ch. ALE AND BEER MEASURE. hhds. gal. gr. 51 3 A.B. fir. gal. 25 2 7 76 57 .. 97 27 22 17 $2 38 3 38.. 1 4 5 3 we take a lesser number from a greater, and find their difference. It is exactly opposite to addition, and is performed by learners in a like manner, beginning at the greater, and reckoning downwards the units of the lesser. The greater is called the minuend, and the lesser the subtrahend. If any figure of the subtrahend be greater than the corresponding figure of the minuend, we add ten to that of the minuend, and, having found and marked the difference, we add one to the next place of the subtrahend. This is called borrowing ten. The reason will appear, lasts. weys. qts. bu. pks. if we consider that, when two numbers are equally increased by adding the same to both, their difference will not be altered. When we proceed as directed above, we add ten to the minuend, and we likewise add one to the higher place of the subtrahend, which is equal to ten of the lower place. 47 1 62 0 2 45 1 78 1 29 1 TIME. .. 15 98 2 18 47 38 h. m. 15 42 41 37 28 21 42 27 When one page will not contain the whole account, we add the articles it contains, and write against their sum carried forward, and we begin the next page with the sum of the foregoing, writing against it brought forward. When the articles fill several pages, and their whole sum is known, which is the case in transcribing accounts, it is best to proceed in the following manner: add the pages, placing the sums in a separate paper: then add the sums, and if the amount of the whole be right, it only remains to find what number should be placed at the foot and top of the pages. For this purpose, repeat the sum of the first page on the same line; add the sums of the first and second, placing the amount in a line with the second; to this add the sum of the third, placing the amount in a line with the third. Proceed in the like manner with the others; and if the last sum corresponds with the amount of the page, it is right. These sums are transcribed at the foot of the respective pages, and tops of the following ones. SUBTRACTION. Subtraction is the operation by which APOTHECARIES WEIGHT. FE. qrs. n. yds. qrs. n. 2 1 LONG MEASURE. 2 3 13 1.. 18 The reason for borrowing is the same as in simple subtraction. Thus in subtracting pence, we add 12 pence, when necessary, to the minuend, and at the next 3 step we add one shilling to the subtrahend. When there are two places in the same denomination, if the next higher contain exactly so many tens, it is best to subtract the units first, borrowing ten when necessary; and then subtract the tens, borrowing, if there is occasion, according to the number of tens in the EE. grs. n. higher denomination. If the value of the 35..2.. 1 higher denomination be not an even number of tens, subtract the units and tens at once, borrowing according to the value of the higher denomination. It is often necessary to place the sums in different columns, in order to exhibit a clear view of what is required. For instance, 14 3 2 yds. ft. in. bar. leag. mi. fur. po. if the values of several parcels of goods 107 2 2 6 58 2 7 .. 10 .. 1 147 78 2 11 2 29 hhd. gal. qts. 42 2 37 2 3 0 17 3 49 3 times 274. 15 71 325 16 L. 8. d. 1 6 Remains to pay 22939 Lent......550156 171 13 74 Received 359 15 " at 475 13 several 527 15 payments 272 150 MULTIPLICATION. In multiplication two numbers are given, and it is required to find how much the first amounts to, when reckon. ed as many times as there are units in the second. Thus 8 multiplied by 5, or 5 times 8, is 40. The given numbers (8 and 5) are called factors; the first (8) the multiplicand; the second (5) the multiplier; and the amount (40) the product. This operation is nothing else than addition of the same number several times repeated. If we mark 8 five times under each other, and add them, the sum is 40: but as this kind of addition is of frequent and extensive use, in order to shorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier. For this purpose, the learner must be thoroughly acquainted with the following multiplication table, which is composed by adding each digit 12 times. TABLE. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 91215 18 21 24 27 30 33 36 4 8121620 24 28 32 36 40 44 48 510 15 20 25 30 35 40 45 50 55 60 6121824 30 6642 48 54 60 66 72 714 21 28 35 4249 56 63 70 77 84 816 24 32 40 48 56 64 72 80 88 96 918 27 36 45 546372 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 1122 3344,55667788 99 110 121 132 (12:24 3648 6072|84|96|108 120132144 In this table the multiplicand figures are in the upper horizontal row; the multipliers are in the left hand column, and the products will be found under the multiplicand, and in the same row with the multiplier; thus 9 times 11 are 99; and 99 will be found in the column under the 11, and in the same horizontal row with the 9, among the multipliers. If both factors be under 12, the table exhibits the product at once. If the mul tiplier only be under 12, we begin at the unit place, and multiply the figures in their order, carrying the tens to the higher place, as in addition. Example. 76859 multiplied by 4 4 Ans. 307436 or, 76859 added 4 times. 76859 76859 76859 Ans. 307436 the same as before. If the multiplier be 10, we annex a cypher to the multiplicand. If the multiplier be 100, we annex two cyphers; and so on. The reason is obvious, from the use of cyphers in notation. If the multiplier be any digit, with one or more cyphers on the right hand, we multiply by the figure, and annex an equal number of cyphers to the product. Thus, if it be required to multiply by 60, we first multiply by 6, and then annex a cypher. It is the same thing as to add the multiplicand 60 times; and this might be done by writing the account at large, dividing the column into 10 parts of 6 lines, finding the sum of each part, and adding these ten sums together. If the multiplier consist of several significant figures, we multiply separately by each, and add the products. It is the same as if we divided a long account of Addition into parts corresponding to the figures of the multiplier. A number, which cannot be produced by the multiplication of two others, is called a prime number; as 3, 5, 7, 11, and many others. A number, which may be produced by the multiplication of two or more smaller ones, is called a composite number. For example, 27, which arises from the multiplication of 9 by 3; and these numbers (9 and 3) are called the component parts of 27. 1. If the multiplier be a composite number, we may multiply successively by the component parts. Example. 7638 by 45, or 5 times 9, 7638 817793024 8. 3721478 by 9 9. 4783219 by 11 10. 4733218 by 12 11. 4783882 by 21 12. 2179929 by 32 25, 26. 68742 27. 5 28. 479327 by 403 4932149 by 3028 4731214 by 3008 49496 by 4009 343710 Because the second product is equal to five times the first, and the first is equal to nine times the multiplicand, it is obvious that the second product must be five times nine, or forty-five times as great as the multiplicand. 2. If the multiplier be 5, which is the half of 10, we may annex a cypher, and divide by 2. If it be 25, which is the fourth part of 100, we may annex two cyphers, and divide by 4. Other contractions of the like kind will readily occur to the learner. 3. To multiply by 9, which is one less than 10, we may annex a cypher, and subtract the multiplicand from the number it composes. To multiply by 99,999, or any number of 9's, annex as many cyphers, and subtract the multiplicand. The reason is obvious; and a like rule may be found, though the unit place be different from 9. Multiplication is proved |