5 divides 15, 3 times then join the 3 by the 0 to obtain 30

Note: The negative value is greater than the positive hence the result must be negative.

5 - 7 = -2

- × + = -

-3 × 3 = -9

is means equal to
Double means two times

P = 2 × r
P = 2r

Nearest 10 of 31 = 30
Nearest 10 of 18 = 20

30 x 20 = 600

Perimeter is the sum of all the sides of a shape

The triangle is an isosceles triangle (two sides are equal)

Let the two equal sides length = x = AB = AC

Perimeter = x + x + 4

Perimeter = 2x + 4

But the perimeter = 22

22 = 2x + 4

22 - 4 = 2x

18 = 2x

Divide both sides by 2

x = $\frac{18}{2}$

x = 9

AB = 9

Note: You are not suppose to repeat any of the orders

17 + 3 = 20
29 – 10 = 19

20 > 19

60 ÷ 2 + 4

Apply BODMAS

BODMAS stands for B - Brackets, O - Order of powers or roots, (in some cases, 'of'), D - Division, M - Multiplication A - Addition, and S - Subtraction.

In the order, division comes before addition, hence simplify the 60 ÷ 2 first

60 ÷ 2 = 30

60 ÷ 2 + 4 = 30 + 4
60 ÷ 2 + 4 = 34

Method I

Simplify in pence and later convert to pounds.

Cereal

If 100 grams = 49p
400 grams = $\frac{\mathrm{400\; grams\; x\; 49p}}{\mathrm{100\; grams}}$

400 grams = 4 x 49p
400 grams = 196p

4 x 49p

Pasta

If 100 grams = 14p
250 grams = $\frac{\mathrm{250\; grams\; x\; 14p}}{\mathrm{100\; grams}}$

250 grams = 5 x 7p
250 grams = 35p

Note:
1. The 0s at the end of the 250 and 100 cancels each other
2. 5 divides 10, 2 times and 25, 5 times
3. 2 divides itself once and 14, 7 times

Total cost in pence = 196p + 35p
Total cost = 231p

196 + 35

100p = £1
231p = $\frac{\mathrm{231p\; x\; \pounds 1}}{\mathrm{100p}}$

231p = £2.31

Total cost in pounds = £2.31

Method II

Convert the pence to pounce before simplifying

49p = £0.49
14p = £0.14

Cereal

If 100 grams = £0.49
400 grams = $\frac{\mathrm{400\; grams\; x\; \pounds 0.49}}{\mathrm{100\; grams}}$

400 grams = 4 x £0.49p
400 grams = £1.96

Pasta

If 100 grams = £0.14
250 grams = $\frac{\mathrm{250\; grams\; x\; \pounds 0.14}}{\mathrm{100\; grams}}$

250 grams = 5 x £0.07
250 grams = £0.35

Note:
1. The 0s at the end of the 250 and 100 cancels each other
2. 5 divides 10, 2 times and 25, 5 times
3. 2 divides itself once and 0.14, 0.07 times

Total cost = £1.96 + £0.35
Total cost = £2.31

Mode: the number which appeared the most
Median: the middle number. If there are two numbers in the middle you find their average (add the numbers and divide by 2)

Since the middle number is 12, the third number must be 12 (the third number in ordered set is the middle of five set of numbers)

Since the mode is 8, it means the first two numbers must be 8

The fourth and fifth numbers can be any numbers greater than 12

Possible Numbers

1. 8,8,12,13,14
2. 8,8,12,13,15
3. 8,8,12,14,15
4. 8,8,12,13,16

Thus your answer should be 8,8,12,...,...

Where the fourth and fifth numbers can be any numbers greater than 12.

Note: The fourth number can't be another 12 because that would make the set a bimodal (set with two modes)

Answer: 159 cm

Note: An outlier is an extremely high or extremely low data point relative to the nearest data point and the rest of the neighboring co-existing values in a data graph or dataset

Total dresses = 14

50% of the dresses are red

Red dresses = $\frac{50}{100}$ x 14

Red dresses = 7

Dresses which are not red = 14 - 7
Dresses which are not red = 7

Note:
1. 50 divides itself, 1 and 100, 2 times
2. 2 divides itself, 1 and 14, 7 times

She gives 5 of her red dresses to a charity shop

Remaining red dresses = 7 - 5
Remaining red dresses = 2

She buys 1 new red dress

Current red dresses = 2 + 1
Current red dresses = 3

Total current dresses = Dresses which are not red + current red dresses
Total current dresses = 7 + 3
Total current dresses = 10

Percentage = $\frac{\mathrm{Number}}{\mathrm{Total}}$ x 100%

Percentage of dresses which are red = $\frac{3}{10}$ x 100%

Percentage of dresses which are red = 30%

Note:
1. 10 divides itself 1 and 100, 10 times or the 0s in 100 and 10 cancel each other
2. 3 x 10 = 30

Since the units are not the same, you must convert the length to have the same units before simplifying.

You can either convert the centimetres to metres or the vice versa.

Method I

Converting from metres to centimetres

100 cm = 1 m

To convert a metre to centimetre, you simply multiple the metres by 100

1.2 m = 1.2 x 100 cm
1.2 m = 120 cm

Note: You move the decimal point to the right by the number of zeros (0). There are two zeros in 100 hence you have to move the decimal to the right two times.

1.2 = 1.20

1.20 x 100 = 1.⌒2⌒0 = 120

1m = 1 x 100 cm
1m = 100 cm

The longest side = 120 cm
The shortest side = 40 cm

$\frac{\mathrm{length\; of\; shortest\; side}}{\mathrm{length\; of\; longest\; side}}$ = $\frac{\mathrm{40\; cm}}{\mathrm{120\; cm}}$

$\frac{\mathrm{length\; of\; shortest\; side}}{\mathrm{length\; of\; longest\; side}}$ = $\frac{1}{3}$

Note:
1. The zeros (0) in 40 and 120 cancels each other
2. 4 divides itself 1 and 12, 3 times

Method II

Converting from centimetres to metres

100 cm = 1 m

To convert a centimetre to metre, you divide by 100

40 cm = $\frac{40}{100}$ m

40 cm = $\frac{4}{10}$ m

Note: The zeros (0) in 40 and 100 cancels each other

$\frac{4}{10}$ = 0.4

40 cm = 0.4 m

The shortest length = 0.4 m and the longest then = 1.2 m

For easy simplification, you can change the 1.2 m to fraction.

0.4 m = $\frac{4}{10}$ m

1.2 m = $\frac{12}{10}$ m

$\frac{\mathrm{length\; of\; shortest\; side}}{\mathrm{length\; of\; longest\; side}}$ = $\frac{4}{10}$ m ÷ $\frac{12}{10}$ m

$\frac{\mathrm{length\; of\; shortest\; side}}{\mathrm{length\; of\; longest\; side}}$ = $\frac{4}{10}$ ÷ $\frac{12}{10}$

Note: The unit (m) cancels each other

To simplify the fraction, change the the division (÷) to multiplication (x) and reciprocate the fraction after the division (÷)

$\frac{4}{10}$ ÷ $\frac{12}{10}$ = $\frac{4}{10}$ x $\frac{10}{12}$

$\frac{4}{10}$ m ÷ $\frac{12}{10}$ m = $\frac{1}{3}$

Note:
1. 10 cancels each other
2. 4 divides itself 1 and 12, 3 times

Sum of angles in a triangle is 180°

x° + y° + 112° = 180°

Substitute the value of x (3y) in the above equation.

3y + y + 112 = 180

4y + 112 = 180

4y = 180 - 112

4y = 68

Divide both sides by 4.

y = $\frac{68}{4}$

y = 17

Difference means subtraction.

Method I

Difference = £12.20 - £8.65
Difference = £3.55

12.20 - 8.65

Note:
1. Since 5 is greater than 0, 1 is borrowed from 2, making 10. 10 - 5 = 5
2. The 2 is now 1 since 1 has already been borrowed. Since 1 is lesser than 6, 1 is borrowed from 2, making 11. 11 - 6 = 5
3. Since 1 is borrowed from 2, the 12 is now 11. 11 - 8 = 3

Method II

Convert the pounds (£) to pence (p) and simplify. Once done, convert the pence back to pounce.

You convert pounds (£) to pence (p) by multiplying the pounds by 100

£12.20 = 12.20 x 100p
£12.20 = 1220p

£8.65 = 8.65 x 100p
£8.65 = 865p

Difference = 1220p - 865p
Difference = 355p

1220 - 865

Note:
1. Since 5 is greater than 0, 1 is borrowed from 2, making 10. 10 - 5 = 5
2. The 2 is now 1 since 1 has already been borrowed. Since 1 is lesser than 6, 1 is borrowed from 2, making 11. 11 - 6 = 5
3. Since 1 is borrowed from 2, the 12 is now 11. 11 - 8 = 3

Convert the pence (p) to pounds (£) by dividing the pence by 100

355p = £$\frac{355}{100}$

355p = £3.55

There are 12 months in a year.

Total Damage coverage for a year = £7.25 x 12

Total Damage coverage for a year = £87

Note:
1. Convert the decimal to whole number x 10^-n
2. Convert the whole number x 10^-n back to decimal

The n depends on how many times you have to move the decimal point to the right to obtain the whole number.

For 7.25, the decimal point has to be moved two times hence the n is 2

7.25 = 725 x 10^-2

£7.25 x 12 = 725 x 10^-2 x 12
£7.25 x 12 = 725 x 12 x 10^-2

725 x 12

Note:
1. 2 x 5 = 10. The 0 is written down and the 1 is carried forward
2. 2 x 2 = 4. The carried forward 1 is added to the 4, making it 5
3. 2 x 7 = 14
4. One space is left, 1 x 725 = 725
5. Add the two results
6. 0 and the empty space, gives 0
7. 5 + 5 = 10, 0 is written down and 1 is carried forward
8. 4 + 2 + 1 = 7
9. 1 + 7 = 8

725 x 12 x 10^-2 = 8700 x 10^-2

8700 = 8700.0 (Every number has .0)

To convert the whole number x 10^-n back to decimal, you have to move the decimal point n times to the left.

8700 x 10^-2 = 8700.0 x 10^-2

The n = 2

8700 x 10^-2 = 87⌒0⌒0.0 x 10^-2

8700 x 10^-2 = 87.0

$\frac{11}{18}$ - $\frac{1}{3}$ = $\frac{11}{18}$ - $\frac{1}{3}$

$\frac{11}{18}$ - $\frac{1}{3}$ = $\frac{\mathrm{11\; x\; 1\; -\; 1\; x\; 6}}{18}$

$\frac{11}{18}$ - $\frac{1}{3}$ = $\frac{\mathrm{11\; -\; 6}}{18}$

$\frac{11}{18}$ - $\frac{1}{3}$ = $\frac{5}{18}$

Succeeding terms are obtained by multiplying preceding terms by 2.

Let n = the first term

Second term = n x 2
Second term = 2n

Third term = second term x 2
Third term = 2n x 2
Third term = 4n

But third term = 46

4n = 46

Divide both sides by 4

n = $\frac{46}{4}$

n = 11.5

First term = 11.5

Succeeding terms are obtained by subtracting k from the preceding terms.

The first term is 34

Second term = 34 - k

Third term = 34 - k - k
Third term = 34 - 2k

Fourth term = 34 - 2k - k
Fourth term = 34 - 3k

But fourth term = 10

34 - 3k = 10
34 - 10 = 3k
24 = 3k

Divide both sides by 3

k = $\frac{24}{3}$

k = 8

You can find the translation by figuring out how much the x- and y-coordinates need to change to map the shape onto the other. You map each point to its corresponding point in the translated shape. Moving left means the translation of x is negative, and moving down means the translation of y is negative.

The translated shape moved right hence the translation of x is positive and moved down, hence the translation of y is negative.

Determine a point and it's corresponding translated point.

(2,4) → (9,2)
(2,5) → (9,3)
(5,4) → (12,2)
(5,5) → (12,3)

The vector coordinate = Image point of the coordinate - Object point of the coordinate.

The x coordinate is 7. Thus

9 - 2 = 7
9 - 2 = 7
12 - 5 = 7
12 - 5 = 7

The y coordinate is -2. Thus

2 - 4 = -2
3 - 5 = -2
2 - 4 = -2
3 - 5 = -2

The translation vector = $\left(\begin{array}{c}7\\ -2\end{array}\right)$

Method I

The total probability is 1. Thus if you sum all the probabilities, you're supposed to get 1.

P(red) + P(blue) + P(green) = 1

P(green) = 1 - [P(red) + P(blue)]

P(green) = 1 - [$\frac{1}{8}$ + $\frac{2}{5}$]

P(green) = 1 - $\frac{\mathrm{1\; x\; 5\; +\; 2\; x\; 8}}{40}$

P(green) = 1 - $\frac{\mathrm{5\; +\; 16}}{40}$

P(green) = 1 - $\frac{21}{40}$

P(green) = $\frac{\mathrm{1\; x\; 40\; -\; 21\; x\; 1}}{40}$

P(green) = $\frac{\mathrm{40\; -\; 21}}{40}$

P(green) = $\frac{19}{40}$

Number of red discs = 10

If P(red) = 10
P(green) = ?

$\frac{1}{8}$ = 10

$\frac{19}{40}$ = $\frac{19}{40}$ x 10 ÷ $\frac{1}{8}$

$\frac{19}{40}$ = $\frac{19}{4}$ ÷ $\frac{1}{8}$

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

$\frac{19}{40}$ = $\frac{19}{4}$ x $\frac{8}{1}$

$\frac{19}{40}$ = 19 x 2

$\frac{19}{40}$ = 38

There are 38 green discs.

Method II

The total probability is 1

If $\frac{1}{8}$ = 10 discs

1 = 1 x 10 ÷ $\frac{1}{8}$

1 = 1 x 10 x $\frac{8}{1}$

1 = 80 discs

Total discs = 80

P(blue) = $\frac{2}{5}$

If 1 = 80

$\frac{2}{5}$ = $\frac{2}{5}$ x 80 ÷ 1

$\frac{2}{5}$ = 2 x 16

Note:
1. 5 divides itself 1 and 80, 16 times
2. Any number divided by 1 is the same number

$\frac{2}{5}$ = 32

There are 32 blue discs

red discs + blue discs + green discs = 80
10 + 32 + green discs = 80
42 + green discs = 80
green discs = 80 - 42
green discs = 38

Method III

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

Let x = total discs

P(red) = $\frac{10}{\mathrm{x}}$

But P(red) = $\frac{1}{8}$

$\frac{1}{8}$ = $\frac{10}{\mathrm{x}}$

Cross multiply.

x x 1 = 8 x 10
x = 80

Total discs = 80

Let y = number of blue discs

P(blue) = $\frac{\mathrm{y}}{80}$

But P(blue) = $\frac{2}{5}$

$\frac{2}{5}$ = $\frac{\mathrm{y}}{80}$

Cross multiply.

5 x y = 2 x 80
5y = 160

Divide both sides by 5

y = $\frac{160}{5}$

y = 32

There are 32 blue discs.

red discs + blue discs + green discs = 80
10 + 32 + green discs = 80
42 + green discs = 80
green discs = 80 - 42
green discs = 38

Calculate the y point for selected values of x on the graph for the equation y = 2x + 1.

y = 2x + 1

When x = 0
y = 2(0) + 1
y = 2 x 0 + 1
y = 0 + 1
y = 1
Point = (0,1)

When x = 1
y = 2(1) + 1
y = 2 x 1 + 1
y = 2 + 1
y = 3
Point = (1,3)

When x = 2
y = 2(2) + 1
y = 2 x 2 + 1
y = 4 + 1
y = 5
Point = (2,5)

When x = 3
y = 2(3) + 1
y = 2 x 3 + 1
y = 6 + 1
y = 7
Point = (3,7)

Plot the points

The solution is the value of x at where the two lines intersect.

To get each value of grid, divide the interval by the number of grids in the interval.

The x-axis has 1 interval and in each intervals, 10 grids.

Each grid = $\frac{1}{10}$

Each grid = 0.1

There are two grids after the 1 at the corresponding x value of the intersection.

The x value = 1 + 2 x 0.1
The x value = 1 + 0.2
The x value = 1.2

The approximate solution of 7 – 3x = 2x + 1 is 1.2

Arcs are portions of the circumference of the circle.

Sectors are sections of a circle that are created by two radii and an arc.

The chord of a circle is a straight line that connects two points on the circumference of a circle.

A segment of a circle is created when a chord is drawn within the circle.

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:

Hourly rate = $\frac{\mathrm{Amount\; Paid}}{\mathrm{Total\; Hours}}$

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

Density = $\frac{\mathrm{Mass}}{\mathrm{Volume}}$

Mass = Density x Volume

Mass = 2 g/cm³ x 300 cm³
Mass = 600 g

x : y is 9 : 5

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

$\frac{\mathrm{2x}}{\mathrm{y}}$ = 2 x $\frac{\mathrm{x}}{\mathrm{y}}$

But $\frac{\mathrm{x}}{\mathrm{y}}$ = $\frac{9}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = 2 x $\frac{9}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = $\frac{\mathrm{2\; x\; 9}}{5}$

$\frac{\mathrm{2x}}{\mathrm{y}}$ = $\frac{18}{5}$