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the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier.

Ex. 3.

OPERATION.

Multiplicand 12345= 6 excess.
Multiplier

2231= 8 excess.
1 2 3 4 5 48=3
37035
24 690

Proof.
2 4 6 90
Product 275 416 95=

3 Note. - This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. If two or more figures in the work should be transposed, or the value of one figure be just as much too great as another is too small, or if a nine be set down in the place of a cipher, or the contrary, the excess of nines will be the same, and still the work may not be correct. Such a balance of errors will not, however, be likely to occur.

EXAMPLES FOR PRACTICE.
4.

5.
Multiply 67895

789 56
By
36

47
407370

552692 203685

315824 Ans. 2444220

3710932 6.

7.
Multiply 89325

47896
By
901

2008
89 3 2 5

383168 803 925

9 5792 Ans. 80 481825

96175168 8. What cost 47 hogsheads of molasses at 13 dollars per hogshead?

Ans. 611 dollars. 9. What cost 97 oxen at 29 dollars each?

Ans. 2813 dollars.

QUESTIONS. - Is this method of proof always a true test of the correctness of an operation? What is the reason for this method of proof?

days?

10. Sold a farm containing 367 acres, at 97 dollars per acre; what was the amount ?

Ans. 35599 dollars. 11. An army of 17006 men receive each 109 dollars as their annual pay; what is the amount paid the whole army?

Ans. 1853654 dollars. 12. If a mechanic deposit annually in the Savings Bank 407 dollars, what will be the sum deposited in 27 years ?

Ans. 10989 dollars. 13. If a man travel 37 miles in 1 day, how far will he travel in 365 days?

Ans. 13505 miles. 14. If there be 24 hours in 1 day, how many hours in 365

Ans. 8760 hours. 15. How many gallons in 87 hogsheads, there being 63 gallons in each ?

Ans. 5481 gallons. 16. If the expenses of the Massachusetts Legislature be 1839 dollars per day, what will be the amount in a session of 109 days?

Ans. 200451 dollars. 17. If a hogshead of sugar contains 368 pounds, how many pounds in 187 hogsheads ?

Ans. 68816 pounds. 18. Multiply 675 by 476.

Ans. 321300. 19. Multiply 679 by 763.

Ans. 518077. 20. Multiply 899 by 981.

Ans. 881919. 21. Multiply 7854 by 1234.

Ans. 9691836. 22. Multiply 3001 by 6071.

Ans. 18219071. 23. Multiply 7117 by 9876.

Ans. 70287492. 24. Multiply 376546 by 407091. Ans. 153288487686. 25. Multiply 7001009 by 7007867.

Ans. 49062139937803. 26. Multiply five hundred and eighty-six by nine hundred and. eight.

Ans. 532088. 27. Multiply three thousand eight hundred and five by one thousand and seven.

Ans. 3831635. 28. Multiply two thousand and seventy-one by seven hun. dred and six.

Ans. 1462126. 29. Multiply eighty-eight thousand and eight by three thousand and seven.

Ans. 264640056. 30. Multiply ninety thousand eight hundred and seven by one thousand and ninety-one.

Ans. 99070437. 31. Multiply ninety thousand eight hundred and seven by nine thousand one hundred and six. Ans. 826888542.

32. Multiply fifty thousand and one by five thousand eight hundred and seven.

Ans. 290355807. 33. Multiply eighty thousand and nine by nine thousand and sistcen.

Ans. 721361144.

OPERATION.

34. Multiply forty-seven thousand and thirteen by eighty thousand eight hundred and seven. Ans. 3798979491.

Art. 41. A COMPOSITE number is the product of two or more factors greater than unity or one; thus 12 is a composite number, since it is the product of 3 X 4; and also 24 is a composite number, since it is the product of 2 X 3 X 4.

A FACTOR of any number is a name given to one of two or more numbers, which, being multiplied together, produce that number; thus 3 and 4 are factors of 12, since 3 X 4 = 12.

Art. 42. To multiply by a composite number.

Ex. 1. A merchant bought 15 pieces of broadcloth at 96 dollars per piece ; how much did he pay for the whole ?

Ans. 1440 dollars.

The factors of 15 are 3 and 96 dolls., price of 1 piece.

5. Now if we multiply the

price of one piece by the factor 3

3, we get the cost of 3 pieces ; 288 dolls., price of 3 pieces.

and then, by multiplying the

cost of 3 pieces by the factor 5

5, it is evident we obtain the 1440 dolls., price of 15 pieces. cost of 15, the number of pieces

bought, since 15 is equal to 5 times 3. Hence we adopt the following

RULE. — Multiply the multiplicand by one of the factors of the multiplier, and this product by another, and so on until all the factors have been used as multipliers, and the last product will be the answer.

Note. The product of any number of factors is the same in whatever order they are multiplied. Thus, 3X4=12, and 4 X3= 12.

EXAMPLES FOR PRACTICE. 2. Multiply 30613 by 25 = 5 X 5.

Ans. 765325. 3. Multiply 1469 by 84 = 7 X 12. Ans. 123396. 4. Multiply 7546 by 81, using its factors. Ans. 611226. 5. Multiply 7901 by 125, using its factors. Ans. 987625.

6. In 1 mile there are 63360 inches; how many inches in 45 miles ?

Ans. 2851200 inches. 7. If in 1 year there are 8766 hours, how many hours in 72 years?

Ans. 631152 hours.

QUESTIONS. - Art. 41. What is a composite number? What is a factor of any number? - Art. 42. What are the factors of 15? How do we multiply by a composite number ? Repeat the rule. In what order must the factors of a composite number be multiplied ?

OPERATION.

8. If sound moves 1142 feet in a second, how far will it move in one minute ?

Ans. 68520 feet. Art. 43. To multiply by 10, 100, 1000, &c.

Ex. 1. In 1 day there are 24 hours; how many hours in 10 days? In 100 days ?

Answers 240, 2400 hours.

By a principle of notation, Multiplicand 24

24 the removal of a figure one Multiplier 10 100

place to the left increases its Product 240 2400

value ten times (Art. 9). If

then we annex a cipher to the Or thus, 240, 2400.

right of 24, the multiplicand, each figure is removed one place to the left, and its value is increased ten times, or multiplied by 10. If two ciphers are annexed, each figure is removed two places to the left, and its value is increased 100 times, or multiplied by 100; every additional cipher increasing the value ten times. Hence the propriety of the following

RULE. — Annex to the multiplicand all the ciphers of the multiplier, and the result will be the product required.

EXAMPLES FOR PRACTICE. 2. Multiply 2356 by 10.

Ans. 23560. 3. Multiply 5873 by 100.

Ans. 587300. 4. Multiply 7964 by 1000.

Ans. 7964000, 5. Multiply 98725 by 100000. Ans. 9872500000.

Art. 44. Method of operation when there are ciphers on the right hand of the multiplier or multiplicand, or both.

Ex. 1. What will 600 acres of land cost at 20 dollars per acre ?

Ans. 12,000 dollars.

In this example the multiplicand may be Multiplicand 600

resolved into the factors 6 and 100, and the Multiplier 20

multiplier into the factors 2 and 10. Now Product 12000

it is evident (Art. 42), if these several fac

tors be multiplied together, they will produce the same product as the original factors, 600 and 20. Thus 6 x 2 12, and 12 x 100 = 1200, and 1200 X 10 - 12000, the same result as in the operation. Hence the following

OPERATION.

QUESTIONS. Art. 43. What is the effect of removing a figure one place to the left? What is the effect of annexing a cipher to any figure or number? Two ciphers ? &c. What is the rule for multiplying by 10, 100, &c. ? Art. 44. How do you arrange the figures for multiplication, when there are ciphers on the right hand of either the multiplier or multiplicand, or both ? Why does multiplying the significant figures and annexing the ciphers produce the true product

Rule. - Set down the significant figures of the multiplier under those of the multiplicand, and multiply them together. Then annex as many ciphers to their product as there are on the right of the multiplicand and multiplier.

EXAMPLES FOR PRACTICE.

2. Multiply 8785324 By

3200 1757 0 648

26355972 Ans. 281130 36800

3. 713378900

70080 57070312 49936523 49993593312000

4. Multiply 8010700 by 9000909.

Ans. 72103581726300. 5. Multiply 700110000 by 700110000.

Ans. 490154012100000000. 6. Multiply 4070607 by 7007000.

Ans. 28522743249000. 7. Multiply 4110000 by 1017010.

Ans. 4179911100000. 8. Multiply twenty-nine millions two thousand nine hundred and nine by four hundred and four thousand.

Ans. 11717175236000. 9. Multiply eighty-seven millions by eight hundred thousand seven hundred.

Ans. 69660900000000. 10. Multiply one million one thousand one hundred by nine hundred nine thousand and ninety. Ans. 910089999000.

11. Multiply forty-nine millions and forty-nine by four hundred and ninety thousand. Ans. 24010024010000.

12. Multiply two hundred millions two hundred by two millions two thousand and two. Ans. 400400800400400.

13. Multiply four millions forty thousand four hundred by three hundred three thousand. Ans. 1224241200000.

14. Multiply three hundred thousand thirty by forty-seven thousand seventy.

Ans. 14122412100.

QUESTION. - What is the rule ?

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