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
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 = 2x. We could have let x represent
Andrea’s age, but we would have to re-think the statement as ‘‘Sarah is half as
old as Andrea.’’ This would mean Sarah’s age would be represented by ½ x.
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 = the
number of nickels Larry has.
x + 8 ¼ the
number of nickels John has.
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
A box’s length is three times its width.
Let x = width
(in the given units).
3x = length (in the given units)
Jack is two-thirds as tall as Jill.
Let x = Jill’s
height (in the given units).
⅔ x = Jack’s
height (in the given units)
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
One-eighth of an employee’s time is spent cleaning his work station.
Let x = the
number of hours he is on the job.
⅛ x = the
number of hours he spends cleaning his work station
$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.
A wire is cut into three pieces of
unequal length. The shortest piece is ¼ the length of the longest piece, and
the middle piece is ⅓ the length of the longest piece.
Let x = length
of the longest piece.
⅓ x = length
of the middle piece
¼ x = length
of the shortest piece
A store is having a one-third off sale
on a certain model of air conditioner.
Let x ¼ regular
price of the air conditioner. Then ⅔ x = sale price of the
air
conditioner.
We cannot
say that the sale price is x – ⅓ because ⅓ is not ‘‘one-third off the price of
the air conditioner;’’ it is simply ‘‘one-third.’’ ‘‘One-third the price of the
air conditioner’’ is represented by ⅓ x. ‘‘One-third off the price of the air
conditioner’’ is represented by
Practice
1. Tony
is three years older than Marie.
Marie’s
age = ________
Tony’s
age = ________
2. Sandie
is three-fourths as tall as Mona.
Mona’s
height (in the given unit of measure) = ________
Sandie’s
height (in the given unit of measure) = ________
3. Michael
takes two hours longer than Gina to compute his taxes.
Number
of hours Gina takes to compute her taxes = ________
Number
of hours Michael takes to compute his taxes = ________
4. Three-fifths
of a couple’s net income is spent on rent
Net
income = ________
Amount
spent on rent = ________
5. A
rectangle’s length is four times its width
Width
(in the given unit of measure) = ________
Length
(in the given unit of measure) = ________
6. Candice
paid $5000 last year in federal and state income taxes.
Amount
paid in federal income taxes = ________
Amount
paid in state income taxes = ________
7. Nikki
has $8000 in her bank, some in a checking account, some in a certificate of
deposit (CD).
Amount
in checking account = ________
Amount
in CD = ________
8. A
total of 450 tickets were sold, some adult tickets, some children’s tickets.
Number
of adult tickets sold = ________
Number
of children’s tickets sold =________
9. A
boutique is selling a sweater for three-fourths off retail.
Retail
selling price = ________
Sale
price = ________
10. A string
is cut into three pieces of unequal length. The shortest piece is ⅕
as long as the longest piece. The mid-length piece is ½ the length of the
longest piece.
Length
of the longest piece (in the given units) = ________
Length
of the shortest piece (in the given units) = ________
Length
of the mid-length piece (in the given units) = ________
Solutions
1. Marie’s
age = x
Tony’s
age = x + 3
2. Mona’s
height (in the given unit of measure) = x
Sandie’s
height (in the given unit of measure) = ¾ x
3. Number
of hours Gina takes to compute her taxes = x
Number
of hours Michael takes to compute his taxes = x + 2
4. Net
income = x
Amount
spent on rent = 3/5 x
5. Width
(in the given unit of measure) = x
Length
(in the given unit of measure) = 4x
6. Amount
paid in federal income taxes = x
11. Amount
paid in state income taxes = 5000 – x
(Or
x = amount paid in state income taxes and 5000 – x = amount paid in federal
taxes)
7. Amount
in checking account = x
Amount
in CD = 8000 – x
(Or
x=amount in CD and 8000 – x = amount in checking account)
8. Number
of adult tickets sold = x
Number
of children’s tickets sold = 450 – x
(Or
x = number of children’s tickets and 450 – x = number of adult tickets)
9. Retail
selling price = x
Sale
price 
10. Length of
the longest piece (in the given units) = x
Length
of the shortest piece (in the given units) = ⅕ x
Length
of the mid-length piece (in the given units) = ½ x
Thanks for reading Word Problems. Please share...!
