Word Problemsjfrabajante.weebly.com/uploads/1/1/5/5/11551779/17_math...3. Form an equation involving...

Mathematics Division, IMSP, UPLB

Word Problems


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


Polya’s Heuristics

George Pólya

lived from 1887 to 1985



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.



“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…



…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.”


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


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


Translating English Statements to

Mathematical Symbols(One Variable

Case)

Steps in solving word problems involving one unknown:

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.


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 _________.



Solution: Let x = smaller number

_____= larger number

_____= quotient when the larger is divided by the smaller


Solve the equation and check: The two numbers are:

_________ and ________.

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 y = amount invested at 5%

_____= amount invested at 3%

_____= income from 5% investment

_____= income from 3% investment


Solve the equation and check: PhP ______ was invested at 5%

while PhP ____ was invested at 3%.

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 1680. How much did he invest at each rate?

_____= total income from both investments



Solution: Let z = number of days it will take them 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


Solve the equation and check: Together, they will finish the job in

_____ days.

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?

_____= amount of work done in one day working together



Solution: Let t = time in hours each car travels before they meet

_____= distance traveled by B

_____= total distance traveled by both


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.

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?

_____= distance traveled by A drawing


A B

280

30 km/hr 40 km/hr

After t hours distances traveled are: d = r x t

30t 40t

Back



Solution: Let s = measure of one side of the original square

_____= area of the original square

_____= area of the new square


Solve the equation and check: The measure of the side of the

original square is _________ cm.

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.

_____= measure of the side of the new square



Solution: Let x= smaller of the two numbers

_____=product of the two numbers


Solve the equation and check: The two numbers are _____ and

____.

7. A positive number exceeds three times another positive number

by 5. The product of the two numbers is 68. Find the numbers.

_____= larger of the two numbers



Solution: Let w= width of the frame

_____= width of the picture


Solve the equation and check: The width of the frame is _______.

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.

_____= length of the picture

_____= area of the picture

Drawing


15

w

w

12

Back



Solution: Let x= actual average speed

_____= time to fly 600 km at actual average speed


Solve the equation and check: The actual speed is ______.

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.

_____= new average speed

_____= time to fly 600 km at new average speed

_____= difference in time using the two speeds


Exercises

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.

Solve the following word problems:

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?


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?


Word Problems involving

inequalities


Exercises

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.



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)?



Exercises

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?


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