To Algebra or Not to Algebra?
When faced with an algebra question on the SAT, you could,
as you might expect, try to solve it by using standard algebra—setting
up and working out the equation. But there are often alternative
ways to attack. You might be able to plug the answer choices back
into the question until one of them works out. You could pick numbers to
substitute into the various expressions given as answer choices.
For problems you know how to solve, using algebra is probably
the quickest method. In contrast, a problem that’s giving you trouble
may suddenly become much easier if you start plugging in numbers.
We’re not telling you to pick just one method and always
use it. Far from it. Flexibility is the key. Some methods work for
some problems, and others work better with others. When you study
your practice tests and look over the algebra questions you got
wrong, think about more than just what the right answer was. Ask
yourself if you approached the question correctly. Did you plug
in answers when you should have used algebra? Did you use algebra
when plugging in answers would have simplified the problem?
Here’s an example of an algebra question. We solve it
using each of the different problem-solving methods, explaining
what you need to know about each one in the process.
man flipped a coin 162 times. The coin landed with heads side up
62 more times than it landed with tails up. How many times did the
coin land on heads?
Solving by Algebra
To answer this problem with algebra, you first have to
translate it into algebra. You have to set up an equation. If you
assign the variable x to stand for
the number of times the coin landed on heads, then tails are represented
by x – 62, since the coin landed on
heads 62 times more times than it landed on tails.
And since the coin was thrown 162 total times,
As you can see, setting up the question takes a little
bit of time and knowledge, but once you’ve set it up, the math is
quick and easy.
Using algebra will only take you longer than plugging
in if you have trouble coming up with the equation x +
(x – 62) = 162. So here’s a quick rule of
thumb to help you decide whether to use algebra or to plug in: If
you can quickly come up with the equation, then use algebra to solve
algebra problems. If you have the sense that it will take you a
while to figure out the equation, then plug in.
Solving by Plugging In
There are two ways to plug in: intelligently and maniacally.
Don’t be a maniac. How can you avoid this? Simple. The answer choices
on the SAT that contain numbers (rather than variables) always appear
in either ascending or descending order. The first answer choice
will be the lowest and the last will be the largest, or vice versa.
Let’s say the answer choices are in ascending order. If
you start by plugging in the middle number, the answer choice for
, then even if that choice doesn’t work, you
can use the outcome to determine whether you need to plug in a smaller or
larger number. If you need a smaller number, move to answer choice B
If you need a larger number, try D
. If you follow this
method, instead of having to check all five answer choices, you
shouldn’t ever have to check more than three. That’ll save you time. (5
– 3) / 5 100 = 40%
of your time, to be exact.
To answer the coin-flip problem by plugging in, pick C,
108, as the first number to try. So, if the coin came up
heads 108 times, then how many times did it land on tails?
It landed on tails 162 – 108 = 54. Are 108 heads 62 more
than 54 tails? No: 108 – 54 = 54. In order
for the problem to work out you need more heads. You can eliminate A and B as
possibilities. Choose D, 112, as your
next plug-in number: 162 – 112 = 50. Does 112
– 50 = 62? Yes.
Picking numbers is a variation of plugging in. It should only
be used when the answer choices contain variables. A modified
version of our original sample question shows what kind of problem
lends itself to picking numbers.
man flipped a coin z times. The coin
landed on heads y more times than
it landed on tails. If the number of times the coin landed heads
is h, then, in terms of h and y,
how many times was the coin flipped?
||z = h + y
||z = h – y
||z = 2h – y
Instead of testing your ability to set up and solve an
equation, this question asks you only to set up an equation based
on a word problem. While using algebra to set up the equation would
be the faster tactic, for some people, thinking in terms of variables
can be confusing. Picking numbers allows you to transform variables
into concrete numbers.
To use the picking numbers method, select numbers and
plug them into the answer choices. It doesn’t matter what specific
numbers you pick for each variable as long as you always plug the
same number in for each variable and follow all guidelines given
by the problem.
In the coin-flip problem, you are given three variables, z, y,
and h. The question asks you to find z in
terms of h and y.
We’ll pick some numbers. Let’s say the coin landed on heads (h) 5 times,
and that it landed on heads on 2 more flips (y)
than it landed on tails. That would mean that the coin landed on
tails 3 times, since 5 – 2 = 3. Since
the coin landed on heads on 5 flips, and on tails on 3 flips, the
coin must have been flipped a total of 5 + 3 = 8 times.
Now plug 5 for h and 2 for y into
all the equations and see which one comes out to 8:
only D, which is the right answer.
In addition to giving you a method for solving tricky
problems, picking numbers is also a good way to check your math
for careless calculations.
Solving by Being an Amazing Genius
It’s quite possible that you just looked at this problem
and said to yourself, “Other than the 62 more heads,
all the other flips were equally heads and tails. So: If I take the 62 out
of the total of 162, then I know that the other 100 flips
were 50 heads and 50 tails. Now I can
just add 62 + 50 = 112. Man, I am an amazing genius!”
Yes, you are. No one knows how to teach other people how
to be an amazing genius, though, and you can rest assured that almost
no one taking the test will be an amazing genius on every question.
The moral of the story: Know that amazing-genius shortcuts
exist, and keep a lookout for them, but don’t stress over them.
Only a fool would waste time looking for shortcuts. And you’re no
Algebra: The Bottom Line
There isn’t any “right way” to answer an SAT algebra question.
Some methods work best for some types of questions, and others for
others. The best way to learn which methods work best for you is
to take and study practice tests.