Algebra versus Arithmetic 14: Compound Interest Formula

Chapter 14
Compound Interest Calculations
(Compound Growth Calculations)

More Chapter Sections: [14 The Formula] [14 Direct Use] [14 Indirect Use I] [14 Indirect Use II] [14 Further Notes]

This chapters uses the compound interest formula to introduce the idea of using formulas directly and indirectly, that is forwards and backwards, and also to introduce and compare arithmetic and algebraic solutions to problems. The ideas identify, put into words, a unifying theme for teen and adult education in mathematics & science. Every formula you meet will be used forwards and backwards. And if you understand the algebraic solution method for one formula, you will be able to understand it for all.

If you understand the algebraic solution methods in this chapter and the next you will, and in the site area site folder Solving Linear Equations we hope, have a good base for algebra. To Learn More about compound interest and consumer mathematics (debts, loans, investments, pension plans) see What's in chapters 22 to 35 Next

Teachers & Tutors: The theme forward and backward use of formulas is expanded in full, or nearly so, in the Algebra teaching & Tutoring How-TOs.

Site Reviews

Magellan, the McKinley Internet Directory, 1996: 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. (Magellan is no longer online)
The World-Wide Web Virtual Library Education by Country - Canada 1, 2005. Why Slopes: Appetizers and Lessons for Math and Reason. This online classroom offers appetizers and lessons for math from arithmetic to calculus or why slopes; for deductive reason (logic) and critical thinking; and for learning in general. Included here are opinions on the communication of skills and mathematics instruction. The logic appetizers are math free. Each appetizer is different. If one is not to your liking try another. Most are from three books on understanding and explaining math and reason.

may encourage a visit to site entrance www.whyslopes.com.

You are now going to meet the compound interest formula. When you meet a formula for the first time, you should wonder what it does or means. You should wonder where it came from or how it was obtained. In this discussion of the compound interest formula, I just want to show you how to use it and how to manipulate or change it to extract other formulas from it. Examples with and without numbers will be given. You should regard the justification or origin of the formula as a problem, one that you should solve by finding an explanation for it.

You need to do more than read and to use formulas as written. You should be able to change them or modify them into another form that might be useful. The authors of mathematics books write for people with this manipulative ability. For such people, modification of formulas as needed is routine. The aim is to provide that ability.

1 Compound Interest - Numerical Introduction

When you place (or invest) money in a bank account, a bank pays you money for keeping your money with it, a form of rent for its use of it. The bank is using your money to make loans or investments. The money you are paid is called interest. The amount of interest paid and how often depends on the type of account.

In a compound interest account, a bank adds the interest to your account at the end of a period. This period may be a day, a month, a quarter year, a half-year or a full-year. In each period, all the money in your account is now earning interest. So you now receive interest or rent not only for your original deposit, but also for interest previously added to the account on the completion of each period. Here interest paid at the end of one period will earn interest in future periods. Your money is said to be earning compound interest, or more briefly, compounding.

Postscript: Understanding the Compound Interest Formula

Teachers: Give numerical examples with say i = 5% and P = 1000 dollars (pounds, yen, whatever currency you like, the bigger the better) to show students how or why the formula works. Have them fill-in the following table, or do it for them.

Period
n

Amount at
Start of Period

Amount of Interest

Amount at end of Period

10³(1.05)ⁿ

1000

1050

52.50

1102.50

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)ⁿto nearest whole number as an approximation to the whole number of individuals present in the population.

When you place an initial amount P into an account, it is called the principal. In a compound interest account the following happens. The money in your account grows to an amount A after n periods. (The number n here identifies the number of periods your money stays in the account without any withdrawals, or deposits, except for interest payments at the end of each period.) The amount A is given by the compound interest formula

A = P(1+i)ⁿ

In this formula, the interest rate per period is given by the quantity i. The formula should only be used when interest is compounded. Again, compounded means the interest is reinvested at the end of each period with no other deposits or withdrawals, Each interest payment deposited in your account then earns interest (rent from the bank) in the following periods.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Chapter 14 Compound Interest Calculations (Compound Growth Calculations)

1 Compound Interest - Numerical Introduction

Chapter 14
Compound Interest Calculations
(Compound Growth Calculations)