MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Version A
(NOTE: Throughout this course you will find examples and support questions with
answers. Dependent upon rounding of values, answers given may differ slightly from
what you calculate.)
MAP4C – Foundations for College Mathematics
Introduction
MAP4C - Foundations for College Mathematics 12
Welcome to the Grade 12 Foundations for College Mathematics, MAP4C. This fullcredit course is part of the new Ontario Secondary School curriculum (2007).
Materials
This course is self-contained and does not require a textbook. You will however, need
the following items:
•
•
•
•
•
**Scientific calculator (this is a must have instrument)
Lined paper
Graph paper
Ruler
Writing utensil (preferably a pencil with an eraser)
Structure of Lessons
Each lesson contains one or two concepts and multiple examples for complete insight.
Definitions will be bolded for emphasis. At the end of each new concept, there are
support questions to complete on your own to check for full understanding. At the end
of each lesson, there are several key questions that cover all concepts learned in the
lesson which need to be submitted for evaluation.
Evaluation
In order to be granted a credit in this course, you must:
•
•
•
Successfully complete the Key Questions for each unit and submit them for
evaluation within the required time frame (40%)
Complete and pass the midterm exam (30%)
Complete and pass the final exam (30%)
Copyright © 2007, Durham Continuing Education
Page 2 of 38
MAP4C – Foundations for College Mathematics
Introduction
Key Questions, the midterm exam and the final exam will be evaluated on the basis of
the following four categories of achievement (outlined by the Ministry of Education):
•
•
•
•
Knowledge & Understanding 40%
Applications 40%
Communication 10%
Thinking & Inquiry 10%
Support Questions
Support Questions will be noted with this pencil icon. These questions will help you
understand and master each new concept and are not to be submitted for evaluation.
•
•
Answer support questions in your notebook and keep them as a reference for
key questions.
Check your answers to support questions by using the “Support Question
Solutions” at the end of each unit.
Key Questions
Key Questions will be noted with this key icon. These questions evaluate your
achievement of the expectations for the lesson and must be submitted at the end of
each unit.
•
•
•
Write your solutions (a solution is a step by step process of how you get to your
final answer) on your own paper.
Write the lesson number on the top of the page and label each question clearly
when you submit your work.
You must try all key questions and complete most of them successfully in order
to pass each unit.
Midterm and Final Examinations
The midterm and final exams are weighted exams. The midterm will cover material
learned in Units #1 - 2 and will be a two hour exam. The final examination will cover
material from Units #1 - 4 and will be a two hour exam as well.
Copyright © 2007, Durham Continuing Education
Page 3 of 38
MAP4C – Foundations for College Mathematics
Introduction
Table of Contents
Unit 1
Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Future Value of an Annuity
Present Value of an Annuity
Mortgages
Amortization Tables
Renting Vs Buying
Unit 2
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Exponents
Linear, Quadratic and Exponential Functions
Scatter Plots
Line of Best Fit and Extrapolation
Uses and Misuses of Sample Data
Unit 3
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Survey Design
The Media and Data Management
Imperial and Metric Measurements and Conversions
Perimeter
Area/Surface Area
Unit 4
Lesson 16
Lesson 17
Lesson 18
Lesson 19
Lesson 20
Volume
Trigonometric Ratios
Law of Sine
Law of Cosine
Trigonometry and Problem Solving
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Page 4 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 1
Future Value of an Annuity
MAP4C – Foundations for College Mathematics
Lesson 1
Lesson One Concepts
¾ Calculating the future value of an annuity with matching payment and
compounding frequencies
Future Value of an Annuity
Annuity is a series of equal deposits made at equal time intervals. Each deposit is
made at the end of each time interval.
R[(1 + i ) − 1]
i
n
FV =
FV = the amount the investment “grows” to
R = the regular deposit made in dollars
i = interest rate, as a decimal, per compounding period OR
= interest rate as a decimal
# of interest periods in 1 year
n = number of compounding periods OR
= # of compounding periods per year multiplied by the # of years
A compounding period means the time frame in which compound interest is
calculated.
Example 1
Noah deposits $1500 at the end of each six months in an account that pays 6%
compounded semi-annually. What is the amount in the account at the end of 15 years.
Solution
R = the regular deposit made in dollars
= 1500
i = interest rate, as a decimal, per compounding period OR
= interest rate as a decimal
# of interest periods in 1 year
.06
=
2
=0.03
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MAP4C – Foundations for College Mathematics
Lesson 1
n = number of compounding periods OR
= # of compounding periods per year multiplied by the # of years
= 2 x 15
= 30
30
1500[(1 + .03 ) − 1]
FV =
.03
= 71 363.12
The amount at the end of 15 years is $71 363.12
Example 2
How much did Noah earn in interest from his annuity in example 1.
Solution
1500 x 2 x 15 = $45 000
Interest Earned = total of annuity at investment – total principal paid into annuity.
= 71 363.12 – 45 000.00
= $26 363.12
Example 3
Steven wants to buy a $5000.00 car when he graduates from high school. He deposits
$85 at the end of each month in an account that pays 9% compounded monthly.
Steven will graduate in 4 years. Will he have enough money by then?
Solution
R = 85
.09
i=
= 0.0075
12
n = 4 x 12 = 48
85[(1 + .0075 ) − 1]
.0075
= 4889 .26
48
FV =
Steven will not have quite enough to purchase his $5000 car.
Copyright © 2007, Durham Continuing Education
Page 7 of 38
MAP4C – Foundations for College Mathematics
Lesson 1
Support Questions
R[(1 + i ) − 1]
. Calculate FV for each set of values.
i
n
1.
Use the formula FV =
a. R = $400, i=0.04, n = 12
.05
b. R = $100, i=
, n = 50
2
.08
c. R = $500, i=
, n = 36
12
2.
Calculate the amount of each annuity.
a. $200 deposited at the end of each year for 5 years at 4% compounded
annually.
b. $1000 deposited every six months for 4 years at 9% compounded semiannually.
1
c. $450 deposited at the end of every three months for 10 years at 4 %
2
compounded quarterly.
3.
The Smith’s are buying a house and deposit $1500 every month into an account
that pays 5% compounded monthly. How much will they have in 2 years?
Key Question #1
R[(1 + i ) − 1]
. Calculate FV for each set of values.
Use the formula FV =
i
n
1.
a. R = $100, i=0.035, n = 10
.08
b. R = $300, i=
, n = 30
2
.10
, n = 48
c. R = $450, i=
4
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MAP4C – Foundations for College Mathematics
Lesson 1
Key Question #1 (continued)
2.
Calculate the amount of each annuity.
1
a. $1000 deposited every six months for 6 years at 5 % compounded semi2
annually.
b. $600 deposited at the end of each year for 10 years at 4.25% compounded
annually.
c. $1500 deposited at the end of every three months for 25 years at
5.75%compounded quarterly.
3.
Brianna deposits $150 at the end of each quarter into an account that pays 8%
compounded quarterly. Calculate the amount in the account at the end of each
time period.
a.
b.
c.
d.
4 years
9 months
6.75 years
20.5 years
4.
Beginning one month after birth of their son, Noah, the Nelsons deposited $100
each month in an annuity for his college fund. The annuity earned interest at an
average rate of 6.8% compounded monthly until his 18th birthday. What was the
amount of Noah’s college fund on his 18th birthday?
5.
Referring to question 4, how much interest did Noah’s college fund earn in total
on his 18th birthday?
Copyright © 2007, Durham Continuing Education
Page 9 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 2
Present Value of an Annuity
MAP4C – Foundations for College Mathematics
Lesson 2
Lesson Two Concepts
¾ Calculating the present value of an annuity with matching payment and
compounding frequencies
Present Value of an Annuity
The principal that must be invested now to provide an annuity is called the Present
Value of an Annuity.
R[1 − (1 + i) ]
i
−n
PV =
PV = the amount of the investment that must be invested now
R = the regular withdrawal made in dollars
i = interest rate, as a decimal, per compounding period OR
= interest rate as a decimal
# of interest periods in 1 year
n = number of compounding periods OR
= # of compounding periods per year multiplied by the # of years
A compounding period means the time frame in which compound interest is
calculated.
Example 1
Joey’s annuity pays $2700 at the end of each year for 4 years, starting 1 year from now.
The annuity earns 6.5% compounded annually. Determine the present value of the
annuity.
Solution
R = the regular withdrawal made in dollars
= 2700
i = interest rate, as a decimal, per compounding period OR
= interest rate as a decimal
# of interest periods in 1 year
.065
=
1
=0.065
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MAP4C – Foundations for College Mathematics
Lesson 2
n = number of compounding periods OR
= # of compounding periods per year multiplied by the # of years
=1x4
=4
−4
2700[1 − (1 + .065 ) ]
PV =
.065
= 9249 .66
The amount to invest now is $9249.66 in order to withdraw $2700 each year for 4 years.
Example 2
How much will Joey earn in interest from his annuity in example 1.
Solution
2700 x 4 = $10 800
Interest Earned = total of annuity at investment – total principal paid into annuity.
= 10 800.00 – 9249.66
= $1550.34
Example 3
Kristen is converting her RRSP into an income fund. She wishes to receive $1500
every six months for the next 20 years, starting 6 months from now. She is guaranteed
an interest rate of 6.25% compounded semi-annually. How much must Kristen deposit
now to pay for the annuity?
Solution
R = 1500
.0625
i=
= 0.03125
2
n = 2 x 20 = 40
1500[1 − (1 + .03125 )
PV =
.03125
= 33982 .11
−40
]
Kristen must deposit $33 982.11 now to have semi-annually payment of $1500.00
Copyright © 2007, Durham Continuing Education
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MAP4C – Foundations for College Mathematics
Lesson 2
Support Questions
R[1 − (1 + i) ]
Use the formula PV =
. Calculate PV for each set of values.
i
−n
1.
2.
a. R = $400, i=0.04, n = 20
.05
b. R = $100, i=
, n = 20
2
.08
c. R = $700, i=
, n = 36
12
Calculate the present value amount of each annuity.
a. $200 withdrawn at the end of each year for 6 years at 4% compounded
annually.
b. $1000 withdrawn every six months for 4 years at 10% compounded semiannually.
1
c. $750 withdrawn at the end of every three months for 10 years at 4 %
2
compounded quarterly.
3.
Cliff has recently sold his business. He sets up an annuity that will pay him
$3000.00 per month for the next 10 years. The first payment is to be made 1
month from now. Cliff can invest his money at 5.5% compounded monthly. How
much does the annuity cost today?
Key Question #2
R[1 − (1 + i) ]
. Calculate FV for each set of values.
i
−n
1.
Use the formula PV =
a. R = $100, i=0.025, n = 15
.08
b. R = $200, i=
, n = 50
4
.10
, n = 36
c. R = $750, i=
4
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MAP4C – Foundations for College Mathematics
Lesson 2
Key Question #2 (continued)
2.
Calculate the amount of each annuity.
1
a. $1200 withdrawn every six months for 5 years at 4 % compounded semi2
annually.
b. $500 withdrawn at the end of each year for 12 years at 4.75% compounded
annually.
c. $1250 withdrawn at the end of every three months for 20 years at 6.25%
compounded quarterly.
3.
Don won the “Cash for Life” lottery and will receive a $1000 per week for the next
25 years. How much must the lottery corporation invest today into an account
that pays 4% compounded weekly to provide Don with the prize?
4.
An annuity pays $1200 per year for 15 years. The money is invested at 5.2%
compounded annually. The first payment is made 1 year after the purchase of
the annuity. Determine the interest earned by the annuity over the 15 years.
5.
Noah wants to buy a 5 year annuity. He has two options.
•
•
Option A pays $1000 at the end of each year, starting one year from
now. It earns interest at 6.25% compounded annually.
Option B pays $500 at the end of every six months, starting six months
from now. It earns interest at 6.25% compounded semi-annually.
Which annuity should Noah choose? Write an explanation that includes
calculations to support your answer.
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Page 14 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 3
Mortgage Payments
MAP4C – Foundations for College Mathematics
Lesson 3
Lesson Three Concepts
¾ Calculating mortgage payments with matching payment and compounding
frequencies
¾ Introducing mortgage terminology
Mortgage Terminology
Mortgage: is a loan to buy property, with the property as security
Mortgagor: the person buying the property
Mortgagee: the lending institution lending the money to the purchaser (mortgagor)
Down payment: a required percent of the purchased price given in cash prior to the
mortgage being given
Amortized: when both the principal and interested on a mortgage is repaid with a
series of equal regular payments
Amortization Period: the length of time a mortgage is repaid
Term: The length of time that a mortgage is must be paid in full or renegotiated under
newly negotiated interest rate
When calculating the mortgage payment “R” the present value formula is used.
R[1 − (1 + i) ]
i
−n
PV =
Example 1
A house is purchased for $150 000 with a 10% down payment. What is the monthly
mortgage payment on the house with a 5 year term at 6% compounded monthly
amortized over 25 years?
Solution
Down payment = 150 000 (.10) = $15 000
PV = amount of mortgage
= 150 000 – 15 000
= 135 000
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MAP4C – Foundations for College Mathematics
Lesson 3
i = interest rate, as a decimal, per compounding period OR
= interest rate as a decimal
# of interest periods in 1 year
.06
=
12
=0.005
n = number of compounding periods OR
= # of compounding periods per year multiplied by the # of years
= 12 x 25
= 300
−300
R[1 − (1 + .005 ) ]
135000 =
.005
135000 = 155.207R
135000 155.207R
=
155.207
155.207
869.81 = R
The monthly payment for the next 5 years will be $869.81 per month.
Example 2
In example 1, if that was the payment amount for the entire 25 years how much was
paid in total and what was the total amount paid in interest for the house?
Solution
Total paid for the house = 869.89 x 300 (total number of payments)
= 260 943 +15 000 (down payment)
= $275 943
Total interest paid = 275 943 – 150 000
= $125 943
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Page 17 of 38
MAP4C – Foundations for College Mathematics
Lesson 3
Support Questions
1.
Determine the monthly payment for each mortgage. The interest is compounded
monthly.
a.
b.
c.
d.
Mortgage amount
$140 000
$200 000
$175 000
$350 000
Amortization period
30 years
25 years
20 years
25 years
Interest rate
6%
5.25%
6.20%
5.95%
2.
Determine the total interest paid for each of questions in question 1.
3.
Elaine purchased a house for $225 000. She made a down payment of 25% of
the purchase price and took out a mortgage for the rest. The mortgage has an
interest rate of 7.25% compounded monthly, and amortization period of 30 years,
and a 3 year term. Calculate Elaine’s monthly payment.
4.
The Nelson’s have a mortgage of $125 000 amortized over 25 years at 7%
compounded monthly. After the original 5 year term, the mortgage is renewed at
6.5% compounded monthly. Calculate the new monthly payment.
Key Question #3
1.
Determine the monthly payment for each mortgage. The interest is compounded
monthly.
Mortgage amount Amortization period Interest rate
a. $185 000
25 years
6.25%
b. $220 000
30 years
5.75%
2.
Determine the total interest paid for each of questions in question 1.
3.
Ashlee purchased a house for $875 000. She made a down payment of 15% of
the purchase price and took out a mortgage for the rest. The mortgage has an
interest rate of 6.95% compounded monthly, and amortization period of 20 years,
and a 5 year term. Calculate Ashley’s monthly payment.
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Page 18 of 38
MAP4C – Foundations for College Mathematics
Lesson 3
Key Question #3 (continued)
4.
The Vaughan’s have a mortgage of $725 000 amortized over 25 years at 4.25%
compounded monthly. After the original 4 year term, the mortgage is renewed at
4.0% compounded monthly. Calculate the new monthly payment.
5.
A bank charges 7.75% compounded monthly on a mortgage. The Markins have
an excellent credit rating. They negotiated a rate of 7% compounded monthly on
a mortgage on $230 000 amortized over 25 years. By how much did the Markins
reduce their monthly payment by negotiating the lower rate of interest?
Copyright © 2007, Durham Continuing Education
Page 19 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 4
Amortization Table
MAP4C – Foundations for College Mathematics
Lesson 4
Lesson Four Concepts
¾ Calculating mortgage payments with matching payment and compounding
frequencies
¾ Introducing the concept of an amortization table and calculating the monthly
interest paid, principal paid and outstanding balance
Amortization Table (Schedule)
An amortization table shows in detail how the mortgage is to be repaid over a
predetermined period of time. It lists each payment, how much of each payment is
interest how much of each payment goes to reduce the mortgage, and the outstanding
principal after each payment.
Example 1
Complete and amortization table for the first six monthly payments given the following
information:
Monthly payment: $ 694.63
Mortgage amount: $90000
Interest rate: 8% compounded monthly
Solution
Payment
Number
0
1
Monthly Interest
Payment Paid
None
None
$694.63
.08
90000(
) = 600
12
2
$694.63
3
$694.63
4
$694.63
5
$694.63
6
$694.63
89905.37(
.08
) = 599.37
12
89810.11(
.08
) = 598.73
12
89714.21(
.08
) = 598.09
12
89617.67(
.08
) = 597.45
12
89520.49(
.08
) = 596.80
12
Copyright © 2007, Durham Continuing Education
Principal
Paid
None
694.63 –
600.00=
$94.63
694.63 –
599.37=
$95.26
694.63 –
598.73=
$95.90
694.63 –
598.09=
$96.54
694.63 –
597.45=
$97.18
694.63 –
596.80=
$97.83
Outstanding
Principal
$90 000
90 000 – 94.63=
$89 905.37
89 905.37 – 95.26=
$89 810.11
89 810.11 – 95.90=
$89 714.21
89 714.21 – 96.54=
$89 617.67
89 617.67 – 97.18=
$89 520.49
89 520.49 – 97.83=
$89 422.66
Page 21 of 38
MAP4C – Foundations for College Mathematics
Lesson 4
Example 2
A mortgage of $200 000 is required to purchase a house. The mortgage will be repaid
with equal monthly payments over 25 years at 10% compounded monthly. What is the
monthly payment and complete an amortization table showing the first three payments.
Solution
⎛ .10 ⎞
R[1 − ⎜1 +
⎟
12 ⎠
⎝
200000 =
.10
12
200000 = 110.047R
−300
]
200000 110.047R
=
110.047
110.047
1817.40 = R
Payment
Number
0
1
Monthly Interest
Payment Paid
$1817.40
.10
200000(
) = 1666.67
12
2
$1817.40
3
$1817.40
199849.27(
.10
) = 1665.41
12
199697.28(
.10
) = 1664.14
12
Copyright © 2007, Durham Continuing Education
Principal
Paid
1817.40 –
1666.67=
$150.73
1817.40 –
1665.41=
$151.99
1817.40 –
1664.14=
$153.26
Outstanding
Principal
$200 000
200 000 – 150.73=
$199 849.27
199 849.27–
151.99=
$199 697.28
199 697.28 –
153.26=
$199 544.02
Page 22 of 38
MAP4C – Foundations for College Mathematics
Lesson 4
Support Questions
1.
Complete the first six payments in an amortization table given the following
information. Show all steps.
Monthly payment: $ 1340.37
Interest rate: 8.25% compounded monthly
Mortgage amount: $170 000
2.
Using question 1, how much interest was paid in the first 6 months?
3.
Using question 1, how much principal was paid off the mortgage after six
payments?
4.
The Nelson’s have a mortgage of $125 000 amortized over 20 years at 8%
compounded monthly. Determine the monthly payment and create an
amortization table for the first six payments.
Key Question #4
1.
Complete the first six payments in an amortization table given the following
information. Show all steps.
Monthly payment: $1590.25
Interest rate: 7% compounded monthly
Mortgage amount: $225 000
2.
Using question 1, how much interest was paid in the first 6 months?
3.
Using question 1, how much principal was paid off the mortgage after six
payments?
4.
A mortgage of $300 000 amortized over 35 years at 7.25% compounded
monthly. Determine the monthly payment and create an amortization table for
the first six payments.
5.
Using question 4, how much interest was paid in the first 6 months?
6.
Using question 4, how much principal was paid off the mortgage after six
payments?
Copyright © 2007, Durham Continuing Education
Page 23 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 5
Housing: Owning Vs Renting
MAP4C – Foundations for College Mathematics
Lesson 5
Lesson Five Concepts
¾ Advantages and disadvantages of buying a home
¾ Advantages and disadvantages of renting a home
Accommodations: Rent vs. Buy
Why Rent?
Choice
Today you’ll find more choice of rental accommodation than ever before – as well as
more amenities and flexible housing styles.
Location
When considering cost, don’t forget the cost of commuting to work. Often rental
properties are more central for jobs, amenities and transit than homes in suburban
communities. And if you want to be right downtown in a major metropolitan area, renting
is almost always more affordable than owning a condo.
Space
Apartment units are typically larger than comparable suite types in new condominium
properties.
Simplicity
Compared to owning a home, rental living is easy:
•
•
•
•
•
Property staff to call if anything goes wrong
Full-time maintenance staff to handle maintenance and repairs
No snow shovelling or lawn maintenance
No worries when you go away on vacation
A single monthly rent cheque pays for all of the above – compared to many
separate bills when you own a house, and a separate condo fee when you own a
condo
Savings
Renting leaves you more disposable income for things like travel, recreation or buying a
vacation home:
•
•
Lower, controlled cost – The cost of renting is a bargain compared to total
home ownership costs that include taxes, utilities, insurance and maintenance.
What’s more, your rental costs are fixed for the term of your lease. By contrast,
condo fees are more unpredictable, and most home ownership costs have been
rising much more quickly than the rate of inflation.
Five-year advantage – People say paying rent is like throwing money away. But
if you plan to move in the next five years, you could say the same about a
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MAP4C – Foundations for College Mathematics
•
Lesson 5
mortgage. In the first five years of ownership, most of your mortgage payments
are applied to interest, with minimal paid to the principal. So you won’t get this
money back when you sell, and you’ll have all the hassles and costs of putting
your house on the market.
Possible deductible – In some situations, your rent may be deductible. Home
ownership costs are not.
Flexibility
When you rent, you can pack up and leave at the end of your lease, without having to
find a buyer and wait for the sale to close.
Why Buy?
Advantages of Buying home
Buying a house has its positive and negative aspects and so before deciding whether
to go ahead and purchase that home, look at the pros and cons in conjunction with
your situation and decision will be appropriate for you.
Advantages of buying
Build Equity
As you are making your mortgage payment, you're building equity. Equity is the portion
of the property that you actually own through your payments, versus the portion that
you still owe the mortgage lender. The longer you stay in your home and the more
mortgage payment you make, the more equity you'll have. This may assist you in using
your equity in purchasing another property or another useful investment.
Appreciation of housing value
Over time housing prices gradually increase although this may fluctuate, in general
housing prices consistently over a long period go up.
Stability and Freedom
By owning your own home you can decorate and renovate your home whichever way
you like. Also staying in a common location for a number of years will provide a stable
environment for children growing up.
Financial Credibility
Owning your own home helps you establish financial credibility with banking
institutions which can help if you intend to finance in the future.
Independence
Provides you with independence and privacy from landlords with inspections of your
home. Landlord limitations on pets, renovations etc will not longer apply.
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MAP4C – Foundations for College Mathematics
Lesson 5
Pride
A house is only a building while a home is when people living within its walls. Owning
your own home provides owners with the sense of pride and satisfaction knowing that
they created, renovated and enjoyed times within their home.
Buy vs. Rent Comparison Chart
Advantages
Considerations
Property builds equity
Responsible for maintenance
Sense of community, stability, and
security
Responsible for property taxes
Buy
Free to change decor and
Possibility of foreclosure and loss
landscaping
of equity
Not dependent on landlord to
maintain property
Little or no responsibility for
maintenance
Rent
Easier to move
Less mobility than renting
No tax benefits
No equity is built up
No control over rent increases
Possibility of eviction
Copyright © 2007, Durham Continuing Education
Page 27 of 38
MAP4C – Foundations for College Mathematics
Lesson 5
Example 1
Brenda and Richard have 2 children in elementary school. The couple earns $81 000.
Richard has a 2 year contract as a consultant with the Ministry of Education and Brenda
works part time as a early childhood educator. They are moving from their small 2
bedroom rental apartment to be closer to where both Brenda and Richard work. They
have $15 000 in savings. The only debt they have is a car payment of $425 per month.
The following below shows two options for their new housing needs. Which option
should be recommended?
Option A: Buy
• 4 bedrooms, 2 bathrooms, average
kitchen, all appliances, average
living room, unfinished basement,
no garage, small yard
• 1400 square feet of living space
• price $140 000
• interest rate 7.5%
• term: 5 years
• amortization period 25 years
• down payment 5%
• mortgage payment $976.75 per
month
• taxes $1800 per year
• utilities $417 per month
Option B: Rent
• 3 bedrooms, 2 bathrooms, large
kitchen, large living room, full
recreation room in basement and 1
car garage, large yard
• 1850 square feet
• $1250 per month rent
• 1 year lease minimum
• responsible landlord that lives in
another town
• must maintain yard to satisfactory
level
• utilities $350 month
Solution
Organize the information for each scenario to make a choice
OPTION A: Buying
PROS
• can build equity
• can finish basement and have more living space
• can resell at any time
• reasonable monthly mortgage payment and taxes
• comes with appliances
CONS
• large financial commitment with contract job
• cost money to finish basement
• extra home repair costs are now owners responsibilities
• small yard for the young children
• upfront cost such as land transfer tax and lawyer fees
Copyright © 2007, Durham Continuing Education
Page 28 of 38
MAP4C – Foundations for College Mathematics
Lesson 5
Monthly calculations:
Monthly Payments = Mortgage + utilities + taxes + car payment
= 976.75 + 417 + 1800/12 + 425
= $1968.75
Monthly Income = 81000/12 = $6750 (Gross income)
OPTION B: Renting
PROS
• only 1 year minimum commitment
• large rooms
• large yard for children
• no property taxes
• minimal yard care needed
• has garage
CONS
• no equity
• tied to at least one year in the lease
• landlord is not local
• rent is more than property tax and mortgage payment in option A
Monthly calculations:
Monthly Payments = rent + utilities + car payment
= 1250 + 350 + 425
= $2025
Rental monthly cost ($2025) exceeds monthly owning costs ($1968.75). Since they will
be building equity and the cost of living in both places is roughly the same and keeping
in mind that the upfront costs could be covered by the savings, then purchasing the
house would likely be the best choice. Despite the purchased property having less
living space the building of the equity by purchasing a home would likely outweigh this
shortcoming.
Copyright © 2007, Durham Continuing Education
Page 29 of 38
MAP4C – Foundations for College Mathematics
Lesson 5
Support Questions
1.
Michelle is a single mom of 3 children in elementary school. She earns $60 000
(net). She has $5 000 in savings. The only debt she has is $2000 owing on a
credit card. The following below shows two options for her new housing needs.
Option A: Buy
• 3 bedrooms, 1 bathrooms, average
kitchen, no appliances, average
living room, finished basement,
garage, average sized yard
• 1750 square feet of living space
• price $210 000
• annual interest rate 7.05%
compounded monthly
• term: 5 years
• amortization period 25 years
• down payment 0%
• taxes $2200 per year
• utilities $400 per month
Option B: Rent
• 4 bedrooms, 1 bathrooms, large
kitchen, large living room,
unfinished basement no garage,
large yard
• 2000 square feet
• $1250 per month rent
• 1 year lease minimum
• responsible landlord that lives next
door
• utilities $375 month
• yard maintained by landlord
a. Calculate the monthly mortgage payment for option A.
b. List 4 pros and 4 cons of both options.
c. Calculate the total monthly payment for both options.
d. Which option would you choose? Justify your choice.
Key Question #5
1.
List 3 advantages of renting and 3 advantages of owning a home.
2.
These are the costs of an apartment. Rent: $1270/month; water: $400/year;
heating: 1600/year; electricity: 1550/year. How much is owed per month?
Copyright © 2007, Durham Continuing Education
Page 30 of 38
MAP4C – Foundations for College Mathematics
Lesson 5
Key Question #5 (continued)
3.
Lester and Evelyn want to either rent an apartment or purchase a house with at
least three bedrooms. Their combined gross income is $120 000. If they buy a
house, they will need a down payment of 5%. They have been approved for a
mortgage of 6.05% compounded monthly for a 7 year term amortized over 20
years. Lester and Evelyn enjoy travelling often and frequent the gym on a
regular basis. They also enjoy eating out at least twice per week at fine dinning
restaurants.
Option A: Buy
• 2 bedrooms, 2 bathrooms, new
kitchen, appliances included,
average living room, finished
basement, 2 car garage, small yard
• 1700 square feet of living space
• price $280 000
• taxes $3500 per year
• utilities $400 per month
Option B: Rent
• 3 bedrooms apartment, 2
bathrooms, large kitchen, large
living room, parking space
• 1200 square feet
• $1870 per month rent
• 1 month to month
• property manager onsite
• utilities $300 month
• parking in secured area
a. Calculate the monthly mortgage payment for option A.
b. List 4 pros and 4 cons of both options.
c. Calculate the total monthly payment for both options.
d. Which option would you choose? Justify your choice.
4.
Explain one advantage for each when renting: choice, location, space, simplicity,
savings and flexibility.
Copyright © 2007, Durham Continuing Education
Page 31 of 38
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Answers to Support Questions
MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 1:
1
a.
b.
50
⎛ .05 ⎞
100[⎜ 1 +
− 1]
2 ⎟⎠
⎝
FV =
.05
2
= $9748.43
400[(1 + .04 ) − 1]
FV =
.04
= $6010.32
12
c.
36
⎛ .08 ⎞
500[⎜ 1 +
− 1]
12 ⎟⎠
⎝
FV =
.08
12
= $20267.78
2
a.
b.
⎛ .09 ⎞
1000[⎜ 1 +
2 ⎟⎠
⎝
FV =
.09
2
= $9380.01
200[(1 + .04 ) − 1]
FV =
.04
= $1083.26
5
4×2
− 1]
c.
10×4
⎛ .045 ⎞
450[⎜ 1 +
4 ⎟⎠
⎝
FV =
.045
4
= $22575.07
− 1]
3.
⎛ .05 ⎞
1500[⎜ 1 +
12 ⎟⎠
⎝
FV =
.05
12
= $37778.88
2×12
− 1]
Therefore the Smiths will have saved $37778.88 in two years.
Copyright © 2007, Durham Continuing Education
Page 33 of 38
MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 2:
1
a.
b.
400[1 − (1 + .04 )
PV =
.04
= $5436.13
−20
⎛ .05 ⎞
100[1 − ⎜ 1 +
2 ⎟⎠
⎝
PV =
.05
2
= $1558.92
]
−20
]
c.
⎛ .08 ⎞
700[1 − ⎜ 1 +
12 ⎟⎠
⎝
PV =
.08
12
= $22338.26
2
−36
]
a.
b.
−8
1000[1 − (1 + 0.05 ) ]
PV =
0.05
= $6463.21
200[1 − (1 + .04 ) ]
.04
= $1048.42
−6
PV =
c.
⎛ 0.045 ⎞
750[1 − ⎜ 1 +
4 ⎟⎠
⎝
PV =
.045
4
= $24051.19
−40
⎛ .055 ⎞
3000[1 − ⎜ 1 +
12 ⎟⎠
⎝
PV =
.055
12
= $276430.75
−10×12
]
3.
]
Cliff will need to invest $276 430.75 now so that he can withdraw $3000 per
month for the next ten years.
Copyright © 2007, Durham Continuing Education
Page 34 of 38
MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 3:
1
a.
⎛ .06 ⎞
R[1 − ⎜ 1 +
12 ⎟⎠
⎝
140000 =
.06
12
140000 = 166.792R
140000 166.792R
=
166.792
166.792
839.37 = R
b.
−12×30
⎛ .0525 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
200000 =
.0525
12
200000 = 166.876R
200000 166.876R
=
166.876
166.876
1198.49 = R
]
c.
⎛ .062 ⎞
R[1 − ⎜ 1 +
12 ⎟⎠
⎝
175000 =
.062
12
175000 = 137.355R
175000 137.355R
=
137.355
137.355
1274.07 = R
2
−12×25
]
d.
−12×20
]
⎛ .0595 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
350000 =
.0595
12
350000 = 155.946R
350000 155.946R
=
155.946
155.946
2244.37 = R
−12×25
]
a. 839.37 x 360 = 302 173.20
302 173.20 – 140 000.00 = $162 173.20 in interest
b. 1198.49 x 300 = 359 547.00
359 547.00 – 200 000.00 = $159 547.00 in interest
c. 1274.07 x 240 = 305 776.80
305 776.80 – 175 000.00 = $130 776.80 in interest
d. 2244.37 x 300 = 673 311.00
673 311.00 – 350 000.00 = $323 311.00 in interest
Copyright © 2007, Durham Continuing Education
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MAP4C – Foundations for College Mathematics
Support Question Answers
3.
down payment = $225 000.00 x 0.25 = $56250.00
mortgage needed = 225 000.00 – 56 250.00 = 168 750.00
⎛ .0725 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
168 750 =
.0725
12
168 750 = 146.590R
168 750 146.590R
=
146.590
146.590
1151.17 = R
−12×30
]
4.
payment for 1st 5 years
⎛ .07 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
125000 =
.07
12
125000 = 141.538R
125000 141.538R
=
141.538
141.538
883.16 = R
balance after 5 years of payments
−12×25
]
⎛ .07 ⎞
883.16[1 − ⎜ 1 +
12 ⎟⎠
⎝
PV =
.07
12
PV = 113947.48
−12×20
]
New monthly payment
⎛ .065 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
113947.48 =
.065
12
113947.48 = 134.125R
113947.48 134.125R
=
134.125
134.125
849.56 = R
−12×20
]
Copyright © 2007, Durham Continuing Education
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MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 4:
1.
a.
Payment
Number
0
1
Monthly Interest
Payment Paid
$1340.37
.0825
170000(
) = 1168.75
12
2
$1340.37
3
$1340.37
4
$1340.37
5
$1340.37
6
$1340.37
169828.38(
.0825
) = 1167.57
12
169655.58(
.0825
) = 1166.38
12
169481.59(
.0825
) = 1165.19
12
169306.41(
.0825
) = 1163.98
12
169130.02(
.0825
) = 1162.77
12
Principal
Paid
1340.37–
1168 .75 =
$171.62
1340.37–
1167 .57 =
$172.80
1340.37–
1166 .38 =
$173.99
1340.37–
1165 .19 =
$175.18
1340.37–
1163 .98 =
$176.39
1340.37–
1162 .77 =
$177.60
Outstanding
Principal
$170 000
170 000 – 171.62=
$169 828.38
169 828.38 –
172.80=
$169 655.58
169 655.58 –
173.99=
$169 481.59
169 481.59–
175.18=
$169 306.41
169 306.41–
176.39=
$169 130.02
169 130.02–
177.60=
$168 952.42
2.
1168.75 + 1167.57 + 1166.38 + 1165.19 + 1163.98 + 1162.77 = $6994.64 in
interest paid.
3.
171.62 + 172.80 + 173.99 + 175.18 + 176.39 + 177.60 = $1047.58 paid to the
principal
4.
⎛ .08 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
125 000 =
.08
12
125 000 = 119.554R
125 000 119.554R
=
119.554
119.554
1045.55 = R
−12×20
]
Copyright © 2007, Durham Continuing Education
Page 37 of 38
MAP4C – Foundations for College Mathematics
4.
Support Question Answers
Continued…
Payment
Number
0
1
Monthly Interest
Payment Paid
$1045.55
.08
125000(
) = 833.33
12
2
$1045.55
3
$1045.55
4
$1045.55
5
$1045.55
6
$1045.55
124787.78(
.08
) = 831.92
12
124574.15(
.08
) = 830.49
12
124359.08(
.08
) = 829.06
12
124142.59(
.08
) = 827.62
12
123924.66(
.08
) = 826.16
12
Principal
Paid
1045.55–
833 .33 =
$212.22
1045.55–
831.92=
$213.63
1045.55–
830 .49 =
$215.06
1045.55–
829 .06 =
$216.49
1045.55–
827 .62 =
$217.93
1045.55–
826 .16 =
$219.39
Outstanding
Principal
$125 000
125 000 – 212.22=
$124 787.78
124 787.78–
213.63=
$124 574.15
124 574.14 –
215.06=
$124 359.08
124 359.08–
216.49=
$124 142.59
124 142.59–
217.93=
$123 924.66
123 924.66–
219.39=
$123 705.27
Lesson 5:
1
a.
⎛ .0705 ⎞
R [1 − ⎜ 1 +
12 ⎟⎠
⎝
210 000 =
.0705
12
210 000 = 140.851R
210 000 140.851R
=
140.851
140.851
1490.94 = R
−12×25
]
b. answers vary
2200
+ 400 = $2074.27
12
Option B: 1250 + 375 = $1625.00
c. Option A: 1490.94 +
d. answers vary
Copyright © 2007, Durham Continuing Education
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