Number of subsets = 2ⁿ

Where n is the number of elements of the set.

The set U has two elements hence n = 2

Number of subsets = 2² = 2 x 2 = 4

222 + 343 + abc = 1000

565 + abc = 1000

abc = 1000 - 565

abc = 435

The rule of the mapping is dividing the preceding by 2 to get the succeeding

a = $\frac{2}{2}$ = 1

The rule of the mapping is dividing the preceding by 2 to get the succeeding

a = $\frac{2}{2}$ = 1

b = $\frac{\mathrm{a}}{2}$ = $\frac{1}{2}$

two hundred thousand = 200000

fifty seven = 57

two hundred thousand and fifty seven = 200000 + 57

two hundred thousand and fifty seven = 200,057

The short hand points to the hours and the long hand points to the minutes. The short hand points to 3. The minute interval is 5 minutes. The longer hand points to the fourth division (4 x 5 minutes = 20 minutes)

Let x = Ebo's share.

Kojo had twice as much as Ebo

Kojo's share = 2 x Ebo's share

Kojo's share = 2 x x = 2x

Ebo also had twice as much as Ama

Ebo's share = 2 x Ama's share

x = 2 x Ama's share

Divide both sides by 2 to get Ama's share

$\frac{\mathrm{x}}{2}$ = Ama's share

Thus half of Ebo's share is Ama's share.

The sum of all the shares should give ₵14,000

Kojo's share + Ebo's share + Ama's share = ₵14,000

2x + x + $\frac{\mathrm{x}}{2}$ = ₵14000

Solve for the value of x

Get rid of the fraction by multiplying both sides by the denominator (2)

2 x 2x + 2 x x + 2 x $\frac{\mathrm{x}}{2}$ = 2 x ₵14000

4x + 2x + x = 2 x ₵14000

7x = 2 x ₵14000

Divide both sides by 7

x = $\frac{\mathrm{2\; x\; ₵14000}}{7}$ = 2 x ₵2000 = ₵4000

Note: 7 divides itself 1 time and 14, 2 times.

∴ Ebo's share is ₵4000

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

Principal = ₵5,000.00

Total amount = ₵7,500

Total amount = Principal + Interest

₵5000 + Interest = ₵7500

Interest = ₵7500 - ₵5000

Interest = ₵2500

Time = 5

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

I = $\frac{\mathrm{P\; x\; T\; x\; R}}{100}$

Make R the subject

Get rid of the fraction by multiplying both sides of the equation by the denominator (100)

100 x I = 100 x $\frac{\mathrm{P\; x\; T\; x\; R}}{100}$

100 x I = P x T x R

Divide both sides by P x T

R = $\frac{\mathrm{100\; x\; I}}{\mathrm{P\; x\; T}}$

Substitute the values to solve for R

R = $\frac{\mathrm{100\; x\; ₵2500}}{\mathrm{₵5000\; x\; 5}}$ = 10%

Note:
1. the zeros (0) at the end of 100 and 5000 cancel each other
2. the zeros (0) at the end of 50 and 2500 cancel each other
3. 5 divides itself 1 time and 250, 50 times
4. the other 5 divides itself 1 time and 50, 10 times

$\frac{1}{3}$ - $\frac{1}{2}$ + $\frac{2}{5}$ = $\frac{1}{3}$ - $\frac{1}{2}$ + $\frac{2}{5}$

$\frac{1}{3}$ - $\frac{1}{2}$ + $\frac{2}{5}$ = $\frac{\mathrm{10\; x\; 1\; -\; 15\; x\; 1\; +\; 6\; x\; 2}}{30}$

$\frac{1}{3}$ - $\frac{1}{2}$ + $\frac{2}{5}$ = $\frac{\mathrm{10\; -\; 15\; +\; 12}}{30}$

$\frac{1}{3}$ - $\frac{1}{2}$ + $\frac{2}{5}$ = $\frac{7}{30}$

Express 8 in the for 2ⁿ

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

8 = 2 x 2 x 2 = 2^{1 + 1 + 1} = 2³

2^x = 2³

Since the bases are the same, the powers are also the same hence x = 3

x = 3

C = $\frac{5}{9}$(F - 32)

C = 40

40 = $\frac{5}{9}$(F - 32)

Get rid of the fraction by multiplying both sides by the denominator (9)

9 x 40 = 9 x $\frac{5}{9}$(F - 32)

9 x 40 = 5(F - 32)

360 = 5F - 160

360 + 160 = 5F

520 = 5F

Divide both sides by 5

F = $\frac{520}{5}$

F = 104

C = $\frac{5}{9}$(F - 32)

C = -40

-40 = $\frac{5}{9}$(F - 32)

Get rid of the fraction by multiplying both sides by the denominator (9)

9 x -40 = 9 x $\frac{5}{9}$(F - 32)

9 x -40 = 5(F - 32)

-360 = 5F - 160

-360 + 160 = 5F

-200 = 5F

Divide both sides by 5

F = $\frac{-200}{5}$

F = -40

Mean/Average = $\frac{\mathrm{Sum\; of\; items}}{\mathrm{Number\; of\; items}}$

Mean = $\frac{\mathrm{2\; +\; 3\; +\; 3\; +\; 3\; +\; 4\; +\; 5\; +\; 5\; +\; 5\; +\; 6}}{9}$

Mean = $\frac{36}{9}$ = 4

The median is the middle number of an ordered distribution. If there are two numbers at the middle, add the two numbers and divide the sum by 2.

The middle age of the ordered distribution is 4 hence the median is 4

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

Number of children who are 5 years old = 3

Total number of children = 9

P(5 years) = $\frac{3}{9}$ = $\frac{1}{3}$

PQ→ = OQ→ - OP→

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

PQ→ = $\left(\begin{array}{c}\mathrm{-2\; -\; 2}\\ \mathrm{3\; -\; 5}\end{array}\right)$

PQ→ = $\left(\begin{array}{c}-4\\ -2\end{array}\right)$

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

Triangle RST is an isosceles triangle. The base angles are the same.

Angle RTS = Angle RST = 73°

Angle SRT + Angle RTS + Angle RST = 180°(Sum of interior angles of a triangle)

Angle SRT + 73° + 73° = 180°

Angle SRT + 146° = 180°

Angle SRT = 180° - 146°

Angle SRT = 34°

Angle PRQ = Angle SRT (Opposite angles are equal)

115° + Angle PQR = 180°(Sum of angles on a straight line)

Angle PQR = 180° - 115°

Angle PQR = 65°

a + Angle PQR + Angle SRT = 180°(Sum of interior angles of a triangle)

a + 65° + 34° = 180°

a + 99° = 180°

a = 180° - 99°

a = 81°

a² – b² = (a + b)(a – b)

9.32² – 0.68² = (9.32 + 0.68)(9.32 - 0.68)

9.32² – 0.68² = (10)(8.64)

9.32² – 0.68² = 10 x 8.64

9.32² – 0.68² = 8.⌒6.4 = 86.4

Method I

Simplifying the indices

6² = 6 x 6 = 36

2² = 2 x 2 = 4

2² x 3 = 4 x 3 = 12

$\frac{6^2}{2^{2}x3}$ = $\frac{36}{12}$ = 3

Method II

Using law of indices.

(a x b)^c = a^c x b^c

6 = 2 x 3

6² = (2 x 3)² = 2² x 3²

$\frac{6^2}{2^{2}x3}$ = $\frac{2^{2}x{3}^2}{2^{2}x3}$

2² cancel each other

$\frac{6^2}{2^{2}x3}$ = $\frac{3^2}{3}$

3 = 3¹

Law of division of indices

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

$\frac{6^2}{2^{2}x3}$ = 3^{2 - 1}

$\frac{6^2}{2^{2}x3}$ = 3¹ = 3

Let x = Kojo's age

Kwaku is 4 years older than Kojo

This means if you add 4 years to Kojo's age, you will get Kwaku's age

Kwaku's age = x + 4

Sum means addition.

Kwaku's age + Kojo's age = 30

x + 4 + x = 30

x + x = 30 - 4

2x = 26

Divide both sides by 2

x = $\frac{26}{2}$

x = 13

∴ Kojo's age is 13

120° + Angle PQR = 180°

Angle PQR = 180° - 120°

Angle PQR = 60°

Angle PQR + Angle QPR + Angle PRQ = 180°

Angle PRQ = 60°

60° + Angle QPR + 60° = 180°

Angle QPR + 60° + 60° = 180°

Angle QPR + 120°= 180°

Angle QPR = 180° - 120°

Angle QPR = 60°

All the angles of triangle PQR are the same (60°) hence the triangle is equilateral triangle.

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

x = -1

y = (-1 + 2)(-1 - 2)

y = (1)(-3)

y = 1 x -3 = -3

Zero (0) times any number is 0.

y will be zero when any of the values in the bracket is 0.

When x + 2 = 0

x + 2 = 0

x = -2

When x - 2 = 0

x - 2 = 0

x = 2

The set of values of x = {-2, 2}

The base is circular

Circumference of a circle= 2 x π x radius

Radius is half the diameter.

radius = $\frac{\mathrm{diameter}}{2}$ = $\frac{\mathrm{4\; cm}}{2}$ = 2 cm

π = $\frac{22}{7}$

The circumference of the base = 2 x $\frac{22}{7}$ x 2 cm

The circumference of the base = $\frac{\mathrm{2\; x\; 22\; x\; 2\; cm}}{7}$

The circumference of the base = $\frac{\mathrm{88\; cm}}{7}$

The base is circular

Area of a circle = π x radius x radius

Radius is half the diameter.

radius = $\frac{\mathrm{diameter}}{2}$ = $\frac{\mathrm{4\; cm}}{2}$ = 2 cm

π = $\frac{22}{7}$

The area of the base = $\frac{22}{7}$ x 2 cm x 2 cm

The area of the base = $\frac{\mathrm{22\; x\; 2\; cm\; x\; 2\; cm}}{7}$

The area of the base = $\frac{88}{7}$ cm²

Volume of a cylinder = π x radius x radius x height

Radius is half the diameter.

radius = $\frac{\mathrm{diameter}}{2}$ = $\frac{\mathrm{4\; cm}}{2}$ = 2 cm

π = $\frac{22}{7}$

The height is the length of the diagram.

Height = 14 cm

The volume of the cylinder = $\frac{22}{7}$ x 2 cm x 2 cm x 14 cm

Note: 7 divides itself 1 time and 14, 2 times.

The volume of the cylinder = 22 x 2 cm x 2 cm x 2 cm

The volume of the cylinder = 176 cm³

Profit = Selling Price - Cost Price

Number of times she sells = $\frac{\mathrm{Total\; oranges}}{\mathrm{Number\; of\; oranges\; per\; sale}}$

Number of times she sells = $\frac{210}{3}$ = 70

Note: 3 divides 21, 7 times.

Total amount sold = Number of times she sells x price per sale

Total amount sold = 70 x ₵20.00 = ₵1400

Alternatively using proportionality

If 3 oranges = ₵20

210 oranges = $\frac{\mathrm{₵20\; x\; 210\; oranges}}{\mathrm{3\; oranges}}$ = ₵20 x 70 = ₵1400

Selling Price = ₵1400

Cost Price = ₵650

Profit = ₵1400 - ₵650

Profit = ₵750

The sum of ratios represent the total pupils in the school

Sum of ratios = 9 + 11 = 20

The ratio for boys is 9

If 20 = 400

9 = $\frac{\mathrm{400\; x\; 9}}{20}$ = 20 x 9 = 180

∴ 180 boys in the school.

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.

Change the decimal to whole number x 10^-n and simplify after which you change to a standard form.

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

162.5 = 162.⌒5. = 1625 x 10^-1

0.5 = 0.⌒5. = 5 x 10^-1

162.5 x 0.5 = 1625 x 10^-1 x 5 x 10^-1

162.5 x 0.5 = 1625 x 5 x 10^-1 x 10^-1

162.5 x 0.5 = 8125 x 10^-1 x 10^-1

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

162.5 x 0.5 = 8125 x 10^{-1 + -1}

162.5 x 0.5 = 8125 x 10^{-1 - 1}

162.5 x 0.5 = 8125 x 10^-2

Change 8125 to standard form

Every whole number has .0 at the end which is not written.

8125 = 8125.0

8125 = 8.⌒1⌒2⌒5.0 = 8.125 x 10³

162.5 x 0.5 = 8.125 x 10³ x 10^-2

162.5 x 0.5 = 8.125 x 10^{3 + -2}

162.5 x 0.5 = 8.125 x 10^{3 - 2}

162.5 x 0.5 = 8.125 x 10¹

Any number raised to the power 1 is the same number.

162.5 x 0.5 = 8.125 x 10

Percentage (%) means over 100

16% = $\frac{16}{100}$

of means multiplication (x)

16% of ₵500,000 = $\frac{16}{100}$ x ₵500,000

Note: the zeros (0) at the end of 100 and 500,000 cancel each other

16% of ₵500,000 = 16 x ₵5000 = ₵80000

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

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

0.24 = 0.⌒2⌒4. = 24 x 10^-2

14.3 = 14.⌒3. = 143 x 10^-1

5.2 = 5.⌒2. = 52 x 10^-1

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = $\frac{24x{10}^{-2}x143x{10}^{-1}}{52x{10}^{-1}}$

Note:
1. 10^-1 cancel each other
2. 4 divides 52, 13 times and 24, 6 times
3. 13 divides itself 1 time and 143, 11 times

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = 6 x 11 x 10^-2

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = 66 x 10^-2

Change the whole number x 10^-n to decimal by moving the decimal point n times to the left.

66 = 66.0

$\frac{\mathrm{0.24\; x\; 14.3}}{5.2}$ = .⌒6⌒6.0 = 0.66

Product means multiplication.

2x x 3 = 138

2 x 3 x x = 138

6x = 138

Divide both sides by 6

x = $\frac{138}{6}$

x = 23

Speed = $\frac{\mathrm{Distance}}{\mathrm{Time}}$

The cyclist started the journey at 3 PM and reached at 4 PM covering a distance of 50 km

Total hours for the trip = 1 hour (1h)

Speed = $\frac{\mathrm{50\; km}}{\mathrm{1\; h}}$ = 50 kmh^-1

h^-1 is read as per hour which means being divided by hour

He rested at 1 PM for half an hour (30 minutes) and resumed his journey until he reached at 2 PM.

He reached town Y at 2:00 pm and started his journey back to town X at 3:00 pm. Hence he spent 1 hour at town Y

1 hour = 60 minutes

Volume of water per each = $\frac{\mathrm{Total\; Volume}}{200}$

Volume of water per each = $\frac{\mathrm{350\; litres}}{200}$

Note:
1. the zeros (0) at the end of 350 and 200 cancel each other
2. 5 divides 35, 7 times and 20, 4 times

Volume of water per each = $\frac{\mathrm{7\; litres}}{4}$ = 1.75 litres

2x + 4 > -6

2x > -6 - 4

2x > -10

Divide both sides by 2

x > $\frac{-10}{2}$

x > -5

The fencing is the perimeter of the rectangular field.

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

A rectangle has two equal lengths and widths

Perimeter of a rectangle = 2 x Length + 2 x Width

Perimeter = 260

Width = 50

Length = x

2 x x + 2 x 50 = 260

2x + 100 = 260

A scalene triangle is a triangle whose all three sides have different lengths, and all three angles are of different measures.