Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined. Bright Students: Top universities want you. While many have high fees: many will lower them, many will provide funds, many have more scholarships than students. Postage is cheap. Apply and ask how much help is available. Caution: some programs are rewarding. Others lead nowhere. After acceptance, it may be easy or not to switch. Are you a careful reader, writer and thinker?
Five logic chapters lead to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics. Early High School Arithmetic
Deciml Place Value  funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6. Early High School Algebra
What is
a Variable?  this entertaining oral & geometric view
may be before and besides more formal definitions  is the view mathematically
correct? Early High School GeometryMaps + Plans Use  Measurement use maps, plans and diagrams drawn to scale.  Coordinates  Use them not only for locating points but also for rotating and translating in the plane.  What is Similarity  another view of using maps, plans and diagrams drawn to scale in the plane and space. Many humanmade objects are similar by design.  7 Complex Numbers Appetizer. What is or where is the square root of 1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of 1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.  Geometric Notions with Ruler & Compass Constructions : 1 Initial Concepts & Terms 2 Angle, Vertex & Side Correspondence in Triangles 3 Triangle Isometry/Congruence 4 Side Side Side Method 5 Side Angle Side Method 6 Angle Bisection 7 Angle Side Angle Method 8 Isoceles Triangles 9 Line Segment Bisection 10 From point to line, Drop Perpendicular 11 How Side Side Side Fails 12 How Side Angle Side Fails 13 How Angle Side Angle Fails 
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 14. Forward and Backward Use of a Formula Next: [Chapter 15. Solving Linear Equations.] Previous: [PostscriptUnifying Theme  A Fourth Skill For Algebra.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Chapter 14 

Here is a numerical examples with i = 5% and P = 1000 to show how or why the formula works. Observe how the amount at the end of a period is the same as the amount at the start of the next period.
Period n 
Amount at Start of Period 
Amount of Interest  Amount at end of Period  10^{3}(1.05)^{n} 
1  1000.00  50.00  1050.00  1050.00 
2  1050.00  52.50  1102.50  
3  1102.50  
4  
5  
Observe how the amout at the end of a period is equals 100% of the initial amount plus 5% of the initial amount. So the amount at the end each period is 105% of 1.05 times the initial amount.  
Fill in this table with the aid of a calculator to the nearest penny (two decimal places). Observe the formula use shortens the calculation. Note how the amount at the end of one period becomes the amount at the start of the next. If you do not like to work with interest calculations, turn this whole chapter into a compound population growth model using the values of A = P(1+i)^{n} to nearest whole number as an approximation to the whole number of individuals present in the population. 
The compound interest formula 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 lefthand 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.
 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 spacesaving 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.
3 Using The Formula
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}
^{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.
3.1 Direct 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.


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 fourfold product was obtained with the aid of a calculator. ^{2}
^{2}This 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].
3.2 Direct Use: Example 2
Problem: The principal amount $1200 is invested for 3^{1}/_{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)

^{3}Suggestion: 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

3.2 Direct Use: Example 2
Problem: The principal amount $1200 is invested for 3^{1}/_{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)

^{3}Suggestion: 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

Using The Formula Backwards
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 2^{1}/_{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.
ARITHMETIC SOLUTION. The principal P is unknown. The principal is
the initial amount in the account which we want to find. Now in the
compound interest formula A =
P(1+i)^{n},
 the final amount A = $1350 is given or known,
 the interest rate i = 9% ×12 = .75% = .0075 is known, and
 the number of compound interest periods (months) in 2.5 years n = 2.5 ×12 = 30.



Now multiplying a quantity P by a nonzero number^{4} (or quantity) and then dividing by the same number yields the quantity P, no matter what P equals. In our situation,

ALGEBRAIC SHORTHAND SOLUTION. There is a pattern in the first solution.
This pattern can be followed if we had the same problem again, but with
different numbers. For instance, how would we solve the above problem if
the 9% was replaced by 8%? We will try to capture the pattern using
shorthand notation. This approach is given next. It requires a little
more work. But it will give a formula for solving many similar problems.
On your first reading of the shorthand solution you may assume the letter have the values given above. On your second reading, pretend A, i and n have values not known to you  for instance they might be hidden in an sealed enveloped. This second viewpoint is a key to algebra.
First, note the following simple idea. The compound interest formula A = P(1+i)^{n} says the symbol A and the expression P(1+i)^{n} both stand for, represent or give the final amount in the account.^{5}
^{5}More precisely we can say:
Here the symbol A and the more complicated expression or symbol P(1+i)^{n} both represent the same quantity. So we take the liberty of using one in place of the other, as convenient.
 The symbol A is shorthand for a quantity.
 The expression P(1+i)^{n} when computed gives the same quantity.
Second, we can assume and use the rule: when B is
a nonzero number or quantity, then

We now use the above ideas. We will apply this rule with B = (1+i)^{n}. That is, the value of B is given by (1+i)^{n} whenever the latter is computed. So


using the formula  number substitution. The substitutions A = $1305 , i = [.09/12] = 0.0075, and n = 30 leads to



Remark (Algebraic Viewpoint). The formula we have found, namely

3 Using The Formula Backward, More Examples
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}
3.4 Indirect Use: Example 4
Problem: Joan places $500 dollars in an investment. Four years later, the investment was worth $645.34. What interest rate, compounded yearly, would give her this amount?
The arithmetic solution given next will be followed by an algebraic solution. The algebraic solution again captures a pattern present in the arithmetic solution.
ARITHMETIC SOLUTION. (Suggestion: look at the shorthand solution first). Here we are given the final amount A = $645.34, the initial amount (principal) P = $500 and the number n = 4 of periods (here years) that the money is earning interest. The compound interest formula says



The foregoing implies


^{7}When the symbol a denotes a positive number, b = a^{[(1)/(4)]} stands for the number b > 0 which satisfies b^{4} = a. Here b = 1+i satisfies b^{n} = a when a = (1+i)^{n}.
This yields


^{8}An equation which always holds is called an identity.
From both together, we get

Next we use a calculator to see


ALGEBRAIC SHORTHAND SOLUTION. The shorthand solution is as follows. The first part of this solution does not care what values have been acquired by the symbols A, n and P. We are about to solve many problems at once. This shows the power of shorthand notation in manipulating equations.
We will start with the compound interest formula



   
A
P 
   
^{1}/_{n} 
= [(1+i)^{n}]^{1/n} = 1+i 
^{9}When a is a positive number, b = a^{[1/(n)]} stands for the number b > 0 which satisfies b^{n} = a.
Therefore
i = (1+i)1 = 
   
A
P 
   
^{1}/_{n} 
1 
i = 
   
A
P 
   
^{1}/_{n} 
1 
To return to arithmetic and the use of numbers, we substitute A =
$645.34, P = $500 dollars, and n = 4 into the forgotten
interest rate formula. This yields
i = 
   
$645.34
$500 
   
¼ 
1 = 
é ź ė 
645.34
500 
   
¼ 
1 
Up to the point of substitution of the given numbers, our reasoning did not depend on the numbers themselves. Any others could be used instead. We can use or employ the forgotten interest rate formula to find i from the equation
A = P(1+i)^{n} 
The examples above show the role of algebraic shorthand notation in describing and in changing the way calculations are done. The compound interest formula has been used and manipulated but the origin of this formula has not been described. Ask a mathematics instructor for an explanation.
4 Review and Further Notes
We will review what we have met. We will also state a formula for the exponent n in the compound interest formula A = P(1+i)^{n}. How this formula for n is obtained from the compound interest formula will not be shown here  another intellectual IOU is created.
In money matters dealing with the compound interest formula, we can ask
for the final compounded amount given by the direct use of the formula,
but we can also ask for the other three quantities. That is, we may solve
for the principal, for the interest rate or for the number of compounding
periods. The compound interest formula can be viewed as one relationship
between four quantities, anyone of which can be solved for or expressed
in terms of the other three. In particular, the compound interest formula
and equation A = P(1+i)^{n} involves
four quantities. When any three are known, the fourth can be found. The
easiest quantity to find is A. Given the three numbers and
quantities P, i and n, you can find the final amount
A by the direct use of the formula. But by indirect use of the
compound interest formula, that is by changing or manipulating it, given
any three of the four quantities A, P, i and
n, we can calculate the fourth. From the compound interest
formula
A = P(1+i)^{n} 

 The socalled present value formula
P = A
(1+i)^{n}  the interest rate formula
i = 

A
P

1/n
[n] /
1 = /
\/
__
A
P
1. 
 A nameless formula for the exponent (or power) n. From the
compound interest formula, we can also get or find a expression for
n, the number of compound periods in terms of the other three
quantities P, A and i. The expression is
n = log A
P
log(1+i)
You have seen the derivation of the first two of the above formulas from the compound interest formula. Explanation of the third is left as anintellectual debt. This chapter has shown the usefulness of algebra and shorthand notation in dealing with the compound interest formula. The further study of powers, roots and logarithms is left to another text.
Further Readings
 Mathematics of Finance, 3rd Edition by P. Zima & R. Brown, McGrawHill Ryerson Ltd, IBSN: 0075494914,
 The chapter Money Computations below.
Postscript: Derivation of formula for n assuming a knowledge of logarithms.
www.whyslopes.com >> Volume 2 Three Skills For Algebra >> Chapter 14. Forward and Backward Use of a Formula Next: [Chapter 15. Solving Linear Equations.] Previous: [PostscriptUnifying Theme  A Fourth Skill For Algebra.] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?
Pattern Based Reason
Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule and patternbased reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a storytelling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.
Site Reviews
1996  Magellan, the McKinley Internet Directory:Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.
2000  Waterboro Public Library, home schooling section:
2001  Math Forum News Letter 14,
2002  NSDL Scout Report for Mathematics, Engineering, Technology  Volume 1, Number 8
2005  The NSDL Scout Report for Mathematics Engineering and Technology  Volume 4, Number 4
Senior High School Geometry

Euclidean Geometry  See how chains of reason appears in and
besides geometric constructions.

Complex Numbers  Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trigformulas for dot and
crossproducts.
LinesSlopes [I]  Take I & take II respectively assume no
knowledge and some knowledge of the tangent function in
trigonometry.
Calculus Starter Lessons
Why study slopes  this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.

Why Factor Polynomials  Online Chapter 2 to 7 offer a light introduction function maxima
and minima while indicating why we calculate derivatives or slopes to linear and nonlinear
curves y =f(x)

Arithmetic Exercises with hints of algebra.  Answers are given. If there are many
differences between your answers and those online, hire a tutor, one
has done very well in a full year of calculus to correct your work. You may be worse than you think.