Two sets A and B are equal if both have the same number of elements and all the elements in A are found in B.

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

On a Venn diagram, it is the region where both sets intersect/join/overlap.

11 – (11 – 4) + 13 = 11 – (11 – 4) + 13

Note: - (-) = - - = +. - (-4) = + 4

11 – (11 – 4) + 13 = 11 – 11 + 4 + 13

11 – (11 – 4) + 13 = 17

An integer is a whole number (not a fractional number) that can be positive, negative, or zero.

Express (2.3 × 82) × 7.9 in the form (23 × 82) × 79

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.

2.3 = 2.⌒3. = 23 x 10^-1

7.9 = 7.⌒9. = 79 x 10^-1

(2.3 × 82) × 7.9 = (23 x 10^-1 x 82) x 79 x 10^-1

The order in multiplication doesn't matter.

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^-1 x 10^-1

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^{-1 + -1}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^{-1 - 1}

(2.3 × 82) × 7.9 = (23 x 82) x 79 x 10^-2

But (23 x 82) x 79 = 148,994

(2.3 × 82) × 7.9 = 148,994 x 10^-2

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

Every whole number has .0 at the end which is not written.

148994 = 148994.0

n is 2 in 10^-2 hence we have to move the decimal point 2 times to the left.

148,994 x 10^-2 = 1489.⌒9⌒4.0 = 1489.94

(2.3 × 82) × 7.9 = 1489.94

$$\begin{array}{cc}\mathrm{}& 206\\ 5& \mathrm{41\; r\; 1}\\ 5& \mathrm{8\; r\; 1}\\ 5& \mathrm{1\; r\; 3}\\ 5& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

206_ten = 1311_five

If the solid is shaded then it is either ≤ or ≥ but the solid is not shaded. Hence the inequality sign is either < or > . The inequality sign should be in the direction of the arrow. The arrow is pointed to the right hence the inequality is >

∴ x > -3

Product means multiplication.

Let n = the third number.

3 x 21 x n = 1197

Divide both sides by 3 x 21

n = $\frac{1197}{\mathrm{3\; x\; 21}}$

n = $\frac{1197}{63}$ = 19

$\left(\begin{array}{c}-2\\ 3\end{array}\right)$ - $\left(\begin{array}{c}1\\ 5\end{array}\right)$ = $\left(\begin{array}{c}-2\\ 3\end{array}\right)$ - $\left(\begin{array}{c}1\\ 5\end{array}\right)$

$\left(\begin{array}{c}-2\\ 3\end{array}\right)$ - $\left(\begin{array}{c}1\\ 5\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{-2\; -\; 1}\\ \mathrm{3\; -\; 5}\end{array}\right)$

$\left(\begin{array}{c}-2\\ 3\end{array}\right)$ - $\left(\begin{array}{c}1\\ 5\end{array}\right)$ = $\left(\begin{array}{c}-3\\ -2\end{array}\right)$

Total cost = 3 x cost of pencil + 4 x cost of eraser

Total cost = 3 x ₵180 + 4 x ₵120

Total cost = ₵540 + ₵480

Total cost = ₵1,020.00

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.

zeros (0) at the end of a decimal number is ignored.

2.40 = 2.4

0.12 = 0.⌒1⌒2. = 12 x 10^-2

0.08 = 0.⌒0⌒8. = 8 x 10^-2

2.4 = 2.⌒4. = 24 x 10^-1

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = $\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = $\frac{12x{10}^{-2}x8x{10}^{-2}}{24x{10}^{-1}}$

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = $\frac{12x8x{10}^{-2}x{10}^{-2}}{24x{10}^{-1}}$

Note:
1. 8 divides itself 1 and 24, 3 times
2. 3 divides itself 1 and 12, 4 times

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x $\frac{{10}^{-2}x{10}^{-2}}{{10}^{-1}}$

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x $\frac{{10}^{\mathrm{-2\; +\; -2}}}{{10}^{-1}}$

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x $\frac{{10}^{\mathrm{-2\; -\; 2}}}{{10}^{-1}}$

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x $\frac{{10}^{-4}}{{10}^{-1}}$

Law of division of indices

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

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x 10^{-4 - -1}

Note: - - = - (-1) = +. - - 1 = + 1

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x 10^{-4 + 1}

$\frac{\mathrm{0.12\; x\; 0.08}}{2.40}$ = 4 x 10^-3

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

Every whole number has .0 at the end which is not written.

4 = 4.0

n is 3 in 10^-3 hence we have to move the decimal point 3 times to the left.

4 x 10^-3 = 0.⌒0⌒0⌒4.0 = 0.004

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

From pythagorean theorem,

The hypothenuse (c) is the longest side.

5² + x² = 13²

5² = 5 x 5 = 25

13² = 13 x 13 = 169

25 + x² = 169

x² = 169 - 25

x² = 144

144 = 12 x 12 = 12²

x² = 12²

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

x = 12

Fraction on food + fraction on transport = total fraction on food and transport

Fraction on food = total fraction on food and transport - fraction on transport

Fraction on food = $\frac{17}{35}$ - $\frac{2}{7}$

Fraction on food = $\frac{\mathrm{1\; x\; 17\; -\; 5\; x\; 2}}{35}$

Fraction on food = $\frac{\mathrm{17\; -\; 10}}{35}$

Fraction on food = $\frac{7}{35}$

Note: 7 divides itself 1 time and 35, 5 times.

Fraction on food = $\frac{1}{5}$

x →3x – 4

–2 →3 x -2 – 4

–2 →-6 – 4

–2 →-10

(2ab²) (3a²b) = (2ab²) (3a²b)

(2ab²) (3a²b) = 2ab² x 3a²b

(2ab²) (3a²b) = 2 x 3 x a x a² x b x b²

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Any number without a power has the power 1 but it's not written.

a = a¹

b = b¹

(2ab²) (3a²b) = 6a^{1 + 2}b^{1 + 2}

(2ab²) (3a²b) = 6a³b³

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

15 men used 48 days.

48 = k x $\frac{1}{15}$

k = 48 x 15

The general equation is:

t = 48 x 15 x $\frac{1}{\mathrm{w}}$

t = $\frac{\mathrm{48\; x\; 15}}{\mathrm{w}}$

Calculate how many men would be needed for 16 days.

time = 16 days

16 = $\frac{\mathrm{48\; x\; 15}}{\mathrm{w}}$

16 x w = 48 x 15

Divide both sides by 16

w = $\frac{\mathrm{48\; x\; 15}}{16}$ = 3 x 15 = 45

Note: 16 divides itself 1 time and 48, 3 times.

∴ it will take 45 men to weed the same plot of land in 16 days.

Change the ratio format to fraction.

$\frac{9}{\mathrm{x}}$ = $\frac{36}{20}$

Cross multiply and solve for the value of x.

36 x x = 9 x 20

Divide both sides by 36.

x = $\frac{\mathrm{9\; x\; 20}}{36}$ = 5

Note:
1. 9 divides itself 1 time and 36, 4 times
2. 4 divides itself 1 time and 20, 5 times

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

Scale factor = $\frac{\mathrm{15\; cm}}{\mathrm{3\; cm}}$ = 5

7a – 3(b – a) = 7a – 3(b – a)

Note: - (-) = - - = +. - 3 (-a) = + 3a.

7a – 3(b – a) = 7a - 3b + 3a

7a – 3(b – a) = 7a + 3a - 3b

7a – 3(b – a) = 10a - 3b

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

Amount deposited = principal

Make principal the subject

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

100 x Simple Interest = Principal x Time x Rate

Divide both sides by Time x Rate

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

Interest = ₵35,000

Rate = 14

Time = 5

Principal = $\frac{\mathrm{100\; x\; ₵35,000}}{\mathrm{5\; x\; 14}}$ = 50 x ₵1000 = ₵50,000

Note:
1. 5 divides itself 1 time and 35, 7 times
2. 7 divides itself 1 time and 14, 2 times
3. 2 divides itself 1 time and 100, 50 times

Arranging/Ordering Fractions

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.

$\frac{3}{4}$ = $\frac{\mathrm{3\; x\; 3\; x\; 5}}{\mathrm{4\; x\; 3\; x\; 5}}$ = $\frac{45}{60}$

$\frac{2}{3}$ = $\frac{\mathrm{2\; x\; 4\; x\; 5}}{\mathrm{3\; x\; 4\; x\; 5}}$ = $\frac{40}{60}$

$\frac{3}{5}$ = $\frac{\mathrm{3\; x\; 3\; x\; 4}}{\mathrm{5\; x\; 3\; x\; 4}}$ = $\frac{36}{60}$

The magnitude of the numerators from the lowest to the highest are 36, 40, 45. Hence the ordered fractions are:

$\frac{3}{5}$, $\frac{2}{3}$, $\frac{3}{4}$

The percentage for the marked price is 100%

The selling % = 100% - 5% = 95%

If 100% = ₵450,000

95 = $\frac{\mathrm{₵450,000\; x\; 95\%}}{\mathrm{100\%}}$ = ₵4500 x 95 = ₵427,500

Note: the zeros (0) at the end of 450,000 and 100 cancel each other.

100% represent the total enrolment in the school

If 60% = 240

100% = $\frac{\mathrm{240\; x\; 100\%}}{\mathrm{60\%}}$ = 4 x 100 = 400

Note: 60 divides itself 1 time and 240, 4 times.

a * b = 2a – b

4 * 3

a = 4

b = 3

4 * 3 = 2 x 4 - 3

4 * 3 = 8 - 3

4 * 3 = 5

The mode is the one which occurs the most/ the one with the highest frequency/number of students

The highest frequency is 8 and it's for the 6 marks

Number of pupils = Total frequency

Number of pupils = 5 + 6 + 8 + 7 + 3 = 29

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

Number of students = 600

Number of boys = 600 - 120 = 480

Probability = $\frac{480}{600}$ = $\frac{4}{5}$

Note
1. the zeros (0) at the end of 480 and 600 cancel each other
2. 12 divides 48, 4 times and 60, 5 times

$\frac{1}{k}$ = $\frac{1}{k_1}$ + $\frac{1}{k_2}$

k₁ = 1 and k₂ = 2

$\frac{1}{k}$ = $\frac{1}{1}$ + $\frac{1}{2}$

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

$\frac{1}{k}$ = $\frac{\mathrm{2\; +\; 1}}{2}$

$\frac{1}{k}$ = $\frac{3}{2}$

Cross multiply

3 x k = 1 x 2

Divide both sides by 3

k = $\frac{2}{3}$

13x – 2(3x + 4) = 22
13x – 6x - 8 = 22
7x = 22 + 8
7x = 30

Divide both sides by 7

x = $\frac{30}{7}$

Volume = length x width x height

2.5 = $\frac{25}{10}$

Volume = 4 m x 3 m x $\frac{25}{10}$ m = 2 m x 3 m x 5 m = 30 m³

Note:
1. 2 divides 4, 2 times and 10, 5 times
2. 5 divides itself 1 time and 25, 5 times

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 = 1

y = 1 + (x - 1)d

The d is the interval between each consecutive y

In this mapping, d = 3 - 1 = 5 - 3 = 7 - 5 = 2

We can substitute our d and simplify the expression for y

y = 1 + (x - 1)2

y = 1 + 2x - 2

y = 2x + 1 - 2

y = 2x - 1

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

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

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

2r - 3s = $\left(\begin{array}{c}4\\ -2\end{array}\right)$ - $\left(\begin{array}{c}-9\\ 6\end{array}\right)$

2r - 3s = $\left(\begin{array}{c}\mathrm{4\; -\; -\; 9}\\ \mathrm{-2\; -\; 6}\end{array}\right)$

Note: - - = - (-) = +

2r - 3s = $\left(\begin{array}{c}\mathrm{4\; +\; 9}\\ -8\end{array}\right)$

2r - 3s = $\left(\begin{array}{c}13\\ -8\end{array}\right)$

Speed = $\frac{\mathrm{Distance}}{\mathrm{Time}}$

kilo means 1000

72 kilometres = 72 x 1000 metres

1 hour = 60 minutes

60 seconds = 1 minute

60 minutes = 60 x 60 seconds = 3600s

1 hour = 3600s

Speed = $\frac{\mathrm{72\; x\; 1000\; m}}{\mathrm{3600s}}$ = 2 x 10 m^-1 = 20 ms^-1

Note
1. the zeros (0) at the end of 1000 and 3600 cancel each other
2. 36 divides itself 1 and 72, 2 times

Circumference = 2 x π x radius

diameter is twice the radius. 2 x radius = diameter

Circumference = π x 2 x radius

Circumference = π x diameter

Circumference = 154 m

π = $\frac{22}{7}$

154 m = $\frac{22}{7}$ x diameter

Get rid of the fraction by multiplying both sides by the denominator (7)

7 x 154 m = 7 x $\frac{22}{7}$ x diameter

7 x 154 m = 22 x diameter

Divide both sides by 22

diameter = $\frac{\mathrm{7\; x\; 154\; m}}{22}$ = 7 x 7 = 49 m

Note: 22 divides itself 1 time and 154, 7 times.

y + 35 + angle BCA = 180°(sum of interior angles of a triangle is 180°)

angle BCA = angle DCE = 102° (Opposite angles are equal)

Note: Opposite angles are the angles directly opposite each other where two lines cross.

y + 35 + 102 = 180
y + 137 = 180
y = 180 - 137 = 43

If in the future you add and if in the past/ago, you subtract.

Ama will be (N + 10) years in ten years time.