The Ultimate ISEE Lower Level Math Formula Cheat Sheet

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

Order of Operations  

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

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

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

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.
\(ax+b=c\)

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)

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).

Rectangles

A rectangle has four right angles and opposite sides of equal length.
Area: \(A=l\times w\)
Perimeter: \(P=2l+2w=2(l+w)\)
Diagonal: \(d=\sqrt{l^2+w^2}\)
Both pairs of opposite sides are parallel.

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

mean:

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

mode:

value in the list that appears most often

Average

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