Sat math Essentials

8.
c.
Since 
BCD
is an equilateral triangle, angles
CBD
,
BDC
, and 
BCD
all measure 60 degrees.
FCD
and 
BCF
are both 30-60-90 right trian-
gles that are congruent to each other. The
side opposite the 60-degree angle of triangle
BCF
, side 
FC
, is equal to 
3
times the length
of the side opposite the 30-degree angle, side
BF
. Therefore,
BF
is equal to 
= 6 cm.
The hypotenuse,
BC
, is equal to twice the
length of side 
BF
. The length of
BC
is 2(6) =
12 cm. Since 
BC
= 12 cm,
CD
and 
BD
are
also 12 cm.
BD
is one side of square 
ABDE
;
therefore, each side of
ABDE
is equal to 12
cm. The perimeter of
ABCDE
= 12 cm +
12 cm + 12 cm + 12 cm + 12 cm = 60 cm.
9.
4
Substitute 2 for 
x
and 5 for 
y
: (3
xy
+ 
x
)
x
y
=
((3)(2)(5) + 2)
2
5
= (30 + 2)
2
5
= 32
2
5
= (
5
32
)
2
=
2
2
= 4. Or, 3(2)(5) = 30, 30 + 2 = 32, the 5th root of
32 is 2, 2 raised to the 2nd power is 4.
t
+ 2
t
+ 4
t
+ 8
t
+ 16
t
5
–
P R A C T I C E T E S T 3
–
2 4 1

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10.
1,014
Of the concert attendees, 41% were between
the ages of 18–24 and 24% were between the
ages of 25–34. Therefore, 41 + 24 = 65% of
the attendees, or (1,560)(0.65) = 1,014 peo-
ple between the ages of 18 and 34 attended
the concert.
11.
43.2
Matt’s weight,
m
, is equal to 
3
5
of Paul’s
weight,
p
:
m
= 
3
5
p
. If 4.8 is added to 
m
, the
sum is equal to 
2
3
of
p
:
m
+ 4.8 = 
2
3
p
. Substi-
tute the value of
m
in terms of
p
into the sec-
ond equation:
3
5
p
+ 4.8 = 
2
3
p
,
1
1
5
p
= 4.8,
p
=
72. Paul weighs 72 pounds, and Matt weighs
3
5
(72) = 43.2 pounds.
12.
1
4
Solve –6
b
+ 2
a
– 25 = 5 for 
a
in terms of
b
:
–6
b
+ 2
a
– 25 = 5, –3
b
+ 
a
= 15,
a
= 15 + 3
b
.
Substitute 
a
in terms of
b
into the second
equation:
15 +
b
3
b
+ 6 = 4,
1
b
5
+ 3 + 6 = 4,
1
b
5
=
–5,
b
= –3. Substitute 
b
into the first equation
to find the value of
a
: –6
b
+ 2
a
– 25 = 5,
–6(–3) + 2
a
– 25 = 5, 18 + 2
a
= 30, 2
a
= 12,
a
= 6. Finally, (
a
b
)
2
= (
–
6
3
)
2
= (–
1
2
)
2
= 
1
4
.
13.
6
If
j
@
k
= –8 when 
j
= –3, then:
–8 = (
–
k
3
)
–3
–8 = (
–
k
3
)
3
–8 = –
2
k
7
3
216 = 
k
3
k
= 6
14.
63
The size of an intercepted arc is equal to the
measure of the intercepting angle divided by
360, multiplied by the circumference of the
circle (2
π
r
, where 
r
is the radius of the circle):
28
π
= (
3
8
6
0
0
)(2
π
r
), 28 = (
4
9
)
r
,
r
= 63 units.
15.
10
Write the equation in slope-intercept form (
y
= 
mx
+ 
b
): 3
y
= 4
x
+ 24,
y
= 
4
3
x
+ 8. The line
crosses the 
y
-axis at its 
y
-intercept, (0,8). The
line crosses the 
x
-axis when 
y
= 0:
4
3
x
+ 8 = 0,
4
3
x
= –8,
x
= –6. Use the distance formula to
find the distance from (0,8) to (–6,0):
Distance = 
(
x
2
– 
x
1
)
2
+ (
y
2
– 
y
1
)
2
Distance = 
((–6) –
0)
2
+ (
0 – 8)
2
Distance = 
6
2
+ (–
8)
2
Distance = 
36 + 64
Distance = 
100
Distance = 10 units.
16.
1
The largest factor of a positive, whole num-
ber is itself, and the smallest multiple of a
positive, whole number is itself. Therefore,
the set of only the factors and multiples of
a positive, whole number contains one
element—the number itself.
17.
52
There is one adult for every four children on
the bus. Divide the size of the bus, 68, by 5:
6
5
8
= 13.6. There can be no more than 13 groups
of one adult, four children. Therefore, there
can be no more than (13 groups)(4 children
in a group) = 52 children on the bus.
18.
25
If the original ratio of guppies,
g
, to platies,
p
,
is 4:5, then 
g
= 
4
5
p
. If nine guppies are added,
then the new number of guppies,
g
+ 9, is
equal to 
5
4
p
:
g
+ 9 = 
5
4
p
. Substitute the value
of
g
in terms of
p
from the first equation:
4
5
p
+ 9 = 
5
4
p
, 9 = 
2
9
0
p
,
p
= 20. There are 20 platies
in the fish tank and there are now 20(
5
4
) = 25
guppies in the fish tank.
Section 3 Answers
1.
b.
Parallel lines have the same slope. When an
equation is written in the form 
y
= 
mx
+ 
b
,
the value of
m
(the coefficient of
x
) is the
slope. The line 
y
= –2
x
+ 8 has a slope of –2.
The line 
1
2
y
= –
x
+ 3 is equal to 
y
= –2
x
+ 6.
This line has the same slope as the line 
y
= –2
x
+ 8; therefore, these lines are parallel.
2.
c.
Six people working eight hours produce
(6)(8) = 48 work-hours. The number of peo-
ple required to produce 48 work-hours in
three hours is 
4
3
8
= 16.
–
P R A C T I C E T E S T 3
–
2 4 2
6
3
3

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3.
c.
The function 
f
(
x
) is equal to –1 every time the
graph of
f
(
x
) crosses the line 
y
= –1. The graph
of
f
(
x
) crosses 
y
= –1 twice; therefore, there are
two values for which 
f
(
x
) = –1.
4.
e.
Write the equation in quadratic form and find
its roots:
x
4
2
– 3
x
= –8
x
2
– 12
x
= –32
x
2
– 12
x
+ 32 = 0
(
x
– 8)(
x
– 4) = 0
x
– 8 = 0,
x
= 8
x
– 4 = 0,
x
= 4
x
4
2
– 3
x
= –8 when 
x
is either 4 or 8.
5.
d.
Factor the numerator and denominator;
x
2
–
16 = (
x
+ 4)(
x
– 4) and 
x
3
+ 
x
2
– 20
x
= 
x
(
x
+ 5)
(
x
– 4). Cancel the (
x
– 4) terms that appear in
the numerator and denominator. The fraction
becomes 
x
(
x
x
+
+
4
5)
, or 
x
x
2
+
+
5
4
x
.
6.
b.
Angles 
OBE
and 
DBO
form a line. Since there
are 180 degrees in a line, the measure of angle
DBO
is 180 – 110 = 70 degrees.
OB
and 
DO
are
radii, which makes triangle 
DBO
isosceles, and
angles 
ODB
and 
DBO
congruent. Since 
DBO
is
70 degrees,
ODB
is also 70 degrees, and 
DOB
is
180 – (70 + 70) = 180 – 140 = 40 degrees. Angles
DOB
and 
AOC
are vertical angles, so the meas-
ure of angle 
AOC
is also 40 degrees. Angle 
AOC
is a central angle, so its intercepted arc,
AC
, also
measures 40 degrees.
7.
e.
The volume of a cylinder is equal to 
π
r
2
h
, where
r
is the radius of the cylinder and 
h
is the height
of the cylinder. If the height of a cylinder with a
volume of 486
π
cubic units is six units, then
the radius is equal to:
486
π
= 
π
r
2
(6)
486 = 6
r
2
81 = 
r
2
r
= 9
A cylinder has two circular bases. The area of a
circle is equal to 
π
r
2
, so the total area of the
bases of the cylinder is equal to 2
π
r
2
, or 2
π
(9)
2
= 2(81)
π
= 162
π
square units.
8.
d.
Cross multiply:
a
20
= 
a
2
20
= 2
180
a
2
4
5
= 2
36
5
2
a
2
5
= 12
5
a
2
= 6
a
= 
6
9.
b.
Since triangle 
DEC
is a right triangle, triangle
AED
is also a right triangle, with a right angle at
AED
. There are 180 degrees in a triangle, so the
measure of angle 
ADE
is 180 – (60 + 90) = 30
degrees. Angle 
A
and angle 
EDC
are congruent,
so angle 
EDC
is also 60 degrees. Since there are
180 degrees in a line, angle 
BDC
must be 90
degrees, making triangle 
BDC
a right triangle.
Triangle 
ABC
is a right triangle with angle 
A
measuring 60 degrees, which means that angle
B
must be 30 degrees, and 
BDC
must be a 30-60-
90 right triangle. The leg opposite the 30-degree
angle in a 30-60-90 right triangle is half the
length of the hypotenuse. Therefore, the length
of
DC
is 
1
2
5
units.