Writing Equations

An equation or a statement of equality is a relationship between quantities. In word problems, you will be looking for relationships or connections.  In simpler, more direct cases, the verb of the sentence can be translated into an equal sign. The verb in each example below is highlighted. Notice that it  translates in each case to an equal sign.

EXAMPLES

• Percent Problems:

34 is what percent of 17?

Let x = the percent (in decimal form)

34 is what percent of 17        

• Number Relations:

Three more than twice a number yields ten more than the number.

Let x = the number

Three more than twice a number yields  ten more than the number.

          2x + 3                               =            x + 10

2x + 3 = x + 10 

x = 7

• Markup or Markdown Problems

A sweater is on sale for $12. This represents a 20% mark down from the original. What original price of the sweater?

Sale Price = Regular Price - Markdown

It would be helpful to paraphrase. The red words represent the same thing:

                   A sweater is on sale for $12. This represents a 20% mark down from the original.
                   IS THE SAME AS
                   $12 represents a 20% mark down from the original.

 $12 represents a 20% mark down from the original. What is the original price of the sweater?

Let x = original price of the sweater
0.20x = markdown

Sale Price = Regular Price - Markdown

    12        =              x       -        0.20x

12 = 0.80x

x = 15

The original price of the sweater was $15.

Sometimes the relationship is not directly stated. You will need to  look for indirect relationships or connections. It is helpful to paraphrase the word problem in a more direct fashion in order to see the relationships clearly. Look for formulas, use common sense, read carefully and don't give up. 

EXAMPLES

• Billing Problems

The town's water department charges $7 per month, plus 42 cents for every 100 gallons of water used. Last month, one homeowner  received a bill for $17.98. How many gallons of water was the homeowner charged for using?

Analysis

The first sentence gives a relationship that can be represented as a formula: 

Let C = monthly charge and 
      W = the number of gallons of water used each month

The town's water department charges $7 per month, plus 42 cents for every 100 gallons of water used. 
C = 7 + 0.42W/100     or     C = 7 + 0.0042W

NOTE: "for every" implied divided since it is a unit rate

The second sentence gives a value for the variable C. C = 17.98
The last sentence identifies the unknown quantity: W

Solution

Substitute 17.98 in the formula

17.98 = 7 + 0.0042W

Solve for W

17.98 = 7 + 0.0042W
-7       -7                 
         10.98 =     0.0042W          
       0.0042      0.0042           
 
2614.2857= W
W = 2614

Rounding answer to the nearest whole:

Answer the question

• More Percent Problems

Three hundred employees were absent from work one day. This was twenty percent of the the total number of people employed by the company. How many people does the company employ?

Analysis

The first two sentences can be combined into one. The word "This" in the second sentence refers to 300 employees. 

Three hundred employees were absent from work one day. This was twenty percent of the the total number of people employed by the company. 

Can be paraphrased to:

300  was twenty percent of the number of people employed.

This new version can be translated directly:
 Let X = number of employees

300  was twenty percent of the number of people employed.

 Solution

Translate  to an equation

Solve the equation

Answer the question

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