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

Q ∩ R = {7, 10}

The average/mean of the two points gives the middle.

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

Mean/Average = $\frac{\mathrm{7.2\; +\; 7.9}}{2}$ = $\frac{15.1}{2}$ = 7.55

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

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 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

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

L.C.M = 2² x 3 x 5 = 4 x 3 x 5 = 60

Prime factors of 72

$$\begin{array}{cc}\mathrm{}& 72\\ 2& 36\\ 2& 18\\ 2& 9\\ 3& 3\\ 3& 1\end{array}$$

Prime factors of 72 = 2 x 2 x 2 x 3 x 3

Prime factors of 72 = 2³ x 3²

3t – 2(t + 12) = 11
3t -2 x t - 2 x 12 = 11
t - 24 = 11
t = 11 + 24
t = 35

Amount still owing = ₵550,000 - ₵150,000 = ₵400,000

When the denominator is 1 joined by a number of zeros such as 10, 100, 1000, ..., 1n0s, you simply move the decimal point of the numerator to the left n times.

Where n is the number of zeros.

If the number is whole number, you can add .0 to the whole number and move it by n times to the left.

$\frac{37}{100}$ x $\frac{7}{10}$ = $\frac{\mathrm{37\; x\; 7}}{\mathrm{100\; x\; 10}}$ = $\frac{259}{1000}$

Note: when multiplying by 1 joined by a number of zeros (1n0s), the result is simply 1 joined by the total number of zeros.

For example 100 x 10 = 1joined by three zeros = 1000

259 = 259.0

$\frac{259}{1000}$ = 0.⌒2⌒5⌒9.0 = 0.259

Note: the zeros in 1000 are three (3) hence we moved the decimal point in 259.0 three (3) times to the left to obtain the 0.259

1.25 = 1 + 0.25

Change the decimal part to fraction.

To change a decimal to fraction, follow the steps below:

1. count how many times you have to move the decimal point to the right to obtain a whole number (n)
2. Divide the whole number obtained by 1nos where n is how many zeros you are to join to 1. The n is the count in step 1

In 0.25, we are to move the decimal point two (2) times to the left. Hence we divide 25 by 100, thus 1 joined by two zeros.

Step 1

0.25 = 0.⌒2⌒5.0 = 25, n is two

Step 2

0.25 = $\frac{25}{100}$

The 25 is divided by 1 joined by two (2) zeros. The two (2) is the number of count obtained in step 1.

Now you can simplify the fraction.

$\frac{25}{100}$ = $\frac{1}{4}$

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

∴ 1.25 = 1$\frac{1}{4}$

If 6 boxes = 270

20 boxes = $\frac{\mathrm{270\; x\; 20\; boxes}}{\mathrm{6\; boxes}}$ = 45 x 20 = 900

Note: 6 divides itself 1 and 270, 45

Per means every

If 1 hour = 45 km

12 hour = $\frac{\mathrm{45\; km\; x\; 12\; hour}}{\mathrm{1\; hour}}$ = 45 km x 12 = 540 km

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

Principal = ₵130,000

Time = 2$\frac{1}{2}$

Rate = 12

Change the mixed fraction to improper fraction.

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

Simple Interest = $\frac{\mathrm{₵130,000}x\frac{5}{2}x12}{100}$ = ₵1300 x 5 x 6 = ₵39,000

Note:
1. The zeros(0) at the end of ₵130,000 and 100 cancel each other
2. 2 divides itself 1 and 12, 6 times

To express a fraction as a percentage, you simply multiply the fraction by 100%.

$\frac{2}{5}$ = $\frac{2}{5}$ x 100% = 2 x 20% = 40%

Note: 5 divides itself 1 and 100, 20 times.

The percentage for the amount sold is 100%.

If 20% = ₵45,000

100% = $\frac{\mathrm{₵45,000\; x\; 100\%}}{\mathrm{20\%}}$ = ₵45,000 x 5 = ₵225,000.00

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.

83000 = 83000.0

83000 = 8.⌒3⌒0⌒0⌒0.0 = 8.3000 x 10⁴ = 8.3 x 10⁴

Note:
1. the power is positive 4 because we had to move the decimal point to the left 4 times.
2. zeros (0) at the end of a decimal number can be ignored

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

The 3 mark occurred/appeared the most (4 times). Hence the mode is 3

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

There are 10 marks.

Mean = $\frac{\mathrm{2\; +\; 3\; +\; 5\; +\; 2\; +\; 3\; +\; 4\; +\; 2\; +\; 3\; +\; 5\; +\; 3}}{10}$ = $\frac{32}{10}$ = 3.2

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

Total balls = 12 + 8 = 20

Number of red balls = 12

P(Red ball) = $\frac{12}{20}$ = $\frac{3}{5}$

Note: 4 divides 12, 3 times and 20, 5 times.

y = $\frac{1}{3}$(x - 2)

Get rid of the fraction by multiplying both sides by the denominator (3).

3 x y = 3 x $\frac{1}{3}$(x - 2)

3y = 1(x - 2)
3y = x - 2
3y + 2 = x

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

6a² × 4a²b² = 6 x 4 x a^{2 + 2}b²

6a² × 4a²b² = 24 x a⁴b²

17 base 10 to base 2

$$\begin{array}{cc}\mathrm{}& 17\\ 2& \mathrm{8\; r\; 1}\\ 2& \mathrm{4\; r\; 0}\\ 2& \mathrm{2\; r\; 0}\\ 2& \mathrm{1\; r\; 0}\\ 2& \mathrm{0\; r\; 1}\end{array}$$

You list the remainders from the bottom to the top.

17_ten = 10001_two

< means less than

8+4 = 12 which is not less than 10
7+4 = 11 which is not less than 10
6+4 = 10 which is not less than 10
5+4 = 9 which is less than 10

-2 < x < 2 is read as -2 is less than x and x is less than 2. Hence the value of x are integers greater than -2 but less than 2.

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 = 4

y = -4 + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = -2-(-4) = 0-(-2) = 2 - 0 = 2

We can substitute our d and simplify the expression for y

y = -4 + (x-1)2
y = -4 + 2x - 2
y = 2x - 2 -4
y = 2x - 6

Factorize 2 out

y = 2(x - 3)

The rule of the mapping is y = 2(x - 3)

When x = 5

y = 2(5 - 3) = 2 x 2 = 4

Alternatively, you can use the interval of y to solve for m. The interval is the same for consecutive numbers.

-2-(-4) = 0-(-2) = 2 - 0 = m - 2 = 2

m - 2 = 2
m = 2 + 2
m = 4

Volume of rectangular container = Length x Width x Height

Volume of rectangular container = 2.5 m x 4 m x 5 m

2.5 = $\frac{25}{10}$

Volume of rectangular container = $\frac{25}{10}$ m x 4 m x 5 m

Volume of rectangular container = $\frac{25}{10}$ x 20 m³

Note:
1. 4 x 5 = 20
2. the zeros (0) in 10 and 20 cancel each other

Volume of rectangular container = 25 x 2 m³

Volume of rectangular container = 50 m³

Since water is filled to the brim, before the water was used, the volume was 50 m³

Volume of water left = Volume of water to the brim - Volume of water used

Volume of water left = 50 m³ - 35 m³

Volume of water left = 15 m³

From pythagorean theorem,

The hypothenuse (c) is the longest side.

5² + |XZ|² = 13²

5² = 5 x 5 = 25
13² = 13 x 13 = 169

25 + |XZ|² = 169

|XZ|² = 169 - 25

|XZ|² = 144

144 = 12 x 12 = 12²

|XZ|² = 12²

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

|XZ| = 12 cm

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

Equilateral Triangle

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°.

a = $\left(\begin{array}{c}2\\ 1\end{array}\right)$ and b = $\left(\begin{array}{c}3\\ -4\end{array}\right)$

2a + b = 2$\left(\begin{array}{c}2\\ 1\end{array}\right)$ + $\left(\begin{array}{c}3\\ -4\end{array}\right)$

2a + b = $\left(\begin{array}{c}\mathrm{2\; x\; 2}\\ \mathrm{1\; x\; 2}\end{array}\right)$ + $\left(\begin{array}{c}3\\ -4\end{array}\right)$

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

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

2a + b = $\left(\begin{array}{c}7\\ \mathrm{2\; -\; 4}\end{array}\right)$

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

f + angle WVU = 180°

Angle WUV = 72°

Angle UWV = 56°

Angle WUV + UWV + WVU = 180°

72° + 56° + WVU = 180°

128° + WVU = 180°

WVU = 180° - 128° = 52°

f + angle WVU = 180°

f + 52° = 180°

f = 180° - 52°

f = 128°

Volume of a cuboid = Length x Width x Height

Volume of the cuboid = 2 cm x P cm x 5 cm
Volume of the cuboid = 2 cm x 5 cm x P cm
Volume of the cuboid = 10P cm³

5(3t + 1) – 6(t – 1) = 5(3t + 1) – 6(t – 1)

Note: - x - = +. - 6 x -1 = + 6

5(3t + 1) – 6(t – 1) = 15t + 5 - 6t + 6
5(3t + 1) – 6(t – 1) = 15t - 6t + 5 + 6
5(3t + 1) – 6(t – 1) = 9t + 11

($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$ = ($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$

($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$ = ($\frac{\mathrm{2\; x\; 2\; -\; 3\; x\; 1}}{6}$) ÷ $\frac{1}{6}$

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

($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$ = ($\frac{1}{6}$) ÷ $\frac{1}{6}$

($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$ = $\frac{1}{6}$ ÷ $\frac{1}{6}$

($\frac{2}{3}$ - $\frac{1}{2}$) ÷ $\frac{1}{6}$ = 1

Note: $\frac{1}{6}$ cancel each other 1 time