CONTRACTIONS. 65. A Composite Number is one that is the product of two or more other numbers. Thus, 18 is a composite number, since 6 × 3 = 18; or, 9 × 2 = 18; or, 3 × 3 × 2 = 18. 66. The Component Factors of a number are the several numbers which, when multiplied by one another, produce the given number. Thus, the component factors of 20 are 10 and 2 (10 × 2 = 20); or, 4 and 5 (4 × 5 = 20); or, 2 and 2 and 5 (2 × 2 × 5 = 20). The pupil must not confound the factors with the parts of a number. Thus, the factors of which 12 is composed, are 4 and 3 (4 × 3 = 12); while the parts of which 12 is composed are 8 and 4 (8+ 4 = 12), or 10 and 2 (10+2 = 12). The factors are multiplied while the parts are added, to produce the number. EXAMPLES. 67. When the multiplier is a composite number. 1. What will 32 horses cost, at $174 apiece? RULE.-I. Separate the composite number into two or more factors. Multiply the multiplicand by one of these factors, and the product by another, and so on until all the factors have been used successively; the last product will be the product required. 68. The product of any number of factors will be the same in whatever order they are multiplied. Thus, 4 x 3 x 5 = 60, and 5 × 4 × 3 = 60, and 3 x 4 x 5 = 60. 2. Multiply 3472 by 48, (6 × 8). 3. Multiply 14761 by 64, (8 × 8). Ans. 166656. 4. Multiply 87034 by 81, (3 × 3 × 9). Ans. 7049754. 5. Multiply 47326 by 120, (6 × 5 × 4). 6. Multiply 60315 by 96. 7. Multiply 291042 by 125. Ans. 5790240. Ans. 36380250. 8. If a vessel sails 436 miles in 1 day, how far will she sail in 56 days? Aus. 24416 miles. 9. What will 72 acres of land cost, at $124 an acre? Ans. $8928. 10. There are 5280 feet in a mile. How many feet are there in 84 miles? Ans. 443520 feet. 11. What will 120 yoke of cattle cost, at $125 a yoke? 69. When the multiplier is 10, 100, 1000, etc. If a cipher is annexed to the multiplicand, each figure is removed one place toward the left, and consequently the value of the whole number is increased tenfold (33). If two ciphers are annexed, each figure is removed two places toward the left, and the value of the number is increased one hundred fold; and every additional cipher increases the value tenfold. RULE. there are -Annex as many ciphers to the multiplicand as will be the product required. ciphers in the multiplier; the number so formed 1. Multiply 347 by 10. 2. Multiply 4731 by 100. Ans. 3470. Ans. 473100. 5. If 1 acre of land costs $36, what will 10 acres cost? Ans. $360. 6. If 1 bushel of corn costs 65 cents, what will 1000 bushels cost? Ans. $650. 70. When there are ciphers at the right hand of one or both of the factors. 1. Multiply 1200 by 60. OPERATION. Multiplicand, Multiplier, 1200 60 72000 SOLUTION. Both multiplicand and mul tiplier may be resolved into their component factors; 1200 into 12 and 100, and 60 into 6 and 10. If these several factors are multiplied by one another they will produce the same product as the given numbers (67). Thus, 12 x 672, and 72 × 100 7200, and 7200 × 10 72000, which is the same result as in the operation. Product, RULE.Multiply the significant figures of the multiplicand by those of the multiplier, and to the product annex as many ciphers as there are ciphers on the right of both factors. 9. Multiply eighty million seven thousand six hun dred, by eight million seven hundred sixty. Ans. 640121605776000. 10. Multiply fifty million ten thousand seventy, by sixty-four thousand. Ans. 3200644480000. 11. Multiply ten million three hundred fifty thousand one hundred, by eighty thousand nine hundred. Ans. 837323090000. EXAMPLES COMBINING ADDITION, SUBTRACTION, AND MULTIPLICATION. 1. A dealer bought 45 cords of wood at $4 a cord, and 9 loads of hay at $13 a load. What was the cost of the wood and hay? Ans. $297. 2. A merchant bought 6 hogsheads of sugar at $31 a hogshead, and sold it for $39 a hogshead. What did he gain? 3. I bought 288 barrels of flour for $1875, and sold the same for $9 a barrel. What was the gain? Ans. $717. 4. If a young man receives $500 year salary and pays $240 for board, $125 for clothing, $75 for books, and $50 for other expenses, how much will he have left at the end of the year? Ans. $10. 5. A farmer sold 184 bushels of wheat at $2 a bushel, for which he received 67 yards of cloth at $4 a yard, and the balance in groceries. What did his groceries cost him? 6. A sold a farm of 320 acres at $36 an acre; B sold one of 244 acres at $48 an acre. Which received the greater sum, and how much? Ans. B, $192. 7. A merchant bought 14 bales of cloth, each bale containing 26 pieces, and each piece 43 yards. How many yards of cloth did he buy? Ans. 15652 yards. 8. If a man has an income of $3700 a year, and his daily expenses are $4, what will he save in a year, or 365 days? Ans. $2240. 9. Two persons start from the same point and travel in opposite directions, one at the rate of 35 miles a day, and the other 29 miles a day. How far apart will they be in 16 days? Ans. 1024 miles. 10. A man sold three houses; for the first he received $2475, for the second $ 840 less than he received for the first, and for the third as much as for the other two. What did he receive for the three? Ans. $8220. 11. A man sets out to travel from Albany to Buffalo, a distance of 336 miles, and walks 28 miles a day. How far is he from Buffalo when he has walked 10 days? 12. Mr. C bought 14 cows at $23 each, 7 horses at $96 each, 34 oxen at $57 each, and 300 sheep at $2 each; he sold them all for $3842. What did he gain? Ans. $310. 13. A drover bought 164 head of cattle at $36 a head, and 850 sheep at $3 a head. What did he pay for all? 14. A banker has an income of $14760 a year; he pays $1575 for house rent, and four times as much for family expenses. What does he save annually? Ans. $6885. 15. A flour merchant bought 936 barrels of flour at $9 a barrel; he sold 480 barrels at $10 a barrel, and the remainder at $8 a barrel. What did he gain or lose? Ans. Gained $24. 16. What will 24 acres of land cost, at $124 an acre? Ans. $2976. 17. How far will a ship sail in 56 weeks, at the rate Ans. 84672 miles. of 1512 miles per week? 18. How many pounds of iron are there in 54 loads, Ans. 155142 pounds. each weighing 2873 pounds? 19. Multiply 2874 by 72. 20. Multiply 8074 by 108. Ans. 206928. Ans. 871992. |