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EMA30S

Gr. 11 Essential Math
by

R Abetria

on 2 June 2015

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Transcript of EMA30S

Interest (
I
) = Principal (
P
) x interest rate (
r
) x time (
t
)
Simple
Interest

Interest and Credit
At a glance

- simple interest
- compound interest
- compounding periods
- credit options
credit cards
loans
lines of credit
What words come to mind when you hear "investments"?
Investments
When you
invest,
you use your savings to earn
extra income
!
I = P r t
Ex. 1
David is a sprinkler-system installer. He invested $1500 in a GIC for 2 years. The interest rate is 2.5%/yr. How much interest will he earn in a year?
5 steps in solving problems

1. write given information
2. formula
3. substitute values into the equation
4. solve
5. answer in sentence form
Ex. 2.
Sue is planning a trip to the U.S. She invested $5000 in a U.S. Foreign Currency Term Deposit. The annual interest rate is 0.5%. The deposit matures in 120 days.
How much money will Sue have for her trip
?
Compound
Interest
Compounding
Periods
interest can be compounded in different ways
Credit
Options

Principal
- the money invested or borrowed
Interest rate
- the percent of the principal that is paid or earned as interest
Time
- length of time for an investment in years
is the interest calculated only on the principal invested or borrowed
Simple Interest Problems
TIP:
Rearrange the formula first
to find the missing variable, before substituting values
Ex. 1 Dan knew the following: he earned $35 in interest in an investment, the annual rate was 1.9% and it was for 42 weeks. How much money did Dan invest?
Ex. 2. Jamie wants to earn $500 in interest so she’ll have enough to buy a used car. She puts $2000 into an account that earns interest. How long will she need to leave her money in the account if the annual rate is 1.75%?
Ex. 3. A local bank is advertising that you can double your money in eight years if you invest with them. Suppose you have $1000 to invest.
What interest rate is the bank offering
?
I
P
r
t
Ex. 1.
P = $5000
r = 1.5% /yr
compounded annually

How much will the savings be worth after 4 years?
$ 5000
I = 5000 x 0.015 x 1 = 75
A = 5000 + 75 = $5075
$ 5075
I = 5075 x 0.015 x 1 = 76.13
A = 5075 + 76.125 = $5151.13
$ 5151.13
I = 5151.13 x .015 x 1 = 77.27
A = 5151.13 + 77.27 = 5228.4
$ 5228.40
I = 5228.40 x .015 x 1 = 78.43
A = 5228.40 + 78.43 = 5306.83
3D Geometry
is the interest calculated on the
principal and the interest
earned
Annually
Semi-annually
Quarterly
Monthly
Weekly
Daily
once per yr
2 times per yr
4 times per yr
12 times per yr
52 times per yr
365 times per yr

You invested $1000
1.4%/yr
compounded weekly
2 yrs

What is the value of the
investment?
Compound Interest Problems
1.3%/yr
compounded daily
$1400 investment
5 yrs

What is the amount of the investment after 5 yrs?
72
The Rule of
can be used to estimate how long it will take for the investment to double
You invested $2500
3.1%/yr
compounded monthly
a. Estimate the doubling time for your investment.
b. Check that your answer is reasonable.
1.
Years to double
= 72
annual interest rate (as a percent)
Compound Interest Formula
A = P ( 1 + r )
n
nt
You invested $1000
2.7%/yr
compounded quarterly
How much interest will you earn after 4 years?
Advantages of having a credit card
Disadvantages
- allows you to borrow money
- can be used anywhere
- on-line purchases
- for convenience and emergencies
- points/rewards
- building a good credit rating
- tendency to overspend
- theft/stolen card
- bad credit
- interest payments
- false idea of having money
What are some of the responsibilities that come with having one or more credit cards?
making timely payments
keep track of your cards/statements
pay full balance or the minimum amount
What are some features of a credit card statement?
balance owing
- payment
+ interest (finance charge)
balance owing
Calculating interest charged on purchases
Ex. You purchased a $100 item on Dec. 27 with an interest rate of 19.5%/yr. The end of statement period is Jan 20.
Step 1
: Calculate how many days from post date to 1 day prior to end of statement period:
Dec 27 - Dec 31: 5 days
Jan 1 - Jan 19: 19 days
>
total: 24 days
Step 2
: Calculate interest
(annual or daily)
for the purchase (use simple interest formula).
I = Prt
grace period
: period of time before the due date

revolving balance
: a balance that goes up or down
Credit
- the ability to borrow money
Loans
in most cases, interest is
calculated daily
and
paid monthly
a secured loan requires

collateral
personal
payday/fast cash advance
student loan
auto loan
RRSP loan
equity
- the difference between the value of a property and the claims against it

Ex. Mr. A. is paying his $21,000 student loan.
interest rate = 3.1%/yr
compounded daily
payable monthly

First Payment: he will pay the interest owing, plus $500 of the principal. How much is his first payment?
What type of loans do you know?
Ex. 1 Doug wants a secured loan and will use his car as collateral. The bank will loan 75% of his car's equity. His car is worth $45,000. He owes $15,000 on it. How much will the bank lend Doug?
Lines of Credit
Sales Promotions
Tess' favourite store is having a sale. Everything is 30% off. She wants to buy a $40 sweater. If she applies for a store credit card, the sweater has an additional 15% off.
a. Calculate the discount for the sweater if she applies for the store credit card.

b. Calculate the cost of the sweater (+ PST & GST).
Surface Area
Volume and Capacity
Composite Objects
Relating nets to Surface area
3D objects
area vs. surface area
volume vs. capacity
imperial and SI systems
composite objects
Do you remember the area of the following 2D shapes?
A = 1 x b x h
2
C = 2 r
= d
A = r
2
A = l x w
net

a composite 2D shape that you fold to create a 3D object
the total area of the surface of a 3D object
Surface area: Prism
Prism
2 congruent bases
lateral area
combined area of the side faces
SA = 2 x (base area) + lateral area
Surface area: Cylinder
cylinder
2 circular bases
1 curved surface
SA = 2 r + 2 rh
radius
height
2
base area
lateral area
Surface area: Pyramid
height
slant height
distance from the top of the pyramid to the centre of any side of base
SA = 1(perimeter of base)(slant height) + base area
2
lateral area
Surface area: Cone
Cone:
circular base
curved surface
SA = rs + r
2
base area
curved surface area
Pythagorean Theorem
a + b = c
2
2
2
Surface area: Sphere
SA = 4 r
2
Dimensional Changes
For any right triangle, the square of the hypotenuse is equal to sum of the squares of the sides.
Sphere
- every point on its surface is the same distance from the centre
8
15
8
4
32
10
6
60
How much did the area increase when you double the 2 dimensions?
32 / 8 = 4 -> 4 times
Volume: Prism
Volume: Pyramid
V = base area x h
V = base area x h
3
Volume: Cylinder
V = r h
2
The V of a pyramid is 1/3 the V of a prism (with same base and height)
area of circular base
Volume: Cone
The V of a cone is 1/3 the V of a cylinder (with same base and height)
V = r h
2
3
Area of regular polygon
Rectangular prism
V = lwh
Rectangular pyramid
V = lwh
3
Volume: Sphere
V = 4 r
3
3
Volume: Hemisphere
V = 2 r
3
3
units:
cubic units
units cube (u )
3
A hemisphere is half of a sphere
A composite object is composed of 2 or more distinct objects
Volume of a composite object
1. identify the 3D objects involved
2. add their volumes
Volume: Dimensional Changes
V = r h
2
3
= (4 )(6)
2
3
= 100.53 cm
3
V = r h
= (
8
)(6)
3
2
2
3
= 402. 12 cm
402.12
100.53
= 4
New volume is 4x the old volume.
Capacity
the amount of material a container holds
1. Do example on p.108
2. Convert the answer to inches.
(NOTE: 1 ft = 12 inches
We can use
unit analysis
to convert units.
Calculate the capacity of this cone to the nearest hundredth:
r = 3 in
h = 7 in
Working with Graphs
Bar
Graphs

Bar Graph
a graph used to compare discrete sets of data
shows data with horizontal or vertical bars
the bars do not touch
Parts of a graph
title
vertical axis
horizontal axis
1
2
3
4
5
scale
the number represented by each unit in a graph
Line Graphs
Circle Graphs
Histograms
1. Describe the data
2. Describe the trend
3. Range
Graphs
and Technology

Graphic Representations
Histogram
summarizes discrete or continuous data
divides up the range of possible values in a data set into groups
Length of each bar equals the range of values in that specific group
the bars have no gaps (for continuous data)
generally used when dealing with large sets of data
A. Describe the data.
B. What is the most common height.
C. What is the range for the height if these trees?
D. Make an inference about the frequency of trees that are "very tall" in this sample?
INTERVALS
CATEGORIES
Frequency table
shows the number of items in each interval
a visual comparison of how two variables are related or vary with each other.
shows data with connected plot points
shows relationships more clearly than tables do
y axis: usually indicated quantity or percentage
x axis: often measures units of time
Interpolation
estimating values within a given set of values
Extrapolation
estimating values beyond a given set of values
Line Graph
40 F = 4.4 C
60 F = 15.5 C
a. What is the trend of the graph?
b. What time of the year was the data taken?
a. Desribe the trend.
a. Did Frank get better or worse at taking tests?
a. What is the car doing at 0 sec?
b. Is the car accelerating?
c. What is the car's speed at 3.5 sec?
Circle graph
show the component parts of a whole
summarizes categorical data
best used when you have less than 6 categories
if greater than 6 categories, use a bar graph
circle is divided into segments which represents a category
How to make a circle graph
1. Calculate the percent shift of each data
2. Represent these values in degrees
Note:
there are 360 degrees in a circle
number
total
x 100
percent
100
x 360
Managing
Money

Interest Review
Kyle invested in a term deposit. He deposited $2010 for 5 years with a rate of 2.75% per yr. How much interest will he earn?
Carrie invested $4000 in a savings account with an interest rate of 3.7%, compounded daily. How much will she have afters?
Financial
Institutions

Bank failure
Why do service charges vary?
some pay a monthly fee that includes a wide range of services
low monthly fee, pays for each ATM or debit charge
avoid the basic monthly fee by keeping a minimum monthly balance
Why do people use financial institutions to hold their funds?
they often pay interest on the money people deposit
accounts are a secure way of holding money
allow us to use cheques and debit cards to pay for goods and services we buy
effective record-keeping system for deposits and withdrawals
happens when a bank can't meet its financial obligations
Canada has a
deposit insurance system
, wher
e
government organizations arrange to pay many types of deposits if an insured institution fails
Banking account checklist
Debit Cards
Online Banking
What are the different purposes of a debit card?
What are the advantages of a debit card?
What are the disadvantages of a debit card?
What are the advantages of online banking?
What are the disadvantages of online banking?
What is a debit card?
Budgets
What are the purposes of a monthly budget?
it compares income and expenses
helps you decide where to reduce spending
helps in achieving goals
Fixed expenses
expenses that don't generally change from month to month
Examples:
Variable expenses
expenses that can easily change
Examples:
Recurring expenses
expenses that occur very frequently
What kinds of expenses a celebrity might have?
Monthly budget forms
monthly income
total monthly expenses: fixed + variable
To find out if you are in a surplus or in a deficit,

income - expenses = _______
Pay yourself first
some financial professionals say that it is the
golden rule
of personal finance
setting aside 10% of your income before you use the money
This 10% can be put in towards:
an investment
a financial/personal goal
10%
Relations and Patterns
Slope
Conversions
Linear
Relations

Scale
What is slope?
Getting started
a measure of steepness of a line
m = rise
run
m = change in y
change in x
grade
: slope expressed as a percent
for every 100m horizontal distance, the elevation increases by 8m
What is the slope of the hill?
What is the steepest grade of this hill?
Ex. 1
Where would you find slopes?
Ex. 2
(1, 5)
(-4, 1)
(4, 1)
Comparing slopes
Horizontal lines
Vertical lines
Problem Solving
ex. The minimum grade for a ramp in a hospital is 10%. What should be the vertical distance if the ramp is 7 m long?
the slope of every horizontal line is 0.
the slope of every vertical line is
undefined
.
The slope of a line is 4/9. The two points are: A(3, 7) and B(8, y). Find the missing coordinate?
Rate of change
Rate
A comparison between two numbers with different units.
It can be written or expressed using the same notation as a ratio.
Examples
Rate:
1) You can type 50 words in one minute:
2) Bananas are 67 cents per pound:
3) In one hour, you are paid $11.75:
4) There are 100 centimetres in one metre:
5) The speed limit is 50 km per hour:
the change in one variable relative to the change in another
It is the slope!!!
Unit Analysis
Ex 2. One of your classmates usually sends 156 text messages in 3 hours. How many messages could she send in one minute?
Angle of elevation
the angle between the horizontal and the line of sight (when looking up at an object)
We use the
tan ratio
to calculate for the
slope
tan x = rise
run
What do you notice?
A birdwatcher sights an eagle 20 m from the tree. The birdwatcher is lying on the ground 50 m from the tree. At what angle must he incline his camera to capture a photo of the eagle?
We use the
inverse tan ratio
to determine the angle of elevation
Try these:
a. tan 45
b. tan 30
c. tan 23.7
d. tan 19.6
e. tan 1
tan
-1
tan x = slope
Conversions within systems
Conversions between systems
Ex 2. A cheetah is capable of speeds up to 114 km/hr. Convert this to m/s.
Imperial system
feet
inches
yard
mile
Ex 1. How many hours would each route take?
1. driving 132 km on a gravel road at 80 km/hr
2. driving 175 km on a paved road at 100 km/hr
Express your answers in hours and minutes.
Example, p.164
A speed of a common snail is 0.003 ft/s. What is the speed in km/hr?
Example 2, p. 169
Time (h)
1
2
3
Sam's snowfall (in)
0.8
1.6
2.4
Winter's snowfall (cm)
17
34
51
Who had the greater rate of snowfall?
EMA30S
Relation
describes how two variables are related
Linear relation
has points that lie on a straight line
independent variable
dependent variable
To connect or not to connect?
# of quarters
0
1
2
3
4
value ($)
0
0.25
0.5
0.75
1.0
- an asset (car or property) that the lender may take if the borrower can't repay the loan
a revolving credit source that a bank or business extends to an individual
- interest is payable only on the credit used
A = P + I
amount
principal
interest
Ex. 3 Robyn saved $600 and invested it for 6 months in a GIC. The interest rate is 1.5%/yr. How much money will Robyn have?
May invests in a savings account each month. The interest rate is 1.3%/yr, compounded annually. After 2 years, her investment is worth $97.49. How much did she invest that month?
2
A = P ( 1 + r )
n
nt
loans can be secured or unsecured
a. What is the equity on Doug's car?
b. What is 75% of the equity in David's car?
Ex 2. Amy is lending Anne $600 to buy a sewing machine to start a small business. Amy charges simple interest at 2% /yr. How much will Anne owe if she repays Amy after 1 year?
interest is
compounded daily
and
paid monthly
Ex 1. Trey has an overdraft line of credit. Each transaction costs $5, plus interest at 21% /yr. On Tuesday, he had 3 transactions for $83. She paid back the money in 10 days. How much did he pay?
Ex 2. Mike needs a $4000 line of credit to buy equipment for his carpet-cleaning business. Interest is 5.75%/yr, compounded daily. His transactions are:
a. P = 3200, t = 98 days
b. P = 900, t = 173 days
c. P = 1450, t = 67 days
How much interest did he pay?
some promotions businesses use:

- a gift for applying for and using a credit card
- bonus frequent-flyer points
- low introductory interest rates
- reductions on prices
- cash back on purposes
c. Calculate the amount if she plans to pay with the store credit card in 6 months. (Interest is 21%/yr, compounded daily). The grace period is 30 days.
lets depositors use money in their account, either at a cash machine or when you buy things
Convenient, but you pay for the service either in a single monthly charge or for each transaction
Secret Passwords
Trigonometry
Design Modelling
Bank account statement
are a simple and effective way to watch for scams and errors
reviewing transactions and the balance each month can help avoid problems
being aware of the current balance will help avoid extra charges and embarrassment for overdrawn accounts or refused cheques and debits
careful record keeping helps people detect identity theft, scams, errors, and overspending
A budget compares expenses with income to show whether income will exceed expenses (a
surplus
) or expenses will exceed income (a
deficit
)
in reality, the less money you have, the more likely it is that a budget can help you!
A budget helps you gain control, not lose it!
Income
the first thing to do is to figure out your monthly income
Ex 1. A radio DJ plays 14 songs per hour. How many songs would she play in a 3.5 shift?
is a way of cancelling units and is based on the assumption that multiplying something by "1" does not change the value
Variation
a relationship between two variables
can be expressed as a formula, a table of values or a graph
Direct Variation
a relationship of the form
y = mx
x and y are variables
m is the slope!
Direct variation may be represented by a straight line graph which passes through the origin (0, 0)
Partial Variation
a relationship of the form
y = mx + b
x and y are variables
m is the slope
b is the y-intercept
Partial variation may be represented by a straight line graph which do not pass through the origin.
Ex 1. One litre of paint covers 80 ft . Express the relationship between the variable for the area to be painted, A, and the variable for the number of litres of paint, p.
a. as a formula
b. a table of values
c. graph
Ex 1. An employee is paid $15 an hour. In addition to his minimum hourly rate, he receives a $50 on-call fee. Express the relationship between his gross pay, P and the number of hours worked, n:
a. as a formula
b. table of values
c. graph
It usually involves a variable cost (i.e. $ per person) and a fixed cost,
Ex 2, You are organizing a banquet, It will cost $250 to rent the hall and will cost $10 for each person's food.
a. What is the type of variation?
b. Write a formula for the total cost C, if n people attend.
c. Complete the table of values.
d. Determine the tota lcost of 100 people attend.
Scatter Plots
can be used to show whether a relationship exists between two variables
Drawing scatter plots
Draw and label the axes. Choose a scale that allows you to plot all the data.
Plot the points in the table of values.
Give the graph a title.
Solving 3D
Triangle Problems
Example 1: From the top of a 90-ft observation tower, a fire ranger observes one fire due west of the tower at an able of depression of 5o , and another fire due south of the tower at an angle of depression of 2o . How far apart are the fires to the nearest tenth of a decimal?
Isometric Drawings
Orthographic Drawings
Exploded View Diagrams
Scale
One-Point
Perspective
Drawings

SOH
CAH
TOA

Angle of Elevation
Solving 2D Triangles
We can use scale to enlarge or reduce drawings
Scale factor
the constant factor by which all dimensions of an object is multiplied by in a scale drawing
Scale Ratio
a ratio, using the same units, that expresses the scale on a drawing
Ex. 1 Jeb's map of Manitoba shows a scale of 1 cm to 10 km. What is the scale ratio?
Ex. 2. Architects draw scale drawings of homes. A common scale is 1 in to 4 ft. The height, length, and width of a home are 6.5 in, 6.5 in, and 10 in on the drawing. What are the actual measurements in feet?
compounding period
If an investment is
compounded annually
, then
n = 1
To find out the compound interest earned, use:



I = A - P
Ex 2.
a. How much money will you earn if you have the following?
P = 850
r = 2.3% /yr
n = 1
t = 4 years

b. How much interest will you earn?
compounding period
(n = 1)
(n = 2)
(n = 4)
(n =12)
(n = 52)
(n = 365)
n = 365
t = 30 days
Shae, who installs floors, buys underpadding for $396.42 and tiling for $1,219.23. He saves 5% if he uses his store credit card. What will she save if she uses the credit card?
to calculate the SA of a 3D object for today: draw the net, calculate the area of each 2D object, then add
REVIEW
: when solving equations, use
opposite operations
to isolate the missing variable.
Types of bank accounts
Writing cheques
Identity theft
Full transcript