SAT Math Hard Practice Quiz Answers
8.
B
(Estimated Difficulty Level: 5)
Solve the equation for
y
. You should get:
y
= 12
z
−
4
x
.
Factor out a 4 from the right-hand side:
y
= 4(3
z
−
x
).
Since
z
and
x
are integers, 3
z
−
x
is an integer, so that
y
is a multiple of 4. Only answer B is not a multiple of
4, so it can’t be a possible value of
y
.
You could also combine the fractions on the left-hand
side of the equation to get:
4
x
+
y
12
=
z.
For
z
to be an integer, 4
x
+
y
must be a multiple of
12. Try plugging in various integers for
x
and
y
to get
multiples of 12; you should find that
y
can take on all
of the values in the answers except for 6.
9.
B
(Estimated Difficulty Level: 4)
Combining the two terms on the left-hand side of the
equation gives us 2
√
72, but that doesn’t give us what
we need, which is
m > n
.
For the SAT test, you should know how to rewrite and
simplify radicals. Here,
√
72 =
√
36
·
2 = 6
√
2, so the
left-hand side is equal to 12
√
2, making
m
= 12 and
n
= 2.
10.
E
(Estimated Difficulty Level: 5)
Hint: if you see a zero in a table problem like this one,
try to use it first! When you plug in 0 for
t
, you get
N
(0) =
k
·
2
−
0
=
k
·
1 =
k
, which means that
k
= 128.
Next, try plugging in 1 for
t
:
N
(1) = 128
·
2
−
a
. From
the table,
N
(1) = 16, so that 128
·
2
−
a
= 16, or 2
−
a
=
1
/
2
a
= 1
/
8. Since 8 = 2
3
,
a
= 3. (Your calculator
may also help here, but try to understand how to do it
without it.)
11.
18
(Estimated Difficulty Level: 4)
Let
a
be Amy’s age and
b
be Bill’s age. The problem
tells us that:
a
=
b
+ 2 and
a
2
=
b
2
+ 36. One way to
do this is to plow ahead, substitute for one variable and
solve for the other (a bit messy!). But SAT questions
are designed to be solved without tedious calculations
and/or messy algebra. Let’s try doing the problem us-
ing the SAT way, not the math teacher way.
First, notice that the second equation can be written as:
a
2
−
b
2
= 36. This is a difference of two squares, and is
the same as: (
a
+
b
)(
a
−
b
) = 36. The first equation can
be written as:
a
−
b
= 2. This means that the second
equation is just (
a
+
b
)
·
2 = 36, so that
a
+
b
= 18. We
don’t know what
a
and
b
are, and we don’t even care!
12.
B
(Estimated Difficulty Level: 4)
You can obtain the graph of
y
=
f
(
−
x
) by “flipping”
the graph of
y
=
f
(
x
) across the
y
-axis. For exam-
ple, if the point (3
,
1) is on the graph of
f
(
x
), then
the point (
−
3
,
1) must be on the graph of
f
(
−
x
), since
f
(
−
(
−
3)) =
f
(3) = 1.
The figure below tells the story. Here, the function
g
(
x
) =
f
(
−
x
) is shown as a dashed line:
1
y
1
x
f
(
x
)
g
(
x
)
From the graph,
g
(
x
) is maximum when
x
=
−
2.
erikthered.com/tutor
pg. 13