£. S. d. From 7849 6 8 6979,666 Remains 869 13 4 869,666 XLVIII. MULTIPLICATION OF DECIMALS. MU RULE. [ULTIPLY the decimals, as if they were whole numbers, and from the product cut off as many decimal places, as there are in both numbers. If there be not fo many places, make them out with cyphers on the left-hand of the product. Note. I have made ufe of the fame figures throughout each of thefe examples; yet the reader will find the values of the products are very different. of CONTRACTIONS. It frequently happens in bufinefs, that one ot both the factors confift many decimal places; so that to work them all would be very troublefome, and when done, but little to the purpose, because a lefs number of places may do the bufinefs as well; therefore ufe the following RULE. I. Tranfpofe all the figures of the multiplier in a contrary order to the common way, viz. let the units place ftand to the left-hand. 2. The units place of the multiplier muft ftand under that place of the multiplicand whofe decimal place you intend to retain the product: 3. Begin as in common multiplication, always having regard to the increafe of that figure on the right-hand, the figure that stands over your multiplier; making ufe of no more places of your multiplier than those which stand even with your multiplicand to the left-hand. E. 1. Let it be required to multiply 3,14159 by 24,8253, and to retain 4 decimal places in the product. 3,14159, 77,9907 Product 77,990919227 Note. As the allowance for what may be carried from the columns neglected is altogether a guefs, we may very often make the product less than it ought to be by 1 or 2, as appears by the above example; to avoid which, make one or two columns more than the number of decimal places you would have in the product, and cut them off at pleasure E. 2. Multiply 75,4678 by 6,05408, fo as to retain only three places of decimals in the product? 75,4678 80450,6 452806 3773 301 E. 3. Multiply 68479 by ,0785 to have 5 decimal places in the product, ,68479 5870,0 4793 547 34 ,05374 From thefe examples it is manifest how advantageous thefe contrac tions are to fhorten the work of long calculations and computations, which the experienced practitioner finds too often occur, in arithmetic, algebra, and geometry. To multiply by 10, 100, 1000, &c. remove the decimal point fo many steps further to the right-hand, as there are cyphers in the multiplier. As 86,564X 100-8656,4; and 45 X 1000450, &c. XLIX. DIVISION of DECIMALS. RULE. IVIDE as if they were whole numbers; then cut off as many in the dividend exceeds the number in the divifor; if there are not fo many in the divifor, prefix fo many cyphers. In dividing a whole number by a whole number, if any thing remains, annex cyphers to the remainder, and continue the divifion as far far as you please ; fo you will have a decimal in the quotient of as many places as you annexed cyphers, and the whole quotient thus found will be a mixed number. There are nine cafes, which take in the following order, by which the learner will eafily acquire a true notion of the ground and nature. of decimals. CASE 1. A whole number given to be divided by a whole number. 579268,)314159265,00000(542,33837, 2896340 2317072 1354545 1960090 Remains 208684 In this example here are five cyphers added to the dividend, which produce five decimal places in the quotient. In the laft example three cyphers are added to the right-hand of the whole number in the dividend, which makes the quotient a whole number; and because there is a remainder, you may go on again, by adding cyphers at pleasure; fo the quotient will be a mixed number. CASE 5, A whole number given to be divided by a decimal fraction. 579268)314159265,000000(542338373 2896340 CASE 6. A mixed number given, to be divided by a decimal fraction. If any whole, mixed, or decimal number, is given to be divided by 10, 100, 1000, &c. you only remove the feparating ppint towards the left-hand so many places as there are cyphers in the divifor, contrary to what was taught in multiplication. Thus, 152310=152,3; and 152310001,523, &c. To work any case of divifion by multiplication, and on the contrary, any cafe of multiplication by divifion; and this in many inftances will be found very ufeful. RULE RULE. Divide a unit with cyphers annexed by the given multiplier, and the quotient is the divifor fought. EXAMPLE. Suppofe I have 7315 to multiply by any other number, as 125; but have a defire to divide the faid number, and to have a quotient equal to the product of thofe two numbers; query, the divifor. Given 125)1,000(008 the divifor fought. And,008)7315,000(914375 Quotient; equal to the 72 Suppofe I have 7315 given, to be divided by any other number,008; but would multiply the faid number, and have a product equal to the quotient of the fame number divided by ,008; query the multiplier. RULE. Divide an unit with cyphers annexed by the given divifor, and the quotient will be the multiplier fought. Thus ;008) 1,000(125 The remainder of the work is only the reverfe of the former, and therefore need not be repeated. From the foregoing examples relating to divifion it may be observed, that the first figure of every quotient muft poffefs the fame place (with refpect to its value) as that figure of the dividend doth, which ftands over the units place of the first figure's product, which is an excellent rule to value quotients, obtained by the following, CONTRACTION. When the divifor confifts of many places of decimal parts, the work may be much abbreviated by the following, RULE. Confider in what place the firft figure of the quotient ought to ftand, and find its value or denomination; taking as many of the lefthand figures as you intend to have figures in the quotient for the first divifor; then take an many figures of the dividend as will answer them. In dividing, omit, or point off one figure at each operation; at the fame time, have a due regard to the increafe, which would arife from the figure or figures fo ommited. 2 D EXAMPLE |