Mathematics GCSE

Multiplication Law:                              Negative Indices:
Yn X Ym = Yn+m                                Y-n  =  Y1/n

Division Law:
Yn / Ym = Yn-m

Multiplying Indices:
(Yn)m  = Ynm

Xo = 1, any number to the power of o is 1.
1n = 1, any number with 1 is 1. 

Adding a negative number you go down.
Adding a positive number you go up.
Taking-away a negative number you go up.

If 2 signs follow each other then:
- - = +    
+ + = +
- + = -
+ - = -

- X - = +
+ X + = +
- X + = -
+ X - = -

- / - = +
+ / + = +
- / + = -
+ / - = + 

Ratios are also used when dividing up amounts. The basic method is:
Simplify the ratio, if possible (not essential, but it makes life easier in the long run).
Add the numbers in the ratio together (to get the total number of parts needed).
Divide the amount by this total number of parts.
Multiply the answer by each of the numbers in the ratio. 

Direct Proportion:

Two quantities are in direct proportion when they increase or decrease in the same ratio. For example, you could increase something by doubling it, or decrease it by halving.

For a number to be in standard from it needs to be a number between 1 and 10, but it can be a decimal.

2500 into standard form = 2.5 x 10 to the power of 3

Multiplying standard form:
(2 x 10 to the power of 3) X (6 x 10 to the power of 5)
(6 x 2) x (10 to the power of 3 x 10 to  the power of 5)
= 12 x 10 to the power of 8.
1.2 x 10
= 1.2 X 10 to the power of 9. 

Dividing standard from:
(6 x 10 to the power of 8) / (3 x 10 to the power of 5)
(6 / 3) x (10 to the power of 8 / 10 to the power of 5) 
= 2 x 10 to the power of 3 

Surd numbers are numbers with a square root that is not whole.
Any number in  a square root that is also a squared will come out. 

Multiplying and dividing surds: 

Simplifying surd numbers:

If the number inside the square root sign was for example 12, then we know that 4 x 3 is 12, and because 4 is the square of 2 it is not a surd number whereas 3 cannot be square rooted into a whole number and is therefore a surd. So the answer would be 3 inside a square root sign.

Lowest Boundary                           Upper Boundary
            4.5                       5                     5.5            

1 and 2 are used when perimeter or money is needed to be found.
3 and 4 aren't used very often.
5 and 6 are used to workout perimeter.
7 and 8 are used for petrol or economy. 

Result

Calculation

1.Upper Boundary

UB + UB

2.Lower Boundary

LB + LB

3.Upper Boundary

UB - LB

4.Lower Boundary

LB - UB

5.Upper Boundary

UB x UB

6.Lower Boundary

LB x LB

7.Upper Boundary

UB / LB

8.Lower Boundary

LB / UB

x/4 + 6 = 11   
Look at the equation and try to get the fraction on its own side.

x/4 + 6 = 11, move the +ve 6 to the same side as the 11 and make it -ve 
x/4 = 11 - 6

Then work out the side with just numbers, and add the answer to the equation.
11 - 6 = 5      
x/4 = 5

Next times the number in the fraction with the answer you've got, this will tell you what x equals.

4 x 5 = 20
x = 20 

2(x+4)/8 = 12    

Move the 8 in the bottom of the fraction and times it by the number on its own, also expand the bracket on the top half of the fraction.
12 x 8 = 96 
2x+8

Then reform the equation.
2x+8 = 96

Then move the 8 from the left and move it to the same side as the 96 and make it -ve.
2x = 96 - 8
2x = 88

Finally move the 2 from in front of the x and divide it by the number on the right to find x.   
88 / 2 = 44                   x = 44 

5(x+2) = 3(x+8)

First expand both brackets.
5x + 10 = 3x + 24

Get all the x's one side and the whole number's the other, don't forget if its positive on its original side it will need to be changed to a negative. Then work it out.
5x - 3x = 24 - 10 
2x = 14

Again take the 2 from in front of the x and divide it by the number on the right to find out what x equals.
14 / 2 = 7         x = 7

The subject must always be on the left.

p = 2x + y        make y the new subject.
move 2x to the same side as P.
p - 2x = y
then flip the equation so y is on the left.
y = p - 2x

s = a+b+c / 2       make a the new subject.
move the 2, then the b and c to the other side.
2s - b - c = a
again flip the equation so the subject is on the left.
a = 2s - b - c 

Bearings go from North clockwise.
Find bearing from P to Q.

 
 
If it asked for the bearing from Q to P you go from the North at point Q and measure from that to the line of QP. 

You can use Pythagoras theorem to work out the hypotenuse, opposite and adjacent edge on a triangle. 

The hypotenuse is the longest diagonal side, and the adjacent side is the side next to the labelled angle and the opposite is the one that is left.

a² + b² = c² 

Trigonometry is used to find out the missing side of a triangle when the angle and one of the sides is given.

sinX = o/h
cosX = a/h
tanX = a/o

When you see the question, I would label the sides h, o and a. Then look at the question and see what it wants you to find out.
Then you put the appropriate rule into the calculator and press =.
Once you have that answer you need to times it  by the number you've been given for the side. 

x2 + 3x + 2 = 0
Factorise into 2 brackets.
(x + 2)(x + 1) = 0
The quadratic is equal to 0.
-1 + 1 = 0  ---> x = -1
x + 1 = 0
x + 2 = 0
-2 + 2 = 0   ---> x = -2
One of the brackets or both of them are equal to 0.
So x +1 = 0 ---> x = -1
and/or x + 2 = 0 ---> x = -2
It means for x = -1 and x = -2, x2 +3x +2 is equal 0. 

r = 5cm
h = 12cm

Terms of π.   V = πr2 x h 
                     V = π x 25 x 12
                     V = 300πcm3 

Numerical term. V = 3.14 x r2 x h
                          V = 3.14 x 25 x 12
                          V = 300 x 3.14 = 942cm3 

r = 5cm
h = 12cm

A = 2πr2 + 2πrh

 2 x π x 25 + 2 x π x 5 x 12
= 50π + 120π
= 170π 

Triangular Prism:
V = area of cross section x height.
area of cross section = height x base / 2

Trapezium Prism:
V = area of cross section x length
area of cross section = top length + bottom length x height / 2

Cuboid:
V = area of cross section x length
area of cross section = height x base

Pyramid:
V = 1/3 x area of base x perpendicular height
(square based pyramid) area of base = height x base  
(triangular based pyramid) area pf base = height x base / 2

Cone:
V = 1/3 of area of base x height or V = 1/3πr2h
area of base = πr2 or 3.14 x r2

Sphere:
V = 4 x 3.14 x r3/3 or 4/3πr3
Surface area of a sphere = 4πr2 

Area of sector = angle at centre/360 x πr2
 

Length of arc = angle at centre/360 x 2πr
     

x<4  Any number less than 4.
x>/ 2 All number bigger than and including 2.
x\< 2 All numbers less than and including 2.

2 < 2x < 4 List all the integers.
1 < x < 2   Divide outside numbers by the number with x in the middle.
There are no integers but are decimals.          

Solve for x: 2x < x + 8 think of it as 2x = x + 8      
2x - x < 8 or 2x - x = 8
x < 8 or x = 8

Solve for x: 4x - 3 < 2x + 5
4x - 2x < 5 + 3
2x < 8 ---> 8/2 = 4
x < 4 

4x + y = 17               Subtract the equations.       When the 2 symbols are the same
2x + y = 9   -             x = 4                                    you take away the equations.
2x = 8                       y = 1

5x - y = 14                Add the equations.               The symbols are different so add 
2x + y = 7                 x = 3                                     them.
7x = 21                     y = 1 

5x + 2y = 4               Add the equations.               You need to make either the x or
4x - y = 11  (x2)        x = 2                                     y the same.
8x - 2y = 22              y = -3
13x = 26                   

Then take one of the original equations and use x to find out what y is.