Step 1

Find the least common multiple (LCM)

LCM for 4, 3 and 12 is 12, thus

Multiples of 4

4,8,12,16,20,24,28,32,36,40,44,48

Multiples of 3

3,6,9,12,15,18,21,24,27,30,33,36

Multiples of 12

Multiples of 12 are numbers 12 can divide without a remainder (Simply the multiplication of 12). If you don't know the recitation of the multiples, simply add 12 to the preceding numbers to get the subsequent numbers starting from 12.

12,12+12=24,24+12 = 36,36+12 = 48,48+12 = 60,60+12 = 72,72+12 = 84,84+12 = 96,96+12 = 108,108+12 = 120,120+12 = 132,132+12 = 144,...

⇒ 12,24,36,48,60,72,84,96,108,120,132,144,...

From the above you can see that the least number which appeared in all the above multiples is 12, hence the LCM for 3,4 and 12 is 12

Step 2

Divide by the LCM and find out how many times the denominator goes into the LCM and multiply that by the numerator and maintain the signs

Thus for ¾ , the numerator is 3 and the denominator is 4 and 4(denominator) goes into 12(the LCM),3 times, hence 3 x 3 (numerator)

Thus for -⅓ , the numerator is 1 and the denominator is 3 and 3(denominator) goes into 12(the LCM),4 times, hence 4 x 1 (numerator) and maintain the negative sign.

Thus for +1⁄12 , the numerator is 1 and the denominator is 12 and 12(denominator) goes into 12(the LCM),1 time, hence 1 x 1 (numerator) and maintain the positive sign.

Now simplify the numerator first and afterward simplify the new fraction

As shown below

x:x is read as x is such that x is ...

N contains the list of factors of 18. A factor is a number or algebraic expression that divides another number or expression evenly, thus no remainder.

Factors of 18 are 1,2,3,6,9 and 18 itself

Thus

1x18 = 18

2x9 = 18

3x6 = 18

Hence 1,2,3,6,9 and 18 are factors of 18

⇒ N = {1,2,3,6,9,18}

Multiples of 12 are numbers 12 can divide without a remainder (Simply the multiplication of 12). If you don't know the recitation of the multiples, simply add 12 to the preceding numbers to get the subsequent numbers starting from 12.

Multiples of 12

12,12+12=24,24+12 = 36,36+12 = 48,48+12 = 60,60+12 = 72,72+12 = 84,84+12 = 96,96+12 = 108,108+12 = 120,120+12 = 132,132+12 = 144,...

⇒ multiples of 12 are 12,24,36,48,60,72,84,96,108,120,132,144,...

⇒ M = {12,24,36,48,60,72,84,96,108,120,132,144,...}

∩ means intersection. The element(s) which you can find in both sets.

Hence N ∩ M should contain element(s) which can be found in both set N and M.

From the above, you can see that there is no element that can be found in N,{1,2,3,6,9,18} and at the same time be found in M,{12,24,36,48,60,72,84,96,108,120,132,144,...}. Hence the intersection of N and M is an empty set {}

Standard form, or standard index form, is a system of writing numbers which can be particularly useful for working with very large or very small numbers.

It is based on using powers of 10 to express how big or small a number is.

Standard form is written in the form of a x 10ⁿ, where a is a number bigger than or equal to 1 and less than 10.

n can be any positive or negative whole number.

For example 1.2 x 10³, 3.14 x 10^-5

The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Thus the decimal point must be behind the first non-zero digit

If you move the decimal point to the left, the power (number of times of movement) of 10 will be positive and if to the right, negative

The first non-zero digit in 4382.93 is 4, hence we have to move the decimal point (.) 3 times to the left before placing it behind the 4. Since the movement is to the left, the power (3) of 10 is positive.

Associative Property

This property states that when three or more are added (or multiplied), the sum (or the ) is the same regardless of the grouping of the (or the multiplicands).

Associative property gets its name from the word "Associate" and it refers to grouping of numbers.

Grouping means the use of parentheses or brackets to group numbers.

Associative property involves 3 or more numbers.

The numbers that are grouped within a parenthesis or bracket become one unit.

Associative property can only be used with addition and multiplication and not with subtraction or division.

Example of Associative Property for Addition

Examples of Associative Property for Multiplication:

The above examples indicate that changing the grouping doesn't make any changes to the answer.

The associative property is helpful while adding or multiplying multiple numbers. By grouping, we can create smaller components to solve. It makes the calculations of addition or multiplication of multiple numbers easier and faster.

Example Addition:

17 + 5 + 3 = (17 + 3) + 5
           = 20 + 5
           = 25

Here, adding 17 and 3 gives 20. Then, adding 5 to 20 gives 25. The grouping helped to find the answer easily and quickly.

Example Multiplication:
3 × 4 × 25 = (25 × 4) × 3
           = 100 × 3
           = 300

Here, multiplying 25 by 4 gives 100. Then, 3 can be easily multiplied by 100 to get 300.

However, we cannot apply the associative property to subtraction or division. When we change the grouping of numbers in subtraction or division, it changes the answer and hence, this property is not applicable.

Example Subtraction:

10 – (5 – 2) = 10 - 3 = 7
(10 – 5) – 2 = 5 – 2 = 3
So, 10 – (5 – 2) ≠ (10 – 5) – 2

Example Division:

(24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3
24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12
So, (24 ÷ 4) ÷ 2 ≠ 24 ÷ (4 ÷ 2)

Commutative Property

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.

The above examples clearly show that we can apply the commutative property on addition and multiplication. However, we cannot apply commutative property on subtraction and division. If you move the position of numbers in subtraction or division, it changes the entire problem.

Therefore, if a and b are two non-zero numbers, then:

The commutative property of addition is:
a + b = b + a

The commutative property of multiplication is:
a × b = b × a

In short, in commutative property, the numbers can be added or multiplied to each other in any order without changing the answer.

Distributive Property

To "distribute" means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

Example

( 5 + 7 + 3 ) x 4 = 15 x 4
                  = 60

This can be solved using the distributive property as:
( 5 + 7 + 3 ) x 4 = 5x4 + 7x4 + 3x4
                  = 20+28+12
                  = 60

Subtract (7x-3) from (5-3x) ⇒ (5-3x) - (7x-3)

(5-3x) - (7x-3) = (5-3x) - 1(7x-3)

NOTE: Every number is multiplied by 1 which is the same number or has no effect, thus - (7x-3) = - 1(7x-3)

Applying the distributive property, a(x+y) = axx+axy

⇒ - 1(7x-3) = -1x7x-1x(-3) = -7x+3

NOTE: - x - = +

(5-3x) - 1(7x-3) = 5-3x -7x+3

Grouping like terms ⇒ 5-3x -7x+3 = 5+3 -3x-7x

⇒ 5-3x -7x+3 = 8 -10x

Let x be the cost of one book

12 books = 54.84

1 book = x

Cross multiplying the equations

⇒ 12 x x = 1 x 54.84

Divide both sides by 12

⇒ x = 54.84 / 12

⇒ x = 4.57

Thus

To simplify the fraction, change the decimal number to a form which has 10 raised to the power by moving the decimal position to the right until you get a whole number.

Multiply that whole number by 10 raised to the power negative of the number of times you moved to the right

Thus 58.84 = 5884 x 10^-2

NOTE:The power is negative if you moved to the right and positive if you move to the left

⇒ x = 54.84 / 12 = 5884 x 10^-2/12

Using long division to simplify,

⇒ 5884 x 10^-2/12 = 457 x 10^-2

Now change 457 x 10^-2 back to decimal.

457 is the same as 457.0, thus every whole number has point zero decimal

Since the 10 is raised to the power -2, we move the decimal point to the left 2 times, which gives us 4.57

Alternatively, you can multiply each of the possible answers by 12 and the one that gives you the total cost of the 12 books (GH₵ 54.84) is the answer.

Plotting inequalities on number lines

Inequalities can be represented on a number line.

Use a hollow dot for:
< and >

Use a solid dot for:
≤ and ≥

Example

This shows x is greater than or equal to : -1

x ≥ -1

Example

This shows x is less than : 2

x < 2

Example

This shows that x is greater than or equal to -1 and x is less than 3 :

x ≥ -1 and x < 3

Which can be written as -1 ≤ x < 3, which reads -1 is less than or equal to x and x is less than 3

Notice that we can change the order of the first part and it still means the same thing.

x ≥ -1 means as -1 ≤ x

Concepts

1. a^x x a^y = a^x+y, thus if the bases are the same and multiplying, add the powers

2. (a^xx b^y)ⁿ = a^{x x n} x b^{y x n}, thus the outside power multiplies all the powers inside

Applying these two concepts,

(2ab²)² = 2^1x2a^1x2b^2x2 = 2²a²b⁴ = 4a²b⁴

NOTE: Every number or letter is raised to the power 1, which has no effect, hence 2 = 2¹ and a = a¹

⇒ (2ab²)² x 3a³b = 4a²b⁴ x 3a³b

Simplifying the numbers and the ones with the same bases

⇒ 4a²b⁴ x 3a³b = 4 x 3 x a²xa³b⁴xb

Applying concept 1,a^x x a^y = a^x+y

⇒ 4 x 3 x a²xa³b⁴xb = 12a²⁺³b⁴⁺¹ = 12a⁵b⁵

NOTE: b = b¹ ⇒ b⁴xb = b⁴xb¹ = b⁴⁺¹ = b⁵

A line of symmetry is a line that cuts a shape exactly in half.

This means that if you were to fold the shape along the line, both halves would match exactly. Equally, if you were to place a mirror along the line, the shape would remain unchanged.

The shape which it is not possible to divide into equal halves is the answer

Kite

Isosceles Triangle

Trapezium

Isosceles Trapezium

Rhombus

Trapezium has no symmetry except isosceles trapezium but since the question didn't specify that the trapezium is isosceles, then the answer is Trapezium

Profit = Selling Price - Cost Price

Profit Percent = Profit/Cost Price x 100

Since Profit = Selling Price - Cost Price, substituting it in the second equation,

Profit Percent = (Selling Price - Cost Price)/Cost Price x 100

Profit percent = 8

Selling Price = 72

Let Cost Price = CP

⇒ 8 = (72 - CP)/CP x 100

Multiplying both sides by CP

8 x CP = (72 - CP) x 100

(72-CP) x 100 = 72 x 100 - CP x 100 = 7200 - 100CP

8 x CP = (72 - CP) x 100 ⇒

8CP = 7200 - 100CP

Moving the -100CP to the left size ⇒

8CP + 100CP = 7200

108CP = 7200

Dividing both sides by 108

CP = 7200/108 = 66.666667 = 67 (Nearest whole number)

Each person's share from the ratio is calculated as:
(The ratio / Sum of ratios) x Total being shared

Vera:Brother = 11:9

Sum of ratios = 11+9 = 20

Vera's share = (11/20) x 60 = 11 x 3 = 33

NOTE: 20 goes into 60, 3 times and the 3 multiplies the 11 at the top and the result is 33

Brother's share = (9/20) x 60 = 9 x 3 = 27

To get how much more Vera got, subtract the brother's share from Vera's share to know how much more she got

Thus 33 - 27 = 6

Get rid of the square by taking square root of both sides of the equation

(x-3)² = 16

√(x-3)² = √16

√(x-3)² = x-3

√16 = 4

NOTE: √a = a^½

⇒ √(x-3)² = (x-3)^2x½ = (x-3)¹ = (x-3) = x-3

(x-3)² = 16

⇒ x-3 = 4

⇒ x = 4+3 = 7

The rule for y -axis reflection is (x,y)→(−x,y), thus reflection in the y-axis, the x is negated.

⇒ (-2,3) reflected in the y-axis will be -(-2),3 but -- = +, hence (2,3)

A reflection is a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane. When reflecting a figure in a line or in a point, the image is congruent to the preimage.

A reflection maps every point of a figure to an image across a fixed line. The fixed line is called the line of reflection.

Some simple reflections can be performed easily in the coordinate plane using the general rules below.

Reflection in the x -axis:

A reflection of a point over the x -axis is shown.

The rule for a reflection over the x -axis is (x,y)→(x,−y), thus reflection in the x-axis, the y is negated.

Reflection in the y -axis:

A reflection of a point over the y -axis is shown.

The rule for a reflection over the y -axis is (x,y)→(−x,y), thus reflection in the y-axis, the x is negated.

Reflection in the line y=x :

A reflection of a point over the line y=x is shown.

The rule for a reflection in the line y=x is (x,y)→(y,x), thus you simply switch their positions.

Reflection in the line y=−x :

A reflection of a point over the line y=−x is shown.

The rule for a reflection in the origin is (x,y)→(−y,−x), thus you simply negate the result of y = x.

Fraction of a total is calculated as the value of what fraction is being calculated over the total

Thus fraction of a value = value/total, then you simplify the fraction.

Before you calculate the fraction, make sure they are of the same units. If one is in a different unit, change the other so they are all in the same unit.

Since the total (3 weeks) is in weeks, change the weeks to days

There are 7 days in 1 week hence 3 weeks will be 3 x 7 days = 21 days

Hence the fraction for the 18 days will be 18 days / 21 days

3 goes into 18, 6 times and into 21, 7 times. Hence the simplification of 18/21 = 6/7

The median is the middle score in a set of given numbers.

The mean is the arithmetic average of a set of given numbers.

The mode is the most frequently occurring score in a set of given numbers.

Median

The median is the middle score of a distribution. To calculate the median
* Arrange your numbers in numerical order.
* Count how many numbers you have.
* If you have an odd number, divide by 2 and round up to get the position of the median number.
* If you have an even number, divide by 2. Go to the number in that position and average it with the number in the next higher position to get the median.

In the question above, there are 12 numbers. 12 is even and so to get the median position, divide 12 by 2, which falls on the 6th number.

The numbers are not ordered so you have to order them from the lowest to the highest before determining the 6th position

⇒ 4,16,13,18,3,20,6,7,15,2,10,12 = 2,3,4,6,7,10,12,13,15,16,18,20

The 6th number from the ordered numbers is 10

Since the total number count (12) is even, the median is the average of the 6th and its next number(7th number)

Average simply means the sum of the two numbers divided by 2

So the median will be the 6th number + the 7th number divided by 2

⇒ the median = (10+12)/2 = 22/2 = 11

If the total number count is odd, the median is simply the middle number in the ordered numbers

Mean

The mean, or average, is calculated by adding up the scores and dividing the total by the number of scores. Consider the following number set: 3, 4, 6, 6, 8, 9, 11.

The mean is calculated in the following manner:
3 + 4 + 6 + 6 + 8 + 9 + 11 = 47
number of scores is 7
⇒ The mean = 47 / 7 = 6.7

The mean (average) of the number set is 6.7.

Mode

Since the mode is the most frequently occurring score in a distribution, simply select the most common score as your mode.

Consider the following number distribution of 2, 3, 6, 3, 7, 5, 1, 2, 3, 9.

The mode of these numbers would be 3 since three(3) is the most frequently occurring number.

In some number sets, there may actually be two modes. This is known as bi-modal distribution and it occurs when there are two numbers that are tied in frequency.

For example, consider the following set of numbers: 13, 17, 20, 20, 21, 23, 23, 26, 29, 30.

In this set, both 20 and 23 occur twice.

If no number in a set occurs more than once, then there is no mode for that set of data.

Get rid of the fraction by multiplying each side of the equation by the LCM of the denominators (3,1 and 1)

Thus 3(w-1) = 3(w-1)/1 (every number is being divided by 1 which has no effect on the number)

-1 = -1/1

Hence the denominators are 3,1 and 1 and the LCM is 3

Multiply both sides of the equation by 3

3 x w/3 = 3 x 3(w-1)-3x1

3 cancels 3 in 3 x w/3 = w

3 x w/3 = 3 x 3(w-1)-3x1

⇒ w = 9(w-1)-3

Applying the concept, a(b+c) = axb+axc, a(b-c) = axb-axc

⇒ 9(w-1) = 9xw - 9x1 = 9w-9

⇒ w = 9w-9-3

Grouping the letters (w) at one side and the numbers at the other side of the equation

Concept, when negative moves to the other side of the equation, becomes positive and positive becomes negative

Option 1, grouping the w at the left side of equation

w = 9w-9-3

⇒ w - 9w = -9-3

- 8w = -12

Divide both sides by -8

-8w/-8 = -12/-8

NOTE: The negative cancels each other

w = 12/8 = 3/2, thus 4 goes into 12, 3 times and into 8, 2 times

Option 2, grouping the w at the right side of equation

w = 9w-9-3

⇒ 9+3 = 9w-w

12 = 8w

Divide both sides by 8

12/8 = 8w/8

w = 12/8 = 3/2, thus 4 goes into 12, 3 times and into 8, 2 times

Concepts

A triangle which has two of its sides equal is called an isosceles triangle

Properties of Isosceles Triangle

1. The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.

2. The side opposite the vertex angle is called the base and base angles are equal.

3. The perpendicular from the vertex angle bisects the base and the vertex angle.

Applying property 2 to redraw the triangle in the question gives us the diagram below:

From the above properties, you will realize that angle CBA = BAC = 56^o

Applying the concept that says that angles on a straight line add up to 180^o

⇒ x+56 = 180

Make x the subject

⇒ x = 180 - 56

x = 124

Concepts

A triangle which has two of its sides equal is called an isosceles triangle

Properties of Isosceles Triangle

1. The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.

2. The side opposite the vertex angle is called the base and base angles are equal.

3. The perpendicular from the vertex angle bisects the base and the vertex angle.

Applying property 2 to redraw the triangle in the question gives us the diagram below:

From the above properties, you will realize that angle CBA = BAC = 56^o

Considering the basic properties of angles in a triangle below:

Applying the concept which says that the sum of all the angles in a triangle should be 180^o

⇒ 56+y+56 = 180

y+56+56 = 180

y+112 = 180

⇒ y = 180 - 112

y = 68^o

speed = distance/time

speed = 18 km/h and time = 8 h

Subsituting speed and time in the speed equation above,

⇒ 18 km/h = distance/8 h

Make distance the subject. Since 8h is dividing, multiply both sides by 8h

⇒ 8h x 18 km/h = 8h x distance/8 h, 8h cancels each other leaving distance alone at the right side of the equation.

⇒ 8 x 18 km = distance

⇒ distance = 144 km

NOTE:The pie chart is in percentage (%) and not in degree (^o).

Since its in percentage, the sum of all the percentages should add up to 100% (Thus percentage means out of 100)

If its in fraction, the sum of all the fractions must add up to 1

To get the degree out of a percentage or fraction, its simply the percentage or fraction x 360^o

Thus to get a value of a percentage or fraction out of a total, its simply the multiplication of the fraction or percentage by the total

Thus the degree for the INSURANCE AND TAXES will be the fraction or percentage of the INSURANCE AND TAXES by 360 (sum of all the angles in a pie chart is 360)

First you have to calculate the % of the INSURANCE AND TAXES

INSURANCE AND TAXES%+11%+3%+5%+16%+28%+22% = 100%

INSURANCE AND TAXES% + 85% = 100%

INSURANCE AND TAXES% = 100%-85%

INSURANCE AND TAXES% = 15%

Applying the rule of calculating for a value of fraction or percentage

Value = Fraction or Percentage x the total

The total angles is 360^o

The percentage of the INSURANCE AND TAXES% = 15%

⇒ INSURANCE AND TAXES^o = 15/100 x 360

NOTE:% means over 100 ⇒ 15% = 15/100

⇒ INSURANCE AND TAXES^o = 15/100 x 360^o = 5400/100 = 54^o

Formula

Value of a fraction or percentage of a total = fraction or percentage x the total

Thus to get a value of a percentage or fraction out of a total, its simply the multiplication of the fraction or percentage by the total

The percentage for clothing is 11%

The amount spent on clothing = Percentage of the clothing x the total amount (income)

⇒ Amount spent on clothing = 11/100 x 40,000 = 11 x 4000 = 44000

NOTE:% means over 100 ⇒ 11% = 11/100

Parallelogram has two sides equal. Which means the other two sides of the parallelogram are also 5.8 m and 8.2 m.

Perimeter is the sum of the length of all sides

⇒ The perimeter of the parallelogram = 2x5.8 + 2x8.2 = 11.6+16.4 = 28.0 m

Interest = Principal x Time x Rate / 100

Interest = 240, Time = 4 and Rate = 20

⇒ 240 = Principal x 4 x 20 / 100

Multiply both sides by 100 and simplify 4 x 20 (80)

⇒ 240 x 100 = 80 x Principal

Divide both sides by 80 so that Principal can be alone at one side of the equation

240 x 100 / 80 = Principal

80 goes into 240, 3 times

⇒ 3 x 100 = Principal

⇒ Principal = 300

Total wage = Total hours worked x wage per hour

½ = 0.5 ⇒ 20½ = 20.5

wage per hour = 2.5

⇒ Total wage = 20.5 x 2.5 = 51.25

Percentage = Value/Total x 100

Value here is the number of boys (480)

The percentage is 80

Let x = total number of pupils in the school

⇒ 80 = 480/x x 100

Get rid of the x dividing (fraction) by multiplying both sides of the equation by x

⇒ 80 x x = 480 x 100

Divide both sides of the equation by 80

⇒ x = 480 x 100/80

80 will go into 480, 6 times

⇒ x = 480 x 100/80 = 6 x 100 = 600

Hence there are 600 pupils in the school

Substitute the value of x in each of the rule to see which one gives you the corresponding y value in all the cases

Only x→ x²+1 gives the corresponding y values

Thus when x = 1 → 1²+1 = 1+1 = 2

When x = 2 → 2²+1 = 4+1 = 5

When x = 3 → 3²+1 = 9+1 = 10

When x = 4 → 4²+1 = 16+1 = 17

Concept

Side C is always the side opposite to the right angle □

⇒ 5²+t² = 13²

25+t² = 169

t² = 169 - 25

t² = 144

Take the square root of both sides to get rid of the power 2

√t² = √144

t = 12

NOTE: The result of a square root is simply the number when multiplied by itself will give the number you are calculating the square root of

Thus √144 = 12 since 12 x 12 gives 144

Make sure they are all of the same number of digits by adding zeros(0) to the ones that are less.

The highest digits in all the above numbers is 4, hence 2.4 becomes 2.400 and 1.12 becomes 1.120, the trailing zeros doesn't have any effect on the numbers but makes the calculation easier

Make sure both sides of the equation are in the same base

4 and 32 can both be expressed in base 2

Applying the law of indices, a^b x a^c = a^b+c, thus you add the powers if they are multiplying and the bases are the same

4 = 2 x 2 = 2¹ x 2¹ = 2¹⁺¹ = 2²

32 = 2 x 2 x 2 x 2 x 2 = 2^1+1+1+1+1 = 2⁵

4^x = 32

⇒ (2²)^x = 2⁵

Applying the law of indices, (aⁿ)^m = a^nxm

⇒ 2^2x = 2⁵

Since the bases are the same, the powers must also be the same

⇒2x = 5

Divide both sides by 2

⇒ x = 5/2 = 2½

Equation of a straight line, y = mx+c

where m is the gradient of the line

m = change in y / change in x =(y₂-y₁)/(x₂-x₁)

NOTE: The first coordinate represents the x-axis and the second the y-axis

Thus in (-3,5), x = -3 and y = 5

In (6,8), x = 6 and y = 8

⇒ The gradient of the line = (8-5)/(6--3) = 3/(6+3) = 3/9 = 1/3

Use any of the points and the gradient to calculate the value of c

Using the point (-3,5)

⇒ 5 = ⅓ x -3 + c

⇒ 5 = -1 + c

⇒ 5 + 1= c

⇒ c = 6

Using the point (6,8)

⇒ 8 = ⅓ x 6 + c

⇒ 8 = 2 + c

⇒ 8 - 2= c

⇒ c = 6

Hence the equation will be y = ⅓^x+6

Members in the club = No. of females + No. of males

Concept

No. of A : No. of B = Ratio of A:B

⇒ No. of females : No. of males = 3:2

No. of females = 15

⇒ 15: No. of males = 3:2

Express the : as fractions

⇒ 15/No. of males = 3/2

Cross multiply (The numerator from one side multiplies the denominator from the other side of the equation)

No. of males x 3 = 15 x 2

Make No. of males the subject by dividing both sides of the equation by 3

No. of males = 15 x 2 / 3

3 goes into 15, 5 times

⇒ No. of males = 5 x 2 = 10

Members in the club = No. of females + No. of males

⇒ Members in the club = 15+10 = 25

Translation

A translation moves a shape up, down or from side to side but it does not change its appearance in any other way.

Translation is an example of a transformation. A transformation is a way of changing the size or position of a shape.

Every point in the shape is translated the same distance in the same direction.

If a point (x,y) is translated by the vector ( ^a _b), the image formed will have the point (x+a , y+b)

From the question the translation vector is (^-2 ₃), the image point is P¹ = (4,-5)

⇒ (4,-5) = (x+-2,y+3)

⇒ for the x coordinate, 4 = x-2

⇒ for the y coordinate, -5 = y+3

Solve for the values of x and y

4 = x-2

4+2 = x ⇒ x = 6

-5 = y+3

-5 - 3 = y ⇒ y = -8

⇒ The point translated = (6,-8)

Concept

- x - = + (When a negative number [-] multiplies another negative number[-], the sign becomes positive [+])

- x + = - (When a negative number [-] multiplies a positive number [+], the sign becomes negative [-])

a² = a x a

2y = 1-3x²+4x

When x = -1, substitute x by -1

2y = 1-3(-1)²+4(-1)

(-1)² = -1 x -1 = + 1

4(-1) = 4 x -1 = -4

2y = 1-3x²+4x

⇒ 2y = 1-3(1)-4

2y = 1 -3 - 4

2y = 1 - 7

2y = - 6

Divide both sides by 2 to get rid of the 2 multiplying y

⇒ y = -6 / 2 = -3

Concept

Probability = No. of possible outcomes / No. of sample space

No. of sample space (Total numbers in the set or under examination) = 12

We have to list the prime numbers of the set p and count them to get the No. of possible outcomes

A prime is a number that is greater than 1 and divisible only by itself and 1 (e.g. 2, 3, 5, 7, 11,...)

Note: 1 is not a prime number

⇒ the prime numbers from the set p are {2,3,5,7,11}

The total possible prime numbers is 5 (when you count the list of prime numbers above, you are to get 5)

⇒ The probability of selecting a prime number = 5 / 12

What will the probability of selecting i)an even number ii) old number?

i) Probability of selecting an even number

An even number is a number divisible by 2

Hence the even numbers from set p are {2,4,6,8,10,12} which when you count you will get 6

⇒ The probability of selecting an even number = 6 / 12 = 1/2

ii) Probability of selecting an odd number

Odd numbers are whole numbers that cannot be divided exactly into pairs. Odd numbers, when divided by 2, leave a remainder of 1.

Hence the odd numbers from set p are {1,3,5,7,9,11} which when you count you will get 6

⇒ The probability of selecting an odd number = 6 / 12 = 1/2

3 - (3x+4) ≤ -4

Concept

a(b+c) = axb+axc

-a(b+c) = -a x b + -axc

-(a+b) = -1(a+b) = -1xa + -1xb

⇒ - (3x+4) = -1(3x+4) = -1 x 3x + -1x4 = -3x-4

3 - (3x+4) ≤ -4

⇒ 3 - 3x-4 ≤ -4

⇒ 3 -4 - 3x ≤ -4

⇒ -1 - 3x ≤ -4

Moving the - 3x to the right side of the equation and the -4 to the left

⇒ -1+4 ≤ 3x

3 ≤ 3x

Divide both sides by 3

3/3 ≤ 3x/3

⇒ 1 ≤ x

You can reverse the order and reverse the inequality sign

⇒ x ≥ 1

Perimeter is the sum of the length of the shape. A rectangle has two equal sides. Hence the total length (Perimeter) of a rectangle = 2L + 2B, where L = Length and B = Breadth

⇒ The perimeter of the rectangle = 2 x 10 + 2x6 = 20+12 = 32

A square has all 4 sides equal. Hence the perimeter of a square = 4 x L, where L means Length

Since the perimeter of the rectangle is the same as the perimeter of the square,

⇒ The perimeter of the square is also 32

⇒ 4L = 32

Divide both sides by 4

4L/4 = 32/4

⇒ L = 8

Area of a square = Length x Breadth

Length = Breadth

⇒ area of a square = Length x Length = L²

Length of the square is 8

⇒ the area of the square = 8 x 8 = 64 cm²

Get rid of the fraction by multiplying both sides of the equation by the LCM of the denominators (1 and x). The LCM is x

NOTE: y = y/1 (Every number or letter is being divided by 1)

⇒ y x x = n - x

Move the x at the right to the left in order for n to stand alone (the subject)

⇒ yx+x = n

Factorize x out at the left side of the equation

⇒ x(y+1) = n

⇒ n = x(y+1)

Concept

(a+b)(c+d) = a x (c+d) + b x (c+d)

a(b+c) = a x (b+c) = a x b + a x c

⇒ (a - 2)(2a+3) = a x (2a+3) - 2 x (2a+3)

a x (2a+3) = a x 2a + a x 3 = 2a² + 3a

NOTE: a x a = a¹ x a¹ = a¹⁺¹ = a²

-2(2a+3) = -2 x 2a + -2 x 3 = -4a - 6

⇒ (a - 2)(2a+3) = 2a²+3a - 4a - 6

⇒ (a - 2)(2a+3) = 2a²- a - 6

NOTE: 3a - 4a = -1a = - a

Concept

a^b x a^c = a^b+c

a^b ÷ a^c = a^b-c

⇒ 2² x 2⁷ ÷ 2⁴ = 2²⁺⁷ ÷ 2⁴

⇒ 2² x 2⁷ ÷ 2⁴ = 2⁹ ÷ 2⁴

⇒ 2² x 2⁷ ÷ 2⁴ = 2^{9 - 4}

⇒ 2² x 2⁷ ÷ 2⁴ = 2⁵

Union (∪) is the arrangement from the lowest to the highest of the elements in both sets. If an element occurs in both sets, its written only once

P = {7,11,13} and Q = {9,11,13}

⇒ P ∪ Q = {7,9,11,13}