

Defeating Word Problems
Before you can solve an equation, you first need an equation
to solve. Word problems give you all the information you need to
answer the question—but in English. On the SAT, you must be able
to translate that information into math.

When you’re turning a word problem into an equation (or
equations) there are four things you have to do:
 Know what the question is asking.
 Assign variables.
 Define mathematical relationships between the variables.
 Show the problem who’s boss.
1. Know what the question is asking.
Well, isn’t it obvious what the question is asking? It
wants to know how many blue marbles are in the sack. It doesn’t
get any more obvious than that. You’re right, but there are reasons
we stress this point so seriously. “How many blue marbles are in the
sack?” is not what the question is asking you. What it’s really
asking is, “How many blue marbles are in a sack that contains 50
total marbles, out of which 20 more are red than blue?” This is
the true question, because it identifies all the relationships and
contains all the information that the question describes. SAT word
problems can make even simple equations sound complicated and messy.
Restating the question is a good way to clean things up.
2. Assign variables.
The question tells you the total number of marbles, and
it says that there are red and blue marbles. You don’t need a variable
for total marbles, since you know the total is 50, but you do need
variables for the red and blue marbles, since those are unknown quantities.
Pick whatever ones you like. How about r and b?
3. Define mathematical relationships between variables.
Okay. You know from the question that the 50 total
marbles are made up of red and blue marbles. So 50 = r + b.
The question also tells you that the sack contains 20 more
red marbles than blue marbles. So the number of red marbles is 20 more
than the number of blue marbles: r = b +
20. Just like that, you’ve taken all the information and
put it into mathematical form.
4. Show the problem who’s boss.
Once you’ve got the word problem translated, solving is
a cinch. You’re looking for how many blue marbles are in a sack
that contains 50 total marbles, out of which 20 more are red than
blue. So, in the equation with total marbles, red marbles, and blue marbles,
you’re looking for blue marbles. A little manipulating does the
trick: b = 50 – r.
And since you know that r = b +
20, you can substitute b + 20 for r to
get b = 50 – (b + 20).
Solve: r = 50 – r – 20; 2r =
30; r = 15.
An Example
Here’s a longer, more complicated problem. Conquer it
with the same fourstep technique:

1. Know what the question is asking.
What is the cost of the paint Gus needs to buy if he needs
to cover a total of x square feet,
when each can costs p dollars and
covers y square feet?
2. Assign variables.
The question assigns almost all of the variables for you: x is
total square feet that need painting in the house, p is
the price of a can of paint, and y is
the square feet that each can of paint can cover. Nice! You just
need a variable for that total cost. How about t?
3. Define mathematical relationships between variables.
Since p is the price of
a can of paint, and you need to find the total cost of buying paint,
you must find the number of cans of paint that Gus has to buy .
. . look at that! You need another variable for numbers of cans
of paint. How about n? (This happens sometimes:
You’ll find out that there is a variable you need that is implied
by the problem but not explicitly mentioned.) So t = np.
Now to find n. Use the information
the problem gives you. It tells you the total square feet (x)
and the square feet covered per can (y),
which means that if you were to divide x by y,
you’d get the number of cans:
4. Show the problem who’s boss.
You’ve got the relationships between the variables mapped
out. Plug the equation for n into
the t = np and you’ve
got the total cost:
Unlike the rule of doing the same thing to both sides
of the equation, there’s no universal rule stating that you must
follow the three steps we’ve just explained. But we think following
these steps will help you get a handle on every SAT world problem, no
matter how long and gnarly it is.
