All members of a subset are found in the main set.

Mathematics is the region in M.

Number of pupils who offer Mathematics = 18 + 7 = 25

Only one subject is the region outside the intersection (7)

Only one subject = 18 + 3 = 21

-- = + or -(-) = + or - x - = +

12 - 7 - (-5) = 12 - 7 + 5 = 10

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

L.C.M of 16 and 20

Prime factors of 16

$$\begin{array}{cc}2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$$

Prime factors of 16 = 2 x 2 x 2 x 2 = 2⁴

Prime factors of 20

$$\begin{array}{cc}2& 20\\ 2& 10\\ 5& 5\\ & 1\end{array}$$

Prime factors of 20 = 2 x 2 x 5 = 2² x 5

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

L.C.M of 16 and 20 = 2⁴ x 5 = 16 x 5 = 80

243_five = 2 x 5² + 4 x 5¹ + 3 x 5⁰

Note: a^o = 1, 5⁰ = 1

243_five = 2 x 25 + 4 x 5 + 3 x 1

243_five = 50 + 20 + 3

243_five = 73

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

35x⁵y³ ÷ 7xy² = $\frac{35{\mathrm{x}}^5{\mathrm{y}}^3}{\mathrm{7x}{\mathrm{y}}^2}$

From the division law of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

Every number or letter is raised to the power 1

x = x¹

$\frac{35{\mathrm{x}}^5{\mathrm{y}}^3}{\mathrm{7x}{\mathrm{y}}^2}$ = $\frac{35{\mathrm{x}}^5{\mathrm{y}}^3}{7{\mathrm{x}}^1{\mathrm{y}}^2}$

$\frac{35{\mathrm{x}}^5{\mathrm{y}}^3}{\mathrm{7x}{\mathrm{y}}^2}$ = 5x^5-1y^3-2

$\frac{35{\mathrm{x}}^5{\mathrm{y}}^3}{\mathrm{7x}{\mathrm{y}}^2}$ = 5x⁴y¹ = 5x⁴y

When their multiples are the same, they will ring simultaneously.

For the first time, that will be their least common multiples (L.C.M)

Multiples of 3 = {3,6,9,12,15,...}
Multiples of 4 = {4,8,12,16,...}

After 12 hours both bells will ring at the same time.

To express a fraction as a percentage, simply multiply the fraction by 100%

$\frac{17}{25}$ = $\frac{17}{25}$ x 100% = 17 x 4% = 68%

Find the equivalent fraction for each of the fractions having the L.C.M of the denominators (5,12,4) as a denominator.

1. L.C.M of 5,12 and 4 is 60

2. Multiply the numerator and the denominator of each fractions by the number of times the denominator divides the L.C.M

$\frac{2}{5}$ = $\frac{\mathrm{2\; x\; 12}}{\mathrm{5\; x\; 12}}$ = $\frac{24}{60}$

$\frac{5}{12}$ = $\frac{\mathrm{5\; x\; 5}}{\mathrm{12\; x\; 5}}$ = $\frac{25}{60}$

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

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

Ascending order means from lowest to highest or smallest to largest.

The order of magnitude of the numerators in ascending = 24,25,45

Substitute the order by their corresponding fractions

$\frac{2}{5}$,$\frac{5}{12}$,$\frac{3}{4}$

Method I

Let x = annual income

of means multiplication.

Rent is 0.3 of his annual income.

1800 = 0.3 x x

Divide both sides of the equation by 0.3

$\frac{1800}{0.3}$ = x

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

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

0.3 = 3 x 10^-1

$\frac{1800}{\mathrm{3\; x}{10}^{-1}}$ = x

Note: 3 divides itself once, and 1800, 600 times

$\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$ = a^-b

$\frac{1}{{10}^{-1}}$ = 10^-(-1) = 10¹

600 x 10¹ = x

6000 = x

Annual income = GH₵6000.00

Method II

The whole income is 1 so you can use proportionality to obtain the income.

0.3 = 1800
1 = $\frac{\mathrm{1\; x\; 1800}}{0.3}$ = 6000

Cost of goods = Total cash given - Change

Note: 15 Gp = 0.15
50 Gp = 0.50

Total cash given = 10 + 0.15 = 10.15
Change = 0.50
Cost of goods = 10.15 - 0.50 = GH₵9.65

Calculations

You can put the numbers in their place value and calculate.

Total initial cost of shirts = 15 x GH₵4.00

The same amount will be used to buy the shirts at the new price.

Number of shirts = $\frac{\mathrm{Amount}}{\mathrm{Cost\; per\; shirt}}$

Number of shirts = $\frac{\mathrm{15\; x\; GH₵4.00}}{\mathrm{GH₵5.00}}$ = 3 x 4 = 12 shirts

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

Change the meters to centimeters.

1 m = 100 cm
8 m = 8 x 100 cm = 800 cm

Scale = 4 cm : 800 cm = 1 : 200

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

Method I

Comparing the two sides of equation, 3.14 and 17.5 are at both sides of the equation so you can equate the ones which are not the same.

3p = 18

Divide both sides by 3

p = $\frac{18}{3}$ = 6

Method II

(3.14 x 18) x 17.5 = 3.14 x (3p x 17.5)

The above equation is the same as 3.14 x 18 x 17.5 = 3.14 x 3 x p x 17.5

You can solve for the value of p by dividing both sides of the equation by 3.14 x 3 x 17.5

p = $\frac{\mathrm{3.14\; x\; 18\; x\; 17.5}}{\mathrm{3.14\; x\; 3\; x\; 17.5}}$

p = 6.0

Note: 3.14 and 17.5 cancel each other and 3 divides itself once and 18, 6 times.

Sum of all the angle of sectors is 360^o

x + 80 + 120 + 95 = 360

x + 295 = 360

x = 360 - 295

x = 65^o

Amount for a sector = $\frac{\mathrm{Angle\; of\; sector}}{360}$ x Total amount

Angle of sector for food = 120^o
Total earnings = GH₵630.00

Amount for food = $\frac{120}{360}$ x GH₵630.00 = GH₵210.00

Note: 120 divides itself once and 360, 3 times. 3 divides itself once and 630, 210 times.

Percentage for a sector = $\frac{\mathrm{Angle\; of\; sector}}{360}$ x 100%

Angle of sector for rent and utilities = 80^o

Percentage for rent and utilities = $\frac{80}{360}$ x 100% = 22.2%

Note: The zeros cancel each other, 4 divides 8, 2 times and 36, 9 times. 2 multiplies the 100 and the 200 over 9 is simplified.

Enlargement either increases or decreases the size of the shape base on the scale factor. A scale factor less than 1 decreases the size of the shape. The shape and angles remains the same.

Kofi:Kojo:Ama = 3:5:4

Ama's ratio = 4

Amount received by a ratio = $\frac{\mathrm{Ratio}}{\mathrm{Sum\; of\; ratios}}$ x Total Amount

Sum of ratios = 3 + 5 + 4 = 12

Amount received by Ama = $\frac{4}{12}$ x GH₵480,000.00 = 4 x 40000 = 160000

Ama received GH₵160,000.00

Note: 12 divides itself once and 48, 4 times.

w = 12, x = 5, y = 6 and z = 4

wx - yz = 12 x 5 - 6 x 4

wx - yz = 60 - 24

wx - yz = 36

When the father was 24 years, the son was 0 year.

Fathers age now will be addition of the son's age to 24 years. Thus when the son is 1 year, the father is 25 year (24 + 1), when the son is 2 years, the father is 26 years (24 + 2), etc.

Let x = son's age now.

When the son is x years, the father is 24 + x years

Now he is three times as old as his son means when you multiply the son's age now by 3, you will get the father's age.

3 x x = father's age now

But the father's age is also 24 + x years now hence we can equate the two expressions and solve for the value of x.

3 x x = 24 + x

3x = 24 + x

3x - x = 24

2x = 24

Divide both sides of the equation by 2

x = $\frac{24}{2}$ = 12

The son is 12 years now.

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

Total number of samples = 20

12 of the balls are blue
Number of green balls = 20 - 12 = 8

Number of possible selection of a green ball = 8

Probability (Green ball) = $\frac{8}{20}$ = $\frac{2}{5}$

Note: 4 divides 8, 2 times and 20, 5 times.

Check if the interval of x is 1 and the interval (difference) between consecutive given numbers of y are the same.

If they are the same, the mapping is linear, hence consecutives are obtained by adding the interval (difference) to preceding numbers.

Interval/difference = 5 - 3 = 11 - 9 = 2

Since the intervals are the same, the succeeding numbers are obtained by adding the difference to their preceding.

p = 5 + 2 = 7
q = 11 + 2 = 13

Perimeter is the sum of each of the lengths of a shape.

Rectangle has two equal lengths and two equal widths.

Let w = width of the rectangle.

Perimeter = 2 x 7 cm + 2 x w

But perimeter = 24 cm

24 cm = 14 cm + 2w

24 cm - 14 cm = 2w

10 cm = 2w

Divide both sides of the equation by 2

w = $\frac{\mathrm{10\; cm}}{2}$ = 5 cm

The angle of a bearing starts from the north in a clockwise direction.

Faces of a cuboid

Area of a circle = πr²

Where r = radius.

Radius is half of a diameter.

Diameter = 28 cm
Radius = $\frac{28}{2}$ = 14 cm

r² = r x r

Area of a circular tray = $\frac{22}{7}$ x 14 cm x 14 cm = 22 x 2 x 14 cm² = 616 cm²

Note: The denominator 7 divides itself once and one of the 14s, 2 times.

Volume of a cylinder = πr²h

Where r = radius and h = height.

r² = r x r

Volume = $\frac{22}{7}$ x 7 cm x 7 cm x 10 cm = 22 x 1 x 7 x 10 cm³ = 1,540 cm³

Angle PRQ = e (Vertically opposite angles)

Sum of angles in a triangle is 180^o

50 + 38 + PRQ = 180
88 + PRQ = 180
PRQ = 180 - 88
PRQ = 92

But angle PRQ = e = 92

Angle PRQ = e (Vertically opposite angles)

Sum of angles in a triangle is 180^o

50 + 38 + PRQ = 180
88 + PRQ = 180
PRQ = 180 - 88
PRQ = 92

But angle PRQ = e = 92

132 + angle RTS = 180 (Sum of angles on straight line is 180^o)

Angle RTS = 180 - 132

Angle RTS = 48

e + d + Angle RTS = 180 (Sum of angles in a triangle is 180^o)

e = 92

92 + d + 48 = 180

d + 140 = 180

d = 180 - 140

d = 40^o

Number of pieces = $\frac{\mathrm{Length\; of\; string}}{\mathrm{Length\; of\; each\; piece}}$

Before you divide make sure they are of the same unit. You can either change meter to centimeters or centimeters to meters then divide.

1 m = 100 cm
3.6 m = 3.6 x 100 cm = 360 cm

Number of pieces = $\frac{\mathrm{360\; cm}}{\mathrm{40\; cm}}$ = 9

Note: 3.6 x 100 = 3.⌒6⌒0. = 360

2x + 10 ≥ $\frac{\mathrm{7x}}{2}$ - 5

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.

x ≤ $\frac{-30}{-3}$

x ≤ 10

y-axis as mirror/reflection in y-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 x is negated.

P$\left(\begin{array}{c}5\\ 4\end{array}\right)$ → P₁$\left(\begin{array}{c}-5\\ 4\end{array}\right)$ = P₁(-5,4)

$\left(\begin{array}{c}4\\ 11\end{array}\right)$ = $\left(\begin{array}{c}\mathrm{x\; -\; 3}\\ 11\end{array}\right)$

When vectors are equal, the top at the left is equal to the top at the right and the down at the left is equal to the down at the right.

x - 3 = 4

Solve for the value of x.

x = 4 + 3

x = 7

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

M(-1,2) and N(6,-3)

Y₂ = -3
Y₁ = 2
X₂ = 6
X₁ = -1

Gradient = $\frac{\mathrm{-3-2}}{\mathrm{6-(-1)}}$ = $\frac{-5}{\mathrm{6+1}}$ = $\frac{-5}{7}$

Check the common difference (interval). If its the same for the given consecutive numbers, then the succeeding term is simply obtained by adding the common difference (interval) to the preceding term.

Common difference = 7 - 11, 3 - 7,-1 - 3 = -4

Each of the common differences are the same hence succeeding terms are obtained by adding the common difference to preceding terms.

First missing term = -1 + -4 = -5
Second missing term = -5 + - 4 = -9