Deitz J.E., Southam J.L. Contemporary Business Mathematics for Colleges

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If the term of the loan is stated as a certain number of days, computing interest involves

dividing the number of days by the number of days in 1 year—either 360 or 365. Before

computers and calculators, interest was easier to compute by assuming that every year had

360 days and that every month had 30 days. The 360-day method, called the ordinary

interest method, is still used by some businesses and individuals.

254 Part 4 Interest Applications

Computing Ordinary Interest

Learning Objective

Compute ordinary simple interest,

using a 360-day year.

EXAMPLE E

Compute the ordinary simple interest on $900 at 9% for 120 days.

I 5 P 3 R 3 T

5 $900 3 0.09 3

5 $27

120

360

The Principal is $4,000, the Rate is 7%, and the Time is 180 days. Compute the ordinary

simple interest.

Use a 360-day year: I 5 P 3 R 3 T 5 $4,000 3 0.07 35$140

180

360

✔

CONCEPT CHECK 13.2

Computing Exact Interest

Learning Objective

Compute exact simple interest, using

a 365-day year.

Banks, savings and loan institutions, credit unions, and the federal government use a

365-day year (366 days for leap years) to compute interest. This method is called the

exact interest method. The computations are the same as for ordinary simple interest,

except that 365 days is used instead of 360 days.

EXAMPLE F

Compute the exact simple interest on $900 at 9% for 120 days.

I 5 P 3 R 3 T

5 $900 3 0.09 3

5 $26.6301, or $26.63

120

365

The 360-day year was very useful before the advent of calculators and computers, so

there is a long tradition of using it. However, the 365-day year is more realistic than the

360-day year. Also, the 365-day year is financially better for the borrower because the

interest amounts are always smaller. (Why? Because a denominator of 365 gives a smaller

quotient than a denominator of 360.)

Reexamine examples E and F. The difference between ordinary interest and exact

interest is only $27.00 2 $26.63, or $0.37. When businesses borrow money, the principal

may be very large and then the difference will be more significant. Example G is similar

to examples E and F, except that the principal is in millions of dollars rather than hundreds.

Chapter 13 Simple Interest 255

The Principal is $4,000, the Rate is 7%, and the Time is 180 days. Compute the exact simple

interest.

Use a 365-day year: I 5 P 3 R 3 T 5 $4,000 3 0.07 35$138.0822, or $138.08

180

365

✔

CONCEPT CHECK 13.3

Compare ordinary simple interest

and exact simple interest.

Learning Objective

Comparing Ordinary Interest and Exact Interest

13.3 Before calculators and computers,

interest calculations were simplified by

using a 360-day year.First,a year could

be divided into twelve 30-day months.

Second,360 days made cancellation

more likely because 360 has so many

divisors.If you discuss cancellation,

have students compare the number of

divisors of 360 and 365.A good hint is

that 360 5 2 3 2 3 2 3 3 3 3 3 5,

whereas 365 5 5 3 73.

13.4 The difference between ordinary

and exact interest is not trivial for large

principal amounts.

EXAMPLE G

Find the difference between ordinary interest and exact interest on $8,000,000 at 9% for

120 days.

Ordinary Interest Exact Interest

I 5 P 3 R 3 TI5 P 3 R 3 T

5 $8,000,000 3 0.09 35$8,000,000 3 0.09 3

5 $240,000 5 $236,712.3288, or $236,712.33

The difference is $240,000.00 2 $236,712.33 5 $3,287.67.

120

365

120

360

The Principal is $6,000,the Rate is 12%, and the Time is 120 days. Find the difference between

the amounts of simple interest calculated by using the ordinary method (360-day year) and

the exact method (365-day year).

Ordinary interest: I 5 P 3 R 3 T 5 $6,000 3 0.12 35$240.00

Exact interest: I 5 P 3 R 3 T 5 $6,000 3 0.12 35$236.7123, or $236.71

Difference 5 Ordinary interest 2 Exact interest 5 $240.00 2 $236.71 5 $3.29

120

365

120

360

✔

CONCEPT CHECK 13.4

COMBINATIONS OF TIME AND INTEREST THAT YIELD 1%

To simplify mental approximations, you can round the rate and time to numbers that are

easy to compute mentally. Also, use 360 days instead of 365 because it cancels more often.

For ordinary interest, several combinations of rate and time are easy to use because their

product is 1%. For example, 12% 3512% 351% and 6% 356% 351%.

360

256 Part 4 Interest Applications

13.5 Estimating may be more

important now than ever before.

When students depend on calculators,

they often quit thinking about the

“reasonableness”of an answer—they

assume that the calculator will be

correct.

13.6 Again,the reason for using

360 days in estimation is that the

various divisors create several possible

shortcut combinations.

Although calculators are used to compute exact interest, approximation remains very

useful. The following calculator solution requires a minimum of 20 key entries.

8 000 000 .09 120 365

The display will show 236,712.3288.

Pressing any one of the 20 keys incorrectly could result in a large error. By making an esti-

mate of the interest in advance, you may spot a significant calculator error.

Estimating Exact Simple Interest

Learning Objective

Estimate exact simple interest

computations.

EXAMPLE H

Approximate the ordinary simple interest on $2,500 at 6.15% for 59 days. Then calculate

the actual ordinary simple interest.

Round 6.15% to 6% and 59 days to 60 days.

Estimate: $2,500 3 0.06 35$2,500 3 0.01 5 $25.00

Actual interest: $2,500 3 0.0615 35$25.1979, or $25.20

OTHER RATES AND TIMES

Table 13-1 shows several combinations of Rate and Time whose products are useful for

estimating interest.

360

Table 13-1: Rate and Time

4% 354% 351% 10% 3510% 351%

6% 356% 351% 12% 3512% 351%

8% 358% 351% 18% 3518% 351%

9% 359% 351% 6% 356% 352%

12% 3512% 352% 12% 3512% 353%

8% 358% 352% 9% 359% 353%

120

360

120

360

ESTIMATING EXACT INTEREST

The goal in approximating interest is just to get an estimate. Even though exact interest

requires 365 days in a year, you can make a reasonable estimate by assuming that the

number of days in a year is 360. This permits the use of all of the shortcut combinations

from Table 13-1.

EXAMPLE I

First, compute the actual exact simple interest on $1,200 at 11.8% for 62 days.

Actual interest: $1,200 3 0.118 35$24.0526, or $24.05

Second, estimate the amount of interest by using 12% instead of 11.8%, 60 days instead

of 62 days, and 360 instead of 365.

Estimate: $1,200 3 0.12 35$1,200 3 0.02 5 $24

The difference is $24.05 2 $24 5 $0.05.

360

365

Chapter 13 Simple Interest 257

✔

CONCEPT CHECK 13.5

The Principal is $3,750, the Rate is 9.1%, and the Time is 39 days. Calculate the actual exact

simple interest.Then make an estimate by using a 360-day year and simpler values for R

and T. Compare the results.

Actual interest: I 5 P 3 R 3 T 5 $3,750 3 0.091 35$36.4623, or $36.46

Estimate: I 5 P 3 R 3 T 5 $3,750 3 0.09 35$3,750 3 0.01 5 $37.50

Difference: Estimate 2 Actual 5 $37.50 2 $36.46 5 $1.04

360

365

Every simple interest problem has four variables: Interest, Principal, Rate, and Time.

Thus far, you have solved for the Interest Amount (I) when the Principal (P), Rate (R),

and Time (T) were all given. However, as long as any three variables are given, you can

always compute the fourth by just changing the formula I 5 P

3 R 3 T into one of its

possible variations, as shown in Table 13-2.

Computing the Interest Variables

Compute the Principal, Rate, and

Time from the basic interest formula.

Learning Objective

13.7 If all the students in a class are

prepared for algebra,this is a natural

place to use it.Students could

substitute all the known values into

the equation I 5 P 3 R 3 T and solve

for the missing variable by dividing

both sides of the equation.

Table 13-2: PRT formulas

To find You must know Use this formula

IP,R,and TI5 P 3 R 3 T

PI,R,and T

RI,P, and T

TI,P,and R

T 5

(P 3 R)

R 5

(P 3 T)

P 5

(R 3 T)

Assume the use of ordinary interest (a 360-day year) unless the use of exact interest

(a 365-day year) is indicated. The stated or computed interest rate is the rate for 1 full

year. Also, the length of time used for computing interest dollars must be stated in terms

of all or part of a year.

FINDING THE INTEREST AMOUNT, PRINCIPAL, RATE,

OR TIME

When any three variables are known, you can solve for the fourth variable, using a

formula from Table 13-2. All rates are ordinary simple interest (360-day year).

258 Part 4 Interest Applications

EXAMPLE J

Find the Principal if the Interest is $75, the Rate is 6%, and the Time is 30 days.

P 5 ?; I 5 $75; R 5 6%; T 5 year

P 5

(R 3 T)

$75

a0.06 3

360

$75

0.005

5 $15,000

360

EXAMPLE K

Find the Rate if the Interest is $22, the Principal is $2,000, and the Time is

30 days.

R 5 ?; I 5 $22; P 5 $2,000; T 5 year

R 5

(P 3 T)

$22

a$2,000 3

360

$22

$166.67

5 0.132, or 13.2%

360

EXAMPLE L

Find the Time if the Interest is $324, the Principal is $4,800, and the Rate is 9%. Express

Time in days, based on a 360-day year.

T 5 ?; I 5 $324; P 5 $4,800; R 5 9%

Based on a 360-day year, 0.75 year 5 0.75 3 360 days 5 270 days.

T 5

(P 3 R)

$324

($4,800 3 0.09)

$324

$432

5 0.75 year

Chapter 13 Simple Interest 259

Each of the following problems gives three of the four variables.Compute the missing variable.All rates are ordinary

simple interest (360-day year). Round P and I to the nearest cent; round R to the nearest %; round T to the nearest

whole day, assuming that 1 year has 360 days. Use one of the four formulas:

a. Principal 5 $1,240; Rate 5 6%; Time 5 270 days

Find Interest:

I 5 P 3 R 3 T 5 $1,240 3 0.06 3

c. Principal 5 $1,280; Interest 5 $64; Rate 5 10%

Find Time:

In a 360-day year, T 5 0.5 year 5 0.5 3 360 days

5 180 days.

T 5

(P 3 R)

$64

($1,280 3 0.10)

5 0.5 year

270

360

5 $55.80

I 5 P 3 R 3 T, P 5

(R 3 T)

, R 5

(P 3 T)

, and T 5

(P 3 R)

✔

CONCEPT CHECK 13.6

b. Principal 5 $8,000; Interest 5 $50;

Time 5 45 days

Find Rate:

d. Interest 5 $90; Rate 5 9%; Time 5 60 days

Find Principal:

P 5

(R 3 T)

$90

a0.09 3

360

5 $6,000

R 5

(P 3 T)

$50

a$8,000 3

360

5 0.05, or 5%

COMPLETE ASSIGNMENTS 13.1 AND 13.2.

commercial paper

down payment

exact interest method

interest

interest period

long-term credit

ordinary interest method

principal

short-term credit

simple interest

term of the loan

time

Chapter Terms for Review

Try Microsoft

Excel

Try working the problems using the Microsoft Excel templates found on your student

CD. Solutions for the problems are also shown on the CD.

260 Part 4 Interest Applications

THE BOTTOM LINE

Summary of chapter learning objectives:

Learning Objective

13.1

Compute simple interest with time in years or months.

Example

Find the simple interest using the basic formula:

Interest 5 Principal 3 Rate 3 Time, or I 5 P 3 R 3 T

1. Principal 5 $3,500; Rate 5 9%; Time 5 2.5 years

2. Principal 5 $975; Rate 5 8%; Time 5 9 months

Answers: 1. $787.50 2. $58.50 3. $125.00 4. $40.27 5. $62.50 2 $61.64 5 $0.86

6. $2,100 3 0.06 35$21 7. $2,400 8. 0.09, or 9% 9. 45 days

360

13.2

Compute ordinary simple interest, using a 360-day year.

3. Find the ordinary simple interest for a 360-day year:

Principal 5 $5,000; Rate 5 6%; Time 5 150 days

13.6

Compute the Principal, Rate, and Time from the basic

interest formula.

Solve for Principal, Rate, and Time using a 360-day year and the formulas

7. Interest 5 $42; Rate 5 6%; Time 5 105 days

8. Principal 5 $1,600; Interest 5 $30; Time 5 75 days

9. Principal 5 $7,200; Interest 5 $135; Rate 5 15%

(express Time in days)

P 5

(R 3 T)

, R 5

(P 3 T)

, and T 5

(P 3 R)

13.5

Estimate exact simple interest computations.

6. Estimate the exact interest by using a 360-day year and simpler

values for Rate and Time: Principal 5 $2,100; Rate 5 5.8%;

Time 5 62 days

13.4

Compare ordinary simple interest and exact simple

interest.

5. Find the difference between ordinary simple interest and exact

simple interest:

Principal 5 $5,000; Rate 5 6%; Time 5 75 days

13.3

Compute exact simple interest, using a 365-day year.

4. Find the exact simple interest for a 365-day year:

Principal 5 $2,800; Rate 5 7%; Time 5 75 days

Chapter 13 Simple Interest 261

In problems 1 and 2, compute the amount of (a) ordinary simple interest and (b) exact simple

interest. Then compute (c) the difference between the two interest amounts.

Ordinary Exact

Principal Rate Time Interest Interest Difference

1 $1,680 6% 270 Days a. b. c.

2 $10,500 8% 60 Days a. b. c.

In problems 3 and 4, first compute (a) the actual exact simple interest. Then, change each rate and

time to the closest numbers that permit use of the shortcuts shown in Table 13-1 and compute

(b) the estimated amount of exact interest. Finally, compute (c) the difference between the actual

and estimated exact interest.

Actual Estimated

Principal Rate Time Exact Interest Exact Interest Difference

3 $12,000 3.8% 92 Days a. b. c.

4 $2,000 9.2% 117 Days a. b. c.

5 Dick Liebelt borrowed money for 240 days at a rate of 9% ordinary simple interest. How much did Dick

borrow if he paid $90 in interest?

6 Linda Rojas loaned $1,000 to one of her employees for 90 days. If the employee’s interest amount was

$12.50, what was the ordinary simple interest rate?

7 Tessa O’Leary loaned $8,000 to a machine shop owner who was buying a piece of used equipment. The

interest rate was 6% ordinary simple interest, and the interest amount was $360. Compute the number of days

of the loan.

8 Kaye Mushalik loaned $2,500 to Fay Merritt, a good friend since childhood. Because of their friendship,

Kaye charged only 3% ordinary simple interest. Two months later, when Fay received her annual bonus, she

repaid the entire loan and all the interest. What was the total amount that Fay paid?

9 Katherine Wu and her sister Madeline have a home decorating and design business. They often buy an-

tiques and fine art objects and then resell the items to their clients. They have a line of credit at their bank

to provide short-term financing for these purchases. The bank always charges exact simple interest, but the

rate varies depending on the economy. Katherine and Madeline need to borrow $22,400 for 90 days to buy

a collection of antique furniture at an estate sale. If the bank charges 5.25%, how much interest would

they pay?

SELF-CHECK

Review Problems for Chapter 13

Answers to the Self-Check can be found in Appendix B at the back of the text.

262 Part 4 Interest Applications

Notes

Chapter 13 Simple Interest 263

Assignment 13.1: Simple Interest

Name

Date Score

A (20 points) Compute the simple interest. If the time is given in months, let 1 month be of a year. If the

time is in days, let 1 year be 360 days. (2 points for each correct answer)

Principal Rate Time InterestPrincipal Rate Time Interest

1. $500 6.0% 1 year 2. $4,000 8% 3 years

3. $1,800 8% 4 months 4. $840 9% 15 months

5. $7,500 5% 180 days 6. $3,600 12% 30 months

7. $12,800 7% 2.5 years 8. $450 5% 3 years

9. $5,200 10% 90 days 10. $20,000 7.5% 8 months

Score for A (20)

$1,000.00$130.00

$78.75

$2,240.00

$1,080.00$187.50

$94.50$48.00

$960.00$30.00

Learning Objectives

3 4 5 6

$500 3 0.06 3 1 $4,000 3 0.08 3 3

$1,800 3 0.08 3 $840 3 0.09 3

$7,500 3 0.05 3 $3,600 3 0.12 3

180

360

$12,800 3 0.07 3 2.5 $450 3 0.05 3 3.5

$5,200 3 0.10 3 $20,000 3 0.075 3

360