Machinery's Handbook, 31st Edition
Decimal Numbers 9 Dividing Decimal Numbers: There are several types of decimal division problems: (1) a whole number divided by a decimal number; (2) a decimal number divided by a whole number; and (3) a decimal number divided by a decimal number. For all situations, if the divisor is a decimal, its decimal point must first be moved right to make it a whole number, and the dividend’s decimal likewise moved, before the operation is performed. Examples of each type are: 18 ÷ 0.3 = 180 ÷ 3 = 60; 1.8 ÷ 3 = 0.6; 1.8 ÷ 0.003 = 1800 ÷ 3 = 600. Ratio and Proportion.— A ratio of quantities a to b is written a : b or as a fraction a / b . For example, the ratio of 12 to 3 is written 12:3 or 12/3. Ratios, like fractions, can be reduced: 12:3 is 4:1. The inverse (or reciprocal) ratio of a : b is b : a . Thus, the inverse ratio of 12:3 is 3:12. When two or more ratios are multiplied, the ratio obtained is a called a compound ratio . The compound ratio of a : b , c : d , and e : f is the ratio ace : bdf . For example, the compound ratio of 8:2, 9:3, and 10:5 is 8 × 9 × 10: 2 × 3 × 5, or 720:30. An equality of ratios, a / b = c / d, is called a proportion , which can be written as a : b :: c : d , read as “ a is to b as c is to d .” Thus, 6:3::10:5 because 6/3 and 10/5 both reduce to 2/1 or 2. In a proportion a : b :: c : d, the first and last terms (which can be variables or numbers) are called the extremes , and the second and third are the means . Note that if a / b = c / d , then the rules of algebra show that ad = bc . Thus, the proportion a : b :: c : d is equivalent to a × d = b × c . So the proportion 6:3::10:5 is equivalent to 6 × 5 = 3 × 10. Often, some part of a proportion is an unknown. For example, in the proportion 2:3:: n :4 (2 is to 3 as n is to 4), n is found by setting up a proportion. According to the basic rules of algebra, 2:3:: n :4 means (2)(4) = 3 n , and hence, 8 = 3 n , so n = 8/3. A full discussion of the rules for solving equations can be found in ALGEBRA . If the second and third terms in a proportion are the same, that term is the mean pro- portional of the other two. Thus, in the proportion 8:4::4:2, 4 is the mean proportional of 8 and 2. The mean proportional of any two numbers may be found by multiplying them and extracting the square root of the prod uct. Thus, the mean proportional of 3 and 12 is 6, because 3 × 12 = 36, which is 6 2 . Example 1, Involving Proportion: If it takes 18 days to assemble 4 lathes, how many days would it take to assemble 14 lathes? Solution: Let x be the number of days to be found. The proportion is written 4:18 :: 14: x , where x is the number of days to be found. Setting this up as an equation and solving: 4 18 14 x =
18 14 × 4 = --------- 63 days =
x
Example 2, Involving Direct (Simple) Proportion: 10 linear meters (32.81 feet) of bar stock are required as blanks for 100 clamping bolts. What total length x of stock, in meters and feet, is required for 912 bolts? Solution: The setup to solve the proportional meters-to-bolts problem comes from the way this proportion is read: “10 meters is to 100 bolts as how many meters is to 912 bolts.” It is solved accordingly:
10 100 912 x =
10 912
× 100
9120 100
10 100 912 : :: : , x
: x x
=
=
=
91 2
. meters
that is,
Solving for
32 81 100 912 . : :: : , feet Likewise, the setup to solve the feet-to-bolts problem comes from reading it as: “32.81 feet is to 100 bolts as how many feet is to 912 bolts.” Thus: 10 100 912 10 100 912 10 912 : :: : , : x x x x that is, Solving for = = × 100 9120 100 912 = = . meters : x x x that is, Solving for = = 2992 . = 29,922.72
3281 100 912 . x = 3281 100 912 . x =
3281 912 100 . × 32 81 912 100 . ×
100
29,922.72
32 81 100 912 . : :: : , x
x
x
=
=
=
that is,
Solving for
:
2992 .
feet
100
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