FraCtiOnS

A fraction is a part of a whole

Slice a pizza, and you will have fractions:

1/2	1/4	3/8
(One-Half)	(One-Quarter)	(Three-Eighths)
The top number tells how many slices you have The bottom number tells how many slices the pizza was cut into.

Numerator / Denominator

We call the top number the Numerator, it is the number of parts you have.
We call the bottom number the Denominator, it is the number of parts the whole is divided into.

Numerator

Denominator

You just have to remember those names! (If you forget just think "Down"-ominator)

Equivalent Fractions

Some fractions may look different, but are really the same, for example:

4/8	=	2/4	=	1/2
(Four-Eighths)		Two-Quarters)		(One-Half)
	=		=

It is usually best to show an answer using the simplest fraction ( ¹/₂ in this case ). That is called Simplifying, or Reducing the Fraction

Adding Fractions

You can add fractions easily if the bottom number (the denominator) is the same:

1/4	+	1/4	=	2/4	=	1/2
(One-Quarter)		(One-Quarter)		(Two-Quarters)		(One-Half)
	+		=		=

Another example:

5/8	+	1/8	=	6/8	=	3/4
	+		=		=

Adding Fractions with Different Denominators

But what if the denominators (the bottom numbers) are not the same? As in this example:

3/8	+	1/4	=	?
	+		=

You must somehow make the denominators the same.

In this case it is easy, because we know that ¹/₄ is the same as ²/₈ :

3/8	+	2/8	=	5/8
	+		=

PeRcentaGes

When you say "Percent" you are really saying "per 100"

	So 50% means 50 per 100 (50% of this box is green)		And 25% means 25 per 100 (25% of this box is green)

Examples:

		100% means all. Example: 100% of 80 is 100/100 × 80 = 80
		50% means half. Example: 50% of 80 is 50/100 × 80 = 40
		5% means 5/100ths. Example: 5% of 80 is 5/100 × 80 = 4

Using Percent Because "Percent" means "per 100" you should think "this should always be divided by 100" So 75% really means 75/100 And 100% is 100/100, or exactly 1 (100% of any number is just the number, unchanged) And 200% is 200/100, or exactly 2 (200% of any number is twice the number) Use the slider on the left and try some different numbers (example, what is 40% of 80?)

A Percent can also be expressed as a Decimal or a Fraction

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

Some Worked Examples

Example: Calculate 25% of 80

25% = ²⁵/₁₀₀

(²⁵/₁₀₀) × 80 = 20

So 25% of 80 is 20

Example: 15% of 200 apples were bad. How many apples were bad?

15% = ¹⁵/₁₀₀

(¹⁵/₁₀₀) × 200 = 15 × 2 = 30 apples

30 apples were bad

Example: if only 10 of the 200 apples were bad, what percent is that?

As a fraction, 10/200 = 0.05
As a percentage it is: (10/200) x 100 = 5%
5% of those apples were bad

Example: A Skateboard is reduced 25% in price in a sale. The old price was $120. Find the new price

First, find 25% of $120:
25% = ²⁵/₁₀₀

(²⁵/₁₀₀) × $120 = $30

25% of $120 is $30

So the reduction is $30

Take the reduction from the original price

$120 - $30 = $90

The Price of the Skateboard in the sale is $90

VoluMe FoRmuLas

(pi = = 3.141592...)

Volume Formulas

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a". "b³" means "b cubed", which is the same as "b" times "b" times "b".

cube = a ³
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r ² h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r ² h
sphere = (4/3) pi r ³
ellipsoid = (4/3) pi r₁ r₂ r₃

 

DiViSioN

Division is splitting into equal parts or groups.

It is the result of "fair sharing".

Example: there are 12 chocolates, and 3 friends want to share them, how do they divide the chocolates?

12 Chocolates		12 Chocolates Divided by 3

Answer: 12 divided by 3 is 4: they get 4 each.

Symbols

÷ /	We use the ÷ symbol, or sometimes the / symbol to mean divide: 12 ÷ 3 = 4 12 / 3 = 4 I will use both symbols here so you get used to it.

Opposite of Multiplying

Division is the opposite of multiplying. If you know a multiplication fact you can find a division fact:

Example: 3 × 5 = 15, so 15 / 5 = 3.
Also 15 / 3 = 5.

Why? Well it is easy to understand if you think of the numbers in rows and columns like in this illustration:

Multiplication... ...Division 3 groups of 5 make 15... so 15 divided by 3 is 5     and also:       5 groups of 3 make 15... so 15 divided by 5 is 3.

So there are four related facts:

• 3 × 5 = 15
• 5 × 3 = 15
• 15 / 3 = 5
• 15 / 5 = 3

Knowing your Multiplication Tables can help you with division!

Example: What is 56 ÷ 7 ?

Searching around the multiplication table you find that 56 is 7 × 8, so 56 divided by 7 must be 8. Answer: 56 ÷ 7 = 8.

Names

There are special names for each number in a division:

dividend ÷ divisor = quotient

Example: in 12 ÷ 3 = 4:

• 12 is the dividend
• 3 is the divisor
• 4 is the quotient

But Sometimes It Does Not Work Perfectly!

Sometimes you cannot divide things up evenly ... there may be something left over.

Example: There are 7 cookies, and 2 people want to share them equally.

But 7 cookies cannot be divided exactly into 2 groups,
each person gets 3 cookies,
but there will be 1 left over:

We call that the Remainder.

Multiplication

Long Multiplication

Long Multiplication is a special method for multiplying larger numbers.

It is a way to multiply numbers larger than 10 that only needs your knowledge of the ten times Multiplication Table.

Let us say we want to multiply

612 × 24

• First we multiply 612 × 4 (=2,448),
• then we multiply 612 × 20 (=12,240),
• and last we add them together (2,448+12,240=14,688).

But we can do better!

When we multiply 612 × 20 we only need to multiply 612 × 2 and place the result one column over (so it is the same as multiplying by 20).

We just have to be careful about lining up the columns.

Subtraction

Subtraction is ...

... taking one number away from another.

		If you have 5 apples, then subtract 2, you will be left with 3.
This would be written: 5 - 2 = 3

Train Yourself

You get good at subtraction with practice...

... so we have developed Math Trainer - Subtraction to help get you trained!

Subtraction Table

You can "look up" answers for simple subtraction using this table:

	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0
1	1	0
2	2	1	0
3	3	2	1	0
4	4	3	2	1	0
5	5	4	3	2	1	0
6	6	5	4	3	2	1	0
7	7	6	5	4	3	2	1	0
8	8	7	6	5	4	3	2	1	0
9	9	8	7	6	5	4	3	2	1	0
10	10	9	8	7	6	5	4	3	2	1	0
11	11	10	9	8	7	6	5	4	3	2	1	0
12	12	11	10	9	8	7	6	5	4	3	2	1	0

Example: Find 8 - 5

• find the row starting with "8"
• move along till you are under the column "5"
• and there is the number "3", so 8 - 5 = 3

Names

Other names used in subtraction are Minus, Less, Difference, Decrease, Take Away, Deduct.
The names of the numbers in a subtraction fact are:

Minuend - Subtrahend = Difference
Minuend: The number that is to be subtracted from. Subtrahend: The number that is to be subtracted. Difference: The result of subtracting one number from another.

Subtracting Larger Numbers

To subtract numbers with more than one digit (such as "42-25"), use any one of these methods, choose whichever you prefer:

	Subtraction with Regrouping (also called "Borrowing") This is the method most people use!
	Quick Subtraction (more complicated, but can be faster)
	Subtraction using Addition (also called the Complements Method)

Addition

Addition is ...

... bringing two or more numbers (or things) together to make a new total.

Here 1 ball is added to 1 ball to make 2 balls:
Using Numbers it is:		1 + 1 = 2
And in words it is:		"One plus one equals two"

Example: If you add 2 and 3 you get 5. You would write it like this:

2 + 3 = 5

Try It

Write this down, with the answer, using numbers:

You should get:   4 + 3 = 7  

Swapping Places

Swapping the position of the numbers you are adding still gets the same result!

3 + 2 = 5
... also ...
2 + 3 = 5

More Examples:

5 + 1 = 1 + 5 = 6

7 + 11 = 11 + 7 = 18

4 + 100 = 100 + 4 = 104