Word Problems. Mathematics Division, IMSP, UPLB

Transcription

1 Word Problems

2 Objectives Upon completion, you should be able to: Translate English statements into mathematical statements Use the techniques learned in solving linear, quadratic and systems of equations in solving word problems Check reasonableness of the answer acquired

3 Polya s Heuristics George Pólya lived from 1887 to 1985

4 Polya s Heuristics 1.Understand the problem. Identify what are given and what are to be found. 2. Devise a plan. 3. Implement the plan. 4. Evaluate the solution.

5 Polya s Heuristics In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language

6 Polya s Heuristics The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.

7 Translating English Phrases to Mathematical Symbols Translate each of the following phrases into an algebraic expression: 1) One more than twice a certain number 2)Three less than five times a certain number 3)Each of two numbers whose sum is 100 4) The sum of three consecutive integers 5)The amount by which 100 exceeds three times a given number

8 Translating English Phrases to Mathematical Symbols Translate each of the following phrases into an algebraic expression: 6)A fraction whose numerator is three less than its denominator 7)The perimeter and area of a rectangle if one side is 4 ft longer than twice the other side 8)The number of quarts of alcohol contained in a tank holding x gallons of a mixture which is 40% alcohol by volume

9 Translating English Statements to Mathematical Symbols(One Variable Steps in solving word problems involving one unknown: Case) 1. Read the problem carefully. Reread it to determine the unknown quantity. 2. Express the unknown in terms of a single variable. 3. Form an equation involving the variable as suggested by the problem. 4. Solve the equation and check your answer.

10 Examples of Problem Solving 1. Ten years ago, John was 4 times as old as Bill. Now he is only twice as old as Bill. Find their present ages. Solution: Let x = Bill s present age = John s age now (in terms of x) = John s age ten years ago =Bill s age ten years ago Form the equation: = Solve the equation and check: Bill s age now is while John s age now is.

11 Examples of Problem Solving 2. The sum of two numbers is 37. If the larger is divided by the smaller, the quotient is 3 and the remainder is 5. Find these numbers. Solution: Let x = smaller number = larger number = quotient when the larger is divided by the smaller Form the equation: = Solve the equation and check: The two numbers are: and.

12 Examples of Problem Solving 3. James invested part of PhP 40,000 at 5% and the remainder at 3% simple interest. The total income per year from these investments is PhP How much did he invest at each rate? Solution: Let y = amount invested at 5% = amount invested at 3% = income from 5% investment = income from 3% investment = total income from both investments Form the equation: = Solve the equation and check: PhP was invested at 5% while PhP was invested at 3%.

13 Examples of Problem Solving 4. Joe can do a job in 3 days, while Jill can do the same job in 6 days. How long will they finish the job if they work together? Solution: Let z = number of days it will take them working together = amount of work done in one day working together = amount of work done by Joe in 1 day = amount of work done by Jill in 1 day = amount of work done by Joe and Jill in 1 day Form the equation: = Solve the equation and check: Together, they will finish the job in days.

14 Examples of Problem Solving 5. Two cars A and B having average speeds of 30 and 40 km/hr respectively, are 280 km apart. They start moving toward each other at 3:00 pm. What time and where will they meet? Solution: Let t = time in hours each car travels before they meet = distance traveled by A = distance traveled by B = total distance traveled by both Form the equation: = Solve the equation and check: The two cars will meet at at a distance km from initial position of A or km from the initial position of B. drawing

15 A 30 km/hr 280 After t hours distances traveled are: d = r x t 30t 40t B 40 km/hr Back

16 Examples of Problem Solving 6. When each side of a given square is increased by 4 cm, the area is increased by 64 sq. cm. Find the dimension of the original square. Solution: Let s = measure of one side of the original square = measure of the side of the new square = area of the original square = area of the new square Form the equation: = Solve the equation and check: The measure of the side of the original square is cm.

17 Examples of Problem Solving 7. A positive number exceeds three times another positive number by 5. The product of the two numbers is 68. Find the numbers. Solution: Let x= smaller of the two numbers = larger of the two numbers =product of the two numbers Form the equation: = Solve the equation and check: The two numbers are and.

18 Examples of Problem Solving 8. A picture frame of uniform width has outer dimensions 12 in by 15 in. Find the width of the frame if 88 sq. inches of the picture shows. Solution: Let w= width of the frame = length of the picture = width of the picture = area of the picture Drawing Form the equation: = Solve the equation and check: The width of the frame is.

19 12 w 15 Back w

20 Examples of Problem Solving 9. A pilot flies a distance of 600 km. He could fly the same distance in 30 minutes less time by increasing his average speed by 40 km/hr. Find his actual average speed. Solution: Let x= actual average speed = new average speed = time to fly 600 km at actual average speed = time to fly 600 km at new average speed = difference in time using the two speeds Form the equation: = Solve the equation and check: The actual speed is.

21 Exercises Solve the following word problems: 1. The sum of the digits of a certain two-digit number is 10. If the digits are reversed, a new number is formed which is one less than twice the original number. Find the original number. 2. How many pounds of a 35% salt solution and how many pounds of a 14% salt solution should be combined so that 50lb of a 20% salt solution are obtained?

22 Exercises Solve the following word problems: 3. One pipe can fill a tank in 45min. and another pipe can fill it in 30min. If these two pipes are open and a third pipe is draining water from the tank, it takes 27min. to fill the tank. How long will it take the third pipe alone to empty a full tank?

23 Word Problems involving inequalities

24 Exercises Solve the following word problems: 1. Part of PhP20,000 is to be invested at 9% and the remainder is to be invested at 12%. What is the amount of money that can be invested at 12% in order to have an income of at least PhP2,250 from the two investments.

25 Solve the following word problems: Exercises 2. A student in Math 17 got a grade of 65 and 70 in the first and second long exams, respectively. He also obtained 80 as his recitation grade. If 60% of his Prefi grade is computed from three long exams and 40% of the Prefi grade is from the recitation grade, then what should the student get in his third long exam to obtain at least 75 in his Prefi grade (so that he will be exempted from the Final exam)?

26 Exercises Solve the following word problems: 3. What is the minimum amount of pure alcohol that must be added to 24 liters of a 20% alcohol solution to obtain a mixture that is at least 30% alcohol?