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

Q ∩ R = {1, 3, 5, 7, 11}

2 ≤ x < 10 is read as 2 is less or equal to x and x is less than 10.

The or equal to(≤) could be equal to hence the number is also inclusive.

2 ≤ x < 10 = {2, 3, 4, 5, 6, 7, 8, 9}

If you pick any number from the list, you will realize it is either greater than 2 or equal to 2 but less than 10.

$\frac{4}{3}$x - $\frac{2}{9}$x = $\frac{4}{3}$x - $\frac{2}{9}$x

$\frac{4}{3}$x - $\frac{2}{9}$x = $\frac{\mathrm{4x\; x\; 3\; -\; 2x\; x\; 1}}{9}$

$\frac{4}{3}$x - $\frac{2}{9}$x = $\frac{\mathrm{12x\; -\; 2x}}{9}$

$\frac{4}{3}$x - $\frac{2}{9}$x = $\frac{\mathrm{10x}}{9}$ = $\frac{10}{9}$x

The sum of the ratios represent the total number of students.

Sum of ratios = 5 + 7 = 12

Ratio for boys = 5

If 5 = 120

12 = $\frac{\mathrm{12\; x\; 120}}{5}$ = 12 x 24 = 288

Note: 5 divides itself 1 and 120, 24 times.

Circumference of a circle = 2 x π x r

Where r = the radius.

Circumferece = 2 x $\frac{22}{7}$ x 3.5 cm = $\frac{22}{7}$ x 7 cm = 22 cm

Note:
1. 2 x 3.5 = 7
2. The 7s cancel each other

Prime numbers are numbers with only two factors (1 and itself). Thus no other number can divide that number except 1 and the number itself.

Note: 1 is not a prime number.

Factors of 12 = {1, 2, 3, 4, 6, 12}

Prime numbers from the list are {2,3}

Prime factors of 12 = {2, 3}

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

Perimeter = x + x + 7
Perimeter = 2x + 7

Significant figures, any of the digits of a number beginning with the digit farthest to the left that is not zero and ending with the last digit farthest to the right that is either not zero or that is a zero but is considered to be exact.

The significant figures in 0.003858 are 3, 8, 5 and 8.

The third (three significant) is 5 but the next to it (the fourth) is more than 4 (8) hence we add 1 to the 5 making 6.

0.003858 = 0.00386

a – 3ab

a = -2 and b = 3

a – 3ab = -2 - 3(-2 x 3)
a – 3ab = -2 - 3(-6)

- x - = +. -3 x -6 = + 18

a – 3ab = -2 + 18 = 16

Number of times of sales = $\frac{\mathrm{Total\; sales}}{\mathrm{Selling\; Price\; per\; sales}}$

Number of times of sales = $\frac{\mathrm{₵100,000}}{\mathrm{₵500}}$ = 200 times of sales

Number of oranges sold = Number of times of sales x 3
Number of oranges sold = 200 x 3 = 600 oranges

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

Speed x Time = Distance

Divide both sides by Speed.

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

Speed = 80km per hour

Distance = 320 km

Time = $\frac{\mathrm{320\; km}}{\mathrm{80\; km\; per\; hour}}$ = 4 hours

In plane Euclidean geometry, a rhombus ( plural : rhombi or rhombuses) is a quadrilateral whose four sides all have the same length.

Difference between a square and a rhombus

All squares are rhombuses. All rhombuses cannot be squares, because rhombuses are not always equiangular. The adjacent sides of the square are perpendicular to each other. The adjacent sides of the rhombus are not always perpendicular to each other.

The sum of the ratios represent the total amount shared.

Sum of ratios = 2 + 3 + 4 = 9

The highest ratio represents the highest share. Highest ratio = 4

If 9 = ₵910,800

4 = $\frac{\mathrm{4\; x\; ₵910,800}}{9}$ = 4 x ₵101, 200 = ₵404, 800

Note: 9 divides itself 1 and 9108, 1012 times.

If you observe the sequence, the next term is obtained by multiplying the preceding by 2 to obtain the next term.

The missing term = 24 x 2 = 48

Prime factors of 350

$$\begin{array}{cc}\mathrm{}& 350\\ 2& 175\\ 5& 35\\ 5& 7\\ 7& 1\end{array}$$

Prime factors of 350 = 2 x 5 x 5 x 7

Prime factors of 350 = 2 x 5² x 7

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.

962 = 962.0

962 = 9.⌒6⌒2.0 = 9.62 x 10²

Note: the power is positive 2 because we had to move the decimal point to the left 2 times.

If the solid is shaded (it is either ≤ or ≥)

The solid on the -2 is not shaded and the numbers to the right (e.g -1, 0, 1, etc.) are greater than -2 hence -2 < p (-2 is less than p).

The solid on the 3 is shaded hence either ≥ or ≤. The values at the left of it (e.g 2, 1, 0, -1, etc.) are lesser than 3 hence p ≤ 3 (p is less than or equal to 3).

When you join the two analysis your inequality should be - 2 < p ≤ 3

$\frac{\mathrm{x\; +\; 2}}{3}$ + 2x = 10

Get rid of the fraction by multiplying both sides by 3.

3 x $\frac{\mathrm{x\; +\; 2}}{3}$ + 3 x 2x = 3 x 10

x + 2 +6x = 30
7x = 30 - 2
7x = 28

Divide both sides by 7

x = $\frac{28}{7}$ = 4

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.

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

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

$\frac{4}{5}$ =$\frac{\mathrm{4\; x\; 3\; x\; 4}}{\mathrm{5\; x\; 3\; x\; 4}}$ =$\frac{48}{60}$

The magnitude of the numerators in ascending orders are 40, 45, 48

The ascending order of the fractions = $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$

87 base 10 to base 5

$$\begin{array}{cc}\mathrm{}& 87\\ 5& \mathrm{17\; r\; 2}\\ 5& \mathrm{3\; r\; 2}\\ 5& \mathrm{0\; r\; 3}\end{array}$$

You list the remainders from the bottom to the top.

87_ten = 322_five

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

Total balls = 6 + 5 = 11

Number of black balls = 5

P(Black ball) = $\frac{5}{11}$

$\frac{2^{3}x{3}^{4}x{3}^3}{2^{2}x2x{3}^5}$

Simplify the numerator and the denominator applying the law of multiplication of indices.

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Note: 2 = 2¹

$\frac{2^{3}x{3}^{4}x{3}^3}{2^{2}x2x{3}^5}$ = $\frac{2^{3}x{3}^{\mathrm{4\; +\; 3}}}{2^{\mathrm{2\; +\; 1}}x{3}^5}$

$\frac{2^{3}x{3}^{4}x{3}^3}{2^{2}x2x{3}^5}$ = $\frac{2^{3}x{3}^7}{2^{3}x{3}^5}$

cancel out 2³ and apply the law of division of indices

Law of division of indices

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

$\frac{2^{3}x{3}^{4}x{3}^3}{2^{2}x2x{3}^5}$ = 3^{7 - 5} = 3² = 3 x 3 = 9

Number of people in the club = 5 + 10 + 6 + 9 = 30

The modal age is the age with the highest frequency. The age with the highest frequency is 9 years (frequency of 10).

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

Number of candidates who failed = 175 - 154 = 21

Percentage who failed = $\frac{21}{175}$ x 100% = 3 x 4% = 12%

Note
1. 7 divides 21, 3 times and 175, 25 times
2. 25 divides itself 1 and 100, 4 times

Alternatively, you can calculate the percentage of candidates who passed and subtract it from 100 to get the percentage who failed.

Percentage who passed = $\frac{154}{175}$ x 100% = 22 x 4% = 88%

Note
1. 7 divides 154, 22 times and 175, 25 times
2. 25 divides itself 1 and 100, 4 times

Percentage of candidates who failed = 100% - 88% = 12%

(6 – x)(6 + y) = (6 – x)(6 + y)
(6 – x)(6 + y) = 6(6 + y) – x(6 + y)
(6 – x)(6 + y) = 36 + 6y -6x - xy
(6 – x)(6 + y) = 36 -6x + 6y - xy

The easiest way of finding highest common factor (HCF) is by using the prime factor method.

Prime factors of 20

$$\begin{array}{cc}\mathrm{}& 20\\ 2& 10\\ 2& 5\\ 5& 1\end{array}$$

Prime factors of 20 = 2 x 2 x 5

Prime factors of 20 = 2² x 5

Prime factors of 12

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

Prime factors of 12 = 2 x 2 x 3

Prime factors of 12 = 2² x 3

Prime factors of 28

$$\begin{array}{cc}\mathrm{}& 28\\ 2& 14\\ 2& 7\\ 7& 1\end{array}$$

Prime factors of 28 = 2 x 2 x 7

Prime factors of 28 = 2² x 7

The Highest Common Factor is the product of the prime factors common in all in their lowest.

The prime number common in all is 2 and it's lowest power is 2, hence the HFC of 20, 12 and 28 = 2² = 2 x 2 = 4

F =$\frac{9}{5}$C + 32

C = 40

F =$\frac{9}{5}$ x 40 + 32

F = 9 x 8 + 32

Note: 5 divides itself 1 and 40, 8 times.

F = 72 + 32 = 104

$\frac{2}{3}$(27 - 12) - 6

Method I

Simplifying the bracket first before expanding.

$\frac{2}{3}$(27 - 12) - 6 = $\frac{2}{3}$(27 - 12) - 6

$\frac{2}{3}$(27 - 12) - 6 = $\frac{2}{3}$(15) - 6

$\frac{2}{3}$(27 - 12) - 6 = $\frac{2}{3}$ x 15 - 6

Note: 3 divides 15, 5 times.

$\frac{2}{3}$(27 - 12) - 6 = 2 x 5 - 6

$\frac{2}{3}$(27 - 12) - 6 = 10 - 6

$\frac{2}{3}$(27 - 12) - 6 = 4

Method II

Expand and simplify.

$\frac{2}{3}$(27 - 12) - 6 = $\frac{2}{3}$(27 - 12) - 6

$\frac{2}{3}$(27 - 12) - 6 = $\frac{2}{3}$ x 27 - 12 x $\frac{2}{3}$ - 6

Note:
1. 3 divides 27, 9 times
2. 3 divides 12, 4 times

$\frac{2}{3}$(27 - 12) - 6 = 2 x 9 - 2 x 4 - 6

$\frac{2}{3}$(27 - 12) - 6 = 18 - 8 - 6

$\frac{2}{3}$(27 - 12) - 6 = 18 - 14

$\frac{2}{3}$(27 - 12) - 6 = 4

Sum of angles on a straight line is 180°

x° + 102° + 2x° = 180°

x° + 2x° + 102° = 180°

3x° = 180° - 102°

3x° = 78°

Divide both sides by 3

x = $\frac{78}{3}$ = 26

22ab – 11ac + 6rb – 3rc = 22ab – 11ac + 6rb – 3rc
22ab – 11ac + 6rb – 3rc = 11a(2b - c) + 3r(2b - c)
22ab – 11ac + 6rb – 3rc = (2b - c)(11a + 3r)

Sum means add.

one decimal place means there should be only 1 number after the decimal point. The number at the one decimal position is 6. The next number is less than 5, hence we don't have to add 1 to the 6

198.635 = 198.6 (one decimal place).

6p³ × p² ÷ 3p⁴ = $\frac{6p^{3}x{p}^2}{3p^4}$

1. 3 divides itself 1 and 6, 2 times
2. Apply the law of multiplication of indices to simplify the numerator

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

6p³ × p² ÷ 3p⁴ = 2$\frac{p^{\mathrm{3\; +\; 2}}}{p^4}$

6p³ × p² ÷ 3p⁴ = 2$\frac{p^5}{p^4}$

Apply the law of division of indices.

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

6p³ × p² ÷ 3p⁴ = 2p^5-4 = 2p¹ = 2p

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

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

Principal = ₵75,000
Time = 3 years
Rate = 20

Simple Interest = $\frac{\mathrm{₵75,000\; x\; 3\; x\; 20}}{100}$ = ₵750 x 60 = ₵45,000.00

Note: the zeros (0) in ₵75,000 and 100 cancels each other.

Note: If the mapping has equal interval (difference) for the y and n 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 U₁, the first value of y

U₁ = 10

y = 10 + (n-1)d

The d is the interval between each consecutive y

In this mapping, d = 21-10 = 32-21 = 43-32 = 54-43 = 11

We can substitute our d and simplify the expression for y

y = 10 + (n-1)d

y = 10 + (n-1)11

y = 10 + 11 x n -1 x 11

y = 10 + 11n -11

y = 11n + 10 -11

y = 11n + -1

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.

The distribution is not ordered and so order it first.

The ordered marks are 8, 10, 12, 15, 18

The middle mark is 12 hence 12 is the median.

Translation by a vector

Translation = Point + Vector

P₁ = P + V

If P(x,y) translated by the vector $\left(\begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array}\right)$

P₁ = $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{x\; +\; a}\\ \mathrm{y\; +\; b}\end{array}\right)$

P$\left(\begin{array}{c}3\\ 4\end{array}\right)$ → $\left(\begin{array}{c}3\\ 4\end{array}\right)$ + $\left(\begin{array}{c}3\\ -2\end{array}\right)$ = P^'$\left(\begin{array}{c}\mathrm{3+3}\\ \mathrm{4-2}\end{array}\right)$ = P^'$\left(\begin{array}{c}6\\ 2\end{array}\right)$ = P^'(6, 2)

Angle KGM is right-angled (90°)

angle of elevation of K from M is angle KMG

Sum of interior angles of a triangle is 180°

62° + 90° + angle KMG = 180°
152° + angle KMG = 180°
angle KMG = 180° - 152°
angle KMG = 28°

a = $\left(\begin{array}{c}-2\\ 3\end{array}\right)$ and vector b = $\left(\begin{array}{c}2\\ -5\end{array}\right)$

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

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

a + 2b = $\left(\begin{array}{c}-2\\ 3\end{array}\right)$ + $\left(\begin{array}{c}4\\ -10\end{array}\right)$

a + 2b = $\left(\begin{array}{c}\mathrm{-2\; +\; 4}\\ \mathrm{3\; +\; -10}\end{array}\right)$

a + 2b = $\left(\begin{array}{c}2\\ \mathrm{3\; -\; 10}\end{array}\right)$

a + 2b = $\left(\begin{array}{c}2\\ -7\end{array}\right)$