Sat Math Hard Practice Quiz Worksheet With Answer Key Page 12
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18SAT Math Hard Practice Quiz Answers
Algebra and Functions
4. C
(Estimated Difficulty Level: 4)
Translate the words into an algebraic equation:
1. E
(Estimated Difficulty Level: 5)
m
m
=
n.
3
2
The answers suggest that there aren’t that many pos-
sibilities. So, make up some even integers, plug them
Multiplying both sides by 6 (the common denominator)
in for m, and see if they work! Since we can’t take the
gives 2m = 3m
6n, or m = 6n. So, m must be a
square root of a negative number, m can’t be less than
√
positive multiple of 6, which means that answer C is
6. Also, if m = 2, then
m + 7 = 3, but any larger
correct.
value of m won’t work. So, the possible values are
6,
4,
2, 0, and 2. (Don’t forget that zero is a perfectly
good even integer.)
5. 16
(Estimated Difficulty Level: 4)
Since (a 4)(b+6) = 0, the possible solutions are: a = 4
2. B
(Estimated Difficulty Level: 4)
and b is anything, or b =
6 and a is anything. Now,
2
2
the expression a
+ b
is made smallest by choosing a
Plug in real numbers for x! You can plug in anything
and b to be close to zero as possible. So, a = 4 and
2
2
other than 1. If you set x = 2, then x
1 = 5
1 = 4.
b = 0 will give us the smallest value of a
+ b
, namely,
2
16. Using the other solution would give a
+ 36, which
Now go through the answers, plugging in 2 for x. You
will always be bigger than 16.
will find that answers A and B are both equal in value
to 4. If this happens, simply plug in another number.
(You don’t need to retry the answers that were wrong.)
If x = 0, then x
1 =
3
1 =
4, and only answer
B is also
4, so that is the correct answer.
6. D
(Estimated Difficulty Level: 5)
If you love algebra, here is how to do it:
This is a tough one. For f (x) to be equal to g(x) for
2
4
all x, we need ax
= bx
. First, notice that if x = 0,
x + 3
1 =
1
both sides are zero, so x = 0 is a solution. If x is not
x
x
1
2
zero, we can divide both sides of the equation by x
to
x + 3
x
1
2
get: a = bx
. Solving for x results in x = ± a/b. This
=
x
1
x
1
makes three solutions total, so answer D is correct. It
x + 3
(x
1)
may help to plug in numbers for a and b to make this
=
x
1
problem more concrete.
4
=
x
1
Be still my beating heart!
7. B
(Estimated Difficulty Level: 5)
To make a b as large as possible, we need to make a as
large as possible and b as small as possible. So, a b has
3. B
(Estimated Difficulty Level: 5)
to be less than 40
50 =
10. To make a
b as small
as possible, we need to make a as small as possible and
2/3
Solve the first equation given for a: a = b
. Then
b as large as possible. So, a
b has to be greater than
√
1/2
1/3
a = a
= b
. (You really need to know your ex-
30
70 =
40. The expression that gives all possible
√
1/3
4/3
ponent rules for this one.) So, b
a = b · b
= b
.
values of a
b is then
40 < a
b <
10.
Using real numbers also works here, but it may be hard
to come up with two that work for a and b (such as
a = 4 and b = 8).
pg. 12
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