Kaplan: Math

Comparing Fractions

Multiply Numerator of first fraction by denominator of second. Multiply numerator of second fractions by denominator of first. Product that is greater is the larger fraction. (i.e 2/5 and 5/8. 2×8=16, 5×5= 25, 5/8 is greater then 2/5)

Converting Fractions to Decimals

Divide numerator by denominator (i.e 8/25= 0.32)

Converting Decimal to Fraction

Place number over 1 (i.e 0.3= 0.3/1) Then move decimal to the farthest right of number and add as many zeros to the denominator as places decimal point has been moved (i.e 0.3/1= 3/10, .68= .68/1=68/100)

Converting Fraction or Decimal to a Percent

Multiply Fraction or Decimal by 100% (Fraction to Percent i.e 4/5×100= 80%) (Decimal to Percent i.e .60×100=60%)

Converting Percent to Fraction or Decimal

Divide Percent by 100% (Percent to Decimal i.e 40%÷100%=.4) (Percent to Fraction i.e 40%÷100%= 40/100= 4/10 = 2/5)

Adding & Subtracting Coefficients with Exponents

(i.e x²+x²= 2x²) Base (x) and exponent (2) of both Coefficients must be the same and can then be added or subtracted. (i.e x²+x³ cannot be combined because exponents are different) (i.e x²+y² cannot be combined because bases are different)

Multiplying Coefficients and Exponents

Multiply the coefficients and add the exponents only if the bases are the same (i.e 2x⁵×(8x⁷)= (2×8)(x⁵+x⁷)=16x¹²)

Dividing Coefficients and Exponents

Divide coefficients and subtract exponents

(i.e 6x⁷÷2x⁵=(6÷2)(x⁷-x⁵)=3x²)

Raising a power to an Exponent

Multiply exponents (i.e (x²)⁴=x²×⁴=x⁸

Adding and Subtracting Radicals

Numbers under radical sign must be the same in order to add the coefficients on the outside (i.e 2√2 + 3√2 = 5√2) If they are not the same and can be simplified to the same number then they can be added, otherwise they can not (i.e 5√3+2√75 = 5√3+2√25.√3 = 2×5√3=5√3+10√3 = 15√3)

Multiplying and Dividing Radicals

Multiply or Divide Coefficients on the outside with each other, and radicals with each other. (i.e 3√2×4√5= 12√10) (i.e 12√10÷3√2=4√5)

Square Roots of Fractions

Take square root of numerator and denominator separately (i.e √16/√25=4/5)

To Find X% of Y

Part=Percent in Decimal×Whole (i.e 44% of 25= .44×25=11) (i.e .8% of 244= .008×244=1.952)

To Find X% in X% of Y=Z

Percent= Part/Whole ×100 (i.e 42 is what Percent of 70? 42÷70= 0.6 × 100= 60%)

Coefficient

The number that comes before the Variable (i.e 6x, 6 is the Coefficient)

Variable

The Variable is the letter that stands for the unknown (i.e 6x, x is the Variable)

Term

The Product of a constant and one or more Variables

Monomial

One Term (i.e 6x)

Polynomial

Two or more Terms (i.e 6x-y)

Trinomial

Three Terms (i.e 6x-y+z)

Adding and Subtracting Algebraic Expressions

In an Expression, you can only combine like Terms(i.e 6a+5a=11a) You cannot combine unlike Terms (i.e 6a+ 5a² cannot be combined) You can simplify unlike Terms(i.e (3a+2b-8a) = (3a-8a+2b) = -5a+2b)

Multiplying & Dividing Algebraic Expressions

Different Terms can be Multiplied and Divided. Multiply the Coefficients of each Terms, add the Exponents of Like Variables, and Multiply Different Variables together (i.e (6a)(4ab) = (6×4)(a×a×b) = (6×4)(a¹+¹×b) = 24a²b)

Use the FOIL (First Outer Inner Last) method to multiply and divide binomials (i.e (y+1)(y+2) = {(y×y)+(y×2)+1×y)+(1×2)} = (y²+2y+y+2) = y²+3y+2)

Equations

To solve an equation isolate the Variable to one side (i.e 7x+2y=3x-19y-16 Find x (7x+2y-2y=3x+10y-16-2y) = (7x=3x+8y-16) = (7x-3x= 3x+8y-16-3x) = (4x=8y-16) = (4x/4=8y/4-16/4) = x=2y-4)

Inequalities(< , >)

Solve an inequality like you would any other equation, except when your multiplying or dividing an inequality by a negative number, you must change the direction of the sign (i.e ⁻5a>10= (⁻5a/⁻5>10/⁻5) = a

Average, Median, Mode

Average= Sum of the Terms/ Number of The Terms (i.e Average of 15, 18, 15, 32, 20 = 100/5= 20)

Median= Middle number ,or sum of the two middle numbers divided by 2 (i.e Median of 15,15, 18, 19, 20, 32 = (18+19)÷2= 18.5)

Mode= value of term that occurs most (i.e Mode of 15, 15, 19, 20= 15)

Proportions

Cross Multiply and solve for the variable (i.e x/6=2/3 3x=12 x=4)

Rates

Distance= Rate×Time

Lines & Segments

There are 180° in a straight line. A Line Segment is a piece of the Line. The point in the middle of the Line is called the Midpoint. (i.e P___Q___R PR=12 and QR=4 (PQ=PR-QR) = (PQ=12-4) = PQ=8)

Angles

A Right Angle is 90°. Lines that intersect to form Right Angles are Perpendicular. Angles that form a straight line add up to 180° An angle greater then 90° is Obtuse, an angle smaller then 90° is Acute.(i.e a°__/__b° a is Obtuse, b is Acute) If two Lines Intersect (i.e X) the adjacent angles are Supplementary and add up to 180°. Angles around the point are 360°.

Triangle Basics

The interior Angles of a Triangle add up to 180°. An exterior Angle of a Triangle is equal to the sum of the Remote Interior Angles (i.e exterior angle y°= Interior Remote Angles 40°+ 95°= 135°) The Length of One Side of a Triangle must be Greater then the Positive Difference, and Less then the Sum of the Lengths of the otehr two sides (i.e Triangle x,y,z y=3° & z=4°, so x must be greater then 4-3=1, and less then 4+3=7)

Triangles Area & Perimeter

The Perimeter of a Triangle is the Sum of the Length of its sides. The Area of a Triangle+ 1/2(base)(height), in which case the Height is the Perpendicular side to the Base.

Similar Triangles

Similar Triangles have the Same Shape, Corresponding Angles are Equal, and Corresponding Sides are Proportional.

Special Triangles

Isosceles Triangles have Two Equal Sides, and the Angles opposite the Equal sides are also Equal.

Equilateral Triangles have all Three Sides Equal, and All Three Angles are Equal.

Right Triangles have a Right Angles. Every Right Angle has 2 Acute Angles. The Sides Opposite the Acute Angles are called Legs. The side Opposite the Right Angle is called the Hypotenuse, and the Longest Side of the Triangle.

Pythagorean Theorem

leg1²+leg2²=(hypotenuse)² (i.e A Right Triangle has legs with lengths of 2 & 3. (2²+3²=c²) = (4+9=c²) = (13=c²) = c=√13 )

Quadrilaterals

A Quadrilateral has 4 sides. The Perimeter of a Quadrilateral is the sum of the length of its sides.

Parallelograms, Squares and Rectangles

A Parallelogram is a Quadrilateral with 2 sets of Parallel sides. Opposite sides are Equal, and Opposite angles.

Area of Parallelogram=(base)(height)

A Rectangle is a Parallelogram with 4 Right Angles, and Opposite sides are equal. Area of a Rectangle= (length)(width).

Perimeter of a Rectangle=2(l+w)

A Square is a Rectangle with 4 equal sides. Area of Square=(side)²

Perimeter of A Square=4s

Similar Rectangles (or Squares)

Corresponding sides of Rectangles (or Squares) are in Proportion. (i.e One Rectangle has sides of 6 &12, Another has sides of 4 & 8, so each side of the larger Rectangle is 1½ times the Corresponding side of the smaller Rectangle)

Circles

A Circle is a figure in which each point is an equal distance from it's center.

The Radius of a Circle is the distance from the center to any point on the circle.

A Chord is a line segment that connects any 2 points on a circle.

The Diameter of a Circle is a chord that passes through the center. All Diameters are Equal to twice the Radius.

The Circumference of a Circle is the distance around it, 2πr or πd.

The Area of a Circle equals π times the square of the radius, or πr². (i.e Diameter= 6, radius=6/2=3 and area= πr² × π(3²)=9π)

Coordinate Geometry

Coordinate Geometry has to do with Plotting Points on a Graph. The Horizontal Line is the X Axis, and the Vertical Line is called the Y Axis. When coordinates are written (0,1) 0 is the X Coordinate and 1 is the Y Coordinate.

Pi:

3.14 or 22/7

Circumference of a circle:

C=2 x Pi x r

Area of a Circle:

A= Pi x r ^2

Radius of a circle:

is the line segment connecting the center of the circle to any point on the circle

Circumference of a circle:

The total distance around the circle

Diameter of a circle:

A line that passes through the center of the circle, connecting any two points

Perimeter of a triangle:

P=a+b+c

Area of a triangle;

A=1/2 b x h

Pythagorean Theorem:

a^2 + B^2 = C^2 Hypotenuse is denoted by letter C

Perimeter of a rectangle:

P=2L x 2W

Area of rectangle:

A = L x W

Perimeter of a square:

P= 4 x s

Area of square:

A = S^2

Volume of a cube or Prism:

V= L x W x H or S^3

Volume of a circular cylinder:

V= Pi x r^2 x H

Ratio:

Comparison of two numbers by division

Proportion:

Two equal ratios

Fraction to decimal:

Divide numerator by denominator

Interest formula:

Principal x rate x time