Compound Interest Formula Evaluation
From Volume 2, Three Skills for Algebra, Chapter 14
The compound interest formula
A = P(1+i)n
gives an example of a calculation described in algebraic shorthand
notation. To use the compound interest formula someone has to explain or
show to you the role of each piece of the shorthand. That is done next.
The final compounded amount A on the left-hand side of the
compound interest formula can be computed when three numbers are given,
namely
- the initial amount, also called the principal P,
- the interest rate i, and
- the number n of compounding periods, possibly months, in which
interest is compounded.
When these numbers or quantities are not given, we can only talk or
write about compound interest calculations and not do them. Examples in
which calculations are done and numbers appear are given below.
- Other people thinking perhaps of the word rate rather than the word
interest in the phrase interest rate, use the letter r
instead of i. The shorthand selected does not matter. Like a play,
only the plot is important. The actors or letters can be changed.
We could try to describe the compound interest calculation in words
alone. This description might be a good essay assignment in a language
course alongside the essay of describing in words alone how to tie a
shoelace. The task is formidable. The task should persuade you that the
algebraic shorthand notation has a few space-saving advantages, even if
it may be difficult to read aloud in an understandable way. Formulas like
pictures need to be seen to be fully appreciated. Often, mathematics is
better written and not spoken.
Examples of how to use the formula directly and indirectly follow. Try to
understand both the numerical (arithmetic) and algebraic solutions.
Using The Formula - Forward Use
The compound interest formula A =
P(1+i)n involves four quantities, namely
A, P, i and n. When any three are known, the
fourth can be found. Properties of arithmetic and algebra say how this is
done. Read on. The easiest quantity to find is A. In the following
examples, we consider the cases where the fourth quantity is A or
P or i. We can also consider the case where the fourth
quantity is n. 1
A formula for this case will be stated at
the end of this chapter. How that formula is obtained or used will not
be explained here. Another intellectual debt is created.
Direct [Forward] Use: Example 1
Problem: Find the final amount A of an investment, if the
initial amount invested is $1500, the interest rate per year is 8% and
the interest is compounded for 4 years.
ARITHMETIC SOLUTION. Here the compounding period is one year. In the
compound interest formula A =
P(1+i)n we then have
- the interest rate i = 8% = 8 ×0.01 = 0.08 since 1% = 0.01 =
[1/100].
- the number n of compounding periods is 4 and,
- the principal P = $1500.
Therefore substitution or replacement yields
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A = P(1+i)n = $1500 (1
+.08)4
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So the final amount (maturity value)
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A = $1500 ×1.36049 = $ 2040.73
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to the nearest penny or [1/100]th of a dollar.
Suggestion: check the above calculations (and those done below) by
hand or with the help of a calculator.
Note: (1+.08)4 = (1.08)4 = 1.08×1.08 ×1.08×1.08 is
the shorthand for the product of the number 1.08 with itself, 4 times.
This four-fold product was obtained with the aid of a calculator.
2
2This product can
be regrouped. It equals (1.08×1.08)2 and so its calculation
involves only two multiplications. Aside: how many multiplications does
the computation of (1.08)16 require? The answer is 15 or 4
depending on how this product is computed. Hint: (1.08)16 =
(1.08)8 ·(1.08)8 = [ (1.08)8
]2.
Note: Rates of interests can be written as percentages, fractions or
decimals. The percentage form can be changed to a decimal form by
replacing the percent sign % by one of its equals 0.01 or ([1/100]). The
fraction or decimal form can also be changed into a percentage by
multiplying by 100% = the percentage representation of the number 1 =
[100/100].
Direct [Forward] Use: Example 2
Problem: The principal amount $1200 is invested for
31/2 years in a compound interest account paying 8%
compounded monthly. Find the final amount in the account. (See the
solutions below for the meaning of this phrase: 8% compounded monthly.)
ARITHMETIC SOLUTION: Note that the interest rate per month is not 8%. It
is instead i = [8%/12] = 2/3% per month.
Also the number of periods (here months) is n = 3.5 ×12 = 42 = the
number of months in 3.5 years. So we can use all this in the compound
interest formula to get by replacement (or substitution)
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A = P(1+i)n = $1200 (1 +
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.08
12
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)(3.5×12) = $1200 (1 +
|
.08
12
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)42
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Remember to do calculations inside parentheses before those
outside.3
3Suggestion: When you replace an
expression by another put the other in parentheses.
With the help of a calculator, the final amount in the account is
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A = $1200 (1.00666667)42 = $1200 ×1.321919 =
$1586.30
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Suggestion: check this with the help of a calculator.
Direct [Forward] Use: Example 2
Problem: The principal amount $1200 is invested for
31/2 years in a compound interest account paying 8%
compounded monthly. Find the final amount in the account. (See the
solutions below for the meaning of this phrase: 8% compounded monthly.)
ARITHMETIC SOLUTION: Note that the interest rate per month is not 8%. It
is instead i = [8%/12] = 2/3% per month.
Also the number of periods (here months) is n = 3.5 ×12 = 42 = the
number of months in 3.5 years. So we can use all this in the compound
interest formula to get by replacement (or substitution)
|
A = P(1+i)n = $1200 (1 +
|
.08
12
|
)(3.5×12) = $1200 (1 +
|
.08
12
|
)42
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|
Remember to do calculations inside parentheses before those
outside.3
3Suggestion: When you replace an
expression by another put the other in parentheses.
With the help of a calculator, the final amount in the account is
|
A = $1200 (1.00666667)42 = $1200 ×1.321919 =
$1586.30
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|
Suggestion: check this with the help of a calculator.
Using The Formula Backwards - preview of a later topic
The compound interest formula A =
P(1+i)n involves four quantities,
namely A, P, i and n. When any three are
known, the fourth can be found. Properties of arithmetic and algebra
say how this is done. Read on. The easiest quantity to find is
A. In the following examples, we consider the cases where the
fourth quantity is A or P or i. We can also
consider the case where the fourth quantity is n.1
3.3 Indirect Use: Example 3
Problem (Finding the principal): Tom Oublier, lucky Tom, finds he has
$1350 in an account today. For the past 21/2 years,
the account has been paying Tom 9% compounded monthly. Tom Oublier has
forgotten the initial amount he had in the account. What was the initial
amount (principal) that he placed or deposited in the account?
There are two ways of getting the result. Both will be given. The
advantages of each will be noted.
The first part of Volume 2, Three Skills for Algebra, Chapter 14, was
included here to illustrate the forward or direct use of a formula. The
last part of Chapter 14 is reproduced in the site section on using
formulas formula forwards and backwards.
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