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 663 per cent. less, or only 333 per cent. as large as the other. In common comparison, 1 is the measuring unit. In percentage, 100 is the measuring unit. Let a principal. samount (sum of the principal and percentage). p = rate per cent. of the percentage. a=s bb÷r=100bp 100s (100+p), = p100r100ba — 100(s — a) — a, What is § of § of † of 24 per cent. of 150 lbs. ? 150 X 12100 18 lbs. What is 23 per cent. of 19 bushels ? Ans. 12X180.45125 bushels. Ans. Bought a job lot of merchandise for $850, and sold it the same day, brokerage, 24 per cent., for $975; what was the net gain? (sr+a) = s(1 − r) — a — 975 — 975 850$100.625. 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? 1226, 600 per cent. Ans. .025 Ans. What part of 5 lbs. is 2 of 2 lbs. ? r = 3 × == 0.27. Ans. 241 per cent. is what per cent. of 36 per cent.? 66 per cent. Ans. For an article that cost $4, $5 were received; what per cent. of $4 was received? p5 × 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(484)÷4=1100 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 = (a - b) a=1 b a 0.913, or 913 per cent. Ans. 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? 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 = 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? 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? 2 X 95 +50 2+1 2a+b 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 37 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? a+a21+.a 2a 100+ 60 =80 per cent. Ans. cent. of the whole were damaged? (100 — 60) ÷ 2 = 20 per cent. Ans. Since 20 per cent. of the apples were damaged, what per cent. less was the number that were damaged than the number that were sound? What per cent. greater was the number that were sound than the number that were damaged? r = 2—(1+2. a) — 2 — 2. a160 per cent. Ans. = Since 80 per cent. of the whole were sound, what per cent. less was the number that were damaged than the number that were I sound? Since the number of damaged ones was 37 per cent. less than the number that were sound, what per cent. of the whole were sound? Since 80 per cent. of the whole were sound, what per cent. greater was the number that were sound than the number that were damaged? =2.a—1= 2 × .80 — 1 — 60 per cent. Ans. Lost 20 per cent. of a cargo of coal by jettison, and 5 per cent. of the remainder by screening, what per cent. of the coal was saved? a - b'— d' ?r=(1r) (1 — r'') = (1.20) (1.20) d' — b'=d'' x .05 (1.20)(1—.05)=76 per cent. Ans. d"-bill = d', &c. = Yesterday drew 12 per cent. of my balance of $4,273 in the bank, and deposited $1,000; and to-day have drawn 314 per cent. of the balance left over, or as it stood last night. What per cent. of the sum of the first-mentioned balance and deposit of yesterday have I drawn? |