Union (U) is a set of the list of elements in both sets. If an element is found in both set, it is written only once

A U B = {1, 2, 3, 4, 5, 6}

The number of members in A U B is 6.

Faces of a cuboid

Change the decimals 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.

0.036 = 0.⌒0⌒3⌒6. = 36 x 10^-3

0.02 = 0.⌒0⌒2. = 2 x 10^-2

$\frac{0.036}{0.02}$ = $\frac{36x{10}^{-3}}{2x{10}^{-2}}$

Note:
1. 2 divides itself 1 and 36, 18 times
2. Apply law of division of indices to simplify the base 10s

Law of division of indices

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

$\frac{0.036}{0.02}$ = 18 x 10^{-3 - (-2)}

Note: - - = +

$\frac{0.036}{0.02}$ = 18 x 10^{-3 + 2}

$\frac{0.036}{0.02}$ = 18 x 10^-1

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

Every whole number has a decimal point at the end which is not written.

18 = 18.0

In 18 x 10^-1, you are to move the decimal point once to the left.

18 x 10^-1 = 1.⌒8.0 = 1.8

Angle formed by one complete revolution is 360°.

Right angle is 90°

4 x 90° = 360°

Note:
1. Up to means the number is also included
2. integers can be both positive and negative. Set P doesn't contain negative integers

Prime number is a number with only two factors (1 and itself). Thus only 1 and the number itself can divide it without a remainder.

-35 – (-15) + (-30)

Note:
1. + - = -
2. - - = - (-) = +

-35 – (-15) + (-30) = -35 – (-15) + (-30)
-35 – (-15) + (-30) = -35 + 15 - 30
-35 – (-15) + (-30) = -50

Steps

1. Locate the place value you are to correct to
2. Locate the next place value. If it is 5 or more, add 1 to the located place value in step 1

Nearest thousand

Step 1

The nearest thousand is on 78,000

Step 2

The next place value is 9 which is more than 5 hence we add 1 to the 8 making 79,000

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$

Method I

Express the bases as a product in powers capable of cancelling each other using the concept of the multiplication of indices.

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

2¹⁰ = 2⁸ x 2²

Note: 10 = 8 + 2

3⁶ = 3⁴ x 3²

Note: 6 = 4 + 2

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = $\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = $\frac{2^{8}x{2}^{2}x{3}^2}{3^{4}x{3}^{2}x{2}^8}$

Cancel out

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = $\frac{2^2}{3^4}$

Method II

Using the following law of indices

Law of division of indices

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

a^-b = $\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = $\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = 2^{10 - 8} x 3^{2 - 6}

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = 2² x 3^-4

3^-4 = $\frac{1}{3^4}$

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = 2² x $\frac{1}{3^4}$

$\frac{2^{10}x{3}^2}{3^{6}x{2}^8}$ = $\frac{2^2}{3^4}$

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 < x (-2 is less than x).

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 x ≤ 3 (x is less than or equal to 3).

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

4xy - 16x + 10y – 40 = 4xy - 16x + 10y – 40
4xy - 16x + 10y – 40 = 4x(y - 4) + 10(y - 4)
4xy - 16x + 10y – 40 = (y - 4)(4x + 10)

x → 2x + 5

When x = 2

2 → 2(2) + 5 = 4 + 5 = 9

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

b = b¹

3a²b³ × 4a³b = 3a²b³ × 4a³b
3a²b³ × 4a³b = 3 x 4 x a^{2 + 3}b^{3 + 1}
3a²b³ × 4a³b = 12a⁵b⁴

2x + 1 < 5

Solve for x

2x < 5 - 1
2x < 4

Divide both sides by 2

x < $\frac{4}{2}$

x < 2

x < is read as x is less than 2

The list of numbers less than 2 in the domain {-1, 0, 1, 2, 3} is {-1, 0, 1}

The easiest way to find the HCF is using the prime factor 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 60

$$\begin{array}{cc}\mathrm{}& 60\\ 2& 30\\ 2& 15\\ 3& 5\\ 5& 1\end{array}$$

Prime factors of 60 = 2 x 2 x 3 x 5

Prime factors of 60 = 2² x 3 x 5

Prime factors of 96

$$\begin{array}{cc}\mathrm{}& 96\\ 2& 48\\ 2& 24\\ 2& 12\\ 2& 6\\ 2& 3\\ 3& 1\end{array}$$

Prime factors of 96 = 2 x 2 x 2 x 2 x 2 x 3

Prime factors of 96 = 2⁵ x 3

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

HCF = 2² x 3 = 2 x 2 x 3 = 4 x 3 = 12

Odd number is a number not divisible by 2.

P = {odd numbers between 20 and 30}

P = {21, 23, 25, 27, 29}

Q = {23, 29}

⊂ stands for subset.

A subset is a set with all its members found in the set it is a subset to.

All the members of Q can be found in P. Hence Q is a subset of P.

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 = 1-(-1) = 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

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

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

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

2a - b = $\left(\begin{array}{c}4\\ 8\end{array}\right)$ - $\left(\begin{array}{c}-3\\ 2\end{array}\right)$

2a - b = $\left(\begin{array}{c}\mathrm{4\; -\; (-3)}\\ \mathrm{8\; -\; 2}\end{array}\right)$

Note: - - = - (-) = +

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

2a - b = $\left(\begin{array}{c}7\\ 6\end{array}\right)$

Sum of interior angles = (n - 2) x 180°

Where n is the number of sides of the polygon.

The above polygon has 5 sides.

Sum of interior angles = (5 - 2) x 180°
Sum of interior angles = 3 x 180°
Sum of interior angles = 540°

2x + 4x + x + 5x + 3x = 540
15x = 540

Divide both sides by 15

x = $\frac{540}{15}$ = 36

The percentage for the original price is 100%

After the increment, the new percentage would be 100% + 10% = 110%

We know the original cost to be ₵60,000

If 100% = ₵60,000

110% = $\frac{\mathrm{₵60,000\; x\; 110\%}}{\mathrm{100\%}}$ = ₵600 x 110 = ₵66,000

Note: the zeros (0) at the end of ₵60,000 and 100 cancel each other.

15 – 2x = 6
15 - 6 = 2x
9 = 2x

Divide both sides by 2

x = $\frac{9}{2}$ = 4.5

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

Number of increment = 17,000,000 - 5,000,000
Number of increment = 12,000,000

Percentage increase = $\frac{\mathrm{12,000,000}}{\mathrm{5,000,000}}$ x 100% = 12 x 20 = 240%

Note:
1. the zeros (0) at the end of 12,000,000 and 5,000,000 cancel each other
2. 5 divides itself 1 and 100, 20 times

The sum of angles of triangle is 180°

The sum of the ratios represent the sum of the angles.

The smallest angle has the smallest ratio.

Sum of ratios = 3 + 2 + 1 = 6

If 6 = 180°

1 = $\frac{\mathrm{1\; x\; 180°}}{6}$ = 30°

The easiest way to find the LCM is using the prime factor method.

Prime factors of 4

$$\begin{array}{cc}\mathrm{}& 4\\ 2& 2\\ 2& 1\end{array}$$

Prime factors of 4 = 2 x 2 = 2²

Prime factors of 6

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

Prime factors of 6 = 2 x 3

The easiest way to find the HCF is using the prime factor method.

Prime factors of 10

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

Prime factors of 10 = 2 x 5

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

L.C.M of 4, 6 and 10 = 2² x 3 x 5 = 4 x 15 = 60

134_five = 1 x 5² + 3 x 5¹ + 4 x 5⁰

Note:
1. Any number raised to the power 0 is 1
2. Any number raised to the power 1 is the same number

134_five = 1 x 5 x 5 + 3 x 5 + 4 x 1
134_five = 25 + 15 + 4
134_five = 44

Area of a circle = π x radius x radius

Radius = 7 cm

Area of the circle =$\frac{22}{7}$ x 7 cm x 7 cm = 22 x 1 cm x 7 cm = 154 cm²

Note: the 7 dividing 22 cancels one of the 7s multiplying.

$\frac{1}{2}$ - $\frac{2}{3}$ + $\frac{3}{4}$ = $\frac{1}{2}$ - $\frac{2}{3}$ + $\frac{3}{4}$

$\frac{1}{2}$ - $\frac{2}{3}$ + $\frac{3}{4}$ = $\frac{\mathrm{6\; x\; 1\; -\; 4\; x\; 2\; +\; 3\; x\; 3}}{12}$

$\frac{1}{2}$ - $\frac{2}{3}$ + $\frac{3}{4}$ = $\frac{\mathrm{6\; -\; 8\; +\; 9}}{12}$

$\frac{1}{2}$ - $\frac{2}{3}$ + $\frac{3}{4}$ = $\frac{7}{12}$

$\frac{1}{2}$x - 4x > 20

Get rid of the fraction by multiplying both sides by 2

2 x $\frac{1}{2}$x - 4x x 2 > 20 x 2

x - 8x > 40
-7x > 40

Divide both sides by -7. When dividing by a negative, the inequality sign changes direction.

x < $\frac{40}{-7}$

x < -5$\frac{5}{7}$

2² × 6³

6 = 2 x 3

2² × 6³ = 2² × (2 x 3)³

Apply the law of indices (a x b)^m = a^m x b^m

2² × 6³ = 2² × 2³ x 3³

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

2² × 6³ = 2^{2 + 3} x 3³

2² × 6³ = 2⁵ x 3³

Volume of water = Length x Breadth x Depth

5 cm x 2 cm x depth = 30cm³

Divide both sides by 5 cm x 2 cm

Depth = $\frac{30{\mathrm{cm}}^3}{\mathrm{5\; cm\; x\; 2\; cm}}$ = 3 cm

Note: 5 x 2 is 10 and 30 divided by 10 is 3

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

-4 occurred the most (3 times) hence the modal temperature is –4°C.

Let x = The younger sister's age

The elder sister is 3 years older than the younger one means when you add 3 years to the younger one's age, you will get the older one's age.

Older sister's age = x + 3

The ratio 4 represent the older sister's age and 3 that of the younger sister.

x + 3 : x = 4 : 3

Change the ratio format to division.

$\frac{\mathrm{x\; +\; 3}}{\mathrm{x}}$ = $\frac{4}{3}$

Cross multiply and solve for x

3(x + 3) = 4 x x
3x + 9 = 4x
9 = 4x - 3x
9 = x

∴ the younger sister's age is 9 years.

n² + 4 = 40

n² = 40 - 4
n² = 36

36 = 6 x 6 = 6²

n² = 6²

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

n = 6

Type of angles

An acute angle measures less than 90° at the vertex.
An obtuse angle is between 90° and 180°.
A right angle precisely measures 90° at the vertex.
An angle measuring exactly 180° is a straight angle.
A reflex angle measures between 180°- 360°.

Since the whole numbers are the same, you compare only the fractions.

The fractions have the same numerator (1) hence you compare by their denominators.

The smaller the denominator, the higher the value of the fraction and the vice versa.

Ascending order means listing from the lowest to the highest.

The fraction with the highest denominator is the smallest fraction followed by the next.

$\frac{1}{4}$ is lesser than $\frac{1}{3}$ which is also lesser than $\frac{1}{2}$

The percentage for the original price is 100%

Since the original price is reduced by 10%, the customer bought the item at:

100% - 10% = 90%

We know the amount at 90% so we can calculate the amount at 100% using proportionality.

If 90% = ₵81,000

100% = $\frac{\mathrm{₵81,000\; x\; 100\%}}{\mathrm{90\%}}$ = ₵9,000 x 10 = ₵90,000

Note:
1. The zeros (0) at the end of 90 and 100 cancel each other
2. 9 divides itself 1 and 81, 9 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.

The distribution is not ordered and so order it first.

3, 4, 4, 5, 7, 8, 8, 8

There are two numbers at the middle, 5 and 7

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

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

Number of 4 marks = 2

Number of samples = 8

Probability = $\frac{2}{8}$ = $\frac{1}{4}$

y = k + ax²

k = $\frac{14}{5}$, a = $\frac{4}{5}$ and x = 2

y = $\frac{14}{5}$ + $\frac{4}{5}$ x 2²

2² = 2 x 2 = 4

y = $\frac{14}{5}$ + $\frac{4}{5}$ x 4

y = $\frac{14}{5}$ + $\frac{\mathrm{4\; x\; 4}}{5}$

y = $\frac{14}{5}$ + $\frac{16}{5}$

y = $\frac{\mathrm{14\; +\; 16}}{5}$

y = $\frac{30}{5}$ = 6

For a triangle to have a right angle (90°), it must meet Pythagorean theorem.

Which states that the square of the longest side (Hypotenuse/side facing the 90°) must be equal to the sum of the square of the other two sides.

13² = 5² + 12²

13² = 13 x 13 = 169

5² = 5 x 5 = 25

12² = 12 x 12 = 144

169 = 25 + 144