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Preparatory Course
Mathematics
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Order of operations
Operations means things like add, subtract, multiply, divide, squatting etc. If it is not a number it
is probably an operation.
But when you see something like…
7 + (6 x 52 + 3)
…what part should you calculate first? Start at the left and go to the right? Or go from right to
left? If you calculate them in the wrong order, you will get a wrong answer!
So, long ago people agreed to follow rules when doing calculations, and they are:
1) Do things in parentheses first:
6 × (5 + 3)
= 6× 8
= 48
6 × (5 + 3)
= 30 + 3
= 33 (wrong)
2) Exponents (i.e. powers, roots) before multiply, divide, add or subtract. Example:
5 × 22
= 5× 4
=
20
5 × 22
= 102
=
100 (wrong)
3) Multiply or divide before you add or subtract. Example:
2+ 5×3
= 2 + 15
= 17
2+5×3
= 7×3
= 21 (wrong)
4) Otherwise just go left to right
30 ÷ 5 × 3
= 6 ×3
= 18
30 ÷ 5 × 3
= 30 ÷ 15
=
How do you remember it all? PEMDAS!
P
E
MD
AS
Parentheses first
Exponents
Multiplication and Division
Addition and Subtraction
2 (wrong)
Example 1:
How do you work out 3 + 6 x 2?
Multiplication before addition:
First 6 x 2 = 12, then 3 + 12 = 15
Example 2:
How do you work out (3 + 6) x 2?
Parentheses first:
First (3 + 6) = 9, then 9 x 2 = 18
Example 3:
How do you work out 7 + (6 x 52 + 3)?
7 + (6 x 52 + 3)
7 + (6 x 25 + 3)
Start inside Parentheses, and then use Exponents first
7 + (150 + 3)
Then Multiply
7 + (153)
Then Add
7 + 153
Parentheses complete
= 160
Done!
Link to video:
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variables-expressions/cc7th-order-of-operations/v/introduction-to-order-of-operations
Exponents
The exponent of a number says how many times to use the
number in a multiplication.
In 82 the "2" says to use 8 twice in a multiplication,
so 82 = 8 × 8 = 64
Examples:
53 = 5 × 5 × 5 = 125
24 = 2 × 2 × 2 × 2 = 16
Exponents make it easier to write and use many multiplications. Example: 96 is easier to write
and read than 9 x 9 x 9 x 9 x 9 x 9. You can multiply any number by itself as many times as you
want using exponents.
What if the exponent is 1 or 0?
- If the exponent is 1, then you just have the number itself (example 9 1 = 9)
- If the exponent is 0, then you get 1 (example 90 = 1)
Multiplication with exponents
When multiplying two exponents with the same base number, just add the exponents.
Example:
32 × 34 =
32+4 = 36
Division with exponents
When dividing two exponents with the same base number, subtract the exponents.
Example:
97 : 9 3 =
97-3 = 94
Add/subtract exponents with different base numbers
If you got two exponents with different base number, you need to calculate each exponent
before you add/subtract.
Example:
23 + 4 2 =
2×2×2+4×4=
8 + 16 =
24
Link to video:
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6thexponents/v/introduction-to-exponents
Fractions
A fraction is a part of a whole.
Slice a pizza, and you will have fractions:
1/
1/
2
(One-Half)
3/
4
(One-Quarter)
8
(Three-Eighths)
The top number tells how many slices you have, the bottom number tells how many slices the
pizza was cut into.
Some fractions may look different, but are really the same, for example:
=
4/8
=
2/4
1/2
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.
A fraction (such as 3/4) has two numbers:
We call the top number the Numerator, it is the number of
parts we have.
We call the bottom number the Denominator, it is the number
of parts the whole is divided into.
Adding fractions:
You can add fractions easily if the bottom number is the same:
1/
1/
+
4
(One-Quarter)
4
(One-Quarter)
+
2/
=
4
=
(Two-Quarters)
=
1/
2
(One-Half)
=
Adding fractions with different denominators:
What if the denominators (bottom numbers) are not the same? You cannot add fractions with
different denominators. As in this example, you need to make the denominators the same:
1/
1/
+
=
?
3
6
What should be the new denominator?
One simple answer is to multiply the current denominators together:
3 x 6 = 18
So, instead of having 3 or 6 slices, we will make both of them have 18 slices.
18 is a lot of slices. Can you do it with fewer slices?
Here is how to find out:
1/
List the multiples of 3:
3
1/
List the multiples of 6:
6
3, 6, 9, 12, 15, 18, 21, ...
6, 12, 18, 24, ...
Then find the smallest number that is the same
Multiples of 3:
3, 6 , 9, 12, 15, 18, 21, ...
Multiples of 6:
6 , 12, 18, 24, ...
The answer is 6, and that is the least common denominator.
So let us try using it! We want both to have 6 slices.
When we multiply top and bottom of ⅓ by 2 we get 2/6. 1/6 already has a denominator of 6.
And our question now looks like:
/6
2
+
/6
1
=
Multiplication with fractions:
(Multiply the tops, multiply the bottoms)
Step 1: Multiply the top numbers (numerators)
Step 2: Multiply the bottom numbers (denominators)
Step 3: If needed, simplify
Example 1:
1 1 1
 
2 2 4
Example 2:
1 3 3
 
5 7 35
/6
3
Division with fractions:
When dividing fractions,
Step 1: Turn the last fraction upside down (change numerator with denominator, it becomes
reciprocal).
Step 2: Multiply the first fraction by that reciprocal (multiply tops and bottoms)
Example:
1 2 1 7 7
:   
5 7 5 2 10
We leave the first fraction, while turning fraction number two. 2/7 becomes 7/2. Then multiply
numerator with numerator and denominator and denominator.
Another example:
7 3 7 2 14
:   
9 2 9 3 27
Link to video:
https://www.khanacademy.org/math/pre-algebra/fractions-pre-alg/understanding-fractions-prealg/v/introduction-to-fractions
Equation
What is an equation?
An equation says that two things are equal. It will have an equals sign “=” like this:
7 + 2 = 10 – 1
That equation says: what is on the left (7 + 2) is equal to what is on the right (10 – 1)
So, an equation is like a statement “this equals that”.
To solve an equation, you apply inverses to each side of the equation until the variable you
want to solve for is alone by itself on one side of the equation, and appears nowhere on the
other side.
What is an inverse? An inverse is an operation that 'undoes' another operation. Suppose you
have something like:
x + 5 = 10
The additive inverse of 5 is -5, so you can 'invert' the addition by adding -5:
x + 5 + -5 = 10 + -5
x=5
If you have a problem like 7x – 10 = 3x + 6 you need to find X. This kind of equation is called a
linear equation, and it usually has just one variable. We will walk you through the simple steps:
1) Check the equation for varying terms and constant terms. Varying terms are
numbers like "7x" or "3x" or "6y" or "10z," where the number changes depending on what
you plug into the variable, or letter. Constant terms are numbers like "10" or "6" or "30,"
where the number never changes.
2) Move the varying terms to one side of the equation. It doesn't matter to which side
you move the varying terms.
a. In this example, 7x - 10 = 3x - 6 can be rearranged by choosing to subtract
either(7x) or (3x) from both sides. Choosing to subtract 7x, you have:
(7x - 7x) - 10 = (3x - 7x) - 6.
-10 = -4x - 6
3) Next, bring all the constant terms onto the other side of the equation. That is: move
the constant terms so that they are on the opposite side of the equation from where the
varying terms are.
a. We see that -6 must be subtracted from both sides:
-10 - (-6) = -4x - 6 - (-6).
-4 = -4x
4) Lastly, to find the value of x, simply divide both sides by the coefficient of x. The
coefficient of x (or y, or z, or any letter) is the number in front of the varying term.
a. The coefficient of x in -4x is -4. So divide both sides by -4 to get the value of x = 1.
b. Our answer to the equation 7x - 10 = 3x - 6 is x = 1. You can check this answer by
plugging 1 back into every "x" variable and seeing if both sides of the equation
equal the same number:
7(1) - 10 = 3(1) - 6
7 - 10 = 3 - 6
-3 = -3
Link to video:
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-solving-equations/cc-8thequation-solutions/v/number-of-solutions-to-linear-equations
Squares and Square Roots
First learn about squares, then square roots are easy.
To square a number, just multiply it by itself…
Example: What is 3 squared?
3 squared = 3 x 3 = 9
Squared is often written as a little 2 like this:
This says "4 Squared equals 16"
(the little 2 says the number appears twice in multiplying).
We can also square negative numbers. What happens when we square (-5)?
(-5) x (-5) = 25 (negative times a negative gives positive)
Square roots:
A square root goes the other way:
3 squared is 9, so square root of 9 is 3
A square root of a number is a value that can be multiplied by itself to give the original number.
A square root of 9 is….
….3, because when 3 is multiplied by itself we get 9.
It is like asking: What can we multiply by itself to get this?
Calculating square roots
It is easy to work out the square root of a perfect square, but it is hard to work out other square
roots.
Example: What is √10?
Well, we know that 3 x 3 = 9 and 4 x 4 = 16, so we can guess the answer is between 3 and 4.
Let’s try 3,5:
Let’s try 3,2:
Let’s try 3,1:
3,5 x 3,5 = 12,25 (too high)
3,2 x 3,2 = 10,24 (slightly high)
3,1 x 3,1 = 9,61 (slightly low)
Getting closer to 10, but it will take a long time to get the correct answer. In the admission tests
3,2 or 3,1 is good enough, but out of a calculator you’ll get 3,16227….. So even the calculator’s
answer is only an approximation!
Algebra
Algebra (meaning “reunion of broken parts) is one of the broad parts of mathematics, together
with number theory, geometry and analysis. Algebra is the study of symbols and the rules for
manipulating symbols and is a unifying thread of all of mathematics. We’re going to make this
part as easy as possible. We’ll now have a look at how to evaluate expressions.
Algebraic expressions may be evaluated and simplified, based on the basic properties of
arithmetic operations (addition, subtraction, multiplication and so on). Here are some rules:
-
-
Added terms are simplified using coefficients.
= 3x (where 3 is the
coefficient)
Multiplied terms are simplified using exponents.
is represented as x3
Similar terms are added together. For example:
o 2x2 + 3ab – x2 + ab is written as x2 + 4ab, because the terms containing x2 are
added together, and, the terms containing ab are added together.
o 2x2 - x2 = x2 and 3ab + ab = 4ab.
Brackets can be “multiplied out”. For example:
o x(2x + 3) can be written as (x × 2x) + (x × 3) which can be written as 2x2 + 3x
Some examples:
a) 4a – 2a = 2a
b) 5b – b = 4b
c) a + 4a – 2a = 3a
d) 2a – 3b + 5a – 3a + 7b =
2a + 5a – 3a – 3b + 7b = 4a + 4b
e) 2ab – 13d – 5ba + 24d=
2ab – 5ba – 13d + 24d= -3ab + 11d
Pythagoras’ Theorem
Over 2000 years ago, there was an amazing discovery about triangles:
When the triangle has a right angle (90 degrees), and squares are made on each of the three
sides, then the biggest square has the exact same area as the other two squares put together!
It’s called “Pythagoras’ Theorem” and can be written in one short equation:
a2 + b2 = c2
Note:
-
c is the longest side of the triangle
a and b are the other two sides
Definition
The longest side of the triangle is called the “hypotenuse”, so the formal definition is:
In a right angled triangle: The square of the hypotenuse is equal to the sum of the squares of
the other two sides.
Let’s see if it really works using an example:
A “3,4,5” triangle has a right angle in it.
32 + 4 2 = 52
3×3=9
4 × 4 = 16
9 + 16 = 25
So why is this useful?
If we know the lengths of two sides of a right angled triangle, we can find the length of the third
side. (Remember it only works on right angled triangles!)
How do we use it?
Write it down as an equation:
a 2 + b 2 = c2
Now you can use algebra to find any missing value, as in the following examples:
Example 1, solve this triangle.
a 2 + b 2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
c2 = 169
c = √169
c = 13
Example 2, solve this triangle.
a 2 + b 2 = c2
92 + b2 = 152
81 + b2 = 225
Take 81 from both sides:
b2 = 144
b = √144
b = 12
Surface area and circumference
What is area?
Area is the size of a surface!
These shapes all have the same area of 9:
It helps to imagine how much paint would cover the shape.
There are special formulas for certain shapes, as you can see above.
Example: What is the area of this rectangle?
The formula is:
Area = w × h
w = width
h = height
The width is 5, and the height is 3, so we know w = 5 and h = 3
Area = 5 × 3 = 15
Example: The circle has a radius of 2,1 meters:
The formula is:
Area = π × r2
Π = the number pi (3,14)
R = radius
The radius is 2,1 m, so:
Area = 3,14 × (2,1m)2
= 3,14 × (2,1m x 2,1m)
= 13,8544…m2
Example 1:
Given the rectangle ABCD below.
a) Calculate the surface area of the rectangle.
Surface area  6 m  2 m=12 m
2
b) Calculate the length of the diagonal AC.
Use Pythagoras sentence to find the diagonal.
 AC 2  62  22
 AC 2  36  4
 AC 2  40
AC  40
AC  6,3
Diagonal AC is about 6,3 meters
c) Calculate the surface area of the triangle ABC
6,0 m  2,0 m 12,0 m2

 6,0 m2
Surface area of ABC 
2
2
Example 2:
Calculate the surface area of the circle below.
Surface area of the circle    r     3,0 cm  28,3 cm
2
2
2
Example 3:
A square with lengths 10,0 cm. Calculate the surface area of the square
All sides in a square have the same lengths.
Surface area of the square: 10,0cm × 10,0cm = 100 cm2