ISEE Middle Level Math Formulas

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

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:\(12=2×2×3\)

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
(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. \(|-22|=22\)
or \(|x| =\begin{cases}x \ for \ x≥0 \\x \ for \ x < 0\end{cases} \)
\(|x|<n⇒-n<x<n\)
\(|x|>n⇒x<-n or x>n\)

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

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

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 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

Tax

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

Distributive Property

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

Polynomial

\(P(x)=a_{0} x^n+ a_{1} x^{n-1}+\)⋯\(+a_{n-2} x^2+a_{n-1} x+an\)

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 is a rule to go from one number (x) to another number (y), usually written \(y=f(x)\). For any given value of x, there can only be one corresponding value y. If \(y=kx\) for some number k (example: \(f(x)= 0.5 x\)), then y is said to be directly proportional to x. If y\(=\frac{k}{x }\) (example: f(x \(=\frac{5}{x}\)), then y is said to be inversely proportional to x. The graph of \(y=f(x )+k\) is the translation of the graph of \(y=f(x)\) by \((h,k)\) units in the plane. For example, \(y=f(x+3)\) shifts the graph of \(f(x)\) by 3 units to the left.

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

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:

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
\((y-y_{1})=m \ (x-x_{1})\).

Intersecting lines:

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

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.

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: \(m \times 10^n\)
Decimal notation:
5
\(-25,000\)
0.5
2,122.456
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:

For any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\): \(a^2+b^2=c^2\)
Solving for the hypotenuse: \(c=\sqrt{a^2+b^2}\)
Solving for a leg: \(a=\sqrt{c^2-b^2}\)
Common Pythagorean triples: \(3,4,5\); \(5,12,13\); \(8,15,17\); \(7,24,25\)

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: \(180°\)

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: \(180°\)

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
\((n-2).180^\circ\).

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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 \(=2πr(r+h)\)

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

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
\(I=prt\)

Powers/ Exponents

\(x^a×x^b=x^{a+b}\)
\(\frac{x^a}{x^b}=x^{a-b}\)
\((x^a)^b=x^{ab}\)
\(x^0=1\)
\(x^{-a}=\frac{1}{x^a}\)
\(x^{\frac{1}{n}}=\sqrt[n]{x}\)

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 an order is NOT important, we use the formula for COMBINATIONS:
\(_{n}C_{r}=\frac{n!}{r!(n-1)!}\)