A factor of a number is a number that divides the given number evenly or exactly, leaving no remainder.

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

A ∩ B = {} because both set don't have common elements

The symbol ∅ is used to represent an empty set {}

Yaw is half as old as Mark

Half means divided by 2.

is means equal to

Yaw's age = $\frac{\mathrm{Mark\text{'}s\; age}}{2}$ = $\frac{30}{2}$ = 15 years

Paul is 10 years older than Yaw

This means if you add 10 years to Yaw's age, you will get Paul's age.

Paul's age = Yaw's age + 10 years

Paul's age = 15 + 10 = 25 years

A line of symmetry is the line that divides a shape or an object into two identical parts.

Prime numbers are numbers with only two factors (1 and itself). Thus only 1 and the number itself can divide the number without a remainder.

Note: 1 is not a prime number.

9 is not a prime number because it has more than two factors. Three numbers can divide 9 without a remainder. These are 1, 3 and 9 itself.

2x + 1 < 7

2x < 7 - 1

2x < 6

Divide both sides by 2

x < $\frac{6}{2}$

x < 3

This is read as x is less than 3.

The truth set is {1, 2}

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

I = $\frac{\mathrm{P\; x\; T\; x\; R}}{100}$

Make T the subject and substitute the values to solve for T

Get rid of the fraction by multiplying both sides of the equation by the denominator (100).

I x 100 = $\frac{\mathrm{P\; x\; T\; x\; R}}{100}$ x 100

I x 100 = P x T x R

Divide both sides by P x R

T = $\frac{\mathrm{I\; x\; 100}}{\mathrm{P\; x\; R}}$

I = ₵1,000

P = ₵5,000

R = 5

T = $\frac{\mathrm{₵1000\; x\; 100}}{\mathrm{₵5000\; x\; 5}}$ = $\frac{100}{25}$ = 4 years

Note:
1. the zeros (0) at the end of 1000 and 5000 cancel each other
2. 5 x 5 = 25

f = $\frac{\mathrm{t}}{\mathrm{s}}$ - m

Move the m to the left and f to the right. m is negative so when it goes to the left becomes positive and f is positive so when it goes to the right becomes negative.

m = $\frac{\mathrm{t}}{\mathrm{s}}$ - f

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.

Law of division of indices

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

15⁹ ÷ 15⁷ = $\frac{{15}^9}{{15}^7}$ = 15^{9 - 7}

15⁹ ÷ 15⁷ = 15² = 15 x 15 = 225

$\frac{1}{\mathrm{x}}$ = 1$\frac{1}{2}$

Change the mixed fraction to improper fraction.

1$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 1\; +\; 1}}{2}$ = $\frac{\mathrm{2\; +\; 1}}{2}$ = $\frac{3}{2}$

$\frac{1}{\mathrm{x}}$ = $\frac{3}{2}$

Cross multiply.

3 x x = 1 x 2

Divide both sides by 3

x = $\frac{\mathrm{1\; x\; 2}}{3}$ = $\frac{2}{3}$

Exterior angle of a regular polygon = $\frac{\mathrm{360°}}{\mathrm{n}}$

Where n is the number of sides of the regular polygon.

Pentagon has five sides.

Exterior angle of a pentagon = $\frac{\mathrm{360°}}{5}$ = 72°

x² + bx + c

Split the bx into two to be in such a way that when you multiply the two, you will get c and when you add the two, you will get b.

x² – 5x + 6

Split the -5x into two to be in such a way that when you multiply the two, you will get 6 and when you add the two, you will get -5

The split is -3 and -2. -3 x -2 = + 6 and -3 + -2 = -5

x² – 5x + 6 = x² – 2x – 3x + 6

Now factorize.

x² – 5x + 6 = x(x - 2) - 3(x - 2)

x² – 5x + 6 = (x - 2)(x - 3)

Respectively means the ratios are in the order of the names mentioned.

The sum of the ratios represents the total amount shared.

Sum of ratios = 1 + 4 + 3 = 8

Atsu's ratio is 4

If 8 = ₵500000

4 = $\frac{\mathrm{₵500000\; x\; 4}}{8}$ = $\frac{\mathrm{₵500000}}{2}$ = ₵250,000

Note:
1. 4 divides itself 1 time and 8, 2 times
2. 2 divides itself 1 time and 50, 25 times

a² – b² = (a + b)(a – b)

83² - 17²

a = 83 and b = 17

83² - 17² = (83 + 17)(83 - 17)

83² - 17² = (100)(66)

83² - 17² = 100 x 66 = 6600

Die is numbered from 1 to 6 {1, 2, 3, 4, 5, 6}

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

Prime numbers are numbers with only two factors (1 and itself). Thus only 1 and the number itself can divide the number without a remainder.

Note: 1 is not a prime number.

The prime numbers from the samples {1, 2, 3, 4, 5, 6} are {2, 3 and 5}

Number of possible selection = number of prime numbers = 3

Number of samples = 6

P(Prime) = $\frac{3}{6}$ = $\frac{1}{2}$

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

Scores = 12, 16, 19, 14, 17, 8, 11, 19

Number of samples = 8

Number of possible selection = number of 19s = 2

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

A as a percentage of B = $\frac{\mathrm{A}}{\mathrm{B}}$ x 100%

25 as a percentage of 75 = $\frac{25}{75}$ x 100% = $\frac{\mathrm{1\; x\; 100\%}}{3}$ = $\frac{\mathrm{100\%}}{3}$ = 33.33%

Note: 25 divides itself 1 time and 75, 3 times.

m = (3⁶ ÷ 3⁴) × 27

Law of division of indices

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

3⁶ ÷ 3⁴ = 3^{6 - 4} = 3²

m = 3² × 27

27 = 3 x 3 x 3

Every number/variable is raised to the power 1

27 = 3¹ x 3¹ x 3¹

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

27 = 3^{1 + 1 + 1} = 3³

m = 3² × 3³

m = 3^{2 + 3}

m = 3⁵

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.

0.0043216 = 0.⌒0⌒0⌒4.3216 = 4.3216 x 10^-3

Note: we moved the decimal point 3 times to the right hence the power is -3.

(2a + b) (a – 2b) = (2a + b) (a – 2b)

a = 2 and b = 3

(2a + b) (a – 2b) = (2 x 2 + 3) (2 – 2 x 3)

(2a + b) (a – 2b) = (4 + 3) (2 – 6)

(2a + b) (a – 2b) = (7) (-4)

(2a + b) (a – 2b) = 7 x -4

(2a + b) (a – 2b) = -28

The solid of ≤ and ≥ are shaded but < and > are not shaded.

The solid on -1 is shaded hence the inequality sign is ≤ but the one on the $\frac{3}{2}$ is not shaded hence the inequality is <

Perimeter is the sum of all the sides of a shape.

Perimeter of the shaded sector = |AC| + |AB| + |BC|

|AC| = |AB| = radius of the circle = 20 cm

|BC| is an arc of the shaded sector.

Length of an arc = $\frac{\mathrm{Angle}}{360}$ x circumference of the circle

Length of an arc = $\frac{\mathrm{Angle}}{360}$ x 2 x π x radius

Angle = 90

π = $\frac{22}{7}$

|BC| = $\frac{90}{360}$ x 2 x $\frac{22}{7}$ x 20 cm

Note:
1. 90 divides itself 1 time and 360, 4 times
2. 4 divides itself 1 time and 20, 5 times

|BC| = $\frac{\mathrm{1\; x\; 2\; x\; 22\; x\; 5\; cm}}{\mathrm{1\; x\; 7}}$ = $\frac{\mathrm{220\; cm}}{7}$ = 31.4 cm

Perimeter of the shaded sector = |AC| + |AB| + |BC|

Perimeter of the shaded sector = 20 cm + 20 cm + 31.4 cm

Perimeter of the shaded sector = 71.4 cm

From pythagorean theorem,

The hypothenuse (c) is the longest side. The side facing the right-angle (∟)

|BD|² + |AD|² = 5²

|AD| = half of |AC| = $\frac{\mathrm{|AC|}}{2}$

|AD| = $\frac{8}{2}$ = 4

|BD|² + 4² = 5²

|BD|² = 5² - 4²

5² = 5 x 5 = 25

4² = 4 x 4 = 16

|BD|² = 25 - 16

|BD|² = 9

9 = 3 x 3 = 3²

|BD|² = 3²

Since the powers are the same, the bases are also the same.

|BD| = 3 cm

The place value after thousand is hundred. The hundred value is 632 which is more than half of thousand hence you round it to thousand and add to the 35 thousand making 36 thousand.

2x + 5 = 3x – 7

5 + 7 = 3x - 2x

12 = x

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\left(\frac{\mathrm{4\; x\; 2\; -\; 3\; x\; 1}}{12}\right)÷\frac{5}{6}$

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\left(\frac{\mathrm{8\; -\; 3}}{12}\right)÷\frac{5}{6}$

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\left(\frac{5}{12}\right)÷\frac{5}{6}$

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\frac{5}{12}$ ÷ $\frac{5}{6}$

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

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\frac{5}{12}$ x $\frac{6}{5}$

Note:
1. the 5s cancel each other
2. 6 divides itself 1 time and 12, 2 times

$(\frac{2}{3}-\frac{1}{4})÷\frac{5}{6}$ = $\frac{1}{2}$

Bearing starts from the north (N) clockwise.

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

Scale factor (k) = -$\frac{G_{1}O}{\mathrm{GO}}$ = -$\frac{H_{1}O}{\mathrm{HO}}$ = -$\frac{G_1{H}_1}{\mathrm{GH}}$

|GG₁| = |GO| + |G₁O|

|GG₁| = 5 cm

|GO| = 2 cm

5 cm = 2 cm + |G₁O|

|G₁O| = 5 cm - 2 cm = 3 cm

Scale factor (k) = -$\frac{G_{1}O}{\mathrm{GO}}$

Scale factor (k) = -$\frac{\mathrm{3\; cm}}{\mathrm{2\; cm}}$ = -$\frac{3}{2}$

14_x = 9_ten

Convert 14_x to base 10 and equate to 9_ten

1 x x¹ + 4 x x⁰ = 9

Any number/variable raised to the 0 is 1

Any number/variable raised to the 1 is the same number/variable.

x¹ = x

x⁰ = 1

1 x x + 4 x 1 = 9

x + 4 = 9

x = 9 - 4

x = 5

Factorize ac – 2bc + ad – 2bd

ac – 2bc + ad – 2bd = ac – 2bc + ad – 2bd

ac – 2bc + ad – 2bd = c(a - 2b) + d(a - 2b)

ac – 2bc + ad – 2bd = (a - 2b)(c + d)

The more the workers, the lesser the time.

Hence time (t) is inversely proportional to the number of workers (w).

t ∝ $\frac{1}{\mathrm{w}}$

Remove the proportionality by introducing a constant (k).

t = k x $\frac{1}{\mathrm{w}}$

Use the known information to solve for k

3 workers used 4 hours.

4 = k x $\frac{1}{3}$

k = 4 x 3

k = 12

The general equation is:

t = 12 x $\frac{1}{\mathrm{w}}$

t = $\frac{12}{\mathrm{w}}$

Calculate how much time it will take the 18 workers.

w = 18

t = $\frac{12}{18}$ = $\frac{2}{3}$ hour

Note: 6 divides 12, 2 times and 18, 3 times.

a = 2b (Opposite angles are the same).

b + 2b + 3b = 180°(Sum of angles on a straight line)

6b = 180°

Divide both sides by 6

b = $\frac{\mathrm{180°}}{6}$ = 30°

a = 2b

a = 2 x 30° = 60°

A cube has 6 faces. When the net is folded, should produce a cube.

The mapping is base 10 to base 5 numeral.

40 base 10 to base 5

$$\begin{array}{cc}\mathrm{}& 40\\ 5& \mathrm{8\; r\; 0}\\ 5& \mathrm{1\; r\; 3}\\ 5& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

40_ten = 130_five

e° = Angle CDH + Angle DCH

Angle CDH = 56° (vertically opposite angles are equal)

Angle DCH = 72° (vertically opposite angles are equal)

Method I

An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

e° = Angle CDH + Angle DCH

e° = 56° + 72° = 128°

Method II

Angle CHD + e° = 180°

Angle CHD + Angle CDH + Angle DCH = 180°(Sum of interior angles of a triangle)

Angle CHD + 56° + 72° = 180°

Angle CHD + 128° = 180°

Angle CHD = 180° - 128°

Angle CHD = 52°

Angle CHD + e° = 180°

52° + e° = 180°

e° = 180° - 52°

e° = 128°

Area of a rectangle = Length x Breadth

Length = x cm + 4 cm = (x + 4) cm

Breadth = 5 cm

Area of the figure = (x + 4) cm x 5 cm

Area of the figure = (5x + 20) cm²

But the area is 60 cm²

(5x + 20) cm² = 60 cm²

5x + 20 = 60

5x = 60 - 20

5x = 40

Divide both sides by 5

x = $\frac{40}{5}$ = 8

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

Number of items = 4

Mean = $\frac{\mathrm{4\; +\; 3\; +\; 3\; +\; x}}{4}$

Mean = $\frac{\mathrm{10\; +\; x}}{4}$

But the mean is 5

$\frac{\mathrm{10\; +\; x}}{4}$ = 5

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

4 x $\frac{\mathrm{10\; +\; x}}{4}$ = 5 x 4

10 + x = 20

x = 20 - 10

x = 10

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

Frequency is the number of times a number occurred/appeared

The number of samples is the sum of the frequency.

Number of samples = 25 + 15 + 8 + 10 + 2 = 60

Number of 5 (possible selection) is the frequency of 5 = 8

P(5) = $\frac{8}{60}$ = $\frac{2}{15}$

Note: 4 divides 8, 2 times and 60, 15 times.

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

Total number of bottles sold = 13 + 21 + 14 + 17 + 7 = 72

Note: 72 divides itself 1 time and 360, 5 times.

Pie chart showing the distribution of sales of soft drinks sold by the Jatokrom JSS canteen in one week.