WHEN IS ONE NOT EQUAL TO ONE?Author(s): J. S. HartzlerSource: The Mathematics Teacher, Vol. 77, No. 4 (April 1984), pp. 274-276Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27964013 .

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Find the value of each: -,

T2 =* -- 3ip >lU^d^^f?* -^ ^3 w ?V ,1,,.-.

7"7 =-^ Tn**-^'t,?-

TlO^-- 7*12 -?

Fig.

= nj-Tj-x.

The lesson ended with students having a chance to evaluate different values of and some differences between these values (fig. 1).

The importance of such an activity is the experience gained from organizing data and searching for patterns. Triangular numbers are interesting, simple to work

with, and are not found in the textbook, which might be some reasons why the stu dents enjoyed them. An extension of the ac

tivity might be to challenge students to de

velop definitions for the sum of two tri

angular numbers.

BIBLIOGRAPHY

Hogben, Lancelot. Mathematics for the Million. New York: W. W. Norton & Co., 1968.

Kenny Pinkerton

Campbell High School Smyrna, GA 30080

WHEN IS ONE NOT EQUAL TO ONE? All students of high school mathematics need to be made aware of the utility of mathematics in the world around them. In connection with the study of exponents and

exponential functions, it is possible to pre sent an interesting lesson relating math ematics to federal banking laws and to

newspaper advertisements placed by banks to attract funds into their certificates of de

posit. This application will require a calcu

lator that can evaluate ex and yx for real values of and y.

Two formulas from economics will be used in the discussion. The first formula

yields the value V of an investment of dollars after t years, if the annual interest rate is i expressed as a decimal and is the number of times that interest is com

pounded each year. This formula for period ic compounding of interest is

The formula for continuous compounding of interest is

where e % 2.718 is the base of the natural

logarithmic function. Students interested in the derivation of

the formula for periodic compounding of in terest should refer to Applied Mathematics

for Business, Economics, and the Social Sci ences by Frank S. Budnick and to problem 7 in

" Compound Growth and Related Situ

ations: A Problem-solving Approach'' by Soler and Schuster, which appeared in the November 1982 issue of this journal. The formula for the continuous compounding of interest follows from the formula for period ic compounding and the definition

by letting h = n/L Students will be interested in dis

covering that as the frequency of com

pounding increases, the value of an invest ment will increase even though the annual interest rate and the number of years the

V=P?\

e = lim (1+7

274 -

??-Mathematics Teacher

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TABLE 1

Impact of Frequency of Compounding on the Value of an Investment

Frequency of compounding

Semiannual

Quarterly

Monthly

Daily

Continuous

2

4

12

365

approaches

infinity

V = 5000?1 + ? J

= $551 2.50

V = 5000 $5519.06

/ 0.10\12(1) V = 5000?1 + ? J

= $5523.57

/ 010\365(1) V = 5000 1 + ?- = $5525.78

V 365/ V= 5000e010(1) = $5525.85

investment is held remain fixed. To illus

trate, suppose the annual rate is 10 percent (? = 0.10) and the principal invested for one year (t = 1) is $5000.00. Table 1 demon strates how the value of V increases as increases.

Ask each student to find a bank's news

paper advertisement that states the annual interest rate, term of investment, com

pounding scheme, and effective annual

yield for various certificates of deposit. For

example, most banks offer thirty-month cer tificates at the maximum federally allowed annual rate, but with different frequencies of compounding. The effective annual yield, of course, means the simple interest rate re

quired to yield the same value at the end of one year as the advertised annual rate and

compounding scheme generate. Here is an example from the 16 February

1983 edition of the Baltimore Sun, The rele vant ?dvertisement is shown in figure 1. The bank offered a money market account at an annual interest rate of 10.40 percent with daily compounding and an effective

yield of 11.12 percent. To check the ef fective annual yield, we apply the formula

/ 0.1040\365(1)

= P(1.10958) = + P(0.10958).

The effective yield appears to be 10.958 per cent.

Your Money Always Grows More at

Old Court

for periodic compounding and i = 0.1040, = 365, and t = 1. We find

set

Fig. 1

April 1984 275

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Has the company overstated the ef fective annual yield? The answer is found in the fact that federal banking regulations permit misuse of the formula by letting

= 360 inside the parentheses and = 365 in the exponent. Thus,

which justifies the advertised effective yield of 11.12 percent. That is, for purposes of cal

culating interest, one year equals ??f 1. Most banks offering daily compounding will provide a statement in the fine print of their certificates indicating whether they are using a 365/300 or 365/365 formula.

Our second example deals with continu ous compounding of interest. Suppose a bank offers to pay 9.5 percent for five years with interest compounded continuously, for an effective annual yield of 10.011 percent.

Here the student would use the formula V = Pelt, with i = 0.095 and t = 1, so that

y _

pe0.095(l)

= P(1.09966) = + P(0.09966).

It appears that the effective annual yield should be 9.966 percent. The bank's adver tised effective annual yield can be obtained

by letting one year equal fff, so that

y _

peO.095(365/360)

= P(l.lOlll) = + P(O.lOlll).

The practice of letting one year equal

ff^ is to the advantage of customers invest

ing in certificates of deposit, since the in

terest paid is greater than when the "cor

rect" value t equals fff. Banks use the ff? method to attract more customers by advertis

ing the highest possible effective annual yield.

J. S. Hartzler

Pennsylvania State University Capital Campus

Middletown, PA 17057

REFERENCES

= P(l.11119) = + P(0.11119),

Budnick, Frank S. Applied Mathematics for Business, Economics and the Social Sciences. New York:

McGraw-Hill, 1979.

Johnson, Alonzo F. "The t in I = Prt." Mathematics Teacher 75 (October 1982) :595-96.

Soler, Francisco deP. and Richard E. Schuster. " Com

pound Growth and Related Situations: A Problem

solving Approach. "Mathematics Teacher 75 (No vember 1982):640-44.

USING A CALCULATOR TO CHECK SOLUTIONS OF QUADRATIC EQUATIONS An important part of solving equations is

checking the solutions obtained by stan dard algebraic methods. It is important that students learn to recognize when their work is correct and when it is wrong. How

ever, the checking of solutions is often ne

glected by students when they deal with

quadratic equations. One reason may be that when the quadratic formulas are used to solve the standard quadratic equation

(1) ax2 + bx + c = 0,

where a, 6, and c are real numbers with a 0, it is often very difficult to check the

proposed solutions by direct substitution. In fact, the check often appears more diffi cult than using the quadratic formulas to obtain the proposed solutions. This observa tion is especially true when the solutioris of the quadratic equation are irrational num

bers or complex numbers. The purpose of this article is to present

efficient methods for checking the solutions of quadratic equations with calculators. Different techniques are used when the solutions are real numbers and when they are complex numbers.

Real-Number Solutions

When the solutions of equation (1) are real

numbers, direct substitution can be used to check the solutions. However, when a cal culator is used to perform the computations in the check, it is often more convenient to use the following rewritten form of equa tion (1), which is called the nested form.

(2) (ax + b)x + c = 0

The nested form is convenient to use on

276 -Mathematics Teacher

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