Equal sets are sets in which the number of elements is the same and all elements are equal.

Since P = Q, all the elements in P are found in Q. Hence the missing number 10 is x

If the set A is a subset (⊂) of B then all the elements of A are found in B.

Y ⊂ V because all the elements (house, tree) of A are found in the set V.

Let n = the missing addend.

2345 + 1045 + n = 5110

3390 + n = 5110

n = 5110 - 3390

n = 1720

A number expressed as a product of an integer by itself is called a perfect square. A perfect square is a special type of number that you get by multiplying another number by itself.

As you can see both 2 and 3 are being multiplied by the same number producing the power 2. If multiplied by 7, the 7 will also have a power of 2.

2² x 3² x 7 x 7 = 2² x 3² x 7²

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.

4687.02 = 4.⌒6⌒8⌒7.02 = 4.68702 x 10³

Decimal numeral is a base 10 representation of a number.

11001_two = 1 x 2⁴ + 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰

Any number/variable raised to the power 0 is 1

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

2⁰ = 1

2¹ = 2

2² = 2 x 2 = 4

2³ = 2 x 2 x 2 = 8

2⁴ = 2 x 2 x 2 x 2 = 16

11001_two = 1 x 16 + 1 x 8 + 0 x 4 + 0 x 2 + 1 x 1

11001_two = 16 + 8 + 0 + 0 + 1

11001_two = 25

Fraction spent on transport and sweets = $\frac{2}{7}$ + $\frac{1}{5}$

Fraction spent on transport and sweets = $\frac{\mathrm{5\; x\; 2\; +\; 7\; x\; 1}}{35}$

Fraction spent on transport and sweets = $\frac{\mathrm{10\; +\; 7}}{35}$

Fraction spent on transport and sweets = $\frac{17}{35}$

Let n = the third number.

3 x 19 x n = 1197

Divide both sides by 3 x 19

n = $\frac{1197}{\mathrm{3\; x\; 19}}$ = 21

Note:
1. 3 divides itself 1 time and 1197, 399 times
2. 19 divides itself 1 time and 399, 21 times

The GCF (HCF) of prime factors is the product (multiplication) of the common prime factors in their lowest powers.

GCF = 2³ × 3²

2³ = 2 x 2 x 2 = 8

3² = 3 x 3 = 9

GCF = 8 x 9 = 72

Since the triangle is an isosceles triangle (two sides equal), the base angles are the same.

∠ CAB = ∠ CBA = 56°

a° + ∠ CAB = 180°(Sum of angles on a straight line).

a° + 56° = 180°

a° = 180° - 56°

a° = 124°

Since the triangle is an isosceles triangle (two sides equal), the base angles are the same.

∠ CAB = ∠ CBA = 56°

b° + ∠ CAB + ∠ CBA = 180°(Sum of interior angles of a triangle).

b° + 56° + 56° = 180°

b° + 112° = 180°

b° = 180° - 112°

b° = 68°

56° + c° + 19° = 180°(Sum of angles on a straight line)

56° + 19° + c° = 180°

105° + c° = 180°

c° = 180° - 105°

c° = 75°

2ab² × 3a²b

Every number/variable is raised to the power 1

a = a¹

b = b¹

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

2ab² × 3a²b = 2 x 3 x a^{1 + 2} x b^{2 + 1}

2ab² × 3a²b = 6 x a³ x b³

2ab² × 3a²b = 6a³b³

v = u+ at

When u = 6, a = 10 and t = 2

v = 6 + 10 x 2

v = 6 + 20

v = 26

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

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.

There are two marks in the middle, 5 and 7

Median = $\frac{\mathrm{5\; +\; 7}}{2}$ = $\frac{12}{2}$ = 6

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

Mean = $\frac{\mathrm{0\; +\; 1\; +\; 3\; +\; 3\; +\; 5\; +\; 7\; +\; 8\; +\; 9\; +\; 9\; +\; 9}}{10}$ = $\frac{54}{10}$ = 5.4

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

Number of possible selection of 3 marks = 2

Number of samples = 10

Probability = $\frac{2}{10}$ = $\frac{1}{5}$

100% represents the total sales

Change the mixed fraction to improper fraction.

12$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 12\; +\; 1}}{2}$ = $\frac{\mathrm{24\; +\; 1}}{2}$ = $\frac{25}{2}$

If $\frac{25}{2}$% = ₵35000

100% = ₵35000 x 100% ÷ $\frac{25}{2}$%

Note: the % cancel each other

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

100% = ₵35000 x 100 x $\frac{2}{25}$ = ₵35000 x 4 x 2 = 280,000

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

If 1 cm on map = 100000 cm on land

4 cm on map = $\frac{\mathrm{100000\; cm\; on\; land\; x\; 4\; cm\; on\; map}}{\mathrm{1\; cm\; on\; map}}$ = 400000 cm on land

100000 cm = 1 km

400000 cm = $\frac{\mathrm{1\; km\; x\; 400000\; cm}}{\mathrm{100000\; cm}}$ = 4 km

Note
1. the zeros (0) at the end of 400000 and 100000 cancel each other
2. cm cancel each other

The sum of ratios represent the amount shared.

Sum of ratios = 3 + 2 = 5

Adjoa's ratio = 3

If 5 = ₵600

3 = $\frac{\mathrm{₵600\; x\; 3}}{5}$ = ₵120 x 3 = ₵360

Note: 5 divides itself 1 time and 60, 12 times.

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

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

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

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

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

a⁶ ÷ a⁴ = 64

Law of division of indices

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

a^{6 - 4} = 64

a² = 64

64 = 8 x 8 = 8²

a² = 8²

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

a = 8

one-hundredth = $\frac{1}{100}$

of means multiplication.

one-hundredth of 1.0756 = $\frac{1}{100}$ x 1.0756 = .⌒0⌒1.0756 = 0.010756

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 = 9 - 7 = 2

We can substitute our d and simplify the expression for y

y = 1 + (x - 1)d

y = 1 + (x - 1)2

y = 1 + 2x - 2

y = 2x + 1 - 2

y = 2x - 1

Circumference = π x diameter

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

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

7 x 440 m = 7 x $\frac{22}{7}$ x diameter

7 x 440 m = 22 x diameter

Divide both sides by 22

diameter = $\frac{\mathrm{7\; x\; 440\; m}}{22}$ = 7 x 20 m = 140 m

Note: 22 divides itself 1 time and 44, 2.

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

6(r + 2s) = 6($\left(\begin{array}{c}3\\ 1\end{array}\right)$ + 2$\left(\begin{array}{c}-2\\ 1\end{array}\right)$ )

6(r + 2s) = 6($\left(\begin{array}{c}3\\ 1\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{-2\; x\; 2}\\ \mathrm{1\; x\; 2}\end{array}\right)$ )

6(r + 2s) = 6($\left(\begin{array}{c}3\\ 1\end{array}\right)$ + $\left(\begin{array}{c}-4\\ 2\end{array}\right)$ )

6(r + 2s) = 6($\left(\begin{array}{c}\mathrm{3\; +\; -4}\\ \mathrm{1\; +\; 2}\end{array}\right)$ )

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

6(r + 2s) = 6($\left(\begin{array}{c}-1\\ 3\end{array}\right)$ )

6(r + 2s) = 6$\left(\begin{array}{c}-1\\ 3\end{array}\right)$

6(r + 2s) = $\left(\begin{array}{c}\mathrm{-1\; x\; 6}\\ \mathrm{3\; x\; 6}\end{array}\right)$

6(r + 2s) = $\left(\begin{array}{c}-6\\ 18\end{array}\right)$

64 cm : 48 cm = 4 : 3

Note:
1. cm cancel each other
2. 16 divides 64, 4 times and 48, 3 times

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

Scale factor = $\frac{\mathrm{1.5\; m}}{\mathrm{3\; m}}$ = $\frac{\mathrm{1\; m}}{\mathrm{2\; m}}$ = $\frac{1}{2}$

Scale factor = $\frac{\mathrm{1.5\; m}}{\mathrm{3\; m}}$ = $\frac{\mathrm{1\; m}}{\mathrm{2\; m}}$ = $\frac{\left|P^1{R}^1\right|}{\mathrm{|PR|}}$ = $\frac{1}{2}$

$\frac{\left|P^1{R}^1\right|}{\mathrm{3.6\; m}}$ = $\frac{1}{2}$

Cross multiply.

|P¹R¹| x 2 = 3.6 m x 1

Divide both sides by 2.

|P¹R¹| = $\frac{\mathrm{3.6\; m}}{2}$ = 1.8 m

tan(y) = $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

tan(y) = $\frac{\mathrm{8\; cm}}{\mathrm{6\; cm}}$ = $\frac{4}{3}$

100% represent the percentage for the marked price

Selling Price% = Marked Price% - Discount%

Selling Price% = 100% - 10% = 90%

If 90% = ₵270000

100% = $\frac{\mathrm{₵270000\; x\; 100\%}}{\mathrm{90\%}}$ = ₵3000 x 100 = ₵300,000

Note:
1. the % cancel each other
2. the zeros (0) at the end of 270000 and 90 cancel each other
3. 9 divides itself 1 time and 27, 3 times

Let x = number of students who study only Government.

The number for the region of both sets is 48

17 + 6 + x = 48

23 + x = 48

x = 48 - 23

x = 25

Number of students who study Government = 6 + x

Number of students who study Government = 6 + 25

Number of students who study Government = 31

Let x = number of students who study only Government.

The number for the region of both sets is 48

17 + 6 + x = 48

23 + x = 48

x = 48 - 23

x = 25

∴ 25 pupils study only Government.

Number of pupils who study only Economics = 23 - 6 = 17

Let n = the number.

(10 - n) x 2 = 4

20 - 2n = 4

20 - 4 = 2n

16 = 2n

Divide both sides by 2

n = $\frac{16}{2}$

n = 8

∴ the number is 8

Bearing starts from the north clockwise.

The bearing of point X from Y = 90° + 90° + 45°

The bearing of point X from Y = 225°

The full length is 100%

If 22% = 55 cm

100% = $\frac{\mathrm{55\; cm\; x\; 100\%}}{\mathrm{22\%}}$ = 5 cm x 50 = 250 cm

Note:
1. 11 divides 55, 5 times and 22, 2 times
2. 2 divides itself 1 time and 100, 50 times