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

The elements that can be found in both sets Q and R are {3, 5, 7}

2 ≤ x ≤ 5 is read as 2 is less than or equal to (≤) x and x is less than or equal to (≤) 5.

or equal to means the number is also inclusive.

So x starts from 2 and end at 5 (2, 3, 4, 5)

Step 1

Locate the digit in front of the place you are to round the number to. Thus if hundred thousand, locate the digit in the ten thousand, if ten thousand, locate the digit in thousands, if thousand, locate the digit in the hundred,etc.

We are to round to the nearest hundred.

The digit in front of the hundred place is 6

Step 2

If the digit in front is 5 or more, add 1 to the digit of the place you are rounding to but if not maintain the digit.

The digit of the place we are rounding to is 4 (the hundred place). Since the digit in front of it is 6, we have to add 1 to the digit at the hundred place (4) making 1+4 = 5.

Step 3

Replace the digit in front of the place you are to round the number to by zeros (0s)

6 and 5 which are in front of the hundred place are replaced by 0s

8921465 to the nearest hundred = 8921500

Prime factors of 98

$$\begin{array}{cc}\mathrm{}& 98\\ 2& 49\\ 7& 7\\ 7& 1\end{array}$$

Prime factors of 98 = 2 x 7 x 7

Prime factors of 98 = 2 x 7²

Method I

Simplifying the bracket first before expanding.

4(8 - 2) + 5(3 - 8) = 4(8 - 2) + 5(3 - 8)
4(8 - 2) + 5(3 - 8) = 4(6) + 5(-5)
4(8 - 2) + 5(3 - 8) = 4 x 6 + 5 x -5
4(8 - 2) + 5(3 - 8) = 24 - 25
4(8 - 2) + 5(3 - 8) = -1

Method II

Expanding before simplifying.

4(8 - 2) + 5(3 - 8) = 4(8 - 2) + 5(3 - 8)
4(8 - 2) + 5(3 - 8) = 4 x 8 - 2 x 4 + 5 x 3 - 8 x 5
4(8 - 2) + 5(3 - 8) = 32 - 8 + 15 - 40
4(8 - 2) + 5(3 - 8) = 47 - 48
4(8 - 2) + 5(3 - 8) = -1

Change the decimal and % to fraction.

0.32, $\frac{2}{5}$, 27%, $\frac{1}{3}$.

0.32 = 0.⌒3⌒2. = $\frac{32}{100}$ = $\frac{8}{25}$

% means over 100

27% = $\frac{27}{100}$

Before you can compare fractions, their denominators must be the same. To do so, follow the steps below.

1. Find the L.C.M of the denominators. The L.C.M of 5, 25, 100 and 3 is 300
2. Multiply both the numerator and the denominator of each of the fractions by the number of times the denominator divides the L.C.M

0.32 = $\frac{8}{25}$ = $\frac{\mathrm{8\; x\; 12}}{\mathrm{25\; x\; 12}}$ = $\frac{96}{300}$

$\frac{2}{5}$ = $\frac{\mathrm{2\; x\; 60}}{\mathrm{5\; x\; 60}}$ = $\frac{120}{300}$

27% = $\frac{27}{100}$ = $\frac{\mathrm{27\; x\; 3}}{\mathrm{100\; x\; 3}}$ = $\frac{81}{300}$

$\frac{1}{3}$ = $\frac{\mathrm{1\; x\; 100}}{\mathrm{3\; x\; 100}}$ = $\frac{100}{300}$

Once the denominators are the same, you can order the fractions by magnitude of their numerators.

Descending means from the highest to lowest

The descending magnitude of the numerators are 120, 100, 96, 81

The descending order of magnitude of the fractions are $\frac{2}{5}$, $\frac{1}{3}$, 0.32, 27%

Write the ratios in form of a fraction. Since they are equivalent, their fraction value is the same.

$\frac{8}{12}$ = $\frac{\mathrm{y}}{9}$

Cross multiply and solve for y.

8 x 9 = 12 x y

Divide both sides by 12

y = $\frac{\mathrm{8\; x\; 9}}{12}$ = 2 x 3 = 6

Note:
1. 3 divides 9, 3 times and 12, 4 times
2. 4 divides itself 1 and 8, 2 times
3. 2 x 3 = 6

0.55 = 0.⌒5⌒5. = $\frac{55}{100}$ = $\frac{11}{20}$

Note: 5 divides 55, 11 times and 100, 20 times.

If the base of the pyramid is in the shape of a square (base with 4 sides), then it is called a square pyramid.

5w + 7p² - 4w + 3p² = 5w + 7p² - 4w + 3p²
5w + 7p² - 4w + 3p² = 5w - 4w + 7p² + 3p²
5w + 7p² - 4w + 3p² = w + 10p²

Selling Price = Cost Price + Profit
Selling Price % = Cost Price % + Profit %
Cost Price % = 100%
Selling Price % = 100% + 20% = 120%

Cost Price is known so you can use proportionality to calculate the selling price.

If 100% = GH₵ 5

120% = $\frac{\mathrm{GH₵\; 5\; x\; 120\%}}{\mathrm{100\%}}$ = GH₵ 6.00

Note:
1. The zeros (0) in 120 and 100 cancels each other
2. 5 divides itself 1 and 10, 2 times
3. 2 divides itself 1 and 12, 6 times

Law of division of indices

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

Law of division of indices

$\frac{2^9}{2^3}$ = 2^{9 - 3} = 2⁶

x → 2(x + 3)
-3 → 2(-3 + 3)
-3 → 2(0)
-3 → 2 x 0
-3 → 0

Perimeter is the sum of the sides of a shape. The football field is rectangular. Rectangle has two equal lengths and two equal widths

Perimeter = 2 x Length + 2 x Width
Perimeter = 2 x 120 m + 2 x 75 m
Perimeter = 240 m + 150 m
Perimeter = 390 m

The fraction for a full tank is 1. You can use proportionality to calculate the volume for the given fraction.

If 1 = 240 litres

$\frac{4}{5}$ = $\frac{4}{5}$ x 240 litres ÷ 1 = 4 x 48 litres = 192 litres

Note:
1. Every number/fraction divided by 1 is the same number/fraction
2. 5 divides itself 1 and 240, 48 times

Number of sheep before sales = 6x
Number of goats before sales = 5y

3x sheep sold.

Remaining sheep = 6x - 3x
Remaining sheep = 3x

2y goats sold.

Remaining goats = 5y - 2y
Remaining goats = 3y

Total remaining animals = Remaining sheep + Remaining goats
Total remaining animals = 3x + 3y

Total rainfall = 281 + 452 + 265
Total rainfall = 998 mm

Mean = $\frac{\mathrm{Sum\; of\; numbers}}{\mathrm{Number\; of\; numbers}}$

Mean = $\frac{\mathrm{96\; +\; 147\; +\; 281\; +\; 452\; +\; 265\; +\; 139}}{6}$ = $\frac{1380}{6}$ = 230 mm

Ending zeros(0) after a decimal number are ignored

4.80 = 4.8

Change the decimal to fraction.

4.8 = 4.⌒8. = $\frac{48}{10}$

Profit % = $\frac{\mathrm{Profit}}{\mathrm{Cost\; Price}}$ x 100%

Dozen means 12 quantities.

To convert GP to GH₵, you have to divide the GP by 100

48 GP = GH₵ $\frac{48}{100}$

The total amount sold = GH₵ $\frac{48}{100}$ x 12 = GH₵ $\frac{144}{25}$

Note:
1. 4 divides 100, 25 times and 12, 3 times
2. 48 x 3 = 144

Profit = Selling Price - Cost Price

Profit = $\frac{144}{25}$ - $\frac{48}{10}$

Profit = $\frac{\mathrm{2\; x\; 144\; -\; 5\; x\; 48}}{50}$ = $\frac{\mathrm{288\; -\; 240}}{50}$ = $\frac{48}{50}$

Profit % = $\frac{48}{50}$ ÷ $\frac{48}{10}$ x 100

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

Profit % = $\frac{48}{50}$ x $\frac{10}{48}$ x 100 = 10 x 2 = 20%

Note:
1. 48 cancels each other
2. 50 divides itself 1 and 100, 2 times
3. 10 x 2 = 20

There are 60 minutes to complete 360° (complete cycle)

Angle per minute = $\frac{\mathrm{360°}}{60}$ = 6°

The minutes between 5.25 p.m. and 5.15 p.m. is 10 minutes.

Angle for 10 minutes = 10 x 6° = 60°

Note: For between, the starting and ending numbers given are not included.

E = {prime numbers between 10 and 20}

Prime numbers are numbers with only two factors. 1 and itself. Thus no other number can divide it except 1 and the number.Note: 1 is not a prime number.

E = {11, 13, 17, 19}

F = {odd numbers between 0 and 16}

Odd numbers are numbers not divisible by 2.

F = {1, 3, 5, 7, 9, 11, 13, 15}

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

E ∩ F = {11, 13}

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

A complete revolution is 360°

Fraction = $\frac{\mathrm{Number}}{\mathrm{Total}}$

Fraction = $\frac{\mathrm{72°}}{\mathrm{360°}}$ = $\frac{1}{5}$

Note: 72 divides itself 1 and 360, 5 times.

Before you express a ratio, make sure both entities are of the same units.

Change the weeks to days.

There are 7 days in 1 week.

3 weeks will have 3 x 7 days = 21 days

6 days : 3 weeks = 6 days : 21 days

Note: 3 divides 6, 2 times and 21, 7 times and the days cancel each other.

6 days : 3 weeks = 2 : 7

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

Vowels

A vowel is a sound such as the ones represented in writing by the letters 'a', 'e', 'i', 'o', and 'u', which you pronounce with your mouth open, allowing the air to flow through it.

Total number of letters in HIPPOPOTAMUS = 12
Total number of vowels in HIPPOPOTAMUS = 5 (I, O, O, A, U)

P(Vowels) = $\frac{5}{12}$

Perimeter is the sum of the sides of a shape.

The corresponding unlabeled parts are the same with the labelled parts.

Perimeter = 2 x 16 cm + 2 x 22 cm
Perimeter = 32 cm + 44 cm
Perimeter = 76 cm

3 < x < 5 is read as 3 is less than x and x is less than 5.

Since there is no or equal to (≤) the solids are not shaded.

x is between 3 and 5

For example if x = 4, the statement would be true because 3 is less than 4 and 4 is also less than 5.

Line of symmetry

A line of symmetry is a line that cuts a shape exactly in half. In symmetrical shapes, like a regular polygon, this means that if you were to fold the shape along the line of symmetry, both halves would match exactly and be mirror images of each other.

p = $\left(\begin{array}{c}4\\ 3\end{array}\right)$ and q = $\left(\begin{array}{c}-2\\ 7\end{array}\right)$

4p - 2q = 4$\left(\begin{array}{c}4\\ 3\end{array}\right)$ - 2$\left(\begin{array}{c}-2\\ 7\end{array}\right)$

4p - 2q = $\left(\begin{array}{c}\mathrm{4\; x\; 4}\\ \mathrm{4\; x\; 3}\end{array}\right)$ - $\left(\begin{array}{c}\mathrm{-2\; x\; 2}\\ \mathrm{2\; x\; 7}\end{array}\right)$

4p - 2q = $\left(\begin{array}{c}16\\ 12\end{array}\right)$ - $\left(\begin{array}{c}-4\\ 14\end{array}\right)$

4p - 2q = $\left(\begin{array}{c}\mathrm{16\; -\; -\; 4}\\ \mathrm{12\; -\; 14}\end{array}\right)$

Note: - - = +

4p - 2q = $\left(\begin{array}{c}\mathrm{16\; +\; 4}\\ \mathrm{12\; -\; 14}\end{array}\right)$

4p - 2q = $\left(\begin{array}{c}20\\ -2\end{array}\right)$

A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. All the shapes are four sided except hexagon which is six sided.

Sum of angles on a straight line is 180°

4x + 5x = 180
9x = 180

Divide both sides by 9

x = $\frac{180}{9}$ = 20

Sum of angles on a straight line is 180°

4x + 5x = 180
9x = 180

Divide both sides by 9

x = $\frac{180}{9}$ = 20

Sum of interior angles of a triangle is 180°

3x + 4x + f = 180°
7x + f = 180°

But x = 20

7 x 20 + f = 180
140 + f = 180
f = 180 - 140
f = 40°

Get rid of the fraction by multiplying both sides by 9

$\frac{\mathrm{4k}}{9}$ = 12

4k = 12 x 9
4k = 108

Divide both sides by 4

k = $\frac{108}{4}$ = 27

(a + 4)(a + 6) = (a + 4)(a + 6)
(a + 4)(a + 6) = a(a + 6) + 4(a + 6)
(a + 4)(a + 6) = a² + 6a + 4a + 24
(a + 4)(a + 6) = a² + 10a + 24

Change the mixed fraction to improper fraction.

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

% means over 100

12$\frac{1}{2}$% = $\frac{25}{2}$%

$\frac{25}{2}$% = $\frac{25}{2}$ ÷ 100

Every whole number is divisible by 1

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

$\frac{25}{2}$% = $\frac{25}{2}$ ÷ $\frac{100}{1}$

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

$\frac{25}{2}$% = $\frac{25}{2}$ x $\frac{1}{100}$ = $\frac{\mathrm{25\; x\; 1}}{\mathrm{2\; x\; 100}}$ = $\frac{25}{200}$

Of means multiplication (x)

12$\frac{1}{2}$% of GH₵ 80.00 = $\frac{25}{200}$ x GH₵ 80 = 5 x GH₵ 2 = GH₵ 10.00

Note:
1. The zeros (0) in 200 and 80 cancels each other
2. 5 divides 20, 4 times and 25, 5 times
3. 4 divides itself 1 and 8, 2 times
4. 5 x 2 = 10

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.

34.1 = 3.⌒4.1 = 3.41 x 10¹ = 3.41 x 10

Note: The power is positive 1 because we had to move to the left once. Any number/variable raised to the power 1 is the same number/variable.

$\frac{30}{\mathrm{5(-2)}}$ = -3

Note:
1. 5 divides itself 1 and 30, 6 times
2. 2 divides itself 1 and 6, 3
3. Any number divided by negative is negative except the numerator is also negative to cancel each other

Factors of 15 = {1, 3, 5, 15}

Factors of 21 = {1, 3, 7, 21}

Highest Common Factor (H.C.F) is the factor seen (common) in each factor = 3

Difference means subtraction.

Since the difference is positive, the smaller number was subtracted from the larger number

Let x = the larger number.

Difference = 168
Smaller number = 113
x - 113 = 168
x = 168 + 113
x = 281

r = $\left(\begin{array}{c}2\\ 5\end{array}\right)$ and t = $\left(\begin{array}{c}-2\\ -3\end{array}\right)$

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

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

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

r + t = $\left(\begin{array}{c}0\\ 2\end{array}\right)$