A prime number is a number that has only two factors: 1 and the number itself.

Finding prime factors

1. Keep on dividing with prime number until you get 1

2. Multiply each of the prime numbers used in the division

Finding prime factors of 30

$$\begin{array}{cc}2& 30\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors of 30 = 2 x 3 x 5

Set of prime factors of 30 = {2,3,5}

Number of subsets = 2ⁿ

Where n = number of elements of the set

The set P has 4 elements.

n = 4

Number of subsets = 2⁴ = 2 x 2 x 2 x 2 = 16

M = {Multiples of 4 between 10 and 25}

M = {12,16,20,24}

N = {even numbers between 11 and 23}

N = {12,14,16,18,20,22}

Union (∪) is the list of each of the elements once in both sets.

M∪N = {12,14,16,18,20,22,24}

Prime factors of 24

$$\begin{array}{cc}2& 24\\ 2& 12\\ 2& 6\\ 3& 3\\ & 1\end{array}$$

Prime factors = 2 x 2 x 2 x 3 = 2³ x 3

Prime factors of 42

$$\begin{array}{cc}2& 42\\ 3& 21\\ 7& 7\\ & 1\end{array}$$

Prime factors = 2 x 3 x 7

Prime factors of 72

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

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

Common prime factors are 2 and 3

The H.C.F is the product of each of the common prime factors in their lowest power = 2 x 3 = 6

120₅ = 1 x 5² + 2 x 5¹ + 0 x 5⁰

120₅ = 1 x 25 + 2 x 5 + 0 x 1

120₅ = 25 + 10 + 0

120₅ = 35

p x q x r = 1197

Make r the subject.

Divide both sides of the equation by p x q

$\frac{\mathrm{p\; x\; q\; x\; r}}{\mathrm{p\; x\; q}}$ = $\frac{1197}{\mathrm{p\; x\; q}}$

r = $\frac{1197}{\mathrm{p\; x\; q}}$

But p = 19 and q = 3

r = $\frac{1197}{\mathrm{19\; x\; 3}}$ = 21

Note: 19 divides 1197, 63 times and 3 divides 63, 21 times.

-4,-3,-2,-1,0,1,2,3,4,5,6

Change the decimals to the form whole number x 10^-n

Where n is the number of times you have to move the decimal point to get a whole number.

1.612 = 1612 x 10^-3
0.4 = 4 x 10^-1

$\frac{1.612}{0.4}$ = $\frac{\mathrm{1612\; x}{10}^{-3}}{\mathrm{4\; x}{10}^{-1}}$ = 403 x 10^-3-(-1) = 403 x 10^-3+1 = 403 x 10^-2

Note: from law of indices, $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

4 divides itself once and 1612, 403 times.

Change the format back to decimal by moving to the left, the number of times of the power (2)

403 = 403.0

403.0 x 10^-2 = 4.03

Find their equivalent fraction with the denominator being the L.C.M of the denominator of the fractions (20,8,10)

Step 1

Find the L.C.M of the denominators.

The L.C.M is 40

Step 2

Multiply both the numerator and the denominator of each fraction by the number of times the denominator divides the L.C.M

$\frac{5}{8}$ = $\frac{\mathrm{5\; x\; 5}}{\mathrm{8\; x\; 5}}$ = $\frac{25}{40}$

$\frac{11}{20}$ = $\frac{\mathrm{11\; x\; 2}}{\mathrm{20\; x\; 2}}$ = $\frac{22}{40}$

$\frac{7}{10}$ = $\frac{\mathrm{7\; x\; 4}}{\mathrm{10\; x\; 4}}$ = $\frac{28}{40}$

Step 3

Since the equivalent fractions have the same denominator, order them using the value of the numerator.

Ascending means from lowest to highest

Order 25, 22 and 28 in ascending order

22,25 and 28

Hence the order of the equivalent fractions in ascending order is:

$\frac{22}{40}$, $\frac{25}{40}$, $\frac{28}{40}$

Step 4

Replace the equivalent fractions by their original fractions.

$\frac{22}{40}$, $\frac{25}{40}$, $\frac{28}{40}$ = $\frac{11}{20}$,$\frac{5}{8}$, $\frac{7}{10}$

Summation of all fraction parts should give the whole (1)

When you sum known fractions and you subtract the result from 1, you will get the rest of the fraction.

Total known fraction spent = $\frac{1}{5}$ + $\frac{4}{7}$ = $\frac{\mathrm{7\; x\; 1\; +\; 5\; x\; 4}}{35}$ = $\frac{\mathrm{7\; +\; 20}}{35}$ = $\frac{27}{35}$

Fraction spent on Gari = 1 - $\frac{27}{35}$

Every number is divisible by 1.

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

Fraction spent on Gari = 1 - $\frac{27}{35}$ = $\frac{1}{1}$ - $\frac{27}{35}$ = $\frac{\mathrm{35\; x\; 1\; -\; 1\; x\; 27}}{35}$ = $\frac{\mathrm{35\; -\; 27}}{35}$ = $\frac{8}{35}$

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

Therefore number of workers is inversely proportional to days to complete work.

Let w = number of workers and d = days

w ∝ $\frac{1}{\mathrm{d}}$

Remove the proportionality by introducing equal to and a constant

Let k = contant

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

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

Since we know the number of workers and days taken to complete the work, we can calculate for the value of k

w = 5 boys
d = 14 days

5 = $\frac{\mathrm{k}}{14}$

Multiply both sides by 14 to get rid of the denominator.

5 x 14 = k

The general equation is therefore w = $\frac{\mathrm{5\; x\; 14}}{\mathrm{d}}$

Now we are given the number of workers to find number of days to complete the work.

Number of workers (w) = 7

7 = $\frac{\mathrm{5\; x\; 14}}{\mathrm{d}}$

Multiply both sides by d to get rid of the denominator

7 x d = 5 x 14

Solve for the value of d by dividing both sides by 7

d = $\frac{\mathrm{5\; x\; 14}}{7}$

d = 5 x 2 = 10 days

Note: 7 divides itself once and 14, 2 times

Days taken for the 7 boys to complete work is 10 days.

Total Amount = Principal Amount + Simple Interest

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

Principal = GH₵800.00
Time = 1
Rate = 5

Simple Interest = $\frac{\mathrm{800\; x\; 1\; x\; 5}}{100}$ = 8 x 1 x 5 = 40

Total Amount = 800 + 40 = GH₵840.00

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

Profit = Selling Price - Cost Price

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

Let c = cost price

Selling price = 60000

Profit % = 20

20 = $\frac{\mathrm{60000\; -\; c}}{\mathrm{c}}$ x 100

Get rid of the fraction by multiplying both sides of the equation by the denominator (c)

c x 20 = (60000 - c) x 100

20c = 60000 x 100 - 100 x c

20c + 100 x c = 60000 x 100

120c = 60000 x 100

Divide both sides by 120

c = $\frac{\mathrm{60000\; x\; 100}}{120}$

c = 50000

Cost price = GH₵50,000

Alternate Solution

The cost price is 100% and the selling price is 100%+Profit %

Once you know each of the percentages, you can use proportionality to calculate the other.

Selling Price % = 100 + 20 = 120%

120 = 60000
100 = $\frac{\mathrm{60000\; x\; 100}}{120}$ = 50000

Express the 1000 in base 10 to a power so that you can compare the power with x

1000 = 10 x 10 x 10 = 10³

10^x = 10³

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

x = 3

For standard form, the decimal point must be after the first non-zero digit

In 625.13, the first non-zero digit is 6, hence we must move the decimal point to the front of 6

Step 1

Count while moving the decimal point.

6.⌒2⌒5.13 = 6.2513 with two moves

Step 2

Multiply the new decimal by 10ⁿ

Where n is the number of times you have to move the decimal point

If you move the decimal point to the left, n is positive but if you move to the right n is negative.

In 625.13, we have to move the decimal point two times to the left, hence n = 2

Standard form of 625.13 = 6.2513 x 10²

Median is the middle number in an ascending ordered (lowest to highest) list.

The numbers in the above question are not ordered so you must first order it in ascending order and select the middle number.

8,12,13,14,15,16,17,18

If there are two numbers in the middle, add the numbers and divide the result by 2 to obtain the median.

In the ordered list, 14 and 15 are in the middle, hence we add 14 and 15 and divide by 2 to obtain the median.

Median = $\frac{\mathrm{14\; +\; 15}}{2}$ = $\frac{29}{2}$ = 14.5

Calculation

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

The ages not less than 5 years old are 5,5,5 and 6

Number of possible selection = 4 (4 children with ages not less than 5)

Total number of samples = 10 children

Probability = $\frac{4}{10}$ = $\frac{2}{5}$

Note: 2 divides 4, 2 times and 10, 5 times

Take each part of the expression in the first bracket to multiply each part in the second bracket and simplify the expressions.

(2x + y)(2x - y) = 2x(2x - y) + y(2x - y)

(2x + y)(2x - y) = 4x² - 2xy + 2xy - y²

(2x + y)(2x - y) = 4x² + 0 - y²

(2x + y)(2x - y) = 4x² - y²

The 3 x n is in the place of the decimal place value.

3 x n = 0.003

Divide both sides of the equation by 3

n =$\frac{0.003}{3}$

0.003 = 0.001 x 3

n = $\frac{\mathrm{0.001\; x\; 3}}{3}$

n = 0.001

Alternate Solution

You can as well solve for the value of n in the equation and obtain the same answer

25.003 = (2 x 10)+(5 x 1) + (3 x n)

2 x 10 = 20
5 x 1 = 5

25.003 = 20 + 5 + (3 x n)

25.003 = 25 + (3 x n)

25.003 - 25 = (3 x n)

0.003 = 3 x n

Divide both sides of the equation by 3

$\frac{0.003}{3}$ = n

0.003 = 0.001 x 3

$\frac{\mathrm{0.001\; x\; 3}}{3}$ = n

0.001 = n

Substitute the value of m into the equation and simplify.

(3m)² - 3m²
m = 2

(3m)² - 3m² = (3 x 2)² - 3 x 2²

(3m)² - 3m² = (6)² - 3 x 2 x 2

(3m)² - 3m² = 6² - 3 x 4

(3m)² - 3m² = 6 x 6 - 12

(3m)² - 3m² = 36 - 12

(3m)² - 3m² = 24

Discount amount = Discount % x Cost Price

Percentage (%) means over 100

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

Discount on each watches = $\frac{2}{100}$ x GH₵2000 = GH₵40.00

Discount on the 20 watches = GH₵40.00 x 20 = GH₵800

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

Analyzing the x and its corresponding y, you will realize that the ys are obtained by squaring the xs.

Hence y → x²

You could as well test each of the rules by substituting the value of x into it and see if you will obtain its corresponding y.

Testing y → x + 2

When x = 1, y = 1+2 = 3

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → x + 2 is not the rule.

Testing y → 2x

When x = 1, y → 2 x 1 = 2

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → 2x is not the rule.

Testing y → x²

When x = 1, y = 1² = 1
When x = 2, y = 2² = 4
When x = 3, y = 3² = 9
When x = 4, y = 4² = 16
When x = 5, y = 5² = 25

As you can see, y → x² is the rule as each of the substituted x gave its corresponding y value.

You can still verify the last rule.

Testing y → 2x + 2

When x = 1, y = 2(1) + 2 = 4

The result obtained isn't the value in the mapping. The expected value for x = 1 is 1. Hence y → 2x + 2 is not the rule.

Vertex is a point where two or more edges meet. Vertices are the corner points.

Vertices of a cuboid

Area of a circle = πr²

Where r = radius of the circle.

Circumference of a circle = 2πr

Circumference = 440 m

440 m = 2πr

π = $\frac{22}{7}$

440 m = 2 x $\frac{22}{7}$ x r

Solve for the value of r radius which is needed to calculate for the area of the circle.

Get rid of the fraction by multiplying both sides of the equation by the denominator (7).

440 x 7 = 2 x 22 x r

Divide both sides of the equation by 2 x 22 to make r the subject

r = $\frac{\mathrm{440\; x\; 7}}{\mathrm{2\; x\; 22}}$

22 divides 440, 20 times and itself once. 2 divides itself once and 20, 10 times.

r = 10 x 7

Substitute the value of r in the area equation to calculate for the area

Area = πr²

r² = r x r

Area = π x r x r

π = $\frac{22}{7}$

r = 10 x 7

Area = $\frac{22}{7}$ x 10 x 7 x 10 x 7

Note: The denominator 7 cancels one of the 7s

Area = 22 x 10 x 7 x 10

Area = 15,400 m²

Classification of triangles

The entire cycle of the 12 hours is 360^o

Each hour angle = $\frac{360}{12}$ = 30^o

At 8 O'clock, the minute hand is on the 12 and the hour hand is on the 8.

The hourly intervals between them is 4 (11,10,9 and 8)

Hence the angle between the hour and the minute hands = 4 x 30^o = 120^o.

The fencing is the perimeter of the rectangular field.

Perimeter is the summation (addition) of the length of the sides of a shape

Rectangle has 4 sides.

The perimeter of the rectangular field = 2 x 50 + 2 x y

But the perimeter is given as 260 m

260 = 2 x 50 + 2 x y

Solve for the value of y

260 = 100 + 2 x y

260 - 100 = 2 x y

160 = 2 x y

Divide both sides by 2

y = $\frac{160}{2}$ = 80

When a point moves in a plane according to some given conditions the path along which it moves is called a locus.

Locus Theorem

Locus Theorem 1: The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius.

Locus Theorem 2: The locus of the points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.

Locus Theorem 3: The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.

Locus Theorem 4: The locus of points equidistant from two parallel lines, l1 and l2, is a line parallel to both l1 and l2 and midway between them.

Locus Theorem 5: The locus of points equidistant from two intersecting lines, l1 and l2, is a pair of bisectors that bisect the angles formed by l1 and l2.

Anticlockwise rotation through 90^o about the origin

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

Thus the cordinates are switched and x cordinate is negated.

$\left(\begin{array}{c}1\\ 5\end{array}\right)$ → $\left(\begin{array}{c}-5\\ 1\end{array}\right)$ = (-5,1)

Total angle = 90^o + 90^o + 90^o = 270^o

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

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

2v - u = 2$\left(\begin{array}{c}2\\ -3\end{array}\right)$ - $\left(\begin{array}{c}-3\\ 5\end{array}\right)$

2v - u = $\left(\begin{array}{c}\mathrm{2\; x\; 2}\\ \mathrm{-3\; x\; 2}\end{array}\right)$ - $\left(\begin{array}{c}-3\\ 5\end{array}\right)$

2v - u = $\left(\begin{array}{c}4\\ -6\end{array}\right)$ - $\left(\begin{array}{c}-3\\ 5\end{array}\right)$

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

2v - u = $\left(\begin{array}{c}\mathrm{4-(-3)}\\ \mathrm{-6-5}\end{array}\right)$

2v - u = $\left(\begin{array}{c}\mathrm{4+3}\\ -11\end{array}\right)$

2v - u = $\left(\begin{array}{c}7\\ -11\end{array}\right)$

Pyramid

A square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.

The summation of the first row gives 42, hence the summation of each row, coloumn and diagonals should be 42.

13+12+17 = 42

12 + F + 16 = 42

F + 12 + 16 = 42

F + 28 = 42

F = 42 - 28

F = 14

A square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.

The summation of the first row gives 42, hence the summation of each row, coloumn and diagonals should be 42.

13+12+17 = 42

13 + E + 11 = 42

E + 13 + 11 = 42

E + 24 = 42

E = 42 - 24

E = 18

A square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same.

The summation of the first row gives 42, hence the summation of each row, coloumn and diagonals should be 42.

13+12+17 = 42

13 + E + 11 = 42

E + 13 + 11 = 42

E + 24 = 42

E = 42 - 24

E = 18

17 + 10 + G = 42

G + 17 + 10 = 42

G + 27 = 42

G = 42 - 27

G = 15

E + G = 18 + 15

E + G = 33

The third side can be calculated using pythagoras theorem

The side facing the right-angled (90^o) is the hypotenuse (c).

From the question hypotenuse = 13

Let x = the third side

x² + 5² = 13²

x² + 25 = 169

x² = 169 - 25

x² = 144

A number when multiplied by itself gives 144. 12 x 12 is 144, hence express 144 as 12 x 12 or 12²

x² = 12²

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

x = 12

The third side = 12

When analyzing the sequence,you will realize the rule for the sequence is:

U_n = U_n-1+3+n

Where n ≥ 2

When n = 2

U₂ = U_2-1+3+2

U₂ = U₁+5

U₁ = 11

U₂ = 11+5 = 16

When n = 3

U₃ = U_3-1+3+3

U₃ = U₂+6

U₂ = 16

U₃ = 16+6 = 22

When n = 4

U₄ = U_4-1+3+4

U₄ = U₃+7

U₃ = 22

U₄ = 22+7 = 29

When n = 5

U₅ = U_5-1+3+5

U₅ = U₄+8

U₄ = 29

U₅ = 29+8 = 37

When n = 6

U₆ = U_6-1+3+6

U₆ = U₅+9

U₅ = 37

U₆ = 37+9 = 46

When n = 7

U₇ = U_7-1+3+7

U₇ = U₆+10

U₆ = 46

U₇ = 46+10 = 56

Convert the meters to centimeters

1 m = 100 cm
20 m = 20 x 100 cm = 2000cm

Scale = 10cm : 2000cm

Simplify the expression. The centimeters cancel each other and 10 divides itself once and 2000, 200 times.

10cm : 2000cm = 1:200