The equation y = mx + c is the general equation of any straight line where m is the gradient of the line (how steep the line is) and c is the y -intercept (the point in which the line crosses the y -axis). The power of x of a straight line equation is 1.

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.

0.28 = 0.⌒2⌒8

0.28 = 28 x 10^-2

0.8 = 0.⌒8

0.8 = 8 x 10^-1

$\frac{0.28}{0.8}$ = $\frac{\mathrm{28\; x}{10}^{-2}}{\mathrm{8\; x}{10}^{-1}}$

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{\mathrm{-2-(-1)}}}{2}$

Note:
1. 4 divides 28, 7 times and 8, 2 times
2. From the law of indices the power dividing can be subtracted from the power at the top since the bases are the same

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{\mathrm{-2+1}}}{2}$

Note: -- = +

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x}{10}^{-1}}{2}$

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

10^-1 = $\frac{1}{{10}^1}$

10^-1 = $\frac{1}{10}$

$\frac{0.28}{0.8}$ = $\frac{7}{2}$ x $\frac{1}{10}$

$\frac{0.28}{0.8}$ = $\frac{\mathrm{7\; x\; 1}}{\mathrm{2\; x\; 10}}$

$\frac{0.28}{0.8}$ = $\frac{7}{20}$

Total percentage for the increment = 100% + 7.5%
Total percentage for the increment = 107.5%

107.5% = $\frac{107.5}{100}$

107.5% = 1.075

If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent by RHS rule.

80 000 000 ÷ 200 = $\frac{\mathrm{80\; 000\; 000}}{200}$

80 000 000 ÷ 200 = 400 000

Note:
1. Two of the zeros (0) at the end of 80 000 000 and 200 cancels each other
2. 2 divides itself 1 and 8, 4 times

Convert the 400 000 to standard form

A standard form should be in the form a.bcde x 10ⁿ

The decimal point must be after the first non-zero digit.

In the 400 000, the decimal point must be after 4.

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 number has .0 behind which is not written.

400 000 = 400 000.0

We have to move the decimal point 5 times to the left hence our n is 5 and positive

400 000.0 = 4⌒0⌒0 ⌒0⌒0⌒0.0

400 000.0 = 4.0 x 10⁵

Law of division of indices

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

$\frac{3^{12}}{3^7}$ = 3^{12 - 7}

$\frac{3^{12}}{3^7}$ = 3⁵

$\frac{3^{12}}{3^7}$ = 3 x 3 x 3 x 3 x 3

$\frac{3^{12}}{3^7}$ = 243

Before you can apply the law of indices, all the bases must be the same.

Change the 8 to base 2.

8 = 2 x 2 x 2

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Every number has a power of 1 which isn't written.

2 = 2¹

8 = 2¹ x 2¹ x 2¹

8 = 2^{1 + 1 + 1}
8 = 2³

8 × 2⁶ × 2⁴ = 2³ × 2⁶ × 2⁴
8 × 2⁶ × 2⁴ = 2^{3 + 6 + 4}
8 × 2⁶ × 2⁴ = 2¹³

1. The circles should be labelled or the labels are missing
2. The numbers add to 99 instead of 98 or 48 should be 47

The correct venn diagram should be:

Overtime pay = £700 - £450
Overtime pay = £250

Overtime hours = 40 - 30
Overtime hours = 10

Basic hourly rate = $\frac{450}{30}$

Basic hourly rate = 15

Note:
1. The zeros (0) at the end of 450 and 30 cancels each other
2. 3 divides itself 1 and 45, 15 times

Overtime hourly rate = $\frac{250}{10}$

Overtime hourly rate = 25

basic hourly rate : overtime hourly rate = 15 : 25
basic hourly rate : overtime hourly rate = 3 : 5

Note: 5 divides 15, 3 times and 25, 5 times.

1. $\frac{3}{5}$ and $\frac{1}{2}$

2. $\frac{6}{10}$ and $\frac{5}{10}$

3. $\frac{2}{5}$ and $\frac{3}{4}$

1. 0.3 and 0.2

2. 0.60 and 0.10

3. 0.5 and 0.12

4. 0.75 and 0.08

1. At B, draw an arc to cut both AB and BC

2. Move the compass point to the point where the arc cuts BC and construct an arc with the same radius of the compass used in step 1

3. Move the compass point to the point where the arc cuts AB and construct an arc with the same radius of the compass used in step 1

4. Draw a line from point B to pass through the point of intersection of the two arcs drawn in step 2 and 3 all the way through the garden

5. Shade the region between AB and the straight line drawn in step 4 as R

Area of a sector = fraction of the circle x π x radius square

Area for section P = $\frac{3}{4}$ x π x 20²

Area for section P = $\frac{3}{4}$ x π x 20 x 20

Area for section P = 3 x π x 20 x 5

Note: 4 divides itself 1 and 20, 5 times.

Area for section P = 300π

Area for section Q = $\frac{1}{3}$ x π x 15²

Area for section P = $\frac{1}{3}$ x π x 15 x 15

Area for section P = 1 x π x 15 x 5

Note: 3 divides itself 1 and 15, 5 times.

Area for section P = 75π

$\frac{\mathrm{Area\; of\; P}}{\mathrm{Area\; of\; Q}}$ = $\frac{\mathrm{300\pi }}{\mathrm{75\pi }}$

$\frac{\mathrm{Area\; of\; P}}{\mathrm{Area\; of\; Q}}$ = 4

Area of P is 4 times bigger than area of Q.

$\frac{\mathrm{2w}}{15}$ = $\frac{4}{5}$

Cross multiply

2w x 5 = 15 x 4
10w = 60

Divide both sides by 10

w = $\frac{60}{10}$

w = 6

The more the workers, the lesser the time.

Hence time (t) is inversely proportional to the number of workers (w).

t ∝ $\frac{1}{\mathrm{w}}$

Remove the proportionality by introducing a constant (k).

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

Use the known information to solve for k

15 workers used 8 days.

8 = k x $\frac{1}{15}$

k = 8 x 15

k = 120

The general equation is:

t = 120 x $\frac{1}{\mathrm{w}}$

t = $\frac{120}{\mathrm{w}}$

Calculate how many workers would be needed for 6 days.

t = 6

6 = $\frac{120}{\mathrm{w}}$

Multiply each side of the equation by the denominator w

6 x w = w x $\frac{120}{\mathrm{w}}$

6w = 120

Divide both sides by 6

w = $\frac{120}{6}$

w = 20

Note: 6 divides itself 1 and 12, 2 times then join the zero (0)

∴ It will take 20 workers to complete in 6 days.

More workers needed = 20 workers - 15 workers
More workers needed = 5 workers

A prism whose bases are n-sided polygons has 2n vertices, n+2 faces and 3n edges.

13 at 50 (years)
30 at 60 (years)
66 at 70 (years)

Differences of readings

50 - 60 years = 30 - 13 = 17
60 - 70 = 66 - 30 = 36

Yes because 36 is more than twice as 17.

1. Expand
2. Group the terms which doesn't contain a, b and c at the left
3. Compare the expressions at the left to the expressions at the right to obtain your values for a, b and c

12x³ + 7x² + 3x – 10 ≡ 2(ax³ + x² + 2x – 5) + x(bx + c)

12x³ + 7x² + 3x – 10 ≡ 2 x ax³ + 2 x x² + 2 x 2x – 2 x 5 + x x bx + x x c

12x³ + 7x² + 3x – 10 ≡ 2ax³ + 2x² + 4x – 10 + bx² + cx

12x³ + 7x² - 2x² + 3x - 4x – 10 + 10≡ 2ax³ + bx² + cx

12x³ + 5x² - x ≡ 2ax³ + bx² + cx

Comparing,

12 = 2a

Divide both sides by 2.

a = $\frac{12}{2}$

a = 6

b = 5

c = -1

Fourth term = xy x y
Fourth term = xy²

Fifth term = xy² x xy
Fifth term = x¹⁺¹y²⁺¹
Fifth term = x²y³

Note:
1. Every letter is raised to the power 1 but not written
2. x = x¹
3. y = y¹
4. a^m x aⁿ = a^{m + m}

For any variable/number with a power which is even, it doesn't matter whether the variable/number is negative or positive as the result will still be positive.

x² = x x x
(-x)² = -x x -x = x²

x has a power of even (8), hence it could either be a negative or positive and the result will still be positive.

In order for the term to be negative, y must there be negative since it has an odd power so that the result is negative to render the product (x⁸y¹³) negative.

y = $\frac{\mathrm{5x\; +\; 9}}{\mathrm{x}}$

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

x x y = x x $\frac{\mathrm{5x\; +\; 9}}{\mathrm{x}}$

xy = 5x + 9

xy - 5x = 9

Factorize x out.

x(y - 5) = 9

Divide both sides by (y - 5)

x = $\frac{9}{\mathrm{y\; -\; 5}}$

Note: If you took the 9 to the left side and the x to the right, your answer should be:

x = $\frac{\mathrm{-\; 9}}{\mathrm{5\; -\; y}}$

DH→ = a + 6b

EG→ = 2a + 15b

ED→ = 3a + b

EH→ = ED→ + DH→

EH→ = 3a + b + a + 6b

EH→ = 4a + 7b

FG→ = 2EH→

FG→ = 2(4a + 7b)

FG→ = 8a + 14b

FE→ = FG→ + GE→

But GE→ = -EG→

GE→ = - (2a + 15b)

GE→ = -2a -15b

FE→ = 8a + 14b -2a -15b

FE→ = 8a - 2a + 14b - 15b

FE→ = 6a - b

Method I

Since both recurring part are at the second decimal places, you can subtract and convert the result to fraction.

0.68. - 0.45. = 0.23.

Convert 0.23. to fraction.

x = 0.23..
10x = 2.33..

10x - x = 2.33.. - 0.23..

9x = 2.1

Divide both sides by 9

x = $\frac{2.1}{9}$

2.1 = $\frac{21}{10}$

$\frac{2.1}{9}$ = $\frac{21}{10}$ ÷ 9

Every number is divisible by 1

9 = $\frac{9}{1}$

$\frac{2.1}{9}$ = $\frac{21}{10}$ ÷ $\frac{9}{1}$

Change the division sign (÷) to multiplication (x) and reciprocate the fraction after the division sign (÷)

$\frac{2.1}{9}$ = $\frac{21}{10}$ x $\frac{1}{9}$

$\frac{2.1}{9}$ = $\frac{\mathrm{21\; x\; 1}}{\mathrm{10\; x\; 9}}$

3 divides 9, 3 times and 21, 7 times.

$\frac{2.1}{9}$ = $\frac{\mathrm{7\; x\; 1}}{\mathrm{10\; x\; 3}}$

$\frac{2.1}{9}$ = $\frac{7}{30}$

Method II

Convert each of the recurring decimals to fraction and simplify.

0.68.

x = 0.68..
10x = 6.88..

10x - x = 6.88.. - 0.68..
9x = 6.2

Divide both sides by 9

x = $\frac{6.2}{9}$

6.2 = $\frac{62}{10}$

$\frac{6.2}{9}$ = $\frac{62}{10}$ ÷ 9

Every number is divisible by 1

9 = $\frac{9}{1}$

$\frac{6.2}{9}$ = $\frac{62}{10}$ ÷ $\frac{9}{1}$

Change the division sign (÷) to multiplication (x) and reciprocate the fraction after the division sign (÷)

$\frac{6.2}{9}$ = $\frac{62}{10}$ x $\frac{1}{9}$

$\frac{6.2}{9}$ = $\frac{\mathrm{62\; x\; 1}}{\mathrm{10\; x\; 9}}$

$\frac{6.2}{9}$ = $\frac{62}{90}$

0.45.

x = 0.45..
10x = 4.55..

10x - x = 4.55.. - 0.45..
9x = 4.1

Divide both sides by 9

x = $\frac{4.1}{9}$

4.1 = $\frac{41}{10}$

$\frac{4.1}{9}$ = $\frac{41}{10}$ ÷ 9

Every number is divisible by 1

9 = $\frac{9}{1}$

$\frac{4.1}{9}$ = $\frac{41}{10}$ ÷ $\frac{9}{1}$

Change the division sign (÷) to multiplication (x) and reciprocate the fraction after the division sign (÷)

$\frac{4.1}{9}$ = $\frac{41}{10}$ x $\frac{1}{9}$

$\frac{4.1}{9}$ = $\frac{\mathrm{41\; x\; 1}}{\mathrm{10\; x\; 9}}$

$\frac{4.1}{9}$ = $\frac{41}{90}$

0.68. - 0.45. = $\frac{62}{90}$ - $\frac{41}{90}$

0.68. - 0.45. = $\frac{\mathrm{62\; -\; 41}}{90}$

0.68. - 0.45. = $\frac{21}{90}$

3 divides 21, 7 times and 90, 30 times.

0.68. - 0.45. = $\frac{7}{30}$

1. Use dashed lines to indicate the >

2. Use solid line to indicate ≤

Plot x = 3 using dashed lines
Plot y = 1 using dashed lines

x + y ≤ 7

1. Make y the subject, y ≤ 7 - x
2. Plot y = 7 - x
3. Pick any x value on the grid to calculate the corresponding y value

When x = 2
y = 7 - 2
y = 5

When x = 4
y = 7 - 4
y = 3

When x = 6
y = 7 - 6
y = 1

Draw a straight line through the points.

Indicate the R region, the region which matches all the three inequalities.

$\frac{6}{\mathrm{a}}$ - $\frac{11}{\mathrm{4a}}$ = $\frac{6}{\mathrm{a}}$ - $\frac{11}{\mathrm{4a}}$

$\frac{6}{\mathrm{a}}$ - $\frac{11}{\mathrm{4a}}$ = $\frac{\mathrm{4\; x\; 6\; -\; 1\; x\; 11}}{\mathrm{4a}}$

$\frac{6}{\mathrm{a}}$ - $\frac{11}{\mathrm{4a}}$ = $\frac{\mathrm{24\; -\; 11}}{\mathrm{4a}}$

$\frac{6}{\mathrm{a}}$ - $\frac{11}{\mathrm{4a}}$ = $\frac{13}{\mathrm{4a}}$

y² - 3y = y(y - 3)

y² - 9 is a difference of two square.

Difference of two square

a² - b² = (a + b)(a - b)

9 = 3 x 3 = 3²

y² - 9 = y² - 3²

y² - 9 = (y + 3)(y - 3)

y² + 10y + 21

Factorizing y² + bx + c

Split b such that two numbers when multiplied would result c and when added would result b

y² + 10y + 21

c = 21
b = 10

Two numbers when multiplied would result 21 and when added would result 10.

The two numbers are 7 and 3. Hence split 10 into 7 + 3

y² + 10y + 21 = y² + 7y + 3y + 21

Factorize out.

y² + 10y + 21 = y(y + 7) + 3(y + 7)

y² + 10y + 21 = (y + 7)(y + 3)

Substitue each of the simplied parts into the expression and cancel out.

(y² – 3y) × $\frac{{\mathrm{y}}^2\mathrm{+\; 10y\; +\; 21}}{{\mathrm{y}}^2\mathrm{-\; 9}}$ = y(y - 3) x $\frac{\mathrm{(y\; +\; 7)(y\; +\; 3)}}{\mathrm{(y\; +\; 3)(y\; -\; 3)}}$

Note:
1. (y - 3) at the left and denominator cancels each other
2. (y + 3) at the numerator and denominator cancels each other

(y² – 3y) × $\frac{{\mathrm{y}}^2\mathrm{+\; 10y\; +\; 21}}{{\mathrm{y}}^2\mathrm{-\; 9}}$ = y(y + 7)

You could leave the answer as y(y + 7) or expand.

(y² – 3y) × $\frac{{\mathrm{y}}^2\mathrm{+\; 10y\; +\; 21}}{{\mathrm{y}}^2\mathrm{-\; 9}}$ = y x y + y x 7

(y² – 3y) × $\frac{{\mathrm{y}}^2\mathrm{+\; 10y\; +\; 21}}{{\mathrm{y}}^2\mathrm{-\; 9}}$ = y² + 7y

If an objects speed remains constant, then its acceleration must be zero.

Halfway through the journey, speed was constant (20 m/s) hence the acceleration is 0

The distance covered is the area of the shape of the graph.

The area could be calculated in two ways:
1. using the area of a trapezium formua
2. Grouping the shapes into 2 triangles and 1 rectangle and using the formula for calculating the area of a triangle and rectangle to calculate each area.

Method I (Area of a trapezium)

Grom the graph,

b = 50
a = 40 - 10
a = 30
h = 20

Total distance travelled = $\frac{1}{2}$ x (50 + 30) x 20

Total distance travelled = $\frac{1}{2}$ x 80 x 20

Total distance travelled = 80 x 10
Total distance travelled = 800

Note: 2 divides itself 1 and 20, 10 times.

Method II (Separate Shapes)

Area of triangle = $\frac{1}{2}$ x base x height

Area of a rectangle = Length x Breadth

Total distance = Area of A + Area of B + Area of C

Area of A

Area of A = $\frac{1}{2}$ x 10 x 20

Note: 2 divides itself 1 and 20, 10 times.

Area of A = 10 x 10
Area of A = 100

Area of B

Length = 40 - 10
Length = 30

Area of B = 30 x 20
Area of B = 600

Area of C

Base = 50 - 40
Base = 10

Area of C = $\frac{1}{2}$ x 10 x 20

Note: 2 divides itself 1 and 20, 10 times.

Area of A = 10 x 10
Area of A = 100

Total distance = 100 + 600 + 100
Total distance = 800

A dice is number 1,2,3,4,5,6

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

The probability of getting any of the numbers is $\frac{1}{6}$

Zoe wins within her next two turns. This means Zoe may win during her first turn or second turn.

Zoe needs to move 3 to get HOME.

Possible scenarios

1. Zoe wins during her first turn by getting 3.
2. Zoe gets 2 during her first turn and 1 during her second turn
3. Zoe gets 1 during her first turn and 2 during her second turn

P(3) = $\frac{1}{6}$

P(2,1) = $\frac{1}{6}$ x $\frac{1}{6}$

P(2,1) = $\frac{1}{36}$

P(1,2) = $\frac{1}{6}$ x $\frac{1}{6}$

P(1,2) = $\frac{1}{36}$

P(win) = $\frac{1}{6}$ + $\frac{1}{36}$ + $\frac{1}{36}$

P(win) = $\frac{\mathrm{1x\; 6\; +\; 1\; x\; 1\; +\; 1\; x\; 1}}{36}$

P(win) = $\frac{\mathrm{6\; +\; 1\; +\; 1}}{36}$

P(win) = $\frac{8}{36}$

Note:
4 divides 8, 2 times and 36, 9 times

P(win) = $\frac{2}{9}$

y = –f(x) means reflection in the x-axis.

In the reflection of the x-axis, the y coordinate is negated.

The point (x,y) is sent to (x,-y)

Pick points on the y = f(x) curve and negate the y coordinate and plot. Use dashes to draw the curve.

(0,1) → (0,-1)
(-1,2.2) → (-1,-2.2)
(2,1) → (2,-1), etc.

cos 30° = $\frac{\sqrt{3}}{2}$

sin 45° = $\frac{\sqrt{2}}{2}$

tan 60° = $\sqrt{3}$

cos 30° × sin 45° × tan 60° = $\frac{\sqrt{3}}{2}$ x $\frac{\sqrt{2}}{2}$ x $\sqrt{3}$

cos 30° × sin 45° × tan 60° = $\frac{\sqrt{\mathrm{3\; x\; 2\; x\; 3}}}{\mathrm{2\; x\; 2}}$

cos 30° × sin 45° × tan 60° = $\frac{\sqrt{\mathrm{18}}}{4}$

$\sqrt{a}$ = ${\mathrm{a}}^{\frac{1}{2}}$

$\sqrt{\mathrm{18}}$ = ${18}^{\frac{1}{2}}$

cos 30° × sin 45° × tan 60° = $\frac{{18}^{\frac{1}{2}}}{4}$

${\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{c}}$ = $\frac{{\mathrm{a}}^{\mathrm{c}}}{{\mathrm{b}}^{\mathrm{c}}}$

(cos 30° × sin 45° × tan 60°)² = $\frac{{18}^{\frac{1}{2}\mathrm{x\; 2}}}{4^2}$

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

Note: The 2s cancels each other.

18¹ = 18

4² = 4 x 4
4² = 16

(cos 30° × sin 45° × tan 60°)² = $\frac{18}{16}$

Note:2 divides 18, 9 times and 16, 8 times.

(cos 30° × sin 45° × tan 60°)² = $\frac{9}{8}$ or 1$\frac{1}{8}$ or 1.125