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 $23 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 ? 3 4 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? 15 X 7 X 4 X 40 14 X 10 X 7 174 bushels. Ans. PERCENTAGE. Pure percentage, or PERCENTAGE, is a rate by the hundred of a part of a quantity or number denominated the principal, or basis. But percentage, considered as a means, and as commonly applied, is mixed and related in an eminent degree; and in this light may be regarded as divided into orders bearing different names. Thus Interest is percentage related to intervals of time in the past. Discount is percentage related to interest, and intervals of time in the future. Profit and Loss is comparative percentage, or percentage related to the positive and negative interests in business, etc., etc. Pure percentage is commonly called BROKERAGE when paid to a broker for services in his line. It is called COMMISSION when paid to or received by a factor or commission merchant for buying or selling goods. It is called PREMIUM by an insurance company, when taken for insuring against loss. It is called PRIMAGE when it is a charge in addition to the freight of a vessel, etc. Comparative percentage relates to the differences of quantities, and is confined always to the idea of more or less. It implies ratio. This description of percentage, though much in practice, seems not to be well understood; and often a quantity is indirectly stated to be many times less than nothing, or many times greater than it is. The difference of two quantities cannot be as great as a hundred per cent. of the greater, however widely unequal the quantities may be, nor as small as no per cent. of the greater or lesser, however nearly equal they may be. No quantity or number can be as small as 1 time less than another quantity or number; and therefore cannot be as small as 100 per cent. less. But, since one quantity may be many by 1 time, or many times greater than another with which it is compared, it may be said to be many by 100 times, or many hundred per cent. greater. When one of two quantities in comparison is stated to be three times less, or three hundred per cent. less, for instance, than the other, the expression is incorrect and absurd. The meaning evidently is, that it is two-thirds less, or only one-third as large as the other, that it is 66 per cent. less, or only 33 per cent. as large as the other. In common comparison, 1 is the measuring unit. In percentage, 100 is the measuring unit. percentage. samount (sum of the principal and percentage). prate per cent. of the percentage. α as b = b — r = 100b÷p=100s ÷ (100+p), b=s p100r1006a100(sa)÷a, d = a b = To find the Percentage. EXAMPLES. What is of 1 per cent. of $200 ? b= =arap 100$0.50. Ans. § of 2 per cent. of 50 is what part of 50? What is of of of 24 per cent. of 150 lbs.? 150 X 12 100 18 lbs. Ans. What is 23 per cent. of 19 bushels ? 12 × 10% = 0.45125 bushels. Ans. Bought a job lot of merchandise for $850, and sold it the same day, brokerage, 21 per cent., for $975; what was the net gain? (sr+a) = s(1 − r) — a — 975 — 975 × .025 - 850 $100.625. Ans. sr as — To find the Rate or Rate Per Cent. EXAMPLES. What per cent. of $20 is $2 ? rba, p=100ba10 per cent. Ans. 12 dozen is equal to what per cent. of 2 dozen? 12 2 6, 600 per cent. Ans. What part of 5 lbs. is 2 of 2 lbs. ? 2 r = 1 × & = 10.27. Ans. 241 per cent. is what per cent. of 36 per cent.? 663 per cent. Ans. For an article that cost $4, $5 were received; what per cent. of $4 was received? p=5 × 1004 125 per cent. Ans. A farmer sowed 4 bushels of wheat, which produced 48 bushels; what per cent. was the increase? 48 is more than 4 by what per cent. of 4? The difference of 48 and 4 is what per cent. of 4? 100(48 — 4) 41100 per cent. Ans. What per cent. would have been the decrease, if he had sowed 48 bushels, and harvested only 4 bushels? 4 is less than 48 by what rate of 48? The difference of 48 and 4 is what per cent. of 48? r = (ab) b a=1— 0.913, or 91 per cent. Ans. a Since water is composed of 8 atoms of oxygen and 1 atom of hydrogen, what per cent. of it is oxygen? 8 is what per cent. of the sum of 8 and 1 ? What per cent. of it is hydrogen? 1 is what per cent. of the sum of 8 and 1 ? How many volumes of water must be added to 100 volumes of 90 per cent. alcohol to reduce it to 50 per cent. alcohol or common proof? 90 is more than 50 by what per cent. of 50? The difference of 90 and 50 is what per cent. of 50? p= (a - b)100 (90 - 50)100 b 50 80. Ans. How many volumes of 50 per cent. alcohol must be added to 100 volumes of 90 per cent. alcohol to produce 80 per cent. alcohol? 90 is more than 80 by what per cent. of the difference of 80 and 50? The difference of 90 and 80 is what per cent. of the difference of 80 and 50? p = (a - b)100 (90 80)100 80 - 50 33. Ans. How many volumes of 90 per cent. alcohol must be added to 100 volumes of 50 per cent. alcohol to raise it to 80 per cent. alcohol? 50 is less than 80 by what per cent. of the difference of 90 and 80? The difference of 80 and 50 is what per cent. of the difference of 90 and 80? (b — b')100 α (80-50)100 300. Ans. If to 2 volumes of 95 per cent. alcohol, 1 volume of 50 per cent. alcohol be added, what per cent. alcohol will be the mixture? The sum of 50 and twice 95 is what per cent. of the sum of 2 and 1? 2a+b 2 X 95 +50 80 per cent. Ans. In a barrel of apples, the number of sound ones was 60 per cent. greater than the number that were damaged. What per cent. less was the number that were damaged than the number that were sound? 60 per cent. is what per cent. of the sum of 100 per cent. and 60 per cent. ? .6 is what rate of 1 +.6 ? Since the number of damaged apples was 371 per cent. less than the number that were sound, what per cent. greater was the number that were sound than the number that were damaged? r = a ÷ (1 − a) — 1 ÷ (1 − a) — 1 60 per cent. Ans. Since the number of sound ones was 60 per cent. greater than the number that were damaged, what per cent. of the whole were sound? What per cent. of the whole were damaged? |