2 = y m + 2(2m + 1) - 3m + 6 = 3m + 4m + 2-3m + 6 = 4m x + 2y Cannot simplify Unit 3.2 Equations: Solving for the unknown 21. Which of these are equations? a. 3x - 2y + 2 b. x - 2 = 16 c. 3a + 2 b is an equation; the other two are expressions (no equal sign). For Problems 22 27, solve for the variable. 22. x - 8 = y + 17 = a = a 3 = 15 3 x = 35 y = 6 a = t_ 8 = _ 7 y = (3x + 5) - 19 = 2(x + 5) 8_ t_ 1 () 8 = 8(3) 7_ 2 () 7 2_ y = 2 7_ () 1 3_ 9x = 2x x - 4 = 2x x -2x t = 24 y = x - 4 = x = 14 7x 7 = Chapter 3 Equations: A Guide to Finding the Unknown x = 2

2 28. Prove your answer to Problem 27 by substituting your answer for the variable. 3(3x + 5) - 19 = 2(x + 5) 3(3(2)+ 5) - 19 = 2(2 + 5) 3(6 + 5) - 19 = 2(7) 3(11) - 19 = = = 14 Unit 3.3 Guideline for solving word problems For Problems 29 32, write phrases using symbols times b 4b more than x x + 12 or 12 + x less than a a Twice the sum of x, y, and z 2(x + y + z) For Problems 33 35, change the phrase to an equation. 33. y equals 4 more than x y = x r plus 2 totals 4 more than t r + 2 = t A number divided by 3 is 18 (use n as the variable) n 3 = Britney and Megan decide to open an accounting business. They will need a total of $65,000 to get the business started. Britney has spent a lot of time getting the business set up, so Megan agrees to contribute $8,000 more than Britney. How much money should each contribute?? = $65,000 B + M = $65,000 B + B + $8,000 = $65,000 Britney $28,500 2B + $8,000 = $65,000 Megan: $28,500 + $8,000 36,500 2B = $57,000 $65,000 B = $28, Forrest Gray is thinking about leasing a new building with 6,100 square feet (sq ft). If Forrest needs 1,300 sq ft for offices and twice as much warehouse space as showroom space, what size (in sq ft) should the warehouse and showroom be? Note: Because the letter O resembles the number zero, let s use F for offices.? = 6,100 F + W + S = 6,100 1, S + S = 6,100 Offices 1,300 sq ft 1, S = 6,100 Showroom 1,600 sq ft 3S = 4,800 Warehouse: 2(1,600 sq ft) 3,200 sq ft S = 1,600 6,100 sq ft 38. Bill and Charlotte McGuire are thinking about buying a vacant lot so they can build their dream home. They want to limit their total investment to $280,000. They are told that in this area the cost of the home (without land) should be about two and one-half times the cost of the land. What is the maximum price Bill and Charlotte can pay for the land?? = $280,000 L + H = $280,000 L + 2.5L = $280,000 Land $ 80, L = $280,000 Home: 2.5($80,000) 200,000 L = $ 80,000 $280,000 Chapter Review Problems 61

3 39. Clayton Holt dies intestate (without a will), leaving an estate of $350,000. Clayton s heirs consist of his wife and two adult children. His state requires that, in the absence of a will, each child is to receive two-thirds as much as a spouse. How much will each receive?? = $350,000 2C + W = $350,000 ( ) 2 2_ 3 W + W = $350,000 4_ 3 W + W = $350,000 4_ 3 W + 3 3_ W= $350, You are making some lemonade from a powdered concentrate. The instructions say to add 7 parts water for each part of concentrate. If you want to make 64 ounces of lemonade, how many ounces of concentrate should you use?? = 64 oz. C + W = 64 oz. C + 7C = 64 oz. Concentrate 8 oz 8C = 64 oz. Water: 7(8oz.) 56 oz C = 8 oz. 64 oz 41. You paid $58 for a coat. If the coat was on a rack marked 3 1 off, what was the original price?? = $58 P - D = $58 P - 3 1P = $ P = $58 Original price $87 Discount: P = 3 2 ($58) 3 () ($87) -29 Amount after discount $58 P = $ Craig Dean owns a ready-mix concrete company. Craig receives an order for 1.5 cubic yards of concrete. This mixture of concrete contains sand, cement, gravel, and water; for each pound (lb) of water, there are 5 lb of sand, 2 lb of cement, and 7 lb of gravel. Find (a) the total weight of the order (in pounds), assuming a cubic yard of concrete weighs 4,000 lb, and (b) how many pounds of each ingredient must be used. a. 1.5 (4,000 lb) = 6,000 lb b.? = 6,000 lb S + C + G + W = 6,000 lb Water 400 lb Sand: 5(400 lb) 2,000 lb 5W + 2W + 7W + W = 6,000 lb 15W = 6,000 lb W = 400 lb Challenge problems 43. Eddie and June just bought a home and the lender requires that they have $198,000 of property insurance. The policy will insure the home for a certain amount, and automatically covers additional structures for 10 1 of the home coverage. What is the minimum amount of coverage required on the home so that the total coverage (home plus additional structures) is $198,000?? = $198,000 H + A = $198,000 H H = $198,000 Home $180, H = $198,000 Additional: $180, , H = $198,000 $198, () 10 H = ($198,000) H = $180, Chapter 3 Equations: A Guide to Finding the Unknown 7_ 3 W = $350,000 Wife $150,000 3_ 7() 3 7_ W = 7 3_ ($350,000) Child #1: $150,000 2_ 3 100,000 W = $150,000 Child #2: $150, _ 100,000 Cement: 2(400 lb) Gravel: 7(400 lb) 800 lb 2,800 lb 6,000 lb $350,000

4 44. You are thinking about opening an appliance business in a new building containing 4,000 sq ft. You need showroom space and warehouse space. The warehouse space rents for $0.50 per sq ft per month, and the finished space (showroom) rents for $0.90 per sq ft per month. What is the maximum showroom space you can have to keep within a budget of $3,000 for rent?? = $3,000 $0.50W + $0.90S = $3,000 $0.50( S) + $0.90S = $3,000 W + S = 4,000 so $2,000 - $0.50S + $0.90S = $3,000 Showroom: 2,500 sq ft $0.90 $2,250 W = 4,000 - S $2,000 + $0.40S = $3,000 Warehouse: 1,500 sq ft $ $0.40S = $1,000 $3,000 S = 2, A certain mixture of nuts contains cashews, almonds, and macadamia nuts. Each container must include three times as many almonds as cashews and twice as many cashews as macadamia nuts. If there are a total of 18 ounces (oz) in each container, how many ounces of each nut must be included?? C + A + M 2M + 3C + M 2M + 3(2M) + M 2M + 6M + M 9M M = 2 oz 46. You are thinking about opening an oil change shop. Your fixed costs will be $13,000 per month. You will charge customers $28 for a lube-oil-filter. The cost of materials is in addition to your fixed costs and is estimated at $8 per customer. How many customers must you have each month in order to break even? Hint: Break even means revenues equal expenses, so we could say Fixed Costs + Variable Costs = Revenues. F + V = R $13,000 + $8C = $28C $13,000 = $20C 650 = C There are twice as many cashews as macadamia nuts Macadamia nuts 2 oz Cashews: 2(2 oz) 4 oz Almonds: 3(4 oz) 12 oz 18 oz Fixed costs $13,000 Variable costs: 650($8) + 5,200 Total costs $18,200 Revenues: 650($28) $18, Refer to Problem 46. If your shop is open an average of 25 days each month, how many customers do you need each day to break even? 650 customers 25 days = 26 customers per day Practice Test means 8 divided by 5. (T or F) False 2. What is the value of 4 3? 4 3 = = Simplify: 2 + 3(7-2 2 ) 2 = 2 + 3(7-4) 2 = 2 + 3(3) 2 = 2 + 3(9) = = Simplify: 5-2(3a -4) + 3a = 5-6a a = -3a Solve: m - 12 = y = y = m = 40 y = 35 () Practice Test 63

5 7. Solve: 2(4p - 2) - (3p - 4) = 2p + 5 8p - 4-3p + 4 = 2p + 5 5p = 2p + 5 3p = 5 p = Change the phrase m plus 4 totals 3 more than n to an equation. m + 4 = n Brandy and Carlos need a total of $30,000 to start an engineering business. Brandy has spent a lot of time getting the business set up, so Carlos agrees to contribute $5,000 more than Brandy. How much money should each contribute?? = $30,000 B + C = $30,000 B + B + $5,000 = $30,000 2B + $5,000 = $30,000 Brandy $12,500 2B = $25,000 Carlos: $12,500 + $5,000 17,500 B = $12,500 $30, You are thinking about opening a copy center. Your fixed costs are $4,800 per month. You will charge customers 6 cents ($0.06) per copy. The cost of materials is in addition to your fixed costs and is estimated at 1.5 cents ($0.015) per copy. How many copies must you make each month in order to break even? Round your answer to the nearest whole number. F + V = R $4,800 + $.015C = $.06C $4,800 = $.045C $4,800 $.045 = C 106,667 C Fixed costs $4,800 Variable costs: 106,667($0.015) +1,600 Total costs $6,400 Revenues: 106,667($0.06) $6, Chapter 3 Equations: A Guide to Finding the Unknown