Engineering Economy
Engineering Economy
✔ Topic 7 – Annuities
✔ Topic 8 – Gradient
✔ Topic 9 – Application of Money Time Relationship
Engineering Economy
Introduction to Gradient
UNIFORM GRADIENT SERIES OF CASH FLOWS
Arithmetic gradient series is a series of payments in
which each payment is greater than or less than
previous one by a constant amount
GEOMETRIC SERIES OF CASH FLOWS
Geometric gradient series is series of payments
where annual payments increase or decrease over
time, by a constant percentage.
Engineering Economy
Arithmetic Gradient Cash Flow
(n-1)G
(n-2)G
4G
3G
2G
G
0
0 1 2 3 4 n-1
P F
Engineering Economy
Geometric Gradient Cash Flow
AN =A1(1+f )N - 1
A3 =A1(1+f )2
A2 =A1(1+f )
A1
0 1 2 3 4 N
End of Period
P
Engineering Economy
Arithmetic Gradient Formula
A = G ( A/G, i%, n)
P = G ( P/G, i%, n)
F = G ( F/G, i%, n)
Engineering Economy
Sample Problem - Arithmetic
The cash flow associated with a strip mining
operation is expected to be $200,000 in year 1,
$180,000 in year 2, and amounts decreasing by
$20,000 per year through 8. At an interest rate of
12% per year, the equivalent annual cash flow is
nearest to ________.
Engineering Economy
Sample Problem - Arithmetic
The cash flow associated with a strip mining operation
is expected to be $200,000 in year 1, $180,000 in year
2, and amounts decreasing by $20,000 per year
through 8. At an interest rate of 12% per year, the
equivalent annual cash flow is nearest to:
Solution:
AT = A 1 + A G
= 200,000 - 20,000 (A/G, 12%, 8)
= 200,000 - 20,000 (2.9131)
= $141,737.12
Engineering Economy
Geometric Gradient Formula
Engineering Economy
Sample Problem - Geometric
A mechanical contractor has four employees
whose combined salaries through the end of this
year are $250,000. If he expects to give an average
raise of 5% each year, calculate the present value of
the employee’s salaries over the next 5 years.
Let i = 12% per year.
Solution:
Engineering Economy
Practice Problem - Arithmetic
1.Find the value of each of the following:
a. (A/G, 14.5%,23)
b. (P/G, 12%,10)
c. (F/G,7.8%,21)
2. Compute for the present value
150
100
50
0
0 1 2 3 4
i= 10%
Engineering Economy
Practice Problem - Arithmetic
200
150
100
3. Compute the 50
value of the
amount of F 0 1 2 3 4
i= 10%
4. Suppose a man receives an initial annual salary
of $60,000, increasing at the rate of $5,000 a
year. If money is worth 10%, determine his
equivalent uniform salary for the period of 8
years.
Engineering Economy
Practice Problem - Arithmetic
5. Suppose that you make a series of annual
deposits into a bank account that pays 10%
interest. The initial deposit at the end of the first
year is $1,200. The deposit amounts decline by
$200 in each of the next four years. How much
would you have immediately after the fifth
deposit?
Engineering Economy
Practice Problem - Geometric
1. Annual maintenance cost for the machine are 1,500
this year and estimated to increase 10% each year
every year. What is the present worth of the
maintenance cost for 6 years if i= 8%
Engineering Economy
Practice Problem - Geometric
2. On Domingo’s 23rd birthday you decide to invest
$4,500 (10% of your annual salary) in a mutual fund
earning 7% per year. You will continue to make
annual deposits equal to 10% of your annual salary
until you retire at age 62 (40 years after you started
your job). You expect your salary to increase by an
average of 4% each year during this time. How
much money will you have accumulated in your
mutual fund when you retire?
Engineering Economy
