Maths 2025
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products and factors
Binomial
An algebraic expression which consists of 2 terms
Eg. 4a + 9, + 5x
Binomial Product
An algebraic expression which consists of 2 or more binomials multiplying together
Eg, (4a + 9)(5a - 3)
Expand
To rewrite an expression without grouping
Eg. 5(3a - 10) = 15a - 50
Factorise
To rewrite expression with grouping by taking out the highest common factor.
Eg. 5+ 125x = 5x(x + 25)
Perfect Square
A square number or expression which represents one
Eg. 64,
Quadratic Expression
An algebraic expression in which the highest power of the variable is 2
Eg.
Quadratic Trinomial
A quadratic expression which consists of 3 terms
Eg.
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________________
trigonometry
Opposite side
In a right angled triangle, the side which is directly
opposite a given angle is the opposite side
Adjacent side
In a right angled triangle, the side which is directly
adjacent a given angle is the adjacent side
Hypotenuse
The hypotenuse is the longest side of a right angled triangle and is directly opposite the 90 degree angle
Angle of Elevation
The angle of looking up, measured from the horizon
Angle of Depression
The angle of looking down, measured from the horizontal
Bearing
The angle used to show the direction of one location from another
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________________
equations
Consecutive
A series of something that come directly after one another
Eg. 3, 4, 5, 6, etc.
Equation
A mathematical statement that 2 quantities are equal, involving algebraic expressions and equal signs (=)
Eg. 2 + 2 = 4, 5x - 10 = 5
Formula
A rule written as an algebraic expression, using variables
Inverse Operation
An opposite used in solving an equation
Eg. The inverse of multiplication would be division
Linear Equation
An equation involving a variable that is not raised by a power
Eg. 2x + 5 = 17
Quadratic Equation
An equation involving a squared variable
Eg.
Simplifying a Surd
Square root
To simplify a square root, find 2 factors of the number within the root where one of those factors is a perfect square. Then take out that perfect square and root it. Put it outside of the square root symbol.
Eg. √20 → √4 x 5 → 2√5
Cube root
To simplify a cube root, it is the same process as a square root, however one of the factors must be a perfect cube, rather than a perfect square.
Eg. 3√81 → 3√27 x 3 → 33√3
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numeracy and calculations
Cost price
The price the item costs the retailer
GST
Goods and Services tax, a 10% tax added to the original price of an item
Loss
The amount lost when selling an item at a lower price to the cost price
Profit
The amount made when selling an item at a higher price to the cost price
Per annum (p.a)
Per year
Principal
An amount of money which is invested or borrowed, on which interest is given or charged
Recurring decimal
A decimal with one or more digits that repeat endlessly
Unitary method
A method of finding a quantity by finding one part or 1% first
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earning money
Allowable deduction
A part of someone’s yearly income which is not taxed, such as work-related expenses or donations to charity
Annual leave loading
Extra payment to a worker based on 17.5% of 4 weeks annual leave
Income tax
A tax paid to the government based on the size of a person’s gross income
Net pay
Pay received after deductions from gross pay; ‘take home’ pay
Overtime
Time worked beyond normal working hours, such as nights or weekends, at a higher rate of pay
Time-and-a-half
Overtime pay which is calculated at 1.5 times the normal pay rate
Salary
A fixed yearly amount of money that is paid weekly, fortnightly or monthly and is not dependent on the amount of hours worked.
Wage
An amount of money paid to people for work, calculated on the number of hours worked
USE 1 YEAR = 52 WEEKS (NOT 52.18)
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Products and Factors
________________
Important stuff to remember
* BODMAS is very important in algebraic expressions especially during solving and factorising.
* Only like terms can be added or subtracted
* If a term has an index, it can only be multiplied or divided with a like term.
* If a term has an index, it can not be subtracted or added.
* To add or subtract algebraic fractions make sure the denominators are the same, and then simply add or subtract the numerators
* To multiply fractions cancel out any common factors, then multiply the numerator and denominator separately
* To divide two fractions, cancel out all common factors and then reciprocate the seconds fraction, then multiply the numerator and denominator separately.
* Expand and simplify by expanding like terms.
* Use HCF (Highest common factor) to factorise 2 coefficients.
* When using 2 negative numbers use negative HCF.
* Binomials are named as such due to the fact that they only contain two terms/expressions
* Trinomials are named as such due to the fact that they only contain three terms/expressions.
* FOIL method - When expanding, multiply first, outside, inside, last terms respectively.
* Factorise the question completely before expanding/factorising
* Even negative indices can NOT be square rooted, the answer is therefore undefined.
1.01
1. 5r + 4s - 7r - 2s
5r - 7r -2s + 4s
-2r + 2s
= 2s - 2r
2.
=
Questions
1. 2fg + 3fg - 4fg
= 5fg - 4fg
= fg
2. =
=
3. =
=
1.02
1. 3d x 4d
= 3 x 4 x d x d
= 12
2. 2g x 3h x (-5f)
= 2 x 3 x (-5) x g x h x f
= -30ghf
1. =
=
2. =
=
1.03
1. =
=
2. =
=
f)
=
=
=
1.04
1. =
2. =
1. =
2. =
=
1.05
1. 3g(h - 2)
= 3gh - 6g
2. r(10 - 4r)
= 10r - 4
1. 4(3p + 2) - 5p
= 12p + 8 - 5p
= 7p + 8
2. 5(2e - 3) - 4(1 - 5e)
= 10e - 15 - 4 + 20e
= 30e - 19
1.06
1. 20 and 15xy
HCF of 20 and 15 is 5
HCF of and xy is x
Therefore, the HCF is 5x.
2. 16 and 12b
HCF of 16 and 12 is 4
Therefore, the HCF is 4
b) 25- 20ab
= 5b(5b - 4a)
3. v(4 + w) + 2(4 + w)
= (4 + w)(v + 2)
1. =
2. -a - ab
= -a( 1 + b)
1.07
1. (x + 5)(x + 9)
= x(x + 5) + 9(x + 5)
= + 5x + 9x + 45
= + 14x + 45
2. (k + 3)(k - 7)
= k(k - 7) + 3(k - 7)
=- 7k + 3k - 21
= - 4k - 21
3. (7 - m)(4 + m)
= 4(7 - m) + m(7 - m)
= 28 - 4m + 7m -
= 28 - 3m -
4. =
=
1. (x - 6)(4x + 2)
= x(4x + 2) - 6(4x + 2)
=
2. (3t - 1)(2t - 5)
= 3t(2t - 5) - 1(2t - 5)
=
=
1.08
1. =
=
2. =
=
1.09
1. (d + 3)(d - 3)
=
=
2. (2 + r)(2 - r)
=
=
3. (7x + 2)(7x - 2)
=
=
4. (4k - 5p)(4k + 5p)
=
=
1.10
1. (4r + 5)(1 - 2r)
= 1(4r + 5) - 2r(4r + 5)
= 4r + 5 - 8 - 10r
=
2. =
=
=
3. (3d - 10)(3d + 10)
=
=
4. (a + 6)(a - 6) + (a + 12)( a + 3)
=
=
=
1.11
1. 3ac + 2bd + 2bc + 3ad
= 3ac + 3ad + 2bd + 2bc
= 3a(c + d) + 2b(c + d)
= (c + d)(3a + 2b)
2. 4km + 6mn - 6kp -9np
= 2m(2k + 3n) - 3p(2k + 3n)
= (2k + 3n)(2m - 3p)
1. =
= (x - 2)(x + 2)
2. =
=
3. =
=
= 5(
4. = y
=
1.12
1. Product: 12
Sum: 7
4, 3
= (a + 4)(a + 3)
2. Product: -6
Sum: 1
3, -2
= (x + 3)(x - 2)
1. Product: -15
Sum: -2
3, -5
= (a + 3)(a - 5)
2. Product: 8
Sum: -6
-4, -2
= (y - 4)(y - 2)
1. = 3(
= 3(g - 2)(g + 6)
2. = -1(
= -(p - 4)(p + 12)
1.13
1. Product: 5 x 4 = 20
Sum: -12
-2, -10
=
= 5k(k - 2) - 2(k - 2)
= (k - 2)(5k - 2)
2. Product: 9 x -4 = -36
Sum: -9
-12, 3
=
= 3m(3m + 1) - 4(3m + 1)
= (3m + 1)(3m - 4)
1.14
1. =
= 3(a - 3)(a + 3)
2. =
3. = 4
= 4()
= 4[5b(b - 2) - 3(b - 2)]
= 4(b - 2)(5b - 3)
4. = (
= (d - 1)(
= (d - 1)(d - 1)(d + 1)
=
1.15
1. =
= 2a + 5b
2. =
=
=
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❌ Trigonometry
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Important stuff to remember
* Pythagoras Theorem (a² + b² = c²)
* Hypotenuse = Longest Side
* Theta (θ) usually represents an unknown angle
* In your working out, as an example, the hypotenuse in this case would be written as ∠WY
* Trigonometry can only be performed on a right angled triangle (as of a Year 9 level)
* The hypotenuse will always be opposite of the right angle. (Important for drawing diagrams for word problems and bearings)
4.02
S
Sin
C
Cos
T
Tan
O
Opposite
A
Adjacent
O
Opposite
H
Hypotenuse
H
Hypotenuse
A
Adjacent
4.04
* Degrees can be broken down into minutes and seconds if in decimal format
* 1 degree = 60 minutes
* 1 minute = 60 seconds (duh)
* E.g, 57.23 degrees may be written as 57° 13’ 48’’
* The button on your calculator to convert a number into minute/second form looks like this → ° ’ ’’
* * When rounding up degrees, minutes, and seconds, you would use the 30 minute/30 second mark. ( <= 29 gets rounded down, >= 30 gets rounded up)
4.05
4.06
* Note that instead of appearing on the numerator, the variable now appears on the denominator meaning the known length will now be divided by the trig ratio instead of being multiplied by the trig ratio
4.07
* When finding an unknown angle over an unknown length you will need to use inverse trig ratios. (sin-1, cos-1, tan-1)
* For example if the opposite of an angle in a right angled triangle was 2cm and the hypotenuse was 5cm, the equation would be the following: sin-1(⅖), the answer to this equation when input to a calculator would read the degrees of the angle opposite of the 2cm side with a hypotenuse of 5cm
4.08
* The angle of elevation is the angle of looking up, measured from the horizontal
* The angle of depression is the angle of looking down from the horizontal.
* NOTE: An angle of elevation is found inside the triangle whilst the angle of depression is the complementary angle of an angle in the triangle
* In each image, theta represents the angle of elevation/depression
* Problems involving angles of elevation and depression usually require the trig ratio of tangent (tan) for a Year 9 level
4.09
* There are two types of bearings, compass bearings and true bearings
* True bearings are an angle measured from true north and MUST be written with 3 digits.
* E.g, south would be written as 180, East would be written as 090, and south west would be written as 225.
* Compass bearings are based on their angle from the two closest MAIN compass points (North, East, South, West)
* They can only be measured from North or South
* E.g, 030 on a true bearing would be N30E on a compass bearing. 205 would be S25W, 095 would be S5W
Hard Bearings
4.10 (Harder Bearings)
Equations
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Important stuff to remember
* Order of operations
* Expand before solving
* Always substitute the variable in the question with your answer to check if it is right or wrong. (if enough time is present)
* Don't forget +/- when working with square roots (or any root of an even number, such as 4 or 6 roots)
* Some questions will not specify to round, however it is important to judge questions, such as if a question asks to find the maximum amount of people, your answer will not be 13.25, it will be 13. You can not round up either, if your answer is 13.75, the maximum amount is still 13 people.
7.01
* For equations with variables on both sides, move variables and numbers onto separate sides of the equation.
* E.g (7x + 7 = 2x + 2) → (7x - 2x = 2 - 7)
7.02
* For equations with brackets (grouping symbols), expand and then solve.
* E.g 1 → 3(a + 3) = 12 → 3a + 9 = 12 → 3a = 3 → a = 1
* E.g 2 → 9(m - 5) = 7(m + 1) → 9m - 45 = 7m + 7 → 9m - 7m = 7 + 45
7.03
* In a word problem involving equations follow these 5 steps clearly:
1. Read the question carefully and determine what needs to be found
2. Define a variable to represent the unknown quantity (Let x be the missing quantity)
3. Write the problem as an equation
4. Solve the question
5. Answer the problem (Therefore the missing quantity is no.)
* Sometimes during step 3, your equation may need to be simplified or factored before being solved
* In some word problems a variable may already have been defined and the missing quantity asked in the question may be a factor, multiple, addition, or subtraction of the predefined variable.
7.04
* For equations where all terms are fractions, multiply both sides by a common multiple of the denominators. For example if two denominators are 3 and 4, the LCM of 3 and 4 is 12, so multiply both sides by 12.
* * An alternate method can be simplifying the LHS first. (Make both denominators the same and then add or subtract just as you would with any other fraction.
7.05
* An equation in which the highest power of the variable is 2 is called a quadratic equation
* Equations in the form X² = C has a solution: x = +/- √c
* Most of the time, unless specified in the question, if a solution to a quadratic equation is a square root that is not a perfect square, it is best to leave the answer in surd form.
7.06
* To solve quadratic equations of the form x² + bx + c, factorise this into a binomial product of the form (x + d)(x + e)
* This is true because the question can be written as →
x² + 3x + 4x + 12 → x(x + 3) + 4(x + 3) → (x + 3)(x + 4)
* Remember that if the product is negative however the sum is positive, this means that one of the 2 numbers is negative. However in vice versa this means that both numbers are negative.
7.07
* An equation in which the highest power of the variable is 3 is called a cubic equation
* The cubic equation x³ + c only has one solution which is →
X = 3√c
* The answer can never be negative.
* If the answer is not a perfect cube, then it is best to leave it in surd form unless specified otherwise.
7.08
* A formula is an algebraic expression that shows the relationship between variables
1. h = 4
C = 250 +75h
C = 250 + 75(4)
C = 250 + 300
C = 550
The cost is $550
2. C = 750
750 = 250 + 75h
500 = 75h
h =
h = 6
The maximum number of whole hours is 6
7.09
* In the formula v = u + at, v is the subject of the formula
* The process of rearranging a formula to change the subject is called changing the subject of the formula
* To change the subject of a formula, use the same rules as for solving an equation
1. b)
Numeracy and Calculation
________________
Important stuff to remember
* Percentages are the same swapped around (x% of y = y% of x)
* To convert a fraction to a decimal divide the numerator by the denominator.
* Irrational numbers that are unknown in exact form (for example they are not symbols such as pi or fractions) can not be turned into a fraction, as of year 9 mathematics.
* To find the percentage symbol on the casio fx-82AU PLUS second edition, it is shift right bracket, for the first edition it is N/A
* Underline or mentally note down keywords, especially in questions about decreasing or increasing by a percentage, look out for “round to nearest cent/dollar/whole”.
* Remember to make sure all values are in the same measurement, for example seconds and minutes or centimetres and metres.
* Unitary method is best for finding an amount with a percentage of the amount.
* For a question where you must increase/decrease by a percentage on multiple occasions you can not add the percentages together and complete the problem in one step, this will result in an incorrect answer and less marks in terms of working out.
* Some questions will not specify to round, however it is important to judge questions, such as if a question asks to find the maximum amount of people, your answer will not be 13.25, it will be 13. You can not round up either, if your answer is 13.75, the maximum amount is still 13 people.
* If the question doesn’t specify whether to round or not, it is best to round off to the nearest 2 decimal points and write in brackets, “nearest 2 d.p”.
* GST = 10%, or, 110% of net price, unless specified otherwise.
* To simplify a ratio, think of it as a fraction, 14:12 → 14/12 which can be simplified to 7/6 which is the same as 7:6.
* For 8:6:4, find the LCM, which is 2, making it 4:3:2, LCM works for all ratios.
* Unitary method works for ratios asw, say 450 was split between 2 people, 4:5, this would add up to 9 parts equalling 450 meaning 1 part is 50 making 4:5 equivalent to 200:250.
* Rates are similar whereas 600km in 12 hours can be written as 600km/12h which is then 50km/h
* Written in the form a/b (pronounced a per b)
* Remember common conversions, always look out for differing conversions in problems
3.01
* Recurring or repeating decimals have numbers that repeat or recur.
* Dots (or lines) are used to indicate the repeating digit
* Terminating decimals do not repeat (terminating means stopping). For example, 0.6, 0.875, 0.057
* Converting fractions to decimals
1. = 0.8333…
= 0.8
Recurring decimal
2. = 0.275
Terminating decimal
Because both terminating and recurring decimals can be written as a fraction, they are both examples of rational numbers.
Irrational numbers have an infinite number of digits but there is no pattern.
* Converting recurring decimals to fractions
Recurring decimals can be converted to fraction using an algebraic method
1.
Let x =
x =
10x = ①
100x = ②
② - ① 100x - 10x = -
90x = 16
x =
x =
2.
Let x =
x = ①
100x = ②
② - ① 100x - x = - ,
99x = 54
x =
x =
3.02
Expressing quantities as percentages
To express one quantity as a percentage of another, write the appropriate fraction and then change it to a percentage:
Percentage =
Or
Percentage = amount whole amount
1. 12% of $90
=
= $10.80
2. % of $540
= 0.055
= $29.70
1. Increase $200 by 7%
Increase = 7% x 200
= $14
New amount = 200 + 14
= $214
OR
New amount = 107% of 200
= 1.07 x 200
= $214
2. Decrease $150 by 12%
Decrease = 12% x 150
= $18
New amount = 150 - 18
= $132
OR
New amount = 88% of 150
= 0.88 x 150
= $132
Unitary Method
Given a percentage of an amount to find the whole amount:
1. Find 1% of the amount by dividing it by the known percentage
2. Multiply by 100 to find the whole 100%
9% of the wage = $86
1% of the wage = 86 9
= $9.555…
Julio’s wage = 9.55… x 100
= 955.555…
$956
3.03
Profit and Loss
* When an item is sold for more, a profit is made
* When an item is sold for less, a loss is made
* The percentage profit and loss are usually calculated as a percentage of the cost price
1. Profit = 680 - 550
= $130
2. Percentage profit =
=
= 23.636363…
23.6%
Selling price = 100% + 10% = 110%
110% of original price = 2695
10% GST = 2695 11
= $245
3.04
* The original amount of money invested or borrowed is called the principal
* Interest is calculated as a percentage of the principal
* This percentage is called interest rate, usually written as a rate per annum (per year) or p.a.
* Simple interest is interest calculated simply on the original principal
SIMPLE INTEREST FORMULA
I = Prn where,
I = Interest
P = Principal
r = interest rate
n = number of years
P Principal = $18 000
r rate = 3.5% p.a → 0.035
n number of years = 5 years
I = Prn
= 18000 x 0.035 x 5
= $3150
Therefore, the interest is $3150
1. P = 8620
r = 2.4% p.a → 0.024
n = 7 months → years
I = Prn
= 8620 x 0.024 x
= $120.68
2. P = 5600
r = 6.25% p.a → 0.0625
n = 220 days → years
I = Prn
= 5600 x 0.0625 x
= $210.96 (to nearest cent)
I investment = $87.36
P principal = $1560
r rate = r
n number of years = 2
Therefore, the rate is 2.8%
I = Prn
87.36 = 1560 x r x 2
87.36 = 3120r
r =
r = 0.028
r = 2.8%
3.05
1. 36 : 48 : 18
= 6 : 8 : 3
2. 425mL to 5 L
= 425 : 5000
= 17 : 200
Method 1: Equivalent ratios
Jeans : Shorts = 3 : 10 = 240 : ______
Since 240 = 3 x 80
Number of shorts = 10 x 80 = 800
Method 2: Unitary method
Jeans : Shorts = 3 : 10
Since 240 jeans were sold, 3 parts = 240
1 part = 240 3 = 80
Number of shorts = 80 x 10 = 800
1. 600 km in 12 hrs = 600 12
= 50 km/hr
2. $6.45 for 15 min = 6.45 15
= $0.43/min
1. Distance = speed x time
= 92 km/hr x 4.5 hours
= 414 km
2. Time =
=
= 8.7 hours
3.06
1. 80 L/kg to L/g
1 kg = 1000 g
= 80 1000
= 0.08 L/g
2. 5 m/s to km/hr
1 hr = 3600 s
= 5 x 3600
= 18000 m/hr
1 km = 1000 m
= 18000 1000
= 18 km/hr
________________
Earning Money
Important stuff to remember
* Underline or mentally note down keywords, especially in questions about decreasing or increasing by a percentage, look out for “round to nearest cent/dollar/whole”.
* Per annum is written as “p.a”
* I = Prn (P = principal {investment}, r = interest rate expressed as a decimal, n = number of years) For example I = 18000 x 0.035 (3.5% p.a) x 5. It is the added investment, your answer will usually be P + I.
* Simple interest (flat rate interest) is added on top of the invested money. ($150K with 5% p.a after 10 years will be $225K, not $75K)
* Rate of Leave Loading is 17.5% on top of normal income. (117.5% of normal income)
* If the question doesn’t specify whether to round or not, it is best to round off to the nearest 2 decimal points and write in brackets, “nearest 2 d.p”, especially because the topic is money.
* GST = 10%, or, 110% of net price, unless specified otherwise.
* Deductions are taken away from the salary before calculating.
8.01
Wages and salaries
* A wage is calculated by the number of hours worked and is usually paid weekly.
* A salary is a fixed annual amount, paid weekly, fortnightly, or rarely monthly
Weekly income = $26.75 x 35
= $936.25
1. 72 600 52.18 b) 2 x 1391.34
= $1391.3376… = $2782.68
= 1391.34 (nearest cent)
c) 72 600 12
= $6050
Sarah → 62 5000 52.18 38 Anh → 1087 35
= $31.5204… = $31.0571…
= $31.52 = $31.06
Therefore Sarah has the higher hourly rate of pay!!!
8.02
Overtime Pay
* Time and a half pay → 1.5x normal hourly rate
* Double time → 2x normal hourly rate
Normal pay = 35 x 28.94 = $1012.90
Overtime earnings = 4 x 28.94 x 1.5 = $173.64
Total earnings = 1012.90 + 173.64 = $1186.54
Normal pay (Monday to Friday) = 38 x 24.50 = $931
Saturday pay = 24.50 x 6 x 1.5 = $220.50
Sunday pay = 24.50 x 4 x 2 = $196
Total earnings = 931 + 220.50 + 196
= $1347.50
Total amount of hours → 38 + (4 x 1.5) + (5 x 2)
= 54
Hourly rate of pay = 1167.48 / 54 = $21.62
8.03
* Commission is earned by salespeople or agents of a business or company. It is calculated as a percentage of the value of items sold or income made.
* A fixed amount called a retainer may also be paid.
* Royalty is income paid to artists, authors and musicians for the sale or use of their work. When their work is sold or played, they receive a royalty, a percentage of the fee or price.
Commission on 1st 100000 = 2% x 100 000
= $2000
Remaining amount = 580 000 - 100 000
= $480 000
Commission on remaining amount = 1.5% x 480 000
= $7200
Corina’s commission = 2000 + 7200
= $9200
1. Earnings: 15% x 18.95 x 343
= $974.98
2. 15% of sales = 2000
1% = 2000 15 = 133.3333…
Total sales = 133.333 x 100 = $13333.333…
Apps sold: 13333.333… 18.95
= 704
Therefore, 704 apps must be sold
ANNUAL LEAVE LOADING
* Annual leave loading or holiday loading is extra pay given during annual holidays. It is paid at the rate of 17.5% of 4 weeks’ normal pay
1. Normal weekly pay: 91485 52
= $1759.33
2. Leave loading: 17.5% x 4 x 1759.33
= $1231.53
3. Total pay: 4 x 1759.33 + 1231.53
= $8268.85
8.04
Income Tax
* Tax paid to the government taken from the pay of workers, or interest from investments.
* Tax deductions are not taxed, these include charity contributions or car-related expenses.
Taxable income = gross income - allowable tax deductions
1. Taxable Income: 65660 - 727 - 259 = $64674
b) Tax: 5092 + 0.325 x (64674 - 4500) = $11486.05
8.05
PAYG tax and Net Pay
* To avoid paying income tax as a huge sum at the end of the financial year, your employer takes out an estimate of the tax from your pay everyday, this is called PAYG (Pay As You Go) tax.
* The total amount of PAYG tax paid over the year is usually more than the actual income tax payable, so at the end of the financial year you will receive the difference as a tax refund. However if the PAYG tax is less than required this will result in tax debt and you having to pay more.
* Gross pay is the total amount a person earns or receives each pay day
* Net pay is the amount of money left after all deductions including tax, superannuation, union fees, health funds etc.
Net pay = gross pay - tax - other deductions
1. In the table, $700.70 falls in the 696 - 704 range making her PAYG tax withheld $158
2. 700.70 - 158 - 45.6 - 35
= $462.10
3. Total deductions
= 158 + 45.6 + 35
= $238.60
Deductions % = x 100%
= 34.1%
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