EXERCISES FOR THE SLATE. ART. 36. Method of operation when the multiplier does not exceed 12. Ex. 1. Let it be required to multiply 175 by 7. OPERATION. Multiplicand 175 Product 7 1225 Ans. 1225. Having written the multiplier under the unit figure of the multiplicand, we multiply the 5 units by 7, obtaining 35, and set down the 5 units directly under the 7, and reserve the 3 tens for the tens' column. We then multiply the 7 tens by 7, obtaining 49, and, adding the 3 tens which were reserved, we have 52 tens, or 5 hundreds and 2 tens. Writing down the 2 tens, and reserving the 5 hundreds, we multiply 1 by 7; and, adding the reserved 5 hundreds, we have 12 hundreds, which we write down in full, and the product is 1225. 9. Multiply 767853 by 9. 10. Multiply 876538765 by 8. 11. Multiply 7654328 by 7. 12. Multiply 4976387 by 5. 13. Multiply 8765448 by 12. 14. Multiply 4567839 by 11. Ans. 7012310120. Ans. 24881935. Ans. 105185376. Ans. 50246229. 15. What cost 8675 barrels of flour at 7 dollars per barrel ? Ans. 60725 dollars. QUESTIONS. Art. 36. How must numbers be written for multiplication? At which hand do you begin to multiply? Why? Where do you write the first or right-hand figure of the product of each figure in the multiplicand? Why? What is done with the tens or left-hand figure of each product? How, then, do you proceed when the multiplier does not exceed 12? 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. ART. 37. Method of operation when the multiplier ex ceeds 12. Ex. 1. Let it be required to multiply 763 by 24. OPERATION. Multiplicand 763 3052 1526 Product 18312 Ans. 18312. We write the multiplicand and multiplier as before, and proceed to multiply the multiplicand by 4, the unit figure of the multiplier, precisely as in Art. 36. We then, in like manner, multiply the multiplicand by the 2 tens in the multiplier, taking care to write the first figure obtained by this multiplication in tens' column, directly under the 2 of the multiplier; and, adding together these partial products obtained by the two multiplications, and placed as in the operation, we have the full product of 763 multiplied by 24, which is 18312. ART. 38. The preceding examples sufficiently illustrate the principle and method of multiplication; hence the following general RULE. Write the multiplier under the multiplicand, arranging units under units, tens under tens, &c. Multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, writing underneath the right-hand figure of each product, and adding the left-hand figure or figures, if any, to the succeeding product. If the multiplier consists of more than one figure, the right-hand figure of each partial product must be placed directly under the figure of the multiplier that produces it. The sum of the partial 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, remembering to set the first figure of the product directly under the figure of the multiplier that produces it. QUESTIONS. Art. 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? Art. 38. What is the general rule for multiplication? When there are ciphers between the significant figures of the multiplier, how do you proceed? ART. 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. The reason of 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. ART. 40. Second Method of Proof. Begin at the left hand of the multiplicand, and add together its successive figures toward the right till the sum obtained equals or exceeds the number nine. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to, or greater than, nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand. Proceed in the same 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. Then proceed to find the excess of nines in the product obtained by the original operation; and, if the work is right, QUESTIONS. Art. 39. How is multiplication proved by the first method? What is the reason for this method of proof? What is the common mode of proof in business? -Art. 40. What is the second method of proving multiplication the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier. 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. Ans. 2813 dollars. 9. What cost 97 oxen at 29 dollars each? 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? 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 cach P 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? 18. Multiply 675 by 476. Ans. 68816 pounds. Ans. 321300. Ans. 518077. Ans. 881919. Ans. 9691836. Ans. 18219071. Ans. 70287492. Ans. 153288487686. Ans. 49062139937803. Ans. 3831635. 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. 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. 30. Multiply ninety thousand eight one thousand and ninety-one. 31. Multiply ninety thousand eight nine thousand one hundred and six. 32. Multiply fifty thousand and one hundred and seven. 33. Multiply eighty thousand and nine sixteen. Ans. 264640056. hundred and seven by Ans. 99070437. hundred and seven by Ans. 826888542. by five thousand eight Ans. 290355807. by nine thousand and Ans. 721361144. |