Starting zeros are not significant and so the number of significant digits starts from 24561. The 3 digits lands on 5 but the digit next to it is more than 5 (6) and so 1 is added to the 5 to make it 6.

NOTE
Zeros between other digits are significant. Only starting zeros are insignificant. For instance 3.052306, all the zeros are significant because they are not starting zeros after a decimal point.

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

Number of students = 600

Number of boys = 600 - 120 = 480

Probability = $\frac{480}{600}$ = $\frac{4}{5}$

Note
1. the zeros (0) at the end of 480 and 600 cancel each other
2. 12 divides 48, 4 times and 60, 5 times

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

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

The perimeter of the equilateral triangle = 16 cm + 16 cm + 16 cm = 48 cm

A square has the same perimeter.

Square has 4 equal sides.

Let L = side of the square.

The perimeter of the square = L + L + L + L = 4L

4L = 48

Divide both sides by 4.

L = $\frac{48}{4}$ = 12

The length of the square = 12 cm

Area of a square = Length x Length
Area of the square = 12 cm x 12 cm = 144 cm²

Concepts

1. a^x x a^y = a^x+y, thus if the bases are the same and multiplying, add the powers

2. (a^xx b^y)ⁿ = a^{x x n} x b^{y x n}, thus the outside power multiplies all the powers inside

Applying these two concepts,

(2ab²)² = 2^1x2a^1x2b^2x2 = 2²a²b⁴ = 4a²b⁴

NOTE: Every number or letter is raised to the power 1, which has no effect, hence 2 = 2¹ and a = a¹

⇒ (2ab²)² x 3a³b = 4a²b⁴ x 3a³b

Simplifying the numbers and the ones with the same bases

⇒ 4a²b⁴ x 3a³b = 4 x 3 x a²xa³b⁴xb

Applying concept 1,a^x x a^y = a^x+y

⇒ 4 x 3 x a²xa³b⁴xb = 12a²⁺³b⁴⁺¹ = 12a⁵b⁵

NOTE: b = b¹ ⇒ b⁴xb = b⁴xb¹ = b⁴⁺¹ = b⁵

Odd number is a number not divisible by 2.

P = {odd numbers between 20 and 30}

P = {21, 23, 25, 27, 29}

Q = {23, 29}

⊂ stands for subset.

A subset is a set with all its members found in the set it is a subset to.

All the members of Q can be found in P. Hence Q is a subset of P.

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.

Gradient = $\frac{\mathrm{Change\; in\; Y}}{\mathrm{Change\; in\; X}}$ = $\frac{{\mathrm{Y}}_2-{\mathrm{Y}}_1}{{\mathrm{X}}_2-{\mathrm{X}}_1}$

y₂ = 8
x₂ = 4
y₁ = 2
x₁ = 3

Gradient = $\frac{\mathrm{8\; -\; 2}}{\mathrm{4\; -\; 3}}$

Gradient = $\frac{6}{1}$

Gradient = 6

n = 2d + 3

n - 3 = 2d

Divide both sides of the equation by 2

$\frac{\mathrm{n\; -\; 3}}{2}$ = d

d = $\frac{\mathrm{n\; -\; 3}}{2}$

Respectively means the ratios are in the order of the names mentioned.

The sum of the ratios represents the total amount shared.

Sum of ratios = 1 + 4 + 3 = 8

Atsu's ratio is 4

If 8 = ₵500000

4 = $\frac{\mathrm{₵500000\; x\; 4}}{8}$ = $\frac{\mathrm{₵500000}}{2}$ = ₵250,000

Note:
1. 4 divides itself 1 time and 8, 2 times
2. 2 divides itself 1 time and 50, 25 times

Volume of a cylinder = πr²h
r² = r x r

Radius = 2 cm
Height = 3 cm

Volume of a cylinder = π x 2 cm x 2 cm x 3 cm
Volume of a cylinder = π x 12 cm³
Volume of a cylinder = 12π cm³

Sum of observations (numbers) = 5+6+7+x

Total number of observations (numbers) = how many numbers are in the observation. In this case 4, thus (5,6,7 and x)

Every number is divisible by 1

Re-write the equation

Simplify like terms and cross multiply to solve for the value of x

1 x (18+x) = 4 x 8

1 times any number is the same number, hence 1 x (18+x) = 18+x

4 x 8 = 32

18+x = 32

x = 32 - 18

x = 14

Intersection (∩) are elements you can find in both sets.

n in n(A∩B) means the number of elements. So you count how many elements are in A∩B once you are done finding the elements.

A = {1, 3, 5, 7, 9, 11, 13, 15}
B = {3, 6, 9, 12, 15}

As you can see 3, 9 and 15 are the only elements that can be seen in both the sets A and B

Hence A∩B = {3,9,15}

Since you are asked how many elements (n), you count how many elements are in {3,9,15} and I believe you've counted three

Hence n(A∩B) = 3

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

Profit = Selling Price - Cost Price

Profit = 1.3 - 1 = 0.3

Profit % = $\frac{0.3}{1}$ x 100 = 30%

Note: 0.3 x 100 = 0.⌒3⌒0. = 30

4(m + 4) = -8

Expand the bracket by multiplying each expression in the bracket by 4

4 x m + 4 x 4 = -8

4 x m + 16 = -8

4 x m = -8 - 16

4 x m = -24

Divide both sides of the equation by 4

m = $\frac{-24}{4}$ = -6

$\frac{1}{3}$ + $\frac{1}{9}$ + $\frac{1}{27}$ = $\frac{\mathrm{1\; x\; 9\; +\; 1\; x\; 3\; +\; 1\; x\; 1}}{27}$

$\frac{1}{3}$ + $\frac{1}{9}$ + $\frac{1}{27}$ = $\frac{\mathrm{9\; +\; 3\; +\; 1}}{27}$

$\frac{1}{3}$ + $\frac{1}{9}$ + $\frac{1}{27}$ = $\frac{13}{27}$

The region outside the two circles represent pupils who don't speak any of the two languages (neither Twi nor Ga)

Number of samples = 40

Number of sticks not spoilt = 40 - 15 = 25

Alternatively

Probability of not = 1 - Probability of is

Probability of is = 1 - Probability of not

Thus when you sum up probability and it's negative, the result should be 1

Now we can substract the probability of the spoilt from 1 to get the probability of not spoilt

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

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

$\left(\begin{array}{c}3\\ 4\end{array}\right)$ + $\left(\begin{array}{c}\mathrm{3k}\\ \mathrm{4k}\end{array}\right)$ = $\left(\begin{array}{c}-3\\ -4\end{array}\right)$

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

The top at the left equals the top at the right and the down at the left equals the down at the right.

3 + 3k = -3

3k = -3 - 3

3k = -6

Divide both sides by 3

k = $\frac{-6}{3}$ = -2

Alternatively using the down,

4 + 4k = -4

4k = -4 - 4

4k = -8

Divide both sides by 4

k = $\frac{-8}{4}$ = -2

7:55 = 7 hours 55 minutes
4:15 = 4 hours 15 minutes

Using the 24 hour system. Afternoon is 12 hours.

4:15 p.m = 12hr + 4hr + 15 min

4:15 p.m = 16hr + 15 min

Hours difference = 16 - 7 = 9 hours
Minutes difference = 15 - 55 = -40 minutes

1 hour = 60 minutes

9 hours = 8 hours + 60 minutes

Total time = 9 hours - 40 minutes
Total time = 8 hours + 60 minutes - 40 minutes
Total time = 8 hours 20 minutes
Total time = 8 hr 20 min

Point + Translation vector = Image point

Point = (4, 5) = $\left(\begin{array}{c}4\\ 5\end{array}\right)$

Image = (3, 1) = $\left(\begin{array}{c}3\\ 1\end{array}\right)$

Let v = Translation vector

$\left(\begin{array}{c}4\\ 5\end{array}\right)$ + v = $\left(\begin{array}{c}3\\ 1\end{array}\right)$

v = $\left(\begin{array}{c}3\\ 1\end{array}\right)$ - $\left(\begin{array}{c}4\\ 5\end{array}\right)$

v = $\left(\begin{array}{c}\mathrm{3\; -\; 4}\\ \mathrm{1\; -\; 5}\end{array}\right)$

v = $\left(\begin{array}{c}-1\\ -4\end{array}\right)$

Sum of angles on a straight line is 180^o

The arc D bisects the 180^o into 2 producing 90^o

The arc C also bisects the 90^o into 2 producing 45^o

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.

356.07 = 3.⌒5⌒6.07 = 3.5607 x 10²

Volume = length x width x height

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

Volume = 4 m x 3 m x $\frac{25}{10}$ m = 2 m x 3 m x 5 m = 30 m³

Note:
1. 2 divides 4, 2 times and 10, 5 times
2. 5 divides itself 1 time and 25, 5 times

Sum of angles on a straight line is 180

65 + y = 180
y = 180 - 65
y = 115

Percentage = $\frac{\mathrm{Number}}{\mathrm{Total}}$ x 100%

Number of oranges left = 100 - 80 = 20

Total oranges = 100

Percentage = $\frac{20}{100}$ x 100% = 20%

Faces of a cuboid

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.

4687.02 = 4.⌒6⌒8⌒7.02 = 4.68702 x 10³

Prime factors of 72

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

Prime factors of 72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

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

Let x = number of girls in the school
The number of girls exceeds the number of boys by 150 → number of boys = x - 150

Number of girls + Number of boys = 940
x + x - 150 = 940
2x = 940 + 150
2x = 1090

Divide both sides by 2 to solve for x

x = $\frac{1090}{2}$ = 545

There are 545 girls in the school.

1090 divided by 2

Triangle PRQ is isosceles triangle. The base angles are the same.

Angle RPQ = Angle RQP = 46°

Angle RPQ + Angle RQP + b = 180°(sum of interior angles of a triangle is 180°)

46° + 46° + b = 180°

92° + b = 180°

b = 180° - 92°

b = 88°

-(-) = -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

y : 28 = 5 : 7

Change the ratios to fraction format.

$\frac{\mathrm{y}}{28}$ = $\frac{5}{7}$

Cross multiply.

y x 7 = 28 x 5

Divide both sides by 7

y = $\frac{\mathrm{28\; x\; 5}}{7}$

Note: 7 divides itself 1 and 28, 4 times.

y = 4 x 5
y = 20

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

Multiplying a vector by a number

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

Addition of vectors

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

Subtraction of vectors

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

r = $\left(\begin{array}{c}-3\\ 4\end{array}\right)$ and s = $\left(\begin{array}{c}1\\ -3\end{array}\right)$

r - 2s = $\left(\begin{array}{c}-3\\ 4\end{array}\right)$ - 2$\left(\begin{array}{c}1\\ -3\end{array}\right)$

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

r - 2s = $\left(\begin{array}{c}-3\\ 4\end{array}\right)$ - $\left(\begin{array}{c}2\\ -6\end{array}\right)$

r - 2s = $\left(\begin{array}{c}\mathrm{-3\; -\; 2}\\ \mathrm{4\; -\; -6}\end{array}\right)$

-- = +

r - 2s = $\left(\begin{array}{c}-5\\ \mathrm{4\; +\; 6}\end{array}\right)$

r - 2s = $\left(\begin{array}{c}-5\\ 10\end{array}\right)$

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$\frac{1}{2}$

The Lowest Common Multiple (LCM) is the multiplication of each of the prime factors in their highest powers.

The prime factors are 2, 3 and 5

The highest power for the prime number 2 is 3
The highest power for the prime number 5 is 2
The highest power for the prime number 3 is 2

Lowest Common Multiple (LCM) = 2³ x 3² x 5²