respective Product underneath that Figure of the Multiplier by which you multiply. 3. Add the several Products together, and the Sum will be the desired, or total, Product. Case 3. When Ciphers are intermixed with the Figures in the Multiplier. RULE. Omit them, and place the first Figure of each particular Product under its respective Multiplier. EXAMPLES. (21) 10746047 40500108 (22) 804700625 Case 4. When there are Ciphers at the right-hand of either, or both the Multiplier and the Multiplicand. RULE. Proceed as before, neglecting the Ciphers until the particular Products are added together, and to that Sum place the Number of Ciphers that are at the end of both Factors, on the right-hand. EXAMPLES. (23) 1460900 (24) 2768000 8700 24600 If it be required to multiply any Number by 10, 100, 1000, &c. it is only annexing the Ciphers of the Multiplier to the right-hand of the Multiplicand, and the work is done. Case 5. When the Multiplier is such a Number that any two Figures in the Table, being multiplied together, will produce it. RULE. Multiply the given Number by one of those Figures, and that Product by the other; which will give the desired Product. EXAMPLES. (26) Mul. 340764 by 28. (25) Multiply 24674 by 16. (27) Mul. 142395 by 56. (29) Mul. 420746 by 72. (31) Mul. 43074 by 144. (28) Mul. 176848 by 81. (30) Mul. 17093 by 63. (32) Mul. 14068 by 132. Case 6. When the Multiplier is any Number between 10 and 20. RULE. Multiply by the Figure in the Unit's Place, and, as you multiply, add to the Product of each single Figure that of the Multiplicand, which stands next on the right-hand. V. DIVISION TEACHETH us to find how often one Number is contained in another, or to divide any Number or Quantity given into any Parts assigned, and serves instead of many Subtractions. In this Rule there are three Numbers real, and a fourth accidental; viz. 1. The Dividend, or Number to be divided. 2. The Divisor, or Number by which you divide. 3. The Quotient, or Number that shows how often the Divisor is contained in the Dividend. 4. The Remainder, which is always less than what vide by. Case 1. When the Divisor is not greater than 12. RULE. you di First seek how often the Divisor is contained in the first Figure of the Dividend, or, in case the first Figure of the Dividend be less than the Divisor, then in the first two Figures of the Dividend, and set the quotient Figure down accordingly; and, if any thing remain, carry it to the next Figure in the Dividend, where it must be reckoned as so many Tens ; that is, if one remain, you call it 10; if two, 20; if five, 50, and so on; bearing in mind the Remainder of each Figure, and adding it to the next, until you have made use of all the Figures in the Dividend. This is called Short Division. PROOF. Multiply the Quotient by the Divisor, and, as you multiply, add the Remainder, if any, or add the whole Remainder to the Product at last, and if it come the same as the Dividend, the Work is right. (7) 8)2768096 (8) 9)6768094 (10) 12)276484 (9) 10)2762764 Case 2. When the Divisor consists of many Places or Figures. RULE. 1. If the Divisor be a less Number than so many Figures taken in the Dividend, see how often the first Figure of the Divisor is contained in the first Figure of the Dividend, and the Figure which expresses it is the first of the Quotient; by which multiply the Divisor, and place the Product under the said Figures of the Dividend, and draw a line underneath it: subtract it therefrom, and to the remainder annex the following Figure of the Dividend, then proceed as before. 2. But if it happen that the Divisor be a greater Number than so many Figures of the Dividend; you must take a Number of places in the Dividend greater by one, and see how often the first Figure in the Divisor is contained in the first two of the Dividend, allowance being made for what you carry from the Figure on the right. 3. If in any Case the Remainder be so small, that when the Figure of the Dividend, joined with it, makes a Sum less than the Divisor, then a Cipher is to be placed in the Quotient, and another Figure brought down, and then proceed as before; this is called Long Division. EXAMPLES. (11) 25)736473575( (16) 7489)1204530760( (12) 84)35730972( (17) 42163)112737328( (13) 648)272357640( (18) 61745)392628787( (14) 759)30891829676( (19) 684573)3233238699( (15) 3065)63463902247( (20) 476085)98839054780( (21) 4728395)27750950255( Case 3. When the Divisor has Ciphers on the right-hand. RULE. Strike them off, and so many of the last Figures in the Dividend: divide by those Figures of the Divisor that are left when the Ciphers are omitted. But when the Division is ended, those Ciphers so omitted in the Divisor, and the Figures cut off in the Dividend, are both to be restored to their own places. (22) 2800)11928248( EXAMPLES. (23) 172000)247004674( Note-When the Dividend has the same Number of O's on the right-hand as the Divisor, strike them off from each, and the Remainder will be so many of what you divide by, without annexing the O's that were struck off. EXAMPLES. (24) 473000)351858000( (25) 6970000)599430000{ Case 4. When the Divisor is such a Number, that any two Figures in the Multiplication Table, being multiplied together, will produce the said Divisor. RULE. Divide the given Number by one of those Figures, and that Quotient again by the other, which will give the Quotient required. Note.-Observe, that if there be a Remainder in the last Division, it will be so many times the first Divisor; which, added to the first Remainder, if any, will give the true one. Case 5. When the learner is pretty well versed in Division, he may subtract each Figure of the Product, as he produces it, and so only write the Remainder, which will shorten the Work, and be much the best way, when the Divisor is small. (34) 17)690489( EXAMPLES. (36) 467)2148686( |