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GED Mathematical Reasoning Formula Sheet
Area of a:
Square 𝐴 = 𝑠
2
Rectangle 𝐴 = 𝑙𝑤
Parallelogram 𝐴 = 𝑏ℎ
Triangle
𝐴 =
1
2
𝑏ℎ
Trapezoid
𝐴 =
1
2
ℎ(𝑏
1
2
)
Circle
Surface Area and Volume of a:
Rectangular 𝑆𝐴 = 2 𝑙𝑤 + 2 𝑙ℎ + 2 𝑤ℎ 𝑉 = 𝑙𝑤ℎ
Right Prism 𝑆𝐴 = 𝑝ℎ + 2 𝐵 𝑉 = 𝐵ℎ
Cylinder 𝑆𝐴 = 2 𝜋𝑟ℎ + 2 𝜋𝑟
2
𝑉 = 𝜋𝑟
2
ℎ
Pyramid
𝑆𝐴 =
1
2
𝑝𝑠 + 𝐵 𝑉 =
1
3
𝐵ℎ
Cone 𝑆𝐴 = 𝜋𝑟 + 𝜋𝑟
2
𝑉 =
1
3
𝜋𝑟
2
ℎ
Sphere 𝑆𝐴 = 4 𝜋𝑟
2
𝑉 =
4
3
𝜋𝑟
3
(𝑝 = perimeter of base 𝐵; 𝜋 = 3. 14 )
Algebra
Slope of a line
𝑚 =
𝑦
2
− 𝑦
1
𝑥
2
− 𝑥
1
Slope-intercept form of the equation of a line 𝑦 = 𝑚𝑥 + 𝑏
Point-slope form of the Equation of a line 𝑦 − 𝑦 1
= 𝑚(𝑥 − 𝑥
1
)
Standard form of a Quadratic equation 𝑦 = 𝑎𝑥
2
Quadratic formula
𝑥 =
−𝑏 ± √𝑏
2
− 4 𝑎𝑐
2 𝑎
Pythagorean theorem 𝑎
2
2
= 𝑐
2
Simple interest 𝐼 = 𝑝𝑟𝑡
(𝐼 = interest, 𝑝 = principal, 𝑟 = rate, 𝑡 = time)
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A Quick Review and the List of all GED Mathematics
Formulas
Place Value
The value of the place, or position,
of a digit in a number.
Example: In 456, the 5 is in “tens”
position.
Fractions
A number expressed in the form
𝑎
𝑏
Adding and Subtracting with the
same denominator:
𝑎
𝑏
𝑐
𝑏
𝑎 + 𝑐
𝑏
𝑎
𝑏
𝑐
𝑏
𝑎 − 𝑐
𝑏
Adding and Subtracting with the
different denominator:
𝑎
𝑏
𝑐
𝑑
𝑎𝑑 + 𝑐𝑏
𝑏𝑑
𝑎
𝑏
𝑐
𝑑
𝑎𝑑− 𝑐𝑏
𝑏𝑑
Multiplying and Dividing Fractions:
𝑎
𝑏
×
𝑐
𝑑
𝑎 × 𝑐
𝑏 × d
𝑎
𝑏
÷
𝑐
𝑑
𝑎
𝑏
𝑐
𝑑
𝑎𝑑
𝑏𝑐
Comparing Numbers Signs
Equal to =
Less than <
Greater than >
Greater than or equal ≥
Less than or equal ≤
Rounding
Putting a number up or down to the
nearest whole number or the
nearest hundred, etc.
Example: 64 rounded to the nearest
ten is 60 , because 64 is closer to 60
than to 70.
Whole Number
The numbers { 0 , 1 , 2 , 3 , … }
Mixed Numbers
A number composed of a whole
number and fraction
Example: 2
2
3
Converting between improper
fractions and mixed numbers:
a
𝑐
𝑏
= a +
𝑐
𝑏
𝑎𝑏+ 𝑐
𝑏
Estimates
Find a number close to the exact
answer.
Decimals
Is a fraction written in a special
form. For example, instead of
writing
1
2
you can write 0. 5.
Factoring Numbers
Factor a number means to break it
up into numbers that can be
multiplied together to get the
original number.
Example: 12 = 2 × 2 × 3
Divisibility Rules
Divisibility means that you are able
to divide a number evenly.
Example: 24 is divisible by 6 ,
because 24 ÷ 6 = 4
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Addition 2 + 𝑎
2 plus 𝑎
Subtraction 𝑦 – 3 𝑦 minus 3
Division
4 divided
by 𝑥
Multiplication 5 𝑎
5 times 𝑎
Distributive Property
Polynomial
0
𝑛
1
𝑛− 1
𝑛− 2
2
𝑛− 1
Systems of Equations
Two or more equations working
together. example: {
Equations
The values of two mathematical
expressions are equal.
Inequalities
Says that two values are not equal
𝑎 ≠ 𝑏 a not equal to b
𝑎 < 𝑏 a less than b
𝑎 > 𝑏 a greater than b
𝑎 ≥ 𝑏 a greater than or equal b
𝑎 ≤ 𝑏 a less than or equal b
Lines (Linear Functions)
Consider the line that goes through
points 𝐴(𝑥
1
1
) and 𝐵(𝑥
2
2
Distance from 𝑨(𝑥
1
1
) to
2
2
√(𝑥
1
− 𝑥
2
)
2
1
− 𝑦
2
)
2
Mid-point of the segment AB:
M (
𝑥
1
+𝑥
2
2
𝑦
1
+𝑦
2
2
Slope of the line:
2
1
2
1
run
Solving Systems of
Equations by Substitution
Consider the system of
equations
Substitute 𝑥 = 1 − 𝑦 in the
second equation
Substitute 𝑦 = 2 in 𝑥 = 1 + 𝑦
Solving Systems of
Equations by Elimination
Example:
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Point-slope form:
Given the slope m and a
point (𝑥
1
1
) on the line, the
equation of the line is
1
1
Slope-intercept form: given
the slope m and the y-
intercept 𝑏, then the
equation of the line is:
Scientific Notation
It is a way of expressing numbers that
are too big or too small to be
conveniently written in decimal form.
In scientific notation all numbers are
written in this form: 𝑚 × 10
𝑛
Decimal
notation
Scientific notation
5 × 10
0
− 25 , 000 − 2. 5 × 10
4
5 × 10
− 1
2 , 122. 456 2 , 122456 × 10
3
Parallel lines
Have equal slopes.
Perpendicular lines (i.e., those
that make a 90° angle where
they intersect) have negative
reciprocal slopes:
1
2
Intersecting Lines
Parallel Lines ( l ‖m )
Intersecting lines: opposite
angles are equal. Also, each pair
of angles along the same line
add to 180°. In the figure
above, 𝑎 + 𝑏 = 180°.
Parallel lines: eight angles are
formed when a line crosses two
parallel lines. The four big
angles (𝑎) are equal, and the
four small angles (𝑏) are equal.
Exponents
Refers to the number of times a
number is multiplied by itself.
8 = 2 × 2 × 2 = 2
3
Factoring
“FOIL”
2
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Triangles
A good example of a right triangle is
one with 𝑎 = 3 , 𝑏 = 4 , and 𝑐 = 5 ,
also called a 3 – 4 – 5 right triangle.
Note that multiples of these
numbers are also right triangles. For
example, if you multiply these
numbers by 2 , you get 𝑎 = 6 , 𝑏 = 8 ,
and
𝑐 = 10 ( 6 – 8 – 10 ), which is also a
right triangle.
Circles
2
Length Of 𝐴𝑟𝑐 = (𝑛°/360°) × 2 𝜋𝑟
= (𝑛°/360°) × 𝜋𝑟
2
1
2
b. h
Angles on the inside of any triangle
add up to 180°.
The length of one side of any triangle
is always less than the sum and more
than the difference of the lengths of
the other two sides.
An exterior angle of any triangle is
equal to the sum of the two remote
interior angles. Other important
triangles:
Equilateral:
These triangles have three equal
sides, and all three angles are 60°.
Isosceles:
An isosceles triangle has two equal
sides. The “base” angles
(the ones opposite the two sides) are
equal (see the 45° triangle above).
Similar:
Two or more triangles are similar if
they have the same shape. The
corresponding angles are equal, and
the corresponding sides are in
proportion. For example, the 3 – 4 – 5
triangle and the 6 – 8 – 10 triangle
from before are similar since their
sides are in a ratio of 2 to 1.
𝑏
n
◦
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Area of a parallelogram:
Area of a trapezoid:
1
2
h (𝑏
1
+ b
2
Solids
Rectangular Solid
Right Cylinder
2
Rectangles
(Square if 𝑙 = 𝑤)
Parallelogram
(Rhombus if 𝑙 = 𝑤)
Regular polygons are n-sided
figures with all sides equal and all
angles equal.
The sum of the inside angles of an
n-sided regular polygon is
Surface Area and Volume of a
rectangular/right prism:
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Probability
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
number of total outcomes
The probability of two different
events A and B both happening is:
as long as the events are
independent (not mutually
exclusive).
Exponents: Multiplying Two Powers
of the SAME Base
When the bases are the same, you
find the new power by just adding
the exponents
𝑎
𝑏
𝑎+𝑏
Multiplying Two Powers of
Different Bases Same Exponent
If the bases are different but the
exponents are the same, then you
can combine them
𝑎
𝑎
𝑎
Powers of Powers
For power of a power: you multiply
the exponents.
𝑎
𝑏
(𝑎𝑏)
Powers, Exponents, Roots
𝑎
𝑏
𝑎+𝑏
𝑥
𝑎
𝑥
𝑏
𝑎−𝑏
1
𝑥
𝑏
−𝑏
𝑎
𝑏
𝑎.𝑏
𝑎
𝑎
𝑎
0
𝑛
= − 1 , if 𝑛 is odd.
𝑛
= + 1 , if 𝑛 is even.
If 0 < 𝑥 < 1 , then
0 < 𝑥
3
< 𝑥
2
< 𝑥 < √
𝑥 < √
3 𝑥 < 1.
Interest
Simple Interest
The charge for borrowing money
or the return for lending it.
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 × 𝑟𝑎𝑡𝑒 × 𝑡𝑖𝑚𝑒
OR
Compound Interest
Interest computed on the
accumulated unpaid interest as
well as on the original principal.
𝑡
𝐴 = amount at end of time
𝑃 = principal (starting amount)
𝑟 = interest rate (change to a
decimal i.e. 50% = 0. 50 )
𝑡 = number of years invested
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Positive Exponents
An exponent is simply shorthand for
multiplying that number of identical
factors. So 4³ is the same as
(4)(4)(4), three identical factors of 4.
And 𝑥³ is just three factors of 𝑥,
Negative Exponents
A negative exponent means to
divide by that number of factors
instead of multiplying.
So 4
− 3
is the same as
1
4
3
and
− 3
3
Dividing Powers
𝑥
𝑎
𝑥
𝑏
𝑎
−𝑏
𝑎−𝑏
The Zero Exponent
Anything to the 0 power is 1.
0
0
0
