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Exam III: Mathematics Summary Material, Lecture notes of Advanced Accounting

A summary of the key concepts covered in exam iii of a mathematics course. Topics include simple and compound interest, time value of money calculations, linear equations, and matrix operations. Examples are included to illustrate the application of these concepts.

What you will learn

  • How does compound interest differ from simple interest?
  • How do you solve a system of linear equations using Gaussian elimination?
  • What is the formula for the present value of an annuity?
  • What is the formula for simple interest?

Typology: Lecture notes

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Exam III Summary Material
F.1
Interest
is a fee charged for the use of someone
else's money (bank charges you when you borrow;
bank charges themselves when you save)
The
simple interest
I=P rt
(
P
=principal,
or initial amount,
r
=interest rate as a decimal,
t
=time)
The
future value
F=P(1 + rt)
F.2
Compound Interest
is interest applied to in-
terest over time (formula:
A=P1 + r
mmt
(
m
=number of compoundings per year)
Continuous Compound Interest
formula:
F=
P ert
Eective Yield
is the actual rate of interest
earned per year when compounded
TVM Solver
:
N=
mt
I%=
r
(as a percent)
PV=
P
PMT=0 (*in this case, since nothing is added
to or removed from the account throughout);
FV=
A
P/Y=C/Y=
m
F.3
A
future value annuity
is an investment where
you make given periodic deposits to an account (in-
terest earned at the same time as each deposit).
(EXAMPLES: IRA, 401(k))
A
sinking fund
is set up (often by a company) to
make periodic deposits into an account to achieve
a given nal amount over a given time period.
(Both cases are handled similarly: In the TVM
Solver, PV=0 (usually); one of PMT or FV is
given, and we solve for the other)
F.4
Present Value Annuities-
Given periodic with-
drawals on an account for a given amount of time,
at least how much money must initially be in the
account to accomplish this? (Given PMT with
FV=0, nd PV)
Amortization
-how much in periodic payments in
order to kill a given debt over a given amount of
time (Given PV with FV=0, nd PMT)
Equity
: The amount you have already paid on
your house (Cost of house
Remaining Balance)
Closing Costs and Points
(
only needed for Take-
home quiz
): When you buy a house, you pay a lot
of extra legal fees up front called
closing costs
.
Points
are extra fees you pay up front to lower
your interest rate slightly. (In both cases ex-
tra means they do NOT change the price of your
house!)
4.3
A
linear equation
is an equation that can be writ-
ten as
A1x1+A2x2+· · · +Anxn=b
The
matrix form
of a linear system is
AX =B
A
is the
coecient matrix
X
is the
variable matrix
B
is the
constant matrix
Gaussian Elimination:
a set of operations you
can perform on a system of equations (rows of the
matrix) to create an equivalent system of equa-
tions:
1.
Switch rows (equations)
2.
Multiply or divide a row by a
nonzero
num-
ber
3.
Mult/Div a row by a (nonzero) number and
add/subtract to another row.
General Strategy (Goal-1's on diagonal, 0's
rest of the column):
1.
Divide the diagonal row by the given number
(if zero, switch rst)
2.
Multiply the diagonal row by the number in
another row and subtract from that row. (do
this for each of the other rows in the matrix)
To use the calculator, put the
augmented matrix
[A|B]
in the calculator, then
rref
the matrix
4.4
If you get a false statement (
0 = 1
) after rref, the
system of equations has no solution
If you get more variables than equations, the sys-
tem of equations has an innite number of solutions
Choose free variables (all columns which do
not have the Goal stated in 4.3 above)
Solve all other variables in terms of the free
variables
5.1-5.2
An
m×n
matrix has
m
rows and
n
columns.
Row matrix:
m= 1
Column matrix:
n= 1
Square matrix:
m=n
1
pf3

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Exam III Summary Material

• F.

– Interest is a fee charged for the use of someone

else's money (bank charges you when you borrow; bank charges themselves when you save)

– The simple interest I = P rt (P =principal,

or initial amount, r=interest rate as a decimal, t=time)

– The future value F = P (1 + rt)

• F.

– Compound Interest is interest applied to in-

terest over time (formula: A = P

r m

)mt

(m=number of compoundings per year)

– Continuous Compound Interest formula: F =

P ert

– Eective Yield is the actual rate of interest

earned per year when compounded

– TVM Solver:

∗ N=mt

∗ I%=r (as a percent)

∗ PV=P

∗ PMT=0 (*in this case, since nothing is added

to or removed from the account throughout);

∗ FV=A

∗ P/Y=C/Y=m

• F.

– A future value annuity is an investment where

you make given periodic deposits to an account (in- terest earned at the same time as each deposit). (EXAMPLES: IRA, 401(k))

– A sinking fund is set up (often by a company) to

make periodic deposits into an account to achieve a given nal amount over a given time period.

– (Both cases are handled similarly: In the TVM

Solver, PV=0 (usually); one of PMT or FV is given, and we solve for the other)

• F.

– Present Value Annuities- Given periodic with-

drawals on an account for a given amount of time, at least how much money must initially be in the account to accomplish this? (Given PMT with FV=0, nd PV)

– Amortization-how much in periodic payments in

order to kill a given debt over a given amount of time (Given PV with FV=0, nd PMT)

– Equity: The amount you have already paid on

your house (Cost of house − Remaining Balance)

– Closing Costs and Points (only needed for Take-

home quiz ): When you buy a house, you pay a lot of extra legal fees up front called closing costs. Points are extra fees you pay up front to lower your interest rate slightly. (In both cases ex- tra means they do NOT change the price of your house!)

– A linear equation is an equation that can be writ-

ten as

A 1 x 1 + A 2 x 2 + · · · + Anxn = b

– The matrix form of a linear system is AX = B

∗ A is the coecient matrix

∗ X is the variable matrix

∗ B is the constant matrix

– Gaussian Elimination: a set of operations you

can perform on a system of equations (rows of the matrix) to create an equivalent system of equa- tions:

1. Switch rows (equations)

2. Multiply or divide a row by a nonzero num-

ber

3. Mult/Div a row by a (nonzero) number and

add/subtract to another row.

– General Strategy (Goal-1's on diagonal, 0's

rest of the column):

1. Divide the diagonal row by the given number

(if zero, switch rst)

2. Multiply the diagonal row by the number in

another row and subtract from that row. (do this for each of the other rows in the matrix)

– To use the calculator, put the augmented matrix

[A|B] in the calculator, then rref the matrix

– If you get a false statement (0 = 1) after rref, the

system of equations has no solution

– If you get more variables than equations, the sys-

tem of equations has an innite number of solutions

∗ Choose free variables (all columns which do

not have the Goal stated in 4.3 above)

∗ Solve all other variables in terms of the free

variables

– An m × n matrix has m rows and n columns.

∗ Row matrix: m = 1

∗ Column matrix: n = 1

∗ Square matrix: m = n

– The Aij element of a matrix is the number in the

ith row and jth column of A.

– Two matrices A and B are equal if Aij = Bij for

all i and j

– Add and subtract matrices by adding/subtracting

the corresponding elements (matrices must be the same size)

– Multiply a matrix by a scalar by multiplying every

element by the scalar

– The transpose of a matrix is formed by interchang-

ing the rows and columns

– The dot product of a row matrix [a 1 a 2 · · · an]

and a column matrix

b 1 b 2 · · · bn

 is given by a 1 b 1 +

a 2 b 2 + · · · + anbn (must be the same length)

– Multiply matrices: ABij is the dot product of the

ith row of A and the jth column of B (number of columns in A must equal the number of rows in B)

– The inverse of a square matrix A is a matrix B

such that AB = BA = I (where I is the n × n identity matrix). We write B = A−^1.

– If a square matrix A does not have an inverse, it is

a singular matrix.

– Given a matrix equation AX = B, if A has an

inverse, then X = A−^1 B

Exam III Summary of Examples in Class

• F.

1. You borrow money at 9% simple interest, and at

the end of 2 years, you pay back $531. How much did you originally borrow?

2. You've bought a house using a $135,000 30-year

mortgage at 3.75%. Later, we will learn that your monthly payments are $625.21. What is the new balance on your loan after one month?

• F.

1. When you were born, your grandparents deposited

$10,000 in an account which earned about 3.5% per year compounded quarterly. How much money was in the account when you turned 18? How much interest did the account earn?

2. What is the eective yield of the account above?

3. Nineteen years ago, estimated annual tuition to

attend A&M was $1,948. Today it is about $10,400 (Sources: www.collegecalc.org and tu- ition.tamu.edu). To the nearest 0.1%, what is the annual increase in tuition?

• F.

1. When you were born, your grandparents deposited

$140/quarter into an account which earned about 3.5% per year compounded quarterly.

(a) How much money was in the account when

you turned 18?

(b) How much interest did the account earn?

2. (Give up this day our daily Starbucks...) You

forego your 3 lattes/week and contribute the $50/month you save into a high-risk fund which earns 7.5% per year compounded monthly.

(a) How much money will you have in the account

when you retire in 45 years?

(b) How much interest will you have earned total

on the account?

(c) How much interest did the account earn dur-

ing the 8th month of the 42nd year?

• F.

1. On Sept 6, the Lotto Texas advertised jackpot was

$8.25M, to be paid in 30 equal annual payments. Their interest factor is 1.4108%. How much money needs to be in their account initially in order to pay a winner?

2. You decide to purchase a $29,000 new car. The

car dealership oers you a no down payment op- tion, nancing the entire amount at 4.9% per year compounded monthly for 6 years.

(a) What are your monthly payments?

(b) How much do you end up paying in interest?

(c) Construct the start of an Amortization

table for the car you bought in the previous example: Period Payment Interest Principal Balance 0 1 2 3

(d) How much of the 20th payment goes toward

the principal?

3. You start working for SuperMegaConglo-

moMonopolCorp.com at 22 with a monthly salary of $4500.

(a) Assuming an average of 2% per year ination,

what will your monthly salary be when you retire in 48 years?

(b) If you want to draw this monthly salary af-

ter retirement for 15 years, how much money should you have in your 401(k) when you re- tire if it is in a conservative fund which earns 3% per year compounded monthly?