Proof. Subtract the remainder (if any) from the dividend, divide the dividend thus leffened, by the quotient, and the answer will be the former divifor. Or, add the remainder, and all the products of the quotient figures, as they stand in the operation; and the fum will be equal to the dividend. The reafon of the rule will appear, by refolving the dividend. and quotient into their feparate values, and comparing the steps: thus, in example ift. Laft remainder, 11 268 fum, or whole quotient. When there is a remainder, the operation is completed by annexing to the question the remainder placed above a line, and the divifor below it. Thus, in the above example, 268 is the answer. As it is fometimes difficult, when the divifor confifts of feveral figures, to difcover how often the divifor is contained in the feveral dividuals; we may try how often the first figure of the divifor may be had in the first figure, or first and second figures of the dividend; that number diminished by one, or two, is generally the quotient figure. When the divifor is 12, or under, we readily find the highest product, by the multiplication table; therefore the fubtractions may be performed mentally, and only the quotient put down. In example ift we proceed thus; the fives in 8, once and 3 over; we then put down 1, and fuppofing 3 prefixed to the next figure, we say the fives in 35, seven times and o over; having put down feven, we fay the fives in 6, once and 1 over; then having put down 1, the fives in 14, twice and 4 over; having put down 2, we fay the fives in 45, nine times and o over; and having put 9, we add a o for the o in the dividend. Examples for Practice. Divide 340608572 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. When the divifor is a compofite number, inftead of dividing by it at once, we may divide by the component parts fucceffively. Note. If the first divifor have a remainder, fet it down; then multiply the remainder from each of the following divifions, fucceffively by each of the divifors preceding its own divifor, and the fum of them is the true remainder. When there are cyphers on the right of the divifor, cut them off, and cut off also the fame number of figures on the right of the dividend; annex these figures to the remainder. To divide by 10, 100, and fo on, cut off as many figures on the right of the dividend, as there are cyphers in the divifor; the figures on the left are the quotient, and those on the right, the remainder. Examples for Practice. 7435169 by 800 747786600 by 9900 =, 9293388 760 75534 3- 8567539200 by 7200 = 1189936 62828, or 34 208 When there is a fraction in the divifor, multiply both divifor and dividend by the under-figure of the fraction, adding the upper-figure to the product of the divifor; then divide the products. As in examples first and fecond. Note. When the multiplier contains a fraction, first, multiply the whole number; then multiply by the upper-figure, and divide by the underfigure, and add the quotient to the former products as in example 3d. Note. Multiplication and Division, being in their nature and tendency directly oppofite, one proves the other. To prove multiplication, divide the product by either factor, and the quotient will be the other factor. And, to prove divifion, multiply the divifor by the quotient, add the remainder, if there be any, and the amount will make up the dividend. Befides the methods of proof already given for multiplication and divifion, there is one more expeditious, though indeed lefs decifive; which is that of cafting out the nines. To prove multiplication. Caft the nines out of both factors, and multiply the remainders together; what remains after cafting the nines out of their product, will be equal to what remains, after cafting the nines out of the total product. To prove divifion. Caft the nines out of the divifor and quotient, and multiply the remainders together; what remains, after cafting the nines out of their products, and adding the remainder, if any, will be equal what remains, after cafting the nines out of the dividend TAN Sect. II. COMPOUND DIVISION. RULE ft. Begin at the highest denomination, and proceed as in Simple Divifion. 2d, Reduce the remainder to the next lower denomination, and add the number belonging to it. 3d, Divide this fum in the fame manner, and fo on to the lowest. (1.) EXAMPLES. (2.) Divide L.1346: 14: 10 by 38. Divide 138 cwt. 2qr. 27 lb. by 41. If the divifor be 12 or under, we may work mentally, as in example 3d. To multiply by 4, divide the multiplicand by 2; thus in example 4th, we take the of 15s. 7d. for the price of yard. If the divifor be a compofite number, we may divide fuccellively by the component parts. H |