Decimals, Fractions and Percentages

Decimals, Fractions and Percentages are just different ways of showing the same value:

A Half can be written...     As a fraction: 1/2 As a decimal: 0.5 As a percentage: 50%

A Quarter can be written...     As a fraction: 1/4 As a decimal: 0.25 As a percentage: 25%

Here, have a play with it yourself:

Example Values

Here is a table of commonly occuring values shown in Percent, Decimal and Fraction form:

Percent	Decimal	Fraction
1%	0.01	1/100
5%	0.05	1/20
10%	0.1	1/10
12½%	0.125	1/8
20%	0.2	1/5
25%	0.25	1/4
331/3%	0.333...	1/3
50%	0.5	1/2
75%	0.75	3/4
80%	0.8	4/5
90%	0.9	9/10
99%	0.99	99/100
100%	1
125%	1.25	5/4
150%	1.5	3/2
200%	2

Conversions

From Percent to Decimal

To convert from percent to decimal: divide by 100, and remove the "%" sign.
The easiest way to divide by 100 is to move the decimal point 2 places to the left:

From Percent	To Decimal
		move the decimal point 2 places to the left, and remove the "%" sign.

From Decimal to Percent

To convert from decimal to percent: multiply by 100, and add a "%" sign.
The easiest way to multiply by 100 is to move the decimal point 2 places to the right:

From Decimal	To Percent
		move the decimal point 2 places to the right, and add the "%" sign.

From Fraction to Decimal

The easiest way to convert a fraction to a decimal is to divide the top number by the bottom number (divide the numerator by the denominator in mathematical language)

Example: Convert ²/₅ to a decimal

Divide 2 by 5: 2 ÷ 5 = 0.4
Answer: ²/₅ = 0.4

From Decimal to Fraction

To convert a decimal to a fraction needs a little more work.

Example: To convert 0.75 to a fraction

Steps	Example
First, write down the decimal "over" the number 1	0.75 1
Multiply top and bottom by 10 for every number after the decimal point (10 for 1 number, 100 for 2 numbers, etc)	0.75 × 100 1 × 100
(This makes a correctly formed fraction)	75 100
Then Simplify the fraction	3 4

From Fraction to Percentage

The easiest way to convert a fraction to a percentage is to divide the top number by the bottom number. then multiply the result by 100, and add the "%" sign.

Example: Convert ³/₈ to a percentage

First divide 3 by 8: 3 ÷ 8 = 0.375,
Then multiply by 100: 0.375 x 100 = 37.5
Add the "%" sign: 37.5%
Answer: ³/₈ = 37.5%

From Percentage to Fraction

To convert a percentage to a fraction, first convert to a decimal (divide by 100), then use the steps for converting decimal to fractions (like above).

Example: To convert 80% to a fraction

Steps	Example
Convert 80% to a decimal (=80/100):	0.8
Write down the decimal "over" the number 1	0.8 1
Multiply top and bottom by 10 for every number after the decimal point (10 for 1 number, 100 for 2 numbers, etc)	0.8 × 10 1 × 10
(This makes a correctly formed fraction)	8 10
Then Simplify the fraction	4 5

Proportions

Proportion says that two ratios (or fractions) are equal.

Example:

So 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

When things are "in proportion" then their relative sizes are the same.

Here you can see that the ratios of head length to body length are the same in both drawings. So they are proportional. Making the head too long or short would look bad!

Working With Proportions

NOW, how do we use this?

Example: you want to draw the dog's head, and would like to know how long it should be:

Let us write the proportion with the help of the 10/20 ratio from above:

?	=	10
42		20

Now we solve it using a special method:

Multiply across the known corners,
then divide by the third number

And you get this:

? = (42 × 10) / 20 = 420 / 20 = 21

So you should draw the head 21 long.

Using Proportions to Solve Percents

A percent is actually a ratio! Saying "25%" is actually saying "25 per 100":

25% =	25
	100

We can use proportions to solve questions involving percents.
First, put what you know into this form:

Part	=	Percent
Whole		100

Example: what is 25% of 160 ?

The percent is 25, the whole is 160, and we want to find the "part":

Part	=	25
160		100

Find the Part:

Example: what is 25% of 160 (continued) ?

Part	=	25
160		100

Multiply across the known corners, then divide by the third number:

Part = (160 × 25) / 100 = 4000 / 100 = 40

Answer: 25% of 160 is 40.

Note: you could have also solved this by doing the divide first, like this:

Part = 160 × (25 / 100) = 160 × 0.25 = 40

Either method works fine.

We can also find a Percent:

Example: what is $12 as a percent of $80 ?

Fill in what you know:

$12	=	Percent
$80		100

Multiply across the known corners, then divide by the third number. This time the known corners are top left and bottom right:

Percent = ($12 × 100) / $80 = 1200 / 80 = 15%

Answer: $12 is 15% of $80

Or find the Whole:

Example: The sale price of a phone was $150, which was only 80% of normal price. What was the normal price?

Fill in what you know:

$150	=	80
Whole		100

Multiply across the known corners, then divide by the third number:

Whole = ($150 × 100) / 80 = 15000 / 80 = 187.50

Answer: the phone's normal price was $187.50

Using Proportions to Solve Triangles

You can use proportions to solve similar triangles.

Example: How tall is the Tree?

Sam tried using a ladder, tape measure, ropes and various other things, but still couldn't work out how tall the tree was.

But then Sam has a clever idea ... similar triangles! Sam measures a stick and its shadow (in meters), and also the shadow of the tree, and this is what he gets:

Now Sam makes a sketch of the triangles, and writes down the "Height to Length" ratio for both triangles: Height ⇒ Shadow Length ⇒   h  =  2.4 m 2.9 m 1.3 m

Multiply across the known corners, then divide by the third number:

h = (2.9 × 2.4) / 1.3 = 6.96 / 1.3 = 5.4 m (to nearest 0.1)

Answer: the tree is 5.4 m tall.
And he didn't even need a ladder!

The "Height" could have been at the bottom, so long as it was on the bottom for BOTH ratios, like this:

Let us try the ratio of "Shadow Length to Height": Shadow Length ⇒ Height ⇒   2.9 m  =  1.3 m h 2.4 m

Multiply across the known corners, then divide by the third number:

h = (2.9 × 2.4) / 1.3 = 6.96 / 1.3 = 5.4 m (to nearest 0.1)

It is the same calculation as before.

A "Concrete" Example

Ratios can have more than two numbers!
For example concrete is made by mixing cement, sand, stones and water.

A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. You can multiply all values by the same amount and you will still have the same ratio. 10:20:60 is the same as 1:2:6 So if you used 10 buckets of cement, you should use 20 of sand and 60 of stones.

Example: if you have just put 12 buckets of stones into a wheelbarrow, how much cement and how much sand should you add to make a 1:2:6 mix?
Let us lay it out in a table to make it clearer:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:			12

You can see that you have 12 buckets of stones but the ratio says 6.
That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.
Here is the solution:

	Cement	Sand	Stones
Ratio Needed:	1	2	6
You Have:	2	4	12

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

Why are they the same ratio? In the 1:2:6 ratio there is 3 times more Stones as Sand (6 vs 2), and in the 2:4:12 ratio there is also 3 times more Stones as Sand (12 vs 4) ... similarly there is twice as much Sand as Cement in both ratios.

That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same.

So the answer is: add 2 buckets of Cement and 4 buckets of Sand.

Mis mejores deseos en este nuevo año y siempre.

Polygons

A polygon is a plane shape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon (straight sides)	Not a Polygon (has a curve)	Not a Polygon (open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or Irregular

If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Regular		Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.
If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)

Convex		Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon (this one's a Pentagon)		Complex Polygon (also a Pentagon)

More Examples

Irregular Hexagon		Concave Octagon		Complex Polygon (a "star polygon", in this case a pentagram)

Play With Them! Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

		If it is a Regular Polygon...
Name	Sides	Shape	Interior Angle
Triangle (or Trigon)	3		60°
Quadrilateral (or Tetragon)	4		90°
Pentagon	5		108°
Hexagon	6		120°
Heptagon (or Septagon)	7		128.571°
Octagon	8		135°
Nonagon (or Enneagon)	9		140°
Decagon	10		144°
Hendecagon (or Undecagon)	11		147.273°
Dodecagon	12		150°
Triskaidecagon	13		152.308°
Tetrakaidecagon	14		154.286°
Pentadecagon	15		156°
Hexakaidecagon	16		157.5°
Heptadecagon	17		158.824°
Octakaidecagon	18		160°
Enneadecagon	19		161.053°
Icosagon	20		162°
Triacontagon	30		168°
Tetracontagon	40		171°
Pentacontagon	50		172.8°
Hexacontagon	60		174°
Heptacontagon	70		174.857°
Octacontagon	80		175.5°
Enneacontagon	90		176°
Hectagon	100		176.4°
Chiliagon	1,000		179.64°
Myriagon	10,000		179.964°
Megagon	1,000,000		~180°
Googolgon	10100		~180°
n-gon	n		(n-2) × 180° / n

Types of Triangles

Triangles can be classified by various properties relating to their angles and sides. The most common classifications are described on this page.

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Classifications of Triangles

Right Triangles
 

A right triangle has one 90° and a variety of often-studied properties including: Right Triangle Calculator Proof of Pythagorean Theorem Pythagorean Triplets Sine, Cosine, Tangent Pictures of Right Triangles 7 24 25 Right Triangle Images 3 4 5 Right Triangles 5 12 13 Right Triangles

The Equilateral triangle shown on the left has three equal sides and three equal angles.
Each angle is 60°

The Isosceles triangle shown on the left has two equal sides and two equal angles.

The Scalene Triangle has no congruent sides. In other words, each side must have a different length..

The Acute Triangle has three acute angles (an acute angle measures less than 90°)

The  Obtuse Triangle has an obtuse angle (an obtuse angle has more than 90°). In the picture on the left, the shaded angle is the obtuse angle that distinguishes this triangle

Since the total degrees in any triangle is 180°, an obtuse triangle can only have one angle that measures more than 90°.

Points, lines and planes

The word ‘geometry’ derives from the Greek words for ‘earth’ (geo) and ‘to measure’ (metron).

A construction is a geometric drawing for which only a compass and a straightedge may be used.

a compass: for making circles; for transferring the distance between two existing points

ruler/straightedge: used for drawing a line between two existing points; no distance markings will be used; no measurements are allowed

A conjecture is an educated guess.
Using specific observations and examples to arrive at a conjecture is called inductive reasoning.
For example, you might make the conjecture that for all real numbers x and y, the distance between them is given by the formula y−x.
A counterexample is a specific example that shows that a conjecture is not always true.
For example, here is a counterexample to the previous conjecture:
Let y=5 and x=7. Then, y−x=5−7=−2, which is not the distance between them.
Deductive reasoning uses logic, and statements that are already accepted to be true, to reach conclusions.
The methods of mathematical proof are based on deductive reasoning.
Point, line, and plane are three undefined terms to get us started in the study of geometry—we will just agree on their meaning.
A point represents an exact location. It is represented with a dot. Capital letters, like P, are frequently used to denote points.

DEFINITIONS space; geometric figure

Space is the set of all points.

A geometric figure is a subset of space.

That is, a geometric figure is any collection of points.
Of course, there are certain important geometric figures (like triangles and circles) that will be studied throughout the course.
A line has length only; it has no width or thickness; it extends forever in both directions.
A line will be denoted using a lowercase script letter, like ℓ.
If A and B are two distinct points, then they determine a unique line which will be denoted by AB←→ or BA←→.


A plane is a flat surface that extends infinitely in all directions; it has length and width only; it has no thickness.
A plane will be denoted using an uppercase script letter, like P.
If A, B, and C are three distinct noncollinear points (see below), then they determine a unique plane which will be denoted by ABC.


Note: In the following definitions, the prefix ‘co’ means ‘same’.

DEFINITIONS collinear/noncollinear/coplanar/noncoplanar points

collinear points: three or more points lying on the same line


noncollinear points: points not lying on the same line
coplanar points: points lying in the same plane
noncoplanar points: points not lying in the same plane

Coplanar can also refer to other geometric figures.
For example, two lines are coplanar if and only if they lie in the same plane.