Factors are numbers which can divide the number whose factor you are asked to find without a remainder. The factors of a number contains 1, the number itself and other numbers which can divide the number withou a remainder.

Factors of 30 = {1, 2, 3, 5, 6, 10, 15, 30}

Odd numbers are numbers which are not divisible by 2.

P = {factors of 30 which are odd}.

You can list the odd numbers from the list of factors of 30.

P = {1, 3, 5, 15}

Union (∪) is a set of the list of elements in both sets. If an element is found in both set, it is written only once

W U W = {g, h, o, p, q, r, s, t}

y = -$\frac{1}{2}$x + 6

when x = 4

y = -$\frac{1}{2}$ x 4 + 6

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

y = -2 + 6
y = 4

The mode is the score that appeared the most. 2 appeared the most (4 times) hence 2 is the mode.

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 distribution is not ordered and so order it first.

2, 2, 2, 2, 3, 5, 7, 9, 9, 10, 11, 12, 18

The middle number of the ordered list is 7 hence the median is 7.

The sum of angle of sectors is 360°

150° + 70° + x° + 100° = 360°
x° + 320° = 360°
x° = 360° - 320°
x° = 40°

In order to know how many more students are in Yellow House than in Blue House, you have to find the number of pupils in each of the houses. The difference (subtraction) gives how many more students are in Yellow House than in Blue House.

Number of a sector = $\frac{\mathrm{Angle\; of\; sector}}{\mathrm{360°}}$ x Total number

Total number of pupils = 360

Number of pupils in Yellow House = $\frac{\mathrm{100°}}{\mathrm{360°}}$ x 360 = 100

Note: 360 cancels each other.

Number of pupils in Blue House = $\frac{\mathrm{70°}}{\mathrm{360°}}$ x 360 = 70

Number of students more in Yellow House than Blue House = 100 - 70 = 30

Alternatively, you can calculate how many more angle in Yellow House than in Blue House and use it to calculate the number of pupils.

More angles in Yellow House than in Blue House = 100° - 70° = 30°

Number of pupils More in Yellow House than Blue House = $\frac{\mathrm{30°}}{\mathrm{360°}}$ x 360 = 30

Prime factors of 242

$$\begin{array}{cc}\mathrm{}& 242\\ 2& 121\\ 11& 11\\ 11& 1\end{array}$$

Prime factors of 242 = 2 x 11 x 11

Prime factors of 242 = 2 x 11²

110011_two = 1 x 2⁵ + 1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰

Note: any number/variable raised to the power 0 is 1

110011_two = 1 x 2 x 2 x 2 x 2 x 2 + 1 x 2 x 2 x 2 x 2 + 0 + 0 + 1 x 2 + 1 x 1

110011_two = 1 x 32 + 1 x 16 + 2 + 1

110011_two = 32 + 16 + 2 + 1

110011_two = 51

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

A die is labelled from 1 to 6 {1, 2, 3, 4, 5, 6}

Number of possible selection of 4 = 1
Number of samples = 6

P(4) = $\frac{1}{6}$

Before you can compare fractions, their denominators must be the same. Once they are the same, you can arrnage the fractions in the magnitude of their numerators.

To make the denominators the same, you must multiply both the numerator and the denominators of each fraction by the product (multiplication) of the denominators of the other fractions.

$\frac{5}{6}$ = $\frac{\mathrm{5\; x\; 5\; x\; 7}}{\mathrm{6\; x\; 5\; x\; 7}}$ = $\frac{175}{210}$

$\frac{4}{5}$ = $\frac{\mathrm{4\; x\; 6\; x\; 7}}{\mathrm{5\; x\; 6\; x\; 7}}$ = $\frac{168}{210}$

$\frac{4}{7}$ = $\frac{\mathrm{4\; x\; 5\; x\; 6}}{\mathrm{7\; x\; 5\; x\; 6}}$ = $\frac{120}{210}$

The order of the numerators from highest to lowest is 175, 168, 120. Hence the order of the fractions is:

$\frac{5}{6}$, $\frac{4}{5}$, $\frac{4}{7}$

3x + 6 ≤ 5x - 2
3x - 5x ≤ -2 -6
-2x ≤ -8

Divide both sides by - 2. When dividing by a negative, the inequality changes direction. Thus ≤ becomes ≥

x ≥ $\frac{-8}{-2}$

x ≥ 4

8x - 3(2x - 4) = 4

Expand the bracket. - x - = +. - 3 x -4 = + 12

8x - 6x + 12 = 4
2x = 4 - 12
2x = -8

Divide both sides by 2.

x = $\frac{-8}{2}$ = -4

Angle NTW = Angle QZT (Corresponding angles)

y = 105°

Supplementary angles are those angles that sum up to 180 degrees.

Angle NTW = Angle QZT (Corresponding angles)

y = 105°

NTZ + y = 180° (sum of angles on a straight line is 180°)

QZT = y

NTZ + QZT = 180°

$\frac{1}{4}$px² + $\frac{1}{8}$px

x² = x x x

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

$\frac{1}{4}$px² + $\frac{1}{8}$px = $\frac{1}{4}$px² + $\frac{1}{8}$px

$\frac{1}{4}$px² + $\frac{1}{8}$px = $\frac{1}{4}$px x x + $\frac{1}{4}$px x $\frac{1}{2}$

$\frac{1}{4}$px is common in both hence factorize out.

$\frac{1}{4}$px² + $\frac{1}{8}$px = $\frac{1}{4}$px(x + $\frac{1}{2}$)

The sum of the ratios represent the total amount shared. Since we know Fiifi's ratio and amount, we can use proportionality to solve for the total amount shared

Sum of ratios = 3 + 4 = 7

Fiifi's ratio = 4

If 4 = ₵ 140, 000

7 = $\frac{\mathrm{₵\; 140,\; 000\; x\; 7}}{4}$ = ₵ 35, 000 x 7 = ₵ 245,000.00

Note: 4 divides itself 1 and 140, 35 times.

6(7a + 4) - 3(8a + 9) = 6(7a + 4) - 3(8a + 9)

Expand and simplify.

6(7a + 4) - 3(8a + 9) = 42a + 24 - 24a - 27

6(7a + 4) - 3(8a + 9) = 42a - 24a + 24 - 27

6(7a + 4) - 3(8a + 9) = 18a - 3

Simple Interest = $\frac{\mathrm{Principal\; x\; Time\; x\; Rate}}{100}$

Principal = ₵ 500,000.00
Rate = 15
Time = 2 years

Simple Interest = $\frac{\mathrm{₵\; 500,000\; x\; 2\; x\; 15}}{100}$ = ₵ 5000 x 30 = 150,000

Note:
1. The zeros (0) in 500,000 and 100 cancels each other
2. 2 x 15 = 30

Amount at the end of the period = Principal + Simple Interest
Amount at the end of the period = ₵ 500,000.00 + 150,000.00
Amount at the end of the period = ₵ 650, 000.00

p = $\frac{\mathrm{3m\; +\; 1}}{\mathrm{m}}$

Get rid of the fraction by multiplying both sides by m

m x p = 3m + 1
m x p - 3m = 1

Factorize the m out.

m(p - 3) = 1

Divide both sides by (p - 3)

m = $\frac{1}{\mathrm{p\; -\; 3}}$

Change the decimal to whole 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.

0.01 = 0.⌒0⌒1. = 1 x 10^-2

200 x 0.01 x 372 = 200 x 1 x 10^-2 x 372
200 x 0.01 x 372 = 74,400 x 10^-2

Note: 200 x 372 = 74,400

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.

74,400 = 74400.0

74400.0 = 7.⌒4⌒4⌒0⌒0.0 = 7.44 x 10⁴

Note
1. The power in 7.44 x 10⁴ is positive because we have to move to the left.
2. The zeros at the end of a decimal number is ignored (7.4400 = 7.44)

74,400 x 10^-2 = 7.44 x 10⁴ x 10^-2

From the law of indices, when the bases are the same and multiplying, the powers are added.

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

74,400 x 10^-2 = 7.44 x 10^{4 + -2}

74,400 x 10^-2 = 7.44 x 10²

Percentage means over 100

of means multiplication

is means equal to (=)

Let x = the percentage

x% = $\frac{\mathrm{x}}{100}$

$\frac{\mathrm{x}}{100}$ x 5 = 0.25

Multiply both sides by 100 to get rid of the fraction.

x x 5 = 0.25 x 100
x x 5 = 25

Divide both sides by 5

x = $\frac{25}{5}$ = 5

∴ 5% of 5 is 0.25

Method I

Using the area of a trapezium formula

Area = $\frac{1}{2}$(12 cm + 18 cm) x 6 cm

Area = $\frac{1}{2}$30 cm x 6 cm = 15 cm x 6 cm = 90 cm²

Note: 2 divides itself 1 and 30, 15 times.

Method II

Treating the figure as two separate shapes (rectangle and triangle)

Area of figure = Area of the rectangle + Area of triange

Area of rectangle = Length x Width

Length = 12 cm
Width = 6 cm

Area of rectangle = 12 cm x 6 cm = 72 cm²

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

Base of the triangle = 18 cm - 12 cm = 6 cm

Height of the triangle = 6 cm

Area of the triangle = $\frac{1}{2}$ x 6 cm x 6 cm = 3 cm x 6 cm = 18 cm²

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

Area of figure = 72 cm² + 18 cm² = 90 cm²

Equidistant means at equal distances.

Change the ratios to fraction form and solve for a.

$\frac{\mathrm{a}}{27}$ = $\frac{6}{18}$

Cross multiply.

a x 18 = 27 x 6

Divide both sides by 18

a = $\frac{\mathrm{27\; x\; 6}}{18}$ = 9

Note:
1. 6 divides itself 1 and 18, 3 times
2. 3 divides itself 1 and 27, 9 times

Ratios = 10 : 6 : 2

Simplify the ratios by dividing both ratios by 2

Ratios = 5 : 3 : 1

The sum of the ratios represent the total amount shared. Since we know the amount shared, we can use proportionality to find the youngest share.

The ratio for the youngest is 1

Sum of ratios = 5 + 3 + 1 = 9

If 9 = ₵ 5, 400

1 = $\frac{\mathrm{₵\; 5,\; 400\; x\; 1}}{9}$ = ₵ 600.00

Note: 9 divides itself 1 and 54, 6 times.

The easiest way of finding the L.C.M is by using the prime factors approach.

Prime factors of 4

$$\begin{array}{cc}\mathrm{}& 4\\ 2& 2\\ 2& 1\end{array}$$

Prime factors of 4 = 2 x 2

Prime factors of 4 = 2²

Prime factors of 5

$$\begin{array}{cc}\mathrm{}& 5\\ 5& 1\end{array}$$

Prime factors of 5 = 5

Prime factors of 6

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

Prime factors of 6 = 2 x 3

The L.C.M is the product of each of the prime factors in their highest powers

L.C.M of 4, 5 and 6 = 2² x 3 x 5 = 4 x 3 x 5 = 60

Method I

Expand and simplify.

(a + 2b)(a - 2b) = (a + 2b)(a - 2b)
(a + 2b)(a - 2b) = a(a - 2b) + 2b(a - 2b)
(a + 2b)(a - 2b) = a² - 2ab + 2ab - 4b²
(a + 2b)(a - 2b) = a² + 0 - 4b²
(a + 2b)(a - 2b) = a² - 4b²

Method II

The product is difference of two square which has the formula below:

(a + b)(a - b) = a² - b²

As you can see in (a + 2b)(a - 2b), both terms in the bracket are the same and the difference is just the + and -

You can therefore apply the difference of two square formula where the a is still a and the b is 2b

(a + 2b)(a - 2b) = (a + 2b)(a - 2b)
(a + 2b)(a - 2b) = a² - (2b)²

From the law of indices, (ab)^c = a^cb^c

(a + 2b)(a - 2b) = a² - (2²b²)
(a + 2b)(a - 2b) = a² - 4b²

Change the mixed fraction to improprer fraction and use proportionality to find the amount for 1 day.

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

If $\frac{9}{2}$ days = ₵180,000

1 day = ₵180,000 x 1 ÷ $\frac{9}{2}$

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

1 day = ₵180,000 x 1 x $\frac{2}{9}$ = ₵20,000 x 2 = ₵40,000.00

Note: 9 divides itself 1 and 18, 2 times.

0.68 = 0.⌒6⌒8. = $\frac{68}{100}$ = $\frac{17}{25}$

Note: 4 divides 68, 17 times and 100, 25 times.

21 < y ≤ 27 is read as 21 is less than y and y is also less or equal to 27. The or equal means the 27 is inclusive.

Integers greater than 21 and lesser than or equal to 27 are {22, 23, 24, 25, 26, 27}

(2⁶ x 3⁴) ÷ (2⁴ x 3²) = $\frac{2^{6}x{3}^4}{2^{4}x{3}^2}$

Law of division of indices

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

(2⁶ x 3⁴) ÷ (2⁴ x 3²) = 2^{6 - 4} x 3^{4 - 2}

(2⁶ x 3⁴) ÷ (2⁴ x 3²) = 2² x 3²

Angle of a vector/bearing starts from the north (N).

Angle NAE = 90°

Angle NAB = Angle NAE + Angle EAB

Angle NAB = 90° + 45° = 135°

AB→ = (5 cm, 135°)

3 + 3y = 1 - 13y

3y + 13y = 1 - 3
16y = -2

Divide both sides by 16

y = $\frac{-2}{16}$ = -$\frac{1}{8}$

Note: 2 divides itself 1 and 16, 8 times.

37 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 37\\ 2& \mathrm{18\; r\; 1}\\ 2& \mathrm{9\; r\; 0}\\ 2& \mathrm{4\; r\; 1}\\ 2& \mathrm{2\; r\; 0}\\ 2& \mathrm{1\; r\; 0}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

37_ten = 100101_two

Perimeter is the sum of the sides of a shape.

A rectangle has two equal lengths and two equal widths.

Perimeter of a rectangle = 2 x Length + 2 x Width

Let w = width of the field.

Length is 5 metres longer than its width.

Lengh = 5 + w

Substitute the length and width into the perimeter equation.

Perimeter = 2 x (5 + w) + 2 x w

But perimeter = 150

2 x (5 + w) + 2 x w = 150
2 x 5 + 2 x w + 2w = 150
10 + 2w + 2w = 150
4w = 150 - 10
4w = 140

Divide both sides by 4

w = $\frac{140}{4}$ = 35 metres

m = $\left(\begin{array}{c}3\\ 2\end{array}\right)$ and n = $\left(\begin{array}{c}-1\\ 3\end{array}\right)$

m - 2n = $\left(\begin{array}{c}3\\ 2\end{array}\right)$ - 2$\left(\begin{array}{c}-1\\ 3\end{array}\right)$

m - 2n = $\left(\begin{array}{c}3\\ 2\end{array}\right)$ - $\left(\begin{array}{c}\mathrm{-1\; x\; 2}\\ \mathrm{3\; x\; 2}\end{array}\right)$

m - 2n = $\left(\begin{array}{c}3\\ 2\end{array}\right)$ - $\left(\begin{array}{c}-2\\ 6\end{array}\right)$

m - 2n = $\left(\begin{array}{c}\mathrm{3\; -\; -2}\\ \mathrm{2\; -\; 6}\end{array}\right)$

Note: - - = +

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

m - 2n = $\left(\begin{array}{c}5\\ -4\end{array}\right)$