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This Lesson (OVERVIEW of my additional lessons on Miscellaneous word problems, section 3) was created by by ikleyn(52754)
  About ikleyn: OVERVIEW of my additional lessons on Miscellaneous word problems, section 3My additional lessons on Miscellaneous word problems, section 3 in this site are - More complicated problems on finding number of elements in finite subsets - Solving problems by the Backward method - Minimax linear problems to solve MENTALLY based on common sense - Solving linear optimization problems without LP-method by reduction to linear function - Solving one special linear minimax problem in 100-D space by the Linear Programming method - Miscellaneous logical problems - Upper class entertainment Math problems for all ages List of lessons with short annotationsMore complicated problems on finding number of elements in finite subsets Problem 1. All 30 students in a class study at least one of the two subjects History and Geography. Twice as many study History as Geography. 8 students study only Geography. Find the number of students who study both History and Geography. Problem 2. All 30 students in a class study at least one of the two subjects History and Geography. Twice as many study History as Geography. 8 students study only Geography. Find the number of students who study both History and Geography. Problem 3. In a city school, 60% of students have blue eyes, 55% have dark hair, and 20% have neither blue eyes nor dark hair. How many students have blue eyes and dark hair? Solving problems by the Backward method Problem 1. Bob and John had $540.00 altogether. Bob gave 1/7 of his money to John. In return John gave 1/4 of his total amount he had to Bob. Then they had both an equal amount of money. How much did Bob have at first ? Problem 2. Andy, Berlin and Cheryl had a total of 6750 stamps. (1) At first, Andy have 50% of his stamps to Berlin. (2) Berlin then gave 1/3 of her stamps to Cheryl. (3) Finally, Cheryl gave 1/6 of her stamps to Andy. In the end, the ratio of the number of Andy’s stamps to the number of Berlin’s stamps became 4:5, and Cheryl had twice the total number of stamps that Andy and Berlin had. How many stamps did Berlin and Cheryl have in total at first? Minimax linear problems to solve MENTALLY based on common sense Problem 1. A farmer has a field of 70 acres in which he plants potatoes and corn. The seed for potatoes costs $20 per acre, the seed for corn costs $60 per acre, and the farmer has set aside $3000 to spend on seed. The profit per acre of potatoes is $150 and the profit per acre of corn is $50. How many acres of each should the farmer plant? What is the maximum profit? Problem 2. A farmer is going to divide her 60 acre farm between two crops. Seed for crop A costs $20 per acre. Seed for crop B costs $40 per acre. The farmer can spend at most $1400 on seed. If crop B brings in a profit of $270 per acre, and crop A brings in a profit of $120 per acre, how many acres of each crop should the farmer plant to maximize her profit? Problem 3. A fertilizer producing company has two plants where the product is made. Plant A can make at most 30 tons per month and plant B can make at most 40 tons per month. The company wants to make at least 50 tons per month. The process produces a particulate matter in the atmosphere as observed over a nearby town. It is found that 20 pounds of the particulate matter for each ton made by plant A and 30 pounds for each ton of the product made at plant B. How many tons should be made each month from each plant to minimize the amount of particulate matter in the atmosphere. Problem 4. A computer company has two type of laptop products, its flagship TURBO and the newer laptop called DELUXE. Every unit of Turbo has a profit of P1,000 while every unit of Deluxe has a profit of P6,000. The demand is limited to at most 200 Turbo and at most 300 of Deluxe. The current workforce can produce not more than 400 unit of both models of laptop. How much of each should it produce to maximize profits? Problem 5. Tala is a semi-professional photographer in her spare time. As a wedding photographer she can earn 10 Jordan Dinars (JOD) per hour. As a studio photographer she can earn 8 JOD per hour. She wants to spend at least 6 hours as a studio photographer. She can only work for a maximum of 20 hours a week. Find her maximum possible weekly earnings. Problem 6. A company makes two types of biscuits: Jumbo and Regular. The oven can cook at most 400 biscuits per day. Each Jumbo biscuit requires 2 oz of flour, each Regular biscuit requires 1 oz of flour, and there is 600 oz of flour available. The profit from each jumbo biscuit is $0.07 and from each regular biscuit is $0.12 . How many of each size biscuit should be made to maximize profit ? What is the maximum profit ? Problem 7. A factory makes two types of beds, type A and type B. Each month, a number of type A and a number of type B are produced. The following constraints control monthly production: No more than 50 beds of ype A and no more than 40 beds of type B can be made. At least 60 beds in all must be made. The maximum number of beds that can be produced is 80. The profit on type A i s Php300 and on type B is Php150. How many beds on both types must be produced to maximize the profit? What is the maximum profit? Problem 8. Sophia is organizing a movie night and wants to provide popcorn and soda to her guests. A bag of popcorn costs her $2 to buy, and a can of soda costs her $1. She only has room to store 24 cans of soda and 50 bags of popcorn at her house. She needs to have at least 60 items total to satisfy her guests. What is the least amount of money she can spend on popcorn and soda? Problem 9. A picnic basket can hold a maximum of 30 apples and 20 sandwiches. Each apple takes up 1 unit of space, and each sandwich takes up 2 units of space. The picnic basket has a total capacity of 60 units of space. What is the maximum number of apples and sandwiches that can be packed in the basket? Problem 10. Sarah wants to plant a combination of tulips and roses in her garden. Each tulip requires 2 square feet of space, and each rose requires 3 square feet of space. The garden has a total area of 30 square feet. Sarah does not want more tulips than roses in her garden. What is the maximum number of tulips and roses? Problem 11. Blue strand wire rope is the top seller of the company. TSD has to make a production schedule. To create product a blue strand wire rope, 3 machines are need. For wire rope #1, it needs 19 coils of wires and 3 hours of production, while for wire rope #2, 14 coils of wires and 4 hours of production. Wire rope #1 is sold for PhP 64,500 while wire rope #2 is sold for PhP 10,500. How many rolls of wire rope #1 and #2 does TSD has to schedule in the production to maximize the monthly sales while being limited to 6 machines, 2,006 coils of wires and 315 hours production? Problem 12. A western shop wishes to purchase 470 felt and 370 straw cowboy hats. Bids have been received from three wholesalers. Texas hatters has agreed to supply not more than 370, Lone Star Hatters not more than 463, and Lariat Ranch Wear not more than 185. The owner of the shop has estimated that his profit per hat sold from Texas Hatters would be $3/felt and $4/straw, from Lone Star Hatters would be $3.80/felt and $3.50/straw and from Lariat Ranch Wear $4/felt and $3.60/straw. Solve this to maximize the owner's profit, and calculate that profit. Problem 13. Maximize P = 3x-y subject to 4x+y <= 16, x+2y <= 9, x, y ≥ 0 using the simplex method. Problem 14. TMA manufactures 37-in. high-definition LCD televisions in two separate locations: Location I and Location II. The output at Location I is at most 5,800 televisions/month, whereas the output at Location II is at most 5,100 televisions/month. TMA is the main supplier of televisions to Pulsar Corporation, its holding company, which has priority in having all its requirements met. In a certain month, Pulsar placed orders for 2,900 and 4,000 televisions to be shipped to two of its factories located in City A and City B, respectively. The shipping costs (in dollars) per television from the two TMA plants to the two Pulsar factories are as follows:
To Pulsar Factories
City A City B
From TMA Location I $5 $5
From TMA Location II $8 $8
TMA will ship x televisions from Location I to City A and y televisions from Location I to City B.
Find a shipping schedule that meets the requirements of both companies while keeping costs, C (in dollars), to a minimum. Problem 15. The dean of Faculty of Management and Information Technology must plan the faculty’s course offerings for the first semester, 2020/2021. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught course the faculty an average of RM2,500 in faculty wages, and each graduate course costs RM3,000. Find the number of undergraduate and graduate courses should be taught in the first semester so that the total faculty salaries are kept to a minimum. Problem 16. An oil refinery refines types A and B of crude oil and can refine as much as 4000 barrels each week. Type A crude has 2 kg of impurities per barrel, type B has 3 kg of impurities per barrel, and the refinery can handle no more than 9000 kg of these impurities each week. How much of each type should be refined in order to maximize profits, if the profit is R25/barrel for type A and R30/barrel for type B? Problem 17. A company has $14,830 available per month for advertising. Newspaper ads cost $190 each and can't run more than 24 times per month. Radio ads cost $590 each and can't run more than 32 times per month at this price. Each newspaper ad reaches 5700 potential customers, and each radio ad reaches 6700 potential customers. The company wants to maximize the number of ad exposures to potential customers. Determine the most profitable / (effective) way to do it. Problem 18. A box company makes small and large wooden boxes. Small boxes require 0.8 square meters of wood, while large ones require 1.4 square meters. All boxes require 0.5 hours of labor, regardless of size. Wood is limited to 42 square meters, and only 24 hours of labor are available. Due to warehouse space limitations, no more than 20 large boxes can be made each day. Also, demand by customers for small boxes is limited to a maximum of 30 boxes. Each small box yields a profit of R42.00 and each large box earns only R14.00. Find the strategy to maximize the profit. Solving linear optimization problems without LP-method by reduction to linear function Problem 1. An oil refinery refines types A and B of crude oil and can refine as much as 4000 barrels each week. Type A crude oil has 2 kg of impurities per barrel, type B has 3 kg of impurities per barrel, and the refinery can handle no more than 9000 kg of these impurities each week. How much of each type should be refined in order to maximize profits, if the profit is R25/barrel for type A and R30/barrel for type B? Problem 2. The dean of Faculty of Management and Information Technology must plan the faculty’s course offerings for the first semester, 2020/2021. Student demands make it necessary to offer at least 30 undergraduate and 20 graduate courses in the term. Faculty contracts also dictate that at least 60 courses be offered in total. Each undergraduate course taught course the faculty an average of RM2,500 in faculty wages, and each graduate course costs RM3,000. Find the number of undergraduate and graduate courses should be taught in the first semester so that the total faculty salaries are kept to a minimum. Solving one special linear minimax problem in 100-D space by the Linear Programming method Problem 1. Let If Miscellaneous logical problems Problem 1. Every day at noon a ship leaves San Francisco for Tokyo, and at the same instant a ship leaves Tokyo for San Francisco. Each trip lasts exactly eight days. How many Tokyo ships will each San Francisco ship meet? Problem 2. Zack, Susie and Peter are standing in a line. All are facing in the same direction. Zack is facing the wall. Susie is behind Zack and Peter is behind Susie. They all know that Mr. Smith has four black and two red hats in a bag. Mr. Smith asks them to close their eyes and randomly puts one hat on each of them. He tells them to open their eyes. He then asks a question, " Tell me the color of the hat you are wearing? " After a few minutes of silence, Zack started jumping in excitement and yelled that he knew a color of his hat. Remember that they can only look in front of them. Which color was Zack’s hat? Problem 3. Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors? Upper class entertainment Math problems for all ages Problem 1. If a simple, connected, graph has 20 vertices, what is the maximum number of edges it can have? (Recall that a simple graph does not have loops and does not have multiple / parallel edges) Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I. Use this file/link ALGEBRA-II - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-II. This lesson has been accessed 288 times. |
