Gradient Series
Lecture No. 8
Chapter 3
Contemporary Engineering Economics
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Linear Gradient Series
Contemporary Engineering Economics, 6e, GE
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Linear Gradient Series
An Increasing Linear Strict Gradient Series
!!! Most
problems do
not
correspond
to this form
Contemporary Engineering Economics, 6e, GE
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Linear Gradient Series
• Gradient Series as a Composite Series of a Uniform Series
of N Payments of A1 and the Gradient Series of Increments
of Constant Amount G
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Example 1: Find P, given A1, G, N, and i
Maintanence cost of a machine is $1,000 during the first year.
As the machine ages, maintanence cost is expected to increase
at a rate of $250 per year over the remaining life time. Life
time of machine is 5 years. You want to open a maintanence
cost account today that earns 12% annual interest. How much
should you deposit now?
Contemporary Engineering Economics, 6e, GE
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Example 2: Find F, given A1, G, N, and i
Given: A1 = $1,200, G = -$200, N = 5 years,
and i = 10% per year
Find: F
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Gradient-to-Equal-Payment Series
Conversion Factor, (A/G, i, N)
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Example 3: Find A, given A1, G, i, and N
John and Barbara open two savings accounts. John puts $1,000
at the end of year 1 and increase this amount by $300 for each
of the next 5 years. Barbara simply deposits the same amount
each year. What should Barbara deposit so that the accounts will
have the same balance?
Contemporary Engineering Economics, 6e, GE
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Geometric Gradient Series
Cash flows that
increase or decrease
over time by a constant
percentage (g%) (e.g.
construction costs)
(g>0 or g<0)
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Example 4: Find P, given A1, g, i, and N
A manufacturer uses a compressor that runs 70% of time for 250
workdays and 24 hours. It requires 260 kW of electricity at a rate
of $0.05/kWh. The company has two options:
Option 1: Operate with current system (running costs will
increase by 7% per year for the next 5 years).
Option 2: Replace the old machine with a new one that costs
$28,570. The new machine uses 23% less power.
Which option to choose (i=12%)?
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Solution
 1  (1  0.07)5 (1  0.12)5 
POld  $54,600 

0.12

0.07


 $222,937
PNew  $54,600(1  0.23)(P / A,12%,5)
 $42,042(3.6048)
 $151,552
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Example 5: Find A1, given F, g, i and N
 Given:
o F = $1,000,000,
o g = 6%,
o i = 8%, and
o N = 20
 Find: A1
Contemporary Engineering Economics, 6e, GE
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