The sequence is a linear sequence with a common difference of 3.

Common difference = Succeeding - Preceding

Succeeding = Preceding + Common difference.

For a linear sequence, the common difference is the same for all consecutive numbers.

Common difference = -2 -- 5 = 1 -- 2 = 3

Notes:

1. -- = +
2. -- 5 = + 5
3. -- 2 = + 2

The missing term = Preceding + Common difference

The preceding number is 1

The common difference = 3

The missing term = 1 + 3 = 4

1. The third (three) significant figure in 56782 is 7

2. The digit/number next to the third significant figure (7) is 8 which is more than 4 and therefore 1 is added to 7, making 8

3. The remaining digits/numbers (8 and 2) are then replaced by zeros (0)

56782 = 56800 (three) significant figures.

1. There are two numbers after the decimal point in 0.24. Hence the number to multiply with, in step 3 will be 100

2. 0.24 = $\frac{0.24}{1}$

3. 0.24 = $\frac{\mathrm{0.24\; x\; 100}}{\mathrm{1\; x\; 100}}$

0.24 x 100 produces whole number, 24

4. 0.24 = $\frac{24}{100}$ = $\frac{6}{25}$

Notes

1. 4 divides 24, 6 times and 100, 25 times
2. The top in a fraction is called numerator
3. The down in a fraction is called denominator

The denominator in $\frac{6}{25}$ is 25

If 500 kg = 20 goats

200 kg = $\frac{\mathrm{200\; kg\; x\; 20\; goats}}{\mathrm{500\; kg}}$ = 8

Notes:

1. The zeros(0) in 200 and 500 cancel each other

2. kg cancel each other

2 x 20 = 40

5 divides itself 1 time and 40, 8 times

Wage = GH₵ 700

Amount spent = GH₵ 560

Amount saved = Wage - Amount spent

Amount saved = GH₵ 700 - GH₵ 560

Amount saved = GH₵ 140

Number = GH₵ 140

Total = GH₵ 700

Percentage saved = $\frac{\mathrm{GH₵\; 140}}{\mathrm{GH₵\; 700}}$ x 100% = 20%

Notes

1. The zeros(0) in 700 and 100 cancel each other

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

Sum of interior angles of a triangle is 180°.

(3x - 10)° + (4x + 20)° + third angle = 180°

3x - 10 + 4x + 20 + third angle = 180

3x + 4x - 10 + 20 + third angle = 180

7x + 10 + third angle = 180

third angle = 180 - 10 - 7x

third angle = 170 - 7x

third angle = (170 - 7x)°

100 GP = 1 cedi

50 GP = 0.5 cedi (half a cedi)

Cost of two toffees = 2 x 0.5 cedi = 1 cedi

Total amount she spent = GH₵ 1.75 + GH₵ 1 = GH₵ 2.75

She had GH₵ 4 before spending

Amount left = GH₵ 4 - GH₵ 2.75 = GH₵ 1.25

Note: If you add 0.25 to 0.75, you will get 1

a = $\left(\begin{array}{c}-2\\ 1\end{array}\right)$ and b = $\left(\begin{array}{c}-5\\ -3\end{array}\right)$

2a - b = 2 x $\left(\begin{array}{c}-2\\ 1\end{array}\right)$ - $\left(\begin{array}{c}-5\\ -3\end{array}\right)$

2a - b = $\left(\begin{array}{c}\mathrm{-2\; x\; 2}\\ \mathrm{1\; x\; 2}\end{array}\right)$ - $\left(\begin{array}{c}-5\\ -3\end{array}\right)$

2a - b = $\left(\begin{array}{c}-4\\ 2\end{array}\right)$ - $\left(\begin{array}{c}-5\\ -3\end{array}\right)$

2a - b = $\left(\begin{array}{c}\mathrm{-4\; -\; -5}\\ \mathrm{2\; -\; -3}\end{array}\right)$

Notes:

1. - - = +

2. - -5 = + 5

3. - -3 = + 3

2a - b = $\left(\begin{array}{c}\mathrm{-4\; +\; 5}\\ \mathrm{2\; +\; 3}\end{array}\right)$

2a - b = $\left(\begin{array}{c}1\\ 5\end{array}\right)$

Note: If the mapping has equal interval (difference) for the y and x interval of 1, you can use the general linear sequence formula to find the rule of the mapping.

U_n = U₁ + (n-1)d

In the mapping, U_n = y and n = t and U₁ is the value of y when t = 1

U₁ = 9

y = 9 + (t-1)d

The d is the interval between each consecutive y

In this mapping, d = 20-9 = 31-20 = 42 - 31 = 11

We can substitute our d and simplify the expression for y

y = 9 + (t-1)d

y = 9 + (t-1)(11)

y = 9 + 11t - 11

y = 11t + 9 - 11

y = 11t - 2

t → 11t - 2

1 hour = 60 minutes

2 hours = 2 x 60 minutes = 120 minutes

Remaining minutes after the first question = 120 minutes - 20 minutes

Remaining minutes after the first question = 100 minutes

Remaining questions = 9 - 1 = 8

Minutes spent on each remaining question = $\frac{\mathrm{100\; minutes}}{8}$ = 12.5 minutes

Notes:

1. 4 divides 8, 2 times and 100, 25 times

2. 2 divides itself, 1 times and 25, 12.5 (12 and half) times.

The distance covered is the circumference of the circular pond.

Circumference of the pond = 2 x π x radius

π = $\frac{22}{7}$

Radius = 28 m

Circumference of the pond = 2 x $\frac{22}{7}$ x 28 = 2 x 22 x 4 m = 176 m

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

Distance covered = 176 m

To apply law of indices, make sure the bases are the same. The least base is 2. Hence change the 8 to the least base 2.

8 = 2 x 2 x 2

Every number is raised to the power 1 but not written.

8 = 2¹ x 2¹ x 2¹

8 = 2^{1 + 1 + 1} = 2³

2^x = 2³ x 2⁰

Apply law of multiplication of indices.

2^x = 2^{3 + 0}

2^x = 2³

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

x = 3

Note: any number/letter raised to the power 0 is 1

Alternatively,

2^x = 2³ x 2⁰

2⁰ = 1

2^x = 2³ x 1

2^x = 2³

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

x = 3

Percentage means over 100

Total percentage of men = 100%

Percentage of men who are vegetarian = 28%

Probability (Vegetarian) = $\frac{\mathrm{28\%}}{\mathrm{100\%}}$ = $\frac{28}{100}$ = $\frac{7}{25}$

Note: 4 divides 28, 7 times and 100, 25 times.

Method I

Using formula

Amount sold = GH₵ 25,000

5% = $\frac{5}{100}$

Commission = GH₵ 25000 x $\frac{5}{100}$ = GH₵ 1,250

Notes:

1. The zeros(0) in 25000 and 100 cancel each other

2. 5 x 250 = 1,250

Method II

Using proportion

100% represents the amount sold.

If 100% = GH₵ 2500

5% = $\frac{\mathrm{5\%\; x\; GH₵\; 2500}}{\mathrm{100\%}}$ = GH₵ 1,250

Notes:

1. When the solid is shaded, it is either ≤ or ≥ base on the direction of the arrow.

2. Since the arrow is pointing left and the solid is shaded, the inequality sign ≤

3. The number is the value the solid is placed. In the above figure, the solid is placed on 3.

Method I

Using selling price percentage

100% represents the cost price.

Selling Price = Cost Price + Profit

Selling Price% = Cost Price% + Profit%

Profit% = 40%

Selling Price% = 100% + 40%

Selling Price% = 140%

Cost Price = GH₵ 25

If 100% = GH₵ 25

140% = $\frac{\mathrm{140\%\; x\; GH₵\; 25}}{\mathrm{100\%}}$ = GH₵ 35

Selling Price = GH₵ 35

Method II

Using profit percentage

100% represents the cost price.

Selling Price = Cost Price + Profit

Profit% = 40%

Cost Price = GH₵ 25

If 100% = GH₵ 25

40% = $\frac{\mathrm{40\%\; x\; GH₵\; 25}}{\mathrm{100\%}}$ = GH₵ 10

Profit = GH₵ 10

Selling Price = GH₵ 25 + GH₵ 10

Selling Price = GH₵ 35

Method III

Using profit percentage formula

Profit% = 40%

Cost Price = GH₵ 25

Profit = $\frac{\mathrm{40\; x\; GH₵\; 25}}{100}$ = GH₵ 10

Selling Price = Cost Price + Profit

Selling Price = GH₵ 25 + GH₵ 10

Selling Price = GH₵ 35

Method IV

Using selling price formula

Profit% = 40%

Cost Price = GH₵ 25

Selling Price = $\frac{\mathrm{GH₵\; 25\; x\; (40\; +\; 100)}}{100}$ = $\frac{\mathrm{GH₵\; 25\; x\; 140}}{100}$ = GH₵ 35

Let the length of the rectangular card = l

Let the breadth/width of the rectangular card = w

Area of a rectangle = Length x Breadth/Width

Area of the rectangular card = l x w

But the area is 15 cm²

l x w = 15 cm²

New dimension by a scale factor of f = Original dimension x f

Scale factor = 3

Enlarged length = l x 3

Enlarged breadth/width = w x 3

Area of the enlarged rectangular card = l x 3 x w x 3

Area of the enlarged rectangular card = l x w x 3 x 3

But l x w = 15 cm²

Area of the enlarged rectangular card = 15 cm² x 3 x 3

Area of the enlarged rectangular card = 15 cm² x 9

Area of the enlarged rectangular card = 135 cm²

Let k = Kofi's age.

Ama is three times as old as Kofi means when you multiply Kofi's age by 3, you will get Ama's age.

Ama's age = 3 x k = 3k

The sum of their ages is 40 means when you add their ages, you will get 40.

Ama's age + Kofi's age = 40

3k + k = 40

4k = 40

Divide both sides by 4.

k = $\frac{40}{4}$ = 10

Kofi is 10 years.

Ama's age = 3 x k = 3 x 10 = 30

Ama is 30 years

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

Notes:

1. - x - = +

2. -3 x -3 = + 9

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

Note: the outside term multiplies each term in the bracket.

2(3a + 1) - 3(4a - 3) = 6a + 2 - 12a + 9

2(3a + 1) - 3(4a - 3) = 2 + 9 + 6a - 12a

2(3a + 1) - 3(4a - 3) = 11 - 6a

0.03 x y = 2.4

Divide both sides by 0.03

y = $\frac{2.4}{0.03}$

Change the decimals to whole numbers by multiplying both the numerator and the denominator by 100

Two zeros(00) are attached to 1 because the highest decimal places is two in 0.03.

y = $\frac{\mathrm{2.4\; x\; 100}}{\mathrm{0.03\; x\; 100}}$

y = $\frac{240}{3}$ = 80

Notes:

1. Multiplying a decimal by 100.. moves the decimal point to the right by the number of zeros times.

2 Multiplying the decimal by 100 moves the decimal point to the right two times.

3. 2.4 x 100 = 2.⌒4⌒ = 240

4. 0.03 x 100 = 0.⌒0⌒3 = 3

Sum of ratios represents the total amount.

Sum of ratios = 4 + 3 = 7

Respectively means the ratios are in the order the names where mentioned/stated.

Helen's name was stated second, hence her ratio is 3.

Total amount contributed was GH₵ 7,000

If 7 = GH₵ 7,000

3 = $\frac{\mathrm{3\; x\; GH₵\; 7,000}}{7}$ = GH₵ 3000.00

Notes:

1. 7 divides itself 1 time and 7000, 1000 times.

2. 3 x 1000 = 3000

3. When multiplying by zeros(0) at the end, multiply the numbers and attach the number of zeros afterward.

4. 3 x 1000 implies 3 x 1 which gives 3 then since there are three zeros at the end in 1000, you attach three zeros to the result 3 making 3000.

Number of samples/balls = 10 + 8

Number of samples/balls = 18

Number of green balls to select from = 10

Probability (Green) = $\frac{10}{18}$ = $\frac{5}{9}$

Note: 2 divides 10, 5 times and 18, 9 times.

4x - 2(x + 5) = -10

Expand the bracket by multiplying the outside term with each term in the bracket.

4x - 2 x x - 2 x 5 = -10

4x - 2x - 10 = -10

2x - 10 = -10

2x = -10 + 10

2x = 0

Divide both sides by 2.

x = $\frac{0}{2}$

Note: 0 divided by any number/letter is 0

x = 0

The whole of a fraction is 1.

She sold half.

Half is represented by $\frac{1}{2}$

Since half is sold, half is also left.

Fraction left after selling half of the cloth = $\frac{1}{2}$

two-fifths is represented by $\frac{2}{5}$

of in mathematics is multiplication.

She used two-fifths of the remaining to sew a dress.

Fraction used for sewing = $\frac{2}{5}$ x the fraction remaining

Fraction remaining was half, $\frac{1}{2}$

Fraction used for sewing = $\frac{2}{5}$ x $\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 1}}{\mathrm{5\; x\; 2}}$ = $\frac{1}{5}$

Note: the 2 in the numerator and denominator cancels each other.

Fraction left = Half of the cloth - fraction used for sewing.

Fraction left = $\frac{1}{2}$ - $\frac{1}{5}$

Fraction left = $\frac{\mathrm{5\; x\; 1\; -\; 2\; x\; 1}}{10}$ = $\frac{\mathrm{5\; -\; 2}}{10}$ = $\frac{3}{10}$

Alternatively,since two-fifths is used, three-fifths is left.

Fraction left = three-fifths of the half

Three-fifths is $\frac{3}{5}$

of means multiplication.

Fraction left = $\frac{3}{5}$ x $\frac{1}{2}$ = $\frac{\mathrm{3\; x\; 1}}{\mathrm{5\; x\; 2}}$ = $\frac{3}{10}$

Notes:

1. Fifths means the remaining half is imaginary divided into 5 parts.

2. Two-fifths used means two out of the 5 imaginary divided parts is used for sewing.

3. Three-fifths remains means three out of the 5 imaginary divided parts remains.

2y = 5 - 3x

When y = 1, substitute y by 1 in the given relation.

2 x 1 = 5 - 3x

2 = 5 - 3x

Take the - 3x to the left side. It is negative (-) so when you move to the other side, it becomes positive (+).

2 + 3x = 5

Take 2 to the right side. It is positive so when it moves, it will become negative.

3x = 5 - 2

3x = 3

Divide both sides by 3.

x = $\frac{3}{3}$ = 1

15 twenty cedi notes = 15 x 20 = 300

14 ten cedi notes = 14 x 10 = 140

15 two cedi notes = 15 x 2 = 30

Total sales = 300 + 140 + 30

Total sales = GH₵ 470

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.

(-2, 3) → (-2, -3)

3(x - 5) > 15 - 4(8 - x)

Expand the brackets. The term outside multiplies each terms in the bracket.

3 x x - 5 x 3 > 15 - 4 x 8 -4 x - x

3x - 15 > 15 - 32 + 4x

Notes:

1. - x - = +

2. - 4 x -x = + 4x

Group the x at the left and the numbers at the right. The signs changes when they move. If positive becomes negative and if negative becomes positive.

3x - 4x > 15 + 15 - 32

-x > 30 - 32

-x > -2

Divide both sides by -1. When dividing by a negative, the inequality signs changes direction.

x < $\frac{-2}{-1}$

x < 2

Note: the negatives at the numerator and denominator cancels each other.

2² = 2 x 2 = 4

3² = 3 x 3 = 9

2² x 3² = 4 x 9 = 36

MN⃗ = ON⃗    -    OM⃗

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

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

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

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

MN⃗ = $\left(\begin{array}{c}3\\ 2\end{array}\right)$

Average score for Science and Mathematics

Average = $\frac{\mathrm{Sum\; of\; Science\; and\; Mathematics\; Scores}}{2}$

But the average is 53.

$\frac{\mathrm{Sum\; of\; Science\; and\; Mathematics\; Scores}}{2}$ = 53

Multiply both sides by the denominator, 2.

Sum of Science and Mathematics Scores = 53 x 2

Sum of Science and Mathematics Scores = 106

Mean of the four subjects

Mean = $\frac{\mathrm{Sum\; of\; Science\; and\; Mathematics\; Scores\; +\; English\; Language\; Score\; +\; Social\; Studies\; Score}}{4}$

But Sum of Science and Mathematics Scores = 106

Mean = $\frac{\mathrm{106\; +\; 50\; +\; 60}}{4}$

Mean = $\frac{216}{4}$ = 54

P = {4, 8, 12, 16, 20} and Q = {2, 4, 6, 8, 10}

P ∩ Q = {4, 8}

Product means multiplication

product of the members of (P ∩ Q) = 4 x 8 = 32

Volume of the container = 3 m x 4 m x 5 m = 60 m³

Two-third is $\frac{2}{3}$

Volume of water = $\frac{2}{3}$ x Volume of the container

Volume of water = $\frac{2}{3}$ x 60 m³ = 40 m³

Notes:

1. 3 divides itself 1 time and 60, 20 times.

2. 2 x 20 = 40

Y₂ = 5
Y₁ = 3
X₂ = -4
X₁ = 2

Gradient = $\frac{\mathrm{5\; -\; 3}}{\mathrm{-4\; -\; 2}}$ = $\frac{2}{-6}$ = -$\frac{1}{3}$

Note: 2 divides itself 1 time and 6, 3 times.

Total bags = 400

Bags used = 120

Bags left = 400 - 120 = 280

Percentage left = $\frac{280}{400}$ x 100% = 70%

Note: numbers on the number line increases from left to right.

The increasing values on the number line are ..., -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Principal = GH₵ 550
Rate = 12
Simple Interest = GH₵ 132

Substitute the values into the formula and solve for the time.

132 = $\frac{\mathrm{550\; x\; Time\; x\; 12}}{100}$

Multiply both sides by the denominator, 100.

132 x 100 = 550 x Time x 12

Divide both sides by 550 x 12

Time = $\frac{\mathrm{132\; x\; 100}}{\mathrm{550\; x\; 12}}$ = 2

Notes:

1. The zero(0) in 550 and 100 cancel each other

2. 12 divides itself 1 time and 132, 11 times

3. 5 divides 55, 11 times and 10, 2 times

4. 11 at the numerator and denominator cancels each other

5. 1 x 2 = 2

5ay - by + 15a - 3b = 5ay - by + 15a - 3b

Bring out the common term and place the rest in bracket.

15 = 5 x 3

5ay - by + 15a - 3b = 5ay - by + 5 x 3a - 3b

5ay - by + 15a - 3b = y(5a - b) + 3(5a -b)

5ay - by + 15a - 3b = (5a - b)(y + 3)

5ay - by + 15a - 3b = (y + 3)(5a - b)

Note: the order of (y + 3) and (5a - b) can be reversed.

Method I

Using difference of two squares formula

(8s - 7)(8s + 7) = (8s + 7)(8s - 7)

(8s - 7)(8s + 7) = (8s)² - 7²

From law of indices, a power outside a bracket, affects each of the numbers.

(8s)² = 8² x s²

8² = 8 x 8 = 64

(8s)² = 64s²

7² = 7 x 7 = 49

(8s - 7)(8s + 7) = 64s² - 49

Method II

Expanding and simplifying

(8s - 7)(8s + 7) = (8s - 7)(8s + 7)

(8s - 7)(8s + 7) = 8s(8s + 7) - 7(8s + 7)

(8s - 7)(8s + 7) = 8s x 8s + 7 x 8s - 7 x 8s - 7 x 7

7 x 8s - 7 x 8s = 0

(8s - 7)(8s + 7) = 8s x 8s + 0 - 49

(8s - 7)(8s + 7) = 8 x 8 x s x s - 49

(8s - 7)(8s + 7) = 64s² - 49