Sunday, July 10, 2016

MEDIAN

How to Find the Median Value

It's the middle of a sorted list of numbers.

Median Value

The Median is the "middle" of a sorted list of numbers.

How to Find the Median Value

To find the Median, place the numbers in value order and find the middle.


Example 1 : find the Median of 12, 3 and 5

Put them in order:
3, 5, 12
The middle is 5, so the median is 5.


Example 2 :

3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

There are fifteen numbers. Our middle is the eighth number:
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

The median value of this set of numbers is 23.

(It doesn't matter that some numbers are the same in the list.)


Two Numbers in the Middle


BUT, with an even amount of numbers things are slightly different.

In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two.


Example 3 :

3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don't have just one middle number, we have a pair of middle numbers:
3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

In this example the middle numbers are 21 and 23.
To find the value halfway between them, add them together and divide by 2:
21 + 23 = 44
then 44 ÷ 2 = 22

So the Median in this example is 22.

(Note that 22 was not in the list of numbers ... but that is OK because half the numbers in the list are less, and half the numbers are greater.)


EXERCISE :

Find the median of the following numbers:

a) 4, 2, 7, 4, 15, 16, 18

b) 5, 2, 2, 3, 4, 11, 12, 12, 14




INEQUALITIES 3

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example 1 :

Solve: x + 3 < 7

If we subtract 3 from both sides, we get:
x + 3 - 3 < 7 - 3    
x < 4
And that is our solution: x < 4
In other words, x can be any value less than 4.

What did we do?

 And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality.

Example 2: 

Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example 3: 

12 < x + 5
If we subtract 5 from both sides, we get:
12 - 5 < x + 5 - 5    
7 < x
That is a solution!
But it is normal to put "x" on the left hand side ...
... so let us flip sides (and the inequality sign!):
x > 7
Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7

 Note: "x" can be on the right, but people usually like to see it on the left hand side.

Exercise :

Solve x + 5 < 12




INEQUALITIES 2

Examples:

How to Solve

Solving inequalities is very like solving equations ... we do most of the same things ...
... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points".

Some things we do will change the direction!
< would become >
> would become <
would become
would become

Safe Things To Do

These are things we can do without affecting the direction of the inequality:

  • Add (or subtract) a number from both sides
  • Multiply (or divide) both sides by a positive number
  • Simplify a side

Example 1: 

3x < 7+3

We can simplify 7+3 without affecting the inequality:
3x < 10

But these things will change the direction of the inequality ("<" becomes ">" for example):

  • Multiply (or divide) both sides by a negative number
  • Swapping left and right hand sides

Example 2: 

2y+7 < 12

When we swap the left and right hand sides, we must also change the direction of the inequality:
12 > 2y+7


Example 3: 

2x+6 < 12

Ans = x < 3



Exercise :

Solve the following
(a) 3x < 12

(b) 2y +4 > 16

PROBABILITY 2

Probability line EXAMPLES:
We can show probability on a Probability Line:








Probability is always between 0 and 1.

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide.

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.
But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Monday, June 20, 2016

SEQUENCE

Sequences

Arithmetic Sequences


An Arithmetic Sequence is made by adding the same value each time. The value added each time is called the "common difference"
Example 1:
1, 4, 7, 10, 13, 16, 19, 22, 25, ...
This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:
Example 2:
3, 8, 13, 18, 23, 28, 33, 38, ...
This sequence has a difference of 5 between each number.
The pattern is continued by adding 5 to the last number each time, like this:
The common difference could also be negative.
Example 3:
25, 23, 21, 19, 17, 15, ...
This common difference is −2
The pattern is continued by subtracting 2 each time, like this:


PROBABILITY

Probability

How likely something is to happen.
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin 

When a coin is tossed, there are two possible outcomes:
  • heads (H) or
  • tails (T)
We say that the probability of the coin landing H is ½.
And the probability of the coin landing T is ½.
Example 1:

The chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)
Total number of outcomes: 6 (there are 6 faces altogether)
So the probability = 16
Example 2:


There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)
Total number of outcomes: 5 (there are 5 marbles in total)
So the probability = 45 = 0.8
Example 3:



There are 7 marbles in a bag: 5 are blue, and 2 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 2 (there are 2 blues)
Total number of outcomes: 7 (there are 7 marbles in total)
So the probability = 27

Exercise:


There are 6 flowers; 1 lavender, 2 roses and 3 sunflower. What is the probability of getting


(a) Lavender roses
(b) sunflower roses

INEQUALITIES

Introduction to Inequalities

Inequality tells us about the relative size of two values.

Mathematics is not always about "equals"! Sometimes we only know that something is bigger or smaller.

Greater or Less Than

The two most common inequalities are:
Symbol
Words
Example Use
>
greater than
5 > 2
<
less than
7 < 9

... Or Equal To!

We can also have inequalities that include "equals", like:
Symbol
Words
Example Use
greater than or equal to
x ≥ 1
less than or equal to
y ≤ 3
Example 1:

Alex and Billy have a race, and Billy wins!

What do we know?
We don't know how fast they ran, but we do know that Billy was faster than Alex:
Billy was faster than Alex
We can write that down like this:
b > a
(Where "b" means how fast Billy was, ">" means "greater than", and "a" means how fast Alex was).
We call things like that inequalities (because they are not "equal").
Example 2:

Alex plays in the under 15's soccer. How old is Alex?

We don't know exactly how old Alex is, because it doesn't say "equals"
But we do know "less than 15", so we can write:
Age < 15
The small end points to "Age" because the age is smaller than 15.
Example 3 is equal to:

You must be 13 or older to watch a movie.

The "inequality" is between your age and the age of 13.
Your age must be "greater than or equal to 13", which is written:
Age ≥ 13

Exercise:

Represent the following sentence into inequalities:-

a) To watch a parental guidance movie, you must be 18 years and above.

b) Ali is twice older than Hisyam