Maths notes- Yearly year 11 - Bored of Studies
Maths notes- Yearly year 11
Statistics in society (DA1)
Types of Data-
1- Categorical (Qualitative)
- This data cannot be measured
- Put this data into different categories e.g. models of cars.
- Distinct categories
- Pets- cats, dogs etc
- This data has no order
- Ordered Categories
- School Certificate bands (6, 5, 4, 3, 2, and 1)
- This data can be placed in order
2- Quantative (Numerical)
- This data can be measured
- Use numbers
- Discrete categories
- Use a counting process e.g. number of students in a class
- Continuous categories
- Using some type of measurement
- The values are on a continuous scale e.g. temp in a day
Target populations and sampling
1- Census
- involves the whole population
2- Sample
- part of the population
a) Random sample- using random numbers
b) Stratified sample- those sampled are chosen in proportion to the whole population
e.g. Adrian is conducting a survey of school students. If Adrian decides to survey 50 students, how many students should he choose from each year?
If there are 160 students in year 7 out of a total number of 800 students
Yr 7- 160/800 X 100
= 20%
= 20/100 X 50 = 10 students from yr 7
c) Systematic sample- a system is used to choose who is in the sample (every 20th pair of shoes on the production line)
Quality control
e.g. A batch of match boxes is tested for its control. If more then 3% of the matchboxes have less then 50 matches in them, the batch is rejected. If 300 matchboxes are tested and 10 have less then 50 matches, is the batch accepted or rejected?
10/300 X 100 = 3.3%
REJECTED!
Maths notes- Yearly year 11
Data collection and Sampling (DA 2)
Estimating Populations
e.g. a group of marine biologists want to estimate the population of fish in a lake. To do this they catch 100 fish, tag them and release them back into the lake. The next day they catch 200 fish and find that 10 of them have been previously tagged. What would be their estimate population?
% of tagged= 10/200 x 100 = 5%
5% of total= 100
1%= 100/5 = 20
100%= 20 x 100= 2000 fish.
Displaying Data (DA 3)
Types of Graphs
1- Sector graph-
(To find each sector-
32/60 x 360 = 210 degrees)
2- Column or bar graphs-
3- Line graph -
4- Divided bar graphs-
Maths notes- Yearly year 11
5- Frequency tables-
6- Dot plots-
7- Frequency Histogram and polygon-
8- Culmative frequency histogram and polygon-
Maths notes- Yearly year 11
9- Stem and Leaf plots-
10- Five number summaries-
- Lower extreme- the lowest score in the data set
- Lower quartile- the score at the 25th percentile (25%)
- Median- the middle score (50%)
- Upper quartile- the score at the 75th percentile (75%)
- Upper extreme- the highest score in the data set
11- Bar and whisker plots-
Measurements of central tendency-
- Mean- Average
- Add up all the scores and then divide by how many there are
- Median- The middle score when the scores are in order of size
- Mode- Most common score (score with highest frequency)
- Range- Highest – lowest score
Maths notes- Yearly year 11
Mean of a distribution table-
Probability (PB1 + PB 2)
Outcomes and Sample Space-
- The sample space is the list of all possible outcomes.
- The # of elements in a sample space is the total # of possible outcomes.
Relative Frequency-
Number of times an event occurred
Number of Trials
Complementary Events-
P(event does not occur)= 1- P(event does occur)
Units of Measurement (M1)
Km
÷ 1000 × 1000
M
÷100 × 100
Cm
÷10 × 10
Mm
T
÷1000 ×1000
Kg
÷ 100 ×100
G
Maths notes- Yearly year 11
K
÷1000 ×1000
L
÷1000 ×1000
ML
Significant Figure’s-
-Non-zero digits are significant- 364 (3 sig fig)
- Zero’s between non-zero digits are significant- 3064 (4 sig fig)
- Zero’s at the end of a number may or may not be significant- 2000 (1 sig fig)
-Zero’s at the beginning of a decimal are not significant- 0.00035 (2 sig fig)
- Zero’s at the end of a decimal are significant- 2.350 (4 sig fig)
Scientific Notation-
-A large number or a small number can be written in Scientific notation.
- Number between 1 + 10 × a power of 10
- 245 000 000 000 = 2.45 × 1011
Rates-
4 Banana’s
20g sugar
10ml lemon juice
250g coconut
2 eggs
20g apricot jam
Ban is having 9 people to dinner. The above ingredients are enough for 6 people. Modify this recipie to serve 9 people.
9÷6= 1.5, so times all of the above ingredients by 1.5g
Speed and fuel consumption-
Speed = distance
—————
time
A car uses 45 L of petrol on a 432km trip. Write the fuel consumption in km/L-
432km/45L= 9.6km/L
Converting Rates-
-Change the units 1 step at a time
6km/h to m/min= 6km/h= 6000m/60mins
60 60
= 100m/min
Ratio Problems-
Bill and bobs height are in the ratio 10:9. if bobs height is 162 cm tall, how tall is bill? 10:9= x:162 ...÷ 9 by 162 = 18, then × 18 by 10 = 180= bills!
Maths notes- Yearly year 11
Area and Volume (M2)-
Area Formula’s-
-Square- A= S²
- Rectangle- A= LB
- Triangle- A= ½bh
- Parallelogram- A= bh
- Rhombus- A= ½xy
- Trapezium- A= ½h(x×y)
- Circle- A= pie r²
Volume of Prisms-
V of cylinder= pie r²h
V of cones and pyramids= ½AH
V of sphere’s= 4/3 pie r³
Similarity (M3)
Using Ratio-
The ratio of boys to girls in a school class is 5:4. if there are 60 boys, how many girls are there?
B : G
5 : 4
60 : n
60÷5×4= 48
60 : 48
Ratio Splitting-
Split $50 into the ratio of 7:3-
10 parts = $50
7 parts= 7/10 × 50
= 35
3 parts= 3/10 × 50
= 15
$50= $35:$15
Right Angle Triangles (M4)
Pythagoras Theorm-
C²=A²+B²
• Write py. Thag for the triangle
• Is the letter out the front?
• If yes, work it out. If no, put the letter out the front of the equation and change the sign to minus.
• Find the square root.
Maths notes- Yearly year 11
Trigonometry-
Hypotenuse- opposite the right angle.
Opposite- Opposite the marked angle.
Adjacent- Next to the marked angle.
SOH CAH TOA!
Sin=opp Cos=adj Tan=opp
Hyp hyp adj
Finding the unknown side-
• Label the sides of the triangle
• Write ratio (sin, cos tan)
• Put the #’s in
• If the letters on the top, then ×
• If the letters on the bottom, then ÷
Financial Mathematics (FM1)
Commission, Piecework and Royalties-
Commission- Percentage of the total amount sold.
Piecework- Earn money for each item made/ produced.
Royalties- Earn payments through a copywrited product.
Bonuses and Allowances-
Bonus- extra money earned for either high quality work or volume.
Allowance- paid to cover costs or to cover difficult working conditions.
Annual leave- usually 17.5% of 4 weeks normal pay.
Overtime-
Normal time- Hours × Rate
Time and a half- 1½ × Hours × Rate
Double time- 2 × Hours × Rate
Investing Money (FM 2)
Simple Interest-
I= PrN
I=Interest
P= Principle $
r= % Rate (÷100)
N= Term
E.g’s $12000 at 9.5% for 5 years=
12000 × 9.5 × 5 = $5700
100
Therefore = 12000 + 5700= 17700
$7600 at 5.2% for 9 months=
7600 × 5.2 × 9= $296.40
100 12
Maths notes- Yearly year 11
Finding Rates and Terms-
What was the interest rate p.a. if $2800 invested for 4½ years earned $1102.50?
1102.50= 2800 × r × 4½ = 12600 × r
12600 12600
= 0.0875 × 100%
= 8.75%
Compound Interest-
A= P (1+r)ⁿ
A= Total amount
P= $, Principle
R= % rate
ⁿ= time periods
e.g. Find the final amount of an investment of $12000 at 7%p.a. for 5 years, where interest is compounded annually.
A=P(1+r)ⁿ 5
= 12000 (1+7/100)
= $16830.62
Changing Rates and Time periods-
2.5%p.a., 2 yrs, interest compounded annually.
2.5% = 0.2083%/month
12
2yrs= 2 × 12 = 24 months
Share dividends-
Dividend= Profit ÷ no. of shares
Dividend yield= dividend/share × 100%
Market price of share
E.g. A company with a share price of %5.42 declares a dividend of 25¢. Calculate the dividend yield correct to 2 dec.pl.
D= 25¢ × 100
5.42
= 4.61%
Inflation and Appreciation-
The cost of a new car is $35000. if the inflation rate is 5%, find the price of a new car after 1 year
5 × 35000= 1750
100
1750+ 35000= 36750
The cost of a T.V. is $800. if the average inflation rate is 4% estimate the cost of the T.V. after 5 years. 5
P(1+r)ⁿ= 800(1+ 4 )
100
= $973.32
Maths notes- Yearly year 11
Taxation (FM 3)
Allowable deductions-
Allowed a deduction of 51.9¢/km. calculate the size of the tax deduction in a year where 2547 km is travelled.
0.519 × 2547= $1321.89
Taxable income-
Taxable income= Total income- deductions
Medicare Levy-
Usually 1.5% of taxable income
Calculate the levy for a person with a taxable income of $44300
Levy= 1.5 × 44300
100
= $664.50
GST-
To calculate the amount of GST from an amount including the tax, divide by 11.
E.g.1 A cricket bat has a pre GST price of $125.50. Calculate the GST payable on the bat- 10 × 125.5 = $12.55
100
E.g.2 The smiths pay $19.80 for a meal which included GST. Calculate the GST that was paid.
19.80 ÷ 11= $1.80
-----------------------
• Spaces are needed between each bar.
• A column graph is the same, just upright
Score
Tally
Frequency
1
2
3
4
5
6
7
8
9
111
11
1111
1
111
11
11
111
1111
3
2
4
1
3
2
2
3
4
24
• Always add up the frequency column
• Space in between first column
• Polygon placed in the middle of each column
• HJ)*‡§©ê
ì
;íÞÍ¿±¿±¢‘±±p¢bPpbpbBbph•eéOJQJ^JmH
sH
#hµj™hÅX>*[pic]OJQJ^JmH
sH
hÅXOJQJ^JThe Polygon is placed at the right side of each column
• Space in between first column
Stem
Leaf
4
5
6
7
5 9
4 5 7 2 5
8 2 1 4 5 6 6
1 0 2
Interquartile range= Upper quartile – lower quartile
Lower
Extreme
Lower Quartile
Median
Upper
Quartile
Upper
Extreme
Score (x)
Frequency (f)
F x X
4
5
6
7
8
9
3
7
11
13
10
6
50
12
35
66
91
80
54
Mean= sum of F x X
sum of f
= 338
50
= 6.76
Length
Mass
Capacity
V=A×H
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