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MATH TRIAL > Formulas/info > Flashcards

Formulas/info Flashcards

1
Q

Simple Interest 1.1

A
  • I = PRN
  • Total owing = principal + interest

Interest: money earned on investment/charged on loan
Simple interest: (flat rate) interest calculated as a percentage of principal (amount invested/borrowed)

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2
Q

Compound Interest 1.2

A
  • FV = PV(1+R)^2
  • I = FV - PV

Compound Interest: interest added to principal + reinvested

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3
Q

Inflation and Appreciation 1.3

A
  • FV = PV(1+R)^n

Inflation: increase in price of goods/services
appreciation: increase in value of an item

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4
Q

Investing in Shares 1.4

A
  • DIVIDEND YIELD = dividend per share ÷ market price per share X 100
  • DIVIDEND = dividend yield X market price

Share: part ownership in a company
Dividend: type of interest earned by share holders
Dividend yield: percentage of market price of share
Brokerage: commission charged by stockbrokers

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5
Q

Share tables and graphs 1.5

A

LAST SALE = market price at end of day
+ OR - = change in price from previous day
NO. SOLD (100s) = no. shares sold (100s)
52 WEEK HIGH/LOW = maximum and minimum price in 52 weeks

Share tables: daily information about shares

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6
Q

Straight Line Depreciation 1.6

A
  • S = Vo-Dn

s - salvage
Vo - initial value
D - depreciation
n - no.periods

Depreciation: loss in value of asset over time
Straight line: item value decreases by same each time

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7
Q

Declining Balance Depreciation 1.7

A
  • S = Vo(1-r)^n

s - salvage (current value)
Vo - initial value
r = depreciation rate
n = number of periods

Declining balance: item value decreased by some percentage each period

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8
Q

Ration problems 2.1

A

Ratio: compares two or more parts of the same type

*simplify ratio, divide both parts by the HCF

105 : 75 (÷ by 15)
7 : 5

0.04 : 0.4 (x100 to make integers)
4 : 40 (÷ by 4)
1 : 10

45cm : 3m (convert to cm)
45cm : 300cm (÷ by 15)
3 : 20

A jar of jellybeans contains one yellow, white + green in ratio 12 : 11 : 22. if there are 600g of yellow, what is the mass of white jellybeans and all jellybeans.
12 : 11 : 22
Y : W : G
Y = 12 parts, 12p=600g
                      1p= 50g

since white=11p
11p = 50g X 11 = 550g
all parts= 12+11+22=45p
45p = 50g X 45=2250g

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9
Q

Dividing quantity in Ratio 2.2

A
  1. Find total number of parts by adding
  2. Find the size of one part by dividing
  3. Find size of each term of ratio by multiplying
  4. Check answers add to original quantity

Tim pays 30% of his weekly income of $1080 in tax. The remaining income is divided into savings + living expenses in the ratio 2:5. How much does Tim use for living expenses?

Saves 100-30=70% X 1080= $756
2 : 5 - total parts = 2+5=7p
7p = $756 (÷ by 7)
1p = $108
living expenses = 5p = 5 X $108 - $540
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10
Q

Rate problems 2.3

A
  • Rate rule
    write unit as fraction and solve using X/÷

e.g. beats/min = beats÷min

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11
Q

Unit Pricing 2.4

A
  • UNIT PRICE = price ÷ no. items/units

Unit price: price of one item or unit

Small can:
$2.85/375g (divide)
$0.0076/g = 0.76c/g

Large can:
$5.85/750g (÷)
$0.0078/g = 0.78c/g
i.e. small can is best buy

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12
Q

Solving Equations 3.1

A
  1. keep equation balanced by performing same on both sides
  2. Aim to have variable on one side of equation + a number on the other side

Equation: contains algebraic expression and equals sign

solve: 5-x÷2 = 2
10-x = 4
-x = -6
x = 6

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13
Q

Formulas and Equations 3.2

A

M = x+y+z÷3

M=22, x=25, y=26

22= 25+26+z÷3
66=51+z
z=15

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14
Q

Changing the subject 3.3

A 
y = mx+c
mx+c = y
c = y-mx
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15
Q

Linear Functions 3.4

A
  • Form y=mx+c
  • Gradient: rate of change y relative to x - m= rise÷run
  • y-intercept = value of y when x = 0

Function: relationship between 2 variables

y= -2x+10
-2 is coefficient of x
10 is constant term

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16
Q

Direct Linear Variation 3.5

A
  • y = kx

y - directly proportional to x
k - constant

solving a linear variation problem

  1. identify 2 variables and form variation equation
  2. substitute values for x and y to find k
  3. Rewrite y=kx using value of k
  4. Substitute a value for x or y into y=kx
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17
Q

Intersection of Lines 3.6

A

Simultaneous equations: equations solved same time

Break even point: when revenue covers the cost exactly

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18
Q

Reducing Balance Loan 2.1

A
  • TOTAL PAID = loan repayment X no. repayment
  • TOTAL PAID = principal + interest

Reducing balance loan: calculated on balance owing on loan original principal borrowed

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19
Q

Credit Cards 4.2

A
  • DAILY INTEREST RATE = annual interest rate ÷ 365
  • FV = PV(1+R)^N
  • I = FV-PV

Credit cards: used to buy goods/services and pay for them later, interest can be flat/compound.

  • if payment not received, interest charged from purchase date
  • Interest usually charged daily using compound interest
  • Monthly statement provides record of spending
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20
Q

Annuities 4.3

A

Using Table

  1. Determine time period and rate of interest
  2. Find intersection of time/rate of interest
  3. Multiply number w/ money contributed

Annuity: form of an investment involving regular contributions e.g. investment into super

  • made at end of compound period
  • earns more interest because regular/equal payments made

Future value: total value at end of term, sum of all contributions plus interest

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21
Q

Loan repayment tables 4.4

A

Loan repayment tables: enable repayments for reducing balance loans to be easily calculated

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22
Q

Repaying Home loans 4.5

A

$560000 purchase price with a $150 000 deposit over 30yrs compounding monthly at 7.2%p.a. with these fees:

  • $600 loan establishment fee
  • 1.4c per $1000 borrowed per month
  • $3.80 per month management fee
  • $720 mortgage discharge fee

Total cost of loan
P = 560 000 - 150 000 = 410 000
therefore PV = 410 000

Monthly repayments
monthly rate = 7.2÷12=0.6%/month, n=30X12 = 360 months

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23
Q

Scatter Plots 5.1

A

Scatterplot: graph of points on plane

- represents bivariate data (2 variables)

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24
Q

Correlation 5.2

A

Correlation: measure of strength of linear association or relationship between 2 variables.

Pearson’s correlation co-efficient: r, is a number between -1, +1 measuring correlation of 2 units

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25
Q

Line of Best fit 5.3

A

Line of best fit: the straight line drawn through the centre of dots with an equal number of dots above and below lone.
- The line can be used to predict values given one variable.

LINE OF BEST FIT - REGRESSION LINE (least squares line of best fit)

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26
Q

Speed + fuel consumption 2.5

A

AVERAGE SPEED = Distance travelled ÷ time taken

Speed: rate at which something is moving/changing

Fuel consumption: Rate at which a vehicle uses fuel measured in L/100km
- depends on: driving urban, towing, using A/C

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27
Q

Converting Rates 2.6

A
  • express as fraction, then divide

A car travels at 75km/h, how far to the nearest 0.1m will it travel in 5sec

28
Q

Least Squares regression 5.4

A
  • y = mx+c
  • m = r X standard deviation of y-scores ÷ standard deviation of x-scores
  • c = mean of y-scores - mean of x-scores

Regression line: line of best fit, representing all points (gradient/y-intercept calculated using formula)

29
Q

Life Expectancy 5.5

A

Life expectancy: average number of years of life remaining to a person of specific age

30
Q

Networks 6.1

A

Graph = diagram of a network, containing vertices (points), joined by lines called edges (arcs)

Loop = edge connecting a vertex to itself

Directed network = arrows on edges, indicate direction

Weighted network = numbers/weights on edges, distance same
- WEIGHT = SUM OF ALL EDGES

Degree: number of edges connected to vertices can be odd/even
- SUM OF ALL DEGREES = TWICE NUMBER OF EDGES

Walk: any route along edges that includes repeat vertexes/edges

Closed: starts/ends at same vertex

Trail: walk with no repeat edges, vertices may be reused

Circuit: Closed trail with no repeat edges

Path: no repeat vertices/edges

Cycle: Closed path with no repeat vertices/edges

Connected network: connected if path between 2 vertices

Networks/tables: networks can be represented using matrix/table

31
Q

Eulerian trails/circuits 6.2

A

Eulerian trail: uses every edge only once
- TRAILS = 2 ODD VERTICES, START AND FINISH AT THOSE VERTICES

Eulerian circuit: uses every edge only once, starts and ends at the same
- CIRCUIT = VERTICES ARE EVEN

32
Q

Minimum spanning trees 6.3

A

Tree: any 2 vertices are connected by one path, no cycles

Spanning tree: tree connecting all vertices

Minimum spanning tree: spanning tree with smallest weight

KRUSKAL’S ALGORITHM

  1. sort edges in increasing order of weight
  2. Diagram of network with all vertices, no edges
  3. Choose edge with smallest weight
  4. Choose next edge with smallest weight
  5. Continue process with no cycles and vertices

PRIM’S ALGORITHM

  1. choose any vertex in network
  2. Choose edge with least weight and correlate to another
  3. Use 2 vertices, select next edge with least weight
  4. Choose next edge with least weight
  5. Repeat step 4 until all connected
33
Q

Shortest path problems 6.4

A

Shortest path: between 2 vertices, can be found by trial and error

FINDING SHORTEST PATH

  1. redraw network with circles at each except start
  2. For all vertices from start write shortest distance
  3. Continue until final vertex reached
  4. Shortest path found by moving back from finish to start
34
Q

Activity tables/forward scanning 6.5

A

Activity table: shows order/time of each project

predecessor: previous activity needing to be completed

Dummy activity: eliminate repeat edges, zero time/weight

Critical path: longest time path start/finish vertices

Forward scanning: calculates EST, depending on completion of predecessor

  1. redraw network w/ empty box on vertex, with 0 at start
  2. select highest point for each 1st box
  3. Continue until end
35
Q

Backward scanning/ critical path 6.6

A
  • FLOAT TIME = LSTnext - ESTnext
  • FLOAT TIME = LSTnext - EST - activity time

*where ESTnext is ESTof next activity (earliest end time), LSTnext is LST of next activity (latest end time)

Backward scanning: calculates LST for each activity (take smallest distance)

Float time: longest time activity can be delayed without decreasing completion

36
Q

Network flow problems 6.7

A

Source: start of network

sink: end of flow network
capacities: weights on edges representing amount it holds

flow capacity: total flow through the network

Inflow: total capacity of all edges entering the vertex

outflow: total capacity of all edges leaving vertex

maximum outflow: smaller of the inflow/outflow of vertex

cut: line drawn through edges of flow network

capacity of cut: amount of flow blocked by cut, found by summing the cut edges passing from source to sink

37
Q

Maximum flow minimum cut 6.8

A

Maximum flow: capacity of the minimum cut

TO FIND
- calculate capacity of each cut and identify the cut that has the minimum capacity

38
Q

Heart Rates 7.1

A

Max heart rate: upper limit of what cardiovascular system handles

  • MHR MALES = 220 - AGE
  • MHR FEMALES = 226 - AGE

Target heart rate: desired heart rate to better aerobic fitness
- THR = I X (MHR - RHR) + RHR

*RHR = resting heart rate, MHR = max heart rate, I = exercise intenity

39
Q

Food and energy Consumption 7.2

A

ENERGY UNITS

  • 1 kilojoule (kj) = 1000 joules (j)
  • 1 megajoule (MJ) = 1 000 000 joules = 1000kj
  • 1 Calorie (cal) = 4.184kj
40
Q

Electricity usage homes 7.3

A
  • WATT = 1w = 1000mW
  • KILOWATT = 1kW = 1000W
  • MEGAWATT = 1MW = 1000kW = 1000000W (10^6W)
  • GIGAWATT = 1GW = 1000MW = 10^9

Kilowatt hours: electricity usage measured in kWh

Cents/kilowatt hours (c/kwh) = electricity usage given

Domestic rate: electricity used during day

Off-peak rate: electricity used late evenings/early morning, charges at cheaper rate

41
Q

Energy consumption cost appliances 7.4

A

RUNNING COST = POWER USED (kWh) X ELECTRICITY COST ($/kWh)

42
Q

Energy efficient housing 7.5

A

Energy efficient: homes making most of natural heating/cooling

Features considered

  • orientation: should face North to capture Winter sun
  • insulation: light-coloured roofing
  • wind/skylight: windows face North
  • Ventillation: cross-ventilation allowing air flow
  • Landscapes/shading: plants/trees can shade walls/windows
  • Building materials: materials that can absorb stone and heat
43
Q

Right-angled trig 8.1

A

SOH CAH TOA

  • Sin 0 = opposite÷hypotenuse
  • Cos 0 = adjacent÷hypotenuse
  • Tan 0 = opposite÷adjacent

Angle of elevation: angle looking up
Angle of depression: angle looking down

44
Q

Bearings and navigation 8.2

A

Bearings: show direction of location from given point

true bearing: (3 figures) between 000 + 360 measured clockwise from North

Compass bearing: angles measured from North-South axis towards East or West

45
Q

Sine Rule 8.3

A
  • SINE RULE - a÷sinA = b÷sinB = c÷sinC

Sine rule: relationship between sides/opposite angles

46
Q

Sine rule unknown angles 8.4

A
  • OBTUSE ANGLE ANSWERS = 180º0
47
Q

Cosine rule 8.5

A

c^2 = a^2+b^2 - 2abcosC

48
Q

Using cosine to find unknown 8.6

A

CosC = a^2+b^2-c^2 ÷ 2ab

49
Q

Area of Triangle 8.7

A

A = 1/2abSinC

50
Q

Problems sine/cosine 8.8

A

Triangle problems involving 2 sides and 2 angles opposite these sides
- SINE RULE

Triangle problems involving 3 sides and 1 angle
- COSINE RULE

51
Q

Scale drawings 9.1

A

Scale drawing: reduction of real object

Scale ratio: compares lengths in scale drawings to actual lengths
- e.g. if scale is 1:3, actual lengths are 3x size scaled length

52
Q

Scale Maps + plans 9.2

A

Scale = map: real life

1:1000 = 1mm:1000mm or 1cm : 1000cm or 1m : 1000m

53
Q

House plans 9.3

A

House plan: drawings drawn to scale

  • the plan: diagram of the floor (floor plan)
  • the elevation: view of front/back/sides of house
54
Q

Offset + radial survey 9.4

A

Right angled triangles: use pythag theorem, the trigonometric ratios and
- A=1/2bh

Non right angled triangles: use sine/cosine rules and
- A=1/2

Offset survey: used for irregular blocks of land e.g. surveyor chooses traverse line measuring offsets to each corner

Compass radial survey: taking true bearing of each corner on field using compass placed in middle of field

55
Q

Volume of tanks and dams 9.5

A

TRAPEZOIDAL RULE
- A=h/2 (df + dl)

*h = distance between successive points, df = first measurement, dl = last measurement

56
Q

Quadratic function 10.1

A

THE PARABOLA y=ax^2+bx+c

  • if positive: concave up
  • if negative: concave down
  • y-intercept is c and vertex is (0,c)
57
Q

Maximum and minimum problems 10.2

A

maximum and minimum quadratic: represented by vertex of its graph, y-value of vertex

  • Concave down: maximum value
  • Concave up: minimum value

(*answer is the y-value)

58
Q

Exponential Function 10.3

A
  • EXPONENTIAL CURVE - y=a^x, y=a^-x
  • exponential curve always above x-axis
  • y-intercept is 1
  • if a>1, y=a^x is increasing
  • if a>1, y=a^-x is decreasing

Exponential growth: any quantity increasing y=b(a^x), where a>1 e.g. population size, investments, bushfire

Exponential decay: any quantity decreasing y=b(a^-x), where a>1 or between 0 and 1 in y=b(a^x) e.g. radioactive decay

EXPONENTIAL GROWTH AND DECAY

  • growth occurs when y=b(a^x) and a>1
  • decay occurs when y=b(a^-x) and a>1, or when y=b(a^x) and a is between 0 and 1)
  • b is initial value of y (when x =0)
  • b is y-intercept of exponential curve
59
Q

Reciprocal function 10.5

A

THE HYPERBOLA y=k÷x

  • 2 seperate branches in opposite quadrants
  • point symmetry (0,0)
  • asymptotes = x,y axis
  • if k is positive, hyperbola decreasing, branches in 1st/3rd quadrant
  • if k is negative, hyperbola increasing, branches in 2nd/4th quadrant
  • higher value of k, further hyperbola from x, y axis

Reciprocal function: non-linear function, when x is denominator

Hyperbola: graph of reciprocals function, 2 seperate functions

60
Q

Inverse variation 10.6

A

SOLVING INVERSE VARIATION PROBLEM

  1. identify two variables and form variation equation
  2. substitute given values of x/y to find k
  3. rewrite equation y=k÷x using k value
  4. sub value for x/y into equation to solve problem

Direct variation: two variables change in same direction e.g. fuel consumption of car, viewing range/height of tower

Inverse variation: two variables change in different directions e.g. time of journey and average speed, or temperature and altitude

61
Q

The normal distribution 11.1

A

Normal distribution: mean, median, mode are equal - bell shaped curve is smooth and symmetrical about the mean

Frequency histogram: large small, bigger = normal curve

62
Q

z-scores

A

Z-SCORES = SCORE-MEAN ÷ STANDARD DEVIATION

z-score: shows position of raw score relative to mean
- e.g. z-score of 1.8 standard deviation is 1.8means, a score of -0.5 means is half standard deviation below the mean

63
Q

Comparing z-scores 11.3

A

since z-scores don’t depend on value of mean, we use them to compare scores from different distributions

64
Q

Measures of central tendency 11.4

A

CENTRAL TENDENCY

  • MEAN = SUM OF SCORES NO. OF SCORES
  • MEDIAN = MIDDLE SCORE

SPREAD

  • RANGE = HIGHEST SCORE - LOWEST
  • IQR = Q3-Q1
  • STANDARD DEVIATION = USING STATISTICS MODE

Measure of central tendency: an average used to indicate the centre of a data set

Measures of spread: used to indicate spread of data

65
Q

Skew of distribution 11.5

A

peaks: high points of distribution
clusters: groups of scores bunched together

+ skew: tail right, mean higher than mode/median

  • skew: tail left, mean lower than mode/median
    symmetry: data balanced
66
Q

Effect of outliers 11.6

A

OUTLIER

  • LESS THAN = Q1 - 1.5XIQR
  • MORE THAN = Q3 + 1.5 X IQR

outlier: very high/low score, clearly seperate from data
- mean is most affected by outlier
- median can be affected but not much
- mode is not affected

67
Q

Comparing data sets and plots 11.7

A

Back-to-back stem and leaf: compares 2 data sets

Box plot: uses five-number summary to show data spread

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