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Chapters and Appendices
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What is a Variable? |
Chapter 14
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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].
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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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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