RULE.First, seek how many times the divisor is contained in as many of the left band figures of the dividend as are just necessary ; (that is, find the greatest figure that the divisor can be multiplied by, 80 as to produce a product that shall not exceed the part of the dividend used ;) when found, place the figure in the quotient; multiply the divisor by this quotient figure ; place the product under that part of the dividend used ; then subtract it therefrom, and bring down the next figure of the dividend to the right hand of the remainder; after which, you must seek, multiply and subtract, till you have brought down every figure of the dividend. Proof. Multiply the divisor and quotient together, and add the remainder, if there be any, to the product; if the work be right, the sum will be equal to the dividend.* EXAMPLES. 1. How many times is 4 4)9391(2347 4 2. Divide 3656 dollars equally among 8 men. Divisor, Div. Quotient. 8)3656(457 13 9388 45 19 9391 Proof. 56 31 3656 Proof by addition. 3 Remainder. * Another method which some make use of to prove division is as follows : viz. Add the remainder and all the products of the several quotient figures multiplied by the divisor together, according to the order in which they stand in the work ; and this sum, when the work is right, will be equal to the dividend. A third method of proof by excess of nines is as follows, viz. 1. Cast the nines out of the divisor, and place the excess on the left hand. 2. Do the same with the quotient, and place it on the right hand. 3. Multiply these two figures together, and add their product to the remainder, and reject the nines, and place the excess at top. 4. Cast the nines out of the dividend, and place the excess at bottom. Note. If the sum is right, the top and bottom figures will be alike. 365)49640(136 365 Divisor. Div. Quotient. 29)15359(529 145 Proof by excess of 9's. 85 5 58 2X7 279 5 261 1314 2190 2190 Remains 18 0 Rem. Divisor. Div. Quotient. 95(85595(901 61)23609(469 736)863256(1172 472251104(532 there remains 664. 9. Divide 1893312 by 912. Ans, 2076, 10. Divide 1893312 by 2076. Ans. 912, 11. Divide 47254149 by 4674. Ans. 10110 617 12. What is the quotient of 330098048 divided by 4207? Ans. 78464. 13. What is the quotient of 761858465 ?ivided by 8465? ! Ans. 90001. 14. How often does 761858465 contain 90001 ? Ans. 8465. 15. How many times 38473 can you have in 119184693 ? Ans. 3097314 16. Divide 280208122081 by 912314. Quotient, 307140 Remainder. 9182 987654(988641654 )---0 CASE II. When there are ciphers at the right hand of the divisor, cut off the ciphers in the divisor, and the same number of figures from the right hard of the dividend ; then divide the remaining ones as usual, and to the remainder (if any) annex those figures cut off from the dividend, and you will have the true remainder 214(00346736)25(218.30 EXAMPLES. 1. Divide 4673625 hy 21400. 214(00)46736)25(21831 true quotient by Restitution. 428 393 1796 8425 true rem. 2. Divide 379432675 by 6500. Ans. 583741974. 3. Divide 421400000 by 49000. Ans. 8600. 4. Divide 11659112 by 89000. Ars. 131,117 5. Divide 9187642 by 9170000. Ans. l.1748 MORE EXAMPLES. Remains. 76478 901000)654347230 )221230 720000)9876540000 )534000 CASE III. Short Division is when the Divisor does not exceed 12. Rule. Consider how many times the divisor is contained in the first figure or figures of the dividend, put the result under, and carry as many tens to the next figure as there are ones over. Divide every figure in the same manner till the whole is finished. EXAMPLES, Divisor. Dividend. 2)113415 3)85494 4)39407 5)94379 Quotient, 56707-1 6)120616 7)152715 8)96872 9)118724 11)6986197 12)14814096 12)570196382 Contractions in Division. When the divisor is such a number, that any two figures in the Table, being multiplied together, will produce it, divide the given dividend by one of those figures ; the quotient thence arising by the other; and the last quotient will be the answer. NOTE. The total remainder is found by multiplying the last remainder by the first divisor, and adding in the first remainder. 8 2258-7 63 first rem. +2 True Quotient 225844. True rem. I 2. Divide 178464 by 16. Ans. 11154. 3. Divide 467412 by 24. Ans. 1947512. 4. Divide 942341 by 35. Ans. 2692435 5. Divide 79638 by 36. Ans. 2212 6. Divide 144872 by 48. Ans. 3018 7. Divide 937387 by 54. Ans. 1735934. 8. Divide 93975 by 84. Ans. 111851. 9. Divide 145260 by 108. Ans. 1345. 10. Divide 1575360 by 144. Ans. 10940. 2. To divide by 10, 100, 1000, &c. RULE.-Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor, and these figures so cut off are the remainder; and the other figures of the dividend are the quotient. EXAMPLES. 1. Divide 365 by 10. Ans. 36 and 5 remains, 2. Divide 5762 by 100. Ans. 57 3. Divide 763753 by 1000. Ans. 763 62 rem. 753 rem. SUPPLEMENT TO MULTIPLICATION. To multiply by a mixt number; that is, a whole number joined with a fraction, as 81, 5., 6., &c. RULE.-Multiply by the whole number, and take 4, 3, 4, &c. of tac multiplicand, and add it to the product. EXAMPLES. Multiply 48 by 23. 2)37 48 234 23 18! 111 24=; 74 869Answer. 3. Multiply 211 by 501 4. Multiply 2464 by 84. 5. Multiply 345 by 191 6. Multiply 6497 by 54. 132 Ans. Ans. 10655. Ans. 20533 Ans. 6598 Ans. 33413.. Questions to exercise Multiplication and Division. 1. What will 9 tuns of hay come to, at 14 dollars a tun ? Ans. $136. 2. If it take 320 rods to make a mile, and every rod coutains 5} yards; how many yards are there in a mile ? Ans. 1760. 3. Sold a ship for 11516 dollars, and I owned of her; what was my part of the money? Ans. 98637. 4. In 276 barrels of raisins, each 3; cwt. how many hundred weight ? Ans. 966 cwt. 5. In 36 pieces of cloth, each piece containing 241 yards; how many yards in the whole ? Ans. 873 yds. 6. What is the product of 161 multiplied by itself? Ans. 25921. 7. If a man spend 492 dollars a year, what is that per calendar month ? Ans. $41. 8. A privateer of 65 men took a prize, which being equally divided among them, amounted to 1191. per man ; Ans. £7735. what is the value of the prize ? D |