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General Math Notes – HSC 2013
Algebraic Skills and Techniques
1) Substitution
 A pronumeral takes the place of a number.
 A collection of pronumerals and numbers connected by operational symbols is called
an expression.
 An algebraic expression can be evaluated by substituting values for pronumerals.
 Replacing the pronumeral in an expression with a value
e.g solve if a=6 and b=2.
√𝑎 − 2𝑏 = √6 − 2(2)
√6 − 4
√2 = 1.41
2) Simplifying algebraic expressions
 Algebraic expressions can be simplified by collecting like terms.
 Like terms are the only terms that can be added or subtracted.
 (Example) Just as ‘5 x 10’ is the same as ‘10 x 5’, ‘pq’ is the same as ‘qp’. ‘x’ and ‘x2’
are not like terms.
 When expanding brackets, multiply by every term inside the brackets.
 Pronumeral x pronumeral = pronumeral2, negative x negative = positive.
e.g
a) Simplify 4x – x2+ 7x = 4𝑥 − 𝑥 2 + 7𝑥 = 11𝑥 − 𝑥 2
b) E.g expand and simplify: 3+2(2-5x)
3 + 2(2 − 5𝑥)
= 3 + 4 − 10𝑥
= 7 − 10𝑥
c) Expand and simplify: x(2-x)-4(x-3)
𝑥(2 − 𝑥) − 4(𝑥 − 3)
= 2𝑥 − 𝑥 2 − 4𝑥 + 12
= −2𝑥 − 𝑥 2 + 12
3) Multiplying and dividing algebraic expressions
 Numbers first then pronumerals
 The algebraic terms do not have to be like terms when multiplying or dividing.
 If there are algebraic fractions then they may be cancelled in the same way as
numerical fractions.
 In division cancel by dividing top and bottom.
e.g
a) simplify: 7x x (-x3) = −74
b) E.g expand and simplify: 3+2(2-5x) = 3 + 2(2 − 5𝑥)
= 3 + 4 − 10𝑥
= 7 − 10𝑥
c) Expand and simplify: x(2-x)-4(x-3) = 𝑥(2 − 𝑥) − 4(𝑥 − 3)
= 2𝑥 − 𝑥 2 − 4𝑥 + 12
= −2𝑥 − 𝑥 2 + 12
4) Multiplying and dividing algebraic expressions
 Numbers first then pronumerals
 The algebraic terms do not have to be like terms when multiplying or dividing.
 If there are algebraic fractions then they may be cancelled in the same way as
numerical fractions.
 In division cancel by dividing top and bottom.
 If no number is in front of the pronumeral, put a 1. If no number is to the power of a
pronumeral it is a 1.
 Add the powers for multiplication and subtract for division.
 Pronumeral x pronumeral = pronumeral2, negative x negative = positive.
 Change decimals into fractions.
 When multiplying/dividing fractions, multiply/divide across
e.g
Multiplication
a) Simplify: 7x x (-x3) = 7𝑥 1 × (−1𝑥 3 ) = 74
b) Simplify: (-7x)2= −7𝑥 × −7𝑥 = 14𝑥
Division
5 15𝑝𝑞
1 3𝑝
3−15𝑎
3
(-15a)/20a = 4 20𝑎2 𝑎2
a) 15pq/3p=
b)
=
3
4𝑎 2
c)
d)
𝒂
𝒃
𝒂𝒃
𝒙 = 𝟐
𝒙
𝒙
𝒙
14𝑏1
7
= 𝑏
2𝑏2
5) Equations reviews
 Addition and subtraction
 Multiplication and division
 Substitution
 Simplification
6) Practical equations
 A formula links two or more pronumerals according to a rule.
 If one pronumeral is expressed in terms of the others, it is called the subject of the
formula.
 If the value to be found id the subject of the formula then it is found directly from
substitution.
 If the value to be found is not the subject, then following substitution, the resulting
equation is solved to find the value.
 When the square root of both sides is taken ± is used
e.g find the value of t if v = 117, u = 5, a=8 and v= u+at
𝑣 = 𝑢 + 𝑎𝑡
117 = 5 + 8𝑡
117 − 5 = 8𝑡
112 = 8𝑡
112
=𝑡
8
𝑡 = 14
7) Changing the subject
 To rearrange formulas
 The subject of the formula is the letter on the left hand side of the equation
followed by the equals sign. The formula p=rx+d has p as the subject because it is
expressed in terms of the variables r, x and d.
e.g
a) Make u the subject of 𝑣 2 = 𝑢2 + 2𝑎𝑠
𝑣 2 − 2𝑎𝑠 = 𝑢2
𝑢 = ± √𝑣 2 − 2𝑎𝑠
b) make x the subject of 𝑝 =
1
ℎ(𝑥
2
+ 𝑦)
1 ℎ (𝑥 + 𝑦)
× ×
2 1
1
ℎ(𝑥 + 𝑦)
2𝑎𝑝 =
×2
2
2𝑝 ℎ(𝑥 + 𝑦)
=
ℎ
ℎ
=
2𝑝
− 𝑦 = (𝑥 + 𝑦) − 𝑦
ℎ
2𝑝
𝑥=
−𝑦
ℎ
8) Scientific notation
 Multiplying by 10n shifts the decimal place n places to the left.
 Mm to m to km, no cm.
 Add the powers when multiplying.
 A x 10n, A = no. between 1-10, n = decimal place shift.
x10-3
When moving from a smaller
measurement to a larger
measurement (TOP) you ‘×’
(or add 3)
x10-3
Distance:
mm
m
When moving from a larger
measurement to a smaller
measurement (BOTTOM) you
‘÷’ (or subtract 3)
km
X103
X103
x10-3
x10-3
Weight:
mg
g
X103
x10-3
kg
X103
t
X103
e.g convert:
a) 6.42 x 104km to mm
6.42 x 104 x 103m =5.42 x 107m (powers added, can write x4+3 since 1000 = 103)
6.42 x 107+3 = 6.42 x 1010mm
b) 8.3 x 104mg to kg
8.3 x 104-3g = 8.3 x 101g
8.3 x 101-3mg = 8.3 x 10-2mg
Further Applications of Area and Volume
1. Area of Circles
 Area of a circle: A = πr2, where r = radius of the circle.
 Half the diameter to get the radius. Always use radius, NOT diameter.
 Circle: full circle shape.
 Sector: part of a circle (looks like a pie piece).
 Annulus: the shaded region, often the outer section of a circle with a piece cut out of
the middle.
Circle
Sector
Annulus
∅
r
d
A = πr2
𝑨=
∅
× 𝝅𝒓𝟐
𝟑𝟔𝟎
A = πR2 – πr2 (R = big) (r = small)
Or
A = π(R2-r2) (HSC Formula)
e.g
a) What fraction of the circle is drawn and the area to the nearest cm2?
6.8cm
1
× 𝜋𝑟 2
4
90
1
=
360
4
1
× 𝜋𝑟 2 × (6.8)2
4
= 36.3 = 36cm2
b)
𝐴 = 𝜋(𝑅 2 − 𝑟 2 )
11
52
𝐴 = 𝜋 (( )2 − ( ))
2
2
2
𝐴 = 75.4cm
5cm
11cm
If the annulus is a half
semicircle shape,
simply divide the end
result by 2.
2. Area of Ellipses
 A plane with a curved boundart is the ellipse. The ellipse does not have a radius, as
there are many different distances from the centre to the boundary.
 We use two different measurements:
 The length of the semi- major axis.
 The length of the semi-minor axis.
 The semi major axis is the longest distance from the centre of the ellipse to the
boundary.
 The semi-minor is the shortest distance from the centre of the ellipse to the
boundary.
 If diameter is given, divide by 2.
α is the length of the semi-major
axis.
b is the length of the semi-minor
axis.
b
a
The area of an ellipse:
A=πab
3. Simpson’s Rule
 If a property being surveyed has an irregular boundary, like a river, then Simpsons
rule is used to find the area.
 If there are 6 vertical line measurements two calculations must be done and added
together.
 If no measurement is given, the measurement is 0.
ℎ
(𝐷𝑓 + 𝐷𝑚 + 𝐷𝑙
3
Df: first measurement
𝐴≑
Dm: middle measurement
Dl: last measurement
dM
dF
dL
h
h
H: equal distance between
successive measurements.
4. Surface Area of a Cylinder
r
h
r
h
Closed cylinder: 𝟐𝝅𝒓𝒉 + 𝟐𝝅𝒓𝟐
A = πr2
Area of closed space + 2 circles
One end open: 𝟐𝝅𝒓𝒉 + 𝝅𝒓𝟐
Open cylinder: 𝟐𝝅𝒓 𝒙 𝒉
5. Surface Area of Spheres
Sphere: 𝟒𝝅𝒓𝟐
Open hemisphere:
4𝜋𝑟 2
2
Closed hemisphere:
6. Volume of composite shapes
 Prism: V=Ah
 Rectangular prism: V=lbh
1
2

Cone: 𝑉 =

Cylinder: 𝑉 = 𝜋𝑟 2 ℎ

Pyramid: 𝑉 = 3 𝐴ℎ

Sphere: 𝑉 =
𝜋𝑟 2 ℎ
1
4
𝜋𝑟 2
3
7. Accuracy of Measurement
 Limit of reading is the smallest unit. E.g 0.1 from 21.2
 A given measurement is accurate to ±0.5 of the smallest division on the scale.
 Absolute error: limit of reading x 0.5
 Upper limit: original measurement + absolute error
 Lower limit: original measurement – absolute error
4𝜋𝑟 2
2
+ 𝜋𝑟 2
Trigonometry
1. Review of Right-angled Triangles
 Right angled triangles use sine, tan and cosine.

Sine = 𝑆𝑖𝑛 = 𝑆𝑂𝐻 =

Cosine = 𝐶𝑜𝑠 = 𝐶𝐴𝐻

Tan = 𝑇𝑎𝑛 = 𝑇𝑂𝐴 =
Opposite
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
= 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Hypotenuse
Adjacent
2. Bearings
 Always 3 digits e.g 042°T.
 Compass bearing: direction.
 True bearing: degrees from North.
 Bearing: Clockwise
3. Area of a Triangle
1
×𝑏×ℎ
2
1
×𝑎×𝑏×
2

𝐴=

𝐴=

Need 2 sides and the included angle
𝑆𝑖𝑛𝐶
4. The Sine rule
 Use brackets.
 Angles all add to 180°
 Non-right angled
 The sine rule works in pairs, you must match a side with the angle opposite. The sine
rule is used to find:
 A side given two angles and one side, or
 An angle given 2 sides and one angle (opposite a side), to find the second
angle.
Side

𝑎
𝑆𝑖𝑛𝐴
𝑏



The letters on top indicate sides and the bottom indicates angles
When finding an angle, flip the formula so the angles are on top.
If you can draw an X it’s sine rule
= 𝑆𝑖𝑛𝐵 . Put the unknown on the left side.
Angle
Use brackets here.
5. Cosine rule
 If you can’t draw an X it’s cosine rule.
 The cosine rule is used to find the 3rd side given 2 sides and the included angle
or an angle given 3 sides.
 Side: 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑎𝑏 × 𝐶𝑜𝑠𝐴

Angle: 𝐶𝑜𝑠𝐴 =
(𝑏2 +𝑐 2 −𝑎 2 )
2𝑏𝑐
6. Radial surveys
 Two types of radial survey:
 Plain table radial survey
 Compass radial survey
 The cosine rule is used to find the length of each boundary and hence the
perimeter.

Credit and Borrowing
1. Flat Rate Loans
 Simple interest loans
𝑃𝑟𝑛
100

𝐼=





𝑝 - principal
R - rate
N – time
Convert months to years dividing by 12.
Non compounding
2. Reducing balance loans
 Reducing balance loans.
 Monthly reducible loans.
 Tables of values.
 Graphs
3. Comparing loans
 Comparing loans with interest rates that are applied in different ways
 Effective interest rate.
 By changing different rates to the effective interest rate, the rate is then expressed
as an annual rate of interest, as if the loan was compounded annually.
 The formula used to convert compound (nominal) interest rate to the effective
interest rate is 𝐸 = (1 + 𝑟)𝑛 − 1
 E = effective rate of interest per annum as a decimal.
 R = stated interest rate per compounding period as a decimal.
 N = number of time periods.
 This expresses rates as an annual rate of interest as if the loan was compounded
annually.
4. Credit cards
 Annual fee charged.
 Minimum payment that must be listed.
 Many banks follow an interest free period and providing the balance is paid by the
due date there are no credit charges on purchases imposed by the bank.
 If the interest free period is exceeded two charges are imposed.
 An initial charge equal to one month’s interest on the amount outstanding.
 A daily compound interest charge from the end of the interest free period.
 Compound interest formula is used.
Annuities and Loan repayments
1) Future value
 An annuity is a f