ALEKS Math Formulas: Study Reference and Examples

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 \( \{0,1,2,3,…\} \)

Estimates  

Find a number close to the exact answer.  

Decimals  

A decimal represents a number using place values in powers of ten. For example, \(0.5=\frac{1}{2}\).

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:\(12=2×2×3\)

Divisibility Rules

Divisibility means that you are able to divide a number evenly. Example: 24 is divisible by 6, because \(24÷6=4\)

Greatest Common Factor

Multiply common prime factors
Example:\( 200=2×2×2×5×5 60=2×2×3×5\)
GCF \((200,60)=2×2×5=20\)

Least Common Multiple  

Check multiples of the largest number
Example: LCM (200, 60): 200 (no),  400 (no), 600 (yes!)

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: Evaluate parentheses and other grouping first, then exponents. Multiply and divide from left to right; then add and subtract from left to right. Multiplication does not always come before division.

Absolute Value

Absolute value is distance from zero, so it is nonnegative: \(|-22|=22\) and \(|0|=0\). For \(n>0\), \(|x|n\) means \(x<-n\) or \(x>n\).

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 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

For a nonzero positive original value, percent change is \(\frac{\text{new}-\text{original}}{\text{original}}\times100\%\). A negative result indicates a decrease; use its magnitude to report the percent decrease.

Markup  

Markup \(=\) selling price \(-\) cost
Markup rate \(=\) markup divided by the cost

Discount  

Multiply the regular price by the rate of discount
Selling price \(=\) original price \(-\) discount

Expressions and Variables  

A variable is a symbol representing a number. Examples: \(2+a\) adds 2 and \(a\); \(y-3\) subtracts 3 from \(y\); \(4/x\) divides 4 by a nonzero \(x\); \(5a\) multiplies 5 by \(a\).

Tax

To find tax, multiply the tax rate by the taxable amount (income, property value, etc.)

Distributive Property  

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

Polynomial

A polynomial has nonnegative integer powers of its variable. For example, \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\), where the coefficients are constants and \(a_n\ne0\).

Systems of Equations  

Two or more equations working together.
example: \( \begin{cases}-2x+2y=4\\-2x+y=3\end{cases} \)

Equations  

The values of the two mathematical expressions are equal.
\(ax+b=c\)

Functions

A function assigns exactly one output to each input in its domain. Direct variation has the form \(y=kx\); inverse variation has the form \(y=k/x\), with \(x\ne0\). The graph of \(y=f(x-h)+k\) shifts \(f(x)\) right by \(h\) and up by \(k\). Thus \(f(x)+3\) shifts up 3, while \(f(x+3)\) shifts left 3.

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  

Solving Systems of Equations by Elimination

Add the equations \(x+2y=6\) and \(-x+y=3\) to eliminate \(x\): \(3y=9\), so \(y=3\). Substitute into the first equation: \(x+6=6\), giving \(x=0\). The solution is \((0,3)\).

Lines (Linear Functions)  

Consider the line that goes through points \(A(x_{1},y_{1}) \) and \(B(x_{2},y_{2})\).  

Distance from A to B:

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

Parallel and Perpendicular lines:  

Distinct parallel nonvertical lines have equal slopes. Perpendicular nonvertical lines have slopes satisfying \(m_1m_2=-1\). Vertical lines are parallel to one another and perpendicular to horizontal lines; a vertical line has undefined slope.

Mid-point of the segment AB:  

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

Slope of the line:  

For two points with \(x_2\ne x_1\), slope is \(m=\frac{y_2-y_1}{x_2-x_1}=\frac{\text{rise}}{\text{run}}\). If \(x_2=x_1\) and the points are distinct, the line is vertical and its slope is undefined.

Point-slope form:  

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

Intersecting lines:

Opposite angles are equal. Also, each pair of angles along the same line add to \(180^\circ\). In the figure above, \(a+b=180^\circ\).

Slope-intercept form:

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

Transversal: 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.

Parabolas:

For \(y=ax^2+bx+c\) with \(a\ne0\), the parabola opens up if \(a>0\) and down if \(a<0\). Its y-intercept is \(c\), and the vertex has x-coordinate \(-b/(2a)\).

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”
\(x^2+(b+a)x+ab=\) \((x+a)(x+b)\)

You can use Reverse FOIL to factor a polynomial by thinking about two numbers a and b which add to the number in front of the x, and which multiply to give the constant. For example, to factor \(x^2+5x+6\), the numbers add to 5 and multiply to 6, i.e.: \(a=2\) and \(b=3\), so that \(x^2+5x+6=(x+2)(x+3)\). To solve a quadratic such as \(x^2+bx+c=0\), first factor the left side to get \((x+a)(x+b)=0\), then set each part in parentheses equal to zero. For example, \(x^2+4x+3= (x+3)(x+1)=0\) so that \(x=-3\) or \(x=-1\).
To solve two linear equations in x and y: use the first equation to substitute for a variable in the second. E.g., suppose \(x+y=3\) and \(4x-y=2\). The first equation gives y=3-x, so the second equation becomes \(4x-(3-x)=2 ⇒ 5x-3=2\) \(⇒ x=1,y=2\).

Exponents:  

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

Scientific Notation:  

Scientific notation is \(m\times10^n\), where \(1\le |m|<10\) and \(n\) is an integer. Examples: \(5=5\times10^0\), \(-25{,}000=-2.5\times10^4\), \(0.5=5\times10^{-1}\), and \(2122.456=2.122456\times10^3\).

Square:  

The square of a real number is that number multiplied by itself. Examples: \(2^2=4\) and \((1/2)^2=1/4\).

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:  

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

Triangles

Area: \(A=\frac{1}{2}bh\) where \(b\) is the base and \(h\) is the height.
Perimeter: \(P=a+b+c\) (sum of all three sides).
Pythagorean Theorem (right triangles): \(a^2+b^2=c^2\) where \(c\) is the hypotenuse.
Sum of interior angles: \(180°\)

Right 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.

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:  

Similar triangles have equal corresponding angles and proportional corresponding sides. A 3-4-5 triangle and a 6-8-10 triangle are similar with corresponding side ratio \(1:2\).

Circles

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

Length Of Arc \(=(n^\circ/360^\circ).2πr\)
Area Of Sector \(=(n^\circ/360^\circ) .πr^2\)
Equation of the circle (above left figure): \((x-h)^2+(y-k)^2=r^2\).

Rectangles

A rectangle has area \(A=lw\) and perimeter \(P=2(l+w)\). It is a square when \(l=w\).

Parallelogram

A parallelogram has area \(A=bh\), where \(h\) is the perpendicular height, not a sloping side. A rhombus is a parallelogram with all four sides equal. For any simple convex n-sided polygon, the interior-angle sum is \((n-2)180^\circ\). A regular polygon also has all sides and angles equal.

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:  

For a right prism, \(SA=ph+2B\) and \(V=Bh\), where \(B\) is base area, \(p\) is base perimeter and \(h\) is perpendicular height.

Surface Area and Volume of a Cylinder:

\(SA =2πrh+2πr^2\)
\(V =πr^2 h  \)

Surface Area and Volume of a Pyramid

For a regular pyramid, \(SA=\frac12p\ell+B\) and \(V=\frac13Bh\), where \(p\) is base perimeter, \(B\) is base area, \(\ell\) is slant height and \(h\) is perpendicular height.

Surface Area and Volume of a Cone  

For a right circular cone, \(SA=\pi r\ell+\pi r^2\) and \(V=\frac13\pi r^2h\), with radius \(r\), slant height \(\ell\) and perpendicular height \(h\).

Surface Area and Volume of a Sphere  

For a sphere of radius \(r\), \(SA=4\pi r^2\) and \(V=\frac43\pi r^3\).

Solids

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

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

Quadratic formula:  

For \(ax^2+bx+c=0\), with \(a\ne0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).

Simple interest:

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

mean:

The mean is \(\bar{x}=\frac{\text{sum of data values}}{\text{number of data values}}\), for a nonempty data set.

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

For equally likely outcomes, \(P(A)=\frac{\text{number of outcomes in }A}{\text{total number of outcomes}}\). For independent events, \(P(A\cap B)=P(A)P(B)\). Independence differs from mutual exclusivity: mutually exclusive events cannot occur together.

Powers, Exponents, Roots

For positive real bases, exponent laws include \(x^ax^b=x^{a+b}\), \(x^a/x^b=x^{a-b}\), \((x^a)^b=x^{ab}\), and \(x^{-b}=1/x^b\). Also \(x^0=1\) for \(x\ne0\). For real square roots, \(\sqrt{xy}=\sqrt{x}\sqrt{y}\) when \(x,y\ge0\). For integer \(n\), \((-1)^n\) is -1 when \(n\) is odd and 1 when it is even. If \(0

Interest

Simple Interest: \(I=Prt\)
where \(P\) = principal, \(r\) = annual interest rate (as a decimal), \(t\) = time in years.
Total amount: \(A=P+I=P(1+rt)\)
Compound Interest: \(A=P(1+\frac{r}{n})^{nt}\) where \(n\) is the number of times interest is compounded per year.

Simple Interest

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

Compound Interest

For annual compounding, \(A=P(1+r)^t\), where \(r\) is the annual rate as a decimal and \(t\) is the number of years. For \(n\) equal compounding periods per year, \(A=P(1+r/n)^{nt}\). These models assume a constant rate and no additional deposits or withdrawals.

Powers/ Exponents

For positive real bases, \(x^ax^b=x^{a+b}\), \(x^a/x^b=x^{a-b}\), \((x^a)^b=x^{ab}\), and \(x^{-a}=1/x^a\). The rule \(x^0=1\) requires \(x\ne0\). For a positive integer \(n\), \(x^{1/n}=\sqrt[n]{x}\); in the real numbers, an even root requires \(x\ge0\).

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  

For a positive integer \(n\), \(n!=n(n-1)\cdots2\cdot1\); by definition, \(0!=1\). Examples: \(8!=40{,}320\), \(5!=120\) and \(2!=2\).

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 power of a 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

For \(x\ne0\), \(x^0=1\). This rule does not assign a value to \(0^0\).

Permutation: 

For \(r\) ordered selections from \(n\) distinct objects without repetition, \(P(n,r)=\frac{n!}{(n-r)!}\), where \(0\le r\le n\). Example: choosing a captain and deputy from 5 people gives \(P(5,2)=5\times4=20\).

Combination:

When order does not matter and repetition is not allowed, \(C(n,r)=\frac{n!}{r!(n-r)!}\), where \(0\le r\le n\). Choosing an unordered pair from 5 people gives \(C(5,2)=10\), half the 20 ordered pairs.

Laws of Logarithm

For \(a>0\), \(a\ne1\) and positive \(M,N\): \(\log_aN=x\) means \(a^x=N\); \(\log_a1=0\) and \(\log_aa=1\). The product, quotient and power rules are \(\log_a(MN)=\log_aM+\log_aN\), \(\log_a(M/N)=\log_aM-\log_aN\), and \(\log_a(N^m)=m\log_aN\). Also \(a^{\log_ax}=x\) for \(x>0\), and \(\log_a(a^x)=x\).

The Six Trig Ratios

For an acute angle in a right triangle, \(\sin\theta=\text{opposite}/\text{hypotenuse}\), \(\cos\theta=\text{adjacent}/\text{hypotenuse}\), and \(\tan\theta=\text{opposite}/\text{adjacent}\). Their reciprocals are \(\csc\theta\), \(\sec\theta\) and \(\cot\theta\), respectively.

Trig Functions relationships:

Where the denominators are nonzero, \(\tan x=\sin x/\cos x\), \(\csc x=1/\sin x\), \(\sec x=1/\cos x\), and \(\cot x=\cos x/\sin x\).