ISEE Upper-Level Math Formulas

Rounding

Whole Number  

Estimates  

Decimals  

Mixed Numbers

Factoring Numbers

Divisibility Rules

Greatest Common Factor

Least Common Multiple  

Integers  

Real Numbers  

Order of Operations  

Absolute Value

Ratios

Percentages

Proportional Ratios

Percent of Change

Markup  

Discount  

Expressions and Variables  

Tax

Distributive Property  

Polynomial

Systems of Equations  

Equations  

Functions

Inequalities

Solving Systems of Equations by Elimination

Lines (Linear Functions)  

Distance from A to B:

Parallel and Perpendicular lines:  

Mid-point of the segment AB:  

Slope of the line:  

Point-slope form:  

Intersecting lines:

Slope-intercept form:

Transversal: Parallel lines:

Parabolas:

Factoring:

Exponents:  

Scientific Notation:  

Square:  

Square Roots:

Pythagorean Theorem:  

Triangles

Right triangles:

All triangles:

Area \(=\frac{1}{2}\) b. h
Angles on the inside of any triangle add up to \(180^\circ\).
The length of one side of any triangle is always less than the sum and more than the difference between 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 \(60^\circ\).

Isosceles:

An isosceles triangle has two equal sides. The “base” angles (the ones opposite the two sides) are equal (see the \(45^\circ\)  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 to.

Circles

Area \(=πr^2\)
Circumference \(=2πr\)
Full circle \(=360^\circ\)

Rectangles

(Square if l=w)
Area=lw

Parallelogram

(Rhombus if l=w)
Area=lh
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
\((n-2).180^\circ\).

Area of a parallelogram:  

 \(A = bh\)

Area of a trapezoid:  

\(A =\frac{1}{2} h (b_{1}+b_{2})\)

Surface Area and Volume of a Rectangular/right prism:  

\(SA=ph+2B\)
\(V=Bh\)

Surface Area and Volume of a Cylinder:

\(SA =2πrh+2πr^2\)
\(V =πr^2 h  \)

Surface Area and Volume of a Pyramid

\(SA=\frac{1}{2} \ ps+b\)
\(V=\frac{1}{3}\ bh\)

Surface Area and Volume of a Cone  

\(SA =πrs+πr^2\)
\(V=\frac{1}{3} \ πr^2 \ h\)

Surface Area and Volume of a Sphere  

\(SA =4πr^2\)
\(V =\frac{4}{3} \ πr^3\)
(p \(=\) perimeter of base B; \(π ~ 3.14 \))

Solids

Rectangular Solid
Volume =lwh
Area =2(lw+wh+lh)

Right Cylinder
Volume \(=πr^2 \ h\)
Area \(=2πr(r+h)\)

Quadratic formula:  

\( x=\frac{-b±\sqrt{b^2-4ac}}{2a}\)

Simple interest:

\(I=prt\)
(I = interest, p = principal, r = rate, t = time)

mean:

mean: \(\frac{sum \ of \ the \ data}{of \ data \ entires}\)

mode:

value in the list that appears most often

range:

largest value \(-\) smallest value

Median  

The middle value in the list (which must be sorted)
Example: median of
\( \{3,10,9,27,50\} = 10\)
Example: median of
\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)

Sum  

average \(×\) (number of terms)

Average

\( \frac{sum \ of \ terms}{number \ of \ terms}\)

Average speed

\(\frac{total \ distance}{total \ time}\)

Probability

\(\frac{number \ of \ desired \ outcomes}{number \ of \ total \ outcomes}\)
The probability of two different events A and B both happening are:
P(A and B)=p(A).p(B)
as long as the events are independent (not mutually exclusive).

Powers, Exponents, Roots

\(x^a.x^b=x^{a+b}\)
\(\frac{x^a}{x^b} = x^{a-b}\)
\(\frac{1}{x^b }= x^{-b}\)
\((x^a)^b=x^{a.b}\)
\((xy)^a= x^a.y^a\)
\(x^0=1\)
\(\sqrt{xy}=\sqrt{x}.\sqrt{y}\)
\((-1)^n=-1\), if n is odd.
\((-1)^n=+1\), if n is even.
If \(0<x<1\), then
\(0<x^3<x^2<x<\sqrt{x}<\sqrt{3x}<1\).

Interest

Simple Interest

The charge for borrowing money or the return for lending it.
Interest = principal \(×\) rate \(×\) time
OR
\(I=prt\)

Compound Interest

Interest is computed on the accumulated unpaid interest as well as on the original principal.
A \(=P(1+r)^t\)
A= amount at end of the time
P= principal (starting amount)
r= interest rate (change to a decimal i.e. \(50\%=0.50\))
t= number of years invested

Powers/ Exponents

Positive Exponents

An exponent is simply shorthand for multiplying that number of identical factors. So \(4^3\) is the same as (4)(4)(4), three identical factors of 4. And \(x^3\) is just three factors of x, \((x)(x)(x)\).

Negative Exponents

A negative exponent means to divide by that number of factors instead of multiplying.
So \(4^{-3}\) is the same as \( \frac{1}{4^3}\) and
\(x^{-3}=\frac{1}{x^3}\)

Factorials  

Factorial- the product of a number and all counting numbers below it.
8 factorial \(=8!=\)
\(8×7×6×5×4×3×2×1=40,320\)
5 factorial \(=5!=\)
\(5×4×3×2×1=120\)
2 factorial \(=2!=2× 1=2\)

Multiplying Two Powers of the SAME Base  

When the bases are the same, you find the new power by just adding the exponents
\(x^a.x^b=x^{a+b }\)

Powers of Powers

For the power of power: you multiply the exponents.
\((x^a)^b=x^{(ab)}\)

Dividing Powers

\(\frac{x^a}{x^b} =x^a x^{-b}= x^{a-b}\)

The Zero Exponent

Anything to the 0 power is 1.
\(x^0= 1\)

Permutation: 

When different orderings of the same items are counted separately, we have a permutation problem:
\(_{n}p_{r}=\frac{n!}{(n-1)!}\)

Combination:

The fundamental counting principle, as demonstrated above, is used any time the order of the outcomes is important.  When selecting objects from a group where the order is NOT important, we use the formula for COMBINATIONS:
The fundamental counting principle, as demonstrated above, is used any time the order of the outcomes is important. When selecting objects from a group where the order is NOT important, we use the formula for COMBINATIONS:
\(_{n}C_{r}=\frac{n!}{r!(n-1)!}\)