Examples Arithmetic Gradient Factors (P/ G and A/ G)


A local university has initiated a logo-licensing program with the clothier Holister, Inc. Esti-
mated fees (revenues) are $80,000 for the fi  rst year with uniform increases to a total of $200,000
by the end of year 9. Determine the gradient and construct a cash fl ow diagram that identifi  es
the base amount and the gradient series.


 The cash fl ow diagram (Figure 2–13) shows the base amount of $80,000 in years 1 through 9 and the $15,000 gradient starting in year 2 and continuing through year 9.


Neighboring parishes in Louisiana have agreed to pool road tax resources already designated for bridge refurbishment. At a recent meeting, the engineers estimated that a total of $500,000 will be deposited at the end of next year into an account for the repair of old and safety-questionable bridges throughout the area. Further, they estimate that the deposits  will increase by $100,000 per year for only 9 years thereafter, then cease. Determine the  equivalent (a) present worth and (b) annual series amounts, if public funds earn at a rate  of 5% per year.


(a) The cash fl ow diagram of this conventional arithmetic gradient series from the perspective of the parishes is shown in Figure 2–16. According to Equation [2.19], two computations must be made and added: the first for the present worth of the base amount PA  and the second for the present worth of the gradient PG. The total present worth PT   occurs in year 0. This is illustrated by the partitioned cash fl  ow diagram in Figure 2–17. In $1000 units, the total present worth is

(b) Here, too, it is necessary to consider the gradient and the base amount separately. The total annual series AT is found by Equation [2.20] and occurs in years 1 through 10.


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