# The Ultimate SSAT Upper Level Math Formula Cheat Sheet

If you’re taking the SSAT Upper-Level Math test in a few weeks or months, you might be anxious about how to remember ALL the different formulas and math concepts and recall them during the test.

The SSAT Upper-Level Math covers a wide range of topics—from as early as elementary school to high school.

While you have probably learned many of these formulas at some point, it may have been a long time since you’ve used them. This is where most test takers have a hard time preparing for the test.

So, what formulas do you need to have memorized for the SSAT Upper-Level Math before the test day?

Following is a quick formula reference sheet that lists all important SSAT Upper-Level Math formulas you MUST know before you sit down for the test.

If you learn every formula in this SSAT Upper-Level Math Formula Cheat Sheet, you will save yourself valuable time on the test and probably get a few extra questions correct.

Looking for a comprehensive and complete list of all SSAT Upper-Level Math formulas? Please have a look at SSAT Upper-Level Math Formulas.

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## SSAT Upper LevelMath Cheat Sheet

### Fractions

A number expressed in the form $$\frac{a}{b}$$
Adding and Subtracting with the same denominator:
$$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$$
$$\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}$$
Adding and Subtracting with the different denominator:
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+cb}{bd}$$
$$\frac{a}{b}-\frac{c}{d}=\frac{ad-cb}{bd}$$
Multiplying and Dividing Fractions:
$$\frac{a}{b} × \frac{c}{d}=\frac{a×c}{b×d}$$
$$\frac{a}{b} ÷ \frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc}$$

### Decimals

Is a fraction written in a special form? For example, instead of writing  $$\frac{1}{2}$$ you can write $$0.5$$.

### Mixed Numbers

A number is composed of a whole number and a fraction. Example: $$2 \frac{2}{ 3}$$ Converting between improper fractions and mixed numbers: $$a \frac{c}{b}=a+\frac{c}{b}= \frac{ab+ c}{b}$$

### Factoring Numbers

Factor a number means breaking it up into numbers that can be multiplied together to get the original number. Example:$$12=2×2×3$$

### Integers

$$\{…,-3,-2,-1,0,1,2,3,…\}$$
Includes: zero, counting numbers, and the negative of the counting numbers

### Real Numbers

All numbers that are on a number line. Integers plus fractions, decimals, and irrationals, etc.) ($$\sqrt{2},\sqrt{3},π$$, etc.)

### Order of Operations

PEMDAS
(parentheses / exponents/ multiply/ divide/ add/ subtract)

### Absolute Value

Refers to the distance of a number from, the distances are positive as the absolute value of a number cannot be negative. $$|-22|=22$$

### Ratios

A ratio is a comparison of two numbers by division. Example: $$3 : 5$$, or $$\frac{3}{5}$$

### Percentages

Use the following formula to find the part, whole, or percent
part $$=\frac{percent}{100}×whole$$

### Proportional Ratios

A proportion means that two ratios are equal. It can be written in two ways:
$$\frac{a}{b}=\frac{c}{d}$$ , $$a: b = c: d$$

### Percent of Change

$$\frac{New \ Value \ – \ Old \ Value}{Old Value}×100\%$$

### Expressions and Variables

A variable is a letter that represents unspecified numbers. One may use a variable in the same manner as all other numbers: Addition: $$2+a$$: $$2$$ plus a
Subtraction: $$y-3$$  : $$y$$ minus $$3$$
Division: $$\frac{4}{x}$$  : 4 divided by x
Multiplication: $$5a$$  : $$5$$ times a

### Distributive Property

$$a(b+c)=ab+ac$$

### Equations

The values of the two mathematical expressions are equal.
$$ax+b=c$$

### Distance from A to B:

$$\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 }$$

### Parallel and Perpendicular lines:

Parallel lines have equal slopes. Perpendicular lines (i.e., those that make a $$90^°$$ angle where they intersect) have negative reciprocal slopes: $$m_{1}$$ .$$m_{2}=-1$$.
Parallel Lines (l $$\parallel$$ m)

### Mid-point of the segment AB:

M ($$\frac{x_{1}+x_{2}}{2} , \frac{y_{1}+y_{2}}{2}$$)

### Slope of the line:

$$\frac{y_{2}- y_{1}}{x_{2} – x_{1} }=\frac{rise}{run}$$

### Point-slope form:

Given the slope m and a point $$(x_{1},y_{1})$$ on the line, the equation of the line is
$$(y-y_{1})=m \ (x-x_{1})$$.

### Slope-intercept form:

given the slope m and the y-intercept b, then the equation of the line is:
$$y=mx+b$$.

### Factoring:

“FOIL”
$$(x+a)(x+b)$$
$$=x^2+(b+a)x +ab$$ “Difference of Squares”
$$a^2-b^2= (a+b)(a-b)$$
$$a^2+2ab+b^2=(a+b)(a+b)$$
$$a^2-2ab+b^2=(a-b)(a-b)$$ “Reverse FOIL”
$$x^2+(b+a)x+ab=$$ $$(x+a)(x+b)$$

### Exponents:

Refers to the number of times a number is multiplied by itself.
$$8 = 2 × 2 × 2 = 2^3$$

### 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: $$m \times 10^n$$
Scientific notation:
$$5×10^0$$
$$-2.5×10^4$$
$$5×10^{-1}$$
$$2,122456×10^3$$

### Square:

The number we get after multiplying an integer (not a fraction) by itself. Example: $$2×2=4,2^2=4$$

### Square Roots:

A square root of $$x$$ is a number r whose square is $$x: r^2=x$$
$$r$$ is a square root of $$x$$

### Pythagorean Theorem:

$$a^2+b^2=c^2$$

### All triangles:

Area $$=\frac{1}{2}$$ b . h
Angles on the inside of any triangle add up to $$180^\circ$$.

### 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 $$=360^\circ$$

(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; $$π ~ 3.14$$)

### 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 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
$$0<x^3<x^2<x<\sqrt{x}<\sqrt{3x}<1$$.

### Simple Interest

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

### 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.
$$x^0= 1$$

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