many remain ? OPERATION 10. A gentleman gave his son 3692 dollars, and his daughter 1212 dollars less than his son ; how much did his daughter receive ? Ans. 2480 dollars. Art. 30. Method of operation when any figure in the subtrahend is greater than the figure above it in the minuend. Ex. 1. If I have 624 dollars, and lose 342 of them, how Ans. 282. In performing this example, we first take Minuend 6 24 the 2 units from the 4, and find the difference Subtrahend 342 to be 2, which we write directly under the figure subtracted We then proceed to take Remainder 282 the 4 tens from the 2 tens above it; but we here find a difficulty, since the 4 is greater than 2, and cannot be subtracted from it. We therefore add 10 to the 2, which makes 12, and then subtract 4 from 12 and 8 remains, which we write directly below. Then, to compensate for the 10 thus added to the 2 in the minuend, we add one to the 3 in the next higher place in the subtrahend, which makes 4, and subtract 4 from 6, and 2 remains. The remainder, therefore, is 282. The reason of this operation depends upon the self-evident truth, That, if any two numbers are equally increased, their difference remains the same. In this example 10 tens, equal to 1 hundred, were added to the 2 tens in the upper number, and I was added to the 3 hundreds in the lower number. Now, since the 3 stands in the hundreds' place, the 1 added was in fact 1 hundred. Hence, the two numbers being equally increased, the difference is the same. Note. – This addition of 10 to the minuend is sometimes called bor. rowing 10, and the addition of 1 to the subtrahend is called carrying 1. Art. 31. From the preceding examples and illustrations in subtraction, we deduce the following general RULE. — 1. Place the less number under the greater, units under units, tens under tens, fc., and draw a line under them. 2. Then, commencing with the units, subtract each figure of the subtrahend from the figure above it in the minuend, and write the difference below. QUESTIONS. Art. 30. How do you proceed when a figure of the subtrabend is larger than the one above it in the minuend? How do you compensate for the 10 which is added to the minuend? What is the reason for this addition to the minuend and subtrahend? How does it appear that the 1 added to the subtrahend equals the 10 added to the minuend? What is the addition of 10 to the minuend sometimes called? The addition of 1 to the subtrahend ? -- Art. 31. What is the general rule for subtraction ? 3. If any figure of the subtrahend is larger than the figure above it in the minuend, add 10 to that figure of the minuend, and from their sum subtract the lower figure; then carry I to the next figure of the subtrahend, and subtract as before, till all the figures of the subtrahend are subtracted, and the result will be the difference or remainder. Art. 32. Second Method of Proof. Subtract the remainder or difference from the minuend, and the result will be like the subtrahend if the work is right. This method of proof depends on the principle, That the smaller of any two numbers is equal to the remainder obtained by subtracting their difference from the greater. EXAMPLES FOR PRACTICE. 3. OPERATION. Minuend 376 OPERATION AND PROOF. OPERATION. OPERATION AND PROOF. 531 389 00000000 14001 14. From 671111 take 199999. Ans. 471112. 15. From 1789100 take 808088. Ans. 981012. 16. From 1000000 take 999999. Ans. 1. 17. From 9999999 take 1607. Ans. 9998392. 18. From 6101507601061 take 3806790989. Ans. 6097700810072. QUESTIONS. — Art. 32. What is the second method of proving subtraction ? What is the reason for this method of proof, or on what principle does it depend? 19. From 8054010657811 take 76909748598. Ans. 7977100909213. 20. From 7100071641115 take 10071178. Ans. 7100061569937. 21. From 501505010678 take 794090589. 501502 Ans. 500710920089. 22. Take 99999999 from 100000000. Ans. 1. 23. Take 44444444 from 500000000. Ans. 455555556. 24. Take 1234567890 from 9987654321. Ans. 8753086431. 25. From 800700567 take 1010101. Ans. 799690466. 26. Take twenty-five thousand twenty-five from twenty-five millions. Ans. 24974975. 27. Take nine thousand ninety-nine from ninety-nine thousand. Ans. 9901. 28. From one hundred one millions ten thousand one hun. dred one take ten millions one hundred one thousand and ten. Ans. 90909091. 29. From one million take nine. Ans. 999991. 30. From three thousand take thirty-three. Ans. 2967. 31. From one hundred millions take five thousand. Ans. 99995000. 32. From 1,728 dollars, I paid 961 dollars; how many remain ? Ans. 767 dollars. 33. Our national independence was declared in 1776; how many years from that period to the close of the last war with Great Britain in 1815 ? Ans. 39 years. 34. The last transit of Venus was in 1769, and the next will be in 1874; how many years will intervene? Ans. 105 years. 35. In 1830, the number of inhabitants in Bradford was 1,856, and in 1840 it was 2,222; what was the increase ? Ans. 366. 36. How many more inhabitants were there in New York city than in Boston, in 1840, there being, by the census of that year, 312,710 inhabitants in the former, and -93,383 in the latter city ? Ans. 219,327 inhabitants. 37. In 1821 there were imported into the United States 21,273,659 pounds of coffee, and in 1839, 106,696,992 pounds; what was the increase ? Ans. 85,423,333 pounds. 38. By the census of 1840, 11,853,507 bushels of wheat were raised in New York, and 13,029,756 bushels in Pennsylvania ; how many bushels in the latter State more than in the former? Ans. 1,176,249 bushels. 39. The real estate of James Dow is valued at 3,769 dollars, and his personal estate at 2,648 dollars; he owes John Smith 1,728 dollars, and Job Tyler 1,161 dollars; how much is Dow worth? Ans. 3,528 dollars. 40. If a man receive 5 dollars per day for labor, and it cost him 2 dollars per day to support his family, what will he have accumulated at the close of one week ? Ans. 18 dollars. 41. The city of New York owes 9,663,269 dollars, and Boston owes 1,698,232 dollars; how much more does New York owe than Boston ? Ans. 7,965,037 dollars. 42. From five hundred eighty-one thousand take three thousand and ninety-six. Ans. 577,904. 43. E. Webster owns 6,765 acres of land, and he gave to his oldest brother 2,196 acres, and his uncle Rollins 1,981 acres ; how much has he left ? Ans. 2,588 acres. 44. It was ascertained by a transit of Venus, June 3, 1769, that the mean distance of the earth from the sun was ninetyfive millions one hundred seventy-three thousand one hundred twenty-seven miles, and that the mean distance of Mars from the sun was one hundred forty-five millions fourteen thousand one hundred forty-eight miles. Required the difference of their distances from the sun. Ans. 49,841,021 miles. S IV. MULTIPLICATION. MENTAL EXERCISES. ART. 33. When any number is to be added to itself several times, the operation may be shortened by a process called Multiplication. Ex. 1. If a man can earn 8 dollars in 1 week, what will he earn in 4 weeks? ILLUSTRATION. - It is evident, if a man can earn 8 dollars in 1 week, in 4 weeks he will earn 4 times as much, and the result may be obtained by addition ; thus, 8 +8+8+8 = 32; or, by a more convenient process, by setting down the 8 but once, and multiplying it by 4, the number of times it is to be repeated; thus, 4 times 8 are 32. Hence in 4 weeks he will earn 32 dollars. The following table must be thoroughly committed to memory before any considerable progress can be made in this rule: MULTIPLICATION TABLE. 1 are 6 times 8 9 times l are 9 6 times 2 are 12 7 times 2 are 14 8 times 2 are 16 9 times 2 are 18 6 times 3 are 18 7 times 3 are 21 8 times 3 are 24 9 times 3 are 27 6 times 4 are 24 7 times 4 are 28 8 times 4 are 9 times 4 are 36 6 times 5 are 30 7 times 5 are 35 8 times 5 are 401 9 times 5 are 45 6 times 6 are 36 7 times 6 are 42 8 times 6 are 45 9 times 6 are 51 6 times 7 are 42 7 times 7 are 49 8 times 7 are 56 9 times 7 are 63 6 times 8 are 48 7 times 8 are 56 8 times 8 are 61 9 times 8 are 72 6 times 9 are 54 7 times 9 are 63 8 times 9 are 72 9 times 9 are 81 6 times 10 are 60 7 times 10 are 70 8 times 10 are 80 9 times 10 are 90 6 times 11 are 66 7 times 11 are 77 8 times 11 are 85 99 6 times 12 are 72 7 times 12 are 84 8 times 12 are 96 9 times 12 are 108 10 times I are 10 10 times 11 are 110 11 times 8 are 8812 times 4 are 48 10 times 2 are 2010 times 12 are 120 11 times 9 are 99 12 times 5 are 60 10 times 3 are 30 11 times 10 are 110 12 times 6 are 72 10 times 4 are 40 11 times l are 11 11 times 11 are 121 12 times 7 are 84 10 times 5 are 5011 times 2 are 22 11 times 12 are 132 12 times 8 are 96 10 times 6 are 60 u times 3 are 33 12 times 9 are 108 10 times 7 are 70 | 11 times 4 are 44 12 times l are 12 12 times 10 are 120 10 times 8 are 80/11 times 5 are 55 12 times 2 are 24 12 times 11 are 132 10 times 9 are 90 11 times 6 are 66 12 times 3 are 36 12 times 12 are 144 10 times 10 are 100 11 times 7 are 77 2. What cost 5 barrels of flour at 6 dollars per barrel ? ILLUSTRATION. — If i barrel of flour cost 6 dollars, 5 barrels will cost 5 times as much ; 5 times 6 are 30. Hence 5 barrels of flour at 6 dollars per barrel will cost 30 dollars. 3. What cost 6 bushels of beans at 2 dollars per bushel ? 4. What cost 5 quarts of cherries at 7 cents per quart ? 5. What will 7 quarts of vinegar cost at 12 cents per quart ? 6. What cost 9 acres of land at 10 dollars per acre ? 7. If a pint of currants cost 4 cents, what cost 9 pints? 8. If in 1 penny there are 4 farthings, how pence? In 7 pence? In 8 pence? In 4 pence?' In 3 many in 9 pence ? 9. If 12 pence make a shilling, how many pence in 3 shillings? In 5 shillings? In 7 shillings? In 9 shillings? |