CONTRACTIONS.

CASE I.

65. When the multiplier is a composite number.

A Composite Number is one that may be produced by multiplying together two or more numbers; thus, 18 is a composite number, since 6 × 3 = 18; or, 9 X 2 = 18; or, 3 X 3X 2 = 18.

66. The Component Factors of a number are the several numbers which, multiplied together, produce the given number; thus, the component factors of 20 are 10 and 2, (10 X 2 = 20 ;) or, 4 and 5, (4 × 5 = 20;) or, 2 and 2 and 5, (2 X 2 X 5 = 20.)

NOTE. 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.

1. What will 32 horses cost, at 174 dollars apiece?

the cost of 8 times 4 horses, or 32 horses, the number bought.

67. Hence we have the following

RULE. I. Separate the composite number into two or more factors.

II. Multiply the multiplicand by one of these factors, and

What are contractions?

Case I is what? Define a composite number. Component factors. What caution is given? Give explanation. Rule, first step? Second?

that product by another, and so on until all the factors have been used successively; the last product will be the product required.

NOTE. The product of any number of factors will be the same in whatever order they are multiplied. Thus, 4 X 3 X 560, and 5 X 4 X 360.

Ans. 36380250.

7. Multiply 291042 by 125. 8. If a vessel sail 436 miles in 1 day, how far will she sail in 56 days? Ans. 24416 miles. 9. How much will 72 acres of land cost, at 124 dollars an acre? Ans. 8928 dollars. 10. There are 5280 feet in a mile; how many feet in 84 miles? Ans. 443520 feet. 11. What will 120 yoke of cattle cost, at 125 dollars a yoke?

CASE II.

68. When the multiplier is 10, 100, 1000, &c.

If we annex a cipher to the multiplicand, each figure is removed one place toward the left, and consequently the value of the whole number is increased tenfold, (32.) 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. 69. Hence the following

RULE. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier; the number so formed will be the product required.

Case II is what? Give explanation. Rule?

6. If 1 bushel of corn cost 65 cents, what will 1000 bushels Ans. 65000 cents.

cost?

CASE III.

70. When there are ciphers at the right hand of one or both of the factors.

1. Multiply 1200 by 60.

OPERATION.

Multiplicand, 1200

Multiplier,

Product,

60 72000

ANALYSIS, Both multiplicand and multiplier may be resolved into their component factors; 1200 into 12 and 100, and 60 into 6 and 10. If these several factors be multiplied together they will produce the same product as the given numbers, (67.) Thus, 12 × 672, and 72 X 100 = 7200, and 7200 X 10-72000, which is the same result as in the operation. Hence the following

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.

4. Multiply 70340 by 800400.

5. Multiply 3400900 by 207000.

Ans. 56300136000.

Ans. 703986300000.

6. Multiply 634003000 by 40020. Ans. 25372800060000. 7. Multiply 10203070 by 50302000.

Ans. 513234827140000. 8. Multiply 30090800 by 600080. Ans. 18056887264000. 9. Multiply eighty million seven thousand six hundred, by eight million seven hundred sixty. Ans. 640121605776000. 10. Multiply fifty million ten thousand seventy, by sixtyfour thousand. Ans. 3200644480000. 11. Multiply ten million three hundred fifty thousand one hundred, by eighty thousand nine hundred.

Ans. 837323090000. 12. There are 296 members of Congress, and each one receives a salary of 3000 dollars a year; how much do they all receive?

EXAMPLES COMBINING ADDITION, SUBTRACTION, AND

MULTIPLICATION.

1. Bought 45 cords of wood at 4 dollars a cord, and 9 loads of hay at 13 dollars a load; what was the cost of the wood and hay? Ans. 297 dollars.

2. A merchant bought 6 hogsheads of sugar at 31 dollars a hogshead, and sold it for 39 dollars a hogshead; how much did he gain?

3. Bought 288 barrels of flour for 1875 dollars, and sold the same for 9 dollars a barrel; how much was the gain? Ans. 717 dollars.

4. If a young man receive 500 dollars a year salary and pay 240 dollars for board, 125 dollars for clothing, 75 dollars for books, and 50 dollars for other expenses, how much will he have left at the end of the year? Ans. 10 dollars.

5. A farmer sold 184 bushels of wheat at 2 dollars a bushel, for which he received 67 yards of cloth at 4 dollars a yard, and the balance in groceries; how much did his groceries cost him?

6. A sold a farm of 320 acres at 36 dollars an acre; B sold one of 244 acres at 48 dollars an acre; which received the greater sum, and how much? Ans. B, 192 dollars.

7. 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. 8. A merchant tailor 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.

9. If a man have an income of 3700 dollars a year, and his daily expenses be 4 dollars; what will he save in a year, or 365 days? Ans. 2240 dollars.

10. A man sold three houses; for the first he received 2475 dollars, for the second 840 dollars less than he received for the first, and for the third as much as for the other two; how much did he receive for the three? Ans. 8220 dollars. 11. A man sets out to travel from Albany to Buffalo, a distance of 336 miles, and walks 28 miles a day for 10 days; how far is he from Buffalo?

12. Mr. C bought 14 cows at 23 dollars each, 7 horses at 96 dollars each, 34 oxen at 57 dollars each, and 300 sheep at 2 dollars each; he sold the whole for 3842 dollars; how much did he gain? Ans. 310 dollars.

13. A drover bought 164 head of cattle at 36 dollars a head, and 850 sheep at 3 dollars a head; how much did he pay for all?

14. A banker has an income of 14760 dollars a year; he pays 1575 dollars for house rent, and four times as much for family expenses; how much does he save annually?

Ans. 6885 dollars.

15. A flour merchant bought 936 barrels of flour at 9 dollars a barrel; he sold 480 barrels at 10 dollars a barrel, and the remainder at 8 dollars a barrel; how much did he gain or lose? Ans. Gained 24 dollars.