The Ultimate ISEE Upper Level Math Formula Cheat Sheet

Mixed Numbers

Factoring Numbers

Integers  

Real Numbers  

Order of Operations  

Absolute Value

Ratios

Percentages

Proportional Ratios

Percent of Change

Expressions and Variables  

Distributive Property  

Equations  

Distance from A to B:

Parallel and Perpendicular lines:  

Mid-point of the segment AB:  

Slope of the line:  

Point-slope form:  

Slope-intercept form:

Factoring:

Exponents:  

Scientific Notation:  

Square:  

Square Roots:

Pythagorean Theorem:  

Triangles

All 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).

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 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 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; \(π ~ 3.14 \))

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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
\( \{3,9,10,27\}=\frac{(9+10)}{2}=9.5 \)

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 is:
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
\(0<x^3<x^2<x<\sqrt{x}<\sqrt{3x}<1\).

Simple Interest

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

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 \(x\), \((x)(x)(x)\).

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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.
\(x^0= 1\)