Introduction to Variables in Word Problems
Often the equations used to solve word problems should have only one variable, and other unknowns must be written in terms of one variable. The goal of this section is to get you acquainted with setting your variable equal to an appropriate unknown quantity, and writing other unknown quantities in terms of the variable.
Examples
Example 1:
Andrea is twice as old as Sarah.
Because Andrea’s age is being compared to Sarah’s, the easiest thing to do is to let x represent Sarah’s age:
Let x = Sarah’s age.
Andrea is twice as old as Sarah, so Andrea’s age = 2 x . We could have let x represent Andrea’s age, but we would have to rethink the statement as “Sarah is half as old as Andrea.” This would mean Sarah’s age would be represented by .
Example 2:
John has eight more nickels than Larry has.
The number of John’s nickels is being compared to the number of Larry’s nickels, so it is easier to let x represent the number of nickels Larry has.
Let x = the number of nickels Larry has.
x + 8 = the number of nickels John has.
Example 3:
A used car costs $5000 less than a new car.
Let x = the price of the new car.
x – 5000 = the price of the used car
Example 4:
A box’s length is three times its width.
Let x = width (in the given units).
3 x = length (in the given units)
Example 5:
Jack is twothirds as tall as Jill.
Let x = Jill’s height (in the given units).
= Jack’s height (in the given units)
Example 6:
From 6 pm to 6 am the temperature dropped 30 degrees.
Let x = temperature (in degrees) at 6 pm.
x – 30 = temperature (in degrees) at 6 am
Example 7:
Oneeighth of an employee’s time is spent cleaning his work station.
Let x = the number of hours he is on the job.
= the number of hours he spends cleaning his work station
Example 8:
$10,000 was deposited between two savings accounts, Account A and Account B.
Let x = amount deposited in Account A.
How much is left to represent the amount invested in Account B? If x dollars is taken from $10,000, then it must be that 10,000 – x dollars is left to be deposited in Account B.
Or if x represents the amount deposited in Account B, then 10,000 – x dollars is left to be deposited in Account A.
Example 9:
A wire is cut into three pieces of unequal length. The shortest piece is the length of the longest piece, and the middle piece is the length of the longest piece.
Let x = length of the longest piece.
Example 10:
A store is having a onethird off sale on a certain model of air conditioner.
Let x = regular price of the air conditioner. Then = sale price of the air conditioner.
We cannot say that the sale price is because is not “onethird off the price of the air conditioner;” it is simply “onethird.” “Onethird the price of the air conditioner” is represented by . “Onethird off the price of the air conditioner” is represented by

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