7 hundreds, and subtract the 7 hundreds from 9 hundreds, and 2 hundreds remain. By adding the 10 tens to the minuend and the 1 hundred to the subtrahend, the two numbers being equally increased, their difference is not changed. (Art . 24, Ax. 9.) The remainder is 262. Note. — The addition of 10 to the minuend is sometimes called borrowing 10, and the addition of 1 to the subtrahend is called carrying 1. Rule. — Place the less number under the greater, so that units of the same order shall stand in the same column. Commencing at the right hand, subtract each figure of the subtrahend from the figure above it. If any figure of the subtrahend ii larger than the figure above it in the minuend, add 10 to that figure of the minuend before subtracting, and then add 1 to the next figure of the subtrahend. 51. First Method of Proof.—Add the remainder and the subtrahend together, and their sum will be equal to the minuend, if the work is right. , This method of proof depends on the principle, that the greater of any two numbers is equal to the less added to the difference between them. 52. 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. OPERATION, OPERATION AND PROOF. OPERATION. OPERATION AND PROOra Minuend 469 469 788 788 Subtrahend 183 183 369 369 4. 5. 6. 7. Miles. Gallons. Minutes. Pecks. From 7654 7116 6178 4567 Take 1978 1997 1769 1978 28. What is the value of 6767851 — 81715? 29. What is the value of 761619161 — 916781? 30. What is the value of 31671675 — 361784? 31. What is the value of 16781321 — 100716? 32. What is the value of 1002007000 — 5971621? 33. Sir Isaac Newton was born in the year 1642, he died in 1727; how old was he at the time of his and de cease? Ans. 85 years. 34. Gunpowder was invented in the year 1330; how long was this before the invention of printing, which was in 1440? Ans. 110 years. 35. The mariner's compass was invented in Europe in the year 1302; how long was this before the discovery of America by Columbus, which happened in 1492? Ans. 190 years. 36. What number is that, to which if 6956 be added, the 37. A man bought an estate for seventeen thousand five hundred and sixty-five dollars, and sold it for twenty-nine thousand three hundred and seventy-five dollars. Did he gain 38. Bonaparte was declared emperor in 1804; how many years since? 39. The union of the government of England and Scotland was in the year 1603; how long was it from this period to 1776, the time of the declaration of the independence of the United States? Ans. 173 years. 40. Jerusalem was taken and destroyed by Titus in the year 70; how long was it from this period to the time of the first Crusade, which was in the year 1096? Ans. 1026 years. 41. From the Creation to the Deluge was 1656 years; thence to the founding of Rome 1595 years; thence to the death of Charlemagne, which took place 814 years after Christ, 1567 years. In what year of the world was Christ born? Ans. 4004. 42. A gentleman 83 years old has two sons; the age of the older son added to his makes 128 years, and the age of the younger son is equal to the difference between the age of the father and that of the older son. How old is each of his sons? Ans. The older, 45 years; the younger, 38 years. 43. During the year 1810, there were manufactured in the United States one hundred and forty-six thousand nine hundred and seventy-four yards of cotton cloth; and during the year 1855, five hundred and twenty million yards. What was the increase? Ans. 519,853,026 yards. 53. Method of subtracting, when there are two or more subtrahends. Ex. 1. From a pile of wheat containing 657 bushels, A is to have 141 bushels, B 244 bushels, C 134 bushels, and D the remainder. How many bushels is D to have? Ans. 138. sum will be one million? Ans. 993044. or lose, and how much? Ans. Gained $11810. Minuend FIR3T OPERATION. Bushels. 65 7 m 244 Minuend Bushels. 657 1 IT 244 134 ». In the first operation, the several subtrahends are added for a single subtrahend to be taken from the minuend. In the second, the subtrahends are subtracted as then are added, at one operation, thus: beginning with units, 4 and 4 and 17 units leaves 8 units; passing to tens, 1 4 and 4 = 12 tens; reserving the left-hand figure to add in with the figures of the subtrahends in the next column, the right-hand figure, 2 tens, which we subtract from the 5 tens of the minuend, and have left 3 tens; and, passing to hundreds, we add in the left-hand figure 1, reserved from the 12 tens, which with the other figures 1 and 2 and 1 = 5 hundreds, which, taken from 6 hundreds, leaves 1 hundred; and 138 is the answer sought. Subtrahend 5 19 Remainder 13 8 1 = 9, which from (carried) and 3 and 2. John Drew has a yearly income of 2,500 dollars; his family expenses are 1,300 dollars, his expenditures in improving his estate 450 dollars, and his contributions to several worthy objects 225 dollars. What remains to lay up or invest? 3. A speculator bought four village lots; for the first he paid 620 dollars; for the second, 416 dollars; for the third, 350 dollars; for the fourth, 225 dollars; and sold the whole for 2,000 dollars. What did he gain? 4. Daniel White, dying, left property to the amount of 27,563 dollars, of which his wife received 9,188 dollars, each of his two daughters, 4,594 dollars, and his only son the balance. What did his son receive? Ans. 9,187 dollars. 5. The United States contain 2,983,153 square miles, of which the Atlantic slope includes 967,576, the Pacific slope 778,266, and the Mississippi Valley the remainder. How many square miles does the Mississippi Valley contain? Ans. 1,237,311. 6. The British North American Provinces contain 3,125,401 square miles; of which 147,832 square miles belong to Canada West; 201,989 to Canada East; 27,700 to New Brunswick; 18,746 to Nova Scotia; 2,134 to Prince Edward's Island; 57,000 to Newfoundland; 170,000 to Labrador; and the remainder to the Hudson's Bay Territory. What number of square miles belong to the Hudson's Bay Territory? Ans. 2,500,000. 7. James Howe has property to the amount of 63,450 dollars, and owes in all three debts; one of 1000 dollars, another of 350 dollars, and another of 12,468 dollars. How much has he after paying his debts? 8. The entire coinage of the mint of the United States, including the coinage of its branches, from 1792 to 1856, amounted in value to $ 498,197,382,.of which $396,895,574 was gold, $ 100,729,602 silver, and the remainder of the amount copper. What was the value of the copper coinage? Ans. $ 572,206. MULTIPLICATION. 54. Multiplication is the process of taking one numbe* as many times as there are units in another number. In multiplication three terms are employed, called the Multiplicand, the Multiplier, and the Product. The multiplicand is the number to be multiplied or taken. The multiplier is the number by which we multiply, and denotes the number of times the multiplicand is to be taken. The product is the result, or number produced by the multiplication. The multiplicand and multiplier together are called Factors, from the product being made or produced by them. When the multiplicand consists of a simple number, the process is termed Multiplication of Simple Numbers. In the following table, the invention of Pythagoras, may be found all the elementary products necessary in performing any operation in multiplication, since the multiplication of numbers, however large, depends upon the product of one digit by another. The products, therefore, of each digit by any other, should be thoroughly committed to memory. Considerable more of the" table, even, may be memorized with fully compensating results. |