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Art. 62. To multiply by any number of 9's.
By adding 1 to any number 4789 6 5300000
composed of nines, we obtain 4789 653
a number expressed by 1 with 4789 6 0510 347 Product.
as many ciphers annexed as
there are nines in the number to which 1 is added. Thus, 999 + 1 1000. Therefore annexing 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.
Ans. 1233332433. 3. Multiply 876543 by 999999. Ans. 876542123457. 4. Multiply 999999 by 999999. Ans. 999998000001.
CONTRACTIONS IN DIVISION.
Art. 63. To divide by 25.
Ans. 49382 60%.
Multiplying the dividend by 4 makes 1 2 3 4 5 6 7
it four times too great; therefore
obtain the true quotient, we must divide 4
by 100, a divisor four times greater than 49382.6 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.
Ans. 395061. 3. Divide 1378925 by 25.
Ans. 55157. 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.
Art. 64. To divide by 331.
Multiplying the dividend by 3 makes 6789543
it three times too great; therefore, to
obtain the true quotient, we must divide 3
by 100, a divisor three times greater than
the true one. 203 6 8 6.2 9 Quotient.
This is done by cutting off two figures on the right.
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.
EXAMPLES FOR PRACTICE.
2. Divide 987654321 by 33.
Art. 65. To divide by 125.
Multiplying the dividend by 8 makes 9874725
it eight times too great; therefore, to
obtain the true quotient, we must divide. 8
by 1000, a divisor eight times greater 7899 7.8 0 0 Quotient.
than the true one. We do this by cutting off three figures on the right.
RULE. - Multiply the dividend by 8, and the product, except the last three figures, is the quotient. The last three figures are thousandths.
EXAMPLES FOR PRACTICE. 2. Divide 1728125 by 125.
Ans. 13825. 3. Divide 478763250 by 125.
Ans. 3830106. 4. Divide 591234875 by 125.
Ans. 4729879. 5. Divide 489648 by 125.
Ans. 3917106 6. Divide 836184 by 125.
QUESTIONS. — Art. 64. What is the rule for dividing by 33j. Give the reason for the rule. - Art. 65. What is the rule for dividing by 125? What is the reason for the rule ?
VII. MISCELLANEOUS EXAMPLES,
INVOLVING THE FOREGOING RULES.
211) 1. A. bolght 73 hogsheads of molasses at 29 dollars per
hogshead, and sold it at 37 dollars per hogshead; what did he gain ?
*Ans. 584 dollars. 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 ?
Ans. 25088 dollars. 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?
Ans. 940 cents. 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 ?
Ans. 1530 cents. 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 ?
Ans. 873 dollars. 6. Bought 17 yards of cotton at 15 cents per yard, 46 gal. lons 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 ?
Ans. 4257 cents. 7. A man travelled 78 days, and each day he walked 27 miles; what was the length of his journey?
Ans. 2106 miles. 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 ?
Ans. 61 miles. 9. What cost a farm of 365 acres at 97 dollars per acre ?
Ans. 35405 dollars. 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 ?
Ans. 42884 dollars.
f 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 ?
Ans. 7665 dollars. 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
one hundred-dollar bill ; how
dollars must Ames return to Peabody?
Ans. 37 dollars. 13. Bought of P. Parker a cooking-stove for 31 dollars, 7 quintals of his best fish at 6 dollars per quintal, 14 bushels of rye at 1 dollar per bushel, and 5 mill-saws at 16 dollars each; in part payment for the above articles, I sold him eight thousand feet of boards at 15 dollars per thousand; how much must I
him to balance the account? Ans. 47 dollars. 14. In 1 day there are 24 hours; how many in 57 days?
Ans. 1368 hours. 15. In one pound avoirdupois weight there are 16 ounces; how many ounces are there in 369 pounds ?
Ans. 5904 ounces. 16. In a square mile there are 640 acres; how many acres are there in a town which contains 89 square miles ?
Ans. 56960 acres. 17. What cost 78 barrels of apples at 3 dollars per barrel?
Ans. 234 dollars. 18. Bought 500 barrels of flour at 5 dollars per barrel, 47 hundred weight of cheese at 9 dollars per hundred weight, and 15 barrels of salmon at 17 dollars per barrel ; what was the amount of my purchase ?
Ans. 3178 dollars. 19. Bought 760 acres of land at 47 dollars per acre, and sold J. Emery 171 acres at 56 dollars per acre, J. Smith 275 acres at 37 dollars per acre, and the remainder I sold to J. Kimball at 75 dollars per acre ; how much did I gain by my sales ?
Ans. 7581 dollars. 20. Bought a hogshead of oil containing 184 gallons at 75 cents per gallon; but 28 gallons having leaked out, I sold the remainder at 98 cents per gallon ; did I gain or lose by my bargain ?
Ans. 1488 cents, gain. 21. Bought a quantity of flour, for which I gave 1728 dollars, there being 288 barrels; I sold the same at 8 dollars per barrel; how much did I gain?
Ans. 576 dollars. 22. Purchased a cargo of molasses for 9212 dollars, there being 196 hogsheads; I sold the same at 67 dollars per hogshead; how much did I gain on each hogshead ?
Ans. 20 dollars. 23. A farmer bought 5 yoke of oxen at 87 dollars a yoke ; 37 cows at 37 dollars each; 89 sheep at 3 dollars apiece. He sold the oxen at 98 dollars a yoke ; for the cows he received 40 dollars each; and for the sheep he had 4 dollars apiece. How much did he gain by his trade ? Ans. 255 dollars.
24. The sum of two numbers is 5482, and the smaller number is 1962; what is the greater?
Ans. 3520. 25. The difference between two numbers is 125, and the smaller number is 1482 ; what is the greater ?
Ans. 1607. 26. The difference between two numbers is 1282, and the greater number is 6948; what is the smaller ? Ans. 5666.
27. If the dividend is 21775, and the divisor 871, what is the quotient ?
Ans. 25. 28. If the quotient is 482, and the divisor 281, what is the dividend ?
Ans. 135412. 29. If 144 inches make 1 square foot, how many square feet in 20736 inches ?
Ans. 144 feet. 30. An acre contains 160 square rods; how many rods in a farm containing 769 acres ?
Ans. 123040 rods. 31. A gentleman bought a house for three thousand fortyseven dollars, and a carriage and span of horses for five hundred seven dollars. He paid at one time two thousand seventeen dollars, and at another time nine hundred seven dollars. How much remains due ?
Ans. 630 dollars. 32. The erection of a factory cost 68,255 dollars; supposing this sum to be divided into 365 shares, what is the value of each ?
Ans. 187 dollars. 33. Bought two lots of wild land; the first contained 144 acres, for which I paid 12 dollars per acre; the second contained 108 acres, which cost 15 dollars per acre. I sold both lots at 18 dollars per acre; what was the amount of gain ?
Ans. 1188 dollars. 34. Sold 17 cords of oak wood at 6 dollars per cord, 36 cords of maple at 3 dollars per cord, and 29 cords of walnut at 7 dollars per cord, What was the amount received ?
Ans. 413 dollars. 35. Daniel Bailey has a fine farm of 300 acres, which cost him 73 dollars per acre.
He sold 83 acres of this farm to Minot Thayer, for 97 dollars per acre; 42 acres to J. Russel, for 87 dollars per acre; 75 acres to J. Dana, at 75 dollars per acre; and the remainder to J. Webster, at 100 dollars per acre. What was his net gain ?
Ans. 5430 dollars. 36. J. Gale purchased 17 sheep for 3 dollars each, 19 cows at 27 dollars each, and 47 oxen at 57 dollars each. He sold his purchase for 3700 dollars. What did he gain?
Ans. 457 dollars. 37. Purchased 17 tons of copperas at 32 dollars per ton. I sold 7 tons at 29 dollars per ton, 8 tons at 36 dollars per ton,