81 Determine the difference in interest earned on S7 SK for

8.1 Determine the difference in interest earned on S7 SK) for 15 years at 6 % simple interest s.2 What is the preseat worth of S10 000 payable in five ycars, it money is thought to he 8.3 How much must be invested now to grow to $30 000 in seven years if the annual interest 84 Compate the unknown values for each of the following cash-flow diagrams (time shown to that earned when the interest is compounded annually worth (a) 5 %, (b) 15%,(c) 25 % rate is 4.0 percent compounded (a) annually, (h) semiannually. (c) moathly? in years and interest compounded annually). la) Figure 8.32 (c) Figure 8.34 Figure 8.35 8.5 Your real estate taxes are $6 800 per year with one-half due April 1 and the remainder due October L each year, What single sum of money raust you place in an account carn ing 5.25 percent interest 1S months before the fiust poyment is due in onder to accumu late enough money to pay cach lax bill? Assurne monthly compounding

Solution

The formula for simple interest is:

SI = P x r x t ;

and,

The formula for annual compound interest, including principal sum, is:
A = P (1 + r/n) ^(nt)

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

Note that this formula gives you the future value of an investment or loan, which is compound interest plus the principal. Should you wish to calculate the compound interest only, you need this:

Total compounded interest = P (1 + r/n) ^(nt) - P

8.1 Here, P= $7500 , t =15 yrs, r= 6%

Simple interest = $ 7500 x 15 x 6/100 = $6750

Compound Interest = $ 7500(1+0.06/1)¹⁵ - $7500 = $ 10474

Difference = $ (10474-6750) = $3724

8.2 Here, amount after 5 yrs is $10,000

So, A = 10,000, t= 5 yrs, P= ?, r= 5%, 15% and 25%

Since, it is not given whther it is compound interest or simple interest, we can assume it as simple interest. However, if it is compounded, the P can be found out easily usng the above formula(in 8.1)

Now, for 5% rate,

(P x 5/100 x 5)+ P = 10,000 (interest + presen value)

1.25P = 10,000

P = $ 8000

Now, for 15% rate,

(P x 15/100 x 5)+ P = 10,000 (interest + presen value)

1.75P = 10,000

P = $ 5714.29

Now, for 25% rate,

(P x 25/100 x 5)+ P = 10,000 (interest + presen value)

2.25P = 10,000

P = $ 4444.44

8.3 A = $ 30,000, t= 7 yrs, r=4%

When compounded i) annually, n=1 in the formula,thus

P(1+r/1)^(1xt) = P x (1+0.04)⁷ = 30,000

or, P = 30,000/1.316 = $22796.35

when compounded semiannually, n=2 (2 times in a year) in the formula, thus

P(1+r/2)^(tx2) = P x (1+0.04/2) ¹⁴ = 30,000

or, P = 30000/1.319 = $22744.5

when compounded monthly, n =12

P(1+r/12)^(tx12) = P x (1+0.04/12)^(7x12)

or, P = 30,000/1.319 = $22744.5

8.4 Diagrams are not available

8.5 Amount to be paid on 1 April = $3400, and on 1 sep = $3400

rate = 5.25%, t =15 months = 15/12 yrs, compounding monthly, so n=12

P (1+0.0525/12)^12x15/12= P(1.004375)¹⁵ = P x 1.054 (for payment on April 1),

Now,remaining amount after paying 3400 on April 1 = P x1.054 -3400

for sep1 payment,

P\' = 1.054P-3400

A = (1.054P-3400)x (1+0.0525/12)^6/12x12 (April to oct =6 months)

(1.054P-3400 )x (1.027)

Now this shoul be equal to 3400

(1.054P-3400)x1.027 = 3400

1.0824P - 3491.8 = 3400

P = $ 6367.15

9.1

Analysis for foam,

initial cost = $37000,r=7%, t=12 yrs

interest for 12 yrs = 37000 x 7/100 x 12 = $ 3108

painting cost (per 3 months) for 12 months = 4*3000 = 12000

total = 37000+3108+12000 = 52108

net savings = 6500*12 - 52108 = $25892

for fiberglass analysis,

installation = 14000, no maintainance, r=7%

interest for 12 yrs = 14000x7/100x12 = 1176

savings for 12 yrs = 2600 x12 = $31200

net savings = 31200 - 14000+1176 = $18376

savings for foam is more

9.2

machine A;

cost = $ 24000

maintainance for 8 yrs = 2600x8 = 20800

interest at 5 % for 8 yrs = 24000x5/100x8 = 9600

total cost in 8 yrs = 24000+20800+9600 = $ 54400

annual cost = 54400/12 = 4533.33

macihne B;

cost = 32000

maintainance for 8 yrs = 1200x8 = 9600

interest for 8 yrs = 32000 x 5/100 x8 = 12800

total cost = 32000+9600+12800 = $54400

annual cost = 54400/12 = $4533.33

Thus, both cost the same and either can be a better buy.