From the sum of each of the following questions, take the sum of the two first terms, and the remainder will be equal to the sum of the two last terms. From the sum of each of the following questions take the sum of the three middle terms, and the remainder will be equal to the sum of the two extreme terms. 42. The sum of two numbers is not altered by taking any amount from one of them, if the same amount be added to the other. For example The difference of any two numbers is not altered by increasing or diminishing one of them, if the other be increased or diminished to the same amount. For example 5841342=6183 5841 - 3425499 The student may satisfy himself, that the above are general principles by taking a number of examples at pleasure, and working them. At this stage, he may not be able, perhaps, to see the use of this, but he will find the advantage of having done so afterwards. PROMISCUOUS EXAMPLES FOR PRACTICE. 43. (1) Two persons commenced business in separate undertakings, the one with £5,600, and the other with £4,860. After trading for a year, the first lost £1,240; the other gained £1,240. What was the sum of their capitals at the beginning, and also at the end of the year? What was each person's capital at the end of the year? And what was the difference of their capitals at the beginning and at the end of the year? (2) Supposing two persons should begin business, the one with £5,480, the other with £3,640; and after trading a year, each should gain £562: what would be the difference of their capitals at the beginning and also at the end of the year? (3) Supposing two persons should begin business, the one with £3,800, and the other with £4,220; and that, after trading a year, each should lose £1,200: what would be the difference of their capitals at the beginning, and also at the end of the year? and what would be the sum of their capitals at the beginning and also at the end of the year? (4) Rome was founded by Romulus 753 years before Christ: how many years is it since the foundation of Rome to the present time (1835)? (5) The Romans invaded Britain 55 years before Christ, and quitted it A.D. 448: how long had they possession of the country? how long is it since they invaded it? and how long is it since they quitted it (1835)? (6) From London to York is 196 miles; from York to Newcastle 81 miles; from Newcastle to Berwick 64 miles, and from Berwick to Edinburgh is 59 miles: how many miles is it from London to Edinburgh? (7) From London to Buxton is 160 miles; from Buxton to Manchester is 25 miles; from Manchester to Lancaster is 53 miles; from Lancaster to Edinburgh is 159 miles. A person travelling from London to Edinburgh took ill at Manchester, and had to be left by the coach: how far had he to travel when he was sufficiently recovered to resume his journey, and how far is it between London and Edinburgh by this line of road? CONCLUDING OBSERVATIONS ON ADDITION AND SUBTRACTION. 44. ADDITION is used for the purpose of collecting into one sum the amount of several separate quantities; but it frequently happens that we want to know the amount of the same quantity several times repeated. For instance, should we want to know the amount of twenty repeated four times: this result we could certainly easily obtain by Addition, viz. 20+20+20+20=80; but suppose we should want to know the amount of twenty, twenty times repeated: to arrive at this result by Addition, would occupy a large space, and the operation would be exceedingly tedious. To obviate this length, and tediousness of process, we make use of MULTIPLICATION; the manner of performing which we shall treat of in the next chapter. 45. With regard to SUBTRACTION it is used for the purpose of showing the difference of two numbers, but we sometimes want to know how many times one number is contained in another one. Should we want to know how many times twenty are contained in eighty-this result we could certainly arrive at by a repeated subtraction of the number twenty from eighty, viz. 80-20-20-20-20-0, but the result is more easily and shortly attained by DIVISION; of the manner of performing which we shall also treat in the next chapter. 46. QUESTIONS FOR MENTAL EXERCISE. If the student be frequently exercised mentally by the following and similar questions, it will tend to make him ex pert in calculation. 16 and 8 are how many? | 48, 19, and 12, 18 6 do. less 36, are how many? ... do. 47. QUESTIONS FOR EXAMINATION. Are Addition and Subtraction adequate to the solution of all questions in Arithmetic ? What advantage does the student derive from being familiar with the Addition and Subtraction Tables? What is Addition ?-What is Subtraction ?-What are their uses? Do Addition and Subtraction mutually prove each other? Give some examples. When several quantities are to be added together, does the manner of their arrangement make any difference in the result? Does the student understand the reason for carrying, and why one only is carried to repay ten that are borrowed? What idea has the student of the composition and decomposition of the sumation and separation of quantities? If all operations may be solved by Addition and Subtraction, why are any other rules made use of in the practice of Arithmetic ? CHAPTER IV. 48. MULTIPLICATION AND DIVISION. Multiplication and Division, like Addition and Subtraction, are the reverse of each other, and like them may be used for the purpose of proving each other. All operations in Multiplication may be performed by Addition, and all operations in Division by Subtraction, Multiplication and Division being only a series of additions and subtractions of the same number. By Addition we join together various quantities; but by Multiplication we can only join together the same quantity so many times. multiply 8 by 3, we join eight together three times, that is, we take 8 as many times as there are units in three, and thus we say three times eight are twenty-four. Το 49. DIVISION is the finding of how many times an amount is contained in another amount. This object may be obtained by subtracting successively the smaller amount from the larger one, until nothing remain, or until the remainder be less than the amount subtracted, as 24—8—8—8=0; or, 26-8-8-8-2. In the one case 8 is contained three times exactly; in the other, three times and 2 over. But by division we arrive at the result at once: we merely say, eight in twenty-four three times, and this, because three times eight are twenty-four, or eights in twenty-six, three times and 2 over. 50. From what has already been observed, the student will perceive, that though all operations in Multiplication may be performed by Addition, and all operations in Division by Subtraction; that though Addition and Subtraction are available to the solution+ of all questions usually performed by Multiplication and Division; yet Multiplication and Division are not equally applicable to the solution of questions usually performed by Addition and Subtraction. Addition and Subtraction are, then, indispensable rules; Multiplication and Division are not absolutely so. We, nevertheless, are enabled, by their aid, to perform operations with more pleasure and with greater facility than we could do, had we only Addition and Subtraction to depend upon for the solution of all questions in Arithmetic. Successively, one after the other, in uninterrupted order; from sub, under, and ced-o. to go, to go under, or to follow. +Solution, separation, explanation; from solv-o, to loose, to melt, to free. |