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16. What cost 25384 tons of hay at 9 dollars per ton ?
Ans. 228456 dollars. 17. If on 1 page in this book there are 2538 letters, how many are there on 11 pages ?
Ans. 27918 letters.
37. When the multiplier exceeds 12.
Ex. 1. Let it be required to multiply 763 by 24.
OPERATION. We write the multiplier under the Multiplicand 7 6 3 multiplicand, and proceed to multiply Multiplier 24
the multiplicand by 4, the unit figure of
the multiplier, as in Art. 36. We then, 3 0 5 2 in like manner, multiply the multiplicand 15 2 6 by the 2 tens in the multiplier, taking
care to write the first figure obtained by Product 18 312
this multiplication in tens' column, di
rectly under the 2 of the multiplier; and, adding the partial products obtained by the two multiplications, we find the whole product of 763 multiplied by 24 to be 18312.
38. RULE. — Write the multiplier under the multiplicand, arranging units under units, tens under tens, &c.
If the multiplier is one figure, multiply each figure of the multiplicand in succession, beginning with the units' figure, by the multiplier, writing the right-hand figure of each product under the figure multiplied, and adding the left-hand figure, if any, to the succeeding product; but observing to write down all the figures of the last product.
If the multiplier contains more than one figure, multiply by each figure separately, writing its product in a separate line, and observing to place the right-hand figure of each line under the figure by which you multiply.
The sum of the several products will be the whole product required.
NOTE. — When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the significant figures only.
37. How do you proceed when the multiplier exceeds 12? Where do you set the first figure of each partial product ?' Why? How is the true product found ? — 38. The general rule for multiplication? When there are ciphers between the significant figures of the multiplier, how do you proceed?
39. First Method of Proof. — Multiply the multiplier by the multiplicand, and if the result is like the first product, the work is supposed to be right.
Note. — This proof depends on the principle, That, when two or more numbers are multiplied together, the product is the same, whatever the order of multiplying them.
Note. — The common mode of proof in business is to divide the product by the multiplier, and, if the work is right, the quotient will be like the multiplicand. This mode of proof anticipates the principles of division, and therefore cannot be employed without a previous knowledge of that rule.
40. Second Method of Proof. — Beginning at the left hand of the multiplicand, add together its successive figures towards the right till the sum obtained equals or exceeds nine. Omit- the nine, and carry the excess, if any, to the next figure. Proceed in this way till all the figures in the multiplicand have been added, and write the final excess at the right hand of the multiplicand.
Proceed in a like manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together, and place the excess of nines in their product at the right.
Find the excess of nines in the product obtained by the original operation ; and, if the work is right, the excess thus found
39. How is multiplication proved by the first method ? What is the rea. son for this method ? What is the common mode of proof in business ?. 40. What is the second method of proving multiplication ?
will be equal to the excess contained in the product of the excesses of the multiplicand and multiplier.
2 2 31 8 excess.
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.
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.
40. Is this method of proof always a true test of the correctness of an operation ? The reason for this method of proof?
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 days?
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 hundred 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 sixteen.
34. Multiply forty-seven thousand and thirteen by eighty thousand eight hundred and seven. Ans. 3798979491.
41. A Composite number is one produced by multiplying together two or more whole numbers greater than unity or one ; thus, 12 is a composite number, since it is the product of 3 X 4; and 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 whole' numbers greater than unity, which, being multiplied together, produce that number; thus, 3 and 4 are factors of 12, since 3 x 4 = 12.
42. To multiply by a composite number.
Ex. 1. Bought 15 pieces of broadcloth, at 96 dollars per piece; how much did I pay for the whole ? Ans. 1440 dollars.
The factors of 15 are 3 and 9 6 dolls., price of 1 piece. 5. Now, if we multiply the 3
price of one piece by the factor
3, we get the price of 3 pieces ; 28 8 dolls., price of 3 pieces. price of 3 pieces by the factor
and then, by multiplying the 5
5, we obtain the price of 15
pieces, the number bought, since 1 4 4 0 dolls., price of 15 pieces. 15 is equal to five times 3.
RULE. - Multiply the multiplicand by one of the factors of the mul- . tiplier, and the product thus obtained by another, and so on until each of the factors has been used as a multiplier. 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, 3 X 4 : 12, and 4 X 3 = 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. 7. If in one year there are 8766 hours, how many hours in 72 years?
Ans. 631152 hours.
41. What is a composite number? A factor of any number ? — 42. What are the factors of 15 ? How do we multiply by a composite number? The rule? In what order may the factors of a composite number be multiplied ?