INTRODUCTION

 

Most students claim that the most difficult problem in math is a word problem. However, those same students can usually solve the problem once it has been converted to an algebraic expression.  Once a word problem is translated into algebra, it is no different than any other algebra problem.

 

 

METHOD

 

Each of the following words or phrases implies addition:

 

Plus, added to, more than, increased by, sum of, total of

 

For example:

 

3 added to a number implies              3 + x

A number increased by 11 implies     x + 11

The sum of two numbers implies        x + y

 

 

Each of the following words or phrases implies subtraction:

 

Minus, subtracted from, less than or fewer than, decreased by, difference of

 

For example:

 

5 subtracted from a number implies     x – 5

2 less than a number implies                 x – 2

The difference of two numbers implies y – x

 

 

Each of the following words or phrases implies multiplication:

 

Times, multiplied by, of, product of,

  

For example:

 

2 times a number implies                     2 * x

½ of 30 implies                                   ½ * 30

The product of two numbers implies    x * y

 

 

Each of the following words or phrases might imply division or a ratio:

per, a, on, in, for

 

For example:

 

55 miles per one hour implies 55 mi./ 1hr

120 miles on 3 gallons of gas implies 120mi/3gal

 

 

Each of the following words or phrases implies equals:

 

Is, was, will be, were, are, gives, results, yields, sells for

 

For example:

 

Alan was 5 years old implies                 a = 5

The sum of two number is 22 implies     x + y = 22

A car sells for $9000 implies                x = 9000

 

 

Other common phrases include:

 

Proportion implies                             ratio or division

Perimeter implies                               sum of the lengths of the sides

Square of a number implies               x²

Cube of a number implies                  x³

Twice or two times implies                2 *

Three times implies                            3 *

Consecutive numbers implies            x, x + 1, x + 2, x + 3, etc.

Consecutive odd numbers implies     x, x + 2, x + 4, x + 6, etc.

Consecutive even numbers implies    x, x + 2, x + 4, x + 6, etc.

 

   

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Example 1:

Problem:

If  6 times a number is decreased by 3, the result is 8 more than 5 times the number. Write the equation that could be used to find this number, x.

 

 

Solution:

Let x represent the number.

If 6 times a number is decreased by 3, the result is 8 more than the 5 times the number.
   6 *           x               -                    3                   = 8 +                     5 * x

Answer:      6x - 3 = 5x + 8

 

Note:  Notice the order of the right-hand side of the equation.  The problem stated "8 more" than 5 times the number.  Although order does not matter in an addition problem, it does in a subtraction problem.  The next example will illustrate that point.

Example 2:

Problem:

If  7 times a number is decreased by 1, the result is 4 less than 2 times the number. Write the equation that could be used to find this number, x.

 

 

Solution:

Let x represent the number.

If  7 times a number is decreased by 1, the result is 4 less than 2 times the number.
    7 *           x               –                    1                 =   2 * x – 4

 

Answer:      7x – 1 = 2x – 4

 

Note:  Notice the order of the right-hand side of the equation.

Example 3:

Problem:
If a skirt costs $40 after a 30% discount, what was the original cost?

 

 

Solution:

Another way to write this problem is:

 

The original cost of a skirt, decreased by 30% of the original cost, is $40.

x                                           – .30 * x                                                = 40

 

x – .30x = 40

1x – .30x = 40

.70x = 40

Answer:  x = $57.14

Example 4:

Problem:
The length of a rectangle is 2 feet more than the width. The perimeter of the rectangle is 20 feet. Find the length.

 

 

Solution:

Let W represent the width of the rectangle and L the length.

L    W

We will have 2 equations since we have 2 sentences.

 

The length of a rectangle is 2 feet more than the width.

L                                       = 2 + W

 

Or: L = 2 + W

 

We can rewrite this sentence:

The perimeter of the rectangle is 20 feet.

 

As this sentence:

The sum of the sides of the rectangle is 20 feet.

W + L + W + L                                    = 20

 

Replace L with W + 2

 

W + (W + 2) + W + (W + 2) = 20
4W + 4 = 20
4W = 16
W = 4

 

Solve for L using W:

L = W + 2
L = 4 + 2
Answer: L = 6

A proportion is a statement that shows one ratio is equal to another ratio.

 

Example 5:

 

Problem:
Identify the proportion listed below that solves this problem.

A car can travel 189 miles on 9 gallons of gasoline. How far can the car travel on 13 gallons?

 

 

Solution:

 

 189  =   x 

   9        13

 

Cross multiply:

189 * 13     =     9 *  x

2457 = 9x

 

Answer: x = 273 miles 

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PROBLEMS

1.      If you burn 475 calories in 2 hours while riding a bicycle, how many calories will you burn in 7 hours?

2.      A car can travel 255 miles on 15 gallons of gasoline. How far can the car travel on 26 gallons?

3.      If 8 times a number is increased by 1, the result is 7 less than 13 times the number. Write the equation that could be used to find this number, x.

4.      If 2 times a number is decreased by 17, the result is 5 more than the number squared. Write the equation that could be used to find this number, x.

5.      If a shirt costs $75 after a 20% discount, what was the original cost?

6.      If a dress costs $158 after a 40% discount, what was the original cost?

7.      The length of a rectangle is 6 feet more than the width. The perimeter of the rectangle is 32 feet. Find the length.

8.      The length of a rectangle is 5 feet less than the width. The perimeter of the rectangle is 90 feet. Find the length.

9.      The label on a can of soup says that there are 150 calories in a 12-ounce serving.  How many calories are there in a 7-ounce serving?

10.  Suppose it costs $5 for 10 pounds of apples.  At this rate, how much does it cost for 7 pounds of apples?