PROPORTION, OR RULE OF THREE. THE RULE OF PROPORTION involves the employment of three terms -a divisor and two factors for forming a dividend-and seeks a quotient, which, when the proposition is written in ratio, bears the same relation to the third term that the second term bears to the first Two of the terms given are of like name or nature, and the other is of the name or nature of the quotient or answer saught. That of the nature of the answer is always one of the factors for forming the dividend, and, if the answer is to be greater than that term, the larger of the remaining two is the other; but if the answer is to be less than that term, the less of the remaining two is the other-the remaining term is the divisor. EXAMPLE.-If $12 buy 4 yards of cloth, how many yards will $108 buy? 4 X 108 108 3 = 36 yards. Ans. EXAMPLE. If 4 yards of cloth cost $12, how many dollars will 36 yards cost? EXAMPLE. If 30 men can finish a piece of work in 12 days, how many men will be required to finish it in 8 days? EXAMPLE.If 45 men require 8 days to finish a piece of work, how many men will finish the same work in 12 days? EXAMPLE. If 8 days are required by 45 men to finish a piece of work, how many days will be required by 30 men to finish the same work? EXAMPLE.-If 12 days are required by 30 men to perform a piece of work, how many days will be required by 45 men to do the same work? EXAMPLE.-I borrowed of my friend $150, which I kept 3 months, and, on returning it, lent him $200; how long may he keep the sum that the interest, at the same rate per cent., may amount to that which his own would have drawn? EXAMPLE. A garrison of 250 men is provided with provisions for 30 days, how many men must be sent out that the provisions may last those remaining 42 days? 250 X 30 42 = 179, and 250 17971. Ans. EXAMPLE.If to the short arm of a lever 2 inches from the fulcrum there be suspended a weight of 100 lbs., what power on the long arm of the lever 20 inches from the fulcrum will be required to raise it? 20 2: 100 10 lbs. Ans. EXAMPLE. At what distance from the fulcrum on the long arm of a lever must I place a pound weight, to equipoise or weigh 20 lbs., suspended 2 inches from the fulcrum at the other end? 12:20 40 inches. Ans. NOTE. If we examine the foregoing with reference to the fact, we shall see that every proposition in simple proportion consists of a term and a half! or, in other words, of a compound term consisting of two factors, and a factor for which another factor is sought that together shall equal the compound. We have only to multiply the factors of the compound together and a little observation will enable us to distinguish it- and divide by the remaining factor, and the work is accomplished. See COMPOUND PROPORTION. COMPOUND PROPORTION, OR DOUBLE RULE OF THREE. COMPOUND PROPORTION, like single proportion, consists of THREE terms given by which to find a fourth — a divisor and two factors for forming a dividend — but unlike single proportion, one or more of the terms is a compound, or consists of two or more factors; and sometimes a portion of the fourth term is given, which, however, is always a part of the divisor. Of the given terms, two are suppositive, dissimilar in their natures, and relate to each other, and to each other only; and upon their relation the whole is made to depend; the remaining term is of the nature of one of the former, and relates to the fourth term, which is of the nature of the other. The object sought is a number, which, multiplied into the factor or factors of the fourth term given, if any, and if not, which of itself, bears the same proportion to the dissimilar term to which it relates, as the suppositive term of like nature bears to the term to which it relates. RULE. Observe the denomination in which the demand is made, and of the suppositive terms make that of like nature the second, and the other the first; make the remaining term the third term; and, if there are any factors pertaining to the fourth term, affix them to the first; multiply the second and third terms together and divide by the first, and the quotient is the answer, term, or portion of a term, sought. EXAMPLE. If 12 horses in 6 days consume 36 bushels of oats, how many bushels will suffice 21 horses 7 days? EXAMPLE. If 12 horses in 6 days consume 36 bushels of oats, how many horses will consume 73 bushels in 7 days? EXAMPLE. 36 12 X 6: 73: 7 × x. 12 × 6 × 73 147 36 X 7 = = 21 horses. Ans. If the interest on $1 is 1.4 cts. for 73 days, (exact interest at 7 per cent.,) what will be the interest on $150.42 for 146 days? EXAMPLE. 73 1.4 :: 150.42 × 146: x. 1.4 X 150.42 X 146 73 = $4.21. Ans. -If the interest on $1 is 1.2 cts. for 73 days, (exact interest at 6 per cent.,) what will be the interest on $125 for 90 days? 73: 1.2: 125 × 90 : x = $1.85. Ans. EXAMPLE. If $100 at 7 per cent. gain $1.75 in 3 months, how much at 6 per cent. will $170 gain in 11 months? $9.77,5. Ans. 100 X 7 X 3: 1.75 :: 170 × 6 × 11.5 : x. 1.75 X 170 X 6 X 11.5 ÷ 100 X 7 X 3 = EXAMPLE. By working 10 hours a day 6 men laid 22 rods of wall in 3 days; how many men at that rate, who work but 9 hours a day, will lay 40 rods of wall in 8 days? 226 X 3 X 10 :: 40: 9 X 8 X x. 6 X 3 X 10 X 40 ÷ 22 X 9 X 8 = 4. Ans. EXAMPLE.If it costs $112 to keep 16 horses 30 days, and it costs as much to keep 2 horses as it costs to keep 5 oxen, how much will it cost to keep 28 oxen 36 days? EXAMPLE. - If 24 men, in 8 days of 10 hours each, can dig a trench 250 feet long, 8 feet wide, and 4 feet deep, how many men, in 12 days of eight hours each, will be required to dig a trench 80 feet long, 6 feet wide, and 4 feet deep? 250 X 8X4: 24 X 8 X 10 :: 80 × 6 × 4: 12 X 8 X x=5—. Ans. EXAMPLE. If 120 men in six months perform a given task, working 10 hours a day, how many men will be required to accomplish a like task in 5 months, working 9 hours a day? Or,I 120 X 6 X 10 :: 1:5 × 9 × x. = 160. Ans. EXAMPLE.-The weight of a bar of wrought iron, 1 foot in length, 1 inch in breadth, and I inch thick, being 3.38 lbs., (and it is so,) what will be the weight of that bar whose length is 12 feet, breadth 3 inches, and thickness of an inch? 1:3.38 12.5 X 3.25 X .75 : x. Or, -13.38 :: 25 × 13 X 2 : x, and 3.38 X 25 X 13 X 3 2 X 4 X 4 102.98 lbs. Ans. EXAMPLE.-The weight of a bar of wrought iron, one foot in length and 1 inch square, being 3.38 lbs., what length shall I cut from a bar whose breadth is 24 inches, and thickness inch, in order to obtain 10 lbs. ? 3.381: 10:× ×x. 1 X 10 X 4 X 2 =2 feet 1 inches. Ans. 3.38 X 11 X 1 CONJOINED PROPORTION, OR CHAIN RULE. THE CHAIN RULE is a process for determining the value of a given quantity in one denomination of value, in some other given denomination of value; or the immediate relationship which exists between two denominations of value, by means of a chain of approximate steps, circumstances, or equivalent values, known to exist, which connect them. In every instance at least five terms or values are employed in the process, and in all instances the number employed will be uneven. A proposition involving but three terms, of this nature, is a question in single proportion. The equivalent values employed are divided into antecedents and consequents, or causes and effects; and the value or quantity for which an equivalent is sought, is called the odd term. RULE. 1. When the value in the denomination of the first antecedent is sought of a given quantity in the denomination of the last consequent. — Multiply all the antecedents and the odd term together for a dividend, and all the consequents together for a divisor; the quotient will be the answer or equivalent sought. RULE. -2. When the value in the denomination of the last consequent is sought of a given quantity in the denomination of the first antecedent. - Multiply all the consequents and the odd term together for a dividend, and all the antecedents together for a divisor; the quotient will be the answer required. EXAMPLE.I am required to give the value, in Federal money, of 5 Canada shillings, and know no immediate connection or relationship between the two currencies that of Canada and that of the United States. The nearest that I do know is that 20 Canada shillings have a value equal to 32 New York shillings, and that 12 New York shillings equal in value 9 New England shillings, and that 15 New England shillings equal $2.50; and with this knowledge will seek the value, in Federal money, of the 5 Canada shillings. EXAMPLE. If $21⁄2 equal 15 New England shillings, and nine shillings in New England equal 12 shillings in New York, and 32 shillings in New York equal 20 shillings in Canada, how many shillings in Canada will equal $1 ? EXAMPLE. — If 14 bushels of wheat weigh as much as 15 bushels of fine salt, and 10 bushels of fine salt as much as 7 bushels of coarse, and 7 bushels of coarse salt as much as 4 bushels of sand, how many bushels of sand will weigh as much as 40 bushels of wheat? |