What is Arithmetic Gradient?

8- Easy Illustration of the Arithmetic Gradient.

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Illustration of the Arithmetic Gradient.

Brief description of the video content.

In this video, I made a review of all of the known relations between P & F& A, the present value, the future value, and the arithmetic series. In the second part, a new subject will be explained which is the arithmetic gradient what is the idea of the arithmetic gradient, and how we can find the value of P/G? or the present value with the given gradient in terms of i% the compound interest and n the number of years.


A solved example 4-8 from Prof. Donald G Newman’s book,  Engineering Economic Analysis.-9th edition.

Example 4.8 A man has purchased a new automobile. He wishes to set aside enough money in a bank account to pay the maintenance on the car for the first 5 years-.-It has been estimated that the maintenance cost of an automobile is as follows:
is as follows For year- 1 the maintenance cost = is $120, for year- 2 the maintenance cost = is $150, for year- 3 the maintenance cost = is $180, for year- 4 the maintenance cost = is $210, for year- 5 the maintenance cost =$240, Assume the maintenance costs occur at the end of each year and that the bank pays 5% interest. How much should the car owner deposit in the bank now? The video has a subtitle and a closed caption in English.

The discussion will be about the Arithmetic gradient.

Our new subject for discussion will be the arithmetic gradient, but we will review the relationship between P and F, P is the present- value, and F is the future value. 

Content of Engineering economy post.

Review of the different types of investments.

Deposit money to get the retirement amount at the end of your investment (P/F) relation.

If you deposit money in a bank for the investment of amount=P, for which, you will receive a compound interest i% to get a future  F-value for your investment=F.

We will find that we have different signs, P is represented by a downward arrow, or minus sign since it is a cash-out but, when you receive an F value it will be cash-in and the value is drawn by an upward arrow. The P-value in terms of F, i%, n, and F in terms of P, i%, and n are shown in the slide.

  For the P-value,  it is P=F*(P/F, i%, n). For F- value it is=F=P*(1+i)^n. P- value= F* (1+i) ^-n. The relation is based on annual compound interest i%, but, if we have changed the annual interest to become a quarterly compounded interest. The new interest rate i2=(i1/4%) and raised to the power of 4*(number of years), or 4n. That was the relation of F with a given P, I, and n.

The relation between Future value F with present value P with known I, n.

Deposit money to buy a car at the end of your investment (P/F) relation.

There is another relation, for instance, if somebody wishes to buy a car, he or she takes a loan from a bank with interest i% and payment F will be at the end of time n, this is from the bank’s point of view.

Cash in will be with a plus sign  + and the money received F is will be cash-out, with a negative sign (-).

There is another type of relationship when someone receives money to buy goods and pays through installment or uniform series of compound interest after one year of receiving the cash-in for the agreed period. The P-value is cash in or with a positive sign.

The cash out with equal amounts will be cash-out with negative signs, its value =A. The value for A in terms of P, i, n, A=iP(1+i)^n/((1+i)^n-1)), where i is the interest rate, and n is the time in years. The values of A are equal, the same as uniformly distributed load with A value as cash in installments in n the bank account, with an interest I and for some time.

The relation between uniform series A with present value P with known I, n.

Deposit money as a uniform series to get insured money at the end of an investment (A/F) relation.

For instance, if somebody wants to receive insured money after some time, he will pay equal installments of value  A with an interest rate of i% and for n time, start after one year, the diagram shows that type, the A values are shown as upward cash in and at the end of time, for instance, time=8 years. The F- value received at the end of year 8, can be estimated based on a compound interest i%, the F- value= A(F/A, i%, n), also F=A/i((1+i)^n-1).

How to find F with given A, i, n?

The arithmetic gradient factors.

We will discuss a new subject, which is the arithmetic gradient factors.
While paying installments, these installments are not of equal value, but the difference between each consecutive payment is constant and equals  G, which is the same difference between the second and third installments also between the third and fourth installments as well as between the fourth and fifth installments.

After one year of the agreement, somebody pays an installment with the value A. The cash-in diagram has a shape of a trapezium shape, for which the smaller value is A and the larger base=A+4G  for the sketch shown where n=5 years. But if we have n of years, the larger base value of the trapezium will be A+(n-1)G. But if we have n of years, the larger base value will be A+(n-1)*G.

Introduction to Arithmetic gradient.
Introduction to Arithmetic gradient.

The previous shape consists of a Trapezium and can be solved by converting it into two shapes.

The first shape is in the form of an equal installment diagram with a value of A for 5 years as cash-in.
The second part is the shape of a triangle with increasing hypotenuse from 0 to 4G in the case of 5 years as shown in the sketch with a slope of 4G/4 =G.
If we want to estimate the future- value in terms of A, i%, n. The relation is F=(A/i)*((1+i)^n-1)). If we want to estimate A in terms of P,  i, n, The relation is F=(i*P)*(1+i)^n/((1+i)^n-1)).

While for the triangular shape, the value for F is in terms of i%, n, G. F=(G/I)*(1+i)^n-1)/I)-n. P/G Find P in terms of G. The( P/G, i,n) has a relation, which is shown in the slide.

How to find F with given arithmetic gradient G, i, n?general expression.

This is the pdf file used in the illustration of this post and the next post.

External resource in Engineering Economy. This is a great link that illustrates different types of economies, and how to make economic decisions. the time value of money, Applying Theory to Practice.

The two coming posts are-What is Capital Recovery in the Economy? and A solved problem for EAW/ Equivalent Annual Worth.

The third post: Arithmetic Gradient-part 2. How to find P, with given I%,n, G?

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