But Here once 5 taken from 6, leaves in thousandths place is 5 in the place of hundredths, and 5 added to the 3 hundredths is >. Taking from 8, once 5, 2 remain. As before, the becomes 5 in the place of tenths, which added to 7 tenths, make 12, which added to the remainder 2 in the preceding place, give 32. From this 5 times 5 are to be taken, and 4 remain. But 4 in the place of tenths, is 45, which added to 5 make 50. From 50 take 9 times 53, and remains. Now in 5 are 11 halves, so that is, which annexed to the quotient, gives the whole quotient. Ex. 3. Divide 347 by 23 Ans 144. 4. Divide 13567 by 134. Supplement to Contractions in Multiplication. I. The shortest method of multiplication, when the multiplier is any even part of 100, 1000, &c. is by division: For if the multiplicand be increased by a number of ciphers equal to the number of places in the multiplier, and a part of that product taken for the same proportion, which the multiplier bears to 1, and the same number of ciphers annexed to it, the quotient will be the true product. 1. Multiply 39756 into 125. 125=1 of 1000, wherefore, 8)39756000 2. Multiply 57638 by 334 4969500 Product. 1921266 Product. 3. Multiply 91378 by 333. Ans. 30459333. The reason of the preceding rule is obvious. II. If any digit, with ciphers annexed, be divided by 9, the quotient will consist, wholly, of such digits, and so many 9ths of an unit over; hence the following method of multiplying by repetends of any of the digits. The reason of the method may be seen by the first example. Thus 80000÷9, or 50000-8888, and by multiplying by 80000 and dividing by nine, you get eight 9ths, of 645 too much, which must of course be subtracted to obtain the true product. Or, thus 80600 088888; then 645×8888-645x20000645x=57333334 -5733-5732760. But as three 9ths, belong to both numbers, and 1. as an equal number of 9ths, will belong to both numbers in any case, it is not necessary to write them, as they balance each other in the subtraction. 9)51600000 9)3236400000 III. When the multiplicand has a fraction belonging to it, such as one fourth, one half, &c. add such a part of the multiplier as the fraction signifies, to the product obtained by multiplying, and the sun is the whole product. When such a fraction belongs to the multiplier, add to the product such a part of the multiplicand as the fraction denotes. 1. Multiply 1531 by 6. 6 918 Product. 3 half of 6. 921 Answer. It is evident that 6 halves or 3, must be added to 153×6 to obtain 6 times 1532. And the same must be true in every similar case. 2. Multiply 638 by 61. 61 3828 Product by 6. As the multiplier is 64, it is evident, that to 6 times 638, there must be added one third of 638, the multiplicand, for the whole product. The third part of 638 is evidently 212, and 2 remainder, or, because is 2 parts of the 3 in the divisor. If Note. If the sum of the products, or quotients, of two or more numbers multiplied or divided by the same quantity, be required; then multiply or divide the sum of the numbers by the common multiplier or divisor, and the product or quotient will be the answer. the difference of the products or quotients be required, then multiply or divide the difference of the numbers, as before, for the answer. 1. Required the sum and difference of the products of 27 and 22, ltiplied each by 3. 2. Required the sum and difference of the quotients, by dividing 68 and 44, each by 4. Now 68+4+44÷4 the sum = 28 = 112÷4-68+44÷4. And 68÷4-44÷4 the diff. 6=24÷4-68---44÷4. The same course may obviously be pursued in any similar case. EXAMPLES FOR PRACTICE. 1. Divide 1292 dolls. equally among 17 men. Ans. 76 dolls. each. 2. Divide 2625 dolls. among 35 soldiers. Ans. 75 dolls. each. 3. If 360 cents are to be divided equally among eighteen poor persons, how much will each have? Ans. 20 cents. 4. If a field of 19 acres produce 513 bushels of wheat, how much is that for one-third of an acre? Ans. 9 bushels. 5. A man receives 1095 pounds a year, what is it for a day? Ans. 3 pounds. Ans. 220 yds. 6. There are 5 yds. in a rod; how many yards are there in 40 rods? 7. What number must you multiply by 47, to produce 29889492%? Ans. 6357534. 8. What number must be multiplied by 379 to produce 53789078? Ans. 9. In a rod are 16 feet; how many feet in 320 rods? 10. If 3650 pounds of bread are to be divided equally among 365 soldiers for 10 days, how much will each receive a day? 11. Multiply 34678 by 250. Ans. 1 pound. Ans. 866950. 12. Multiply 125 by 77777. * The following account is abstracted from the "Act establishing a Mint, and regulating the Coins of the United States," passed April 2d, 1792. The money of account of the United States shall be expressed in dollars or units, dismes or tenths of a dollar, cents or hundredths of a dollar, and mills or thousandths of a dollar. The coins of gold, silver, and copper of the U. S. shall be of the following de nominations, viz. The standard for all gold coins of the U. S. shall be eren parts of pure gold one pet of tley in hede. parts of the coia. The alloy is to be silver and copbut fac silver is not to exered one half in the alloy Ugle shell cont in two hunded and forty sever, and a half grains of pure ortwo hundred and seventy grains of standard gold; and the other gold coins este propertion. standard for all silver coirs of the U. S. shall be one thousand four hundred and lubly five parts of pore silver and one hundred and saver ty-nine parts alloy, e aday shall be pure copper. Dollar shall contia free buadred and seventy one and one fourth grains silver, or of four Landred and sixteen grains of stand rd silver; and the iver ceine by the same proportion Cooper coins are to be pure copyr. The Cent is to co tain eleven pennycopper; and the Pdf C'en' in pro ortion. cpárt mal ve hue of gold to save in coins emprent by law in the U. S. stod ei teen to one, or fferm pounds weight of pure slavez shall be equal to ene pe and weight of pure gold. 221eal as of go' Tard skver, i-sne! Gem Se Milf of 4. shall be a lawful te tecla ad payo cals of bio preceding yours when of full weight, and if not of weight, of pre rôn d voli, s. d1, Sik 1, Jewel, Chete ris, and all liquors. Cats, e Federts, and an cunce of silver. therefore, to disting is fineness of metals, such gold as will abide the fire without loss, is accounted 24 carats fine: If it lose 2 carats in trial, it is called 22 carats fine, &c. A pound of silver which loses nothing in trial, is 12 ounces fine; but, if it lose 3 pennyweights, it is 11 oz. 17 pwts. fine, &c. Alloy is some base metal with which gold or silver is mixed to abate its fineness; 22 carats of gold, and 2 carats of copper, are esteemed the true standard for gold coin in England, the alloy being one eleventh part of the fine gold: and 11 oz. 2. pwts. of fine silver, melted with 18 pwts, of copper, make the true standard for silver coin. NOTE. 175 Troy ounces, are precisely equal to 192 Avoirdupois ounces, and 175 Troy pounds are equal to 144 Avoirdupois. 1 lb. Troy 5760 grains, and 1 lb. Avoirdupois = 7000 grains. * = By Avoirdupois are weighed all coarse and drossy goods, grocery and chandiery wares; bread, and all metals, except gold and silver. A barrel of pork weighs 200 lb. A barrel of beef, 200 lb. A quintal of fish 1 Cwt. Avoirdupois. 12 particular things make one dozen; 12 dozen 1 gross, and 144 dozen 1 great gross. 20 particular things make 1 score, A Stone of Iron, shot, lb. A gallon of Train Cil 71 A Weigh A Sack A last 28 182 364 4368 All the weights now used by Apothecaries, above grains, are Avoirdupois. The Apothecaries' pound and ounce, and the pound and ounce Troy are the me, only differently divided and subdivided. |