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31. 57902468 X 5008. 32. 57902468 X 5080., 33. 57902468 X 5800. 34. 12481632 X 1509. 35. 79068025 X 1386. 36. 92948789 X 7043.

37. 58763718 X 6754.
38. 73084163 X 7584.
39. 144x144X144.
40. 3851X3851X3851.
41. 79094451 X 764094,
42. 89548050 X 972800.

CONTRACTIONS IN MULTIPLICATION.

94. The general rule is adequate to the solution of all examples that occur in multiplication. In many instances, however, by the exercise of judgment in applying the preceding principles, the operation may be very much abridged.

95. Any number which may be produced by multiplying two or more numbers together, is called a Composite Number.

Thus, 4, 15, 21, are composite numbers; for 4=2X2; 155 5X3; 21=7X3.

Obs. 1. The factors which, being multiplied together, produce a composite number, are sometimes called the component parts of the number.

2. The process of finding the factors of which a given number is composed, is called resolving the number into factors.

Ex. 1. Resolve 9, 10, 14, 22, into their factors.
2. What are the factors of 35, 54, 56, 63 ?
3. What are the factors of 45, 72, 64, 81, 96 ?

96. Some numbers may be resolved into more than two factors; and also into different sets of factors. Thus, 12=2X2X3; also 12=4X3=6X2.

4. What are the different factors and sets of factors of 8, 16, 18, 20,

24 ? 5. What are the different factors and sets of factors of 27, 32, 36, 40, 48?

96. a. We have seen that the product of any two numbers is the same, whichever factor is taken for the multiplier. (Art. 83.) In like manner, it may be shown that the product of any

three or

QUEST.-95. What is a composite number? Obs. What are the factors which produce it sometimes called? What meant by resolving a number into factors ? 96. Are numbers ever composed of more than two factors ? 96. a. When three or more factors are to be multiplied together, does it make any difference in what order they are taken?

more factors will be the same, in whatever order they are multiplied. For, the product of two factors may be considered as one number, and this may be taken either for the multiplicand, or the multiplier. Again, the product of three factors may be considered as one number, and be taken for the multiplicand, or the multiplier, &c. Thus, 24=3X2X2X2=6X2X2=12X2=6X4= 4X2X3=8X3.

CASE 1.-- When the multiplier is a composite number, 6. What will 27 bureaus cost, at 31 dollars apiece ?

Analysis.-Since 27 is three times as much as 9 ; that is, 27=9 X3, it is manifest that 27 bureaus will cost three times as much as 9 bureaus.

Operation. Dolls. 31 cost of 1 B. Having resolved 27 into the factors 9

9 and 3, we find the cost of 9 bureaus, Dolls. 279 cost of 9 B. then multiplying that by 3, we have 3

the cost of 27 bureaus. Dolls. 837 cost of 27 B.

7. What will 36 oxen cost, at 43 dollars per head ? Solution.-36=9X4; and 43 X 9 X4=1548 dolls. Ans. Or, 36=3X3X4; and 43X3X3 X4=1518 dolls. Ans. Hence, 97. To multiply by a composite number.

Resolve the multiplier into two or more factors ; multiply the multiplicand by one of these factors, and this product by another factor, and so on till you have multiplied by all the factors. The last product will be the answer required.

Obs. The factors into which a number may be resolved, must not be confounded with the parts into which it may be separated. (Art. 53.) The former have reference to multiplication, the latter to addition ; that is, factors must be multiplied together, but parts must be added together to produce the given number. Thus, 56 may be resolved into two factors, 8 and 7; it separated into two parts, 5 tens or 50, and 6. Now, 8X7=56, and 50+6=56.

8. What will 24 horses cost, at 74 dollars a head ?

may be

QUEST.-97. When the multiplier is a composite number, how do you procced ? Obs. "Vhat is the difference between the factors into which a number may he resolved, and the parts into which it may be separated ?

9. What cost 45 hogsheads of tobacco, at 128 dollars a hogs. head?

10. What cost 54 acres of land, at 150 dollars per acre ?

11. At 118 shillings per week, how much will it cost a family to board 49 weeks ?

12. If a man travels at the rate of 372 miles a day, how far will he travel in 64 days ?

13. At 163 dollars per ton, how much will 72 tons of lead cost ? 14. What cost 81 pieces of broadcloth, at 245 shillings apiece? 15. What cost 84 carriages, at 384 dollars apiece? CASE II.- When the multiplier is 1 with ciphers annexed to it.

98. It is a fundamental principle of notation, that each removal of a figure one place towards the left, increases its value ten times ; (Art. 36 ;) consequently, annexing a cipher to a number will increase its value ten times, or multiply it by 10; annexing two ciphers will increase its value a hundred times, or multiply it by 100; annexing three ciphers will increase it a thousand times, or multiply it by 1000, &c. Thus, 15 with a cipher annexed, becomes 150, and is the same as 15X10; 15 with two ciphers annexed, becomes 1500, and is the same as 15 X 100; 15 with three ciphers annexed, becomes 15000, and is the same as 15X1000, &c. Hence,

99. To multiply by 10, 100, 1000, &c.

Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus formed will be the product required.

Note.-To annex means to place after, or at the right hand.

16. What will 10 boxes of lemons cost, at 63 shillings per box? Ans. 630 shillings. 17. How

many bushels of corn will 465 acres of land produce, at 100 bushels per acre ?

Quest.–98. What is the effect of annexing a cipher to a number? Two ciphers ? Three ? Four? 99. How do you proceed when the multiplier is 10, 100, 1000, &c. ? Note. What is the meaning of the term annex ?

18. Allowing 365 days for a year, how many days are there in 1000 years

? 19. Multiply 153486 by 10000. 20. Multiply 3120467 by 100000. 21. Multiply 52690078 by 1000000. 22. Multiply 689063457 by 10000000. 23. Multiply 4946030506 by 100000000. 24. Multiply 87831206507 by 1000000000. 25. Multiply 67856005109 by 10000000000. CASE III.-When the multiplier has ciphers on the right hand. 26. What will 30 wagons cost, at 45 dollars apiece ? Note.--Any numbrr with ciphers on its right hand, is obviously a composite aumber; the significant figure or figures being one factor, and 1, with the given ciphers annexed to it, the other factor. Thus, 30 may be resolved into che factors 3 and 10. We may therefore first multiply by 3 and then by 10, by annexing a cipher as above.

Solution.—45X3=135, and 135x10=1350 dolls. Ans.

27. How many acres of land are there in 3000 farms, if each farm contains 475 acres ?

Analysis.---3000=3X1000. Now 475 X Operation. 3=1425; and adding three ciphers to this

475 product, multiplies it by 1000. (Art. 99.)

3 Hence,

Ans. 1425000 acres, 100. When there are ciphers on the right of the multiplier.

Multiply the multiplicand by the significant figures of the multiplier, and to this product annex as many ciphers, as are found on the right of the multiplier.

OBs. It will be perceived that this case combines the principles of the two preceding cases; for, the multiplier is a composite number, and one of its factors is 1 with ciphers annexed to it.

28. How much will 50 hogs weigh, at 375 pounds apiece?

29. If i barrel of flour weighs 192 pounds, how much will 500 barrels weigh?

30. Multiply 14376 by 25000.

QUEST.–100. When there are ciphers on the right of the multiplier, how do you pro teed ? Obe. What principles does this case combine ?

31. Multiply 350634 by 410000. 32. Multiply 4630425 by 6200000. CASE IV.- When the multiplicand has ciphers on the right hand 33. What will 37 ships cost, at 29000 dollars apiece ?

Analysis.-29000=29X1000. But the Operation. product of two or more factors is the same

29000 in whatever order they are multiplied.

37 (Art. 96. a.) We therefore multiply 29 203 by 37, and this product by 1000 by adding 87 three ciphers to it.

Ans. 1073000 dolls. Proof.—29000X37=1073000, the same as before. Hence, 101. When there are ciphers on the right of the multiplicand.

Multiply the significant figures of the multiplicand by the multiplier, and to the product annex as many ciphers, as are found on the right of the multiplicand.

Obs. When both the multiplier and multiplicand have ciphers on the right, multiply the significant figures together as if there were no ciphers, and to their product annex as many ciphers, as are found on the right of both factors.

34. Multiply 2370000 by 52.
35. Multiply 48120000 by 48.
36. Multiply 356300000 by 74.
37. Multiply 1623000000 by 89.
38. Multiply 540000 by 700.
Analysis.-.540000=54X10000, and

Operation. 700=7X100; we therefore multiply the

540000 significant figures, or the factors 54 and

700 7 together, (Art. 96. a,) and to this pro- Ans. 378000000 duct annex six ciphers. (Art. 99.)

39. Multiply 1563800 by 20000. 40. Multiply 31230000 by 120000. 41. Multiply 5310200 by 3400000. 42. Multiply 82065000 by 8100000. 43. Multiply 210909000 by 5100000. QUEST.-101. When there are ciphers on the right of the multiplicand, how proceed ? Obs. How, when there are ciphers on the right both of the multiplier and multiplicand ?

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