# The Ultimate ACCUPLACER Math Formula Cheat Sheet

If you’re taking the ACCUPLACER 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.

ACCUPLACER 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 ACCUPLACER Math before the test day?

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

If you learn every formula in this **ACCUPLACER** **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 ACCUPLACER Math formulas? Please have a look at **ACCUPLACER** **Math Formulas**.

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**ACCUPLACER Math Cheat Sheet**

*Fractions*

*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*

*Decimals*

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

*Mixed Numbers*

*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*

*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*

*Integers*

\( \{…,-3,-2,-1,0,1,2,3,…\} \)

Includes: zero, counting numbers, and the negative of the counting numbers

*Real 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*

*Order of Operations*

PEMDAS

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

*Absolute Value*

*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*

*Ratios*

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

*Percentages*

*Percentages*

Use the following formula to find the part, whole, or percent

part \(=\frac{percent}{100}×whole\)

*Proportional Ratios*

*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*

*Percent of Change*

\(\frac{New \ Value \ – \ Old \ Value}{Old Value}×100\%\)

*Expressions and Variables*

*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*

*Distributive Property*

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

*Equations*

*Equations*

The values of the two mathematical expressions are equal.

\(ax+b=c\)

*Distance from A to B:*

*Distance from A to B:*

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

*Parallel and Perpendicular lines:*

*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:*

*Mid-point of the segment AB:*

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

*Slope of the line:*

*Slope of the line:*

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

*Point-slope form:*

*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:*

*Slope-intercept form:*

given the slope m and the y-intercept b, then the equation of the line is:

\(y=mx+b\).

*Factoring:*

*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:*

*Exponents:*

Refers to the number of times a number is multiplied by itself.

\(8 = 2 × 2 × 2 = 2^3\)

*Scientific Notation:*

*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:*

*Square:*

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

*Square Roots:*

*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:*

*Pythagorean Theorem:*

\(a^2+b^2=c^2\)

*Triangles*

*Triangles*

**All triangles:**

**All triangles:**

Area \(=\frac{1}{2}\) b . h

Angles on the inside of any triangle add up to \(180^\circ\).

*Equilateral:*

*Equilateral:*

These triangles have three equal sides, and all three angles are \(60^\circ\).

*Isosceles:*

*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*

*Circles*

Area \(=πr^2\)

Circumference \(=2πr\)

Full circle \(=360^\circ\)

**Rectangles**

**Rectangles**

(Square if *l=w*)

Area=*lw*

**Parallelogram**

**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:*

*Area of a trapezoid:*

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

*Surface Area and Volume of a Rectangular/right prism:*

*Surface Area and Volume of a Rectangular/right prism:*

\(SA=ph+2B\)

\(V=Bh\)

**Surface Area and Volume of a Cylinder:**

**Surface Area and Volume of a Cylinder:**

\(SA =2πrh+2πr^2\)

\(V =πr^2 h \)

**Surface Area and Volume of a Cone**

**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**

**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**:

**Simple interest**

\(I=prt\)

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

**mean**:

**mean**

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

**mode:**

**mode:**

value in the list that appears most often

**range:**

**range:**

largest value \(-\) smallest value

*Median*

*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*

*Average*

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

*Average speed*

*Average speed*

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

*Probability*

*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*

*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*

*Simple Interest*

The charge for borrowing money or the return for lending it.

Interest = principal \(×\) rate \(×\) time

OR

\(I=prt\)

*Powers/ Exponents*

*Powers/ Exponents*

*Positive 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*

*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**

**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**

**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*

*Powers of Powers*

For the power of power: you multiply the exponents.

\((x^a)^b=x^{(ab)}\)

*Dividing Powers*

*Dividing Powers*

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

*The Zero Exponent*

*The Zero Exponent*

Anything to the 0 power is 1.

\(x^0= 1\)

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