9. Multiply 32746 by 5. 10. Multiply 840371 by 7. 11. Multiply 137629 by 8. 12. Multiply 93762 by 3. 13. Multiply 543272 by 4. 14. Multiply 703164 by 9.

Ans. 163730.
Ans. 5882597.
Ans. 1101032.

Ans.

281286.

Ans. 2173088.

Ans. 6328476.

15. What will be the cost of 344 cords of wood at 4 dollars a cord? Ans. 1376.

16. How much will an army of 7856 men receive in one week, if each man receive 6 dollars? Ans. 47136 dollars. 17. In one day are 86400 seconds; how many seconds in 7 days? Ans. 604800 seconds. 18. What will 7640 bushels of wheat cost, at 9 shillings a bushel? Ans. 68760 shillings. 19. At 5 dollars an acre, what will 2487 acres of land cost? Ans. 12435 dollars. 20. In one mile are 5280 feet; how many feet in 8 miles? Ans. 42240 feet.

ANALYSIS. Writing the multiplicand and multiplier as in Case I, we first multiply each figure in the multiplicand by the unit figure of the multiplier, precisely as in 2 tens times 6 units, or 6

Case I. We then multiply by the 2 tens. times 2 tens, are 12 tens, equal to 1 hundred, and 2 tens; we place the 2 tens under the tens figure in the product already obtained, and add the 1 hundred to the next hundreds produced. 2 tens times 4 tens are 8 hundreds, and the 1 hundred of the last product added are 9 hundreds; we write the 9 in hundreds' place in the product. 2 tens

Case II is what? Give explanation.

times 7 hundreds are 14 thousands, equal to 1 ten thousand and 4 thousands, which we write in their appropriate places in the product. Then adding the two products, we have the entire product, 17158.

NOTES. 1. When the multiplier contains two or more figures, the several results obtained by multiplying by each figure are called partial products.

2. When there are ciphers between the significant figures of the multiplier, pass over them, and multiply by the significant figures only.

63. From the preceding examples and illustrations we deduce the following general

RULE. I. Write the multiplier under the multiplicand, placing units of the same order under each other.

II. Multiply the multiplicand by each figure of the multiplier successively, beginning with the unit figure, and write the first figure of each partial product under the figure of the multiplier used, writing down and carrying as in addition.

III. If there are partial products, add them, and their sum will be the product required.

64. PROOF. 1. Multiply the multiplier by the multiplicand, and if the product is the same as the first result, the work is correct. Or,

2. Multiply the multiplicand by the multiplier diminished by 1, and to the product add the multiplicand; if the sum be the same as the product by the whole of the multiplier, the work is correct.

What are partial products? When there are ciphers in the multiplier, how proceed? Rule, first step? Second? Third? Proof, first method? Second?

(5.) 7648 328

(6.) 81092

194

(7.) 37967

426

8. How many yards of linen in 759 pieces, each piece containing 25 yards? Ans. 18975 yards.

9. Sound is known to travel about 1142 feet in a second of time; how far will it travel in 69 seconds?

10. A man bought 36 city lots, at 475 dollars each; how much did they all cost him? Ans. 17100 dollars. 11. What would be the value of 867 shares of railroad stock, at 97 dollars a share? Ans. 84099 dollars. 12. How many pages in 3475 books, if there be 362 pages in each book? Ans. 1257950 pages.

13. In a garrison of 4507 men, each man receives annually 208 dollars; how much do they all receive?

19. Multiply 79094451 by 76095. Ans. 6018692248845. 20. Multiply five hundred forty thousand six hundred nine, by seventeen hundred fifty. Ans. 946065750. 21. Multiply four million twenty-five thousand three hundred ten, by seventy-five thousand forty-six.

Ans. 302083414260. 22. Multiply eight hundred seventy-seven million five hundred ten thousand eight hundred sixty-four, by five hundred forty-five thousand three hundred fifty-seven.

Ans. 478556692258448. 23. If one mile of railroad require 116 tons of iron, worth 65 dollars a ton, what will be the cost of sufficient iron to construct a road 128 miles in length? Ans. 965120 dollars.

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 com posite number, since 6 × 3 = 18; or, 9 × 2 = 18; or, 3 X 3 X 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 X 5 = 20;) or, 2 and 2 and × 5, (2 × 2 × 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 × 3 × 560, and 5 X 4 X 360.

EXAMPLES FOR PRACTICE.

2. Multiply 3472 by 486 X 8.
3. Multiply 14761 by 648 X 8.
4. Multiply 87034 by 81
5. Multiply 47326 by 120

6. Multiply 60315 by 96.

Ans. 166656.

3 X 3 X 9.

Ans. 7049754.

6 X 5 X 4.

Ans. 5700240.

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?