Sat math Essentials

10.
d.
p
percent of
q
is equal to 
q
(
1
p
00
), or 
1
p
0
q
0
. If
q
is
decreased by this amount, then the value of
q
is
1
p
0
q
0
less than 
q
, or 
q
– 
1
p
0
q
0
.
11.
e.
A fraction with a negative exponent can be
rewritten as a fraction with a positive expo-
nent by switching the numerator with the
denominator.
(
a
b
)
2
(
a
b
)
–2
(
1
a
)
–1
= (
a
b
)
2
(
a
b
)
2
(
1
a
)
1
= (
a
b
2
2
)(
a
b
2
2
)(
a
) = 
a
b
4
5
.
12.
c.
If
d
is the distance Warrick drives and 
s
is the
speed Warrick drives, then 30
s
= 
d
. Gil drives
five times farther, 5
d
, in 40 minutes, traveling 45
miles per hour: 5
d
= (40)(45). Substitute the
value of
d
in terms of
s
into the second equation
and solve for 
s
, Warrick’s speed: 5(30
s
) =
(40)(45), 150
s
= 1,800,
s
= 12. Warrick drives
12 mph.
2
180
a
–
P R A C T I C E T E S T 3
–
2 4 3

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13.
c.
There are ten coins in the bank (1 penny + 2
quarters + 4 nickels + 3 dimes). The two quar-
ters and three dimes are each worth more than
five cents but less than 30 cents, so the proba-
bility of selecting one of these coins is 
1
5
0
or 
1
2
.
14.
b.
The 
y
-axis divides the rectangle in half. Half of
the width of the rectangle is 
a
units to the left of
the 
y
-axis and the other half is 
a
units to the
right of the 
y
-axis. Therefore, the width of the
rectangle is 2
a
units. The length of the rectangle
stretches from 3
b
units above the 
x
-axis to 
b
units below the 
x
-axis. Therefore, the length of
the rectangle is 4
b
units. The area of a rectangle
is equal to 
lw
, where 
l
is the length of the rec-
tangle and 
w
is the width of the rectangle. The
area of this rectangle is equal to (2
a
)(4
b
) = 8
ab
square units.
15.
a.
Set 
M
contains the positive factors of 8: 1, 2, 4,
and 8. Set 
N
contains the positive factors of 16:
1, 2, 4, 8, and 16. The union of these sets is
equal to all of the elements that are in either set.
Since every element in set 
M
is in set 
N
, the
union of
N
and 
M
is the same as set 
N
: {1, 2, 4,
8, 16}.
16.
b.
The area of a square is equal to 
s
2
, where 
s
is the
length of one side of the square. A square with
an area of 100 cm
2
has sides that are each equal
to 
100
= 10 cm. The diagonal of a square is
equal to 
2
times the length of a side of the
square. Therefore, the lengths of diagonals 
AC
and 
BD
are 10
2
cm. Diagonals of a square
bisect each other at right angles, so the lengths
of segments 
OB
and 
OC
are each 5
2
cm. Since
lines 
BC
and 
EF
are parallel and lines 
OC
and
OB
are congruent, lines 
BE
and 
CF
are also con-
gruent. The length of line 
OF
is equal to the
length of line 
OC
plus the length of line 
CF
:
5
2
+ 3
2
= 8
2
cm. In the same way,
OE
=
OB
+ 
BE
= 5
2
+ 3
2
= 8
2
cm. The area of
a triangle is equal to 
1
2
bh
, where 
b
is the base of
the triangle and 
h
is the height of the triangle.
EOF
is a right triangle, and its area is equal to
1
2
(8
2
)(8
2
) = 
1
2
(64)(2) = 64 cm
2
. The size of
the shaded area is equal to the area of
EOF
minus one-fourth of the area of
ABCD
: 64 –
1
4
(100) = 64 – 25 = 39 cm
2
.
–
P R A C T I C E T E S T 3
–
2 4 4

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absolute value
the distance a number or expression is from zero on a number line
acute angle
an angle that measures less than 90°
acute triangle
a triangle with every angle that measures less than 90°
adjacent angles
two angles that have the same vertex, share one side, and do not overlap
angle
two rays connected by a vertex
arc
a curved section of a circle
area
the number of square units inside a shape
associative property of addition
when adding three or more addends, the grouping of the addends does not affect
the sum.
associative property of multiplication
when multiplying three or more factors, the grouping of the factors does
not affect the product.
average
the quantity found by adding all the numbers in a set and dividing the sum by the number of addends;
also known as the 
mean
base
a number used as a repeated factor in an exponential expression. In 5
7
, 5 is the base.
binomial
a polynomial with two unlike terms, such as 2
x
+ 4
y
bisect
divide into two equal parts
central angle
an angle formed by an arc in a circle 
chord
a line segment that goes through a circle, with its endpoints on the circle
circumference
the distance around a circle
coefficient
a number placed next to a variable
combination
the arrangement of a group of items in which the order doesn’t matter
common factors
the factors shared by two or more numbers