Union is simply listing each of the elements from both sets. If an element appeared in both sets, it is listed only once.

R ∪ S = {1, 2, 3, 4, 5, 6, 7, 8}

The set is a list of multiples of 5 less than 45

The set is {5, 10, 15, 20, 25, 30, 35, 40}

The commutative property states that we can swap numbers over and still get the same answer.

The associative property states that it doesn't matter how we group the numbers (i.e. which we calculate first).

The distributive property states that multiplying the sum of two or more addends by a number is the same as multiplying each addend individually by the number and then adding the products together.

The easiest way of finding the Highest (Greatest) Common Factor is using the prime factors method.

Prime factors of 63

$$\begin{array}{cc}\mathrm{}& 63\\ 3& 21\\ 3& 7\\ 7& 1\end{array}$$

Prime factors of 63 = 3 x 3 x 7

Prime factors of 63 = 3² x 7

Prime factors of 81

$$\begin{array}{cc}\mathrm{}& 81\\ 3& 27\\ 3& 9\\ 3& 3\\ 3& 1\end{array}$$

Prime factors of 81 = 3 x 3 x 3 x 3

Prime factors of 81 = 3⁴

The highest (greatest) common factor (H.C.F) is the product of the common prime factors in their lowest power.

The common prime factor is 3 and it's lowest power is 2.

H.C.F of 63 and 81 = 3² = 3 x 3 = 9

124_five = 1 x 5² + 2 x 5¹ + 4 x 5⁰

Any number raised to the power 0 is 1.

Any number raised to the power 1 is the same number.

5² = 5 x 5 = 25

124_five = 1 x 25 + 2 x 5 + 4 x 1

124_five = 25 + 10 + 4

124_five = 39

Average = $\frac{\mathrm{Sum\; of\; items}}{\mathrm{Number\; of\; items}}$

Number of books = 4

Average = ₵2,500

₵2,500 = $\frac{\mathrm{Sum\; of\; the\; four\; books}}{4}$

Get rid of the fraction by multiplying both sides by the denominator (4)

₵2,500 x 4 = sum of the four books

sum of the four books = ₵10,000

sum of the four books = sum of the three books + the fourth book

₵10,000 = ₵6,500 + cost of the fourth book

cost of the fourth book = ₵10,000 - ₵6,500

cost of the fourth book = ₵3,500

Even numbers are numbers divisible by 2.

Note: for between, the starting and ending numbers are excluded.

Even numbers between 70 and 80 are {72, 74, 76, 78}

Sum means addition.

72 + 74 + 76 + 78 = 300

$\frac{5}{6}$ - $\frac{3}{4}$ = $\frac{5}{6}$ - $\frac{3}{4}$

$\frac{5}{6}$ - $\frac{3}{4}$ = $\frac{\mathrm{2\; x\; 5\; -\; 3\; x\; 3}}{12}$

$\frac{5}{6}$ - $\frac{3}{4}$ = $\frac{\mathrm{10\; -\; 9}}{12}$

$\frac{5}{6}$ - $\frac{3}{4}$ = $\frac{1}{12}$

0.25 x 0.006

Change the decimal to whole number x 10^-n.

Where n is the number of times you have to move the decimal point to the right to obtain a whole number.

0.25 = 0.⌒2⌒5. = 25 x 10^-2

0.006 = 0.⌒0⌒0⌒6. = 6 x 10^-3

0.25 x 0.006 = 25 x 10^-2 x 6 x 10^-3

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

0.25 x 0.006 = 25 x 6 x 10^-2 x 10^-3

0.25 x 0.006 = 150 x 10^{-2 + -3}

0.25 x 0.006 = 150 x 10^{-2 - 3}

0.25 x 0.006 = 150 x 10^-5

Change it back to decimal by moving the decimal point to the left by the number of time of the power.

0.25 x 0.006 = 0.⌒0⌒0⌒1⌒5⌒0.0 = 0.0015 = 0.002 (three decimal places)

Sum of angles at the center of a circle is 360°

30° + 70° + 120° + 2x + 5x = 360°

220° + 7x = 360°

7x = 360° - 220°

7x = 140°

Divide both sides by 7

x = $\frac{140}{7}$ = 20

4.5 km = $\frac{45}{10}$ km

If 1 cm = $\frac{45}{10}$ km

4 cm = $\frac{45}{10}$ km x 4 cm ÷ 1 cm = $\frac{\mathrm{45\; km\; x\; 4\; cm}}{\mathrm{10\; x\; 1\; cm}}$ = $\frac{\mathrm{180\; km}}{10}$ = 18 km

2y = 6 – 3x, find y

When x = 0

2y = 6 – 3 x 0
2y = 6 - 0
2y = 6

Divide both sides by 2

y = $\frac{6}{2}$ = 3

Let Kofi's age now be x

In the next ten years Kofi will be x + 10 years.

Five years ago Kofi's age was x - 5

four times means 4 x

Will be means equal to.

Kofi's age in the next ten years will be four times his age five years ago

x + 10 = 4 x (x - 5)

x + 10 = 4x - 20
10 + 20 = 4x - x
30 = 3x

Divide both sides by 3

x = $\frac{30}{10}$ = 10 years

90 anticlockwise rotation

(x,y) becomes (−y,x)

Thus the coordinates are switched and the x coordinate is negatived.

P(3, 4) → P₁(-4, 3)

A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles.

A standard form should be in the form a.bcde x 10ⁿ

The decimal point must be after the first non-zero digit.

The n is the number of times the decimal point has to be moved to be right after the first non-zero digit.

If it is moved to the left, the n is positive and if to the right, n is negative.

Every whole number has .0 at the end but not written.

2345 = 2345.0

2345 = 2.⌒3⌒4⌒5 = 2.345 x 10³

3(x – 1) ≤ 12

3x - 3 ≤ 12
3x ≤ 12 + 3
3x ≤ 15

Divide both sides by 3

x ≤ $\frac{15}{3}$

x ≤ 5

Sum of ratios represents the total amount shared.

Sum of ratios = 3 + 2 = 5

Kofi's ratio is 3

If 3 = ₵60,000

5 = $\frac{\mathrm{₵60,000\; x\; 5}}{3}$ = ₵20,000 x 5 = ₵100,000

Note: 3 divides itself 1 time and 6, 2 times.

Change the mixed fraction to improper fraction.

4$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 4\; +\; 1}}{2}$ = $\frac{\mathrm{8\; +\; 1}}{2}$ = $\frac{9}{2}$

Percentage means over 100

$\frac{9}{2}$% = $\frac{9}{2}$ ÷ 100

Every number is being divided by 1 but the result is still the same.

100 = $\frac{100}{1}$

$\frac{9}{2}$% = $\frac{9}{2}$ ÷ $\frac{100}{1}$

Reciprocate the fraction (numerator becomes denominator and denominator becomes numerator) at the right and change the division (÷) to multiplication (x) sign.

$\frac{9}{2}$% = $\frac{9}{2}$ x $\frac{1}{100}$ = $\frac{\mathrm{9\; x\; 1}}{\mathrm{2\; x\; 100}}$ = $\frac{9}{200}$

of means multiplication.

$\frac{9}{2}$% of ₵120,000 = $\frac{9}{200}$ x ₵120,000 = 9 x 600 = ₵5,400

Note:
1. the zeros (0) at the end of 120,000 and 200 cancel each other
2. 2 divides itself 1 time and 12, 6

The percentage for the actual price is 100%

The percentage for the new price = 100% - 10% = 90%

If 90% = ₵540,000

100% = $\frac{\mathrm{₵540,000\; x\; 100\%}}{\mathrm{90\%}}$ = ₵6000 x 100 = ₵600,000

Note:
1. the zeros (0) at the end of 540,000 and 90 cancel each other
2. 9 divides itself 1 and 54, 6 times

a = p x q

p = 13
q = 15

a = 13 x 15 = 195

n → 2n – 3

-2 → 2 x -2 – 3

-2 → -4 – 3

-2 → -7

4p + 6p² – 2p + 2p² = 4p + 6p² – 2p + 2p²

4p + 6p² – 2p + 2p² = 4p – 2p + 6p² + 2p²

4p + 6p² – 2p + 2p² = 2p + 8p²

w = $\frac{\mathrm{n\; -\; q}}{\mathrm{q}}$

Get rid of the fraction by multiplying both sides of the equation by the denominator (q).

q x w = $\frac{\mathrm{n\; -\; q}}{\mathrm{q}}$ x q

qw = n - q
qw + q = n

Factorize q out.

q(w + 1) = n

Divide both sides by w + 1

q = $\frac{\mathrm{n}}{\mathrm{w\; +\; 1}}$

E = $\frac{1}{2}$mc²

c² = c x c

m = 10 and c = 4

E = $\frac{1}{2}$ x 10 x 4 x 4 = 5 x 16 = 80

Note:
1. 2 divides itself 1 time and 10, 5 times
2. 4 x 4 = 16

The median is the middle number of an ordered distribution. If there are two numbers at the middle, add the two numbers and divide the sum by 2.

2, 3, 4, 5, 7, 8, 8, 9

There are two numbers at the middle, 5 and 7

Median = $\frac{\mathrm{5\; +\; 7}}{2}$ = $\frac{12}{2}$ = 6

Mean/Average = $\frac{\mathrm{Sum\; of\; items}}{\mathrm{Number\; of\; items}}$

Number of scores = 8

Mean = $\frac{\mathrm{2\; +\; 3\; +\; 4\; +\; 5\; +\; 7\; +\; 8\; +\; 8\; +\; 9}}{8}$

Mean = $\frac{46}{8}$ = 5.75

Probability = $\frac{\mathrm{Number\; of\; possible\; selection}}{\mathrm{Number\; of\; samples}}$

Number of pupils who scored 8 = 2

Total number of pupils = 8

Probability = $\frac{2}{8}$ = $\frac{1}{4}$

Note: 2 divides itself 1 time and 8, 4 times.

Gradient = $\frac{\mathrm{Change\; in\; Y}}{\mathrm{Change\; in\; X}}$ = $\frac{{\mathrm{Y}}_2-{\mathrm{Y}}_1}{{\mathrm{X}}_2-{\mathrm{X}}_1}$

Pick any two points on the line and use it to calculate for the gradient.

Y₂ = 4
Y₁ = 1
X₂ = 3
X₁ = 0

Gradient = $\frac{\mathrm{4\; -\; 1}}{\mathrm{3\; -\; 0}}$ = $\frac{3}{3}$ = 1

If you analyze the mapping, you will realize that y is obtained by squaring the x value. Square means the number multiplied by the same number.

1 x 1 = 1² = 1
2 x 2 = 2² = 4
3 x 3 = 3² = 9
4 x 4 = 4² = 16
5 x 5 = 5² = 25

Area of a rectangle = Length x Breadth

Area = 18 cm²

Length = 9 cm

9 cm x Breadth = 18 cm²

Divide both sides by 9 cm

Breadth = $\frac{18}{9}$ cm = 2 cm

Perimeter is the sum of all the sides of a shape. A rectangle has two equal lengths and two equal breadths.

Perimeter = 2 x Length + 2 x Breadth

Length = 9 cm
Breadth = 2 cm

Perimeter = 2 x 9 cm + 2 x 2 cm
Perimeter = 18 cm + 4 cm
Perimeter = 22 cm

Method I

Using the area of a trapezium formula

a = 15 cm

b = 15 cm + 10 cm = 25 cm

h = 9 cm

Area = $\frac{1}{2}$ x (15 cm + 25 cm) x 9 cm

15 cm + 25 cm = 40 cm

Area = $\frac{1}{2}$ x 40 cm x 9 cm = 20 cm x 9 cm = 180 cm²

Note: 2 divides itself 1 and 40, 20 times.

Method II

Solving for the area of the rectangle PQST and the triangle QRS separately and adding them.

Area of a rectangle = Length x Breadth

Area of PQST = 9 cm x 15 cm = 135 cm²

Area of a triangle = $\frac{1}{2}$ x base x height

Base of triangle = 10 cm
Height of triangle = 9 cm

Area of QRS = $\frac{1}{2}$ x 10 cm x 9 cm = 5 x 9 = 45 cm²

Note: 2 divides itself 1 and 10, 5 times.

Area PQRST = Area of PQST + Area of QRS

Area PQRST = 135 cm² + 45 cm² = 180 cm²

The size increases or decreases base on the value of the scale factor.

105 + angle MNO = 180 (Sum of angles on a straight line is 180)

angle MNO = 180 - 105 = 75

Angle MNO + Angle OMN + Angle MON = 180 (Sum of interior angles of a triangle is 180)

75 + 57 + Angle MON = 180
132 + Angle MON = 180
Angle MON = 180 - 132 = 48

Note: Alternate angles look more like a Z shape.

u = $\left(\begin{array}{c}1\\ 3\end{array}\right)$ and v = $\left(\begin{array}{c}4\\ -2\end{array}\right)$

2u + 3v = 2$\left(\begin{array}{c}1\\ 3\end{array}\right)$ + 3$\left(\begin{array}{c}4\\ -2\end{array}\right)$

2u + 3v = $\left(\begin{array}{c}\mathrm{1\; x\; 2}\\ \mathrm{3\; x\; 2}\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{4\; x\; 3}\\ \mathrm{-2\; x\; 3}\end{array}\right)$

2u + 3v = $\left(\begin{array}{c}2\\ 6\end{array}\right)$ + $\left(\begin{array}{c}12\\ -6\end{array}\right)$

2u + 3v = $\left(\begin{array}{c}\mathrm{2\; +\; 12}\\ \mathrm{6\; +\; -6}\end{array}\right)$

2u + 3v = $\left(\begin{array}{c}14\\ \mathrm{6\; -\; 6}\end{array}\right)$

2u + 3v = $\left(\begin{array}{c}14\\ 0\end{array}\right)$