Finite sets are sets having a finite or countable number of elements. It is also known as countable sets as the elements present in them can be counted.

As you can see, its only the set {3,6,9,12} which has countable number of elements, that is 4. The rest are infinite as the set elements number end or have boundaries.

Number of subsets = 2ⁿ, where n = number of elements.

M={a,b,c}

The number of elements in set M is 3

Number of subsets = 2³ = 2 x 2 x 2 = 8

"∩" means intersection. The set of intersection are elements that can be found in both sets.

2,3 and 4 can be found in both the sets P and Q.

Cost for the 3 pairs of socks = 3 x GH₵17.50 = GH₵52.5

The boy paid with two GH₵50.00 notes, hence amount given out is 2 x GH₵50.00 = GH₵100.00

Change = GH₵100.00 - GH₵52.5 = GH₵47.5

Step 1

Write down the prime factors

5 = 5
10 = 2 x 5
12 = 2 x 2 x 3 = 2² x 3

Step 2

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

5 = 5
2 = 2²
3 = 3

L.C.M of 5,10 and 12 = 2² x 3 x 5

Rounding to the nearest place value

Place value chart

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, etc.

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.

Step 3: Replace the digit in front of the place you are to round the number to by zeros (0s)

If there is a decimal part, it is ignored.

The digit in front of the hundreds is 4 which is less than 5, hence the digit at the hundredth place value is maintained and the tens and ones are replaced by 0s.

48,947.2547 = 48,900

Place the numbers in their place value and add

Note: The numbers in red are the carry-overs.

$\frac{4}{5}$ - $\frac{1}{3}$ + $\frac{2}{9}$

1. Find the L.C.M of the denominators to use as the general denominators for each of the fractions

The L.C.M for 5, 3 and 9 is 45

2. Multiply the numerators of each fractions by the number of times the denominator can divide the L.C.M (general denominator)

3. Simplify

$\frac{4}{5}$ - $\frac{1}{3}$ + $\frac{2}{9}$ = $\frac{\mathrm{9\; x\; 4\; -\; 15\; x\; 1\; +\; 5\; x\; 2}}{45}$

$\frac{4}{5}$ - $\frac{1}{3}$ + $\frac{2}{9}$ = $\frac{\mathrm{36\; -\; 15\; +\; 10}}{45}$

$\frac{4}{5}$ - $\frac{1}{3}$ + $\frac{2}{9}$ = $\frac{31}{45}$

The higher the negative value, the smaller (least) the number

Method I

Since 102 is common in both sides, you can factorize it out and simplify

(46 x 102) + (102 x 54) = 102(46 + 54)

46 + 54 = 100

102(46 + 54) = 102(100) = 102 x 100

When multiplying by a number with zeros at the end, multiply the non-zero numbers and join the zeros at the end to the result obtained.

102 x 100 = 102 x 1 and result joined by two zeros

102 x 100 = 10200 = 10,200

Method II

Multiply each in the bracket and add the result in the brackets.

46 x 102 = 4,692
102 x 54 = 5,508

(46 x 102) + (102 x 54) = 4,692 + 5,508 = 10,200

5178.3426

The second digit after the decimal, the hundredth is 4

The number after the hundredth is 2 which is less than 5, hence we maintain the hundredth value.

5178.3426 to two decimal places = 5178.34

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

Principal = GH₵120,000.00
Time = 5 months
Rate = 12

The simple interest is per annum (year), hence convert the months to years by dividing by 12 months

5 months = $\frac{5}{12}$ annum

Simple Interest = $\frac{\mathrm{120000\; x\; 5\; x\; 12}}{\mathrm{100\; x\; 12}}$

The denominator 12 in $\frac{5}{12}$ moves download

Cancel out

1. The zeros in 100 cancels out the zeros in 120000
2. 12 cancels each other out

Simple Interest = 1200 x 5 = GH₵6000

The more the workers, the lesser time it takes to complete the work

Therefore number of workers is inversely proportional to time.

Let w = number of workers and t = time

w ∝ $\frac{1}{\mathrm{t}}$

Remove the proportionality by introducing equal to and a constant

Let k = contant

w = k x $\frac{1}{\mathrm{t}}$

w = $\frac{\mathrm{k}}{\mathrm{t}}$

Since we know the number of workers and time taken, we can calculate for the value of k

w = 15
t = 12

15 = $\frac{\mathrm{k}}{12}$

Multiply both sides by 12 to get rid of the denominator.

15 x 12 = k

The general equation is therefore w = $\frac{\mathrm{15\; x\; 12}}{\mathrm{t}}$

Now we are given the number of workers to find time.

Number of workers (w) = 9

9 = $\frac{\mathrm{15\; x\; 12}}{\mathrm{t}}$

Multiply both sides by t to get rid of the denominator

9 x t = 15 x 12

Solve for the value of t by dividing both sides by 9

t = $\frac{\mathrm{15\; x\; 12}}{9}$

t = 5 x 4 = 20

Note: 3 divides 12, 4 times and 9, 3 times. 3 divides itself once and 15, 5 times

Time taken for the 9 boys is 20 hours.

Discount = 9% of the cost

Percentage means over 100

9% = $\frac{9}{100}$

of means multiplication

Discount = $\frac{9}{100}$ x 12500 = 9 x 125

Note: The zeros cancel each other

Discount = 1125

Amount paid = cost - discount

Amount paid = 12500 - 1125
Amount paid = GH₵11,375

Subtraction

Note: When one (1) is borrowed from 5, it is 10 and when one is also borrowed from it (the 10), it becomes nine (9). Hence 9 - 2 gives 7

The cost when payment is made by cash = GH₵11,375

From law of indices, when the bases are the same and multiplying, you add the powers.

a^m x aⁿ = a^{m + n}

Note: Every number has a power of 1 which is not written.

2 = 2¹

5² x 2² x 5² x 2 = 5² x 2² x 5² x 2¹

5² x 2² x 5² x 2 = 2²⁺¹ x 5²⁺²

5² x 2² x 5² x 2 = 2³ x 5⁴

The modal mark is the mark which appeared/occurred the most or the mark with the highest frequency

The mark 6 has frequency of 7 which is the highest frequency, hence the modal mark is 6

Add the number of students who got mark less than 4 (students who got the 0,1,2,3 mark)

Number of students who failed = 3+4+5+4 = 16

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

A die has the possible numbers 1,2,3,4,5 and 6

Total samples = 6

There is only one (1) 4, hence the number of selection = 1

Probability of obtaining 4 = $\frac{1}{6}$

Get rid of the denominator by multiplying both sides of the equation by 2

2 x R = P + Q

2R - Q = P

t = p² + 1

t = 10

10 = p² + 1

10 - 1 = p²

9 = p²

Square of a number is a number multiplied by itself.

9 = 3 x 3 = 3²

3² = p²

Since the powers are the same (2), the bases too must be the same.

p = 3

Expand the brackets and simplify

4(x + 2)-3(x +1) = 4 x x + 4 x 2 -3 x x -3 x 1

4(x + 2)-3(x +1) = 4x + 8 -3x -3

4(x + 2)-3(x +1) = 4x-3x + 8-3

4(x + 2)-3(x +1) = x + 5

Let x = the number

Doubled means 2 times the number

Doubled = 2 x x = 2x

The result (2x) is decreased by 9, means 9 is subtracted from the result

2x - 9

The answer is 19.

2x - 9 = 19

Solve for x

2x = 19+9

2x = 28

Divide both sides by 2

$\frac{\mathrm{2x}}{2}$ = $\frac{28}{2}$

x = 14

Get rid of the fraction by multiplying both sides of the equation by the denominator 2

2 x 2x + 2 x 10 ≥ 7x - 2 x 5

4x + 20 ≥ 7x - 10

4x - 7x ≥ -10 -20

-3x ≥ -30

Divide both sides by -3. When dividing by a negative, the sign changes direction

$\frac{\mathrm{-3x}}{-3}$ ≤ $\frac{-30}{-3}$

x ≤ 10

Substitute 5 into the rule of the mapping.

Image of 5 = 4(5) - 7
Image of 5 = 4 x 5 - 7
Image of 5 = 20 -7
Image of 5 = 13

TYPES OF ANGLES

LINES OF SYMMETRY

Perimeter is the sum of all the sides of a shape.

A triangle has three sides.

Isosceles triangle has two of the sides equal.

Let x = third side

Equal side = 14 cm

Perimeter = 14 cm + 14 cm + x cm

Perimeter = 28 cm + x cm

But perimeter = 45 cm

45cm = 28 cm + x cm

45 - 28 = x

17 = x

The third side is 17 cm

Area of a circle = πr²

Area of a circle = π x r x r

Radius is half of the diameter

Radius = $\frac{7}{2}$

π = $\frac{22}{7}$

Area of circle = $\frac{22}{7}$ x $\frac{7}{2}$ x $\frac{7}{2}$

Note: 2 divides itself once and 22, 11 times and the 7 at the denominator cancels one of the 7s at the numerator

Area of circle = $\frac{\mathrm{11\; x\; 7}}{2}$ = $\frac{77}{2}$ = 38$\frac{1}{2}$

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

Let x = the third number.

Sum of the numbers = 14 + 16 + x

Sum of the numbers = 30 + x

Number of numbers = 3

Mean = $\frac{\mathrm{30\; +\; x}}{3}$

But mean = 12

12 = $\frac{\mathrm{30\; +\; x}}{3}$

Get rid of the fraction by multiplying both sides of the equation by the denominator 3

12 x 3 = 30 + x

36 = 30 + x

36 - 30 = x

6 = x

Sum of interior angles of a regular polygon = (n-2)x 180^o

Where n = number of sides of the polygon.

Sum of interior angles of a regular polygon = 540

540 = (n-2)x 180

Divide both sides by 180

$\frac{540}{180}$ = $\frac{\mathrm{(n-2)}}{180}$ x 180

3 = n-2

3 + 2 = n

5 = n

Concept 1

Sum of angles on a straight line is 180^o

Angle SRP + Angle PRS = 180

Angle SRP + 2x-10 = 180

Angle SRP = 180 - 2x + 10

Angle SRP = 180 + 10 - 2x

Angle SRP = 190 - 2x

Concept 2

The interior angles of equilateral triangle are the same.

Angle SRP = Angle RPQ = Angle PQR = 190 - 2x

Concept 3

The sum of interior angles of a triangle gives 180^o

Angle SRP + Angle RPQ + Angle PQR = 180

(190 - 2x) + (190 - 2x) + (190 - 2x) = 180

3 x (190 - 2x) = 180

Divide both sides by 3

190 - 2x = $\frac{180}{3}$

190-2x = 60

-2x = 60 - 190

-2x = - 130

Divide both sides by -2

$\frac{\mathrm{-2x}}{-2}$ = $\frac{-130}{-2}$

x = 65

Since there is right angle (90) degrees, pythagoras theorem could be applied.

The side facing the right angle (90) degrees is the hypotenuse (c).

10² = 8² + b²

10 x 10 = 8 x 8 + b²

100 = 64 + b²

100 - 64 = b²

36 = b²

Square is a number multiplying itself

6 x 6 = b²

6² = b²

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

6 = b

The breadth = 6 cm.

Enlargement Scale factor = $\frac{\mathrm{Image\; Length}}{\mathrm{Object\; Length}}$

Scale factor = $\frac{\mathrm{8\; cm}}{\mathrm{24\; cm}}$ = $\frac{1}{3}$

Series of odd numbers are obtained by adding 2 to an odd number to get its consecutive odd number.

In the first row, the odd numbers are obtained as follows:

13, 13+2 = 15, 15+2 = 17, 17+2 = 19

b = 15 and c = 17

In the second row, the odd numbers are obtained as follows:

7, 7+2 = 9, 9+2 = 11

a = 11

13 + b + c + 19 = 13 + 15 + 17 + 19 = 64

Series of odd numbers are obtained by adding 2 to an odd number to get its consecutive odd number.

In the first row, the odd numbers are obtained as follows:

13, 13+2 = 15, 15+2 = 17, 17+2 = 19

b = 15 and c = 17

In the second row, the odd numbers are obtained as follows:

7, 7+2 = 9, 9+2 = 11

a = 11

a + b + c = 11 + 15 + 17 = 43

Addition of a vector

$\left(\begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{c}\\ \mathrm{d}\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{a\; +\; c}\\ \mathrm{b\; +\; d}\end{array}\right)$

$\left(\begin{array}{c}-3\\ 5\end{array}\right)$ + $\left(\begin{array}{c}2\\ -7\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{-3+2}\\ \mathrm{5+-7}\end{array}\right)$ = $\left(\begin{array}{c}-1\\ -2\end{array}\right)$

Substitute the values and simplify

a = -4 and b = 3

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{\mathrm{3\; x\; (-4)\; +\; 2\; x\; (3)}}{\mathrm{(-4)\; x\; (3)}}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{\mathrm{-12\; +\; 6}}{-12}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{-6}{-12}$

$\frac{\mathrm{3a\; +\; 2b}}{\mathrm{ab}}$ = $\frac{1}{2}$

x-axis as mirror/reflection in x-asis

$\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\end{array}\right)$ → $\left(\begin{array}{c}\mathrm{x}\\ \mathrm{-y}\end{array}\right)$

Thus the y is negated.

$\left(\begin{array}{c}-3\\ 7\end{array}\right)$ → $\left(\begin{array}{c}-3\\ -7\end{array}\right)$ = (-3,-7)

Geometric Tools and Their Uses

Pair of compasses

It can be used to draw circles and trace arcs. The distance between the pointer and the pencil is adjustable and can be changed to line up with measurements on a ruler, allowing us to create circles with specific dimensions.

Set square

A set square is a piece of equipment that can be used to draw lines and shapes. It’s a great way to see whether a shape has a right angle, as a set square always has a corner with 90 degrees.

Protractor

The protractor is a semicircular disk that is used to measure the angles of shapes. It is graduated from zero to 180 degrees and therefore has the ability to measure the angle of any two intersecting lines or vertices within this range.

Pair of dividers

It looks similar to the compasses with its ‘V’-shaped structure. However, it has pointers on both ends of the ‘V’. The distance between them is adjustable and it is used to measure and compare lengths.