b. When there are decimal points, they are ignored until the product has been found. Then the decimal point is inserted in the product according to the following rule: Count off the number of figures to the right of the decimal point in each factor. Then the number of figures to the right of the decimal point in the product is equal to the sum of the number of figures after the decimal point in each factor. Example: Multiply 16.53 by 24.7.

c. When the lower factor contains zeros, the partial products corresponding to these zeros need not all be written down. Only the right hand zero is written down. However, care must be exercised to have the right hand figures of all the partial products directly below their corresponding figures in the second factor. Example: Multiply 16.53 by 24.07.

Solution:

16.5 3

2407

1157

66120

3306

39 7.8771 Answer.

PRODUCT

FIGURE 7.

d. Although there is no very simple way to check a multiplication, it is good practice to anticipate the approximate size of the product before beginning a long multiplication. This is done by "rounding off" the factors to permit easy mental multiplication. Although by no means an accurate check, this will frequently catch mistakes in addition or in the location of the decimal point which would otherwise result in nonsensical answers.

Example: What is the approximate product of 15.73 multiplied by 187.04?

Solution: Round off 15.73 to 15, and 187.04 to 200. Then the

product is roughly 15 by 200-3,000. It is clear then that the product of 15.73 and 187.04 cannot be 150.6 or 6,030.3745, for example.

e. Symbols and units.-The more common symbol for multiplication is X. However, it is quite common simply to write the numbers in parentheses next to each other: (3.04) (17.78)=3.04×17.78=54.0512, for example.

(1) When the same number is to be multiplied by itself, for example 3.04×3.04, this is usually indicated by a small "2" placed above and to the right of the number: 3.04X3.04-3.042. This is read as "3.04 squared," and 3.042 is the "square of 3.04." If the number is to be used as a factor 3 times, then a small "3" is used: 3.04X3.04X3.04= 3.043. This is read as "3.04 cubed," and 3.043 is the "cube of 3.04." (2) Unlike addition and subtraction, multiplication of different units can be performed. The product is then expressed in a unit which is itself the product of the units of the factors.

(a) Example: Multiply 5 lb. by 7 ft.

=

Solution: (5 lb.) (7 ft.)=35 lb.-ft. 35 ft.-lb. (b) Example: Multiply 9 ft. by 17 ft.

Answer

Solution: (9 ft.) (17 ft.)=153 ft. Xft.=153 (ft.)2=153 ft.2 (ft.2 is read "square feet")

Answer.

f. In arithmetic, as in other operations, there are many tricks which often simplify the work. One such trick which is useful and easy to remember is the following: To multiply any number by 25, move over the decimal point in the given number two places to the right; then divide by 4.

Example: Multiply 16.53 by 25.

Solution: Moving the decimal point over two places, 16.53 becomes 1653. Dividing this by 4: 1653/4-413.25. Therefore 16.53×25= 413.25

Answer.

g. Exercises. To each of the following exercises three answers have been given. Eliminate the answers which are obviously wrong by rounding off the factors and finding the approximately correct answers mentally.

(1) 600.3X42.7=1,200.62

25,632.81

[4,273.21

30,740

(2) 180X76=13,680

25,580

400.785

(3) 12.45X18.360.785

227, 835

7. Division.-a. Division is the process of finding how many times one number is contained in another. The number to be divided is called the dividend, and the number by which it is divided is called the divisor. The result of the operation, or the answer, is called the quotient.

b. When the divisor contains but one figure, the method commonly used is known as short division. To perform short division, place the divisor (one figure) to the left of the dividend, separated by a vertical line (see example below). Then place a horizontal line over the dividend. Divide the first or the first two figures of the dividend, as is necessary, by the divisor and place the quotient over the line. If the divisor does not go an even number of times, the remainder is prefixed to the next figure in the dividend and the process is repeated. Example: Divide 4,644 by 6.

c. When the divisor contains two or more figures, the method used is known as long division. This is performed as follows: Place the divisor to the left of the dividend, separated by a line, and place the quotient above the dividend, as in short division. Using the divisor, divide the first group of figures of the dividend which gives a number as large or larger than the divisor (see example below). Place the first figure of the quotient above the dividend. Then multiply this figure by the divisor, and place the product below the figures of the dividend which were used for this division. Then subtract this product from the figures directly above it. The next figure in the original dividend is brought down to form a new dividend. This is repeated until all the figures of the original dividend have been used. Example: Divide 6,646,250 by 10,634.

d. It is not very common in either short or long division to have the divisor go into the last trial dividend a whole number of times. When the last trial remainder is not zero, it must be indicated in the answer.

e. Symbols.-"4,647 divided by 6" may be indicated in symbols in several ways. The division sign may be used: 4,647÷6. The "stroke" is more convenient to use on the typewriter: 4,647/6. Finally, the division may be indicated as a fraction: The fact that 4,647 divided by 6 is 774 with a remainder of 3 may be written

3,,

as "4647÷6=774+ or "4647/6=774+" or

3,, 6

4647
6

f. Decimal point. To locate the decimal point in the quotient when decimal points are present in either the divisor or the dividend, move the decimal point in the divisor to the right of the right hand figure. Then move the decimal point in the dividend to the right the same number of places that the point was moved in the divisor. When dividing, be careful to place the quotient so that each figure of the quotient is directly above the right hand figure of the group of figures which were used in the dividend. Then the decimal point in the quotient will be directly above the new position of the decimal point in the dividend. It will also be helpful to remember that the number of decimal places in the quotient is equal to the difference between the number of decimal places in the dividend and divisor. (1) Example: Divide 4.644 by .06.