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An amount of $500 is borrowed for 6 months at a rate of 12%. Make an amortization schedule showing the monthly payment, the monthly interest on the outstanding balance, the portion of the payment contributing toward reducing the debt, and the outstanding balance.

The reader can verify that the monthly payment is $86.27.

The first month, the outstanding balance is $500, and therefore, the monthly interest on the outstanding balance is

outstanding balance the monthly interest rate = $ 500 . 12 / 12 = $ 5 size 12{ left ("outstanding balance" right ) left ("the monthly interest rate" right )= left ($"500" right ) left ( "." "12"/"12" right )=$5} {}

This means, the first month, out of the $86.27 payment, $5 goes toward the interest and the remaining $81.27 toward the balance leaving a new balance of $ 500 $ 81 . 27 = $ 418 . 73 size 12{$"500" - $"81" "." "27"=$"418" "." "73"} {} .

Similarly, the second month, the outstanding balance is $418.73, and the monthly interest on the outstanding balance is $ 418 . 73 . 12 / 12 = $ 4 . 19 size 12{ left ($"418" "." "73" right ) left ( "." "12"/"12" right )=$4 "." "19"} {} . Again, out of the $86.27 payment, $4.19 goes toward the interest and the remaining $82.08 toward the balance leaving a new balance of $ 418 . 73 $ 82 . 08 = $ 336 . 65 size 12{$"418" "." "73" - $"82" "." "08"=$"336" "." "65"} {} . The process continues in the table below.

Payment # Payment Interest Debt Payment Balance
1 $86.27 $5 $81.27 $418.73
2 $86.27 $4.19 $82.08 $336.65
3 $86.27 $3.37 $82.90 $253.75
4 $86.27 $2.54 $83.73 $170.02
5 $86.27 $1.70 $84.57 $85.45
6 $86.27 $0.85 $85.42 $0.03

Note that the last balance of 3 cents is due to error in rounding off.

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Most of the other applications in this section's problem set are reasonably straight forward, and can be solved by taking a little extra care in interpreting them. And remember, there is often more than one way to solve a problem.

Classification of finance problems

We'd like to remind the reader that the hardest part of solving a finance problem is determining the category it falls into. So in this section, we will emphasize the classification of problems rather than finding the actual solution.

We suggest that the student read each problem carefully and look for the word or words that may give clues to the kind of problem that is presented. For instance, students often fail to distinguish a lump-sum problem from an annuity. Since the payments are made each period, an annuity problem contains words such as each, every, per etc.. One should also be aware that in the case of a lump-sum, only a single deposit is made, while in an annuity numerous deposits are made at equal spaced time intervals.

Students often confuse the present value with the future value. For example, if a car costs $15,000, then this is its present value. Surely, you cannot convince the dealer to accept $15,000 in some future time, say, in five years. Recall how we found the installment payment for that car. We assumed that two people, Mr. Cash and Mr. Credit, were buying two identical cars both costing $15, 000 each. To settle the argument that both people should pay exactly the same amount, we put Mr. Cash's cash of $15,000 in the bank as a lump-sum and Mr. Credit's monthly payments of x size 12{x} {} dollars each as an annuity. Then we make sure that the future values of these two accounts are equal. As you remember, at an interest rate of 9%

the future value of Mr. Cash's lump-sum was $ 15 , 000 1 + . 09 / 12 60 size 12{$"15","000" left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } } {} , and

the future value of Mr. Credit's annuity was x 1 + . 09 / 12 60 1 . 09 / 12 size 12{ { {x left [ left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } - 1 right ]} over { "." "09"/"12"} } } {} .

To solve the problem, we set the two expressions equal and solve for x size 12{x} {} .

The present value of an annuity is found in exactly the same way. For example, suppose Mr. Credit is told that he can buy a particular car for $311.38 a month for five years, and Mr. Cash wants to know how much he needs to pay. We are finding the present value of the annuity of $311.38 per month, which is the same as finding the price of the car. This time our unknown quantity is the price of the car. Now suppose the price of the car is y size 12{y} {} , then

the future value of Mr. Cash's lump-sum is y 1 + . 09 / 12 60 size 12{y left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } } {} , and

the future value of Mr. Credit's annuity is $ 311 . 38 1 + . 09 / 12 60 1 . 09 / 12 size 12{ { {$"311" "." "38" left [ left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } - 1 right ]} over { "." "09"/"12"} } } {} .

Setting them equal we get,

y 1 + . 09 / 12 60 = $ 311 . 38 1 + . 09 / 12 60 1 . 09 / 12 y 1 . 5657 = $ 311 . 38 75 . 4241 y 1 . 5657 = $ 23 , 485 . 57 y = $ 15 , 000 . 04 size 12{ matrix { y left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } = { {$"311" "." "38" left [ left (1+ "." "09"/"12" right ) rSup { size 8{"60"} } - 1 right ]} over { "." "09"/"12"} } {} ## y left (1 "." "5657" right )= left ($"311" "." "38" right ) left ("75" "." "4241" right ) {} ##y left (1 "." "5657" right )=$"23","485" "." "57" {} ## y=$"15","000" "." "04"} } {}

We now list six problems that form a basis for all finance problems. Further, we classify these problems and give an equation for the solution.

Classification of problems and equations for solutions

If $2,000 is invested at 7% compounded quarterly, what will the final amount be in 5 years?

Classification: Future Value of a Lump-sum or FV of a lump-sum.

Equation: FV = $ 2000 1 + . 07 / 4 20 size 12{ ital "FV"=$"2000" left (1+ "." "07"/4 right ) rSup { size 8{"20"} } } {} .

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How much should be invested at 8% compounded yearly, for the final amount to be $5,000 in five years?

Classification: Present Value of a Lump-sum or PV of a lump-sum.

Equation: PV 1 + . 08 5 = $ 5, 000 size 12{ ital "PV" left (1+ "." "08" right ) rSup { size 8{5} } =$5,"000"} {}

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If $200 is invested each month at 8.5% compounded monthly, what will the final amount be in 4 years?

Classification: Future Value of an Annuity or FV of an annuity.

Equation: FV = $ 200 1 + . 085 / 12 48 1 . 085 / 12 size 12{ ital "FV"= { {$"200" left [ left (1+ "." "085"/"12" right ) rSup { size 8{"48"} } - 1 right ]} over { "." "085"/"12"} } } {}

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How much should be invested each month at 9% for it to accumulate to $8,000 in three years?

Classification: Sinking Fund Payment

Equation: m 1 + . 09 / 12 36 1 . 09 / 12 = $ 8, 000 size 12{ { {m left [ left (1+ "." "09"/"12" right ) rSup { size 8{"36"} } - 1 right ]} over { "." "09"/"12"} } =$8,"000"} {}

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Keith has won a lottery paying him $2,000 per month for the next 10 years. He'd rather have the entire sum now. If the interest rate is 7.6%, how much should he receive?

Classification: Present Value of an Annuity or PV of an annuity.

Equation: PV 1 + . 076 / 12 120 = $ 2000 1 + . 076 / 12 120 1 . 076 / 12 size 12{ ital "PV" left (1+ "." "076"/"12" right ) rSup { size 8{"120"} } = { {$"2000" left [ left (1+ "." "076"/"12" right ) rSup { size 8{"120"} } - 1 right ]} over { "." "076"/"12"} } } {}

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Mr. A has just donated $25,000 to his alma mater. Mr. B would like to donate an equivalent amount, but would like to pay by monthly payments over a five year period. If the interest rate is 8.2%, determine the size of the monthly payment?

Classification: Installment Payment.

Equation: m 1 + . 082 / 12 60 1 . 082 / 12 = $ 25 , 000 1 + . 082 / 12 60 size 12{ { {m left [ left (1+ "." "082"/"12" right ) rSup { size 8{"60"} } - 1 right ]} over { "." "082"/"12"} } =$"25","000" left (1+ "." "082"/"12" right ) rSup { size 8{"60"} } } {} .

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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learn
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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Damian
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LITNING Reply
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
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what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan Reply
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
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At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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