CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 327Lesson 7: Word Problems
How to Attack Word ProblemsDon’t be afraid of word problems—they’re eas-
ier than they look. In word problems, the facts
about the unknowns are written as sentences
instead of equations. So all you have to do is
name the unknowns and translate the sen-
tences into equations. Then it’s all algebra.Step 1: Read the problem carefully, and try to get “the
big picture.” Note carefully what the question asks
you to find.
Step 2:Ask: what are the unknowns? Call them xor n
or some other convenient letter. Don’t go overboard.
The fewer the unknowns, the simpler the problem.
For instance, if the problem says, “Dave weighs twice
as much as Eric,” rather than saying d= 2e(which
uses two unknowns), it might be simpler to say that
Eric weighs xpounds and Dave weighs 2xpounds
(which only uses one unknown).
Step 3: Translate any key sentence in the question into
an equation. If your goal is to solve for each unknown,
you’ll need the same number of equations as you have
unknowns.Use this handy translation key to translate
sentences into equations:
percent means ÷100
of means times
what means x
is means equals
per means ÷
x less than y means y– x
decreased by means –
is at least means
is no greater than means Step 4:Solve the equation or system. Check the ques-
tion to make sure that you’re solving for the right thing.
Review Lessons 1 and 2 in this chapter if you need
tips for solving equations and systems.
Step 5:Check that your solution makes sense in the
context of the problem.
Example:
Ellen is twice as old as Julia. Five years ago, Ellen
was three times as old as Julia. How old is Julia
now?
Let’s say that this is a grid-in question, so you can’t
just test the choices. Guessing and checking might
work, but it also may take a while before you guess
the right answer. Algebra is quicker and more reli-
able. First, think about the unknowns. The one you
really care about is Julia’s currentage, so let’s call it j.
We don’t know Ellen’s current age either, so let’s call
it e. That’s two unknowns, so we’ll need two equa-
tions. The first sentence, Ellen is twice as old as Julia,
can be translated as e = 2 j.The next sentence, Five
years ago, Ellen was three times as old as Julia, is a
bit trickier to translate. Five years ago, Ellen was
e– 5 years old, and Julia was j– 5 years old. So the
statement translates into e – 5 = 3(j – 5). Now solve
the system:
e – 5 = 3(j – 5)
Distribute: e – 5 = 3j – 15
Add 5: e = 3 j– 10
Substitute e= 2j:2j= 3j– 10
Subtract 2j:0 = – 10 j
Add 10: 10 = j
Now reread the problem and make sure that the
answer makes sense. If Julia is 10, Ellen must be
20 because she’s twice as old. Five years ago, they
were 5 and 15, and 15 is three times 5! It works!