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

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

(23 × 82) × 79 = 148,994

Express (23 × 82) × 79 = 148,994 in the form (2.3 × 82) × 7.9 and make (2.3 × 82) × 7.9 the subject.

23 = 2.⌒3 = 2.3 x 10¹ = 2.3 x 10

79 = 7.⌒9 = 7.9 x 10¹ = 7.9 x 10

(23 × 82) × 79 = (2.3 x 10 × 82) × 7.9 x 10

(23 × 82) × 79 = (2.3 x 82) x 7.9 x 10 x 10

(23 × 82) × 79 = (2.3 x 82) x 7.9 x 100

But (23 × 82) × 79 = 148994

(2.3 x 82) x 7.9 x 100 = 148994

Divide both sides by 100

(2.3 x 82) x 7.9 = $\frac{148994}{100}$ = 1489.⌒9⌒4 = 1489.94

25 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 25\\ 2& \mathrm{12\; r\; 1}\\ 2& \mathrm{6\; r\; 0}\\ 2& \mathrm{3\; r\; 0}\\ 2& \mathrm{1\; r\; 1}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

25_ten = 11001_two

(3.14 × 18) × 17.5 = 3.14 × (3a × 17.5)

3.14 × 18 × 17.5 = 3.14 × 3a × 17.5

The common numbers cancel each other.

18 = 3a

Divide both sides by 3

a = $\frac{18}{3}$ = 6

Oranges received by each = $\frac{26039}{13}$ = 2003

Total fractions given to children = $\frac{1}{2}$ + $\frac{1}{3}$

Total fractions given to children = $\frac{\mathrm{3\; x\; 1\; +\; 2\; x\; 1}}{6}$

Total fractions given to children = $\frac{\mathrm{3\; +\; 2}}{6}$

Total fractions given to children = $\frac{5}{6}$

The fraction for the money taken = 1

Fraction left = 1 - $\frac{5}{6}$

Every whole number is being divided by 1

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

Fraction left = $\frac{1}{1}$ - $\frac{5}{6}$

Fraction left = $\frac{\mathrm{6\; x\; 1\; -\; 1\; x\; 5}}{6}$

Fraction left = $\frac{\mathrm{6\; -\; 5}}{6}$

Fraction left = $\frac{1}{6}$

If $\frac{1}{6}$ = ₵500

1 = ₵500 x 1 ÷ $\frac{1}{6}$

1 = ₵500 ÷ $\frac{1}{6}$

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

1 = ₵500 x $\frac{6}{1}$ = ₵500 x 6 = ₵3000

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ )

Change the mixed fraction to improper fraction.

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

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{3}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ )

From BODMAS, division comes first hence simplify the division first.

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

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{3}{2}$ + $\frac{3}{4}$ x $\frac{4}{1}$ )

Note: the 4 cancel each other.

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{3}{2}$ + $\frac{3}{1}$ )

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{\mathrm{1\; x\; 3\; +\; 2\; x\; 3}}{2}$ )

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{\mathrm{3\; +\; 6}}{2}$ )

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$($\frac{9}{2}$ )

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{1}{2}$ x $\frac{9}{2}$

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{\mathrm{1\; x\; 9}}{\mathrm{2\; x\; 2}}$

$\frac{1}{2}$(1$\frac{1}{2}$ + $\frac{3}{4}$ ÷ $\frac{1}{4}$ ) = $\frac{9}{4}$ = 2$\frac{1}{4}$

21 : 2x = 7 : 10

Change the ratios to fraction format

$\frac{21}{\mathrm{2x}}$ = $\frac{7}{10}$

Cross multiply.

2x x 7 = 21 x 10

2 x x x 7 = 21 x 10

Divide both sides by 2 x 7

x = $\frac{\mathrm{21\; x\; 10}}{\mathrm{2\; x\; 7}}$

Note:
1. 2 divides itself 1 time and 10, 5 times
2. 7 divides itself 1 time and 21, 3 times

x = 3 x 5 = 15

|XB| = 10 cm - 4 cm = 6 cm

|AX| : |XB| = 4 cm : 6 cm

Note:
1. cm cancel each other
2. 2 divides 4, 2 times and 6, 3 times

|AX| : |XB| = 2 : 3

100% represent the total candidates.

Percentage that failed = 100% - 60% = 40%

If 60% = 240

40% = $\frac{\mathrm{240\; x\; 40\%}}{\mathrm{60\%}}$ = 4 x 40 = 160

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

100% represents the cost price.

Profit = Selling Price - Cost Price

Profit = ₵3,000 - ₵2,400 = ₵600

If ₵2400 = 100%

₵600 = $\frac{\mathrm{100\%\; x\; ₵600}}{\mathrm{₵2400}}$ = 25%

Note:
1. the zeros (0) at the end of 2400 and 600 cancel each other
2. 6 divides itself 1 time and 24, 4 times
3. 4 divides itself 1 time and 100, 25 times

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Law of division of indices

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

a = 2² × 2³ ÷ 2⁴

a = 2^{2 + 3 - 4}

a = 2¹

a = 2

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

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.

12875 = 12875.0

12875 = 1.⌒2⌒8⌒7⌒5.0 = 1.2875 x 10⁴

Note: the decimal point is moved to the left hence the power is positive.

Fraction = $\frac{\mathrm{Number}}{\mathrm{Total}}$

The total angle of sectors of a circle/pie chart is 360°

Fraction = $\frac{\mathrm{Angle\; of\; sector\; for\; food}}{360}$ = $\frac{120}{360}$ = $\frac{1}{3}$

Note: 120 divides itself 1 time and 360, 3 times.

The total angle of sectors of a circle/pie chart is 360°

The angle of sector for rent = 90°(right-angle triangle)

If 360° = ₵18000

90° = $\frac{\mathrm{₵18000\; x\; 90°}}{\mathrm{360°}}$ = ₵4,500

Note:
1. 90 divides itself 1 time and 360, 4 times
2. 4 divides itself 1 time and 180, 45 times

The total angle of sectors of a circle/pie chart is 360°

Angle of sector for savings = 360° - (90° + 60° + 50° + 120°)

Angle of sector for savings = 360° - 320°

Angle of sector for savings = 40°

Let n = the missing addend.

2045 + n + 1918 = 4430

n + 2045 + 1918 = 4430

n + 3963 = 4430

n = 4430 - 3963

n = 467

a – 2(b – 3c) = a – 2(b – 3c)

Note: - x - = +. -2 x -3c = + 6c

a – 2(b – 3c) = a – 2b + 6c

$\frac{\mathrm{2a}}{3}$ - $\frac{\mathrm{a\; -\; b}}{2}$ = $\frac{\mathrm{2a}}{3}$ - $\frac{\mathrm{a\; -\; b}}{2}$

$\frac{\mathrm{2a}}{3}$ - $\frac{\mathrm{a\; -\; b}}{2}$ = $\frac{\mathrm{2\; x\; 2a\; -\; 3(a\; -\; b)}}{6}$

- x - = -(-) = +. -3 x -b = + 3b.

$\frac{\mathrm{2a}}{3}$ - $\frac{\mathrm{a\; -\; b}}{2}$ = $\frac{\mathrm{4a\; -\; 3a\; +\; 3b}}{6}$

$\frac{\mathrm{2a}}{3}$ - $\frac{\mathrm{a\; -\; b}}{2}$ = $\frac{\mathrm{a\; +\; 3b}}{6}$

q = ut + $\frac{1}{2}$ ft

When u = 20, t = 10 and f = 15

q = 20 x 10 + $\frac{1}{2}$ x 15 x 10

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

q = 200 + 15 x 5

q = 200 + 75

q = 275

Equivalent means the value is the same hence you can equate the two fractions.

$\frac{3}{15}$ = $\frac{45}{\mathrm{a}}$

Cross multiply

a x 3 = 45 x 15

Divide both sides by 3

a = $\frac{\mathrm{45\; x\; 15}}{3}$

Note: 3 divides itself 1 time and 15, 5 times.

a = 45 x 5 = 225

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

Prime factor of 7 = 7

Prime factors of 14

$$\begin{array}{cc}\mathrm{}& 14\\ 2& 7\\ 7& 1\end{array}$$

Prime factors of 14 = 2 x 7

Prime factors of 18

$$\begin{array}{cc}\mathrm{}& 18\\ 2& 9\\ 3& 3\\ 3& 1\end{array}$$

Prime factors of 18 = 2 x 3 x 3

Prime factors of 18 = 2 x 3².

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

L.C.M of 7, 14 and 18 = 2 x 3² x 7 = 2 x 9 x 7 = 126

Method I

|YZ| is the base of the triangle.

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

Base of the triangle = |YZ|
Height of the triangle = 12 cm

Area of the triangle = 30 cm².

$\frac{1}{2}$ x |YZ| x 12 = 30

Note: 2 divides itself 1 time and 12, 6 times.

|YZ| x 6 = 30

Divide both sides by 6

|YZ| = $\frac{30}{6}$

|YZ| = 5 cm

Method II

From Pythagorean theorem,

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

|YZ|² + 12² = 13²

12² = 12 x 12 = 144

13² = 13 x 13 = 169

|YZ|² + 144 = 169

|YZ|² = 169 - 144

|YZ|² = 25

25 = 5 x 5 = 5²

|YZ|² = 5²

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

|YZ| = 5 cm

1204_five = 1 x 5³ + 2 x 5² + 0 x 5¹ + 4 x 5⁰

Any number/variable raised to the power 0 is 1.

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

5⁰ = 1

5¹ = 5

5² = 5 x 5 = 25

5³ = 5 x 5 x 5 = 125

1204_five = 1 x 125 + 2 x 25 + 0 x 5 + 4 x 1

1204_five = 125 + 50 + 0 + 4

1204_five = 179

(2x + y) by (2x – y) = (2x + y) x (2x – y)

(2x + y) by (2x – y) = 2x(2x – y) + y(2x – y)

(2x + y) by (2x – y) = 4x² - 2xy + 2xy - y²

(2x + y) by (2x – y) = 4x² + 0 - y²

(2x + y) by (2x – y) = 4x² - y²

February has either 28 days or 29 days in a leap year.

1988 is a leap year with 29 days.

If 1 day = ₵250

29 days = $\frac{\mathrm{₵250\; x\; 29}}{1}$ = ₵7,250

Volume left = 250 - 96 = 154 litres

100% represents the original volume

If 250 litres = 100%

154 litres = $\frac{\mathrm{100\%\; x\; 154\; litres}}{\mathrm{250\; litres}}$ = $\frac{\mathrm{4\%\; x\; 154}}{10}$ = $\frac{\mathrm{616\%}}{10}$ = 61.⌒6 = 61.6%

Note: 25 divides 100, 4 and 250, 10 times.

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ )

Simplify the bracket first.

Change the mixed fractions to improper fractions.

3$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 3\; +\; 1}}{2}$ = $\frac{\mathrm{6\; +\; 1}}{2}$ = $\frac{7}{2}$

1$\frac{1}{5}$ = $\frac{\mathrm{5\; x\; 1\; +\; 1}}{5}$ = $\frac{\mathrm{5\; +\; 1}}{5}$ = $\frac{6}{5}$

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ ($\frac{7}{2}$ + $\frac{6}{5}$ )

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ ($\frac{\mathrm{5\; x\; 7\; +\; 2\; x\; 6}}{10}$ )

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ ($\frac{\mathrm{35\; +\; 12}}{10}$ )

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ ($\frac{47}{10}$ )

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 ÷ $\frac{47}{10}$

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

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 10 x $\frac{10}{47}$

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = $\frac{\mathrm{10\; x\; 10}}{47}$

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = $\frac{100}{47}$

10 ÷ (3$\frac{1}{2}$ + 1$\frac{1}{5}$ ) = 2$\frac{6}{47}$

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

Number of marbles = 24

Number of green marbles = 24 - 10 = 14

Probability = $\frac{14}{24}$ = $\frac{7}{12}$

Note: 2 divides 14, 7 times and 24, 12 times.

The solid on -3 is not shaded hence the inequality is neither ≤ nor ≥. The direction of the arrow is facing right (>) hence the inequality is >. Hence n > -3

The rule of the mapping is x → 2x → 2x x 3.14

When x = 3

The image of 3 = 2 x 3 x 3.14 = 18.84

The rule of the mapping is x → 2x → 2x x 3.14

2x = 10

Divide both sides by 2

x = $\frac{10}{2}$ = 5

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

Square has all 4 sides equal.

Let L = the length of the square.

Perimeter of the square = L + L + L + L

Perimeter of the square = 4L

Area of a square = Length x Length

Area of the square with length, L = L x L = L²

But the area = 49 cm²

L² = 49

49 = 7 x 7 = 7²

L² = 7²

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

L = 7

The length of the square = 7 cm

Perimeter of the square = 4L

Perimeter of the square = 4 x 7 cm

Perimeter of the square = 28 cm

The least means the smallest number.

least number in a set of real numbers is 24 means 24 is also included and any other number greater than 24.

Hence the real number is greater than or equal to 24 (x ≥ 24) or 24 is less than or equal to the real number (24 ≤ x).

The greatest is 30 means the number is 30 or less but not more than 30.

Hence the real number (x) is less than or equal to 30 (x ≤ 30)

The interpretation is 24 ≤ x ≤ 30 which is read as 24 is less than or equal to x and x is also less than or equal to 30.

Method I

Using the formula.

Area of a trapezium = $\frac{1}{2}$ x (a + b) x h

Area of a trapezium = $\frac{1}{2}$ x (sum of parallel sides) x h

Area of the trapezium MNOP = $\frac{1}{2}$(12 cm + 18 cm) x 4 cm

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

Area of the trapezium MNOP = 30 cm x 2 cm

Area of the trapezium MNOP = 60 cm²

Method II

Treating the shape as a rectangle and a triangle and finding each area and total area.

Area of a rectangle = Length x Breadth

Area of the rectangle = 12 cm x 4 cm = 48 cm²

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

Base of the triangle = (18 - 12) cm = 6 cm
Height of the triangle = 4 cm

Area of the triangle = $\frac{1}{2}$ x 6 cm x 4 cm

Area of the triangle = 6 cm x 2 cm = 12 cm²

Total area of the shape = 48 cm² + 12 cm²

Total area of the shape = 60 cm²

The comparism must be in the same unit on the map as well as on the ground/land.

Convert the km to mm

1 km = 1000000 mm

1.2 km = $\frac{\mathrm{1000000\; mm\; x\; 1.2\; km}}{\mathrm{1\; km}}$ = 1.⌒2⌒0⌒0⌒0⌒0⌒0. = 1200000 mm

Scale = map : ground/land

Scale = 40 mm : 1200000 mm

Note: 40 mm divides itself 1 time and 1200000 mm, 30000 times.

Scale = 1 : 30000

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

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

Note: - - = +. --1 = + 1

$\left(\begin{array}{c}2\\ -3\end{array}\right)$ - $\left(\begin{array}{c}-1\\ 5\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{2\; +\; 1}\\ -8\end{array}\right)$

$\left(\begin{array}{c}2\\ -3\end{array}\right)$ - $\left(\begin{array}{c}-1\\ 5\end{array}\right)$ = $\left(\begin{array}{c}3\\ -8\end{array}\right)$