RULE. Cut off the ciphers from the divisor, and the same number of figures from the right hand of the dividend. Then divide the remaining figures of the dividend by the remaining figures of the divisor, and the result will be the quotient. To complete the work, annex to the last remainder found by the operation the figures cut off from the dividend, and the whole will form the true remainder. EXAMPLES FOR PRACTICE. 2. Divide 3594 by 80. 3. Divide 79872 by 240. 4. Divide 467153 by 700. 5. Divide 13112297 by 8900. 6. Divide 71897654325 by 700000000. 7. Divide 3456789123456787 by 990000. 8. Divide 4766666000000 by 55550000000. Rem. 74 192 253 *2597 497654325 306787 44916000000 RULE. - Annex two ciphers to the multiplicand, and divide it by 4, and the quotient is the product required. *If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his progress, they may be omitted for the present, and attended to when he is further advanced. QUESTIONS. What is the rule for dividing when there are ciphers on the right of the divisor? - Art. 59. What is the rule for multiplying by 25? What is the reason for the rule ? EXAMPLES FOR PRACTICE. 2. Multiply 76589658 by 25. 3. Multiply 567898717 by 25. 4. Multiply 123456789 by 25. ART. 60. To multiply by 33. OPERATION. 3) 8767896300 2922632100 Product. Ans. 2922632100. We first multiply by 100, as before, and since 33, the multiplier, is only one third of 100, we divide by 3 to obtain the true product. RULE. - Annex two ciphers to the multiplicand, ånd divide it by 3, and the quotient is the product required. EXAMPLES FOR PRACTICE. 2. Multiply 356789541 by 33. ART. 61. To multiply by 125. OPERATION. 8) 789 6538000 987067250 Product. Ans. 987067250. We multiply by 1000, by annexing three ciphers to the multiplicand, and since 125, the multiplier, is only one eighth of 1000, we divide by 8 to obtain the true product. RULE. - Annex three ciphers to the multiplicand, and divide by 8, and the quotient is the product. EXAMPLES FOR PRACTICE. 2. Multiply 7965325 by 125. 3. Multiply 1234567 by 125. 4. Multiply 3049862 by 125. QUESTIONS. Art. 60. What is the rule for multiplying by 333? What is the reason for this rule? - Art. 61. What is the rule for multiplying by 125 ? Give the reason for the rule ? ART. 62. To multiply by any number of 9's. Ex. 1. Multiply 4789653 by 99999. OPERATION. 478965300000 4789653 4789 60510347 Product. Ans. 478960510347. By adding 1 to any number composed of nines, we obtain a number expressed by 1 with as many ciphers annexed as there are nines in the number 1000. Therefore annex = to which 1 is added. Thus, 999 +1 ing to the multiplicand as many ciphers as there are nines in the multiplier is the same thing as multiplying the number by a multiplier too large by 1, and subtracting the number to be multiplied from this enlarged product will give the true product. RULE. — Annex as many ciphers to the multiplicand as there are 9's in the multiplier, and from this number subtract the number to be multiplied, and the remainder is the product required. EXAMPLES FOR PRACTICE. 2. Multiply 1234567 by 999. CONTRACTIONS IN DIVISION. ART. 63. To divide by 25. Ex. 1. Divide 1234567 by 25. OPERATION. 1234567 4 Ans. 49382 6. makes Multiplying the dividend by it four times too great; therefore, to obtain the true quotient, we must divide by 100, a divisor four times greater than 4 93 82.6 8 Quotient. the true one. This we do by cutting off two figures on the right. RULE.-Multiply the dividend by 4, and the product, except the last two figures at the right, is the quotient. The last two are hundredths. EXAMPLES FOR PRACTICE. 2. Divide 9876525 by 25. 3. Divide 1378925 by 25. 4. Divide 899999 by 25. QUESTIONS. Art. 62. What is the rule for multiplying by any number of 9's? What is the reason for the rule?-Art. 63. What is the rule for dividing by 25? Give the reason for the rule. RULE. Multiply the dividend by 3, and the product, except the last two figures at the right, is the quotient. The last two are hundredths. RULE.-Multiply the dividend by 8, and the product, except the last three figures, is the quotient. The last three figures are thousandths. QUESTIONS. Art. 64. What is the rule for dividing by 33. reason for the rule. -Art. 65. What is the rule for dividing by 125? the reason for the rule? Give the § VII. MISCELLANEOUS EXAMPLES, INVOLVING THE FOREGOING RULES. 1. A. bought 73 hogsheads of molasses at 29 dollars per hogshead, and sold it at 37 dollars per hogshead; what did he gain? 2. B. bought 896 acres of wild land at 15 dollars per acre, and sold it at 43 dollars per acre; what did he gain? 3. N. Gage sold 47 bushels of corn at 57 cents per bushel, which cost him only 37 cents per bushel; how many cents did he gain? 4. A butcher bought a lot of beef weighing 765 pounds at 11 cents per pound, and sold it at 9 cents per pound; how many cents did he lose? 5. A taverner bought 29 loads of hay at 17 dollars per load, and 76 cords of wood at 5 dollars a cord; what was the amount of the hay and the wood? 6. Bought 17 yards of cotton at 15 cents per yard, 46 gallons of molasses at 28 cents per gallon, 16 pounds of tea at 76 cents a pound, and 107 pounds of coffee at 14 cents a pound; what was the amount of my bill? 7. A man travelled 78 days, and each day he walked 27 miles; what was the length of his journey? 8. A man sets out from Boston to travel to New York, the distance being 223 miles, and walks 27 miles a day for 6 days in succession; what distance remains to be travelled? 9. What cost a farm of 365 acres at 97 dollars per acre? 10. Bought 376 oxen at 36 dollars per ox, 169 cows at 27 dollars each, 765 sheep at 4 dollars per head, and 79 elegant horses at 275 dollars each; what was paid for all? 11. J. Barker has a fine orchard, consisting of 365 trees, and each tree produces 7 barrels of apples, and these apples will bring him in market 3 dollars per barrel; what is the income of the orchard? 12. J. Peabody bought of E. Ames 7 yards of his best broadcloth at 9 dollars per yard, and in payment he gave Ames a |