Prime number is a number that is divisible only by itself and 1 (e.g. 2, 3, 5, 7, 11)

To express a number as a product of prime factors express the number as a multiple of prime numbers and simplify in a form of indexes

108 = 2 x 54

108 = 2 x 2 x 27

108 = 2 x 2 x 3 x 9

108 = 2 x 2 x 3 x 3 x 3

Now apply the law of indexes

a^m x aⁿ = a^{m + n}

Thus if the bases are the same and are multiplying, you can add the powers

a = a¹

Thus every number is raised to the power 1

108 = 2 x 2 x 3 x 3 x 3 = 2¹ x 2¹ x 3¹ x 3¹ x 3¹

108 = 2¹⁺¹ x 3¹⁺¹⁺¹

108 = 2² x 3³

1 hour after 2.20 p.m. will be 3.20 p.m.

We are then left with 50 minutes to add to the 3.20 p.m.

If you add 40 minutes to the 20 minutes after the 3 p.m. [3.20 p.m.] that will be 1 hour [60 minutes make an 1 hour] and at that time the time will be 4 p.m. [1 hour after 3.00 p.m.] and the 50 minutes will be left with 10 minutes.

10 minutes after 4.00 p.m. is 4.10 p.m.

Hence the match will end at 4.10 p.m.

ALTERNATIVELY

To add time, add the hours separately and the minutes separately

If the minutes is 60 or more, convert to hours and minutes

2.20 p.m. = 2 hours and 20 minutes

Adding 1 hour 50 minutes

Adding Hours

2 + 1 = 3

Adding Minutes

20 minutes + 50 minutes = 70 minutes

60 minutes make an hour, hence 70 minutes is 1 hour 10 minutes

Add the 1 hour to the previous sum of the hours

Total hours = 3 + 1 = 4

Minutes left is 10 minutes

Hence the match ends at 4.10 p.m.

A quadrilateral is a four-sided two-dimensional shape. Examples: square, rectangle, rhombus, trapezium, parallelogram and kite.

Triangle on the other hand has three sides and three angles also known as trigon. The root word "tri" means "three" and the root word "quad" means "four". Hence, a triangle is not a quadrilateral.

Expand the bracket

- (7x - 3) = -1 x (7x - 3)

Apply a(b - c) = a x b + a x - c

-1 x (7x - 3) = -1 x 7x -1 x - 3

- x - = + hence -1 x - 3 = + 3

-1 x (7x - 3) = -1 x 7x -1 x - 3 = - 7x + 3

5x - (7x - 3) ≤ 9

5x - 7x + 3 ≤ 9

Simplify

-2x ≤ 9 - 3

-2x ≤ 6

Divide both sides by - 2

NOTE: When you divide by negative, the signs changes to the opposite direction

Cancel out

x ≥ -3

Change the mixed fractions to improper fractions

Rule

Apply the division of fraction rule

Thus you reciprocate (down goes top and top down) the fraction at the right and change the division sign to multiplication

Now cancel out and apply the principle of multiplying fractions

3 goes into 6, 2 times and 21, 7 times

5 goes into itself, 1 time and 10, 2 times

If AB is a vector from A to B, then its negative vector is BA and it is from B to A. i.e., just multiplying by a negative sign changes the direction of the vector. In this case, we can say AB = - BA and BA = - AB. Also, |AB| = |BA|. Hence, AB and BA are the negatives of each other.

Negating A Vector

NOTE: - x - = +

Reflection in axis

Reflection in the x axis

The rule for a reflection over the x -axis is (x, y)→(x, −y).

In the reflection in the x, y is negative

Reflection in the y axis

The rule for a reflection over the y -axis is (x, y)→(−x, y).

Reflection in the line y=x :

The rule for a reflection in the line y=x is (x, x)→(y, x)

Reflection in the line y=−x :

The rule for a reflection in the origin is (x, y)→(−y, −x) .

Reflection of the point (x, y) = (-x, y)

Reflection of (4, 3) in the y-axis → S^' = (-4, 3)

Get rid of the denominator (2) by multiplying each part of the equation by 2

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

Expand the bracket

2 x x - 1 x 2 = 1 x x + 1 x 2

2x - 2 = x + 2

2x - x = 2+2

x = 4

The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle.

Diameter is the longest chord of a circle which passes through centre joining the two points on the circumference of a circle.

Group them and factorize like terms out

xy - xm - my + m²

x(y-m) - m(y - m)

NOTE

- x - = + and + = - x -

m² = -m x -m

x(y-m) - m(y - m)

(x - m)(y-m)

Gradient of a line = change in y / change in x =(y₂-y₁)/(x₂-x₁)

The general point of a line is in the form (x,y), thus the first cordinate is x and the second cordinate is y

P(3,-2) and Q(-2,4)

y₂ = 4 and y₁ = -2

x₂ = -2 and x₁ = 3

Gradient = (4--2)/(-2-3)

-- = +

Gradient = (4+2)/(-2-3) = ⁶⁄_-5 = -⁶⁄₅

Change the decimal to fraction by multiplying both the numerator and the denominator by 1n0s where n0s means number of zeros to attach to 1 for the multiplication

n is the number of times to move the decimal point to get a whole number

To change 10.5 to 105, you will realize we have to move the decimal to the right once (1 time)

Hence we can multiply 10.5 by 10/10

10/10 = 1 so doesn't affect the figure but for the sake of simplication that's why we put it so

10.5 = 10.5 x 10/10 = 105/10

The updated equation will now be:

Now cross multiply

n x 10 = 105 x 100

Divide both sides by 10 to make n the subject

10 cancels each other and the zeros cancel each other

n = 105 x 10 = 1050

Substitute x by 2 in the equation and solve for the expression

x → 2x² - 1

2(2)² - 1 = 2 x 4 - 1 = 8 - 1 = 7

NOTE: x² = x x x

(2)² = 2² = 2 x 2 = 4

Make sure each of the fractions have the same denominator

Steps

1. Look for the L.C.M for the denominators (5,4 and 3)

2. Multiply the numerator and the denominator by the number that when you multiply the denominatory with you will get the L.C.M

3. Once the denominators are the same, arrange the fraction using the values of the numerators

STEP 1

The L.C.M of 5, 4 and 3 is 60

The L.C.M is the least number you can find in the multiplication of each of the numbers you are looking for the L.C.M

Multiplication of 5

5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,...

Multiplication of 4

4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,...

Multiplication of 3

3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,...

As you can see the least number you can find in the multiplication table of 5,4 and 3 is 60

STEP 2

Denominator 5

5 x 12, gives 60, hence we have to multiply the numerator (3) and the denominator (5) by 12

Denominator 4

4 x 15, gives 60, hence we have to multiply the numerator (1) and the denominator (4) by 15

Denominator 3

3 x 20, gives 60, hence we have to multiply the numerator (2) and the denominator (3) by 20

STEP 3

Arrange the numerators in order being requested. In this question we are asked to arrange from the lowest to the highest

NOTE: since the duration was given in years, you have to convert the years to months before multiplying the cost per month

There are 12 months in a year

Half a year is 6 months

Hence 1½ years = 12 months + 6 months = 18 months

Method I

Using proportionality.

Per month means every (1) month.

If 1 month = GH₵ 9

18 months = $\frac{\mathrm{GH₵\; 9\; x\; 18\; months}}{\mathrm{1\; month}}$ = GH₵ 9 x 18 = GH₵ 162.00

Method II

Total Amount paid = Total Months x Cost per month

Total amount paid = 18 months x GH₵ 9.00 per month

Total amount paid = 18 x 9 = GH₵ 162.00

Let x be the number

A certain number (x) is subtracted from 12 → result = 12 - x

The result is multiplied by 3 → (12 - x) x 3

The answer is 21 → (12 - x) x 3 = 21

Now solve for x

(12 - x) x 3 = 21

a(b - c) = a x b - a x c

(12 - x) x 3 = 12 x 3 - 3 x x

12 x 3 - 3 x x = 36 - 3x

36 - 3x = 21

Move the - 3x to the right side of the equation and 21 to the left

- 3x becomes 3x at the right (when negative crosses the equal to sign becomes positive

21 becomes -21 at the left

36 - 21 = 3x

15 = 3x

Divide both sides of the equation by 3 to make x the subject

15/3 = 3x/3

5 = x

∴ x = 5

You can verify

12 - 5 = 7 (Result)

7 x 3 = 21 (Answer)

You have to use the property of alternate angles of parallel lines

Alternate angles are equal:

Sometimes called Z angles

We can often spot interior alternate angles by drawing a Z shape

From the diagram above, angle ABC = angle CDE = x (Alternate angles)

Angle BAC = y = DEC (Alternate angles)

Angle ABC = angle CDE = x (Alternate angles)

Angle ABC = 81^o, ∴ x = 81^o

You have to use the property of alternate angles of parallel lines

Alternate angles are equal:

Sometimes called Z angles

We can often spot interior alternate angles by drawing a Z shape

From the diagram above, angle ABC = angle CDE = x (Alternate angles)

Angle BAC = y = DEC (Alternate angles)

Angle ABC = angle CDE = x (Alternate angles)

Angle ABC = 81^o, ∴ x = 81^o

Since we know the angles DCE and CDE, we can calculate the angle DEC using the concept that angles in a triangle add up to 180^o

x = 81^o and DCE = 20^o

Sum of angles in triangle add up to 180^o

81 + 20 + angle DEC = 180

101 + DEC = 180

DEC = 180 - 101

DEC = 79

Since BAC = y = DEC, y = 79^o

The Cardinal Point

We can draw the diagram below from the above information

We can apply pythagoras theorem to calculate how far she was from her starting point

NOTE: the c is the longest part or the side faceing the right angle (90^o)

We can redraw our diagram to:

Now solve for the value of x

Take the square root of both sides to get ride of the square

Whole numbers

Whole numbers is a collection of all the basic counting numbers and 0. In mathematics, counting numbers are called natural numbers. So, we can define the whole number as a collection of all natural numbers and 0. Whole numbers also include all positive integers along with zero.

Composite numbers

In math, composite numbers can be defined as numbers that have more than two factors. Numbers that are not prime are composite numbers because they are divisible by more than two numbers.

Natural numbers

Natural numbers are all positive integers from 1 to infinity. They are also called counting numbers as they are used to count objects. Natural numbers do not include 0 or negative numbers. We need numbers in our everyday life, be it for counting objects, telling time, or numbering houses.

Integers

Is a whole number (not a fractional number) that can be positive, negative, or zero.

As you can see from the definitions, an integer could be a whole number, could be a composite number or natural number. Whole numbers, composite numbers and natural numbers are all subset of integers

A standard form should be in the form a.bcdef x 10^g

Thus the decimal must be after the first non-zero number.

The power is the number of times you have to move the decimal point so that it's positioned after the first non-zero number

If you move to the left, the power is positive and if you move to the right, the power is negative

From 1930.54 you can see that the decimal must be after 1. We have to move the current decimal point three (3) times to the left in order to position it after the 1, hence the power is positive 3

1930.54 = 1.93054 x 10³

a x a = a²

- x - = +

-a x -a = +a²

(2a - b)(a - b) = 2a(a - b) - b(a - b)

2a(a - b) - b(a - b) = 2a x a - 2a x b - b x a + b x b

2a(a - b) - b(a - b) = 2a² - 2ab - ab + b²

2a(a - b) - b(a - b) = 2a² - 3ab + b²

Write the numbers in ascending order from the stem and leaf by joining the stem and each of the leaves. Thus 4|0 = 40, 4|5 = 45, 4|7 = 47, 4|9 = 49, 5|1 = 51, 5|3 = 53 etc.

40,45,47,47,51,53,54,55,57,58,60,62,63,64,64,64,64,65,66,68,71,72,73,74,75,78,78,79,82,83,85,86,89,94,95

Now count the numbers. You can as well count the leaves to know how many items are in the plot

When you count, you will get 35

Write the numbers in ascending order from the stem and leaf by joining the stem and each of the leaves. Thus 4|0 = 40, 4|5 = 45, 4|7 = 47, 4|9 = 49, 5|1 = 51, 5|3 = 53 etc.

40,45,47,47,51,53,54,55,57,58,60,62,63,64,64,64,64,65,66,68,71,72,73,74,75,78,78,79,82,83,85,86,89,94,95

The median is the middle number of a set of data.

From the data, 65 is the middle weight

Write the numbers in ascending order from the stem and leaf by joining the stem and each of the leaves. Thus 4|0 = 40, 4|5 = 45, 4|7 = 47, 4|9 = 49, 5|1 = 51, 5|3 = 53 etc.

40,45,47,47,51,53,54,55,57,58,60,62,63,64,64,64,64,65,66,68,71,72,73,74,75,78,78,79,82,83,85,86,89,94,95

The mode/modal is the number that occurs the most in a set of data.

From the weights, 64 occurs the most

Applying the addition rule

Now add a and b

Convert the 81 to 3^y, where y is the power that will produce 81

81 = 3 x 3 x 3 x 3

From the law of indices, a^b x a^c = a^b+c

81 = 3 x 3 x 3 x 3 = 3^1+1+1+1 = 3⁴

Now replace 81 by 3⁴ in the equation

3^x = 81

3^x = 3⁴

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

Hence x = 4

-(-) = -1 x - = +

+(-) = +1 x - = -

-15 - (-20) + (-10) = -15 + 20 - 10

-15 - (-20) + (-10) = -15 - 10 + 20

-15 - (-20) + (-10) = -25 + 20

-15 - (-20) + (-10) = -5

To convert the fraction to decimal write the numbers in the long division with 3 as the dividend and 8 as the divisor.

As 3 cannot be divided by 8, we can add 0 to 3 to make it 30 and add a decimal point in the quotient.

Continue the division, if you get a remainder that is smaller than 8, you can add 0 to it to make it divisible by 8.

Perform the division till you get the remainder 0 or till the remainder gets repeated.

Area of a square = Length x Length = L²

225 = L²

Square means a number multiplied by itself

What number when it multiply by itself will produce 225?

That will be 15, thus 15 x 15 = 225 or 15² = 225

Hence the length is 15.00 cm

Samples = {1,2,3,4,5,6}

Number of samples = 6

Samples of numbers greater than 5 = {6}

Number of samples of number greater than 5 = 1

Probability = 1/6

Since the number outside the bracket are the same in both sides of the equation, the values inside the bracket are also the same

Thus if a x b = a x c, implies that b = c

12+18 = 15+k

30 = 15 + k

30 - 15 = k

15 = k

Total pocket money given = 3 x 1.5 = GH₵ 4.5

Amount left = Initial Amount - Amount Given = 5 - 4.5 = GH₵ 0.5

Speed = Distance/Time

Since the unit is metres per second, convert the kilometres to meters and the hour to seconds

Kilo = 1000, kilometres = 1000 metres

36 kilometres = 36 000 metres

60 minutes = 1 hour

1 minute = 60 seconds

60 minutes = 60 x 60 seconds = 3600 seconds = 1 hour

Speed = 36 000m / 3600s = 10 ms^-1

Percentage = Amount/Total Amount x 100

Percentage left = Amount Left/Total Amount x 100

Amount Left = GH₵ 200.00 - GH₵ 83.00 = GH₵ 117

Percentage Left = 117/200 x 100% = 58.5%

Profit Percentage = Profit/Cost Price x 100

Selling Price = Cost Price + Profit

Profit = Selling Price - Cost Price

Let x = Cost Price

Profit = Selling Price - x

Selling Price = 44

Profit = 44 - x

Profit Percentage = Profit/Cost Price x 100

Profit Percent = 10%

Multiply both sides of the equations by x

10 x x = (44 - x) x 100

10 x x = 44 x 100 - x x 100

10x = 4400 - 100x

Solve for x

10x + 100x = 4400

110x = 4400

Divide both sides of the equation by 110

x = 40

Cost price = GH₵ 40.00

ALTERNATIVELY

Cost Price Percentage = 100%

Selling Price Percentage = Profit % + Cost Price Percentage

Profit % = 10%

Selling Price % = 10% + 100% = 110%

Selling Price = 44

Let x = Cost Price

110% (Selling Price Percentage) = 44

100% (Cost Price Percentage) = x

Cross multiply and solve for the value of x

x x 110 = 44 x 100

Divide both sides by 110

x = 4400/110

x = 40

Cost Price = GH₵ 40.00

Since the first part of the ratio 1 : n is 1, divide each parts of the ratio of 15 : 12 by 15 (15/15 = 1)

15/15 : 12/15 = 1:0.8

13 for farmers and 11 for children

Farmers = 312

Let x = number of children

13 = 312

11 = x

Cross multiply and solve for the value of x

13 x x = 312 x 11

Divide both sides by 13

x = (312 x 11)/13

13 goes into 312, 24 times

x = 24 x 11

x = 264

Volume of a cylinder = Πr²h

Volume = 40Π and height = 10

40Π = Πr² x 10

40Π = 10Πr²

Divide both sides by 10Π

4 = r²

Take square root of both sides

r = 2