1.10  Introduction to Word Problems  Part I

 from Basic Algebra: One Step at a Time © 2002

P. 63-83

Dr. Robert J. Rapalje

Seminole Community College

 

ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE

 

To see selected solutions in Living C O L O R  click here!

 

In this section, the following "categories" of word problems will be considered:

                                     I. Number problems

                                   II. Consecutive number problems

                                  III. Perimeter problems

                                  IV. Coin problems

                                   V. Mixture problems (Optional)

Before describing these categories of word problems, it will be helpful to identify five steps in setting up and solving word problems:

 

GUIDELINES TO SOLVING WORD PROBLEMS

 

 

STEP 1:    IDENTIFY THE VARIABLE.  State exactly what it is that the variable represents.  For example, "Let x = the number” or "Let x = the smaller of two numbers” or “Let x = the width of a rectangle” or “Let x = the number of dimes”  Then express all other quantities to be used in the problem in terms of x.  This is the most important, often the most difficult, and usually the most overlooked step of the problem.

 

 STEP 2:    WRITE THE EQUATION.  Having completed Step 1, use this step in writing the equation.  This is often no more than translating a sentence of the problem into an equation.  Read the problem carefully, paraphrasing as necessary.

 

STEP 3:     SOLVE THE EQUATION.  This is usually the easy part!

 

STEP 4:    ANSWER THE QUESTION.  After solving for x, there may be other quantities to be determined.  Be sure you have answered the question before going on to the next exercise.

 

STEP 5:  CHECK.  Check the answers in the worded problem itself and make sure the solution actually works.  Reject any extraneous (i.e., inappropriate) answers.

 

 

I.    NUMBER PROBLEMS.  

Probably the simplest type of application is the number problem.  If only one number involved, then begin by writing, “Let x = the number.”  If more than one number is involved, then begin “Let x = one of the numbers,” “Let x = first number,” “Let x = the smaller number,”  “Let x = the larger number,” or something like this.  How to decide what to let x equal will be explained later.

EXAMPLE  1.    Four times a number plus 12 is equal to 40.  Find the number.

Solution:

Step 1  Let x= the number (since there is only one number)

Step 2  Write the equation, (translate the sentence into math!)

                       4x + 12 = 40

Step 3  Solve the equation, (as in the previous section!).

                       4x + 12 = 40

                             - 12   -12

                          4x      =  28

                                 x = 7

Step 4  Answer the question: The number is 7.

Step 5  Check:

           Substitute 7 back into the original wording of the problem to see if it works: 4(7) + 12 = 40.

 

EXAMPLE  2.   Six less than twice a number is ten more than the number.  Find the number.

Solution:

Step 1  “Let x = the number”

Step 2  Write the equation:   2x -  6 = x + 10

Step 3  Solve the equation:   2x -  6 = x  + 10

                                             -x + 6   -x  +  6

                                                x       =       16                    

Step 4  Answer the question: The number is 16.

Step 5  Check                     2(16) - 6 = 16 + 10

                                              32  - 6 = 26  (This checks!)

EXERCISES.  Write equations for and solve each of the following word problems.

1.        Five times a number is 20 more than the number.  Find the number.

           Step 1    “Let x = __________”

           Step 2    Write the equation:  

           Step 3    Solve the equation: 

                             

           Step 4    Answer the question:

           Step 5    Check

 

2.        Twice a number minus ten is equal to 22 more than the number.

           Step 1    “Let x = __________”

           Step 2    Write the equation:  

           Step 3     Solve the equation:  

                              

           Step 4     Answer the question:

           Step 5     Check

 

3.        Three times a number plus 8 is equal to five times the number.  Find the number.

            Step 1      “Let x = ___________

            Step 2       Write the equation:  

            Step 3        Solve the equation:                              

 

             Step 4      Answer the question:

             Step 5      Check

 

 

4.         Six more than four times a number is equal to 18 less than the number.  Find the number.

            Step 1  

            Step 2      

            Step 3    

                             

            Step 4  

            Step 5                 

 

5.       Three less than four times a number is equal to 27 more than the number.  Find the number.

 

 

 

 

 

6.      Five more than four times a number is equal to 35 less than twice the number.  Find the number.

 

 

 

 

 

7.         Five more than four times a number is equal to 35 less twice the number.  Find the number. (NOTE: this looks like #6, but read the problem carefully!!)

 

 

 

 

 

 

 8.     Seven times a number, minus 5, is equal to three times the number plus 19.  Find the number.

 

 

 

 

 

 

 9.      Four less than three times a number is equal to 28 less than the negative of the number.  Find the number.

 

 

 

 

 

 

10.        Twice a number less 10 is equal to the negative of the number plus 14.  Find the number.

 

 

 

 

 

EXAMPLE  3.           The larger of two numbers is twice the smaller number.  Their sum is 243.  Find the numbers. [Hint: The first sentence tells about the larger number, but it assumes you already know the smaller.  Therefore, let x = the smaller number.]

Solution:

Step 1   “Let x = the smaller number

                   2x = the larger number

Step 2   Write the equation:   x + 2x = 243

Step 3   Solve the equation:   x + 2x = 243

                                                  3x   = 243

                                                      x =  81

Step 4   Answer the question: The smaller number is x = 81.

                                               The larger number is 2x = 162.

 

Step 5   Check (optional!)  The sum of the numbers (81 + 162) is 243.   (This checks!)

 

EXERCISES.

11.      Two numbers are such that the larger number is three times the smaller number.  The sum of the numbers is 60.  Find the numbers.

Step 1   “Let x  = the _______ number

               _____= the _______ number

Step 2   Write the equation: 

 

Step 3   Solve the equation: 

 

                             

Step 4   Answer the question:

 

Step 5   Check (optional!)

 

12.      Two numbers are such that the larger is 6 more than twice the smaller number.  The sum of the numbers is 60. Find the numbers.

 

  

 

 

 

13.      Two numbers are such that the larger is 6 less than twice the smaller number.  The sum of the numbers is 60. Find the numbers.

 

 

  

 

 

14.      The smaller of two numbers is 12 less than the larger.  The sum of the numbers is 60. Find the numbers.    [Hint: Let x = the larger number (see end of first sentence!)]

 

 

 

 

 

15.       The smaller of two numbers is 60 less than twice the larger.  The sum of the numbers is 60. Find the numbers.

 

 

 

 

 

16.       The smaller of two numbers is 30 less than three times the larger.  The sum of the numbers is 22. Find the numbers.

 

 

 

 

 

EXAMPLE 4.            Three numbers are such that the second is five more than the first, and the third is four less than three times the first.  The sum of the numbers is 36.  Find the numbers.

Solution:  Let       x = first number

                                              x + 5 = second number

                     3x - 4 = third number

 

Equation:   x + (x + 5) + (3x - 4) = 36      

                           5x    +      1  =   36

                                  5x         =   35

                                            x = 7

 

Answer the question:          x  =   7     first number

                                     x + 5  = 12    second number

                                    3x - 4  = 17    third number

Check:                 7 + 12 + 17 = 36

 

EXERCISES.

17.    Three numbers are such that the second number is 5 more than the first, and the third number is 4 more than three times the second.  The sum of the three numbers is 134.  Find the numbers.

Solution:  Let       x = first number

             _________ = second number

             _________ = third number

 

Equation:        

 

 

 

                      

                         

Answer the question:          x  = _____  first number

                                    _____ = _____  second number

                                    _____ = _____ third number

Check:           

 

18.     Three numbers are such that the second number is 3 less than  the first, and the third is twice the second number.  The sum of the numbers is 91.  Find the numbers. 

 

 

 

  

19.      Three numbers are such that the second number is 4 more than the first, and the third number is equal to the sum of the first two numbers.  The sum of the three numbers is 256.  Find the numbers.

 

 

 

 

20.      Three numbers are such that the second number is 6 less than twice the first, and the third number is five more than the sum of the first two.  The sum of the numbers is 293.  Find the numbers. 

 

 

 

 

 

 

II.       CONSECUTIVE INTEGER PROBLEMS

Word problems frequently refer to consecutive numbers or consecutive integers.  As examples, 5, 6, 7, and 23, 24, 25 are two sets of consecutive integers.  Whatever the first integer may be, you must add 1 to the first integer to get the second integer, and you must add 2 to the first integer to get the third integer.  In general, if x represents the first integer, then the second integer is x + 1, and the third integer is x + 2.

Examples of consecutive odd integers are 7, 9, 11 or 29, 31, 33.  Notice that beginning with the first integer, you must add 2 to obtain the second integer, and you must add 4 to the first integer to obtain the third integer.  In general, if x represents the first integer, then the second integer is x + 2, and the third integer is x + 4.

Examples of consecutive even integers are 6, 8, 10 or 28, 30, 32.  Notice that, like with the consecutive odd integers, beginning with the first integer, you must add 2 to obtain the second integer, and you must add 4 to the first integer to obtain the third integer.  In general, if x represents the first integer, then the second integer is x + 2, and the third integer is x + 4.

 

Consecutive integers: x, x+1, x+2

Consecutive odd integers: x, x+2, x+4

Consecutive even integers: x, x+2, x+4

 

 EXAMPLE 5.             Find two consecutive integers whose sum is 75.

Solution:                     Let x = the first integer

                                    x + 1 = the second integer

 

Equation:                      x + x + 1 =  75

Solve:                               2x + 1  =  75

                                               2x  =  74

                                                x   =  37

                                           x + 1  =  38

 

Check:                              37 + 38 =  75

 

EXAMPLE 6.           Find three consecutive even integers whose sum is 78.

Solution:                   Let  x   = the first integer

                                     x + 2 = the second integer

                                     x + 4 = the third integer           

 

Equation:                      x + x + 2 + x + 4 = 78

Solve:                                     3x  +  6     = 78

                                                        3x    = 72

                                                          x    = 24

                                                      x + 2  = 26

                                                      x + 4  = 28

            

Check:                                 24 + 26+ 28 = 78

 

EXAMPLE 7.            Find three consecutive odd integers whose sum is 123.

 

Solution:                    Let  x   =  the first integer

                                      x + 2 =  the second integer

                                      x + 4 =  the third integer         

 

Equation:                     x + x + 2 + x + 4 = 123

Solve:                                 3x   +   6      = 123

                                                      3x     = 117

                                                       x      =  39

                                                      x + 2 =  41

                                                      x + 4 =  43

            

Check:                                 39 + 41+ 43 = 123