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

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About half of all computations used in business involve addition. The more skilled you

become in adding, the more rapidly you will get accurate answers. Addition is the

process of finding the sum (total) of two or more addends (any of a set of numbers to

be added).

NUMBER COMBINATIONS

Certain aids can help you add more accurately and rapidly. One of the most helpful is to

combine any two numbers that total 10. The following combinations total 10. Practice

the combinations until you can identify them instantly.

123459876

8 7 6 5 1 2 3 4

When these combinations are found sequentially in any column of numbers, you

should add them as 10. In example A, by using the combinations of 10, you can simply

add down the column by saying “9 plus 10 is 19, plus 10 is 29, plus 8 is 37” (or “9, 19,

29, 37”).

The number 3 is carried over to the top of the next column and written in a small fig-

ure above the number 7. The combinations of 10 are used in adding the center column

by simply saying “10, 20, 30.”

In adding the tens and hundreds, you carry over the number 3 from the tens and

hundreds total. You can simply say “8, 18, 28, 32.”

EXAMPLE A

579

424

686

903

157

458

3, 207

Also, you should learn to recognize the combinations of three numbers that total 10.

11112223

12342343

8 7 6 5 6 5 4 4

When three numbers totaling 10 appear in sequence in a column, you should com-

bine them and add them as 10. In example B, you might add the numbers in the ones

column as you add down the column, “10, 18, 28, 38, 41.” Write the number 4, which is

carried over as a small figure above the 1 in the tens column. Then use the combinations

of 10 in adding the tens column by saying “5, 15, 25, 35, 43.”

4 Part 1 Fundamental Review

Addition

Learning Objective

Use shortcuts to add rapidly and

accurately.

1.1 Emphasize to students that in

business both speed and accuracy are

important in making mathematical

calculations.Although computers and

calculators are used in business,they

aren’t always available,and a person

working in business must be able to

handle basic business calculations

without them.

Combining vertical and horizontal

addition (and subtraction) provides an

excellent self-checking process,which

is always encouraged in business

applications.

Chapter 1 Fundamental Processes 5

EXAMPLE B

(5)

(15) 41(10)

28(18)

(25) 65(28)

(35) 32(38)

(43) 83(41)

431

REPEATED DIGITS

When you’re adding a column in which many of the digits are the same, it is often

quicker to count the number of repeated digits and then multiply the digit by that num-

ber. In example C, the ones column totals 33: 10 1 10 1 7 1 6. The tens column shows

five 4s, equaling 20: 5 3 4 5 20. The 3 that was carried over and the 5 are then added to

the 20 for a total of 28 in the tens column. The total for the problem is 283.

EXAMPLE C

49(10)

(5 3 4)

42(10)

56(13)

283(33)

ADDING FROM LEFT TO RIGHT

(COLUMNS OF TWO-DIGIT NUMBERS)

When adding columns of two-digit numbers, you can easily count by tens and add the

ones column to your total.

EXAMPLE D With practice, you can learn to count the

Count: tens by their number:

12 12 12 12

24 22, 32 1 4 5 36 24 32 1 4 5 36

51 46, 56, 66, 76, 86 11 5 87 51 86 1 1 5 87

43 97, 107, 117, 127 1 3 5 130 43 127 1 3 5 130

32 140, 150, 160 1 2 5 162 32 160 1 2 5 162

162 162

CHECKING ADDITION

You should always check the accuracy of your addition. To do so, add the columns again

in the opposite direction—that is, if you added down, add up for the check.

6 Part 1 Fundamental Review

HORIZONTAL ADDITION

When using business records, you may need to add numbers horizontally. You may

check several horizontal additions by adding the columns vertically and then adding

these totals horizontally. This method is called cross-checking. The sums obtained by

adding the totals horizontally and vertically should be the same.

EXAMPLE E

282 1 346 1 723 1 409 1 716 5 2,476

113 1 806 1 629 1 916 1 620 5 3,084

240 1 318 1 718 1 312 1 309 5 1,897

716 1 501 1 423 1 716 1 114 5 2,470

872 1 417 1 909 1 704 1 472 5 3,374

2,223 1 2,388 1 3,402 1 3,057 1 2,231 5 13,301

Add horizontally and vertically; compare horizontal and vertical totals to verify accuracy. Use combinations to

simplify addition.

11 1

2 4 1 7 6 1 6 3 5 163 (4 1 6)

3 6 1 2 4 1 2 5 5 85 (6 1 4) (Note horizontal combinations)

2 7

1 04 3 1 01 2 5 082

(7 1 3)

8 7 1 1 4 3 1 1 0 0 5 330

COMPLETE ASSIGNMENT 1.1.

✔

CONCEPT CHECK 1.1

Chapter 1 Fundamental Processes 7

Subtraction is the process of finding the difference between the minuend (number from

which subtraction is being made) and the subtrahend (number being subtracted); the

result is the difference. When the subtrahend is greater than the minuend, the result is

a negative difference. In business, a negative difference may be called a credit balance.

A credit balance is frequently shown in parentheses.

EXAMPLE F

Negative Difference

Positive Difference (Credit Balance)

$18.88 Minuend $12.00

3.63 2Subtrahend 213.50

$15.25 Difference ($ 1.50)

CHECKING SUBTRACTION

To check subtraction, use addition. If 209 is subtracted from 317, the difference is 108.

You can check this result by adding the difference (108) to the subtrahend (209). The

sum is 317. You can use the same procedure to check subtraction with a negative differ-

ence (credit balance).

EXAMPLE G EXAMPLE H

Subtr

act: Check: Subtr

act: Check:

317 108 $21.10 ($ 3.40)

2209 1209 224.50 124.50

108 317 ($ 3.40) $ 21.10

HORIZONTAL SUBTRACTION

When using some business forms, you may have to subtract numbers horizontally.You

can check a number of horizontal subtractions by adding the columns vertically and

then subtracting these totals horizontally. This answer should equal the total of the dif-

ferences in the column at the right.

EXAMPLE I

Minuend Subtrahend Difference

$ 120 2 $ 20 5 $100

283 2 10 5 273

440 2 110 5 330

$ 269 2 $149 5 $120

$1,112 2 $289 5 $823

Subtraction

Learning Objective

Use shortcuts to subtract rapidly

and accurately.

INTERNATIONAL

Multiplication, stated simply, is repeated addition. When two numbers (called factors)

are multiplied, one number is repeated as many times as there are units in the other. The

factor that is multiplied is called the multiplicand. The factor that indicates how many

times to multiply is the multiplier. The result is the product.

8 Part 1 Fundamental Review

✔

CONCEPT CHECK 1.2

Multiplication

to Multiply Two Numbers

1. Make the smaller factor the multiplier.

2. Multiply from right to left.

3. Add the products to get the final product.

STEPS

EXAMPLE J

456 (multiplicand) In other words:

237 (multiplier) 7 3 456 5 3,192

3 192 (product) 30 3 456 5 13,680

13 680 (product) 200 3 456 5 91,200

91 200 (product) 237 3 456 5 108,072

108,072 (final product)

STEP 3

STEP 2

STEP 1

Subtract

horizontally:

Check by

Subtr

act: Check: comparing totals

276 142 27 2 13 5 14

2134 1134

24 2 11 5 13

2 10 5 26

142 276 87 2 34 5 53

COMPLETE ASSIGNMENT 1.2.

Learning Objective

Use shortcuts to multiply rapidly and

accurately.

Chapter 1 Fundamental Processes 9

1.2 237 3 456 can be illustrated by

the addition process.

7 3 456: 456

456

30 3 456: 4,560

4,560

200 3 456: 45,600

45,600

108,072

We use three-factor multiplication

later in such business applications as

chain discounts and computing cubic

volume for storage and room size (i.e.,

length 3 width 3 height).

Here is an example of a business-

type calculation requiring multiplying

the product of two factors:Total sales

for a car wash that washes 200 cars per

day at $6 per car for 7 days is 200 3

$6 3 7 5 $8,400.

to Multiply Numbers with Zeros

1. Make the multiplier the factor with the smaller number of digits after

ignoring zeros at the right-hand side of the number.

2. Ignore the right-hand zeros and multiply the remaining numbers.

3. Insert the zeros ignored in Step 2 to the right-hand side of the product.

STEPS

EXAMPLE L EXAMPLE M

370 3 200: Make 2 1,200 3 160,800: Make 12

the multiplier. the multiplier.

Ignored: Ignored:

37 (1 zero) 1,608 (2 zeros)

32 (2 zeros) 312 (2 zeros)

74 (3 zeros) 3 216

74 000 5 74,000 16 08

19,296 (4 zeros)

19,296 0000 5 192,960,000

MULTIPLYING WHEN THE MULTIPLIER CONTAINS

ZERO NOT ON THE END

Often a zero appears in the center of the multiplier rather than at the end. To multiply

42,674 by 401, first multiply the multiplicand by 1 and write down the product. Then

multiply by 4 (which is really 400) and write the result two places, instead of one, to the

left. In other words, one extra place is left for each zero in the multiplier.

EXAMPLE N

42,674

3 .401

42 674

17 069 6 (2 places)

17,112,274

STEP 3

STEP 2

STEP 3

STEP 2

STEP 1STEP 1

CHECKING MULTIPLICATION

The best method of checking multiplication is to divide the product by the multiplier to

obtain the multiplicand. Example K shows the relationship between multiplication and

division.

EXAMPLE K

Multiplicand 22

Multiplier 3

Product 132

MULTIPLYING NUMBERS ENDING IN ZERO

To multiply a number by 10, simply add a zero to the end of the number. To multiply a

number by 100, add two zeros to the end:1

0 3 46 5 460; 7,689 3 100 5 768,900.

6q132

10 Part 1 Fundamental Review

Whenever more than one zero appears in the multiplier, the multiplication process is

similar. To multiply 33,222 by 2,004, as in example O, first multiply 33,222 by 4. Then

multiply 33,222 by 2, writing the answer three places to the left. Remember, extra places

must be left for the two zeros (1 place 1 2 extra places 5 3 places).

EXAMPLE O

33,222

3 2,004

132 888

66 444

(3 places)

66,576,888

MULTIPLYING THE PRODUCT OF TWO FACTORS

Sometimes in business you will need to multiply two factors and then multiply the prod-

uct of those factors by a third factor. As shown in example P, you begin by multiplying

the first two factors and then multiplying that product by the third factor.

EXAMPLE P

21 3 30 3 15 5 9,450 21 630

330 315

630 3150

630

9,450

MULTIPLYING BY 25

A shortcut for multiplying by 25 is to multiply by 100 (increase by two zeros) and divide

by 4.

EXAMPLE Q EXAMPLE R

321 3 25 828 3 25

32,100 4 4 5 8,025 82,800 4 4 5 20,700

MULTIPLYING BY 50

A shortcut for multiplying by 50 is to multiply by 100 (increase by two zeros) and divide

by 2.

EXAMPLE S

732 3 50

73,200 4 2 5 36,600

Division is the process of finding how many times one number (the divisor) is con-

tained in another (the dividend). The result is called the quotient. If anything remains

after the division is completed, it is called the remainder. In example T, 47 4 2 5 23

with a remainder of 1, 47 is the dividend, 2 is the divisor, 1 is the remainder, and 23 with

a remainder of (1) is the quotient.

EXAMPLE T

23 (1)

Chapter 1 Fundamental Processes 11

✔

CONCEPT CHECK 1.3

Division

1.3 Note that in the example the re-

mainder is enclosed in parentheses

rather than shown as remainder over

divisor.The reason is that we have not

introduced fractions at this time.If you

feel that it is better to introduce the

concept of remainder over divisor at

this time,feel free to do so.

Emphasize the placement of the

zeros in the quotient,as shown in

example V.

in Long Division

1. Write the divisor in front of and the dividend inside of a division bracket

().

2. Use only as many digits at the left of the dividend as you need as the first

partial dividend in order to have a number that is equal to or larger than

the divisor.

3. Write the number of times the divisor will go into the partial dividend

selected in Step 2.

4. Multiply the divisor by this answer, write the product under the partial

dividend,and subtract.

5. Next to the remainder thus obtained, bring down the next digit of the

dividend to form the second partial dividend.

6. Divide as before and repeat the process until all the digits of the divi-

dend have been used.

STEPS

Multipl

y: Check:

214 multiplicand

3102 multiplier

428

..21 4

(two places)

21, 828 product

102q21,828

214

Learning Objective

Use shortcuts to divide rapidly

and accurately.

To multiply by 10, add one zero to the

end of the number:

36 3 10 5 360

To multiply by 100, add two zeros to the

end of the number:

36 3 100 5 3,600

COMPLETE ASSIGNMENT 1.3.

12 Part 1 Fundamental Review

STEP 3

EXAMPLE U

16 4

12 13

11 48

656

When the partial dividend is smaller than the divisor, a zero must be placed in the

quotient above that digit. This process is continued until the partial dividend is at least

as large as the divisor. Then continue the long division steps, as shown in example V.

EXAMPLE V

13 6

085

068

CHECKING DIVISION

To check division, simply multiply the quotient by the divisor and add any remainder to

the product. The result w ill equal the original dividend. (Examples W and X provide

checks for examples U and V.)

EXAMPLE W EXAMPLE X

174 quotient 5 20,402

3164 divisor 53 34

696 81 608

10 44 612 06

17 4 693,668

28,536 remainder 5117

dividend 5 693,685

Note: Division is the reverse process of multiplication.

DIVIDING BY 10

To divide by 10, drop the digit at the extreme right of the dividend; the dropped digit

will be the remainder.

EXAMPLE Y EXAMPLE Z

790 4 10 5 79 (0 remainder) 3,652 4 10 5 365 (2 remainder)

693,685

20,402 (17)

STEP 6

STEP 5

STEP 4

STEPS 1 & 2

164

28,536

174

DIVIDING BY 100

To divide by 100, drop the two right-hand digits of the dividend—they will be the

remainder.

EXAMPLE AA EXAMPLE BB

81,400 4 100 5 814 (0 remainder) 257,948 4 100 5 2,579 (48 remainder)

DIVIDING WHEN THE DIVISOR AND DIVIDEND

END WITH ZEROS

When a divisor and dividend both end with zeros, a division shortcut is to delete

the ending zeros common to both and then divide.

EXAMPLE CC

Both Divisor Zeros Common to

and Dividend Divisor and Dividend

End with Zer

os Have Been Dropped Answer

8,400 4 200 84 4 242

46,000 4 2,300 460 4 23 20

42,000 4 100 420 4 1 420

20,000,000 4 4,000 20,000 4 4 5,000

2,760 4 270 276 4 27 10 (6 remainder)

3,200 4 1,000 32 4 10 3 (2 remainder)

Chapter 1 Fundamental Processes 13

✔

CONCEPT CHECK 1.4

Divide: Check:

quotient

divisor dividend divisor 3 quotient 1 remainder 5 dividend

43 32 3 21 1 11 5 683

11 remainder

Dividing by 10: 860 4 10 5 86

863 4 10 5 86 (3 remainder)

Dividing by 100: 19,300 4 100 5 193

19,346 4 100 5 193 (46 remainder)

COMPLETE ASSIGNMENT 1.4.

683

21 (11)