Assumed Material From Math 147
Please send requests for more information, suggestions, and corrections to Meredith LaFlesh
mlaflesh@tacomacc.edu
Math 111 is a prerequisite to Math 112. If a student is in Math 112, it is assumed that the
student will know the following information.
Chapter 1
Cost, Revenue, and Profit
Assume x items are produced and sold and that cost, revenue, and profit are linear.
The cost of producing the items is given by C(x) = mx + b.
Here, m is the marginal cost or the cost of making one more item.
mx is the variable cost or the cost that depends on the number of items that are
produced and sold. For example, leather and rubber needed to make shoes.
b is the fixed cost or the cost that remains the same no matter how many items are
produced. The cost of machinery, rent, loan payments, and insurance are fixed costs.
The revenue when the items are sold is given by R(x) = nx.
Here n is the selling price of each item.
Another way to describe revenue is:
revenue = price x quantity
The profit on the items is given by P(x) = R(x) – C(x).
Here R(x) and C(x) are as above.
Remember to distribute the subtraction through the parentheses.
R(x) – C(x) = nx – (mx + b)
An Application of Cost, Revenue, and Profit
A company is said to break even when cost equals revenue or when profit equals zero.
You can solve C(x) = R(x) or P(x) = 0 to get the number of items the company needs to sell in
order to break even.
Supply and Demand
For simplicity, we often say supply and demand are linear.
The quantity a producer (or manufacturer) is willing to supply to the market depends on the
price she can charge. The higher the price is, the more the producer is willing to supply.
Let
q be the quantity
p be the price
We write the supply function as q = mp + b.
The quantity a consumer is willing to buy (or demand) in the market depends on the price she
must charge. The higher the price is, the less the consumer will demand.
Let
q be the quantity
p be the price
We write the demand function as q = mp + b.
You can tell the difference between the supply and the demand functions by the slopes or
rates of change. Supply has a positive slope, while demand has a negative slope.
Example:
Fjord Motors produces a vehicle called the Midnight Sun. The marketing department at Fjord
has done research that shows that if the sale price is $21,000, the company can expect to
sell 15,000 cars a month in the U.S.; whereas if the price is $27,000, Fjord will not be able to
sell any of this model. Assuming demand is linear, write the demand function for Midnight
Suns.
We are given ordered pairs (p, q): (21,000, 15,000) and (27,000, 0). Finding the slope in the
usual way:
0  15,000
5

27,000  21,000
2
Then q = –2.5p + b.
Use any method you like to find b. I show my students how to substitute one ordered pair
into this: 0 = –2.5(27,000) + b
b = 67,500
The demand function is q = –2.5p + 67,500.
An Application of Supply and Demand
Market equilibrium occurs when supply equals demand.
Example: Say the demand for an item is given by q = –2p + 20 and the supply function is q =
8p – 3. You can find the equilibrium price by solving
Supply = Demand
8p – 3 = –2p + 20
p = 2.3
So, the equilibrium price of this item is $2.30.
Chapter 10
Maximizing Profit: An Application of Demand
Since demand is a quantity, we can use it to find a revenue function and, then, a profit
function.
Recall: revenue = price x quantity
If R = pq, and q = mp + b is the demand, we can substitute mp + b for p to get
R = p(mp + b)
R = mp2 + bp
Example:
In this case revenue is a quadratic function, so it can be maximized.
The demand function for World Wide Widgets gives the quantity of widgets in thousands as a
function of price in dollars. It is given by q = –9p + 108 where p is the price, in dollars, World
Wide charges per box of widgets. What price maximizes revenue?
We need to write revenue as a function of price.
We know that revenue = price x quantity, so
R(p) = pq
R(p) = p(–9p + 108) or R = –9p2 + 108p
To maximize this, you can graph it and use the maximum function in a graphing calculator or
- 108
use the vertex formula: p 
 6.
2(-9)
Thus, WWW should charge $6.00 a piece in order to maximize its profit on these widgets.
A logistic function has the form f  x  
N
for some constants N, A, and b, b > 0, b ≠ 1.
1  Ab  x
Quick example:
N = 8, A = 3, b = 1.4 gives f  x  
Note: f 0 
8
x .
1  31.4 
8
8
 2
0
1  31.4  4
Thus the initial value is 2.
In general, the vertical intercept is
N
.
1 A
To see the long term behavior of f(x), let x = 1000:
As x   , f  x  
8
8
1 0
In general, the long term behavior of f(x) is N .
Chapter 5
Facts About Interest
Interest is a fee for borrowing money. When someone deposits money into a savings
account, it is like lending the money to the bank, so the bank pays interest.
Simple interest is interest that is paid to the investor at the end of a period of time.
Let
PV be the initial or principal value of the investment,
r be the interest rate as a decimal,
t be the number of years the money is invested,
FV be the future value or value after t years.
The formula we use is:
FV = PV(1 + rt)
Given any three values, you can solve for the fourth.
This formula can also be used to describe any quantities that change the way an investment
does.
Example: The Walt Disney Company had an income of $825 million in 1990 and their income
dropped to $630 million in 1991.
Find the rate of change in Disney’s income in that year.
We have: PV = 825, FV = 630, and t = 1.
So, substitute these into the formula and solve for r:
630 = 825(1 + r*1)
630/825 = 1 + r
–.23636… = r
“Disney’s income decreased 23.6% in 1991”.
Example: T-Bills
Definition: A one year T-bill sells at a discount of D percent (as a decimal) meaning the
selling price is D% less than their value at maturity. So PV = FV – (Discount)(FV) or
PV = FV – DFV.
At auction on 12/10/98, one-year T-bills were sold at a discount of 4.305%. What was the
annual yield?
We want r.
The discount rate is 4.305% , so
PV = FV – 0.04305FV
It is a 1-year T-bill, so t = 1.
Substitute these into FV = PV(1 + rt) to get
FV = (FV – 0.04305FV)(1 + r)
Solve for r.
FV
1 r
FV  0.04305FV
Factor and cancel FV.
1
1 r
1  0.04305
Or r = 0.0449866...
Round to 3 places.
So the yield is 4.499%.
Compound Interest is interest that is added to the principal so that the interest earns
interest.
Let
PV be the initial or principal value of the investment.
r be the interest rate as a decimal,
t be the number of years the money is invested,
FV be the future value or value after t years,
m be the number of periods in a year (for example, when interest is compounded
quarterly, m = 4).
The formula we use is:
r

FV  PV 1  
 m
mt
If you know any four of these values you can solve for the fifth.
Example: Effective Interest
Suppose you want to compare two investments at two different interest rates.
One account offers 12% compounded annually, and another offers 11.75% compounded
monthly.
To compare these, we need the amount and the time to be the same. It is customary to say
we are investing $1 for one year. We are comparing the return.
With the 1st account:
FV = 1(1 + .12)1
= 1.12, a return of 12%
With the 2nd:
FV=1(1+.1175/12)12
=1.124, a return of 12.4%
So, the 2nd investment yields more interest. 11.75% is called the nominal interest rate.
Banks call this APY for annual percentage yield.
In general, the effective interest rate reff of an investment paying a nominal interest rate of rnom
compounded m times per year is
m
 r 
reff  1  nom   1
m 

Using the formula: The effective interest rate of an investment that pays 6% compounded
quarterly is:
4
 .06 
reff  1 
 1
4 

 6.136%
Example: Bonds
A zero coupon bond pays no interest during its life but promises to pay the maturity value
when it reaches maturity.
Suppose that in 1985 you bought a 30-year zero coupon bond with a maturity value of
$10,000 and a yield of 12% annually. How much would you pay for the bond?
Remember the yield is the amount of interest compounded annually necessary to turn your
initial investment into $10,000 in 30 years.
You can use the formula for compound interest to solve this problem. In that case,
r

FV  PV 1  
 m
mt
1( 30)
 0.12 
becomes 10,000  PV 1 

1 

and you solve for PV.
Another way to solve this is by using the (Time Value of Money) TVM-Solver, a Finance
Application in the TI-83 and TI-84 type calculators.
Press Apps then Enter twice to get to the TVM Solver.
N
Number of payment periods for annuities = m * t
Number of years for an investment earning annual interest = t
I%
Annual interest rate as a percent (not a decimal)
PV
Present value
Starting value if many times are involved.
PMT
= 0 for single investment
Payment amount for annuities
FV
Future value
Final value if many times are involved.
P/Y
P/Y = Number of payment periods per year
C/Y
Number of compounding periods per year, usually the same as P/Y
PMT: END BEGIN END should be highlighted for most applications. It means that the
payment is being made at the end of each payment period.
BEGIN should only be highlighted if the situation is an annuity due.
In this case the payment is made at the beginning of the period and
earns interest for one more period than in an ordinary annuity. Rent
is an example of an annuity due for the benefit of the lessor.
►Incoming cash amounts are positive, and outgoing cash amounts are negative.
►Enter all the information you know, then move the cursor to the value you want to find, and
hit ALPHA Enter .
The TVM-Solver arguments for this problem are:
N = 30
I% = 12
PV = ?
FV = 10,000
P/Y = C/Y = 1
ALPHA Enter tells you that you paid $333.78.
Once bonds are purchased they can be resold in the bond market. The value of a bond
depends on the prevailing interest rate.
Suppose you decide you need money now and you decide to sell it. There are 11 years
remaining till maturity. Treasury bonds of similar length are about 4.07%. (as of 1/27/04)
You are now selling an 11-year bond that yields 4.07%. How much would an investor be
willing to pay for it?
$6447.91.
You made 6447.91 – 333.78 = $6,114.13.