Chapters and Appendices

Book Entrance

14 The Formula
14. Direct Use - First Example
14. Direct Use, Second Example
14 Indirect Use - First Example I
14 Indirect Use - Second Example
14 Going Further

Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6  Old Language
5 Knowledge Islands [2]
7  Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I.  Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning

Would you like to show yourself or others how to be  algebra power users?

What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Chapter 14 
Compound Interest Calculations
(Compound Growth Calculations)

Previous: Statement of  Compound Interest Formula,

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

¹A 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 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].

3.2  Direct 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

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

More Chapter Sections: [ Up ] [ 14 The Formula ] [ 14. Direct Use - First Example ] [ 14. Direct Use, Second Example ] [ 14 Indirect Use - First Example I ] [ 14 Indirect Use - Second Example ] [ 14 Going Further ]

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