Union (∪) 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 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.

2340000 = 2340000.0

2340000 = 2.⌒3⌒4⌒0⌒0⌒0⌒0.0 = 2.34 x 10⁶

Note: The power is positive 6 because we had to move the decimal point to the left 6 times.

Explanations

1. We borrowed 1 from 9 making 10. 10 - 7 = 3
2. 9 became 8 after the borrowing. 8 - 4 = 4
3. We borrowed 1 from 2 and the 0 became 10. We then borrowed 1 to the 3 making 13 (10 + 3). 13 - 5 = 8
4. The borrowed 1 from 2 to the 0 became 9 (10 - 1) after borrowing to 3. 9 - 2 = 7
5. 1 has already been borrowed from 2 to become 1. 1 - 1 = 0 but there is no need to write since it's at the beginning

(5m + 3n)-(2m-n) = (5m + 3n)-(2m-n)

Expand and simplify.

Note: - (-) = - - = +

(5m + 3n)-(2m-n) = 5m + 3n -2m + n

(5m + 3n)-(2m-n) = 5m -2m + 3n + n

(5m + 3n)-(2m-n) = 3m + 4n

Two sets are equal when they have the same elements and the number of elements is also the same.

48% male
Percentage female = 100% - 48%
Percentage female = 52%

Percentage female more than male = 52% - 48%
Percentage female more than male = 4%

Percentage means over 100

Number of females more than males = $\frac{4}{100}$ x 42,800

Note:
1. The zeros(0) in 100 and 42,800 cancels each other
2. 4 x 428 = 1,712

Number of females more than males = 1,712

The inequality sign must face the direction of the arrow. Since the dot is shaded, it means it's also equal to so the inequality must also have the equal to below the inequality.

Sum of the angle of the sectors must be 360^o

Angle for French sector = 360 - (120 + 90 + 55 + 25)
Angle for French sector = 360 - 290
Angle for French sector = 70^o

Use the angle of sectors and proportionality to calculate

Angle of sector for Social Studies = 120^o
Social Studies score = 60%

Angle of sector for Economics = 90^o

If 120^o = 60%
90^o = $\frac{\mathrm{60\%\; x\; 90°}}{\mathrm{120°}}$

Note:
1. 60 divides itself 1 and 120, 2 times
2. 2 divides itself 1 and 90, 45 times

90^o = 45%

Kate scored 45% in Economics

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

Total number of teachers = 6 + 2 = 8
Number of female teachers = 2

P(Female teacher) = $\frac{2}{8}$

P(Female teacher) = $\frac{1}{4}$

2x -3(x - 1) = 6
Note: - - = +
2x - 3x + 3 = 6
-x = 6 - 3
-x = 3

Divide both sides by - 1

x = $\frac{3}{-1}$

x = -3

$\frac{1}{2}$(3m - n)

m = 3 and n = -3

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

Note: - - = -(-) = +

$\frac{1}{2}$(3m - n) = $\frac{1}{2}$(9 + 3)

$\frac{1}{2}$(3m - n) = $\frac{1}{2}$(12)

$\frac{1}{2}$(3m - n) = $\frac{1}{2}$ x 12

Note:
1. 2 divides itself 1 and 12, 6
2. 1 x 6 = 6

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

Properties of right angle triangle

1. One angle is always 90° or right angle
2. The side opposite angle of 90° is the hypotenuse
3. The hypotenuse is always the longest side
4. The sum of the other two interior angles is equal to 90°
5. The other two sides adjacent to the right angle are called base and perpendicular

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, the first value of y

U₁ = $\frac{1}{4}$

Note: First term starts from 1

y = $\frac{1}{4}$ + (x-1)d

The d is the interval between each consecutive y

In this mapping, d = $\frac{1}{4}$ - 0 = $\frac{1}{2}$-$\frac{1}{4}$ = $\frac{3}{4}$ - $\frac{1}{2}$ = 1 -$\frac{3}{4}$ = $\frac{1}{4}$

We can substitute our d and simplify the expression for y

y = $\frac{1}{4}$ + (x-1)$\frac{1}{4}$

y = $\frac{1}{4}$ + $\frac{1}{4}$ x x - 1 x $\frac{1}{4}$

y = $\frac{1}{4}$ + $\frac{\mathrm{x}}{4}$ - $\frac{1}{4}$

Note: $\frac{1}{4}$ - $\frac{1}{4}$ = 0

y = $\frac{\mathrm{x}}{4}$

Perimeter is the sum of the sides of a shape.

A rectangle has 2 equal lengths and 2 equal widths.

Perimeter of a rectangle = 2 x Length + 2 x Width

Area of a rectangle = Length x Width

Area = 36 cm²
Width = 3 cm

36 cm² = Length x 3 cm

Divide both sides by 3 cm

Length = $\frac{36{cm}^2}{\mathrm{3\; cm}}$

Length = 12 cm

Perimeter = 2 x 12 cm + 2 x 3 cm
Perimeter = 24 cm + 6 cm
Perimeter = 30 cm

Every number/letter raised to the power 0 is 1.

4 + x⁰ = 4 + x⁰
4 + x⁰ = 4 + 1
4 + x⁰ = 5

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

Time = 1$\frac{3}{4}$ hours

Change the mixed fraction time to improper fraction.

1$\frac{3}{4}$ hours = $\frac{\mathrm{4\; x\; 1\; +\; 3}}{4}$ hours

1$\frac{3}{4}$ hours = $\frac{\mathrm{4\; +\; 3}}{4}$ hours

1$\frac{3}{4}$ hours = $\frac{7}{4}$ hours

Distance = 154 km

Speed = 154 km ÷ $\frac{7}{4}$ hours

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

Speed = 154 km x $\frac{4}{\mathrm{7\; hours}}$

Note:
1. 7 divides itself 1 and 154, 22
2. 22 x 4 = 88

Speed = 88 km/h

Sum of angles on a straight is 180°

w° + 127° = 180°
w° = 180° - 127°
w° = 53°

x + 5 = -7
x = -7 - 5
x = -12

$\frac{\mathrm{x}}{6}$ = $\frac{-12}{6}$

Note: 6 divides itself 1 and 12, 2 times.

$\frac{\mathrm{x}}{6}$ = -2

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

Scale factor = $\frac{\mathrm{OA1}}{\mathrm{OA}}$ = $\frac{\mathrm{A1B1}}{\mathrm{AB}}$ = $\frac{\mathrm{B1C1}}{\mathrm{BC}}$ = $\frac{\mathrm{OC1}}{\mathrm{OC}}$

OA₁ = A₁A + OA = 1 cm + 2 cm = 3 cm

Scale factor = $\frac{3}{2}$ = 1.5

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

Scale factor = $\frac{\mathrm{OA1}}{\mathrm{OA}}$ = $\frac{\mathrm{A1B1}}{\mathrm{AB}}$ = $\frac{\mathrm{B1C1}}{\mathrm{BC}}$ = $\frac{\mathrm{OC1}}{\mathrm{OC}}$

OA₁ = A₁A + OA = 1 cm + 2 cm = 3 cm

Scale factor = $\frac{3}{2}$ = $\frac{\mathrm{OC1}}{\mathrm{OC}}$

OC = 5 cm

$\frac{\mathrm{3\; cm}}{\mathrm{2\; cm}}$ = $\frac{\mathrm{OC1}}{\mathrm{5\; cm}}$

Cross multiply and solve for OC₁

2 x OC₁ = 3 x 5

Divide both sides by 2

OC₁ = $\frac{\mathrm{3\; cm\; x\; 5\; cm}}{\mathrm{2\; cm}}$

OC₁ = $\frac{\mathrm{15\; cm}}{2}$

OC₁ = 7.5 cm

If 1 cm = 3 m

5 cm = $\frac{\mathrm{3\; m\; x\; 5\; cm}}{\mathrm{1\; cm}}$ = 15 m

ax + 3x + a + 3 = ax + 3x + a + 3
ax + 3x + a + 3 = x(a + 3) + 1(a + 3)
ax + 3x + a + 3 = (a + 3)(x + 1)

y = $\frac{\mathrm{x\; +\; 8}}{\mathrm{x\; -\; 4}}$

x = 2

y = $\frac{\mathrm{2\; +\; 8}}{\mathrm{2\; -\; 4}}$

y = $\frac{10}{-2}$

y = -5

The easiest way of finding Greatest Common Factor (GCF)/Highest Common Factor (GCF) is using prime factor method.

Prime factors of 90

$$\begin{array}{cc}\mathrm{}& 90\\ 2& 45\\ 3& 15\\ 3& 5\\ 5& 1\end{array}$$

Prime factors of 90 = 2 x 3 x 3 x 5

Prime factors of 90 = 2 x 3² x 5

Prime factors of 126

$$\begin{array}{cc}\mathrm{}& 126\\ 2& 63\\ 3& 21\\ 3& 7\\ 7& 1\end{array}$$

Prime factors of 126 = 2 x 3 x 3 x 7

Prime factors of 126 = 2 x 3² x 7

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

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

GCF of 90, 126 and 72 = 2 x 3² = 2 x 9 = 18

Write the ratios in the form of fraction and solve for x.

$\frac{1}{3}$ = $\frac{\mathrm{x}}{21}$

Cross multiply

3 x x = 1 x 21

Divide both sides by 3

x = $\frac{21}{3}$

x = 7

Note: In base 5, the value of 1 borrowed is 5.

Explanations

1. We borrowed 1 from 3 making 7 (2 + 5). 7 - 3 = 4
2. 3 became 2 after the borrowing the 1. We borrowed 1 from 4 making 7 (5 + 2). 7 - 4 = 3
3. 4 became 3 after borrowing the 1 from it. 3 - 1 = 2

Method I

Solve the whole number and the fraction parts differently

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = (14 - 2 + 5)[$\frac{1}{3}$ - $\frac{3}{8}$ + $\frac{7}{12}$]

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = (17)[ $\frac{\mathrm{8\; x\; 1\; -\; 3\; x\; 3\; +\; 2\; x\; 7}}{24}$]

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = (17)[ $\frac{\mathrm{8\; -\; 9\; +\; 14}}{24}$]

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = (17)[ $\frac{13}{24}$]

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = 17 $\frac{13}{24}$

Method II

Changing the mixed fractions to improper fractions and solving them.

14$\frac{1}{3}$ = $\frac{\mathrm{3\; x\; 14\; +\; 1}}{3}$

14$\frac{1}{3}$ = $\frac{\mathrm{42\; +\; 1}}{3}$

14$\frac{1}{3}$ = $\frac{43}{3}$

2$\frac{3}{8}$ = $\frac{\mathrm{8\; x\; 2\; +\; 3}}{8}$

2$\frac{3}{8}$ = $\frac{\mathrm{16\; +\; 3}}{8}$

2$\frac{3}{8}$ = $\frac{19}{8}$

5$\frac{7}{12}$ = $\frac{\mathrm{12\; x\; 5\; +\; 7}}{12}$

5$\frac{7}{12}$ = $\frac{\mathrm{60\; +\; 7}}{12}$

5$\frac{7}{12}$ = $\frac{67}{12}$

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ =$\frac{43}{3}$ - $\frac{19}{8}$ + $\frac{67}{12}$

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = $\frac{\mathrm{8\; x\; 43\; -\; 3\; x\; 19\; +\; 2\; x\; 67}}{24}$

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = $\frac{\mathrm{344\; -\; 57\; +\; 134}}{24}$

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = $\frac{421}{24}$

14$\frac{1}{3}$ - 2$\frac{3}{8}$ + 5$\frac{7}{12}$ = 17$\frac{13}{24}$

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

4$\frac{1}{2}$ = $\frac{\mathrm{8\; +\; 1}}{2}$

4$\frac{1}{2}$ = $\frac{9}{2}$

If 9 = $\frac{9}{2}$ litres

x = x x $\frac{9}{2}$ litres ÷ 9

Every whole number is being divided by 1

9 = $\frac{9}{1}$

x = x x $\frac{9}{2}$ litres ÷ $\frac{9}{1}$

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

x = x x $\frac{9}{2}$ litres x $\frac{1}{9}$

Note: 9 cancels each other.

x = $\frac{1}{2}$x litres

Method I

Using proportionality.

The amount sold is 100%

5% gives GH₵ 15.00

If 5% = GH₵ 15.00

100% = $\frac{\mathrm{GH₵\; 15.00\; x\; 100\%}}{\mathrm{5\%}}$

Note: 5 divides itself 1 and 15, 3 times.

100% = GH₵ 3 x 100 = GH₵ 300

She has to sell GH₵ 300

Method II

Let x = amount sold

Commission Amount = $\frac{\mathrm{Commission}}{100}$ x amount sold

Commission Amount = GH₵ 15.00

GH₵ 15.00 = $\frac{5}{100}$ x x

Get rid of the fraction by multiplying both sides by 100

GH₵ 15.00 x 100 = 5 x x

Divide both sides by 5

x = $\frac{\mathrm{GH₵\; 15.00\; x\; 100}}{5}$

x = GH₵ 3 x 100
x = GH₵ 300

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 12, 5, 15 and 4 is 60
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

$\frac{7}{12}$ = $\frac{\mathrm{7\; x\; 5}}{\mathrm{12\; x\; 5}}$ = $\frac{35}{60}$

$\frac{3}{5}$ = $\frac{\mathrm{3\; x\; 12}}{\mathrm{5\; x\; 12}}$ = $\frac{36}{60}$

$\frac{7}{15}$ = $\frac{\mathrm{7\; x\; 4}}{\mathrm{15\; x\; 4}}$ = $\frac{28}{60}$

$\frac{3}{4}$ = $\frac{\mathrm{3\; x\; 15}}{\mathrm{4\; x\; 15}}$ = $\frac{45}{60}$

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

Ascending means from the lowest to highest.

The numerators in ascending orders are 28, 35, 36 and 45

The fractions in ascending orders are: $\frac{7}{15}$, $\frac{7}{12}$, $\frac{3}{5}$, $\frac{3}{4}$

Product means multiplication.

Change the decimal to whole number x 10^-n and multiply. Where n is the number of times you have to move the decimal point to the right in order to obtain whole number.

4.2 = 4.⌒2. = 42 x 10^-1

0.2 = 0.⌒2. = 2 x 10^-1

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

4.2 x 0.2 = 42 x 10^-1 x 2 x 10^-1
4.2 x 0.2 = 84 x 10^{-1 + -1}
4.2 x 0.2 = 84 x 10^-2

Change the whole number x 10^-n back to a decimal.

84 x 10^-2 = .⌒8⌒4.0 = 0.84

Add the 2.5 to the 0.84

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

Age 7 has the highest frequency/number of children (6). Hence 7 is the modal age.

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

Number of children who are 4 years = 5

Total number of children = 3 + 4 + 2 + 5 + 4 + 4 + 6 + 4 + 2 + 1
Total number of children = 35

P(4 years) = $\frac{5}{35}$ = $\frac{1}{7}$

Note: 5 divides itself 1 and 35, 7 times.

7 or more years old = 6 + 4 + 2 + 1
Total number of children = 13

Method I

Simplifying the bracket before multiplying

5 - 7 + 2(3 - 8) = 5 - 7 + 2(3 - 8)
5 - 7 + 2(3 - 8) = - 2 + 2(-5)
5 - 7 + 2(3 - 8) = -2 + 2 x -5
5 - 7 + 2(3 - 8) = -2 - 10
5 - 7 + 2(3 - 8) = -12

Method II

Expanding the bracket and simplifying.

5 - 7 + 2(3 - 8) = 5 - 7 + 2(3 - 8)
5 - 7 + 2(3 - 8) = -2 + 2 x 3 + 2 x -8
5 - 7 + 2(3 - 8) = -2 + 6 - 16
5 - 7 + 2(3 - 8) = -12

r = $\left(\begin{array}{c}-10\\ -3\end{array}\right)$ and n = $\left(\begin{array}{c}8\\ -6\end{array}\right)$

n + r = $\left(\begin{array}{c}8\\ -6\end{array}\right)$ + $\left(\begin{array}{c}-10\\ -3\end{array}\right)$

n + r = $\left(\begin{array}{c}\mathrm{8\; +\; -10}\\ \mathrm{-6\; +\; -3}\end{array}\right)$

n + r = $\left(\begin{array}{c}\mathrm{8\; -\; 10}\\ \mathrm{-6\; -\; 3}\end{array}\right)$

n + r = $\left(\begin{array}{c}-2\\ -9\end{array}\right)$