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The Basics
PCAT Review
July 9-10, 2011
Laws of Operation
 Operations, addition and multiplication, are both commutative. You can switch the order of
the numbers and not affect the result.
a+b=b+a
a*b=b*a
 Operations, addition and multiplication, are both associative. You can group numbers any way
and not affect the result.
(a+b)+c=a+(b+c)
(a*b)*c=a*(b*c)
NOTE: Division and Subtraction are NOT commutative or associative!
 The Distributive Law allows you to “distribute” a factor over a group of numbers being added
or subtracted.
Example:

2*(3x-4y)=2*(3x)-2*(4y)
The acronym PEMDAS (or Please Excuse My Dear Aunt Sally) guides you in the correct order of
operation. Parentheses, Exponents, Multiplication and Division (from left to right), Addition
and Subtraction (from left to right)
*Note: If there are parentheses within parentheses, work from the innermost, then outermost.
Example:
30  5  4  (7  3) 2  8
30  5  4  4  8
30  5  4  16  8
First perform Parentheses
Next, do Exponents
30  20  2
Next, multiply and divide in order from left to right
10  2
Finally, add and subtract from left to right
12
Fractions
 For the fraction
1
, 1 is the numerator and 2 is the denominator
2
 A mixed number is a whole number with a fraction.
The Basics
Example:
To convert 2
PCAT Review
July 9-10, 2011
3
3 8 3 11
to one fraction, we do 2    
4
4 4 4 4
Adding/Subtracting Fractions
 For fractions with different denominators, change the fractions so that their
denominators agree, then add/subtract the numerators.
Example:
1 1 5
4
1
 


4 5 20 20 20
Multiplying Fractions
 When multiplying two fractions, multiply the numerators together and multiply the
denominators together.
Example:
5 2 10 5
 

6 3 18 9
Dividing Fractions
 When dividing two fractions, reverse the numerator and denominator of the dividing
fraction (2nd fraction), and multiply this to the divided fraction (1st fraction).
Example:
5 2 5 3 15 5
1
   
 1
6 3 6 2 12 4
4
Multiplying Decimals
Example:
0.342  0.25
1. Multiply the two numbers, ignoring the decimals.
0.342  0.25  8550
2. Find the number of digits behind the decimal in each number and add them together.
3 digits (0.342) + 2 digits (0.25) = 5 digits
3. Take the result of your answer in Step 1 and move the decimal to the left by the number of
digits given in the result of Step 2.
0.08550 (since there are only four digits in 8550, we need to add a 0 behind the decimal)
The Basics
PCAT Review
July 9-10, 2011
Dividing Decimals
Example:
0.475  0.25
1. Find the number of digits behind the decimal in each number and pick the larger of the two.
3 digits (0.475), 2 digits (0.25). Therefore, 3.
2. Move both decimals the number of places from Step 1.
0.475  0.25 will be changed to 475  250
3. Use long division to determine the decimal answer: 475  250  19
.
Decimals to Fractions
Example:
0.1337
1. Find the place value of the last digit in the decimal.
2. Write the digits from the decimal in the numerator and the place value of the decimal in the
denominator.
01337
.

1337
10,000
Decimals to Percents
 To convert a decimal to a percent, multiply the decimal by 100. This is equivalent to moving the
decimal 2 places to the right in the decimal.
Example: 01337
.
 (01337
.
 100)%  1337%
.
Percents
 x % is equivalent to the fraction
 x % of a =
x
a
100
x
Part
(or Percent 
)
100
Whole
The Basics
Examples:
PCAT Review
July 9-10, 2011
72
 0.72
100
1.
72% 
2.
60% of 20 =
60
 20  12
100
Common Percents to Know
Fraction
Decimal
Percent
1/1
1.0
100
3/4
0.75
75
2/3
0.66
66.6
1/2
0.5
50
1/3
0.33
33.3
1/5
0.25
25
1/5
0.2
20
1/8
0.125
12.5
1/10
0.1
10
1/20
0.05
5
Percent Formulas
%Increase=Amount of increase/Original whole * 100
%Decrease=Amount of decrease/Original whole *100
New Whole= Original whole  Amount of Change
Exponents
The Basics
PCAT Review
July 9-10, 2011
 Multiplying powers with the same base: Keep the base the same and add exponents
x 3  x 4  x 3 4  x 7
Example:
 Dividing power with the same base: Keep the base the same and subtract exponents
x10  x 3  x103  x 7
Example:
 Raising a power to an exponent: keep the bas the same and multiply the exponents
Example:
( x 3 ) 5  x 35  x15
 Multiplying powers with same exponent: Multiply the base and keep exponent the same
Example:
(3 x )( 4 x )  (3  4) x  12 x
 Dividing powers with the same exponent : Divide the base and keep the exponent the same
Example:
10 x
 2x
5x
Parallelograms and Rectangles
 Four sided figure, opposite sides parallel
 Area = base/width • height
 Perimeter (P) is the sum of the lengths of the sides
Triangles
 Area =
1
1
• base • height = bh
2
2
 Sum of the three angles equal 180º
The Basics
Similar Triangles
 Two triangles that have the exact same shape.
 Corresponding angles are equal between the triangles.
 Corresponding sides are proportional between the triangles.
Right Triangles
 One angle is 90º
 The sides of the triangle satisfy the Pythagorean Theorem,
 a2 + b2 = c2 , where c is the longest side of the triangle
 Common Pythagorean Theorem values: (3,4,5) and (5,12,13)
Isosceles Triangles
 One side is the base, the other two sides have equal length
 The two angles at the base are equal
Equilateral Triangles
 Each side has the same length.
 Each angle is 60º
PCAT Review
July 9-10, 2011
The Basics
Circles
 r is the radius of a circle
 d is the circumference
 Area =  r 2 (  314
. )
 Circumference = 2r or d
Slice of a Circle
 Slice of a circle, radius r and angle x
 Area of a slice =
x
( r 2 )
360
 Length of an arc =
x
(2 r )
360
Uniform Solid Figure
 Each cross-section is identical
 Volume = Ah = area of the base • height
Examples:
PCAT Review
July 9-10, 2011
The Basics
Suppose the cube has side s = 2.
1. What is the total surface area and volume?
2. What are the lengths of a and d?
Solution to #1:
 The area of one side is s 2  2 2  4
 The total surface area is made up of 6 sides:
SA = 6  s 2  6  4  24
 The volume of the cube is s 3  2 3  8
Solution to #2:
 From the two right triangles in the cube, we have 2 2  2 2  a 2  8
So a  8  2 2
 We also have a 2  2 2  d 2  8  4  12
So d  12  2 3
Spheres
 Volume =
4
 r3
3
 Surface Area = 4  r 2
Cones
PCAT Review
July 9-10, 2011
The Basics
 Volume =
1
 r 2h
3
 Total Surface Area = ( r 2 ) + ( rs )
 Due to right triangle and Pythagorean Theorem, s 2  r 2  h 2
Pyramids
 Area of the base = A
A depends on the shape
 Volume =
1
Ah
3
PCAT Review
July 9-10, 2011