Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE

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MTH001 Elementary Mathematics

LECTURE # 18

Applications of Basic Mathematics

OBJECTIVES

The objectives of the lecture are to learn about:

Scope of Module 2

Annuity

Accumulated value

Accumulation Factor

Discount Factor

Discounted value

Algebraic operations

Exponents

Solving Linear equations

Annuity

Let us look at an example to understand what is annuity. Suppose

that you want to buy electric equipment on installments. The value of the

equipment is Rs. 4,000. The company informs you that you must pay Rs. 1,000

at the time of purchase (down payment = 1,000). The rest of the payments are to

be made in 20 installments of 200 rupees each. You are wondering about the

total number and sequence of periodic payments. The sequence of payments at

equal interval of time is called Annuity. The time between payments is called the

Time Interval.

NOTATIONS

The following notations are used in calculations of Annuity:

R = Amount of annuity

N = Number of payments

I = Interest rater per conversion period

S = Accumulated value

A = Discounted or present worth of an annuity

ACCUMULATED VALUE

The accumulated value S of an annuity is the total payments made

including the interest. The formula for Accumulated Value S is as follows:

S = r ((1+i)^n 1)/i

Accumulation factor for n payments = ((1 + i)^n 1) / i

It may be seen that:

Accumulated value = Payment per period x Accumulation factor for n payments

The discounted or present worth of an annuity is the value in today's rupee

value. As an example if we deposit 100 rupees and get 110 rupees (100 x 1.1)

after one year, the Present Worth of 110 rupees will be 100. Here 110 will be

future value of 100 at the end of year 1. The amount 110, if invested again, can

be Rs. 121 after year 2. The present value of Rs. 121, at the end of year 2, will

also be 100. Thus, the total present worth of payments made in year 1 and 2

(100+110 = 210) will be 200. The Future Value of this present worth is 210.

(110x1.1)

DISCOUNT FACTOR AND DISCOUNTED VALUE

When future value is converted into present worth, the rate

at which the calculations are made is called Discount rate. In the previous

example 10% was used to make the calculations. This rate is called Discount

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Rate. The present worth of future payments is called Discounted Value. The

above example may be restated as follows:

The future value of Annuity in year 1 and 2 is 100 and 110 respectively. The

Discount rate is 10%. The Accumulation Factor after year 1 is 100+(10/100) =

100.1. The Accumulation Factor after year 2 will be 110+11/100=1.21.

The Accumulation Factor can also be calculated by treating the value at the end

of year 1 as 1 plus interest on 1. After year 1, the Accumulation Factor will be

1+0.1=1.1. Here we treated 10% of 1 as 0.1.

Obviously the Discounted Value at the beginning of year 1 can be calculated as

(1+0.1)/1.1 =1. Here 1/1.1=0.9 is the Discount Factor. If you multiply the Future

Value or Payment in year 2 (1.1) by the Discount Factor (0.9), you get the

discounted value (1.1 x 0.9 = 1).

Thus, we can write down the formula for Discounted Value as follows:

Discounted value= Payment per period x Discount factor

The formula can be written as follows:

A = r ((1- 1/(1+i)^n)/i)

EXAMPLE 1. ACCUMULATION FACTOR (AF) FOR n PAYMENTS

Calculate Accumulation Factor and Accumulated value when:

t rate of interest i = 4.25 %

Number of periods n = 18

Amount of Annuity R = 10,000 Rs.

Accumulation Factor AF = ((1 + 0.0425)^18-1)/0.0425 = 26.24

Accumulated Value S = 10,000x 26.24 = 260,240 Rs

EXAMPLE 2. DISCOUNTED VALUE (DV)

In the above example calculate the value of all payments at the beginning of

term of annuity i-e present value or discounted value.

Discount rate = 4.25%

Number of periods = 18

Amount of annuity= 10000 Rs

Value of all payments at the beginning of term of Annuity or discounted value

= Payment per period x Discount Factor (DF)

Formula for Discount Factor = ((1-1/(1+i)^n)/i)

= ((1-1/(1+0.0425)^18))/0.0425) = 12.4059

= 6.595

Discounted value = 10000 × 12.4059 = 124059 Rs

EXAMPLE 3. ACCUMULATED VALUE (S)

How much money deposited now will provide payments of Rs. 2000 at the end of

each half-year for 10 years if interest is 11% compounded six-monthly.

Amount of annuity = 2000Rs

Rate of interest = i = 11% / 2 = 0.055

Number of periods = n = 10 × 2 = 20

calculate the Accumulated Value S.

ACCUMULATED VALUE = 2,000 x ((1-1 / (1+0.055)^20) / 0.055)

= 2,000 x11.95

=23,900.77

ALGEBRAIC OPERATIONS

Algebraic Expression indicates the mathematical operations to be carried out on

combination of NUMBERS and VARIABLES.

The components of an algebraic expression are separated by Addition and

Subtraction.

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In the expression 2x2 3x -1 the components 2x2, 3x and 1 are separated by

minus "-" sign.

In algebraic expressions there are four types of terms:

Monomial, i.e. 1 term (Example: 3x2)

Binomial, i.e. 2 terms (Example: 3x2+xy)

Trinomial, i.e. 3 terms (Example: 3x2+xy-6y2)

Polynomial, i.e. more than 1 term (Binomial and trinomial examples are

also polynomial)

Algebraic operations in an expression consist of one or more FACTORs

separated by MULTIPLICATION or DIVISION sign.

Multiplication is assumed when two factors are written beside each other.

Example: xy = x*y

Division is assumed when one factor is written under an other.

Example: 36x2y / 60xy2

Factors can be further subdivided into NUMERICAL and LITERAL coefficients.

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There are two steps for Division by a monomial.

Identify factors in the numerator and denominator

Cancel factors in the numerator and denominator

Example:

36x2y / 60xy2

36 can be factored as 3 x 12.

60 can be factored as 5 x 12

x2y can be factored as (x)(x)(y)

xy2 can be factored as (x)(y)(y)

Thus the expression is converted to: 3 x 12(x)(x)(y)/ 5 x 12(x)(y)(y)

12x(x)(y) in both numerator and denominator cancel each other. The result is:

3(x)/5(y)

Another example of division by a monomial is (48a2 32ab)/8a.

Here the steps are:

Divide each term in the numerator by the denominator

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Cancel factors in the numerator and denominator

48a2 / 8a = 8x6(a)(a) / 8a = 6(a)

32(a)(b) / 8(a) = 4x8(a)(b) / 8(a) = 4(b)

The answer is 6(a) 4(b).

How to multiply polynomials? Look at the example x(2x2 3x -1). Here each

term in the trinomial 2x2 3x -1 is multiplied by x.

= (-x)(2x2) + (-x)(-3x) + (-x)(-1)

= -2x3+ 3x2

Please note that product of two negatives is positive.

(3x6y3 / x2z3)2

Exponent of a term means calculating some power of that term. In the following

example we are required to work out exponent of 3x6y3 / x2z3 to the power of 2.

The steps in this calculation are:

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Simplify inside the brackets first.

Square each factor

Simplify

In the first step, the expression 3x6y3 / x2z3 is first simplified to (3x4)(y3)/z3.

In the next step we take squares. The resulting expression is: (32)(x4*2)(y3*2)/z3*2 =

9x8 y6 /z6

LINEAR EQUATION

If there is an expression A + 9 = 137, how do we calculate the value of A?

A = 137 9 = 128

As you see the term 9 was shifted to the right of the equality.

To solve linear equations:

Collect like terms

Divide both sides by numerical coefficient.

Step 1: x = 341.25 + 0.025x

x 0.025x = 341.25

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x(1-0.025) = 341.25

0.975x = 341.25

Step 2.

x = 341.25/0.975 = 350

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