Note: for between, the starting and ending numbers are not included.

M = {12, 15, 18}
N = {12, 14, 16, 18}.

Intersection (∩) are elements that can be found in both sets

M ∩ N = {12, 18}

Prime factors of 24

$$\begin{array}{cc}\mathrm{}& 24\\ 2& 12\\ 2& 6\\ 2& 3\\ 3& 1\end{array}$$

Prime factors of 24 = 2 x 2 x 2 x 3

Prime factors of 24 = 2³ x 3

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 which is not written.

1342 = 1342.0

1342 = 1.⌒3⌒4⌒2.0 = 1.342 x 10³

Note: the power is positive 3 because we had to move the decimal point to the left 3 times.

10111_two = 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 1 x 2⁰

Any number raised to the power 0 is 1

Any number raised to the power 1 is the same number

2² = 2 x 2 = 4

2³ = 2 x 2 x 2 = 8

2⁴ = 2 x 2 x 2 x 2 = 16

10111_two = 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 1 x 1

10111_two = 16 + 0 + 4 + 2 + 1

10111_two = 23

Sum means addition.

Y + Z = 595 + 7071 = 7666

$\frac{1}{2}$ - $\frac{1}{4}$ + $\frac{1}{8}$ = $\frac{1}{2}$ - $\frac{1}{4}$ + $\frac{1}{8}$

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

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

$\frac{1}{2}$ - $\frac{1}{4}$ + $\frac{1}{8}$ = $\frac{3}{8}$

Number of pieces = $\frac{\mathrm{Length\; of\; ribbon}}{\mathrm{Sice\; of\; a\; piece}}$

Change the metres to centimetres.

1 m = 100 cm

4 m = 400 cm

Sice of a piece 30 cm

Number of pieces = $\frac{\mathrm{400\; cm}}{\mathrm{30\; cm}}$ = 13.33 ≈ 13

5 < x < 9 is read as 5 is less than x and x is lesser than 9. Hence x is the list of numbers greater than 5 but lesser than 9.

0.61 ÷ 0.8 = $\frac{0.61}{0.8}$

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.

0.61 = 0.⌒6⌒1. = 61 x 10^-2

0.8 = 0.⌒8. = 8 x 10^-1

0.61 ÷ 0.8 = $\frac{61x{10}^{-2}}{8x{10}^{-1}}$

0.61 ÷ 0.8 = $\frac{61}{8}$ x $\frac{{10}^{-2}}{{10}^{-1}}$

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

0.61 ÷ 0.8 = $\frac{61}{8}$ x 10^{-2 - -1}

- - = +. - - 1 = + 1

0.61 ÷ 0.8 = $\frac{61}{8}$ x 10^{-2 + 1}

0.61 ÷ 0.8 = $\frac{61}{8}$ x 10^-1

0.61 ÷ 0.8 = 7.625 x 10^-1

Change the whole number x 10^-n back to decimal by moving the decimal point to the left n times.

0.61 ÷ 0.8 = .⌒7.625 = 0.7625 = 0.76 (two decimal places)

2 × 3² × 3⁴ = 2 × 3² × 3⁴

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

2 × 3² × 3⁴ = 2 × 3^{2 + 4}

2 × 3² × 3⁴ = 2 × 3⁶

Since the numerators are the same, you can compare the fractions by their denominators. The lower the denominator, the higher the fraction.

The sum of the ratios represent the sum of their ages

Sum of ratios = 5 + 4 = 9

The younger person has the lower ratio, 4.

If 9 = 81

4 = $\frac{\mathrm{81\; x\; 4}}{9}$ = 9 x 4 = 36

Note: 9 divides itself 1 time and 81, 9 times.

Per hour means every 1 hour

Change the mixed fraction to improper fraction.

2$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 2\; +\; 1}}{2}$ = $\frac{\mathrm{4\; +\; 1}}{2}$ = $\frac{5}{2}$

If 1 hour = 40 km

$\frac{5}{2}$ hour = 40 km x $\frac{5}{2}$ hour ÷ 1 hour

Any number divided by 1 is the same number.

$\frac{5}{2}$ hour = 40 km x $\frac{5}{2}$ = 20 km x 5 = 100 km

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

Percentage = $\frac{\mathrm{Number}}{\mathrm{Total}}$ x 100%

Total number of students = 6 + 18 = 24

Number of girls = 6

Percentage of girls = $\frac{6}{24}$ x 100% = 25%

Note:
1. 6 divides itself 1 time and 24, 4 times
2. 4 divides itself 1 time and 100, 25 times

The sequence is multiples of 4 less than 32.

The percentage for the previous price is 100%

New price selling price% = 100% + 10% = 110%

If 100% = ₵7,500

110% = $\frac{\mathrm{₵7,500\; x\; 110\%}}{\mathrm{100\%}}$ = ₵75 x 110 = ₵8,250

Note: the zeros (0) at the end of 7,500 and 100 cancel each other.

7(y + 1) – 2(2y + 3) = 7(y + 1) – 2(2y + 3)
7(y + 1) – 2(2y + 3) = 7y + 7 - 4y - 6
7(y + 1) – 2(2y + 3) = 7y - 4y + 7 - 6
7(y + 1) – 2(2y + 3) = 3y + 1

Total rainfall = 145 + 227 + 267 = 639 mm

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.

The numbers are not ordered. The higher the negative number, the lower the number. Numbers decreases as you move from right to the left of the number line.

The ordered numbers are: -5, -4, -1, 1, 3. The middle number is -1 hence the median is -1.

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

Total number of balls = 15 + 25 = 40

Number of black balls = 25

P(Black) = $\frac{25}{40}$ = $\frac{5}{8}$

Note: 5 divides 25, 5 times and 40, 8 times.

Scale factor = $\frac{\mathrm{Enlarged\; Side}}{\mathrm{Corresponding\; Object\; Side}}$

Enlarged side = 3 m + 6 m = 9 m

Corresponding object side = 3 m

Scale factor = $\frac{9}{3}$ = 3

Scale factor = $\frac{\mathrm{Enlarged\; Side}}{\mathrm{Corresponding\; Object\; Side}}$

Side 1

Enlarged side = 3 m + 6 m = 9 m

Corresponding object side = 3 m

Side 2

Enlarged side = x

Corresponding object side = 1.5

Scale factor = $\frac{9}{3}$ = $\frac{\mathrm{x}}{1.5}$ = 3

$\frac{\mathrm{x}}{1.5}$ = 3

Get rid of the fraction by multiplying both sides by the denominator (1.5)

x = 3 x 1.5 = 4.5

100% represents total sales

If 25% = ₵180,000

100% = $\frac{\mathrm{₵180,000\; x\; 100\%}}{\mathrm{25\%}}$ = ₵180,000 x 4 = ₵720,000

Note: 25 divides itself 1 time and 100, 4 times.

The fraction for the entire journey is 1.

Fraction for train = 1 - [Fraction by lorry + fraction by taxi]

Fraction for train = 1 - $[\frac{3}{10}+\frac{2}{5}]$

Fraction for train = 1 - $\left[\frac{\mathrm{1\; x\; 3\; +\; 2\; x\; 2}}{10}\right]$

Fraction for train = 1 - $\left[\frac{\mathrm{3\; +4}}{10}\right]$

Fraction for train = 1 - $\left[\frac{7}{10}\right]$

Fraction for train = 1 - $\frac{7}{10}$

Every number is being divided by 1 but it is not written.

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

Fraction for train = $\frac{1}{1}$ - $\frac{7}{10}$

Fraction for train = $\frac{\mathrm{10\; x\; 1\; -\; 1\; x\; 7}}{10}$

Fraction for train = $\frac{\mathrm{10\; -\; 7}}{10}$

Fraction for train = $\frac{3}{10}$

4(m + 4) = 18
4m + 16 = 18
4m = 18 - 16
4m = 2

Divide both sides by 4

m = $\frac{2}{4}$

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

m = $\frac{1}{2}$

Note: If the mapping has equal interval (difference) for the y and x interval of 1, you can use the general linear sequence formula to find the rule of the mapping.

U_n = U₁ + (n-1)d

In the mapping, U_n = y and n = x and U₁ is the value of y when x = 1

U₁ = 3

y = 3 + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = 5 - 3 = 7 - 5 = 9 - 7 = 2

We can substitute our d and simplify the expression for y

y = 3 + (x-1)2

y = 3 + 2x - 2

y = 2x+ 3 - 2

y = 2x+ 1

Note: If the mapping has equal interval (difference) for the y and x interval of 1, you can use the general linear sequence formula to find the rule of the mapping.

U_n = U₁ + (n-1)d

In the mapping, U_n = y and n = x and U₁ is the value of y when x = 1

U₁ = 3

y = 3 + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = 5 - 3 = 7 - 5 = 9 - 7 = 2

We can substitute our d and simplify the expression for y

y = 3 + (x-1)2

y = 3 + 2x - 2

y = 2x+ 3 - 2

y = 2x+ 1

When x = 7

y = 2 x 7 + 1

y = 14 + 1

y = 15

Area of a square = Length x Length

Area of the square = 6 cm x 6 cm = 36 cm²

Area of a rectangle = Length x Breadth

Area of the square is the same as the area of the rectangle.

Length = 9 cm

9 cm x Breadth = 36 cm²

Divide both sides by 9 cm

Breadth = $\frac{36}{9}$ cm = 4 cm

3r²s – 9rs² = 3r²s – 9rs²

r² = r x r

s² = s x s

3r²s – 9rs² = 3r x r x s - 9r x s x s

3r²s – 9rs² = 3rs(r - 3s)

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

45 = $\frac{\mathrm{Sum\; of\; mass\; of\; the\; 4\; boys}}{4}$

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

Sum of mass of the 4 boys = 45 x 4 = 180

Average mass of the 5 boys = $\frac{\mathrm{Sum\; of\; the\; mass\; of\; the\; 4\; boys\; +\; Mass\; of\; the\; fifth\; boy}}{5}$

Let x = mass of the fifth boy.

40 = $\frac{\mathrm{180\; +\; x}}{5}$

Get rid of the fraction by multiplying both sides of the equation by the denominator (5)

180 + x = 40 x 5

180 + x = 200

x = 200 - 180

x = 20

∴ mass of the fifth boy = 20 kg

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

Y₂ = 1
Y₁ = 2
X₂ = 2
X₁ = 4

Gradient = $\frac{\mathrm{1\; -\; 2}}{\mathrm{2\; -\; 4}}$ = $\frac{-1}{-2}$ = $\frac{1}{2}$

L = $\left(\begin{array}{c}3\\ -1\end{array}\right)$ and K = $\left(\begin{array}{c}-4\\ 2\end{array}\right)$.

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

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

L + K = $\left(\begin{array}{c}\mathrm{3\; -\; 4}\\ 1\end{array}\right)$

L + K = $\left(\begin{array}{c}-1\\ 1\end{array}\right)$

v = u (Alternate angles)

148° + u = 180°(sum of angles on a straight line is 180°)

u = 180° - 148°

u = 32°

v = u = 32°

$\frac{1}{\mathrm{y}}$ = $\frac{1}{\mathrm{c}}$ + $\frac{1}{\mathrm{x}}$

$\frac{1}{\mathrm{y}}$ = $\frac{\mathrm{x\; x\; 1\; +\; c\; x\; 1}}{\mathrm{cx}}$

$\frac{1}{\mathrm{y}}$ = $\frac{\mathrm{x\; +\; c}}{\mathrm{cx}}$

$\frac{1}{\mathrm{y}}$ = $\frac{\mathrm{c\; +\; x}}{\mathrm{cx}}$

Cross multiply

y x (c + x) = 1 x cx

y x (c + x) = cx

Divide both sides by (c + x)

y = $\frac{\mathrm{cx}}{\mathrm{c\; +\; x}}$

Cone

68° + 72° + ONM = 180°(Sum of interior angles of a triangle is 180°)

140° + ONM = 180°

ONM = 180° - 140°

ONM = 40°

ONM + ONP = 180°

40° + ONP = 180°

ONP = 180° - 40°

ONP = 140°