P = {prime numbers less than 20}

Q = {odd numbers less than 10}

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.

P = {2, 3, 5, 7, 11, 13, 17, 19}

Odd numbers are numbers not divisible by 2. Thus there is a remainder when divided by 2.

Q = {1, 3, 5, 7, 9}

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

P ∩ Q = {3, 5, 7}

Binary means base 2

104_ten to a binary numeral

$$\begin{array}{cc}\mathrm{}& 104\\ 2& \mathrm{52\; r\; 0}\\ 2& \mathrm{26\; r\; 0}\\ 2& \mathrm{13\; r\; 0}\\ 2& \mathrm{6\; r\; 1}\\ 2& \mathrm{3\; r\; 0}\\ 2& \mathrm{1\; r\; 1}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

104_ten = 1101000_two

The easiest way of finding the Highest Common Factor (H.C.F) is using the prime factors method.

Prime factors of 48

$$\begin{array}{cc}\mathrm{}& 48\\ 2& 24\\ 2& 12\\ 2& 6\\ 2& 3\\ 3& 1\end{array}$$

Prime factors of 48 = 2 x 2 x 2 x 2 x 3

Prime factors of 48 = 2⁴ x 3

Prime factors of 30

$$\begin{array}{cc}\mathrm{}& 30\\ 2& 15\\ 3& 5\\ 5& 1\end{array}$$

Prime factors of 30 = 2 x 3 x 5

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 highest common factor (H.C.F) is the product (multiplication) of the common prime factors in their lowest powers.

H.C.F of 48, 30 and 18 = 2 x 3 = 6

1 metre = 1000 millimetres

1 centimetre = 10 millimetres

34 m = 34 x 1000 mm = 34000 mm

5 cm = 5 x 10 mm = 50 mm

34m 5cm 6mm = 34000 mm + 50 mm + 6 mm = 34056 mm

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.

356.07 = 3.⌒5⌒6.07 = 3.5607 x 10²

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $(1\frac{1}{2}+\frac{1}{4})$ ÷ $(1\frac{1}{2}-\frac{1}{4})$

Simplify the fractions in the bracket first

Change the mixed fraction to improper fraction.

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

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

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\left(\frac{\mathrm{2\; x\; 3\; +\; 1\; x\; 1}}{4}\right)$ ÷ $\left(\frac{\mathrm{2\; x\; 3\; -\; 1\; x\; 1}}{4}\right)$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\left(\frac{\mathrm{6\; +\; 1}}{4}\right)$ ÷ $\left(\frac{\mathrm{6\; -\; 1}}{4}\right)$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\left(\frac{7}{4}\right)$ ÷ $\left(\frac{5}{4}\right)$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{4}$ ÷ $\frac{5}{4}$

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

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{4}$ x $\frac{4}{5}$

Note: the 4s cancel each other 1 time.

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{1}$ x $\frac{1}{5}$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{\mathrm{7\; x\; 1}}{\mathrm{1\; x\; 5}}$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = $\frac{7}{5}$

$(1\frac{1}{2}+\frac{1}{4})$ divided by $(1\frac{1}{2}-\frac{1}{4})$ = 1$\frac{2}{5}$

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.

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

4 occurred the most. Hence 4 is the mode.

If you analyze the corresponding values of y, you will realize, the value is just 2 raised to the power x.

x = 1 → 2¹ = 2

x = 2 → 2² = 2 x 2 = 4

x = 3 → 2³ = 2 x 2 x 2 = 8

x = 4 → 2⁴ = 2 x 2 x 2 x 2 = 16

x = 5 → 2⁵ = 2 x 2 x 2 x 2 x 2 = 32

4x – 6 < -2

4x < -2 + 6

4x < 4

Divide both sides by 4.

x < $\frac{4}{4}$

x < 1

Percentage = $\frac{\mathrm{Number}}{\mathrm{Total}}$ x 100%

Number of pupils who passed = 64

Number of pupils who wrote the exams = 80

Percentage who passed = $\frac{64}{80}$ x 100% = 80%

Note:
1. the zeros (0) at the end of 80 and 100 cancel each other
2. 8 divides itself 1 time and 64, 8 times
3. 8 x 10 = 80

Interior angle of a regular polygon = $\frac{\mathrm{(n\; -\; 2)\; x\; 180°}}{\mathrm{n}}$

Where n = the number of sides of the polygon

120° = $\frac{\mathrm{(n\; -\; 2)\; x\; 180°}}{\mathrm{n}}$

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

120 x n = (n - 2) x 180

120n = 180n - 360

360 = 180n - 120n

360 = 60n

Divide both sides by 60

n = $\frac{360}{60}$

n = 6

∴ the number of sides of the polygon is 6.

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

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

Number of possible selection of 2 is 1 since there is only 1 numbered 2.

Number of samples/numbers is 6.

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

x → $\frac{1}{2}$x - 2

-4 → $\frac{1}{2}$ x -4 - 2

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

-4 → 1 x -2 - 2

-4 → -2 - 2

-4 → -4

30° + p° + Angle ABC = 180°(Sum of interior angles of a triangle)

Triangle BCD is an isosceles triangle hence the base angles are the same.

Angle CBD = 70°(Base angles are the same)

Angle ABC + Angle CBD = 180° (Sum of angles on a straight line)

Angle ABC + 70° = 180°

Angle ABC = 180° - 70°

Angle ABC = 110°

30° + p° + 110° = 180°

p° + 30° + 110° = 180°

p° + 140° = 180°

p° = 180° - 140°

p° = 40°

p = 40

P = 2(a + b)

P = 2a + 2b

P - 2b = 2a

Divide both sides by 2

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

5^x = 125

125 = 5 x 5 x 5 = 5³

5^x = 5³

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

x = 3

QR = QN + NR

From pythagorean theorem,

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

QN² + 12² = 13²

12² = 12 x 12 = 144

13² = 13 x 13 = 169

QN² + 144 = 169

QN² = 169 - 144

QN² = 25

25 = 5 x 5 = 5²

QN² = 5²

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

QN = 5

NR² + 12² = 15²

12² = 12 x 12 = 144

15² = 15 x 15 = 225

NR² + 144 = 225

NR² = 225 - 144

NR² = 81

81 = 9 x 9 = 9²

NR² = 9²

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

NR = 9

QR = 5 + 9 = 14

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

The cards are numbered from 1 to 5. Samples to select from {1, 2, 3, 4, 5}

Even numbers are numbers divisible by 2

The even numbers from the samples are {2, 4}. There are 2 even numbers.

Number of possible selection = 2

Number of samples = 5

P(even) = $\frac{2}{5}$

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

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

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

3u + 2v = $\left(\begin{array}{c}6\\ 3\end{array}\right)$ + $\left(\begin{array}{c}-2\\ 6\end{array}\right)$

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

3u + 2v = $\left(\begin{array}{c}\mathrm{6\; -\; 2}\\ 9\end{array}\right)$

3u + 2v = $\left(\begin{array}{c}4\\ 9\end{array}\right)$

$\frac{3}{4}$, $\frac{5}{8}$, $\frac{4}{5}$, $\frac{13}{20}$

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 denominator the same, express the denominator of each of the fractions to be the L.C.M of the denominators by multiplying the numerator and the denominator by the number of times the denominator can divide the L.C.M for each of the fractions.

The L.C.M of the denominators (4, 8, 5, 20) is 40.

$\frac{3}{4}$ = $\frac{\mathrm{3\; x\; 10}}{\mathrm{4\; x\; 10}}$ = $\frac{30}{40}$

$\frac{5}{8}$ = $\frac{\mathrm{5\; x\; 5}}{\mathrm{8\; x\; 5}}$ = $\frac{25}{40}$

$\frac{4}{5}$ = $\frac{\mathrm{4\; x\; 8}}{\mathrm{5\; x\; 8}}$ = $\frac{32}{40}$

$\frac{13}{20}$ = $\frac{\mathrm{13\; x\; 2}}{\mathrm{20\; x\; 2}}$ = $\frac{26}{40}$

Now arrange the fractions by magnitude of their numerators.

Descending means from the highest to the lowest.

Descending order of magnitude of the numerators are 32, 30, 26, 25

Hence the descending order of the fractions are $\frac{4}{5}$, $\frac{3}{4}$, $\frac{13}{20}$, $\frac{5}{8}$

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

(x, y) → (x, 2y), find the image of (2$\frac{1}{2}$, -$\frac{1}{4}$ )

As you may have realized, the y coordinates are doubled (multiplied by 2)

-$\frac{1}{4}$ x 2 = -$\frac{1}{2}$

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

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

Average = $\frac{\mathrm{14\; +\; 14.5\; +\; 15\; +\; 12\; +\; 11.5\; +\; 13\; +\; 10.5\; +\; 13.5}}{8}$

Note: 0.5 is half. 0.5 + 0.5 = 1.0

Average = $\frac{104}{8}$

Average = 13

133_five = 1 x 5² + 3 x 5¹ + 3 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

133_five = 1 x 25 + 3 x 5 + 3 x 1

133_five = 25 + 15 + 3

133_five = 43

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

Principal = ₵15,000

Rate = 20

Time = 5

Simple Interest = $\frac{\mathrm{₵15000\; x\; 5\; x\; 20}}{100}$ = ₵150 x 100 = ₵15,000

Note:
1. the zeros (0) at the end of 15000 and 100 cancel each other
2. 5 x 20 = 100

{x: 2 < x < 8, x is an integer}

This is read as x is such that 2 is less than x and x is also less than 8

2 is less than x means x is greater than 2 but x is also lesser than 8.

Numbers greater than 2 but lesser than 8 are {3, 4, 5, 6, 7}

Note: it is stated that x is an integer hence x is neither a fraction nor decimal number.

Difference means subtraction.

(2d)² and 2d² when d = 3

(2d)² = (2 x 3)² = 6² = 6 x 6 = 36

2d² = 2 x 3² = 2 x 3 x 3 = 2 x 9 = 18

Difference → 36 - 18 = 18

The lesser ratio represents the shorter boy and the higher ratio represents the taller boy.

If 4 = 80 cm

5 = $\frac{\mathrm{80\; cm\; x\; 5}}{4}$ = 20 cm x 5 = 100 cm

Note: 4 divides 80, 20 times.

5xy + 10ny

10 = 5 x 2

5xy + 10ny = 5y x x + 5y x 2n

Bring out the common term

5xy + 10ny = 5y(x + 2n)

13x – 12 = 5x + 60

13x – 5x = 60 + 12

8x = 72

Divide both sides by 8

x = $\frac{72}{8}$ = 9

L = (74 + 15n) mm

n = 1.2 = $\frac{12}{10}$

L = (74 + 15 x $\frac{12}{10}$) mm

Note:
1. 5 divides 15, 3 times and 10, 2 times
2. 2 divides itself 1 time and 12, 6 times

L = (74 + 3 x 6) mm

L = (74 + 18) mm

L = 92 mm

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

Scale factor = $\frac{\mathrm{|AB|}}{\mathrm{|AD|}}$ = $\frac{\mathrm{|BC|}}{\mathrm{|DE|}}$ = $\frac{\mathrm{|AC|}}{\mathrm{|AE|}}$

|AC| = |AE| + |EC|

|AC| = 20 cm + 10 cm

|AC| = 30 cm

Scale factor = $\frac{\mathrm{|AC|}}{\mathrm{|AE|}}$

Scale factor = $\frac{\mathrm{30\; cm}}{\mathrm{20\; cm}}$ = $\frac{3}{2}$

For ≥ and ≤ , the solids are shaded but for < and > , the solids are not shaded.

If two inequalities are used, join the solids by a line to represent the range of values of x.

2 < x ≤ 6 is read as 2 is less than x and x is also less than or equal to 6.

In order to indicate that the 6 is also included (or equal to), the solid on top is shaded.

Since 2 is less than x, x is greater than 2 but less than or equal to 6

Numbers greater than 2 but less than or equal to 6 are {3, 4, 5, 6} as indicated in the correct option.

The solid on 2 is not shaded because 2 is not included (is less than x).

Volume of a cylinder = π x radius x radius x height

π = $\frac{22}{7}$

radius = 6 cm

height = 7 cm

Volume of the cylinder = $\frac{22}{7}$ x 6 cm x 6 cm x 7 cm

Note: the 7s cancel each other 1 time.

Volume of the cylinder = 22 x 6 cm x 6 cm x 1 cm

Volume of the cylinder = 792cm³

1.2 km in 30 minutes

1.2 km = $\frac{12}{10}$ km

If 30 minutes = $\frac{12}{10}$ km

55 minutes = $\frac{12}{10}$ km x 55 minutes ÷ 30 minutes

Note:
1. minutes cancel each other
2. 5 divides 10, 2 times and 55, 11 times
3. 2 divides itself 1 time and 12, 6 times
4. 6 x 11 = 66

55 minutes = 66 km ÷ 30

55 minutes = $\frac{\mathrm{66\; km}}{30}$ = $\frac{\mathrm{11\; km}}{5}$ = 2.2 km

Note: 6 divides 66, 11 times and 30, 5 times.

Since triangle ABC is an isosceles triangle, the base angles of triangle ABC are the same.

∠ABC = ∠ACB

108° + ∠ABC = 180° (Sum of angles on a straight line)

∠ABC = 180° - 108°

∠ABC = 72°

∠ABC + ∠ACB + y = 180 (Sum of interior angles of a triangle)

But ∠ABC = ∠ACB = 72°

72° + 72° + y° = 180°

144 + y = 180

y = 180 - 144

y = 36