3. x£ 12 yards yards 4 3£ Using x, as before, and dividing the colunins by a perpendicular line. 4. x£ 12 yards Adopting any of the above modes, and placing only £ at x and yards at the right column, without repeating the signs of yards and £. Beginners will do well to repeat the signs, though, in general, this is unnecessary. Instead of x, a point of interrogation may be used; the latter, however, from its similarity to the figure 2, often causes mistakes in the working. The reading of this example, is thus: How many £ are equal to 12 yards, as many times as 4 yards are equal to £3. Or thus: How many £ for 12 yards, Of the two modes of reading a question, the first is expressed according to arithmetical science; the latter to that of a more common and less systematical form. Whatever may be the method of writing this, or any other question, the principle remains the same, viz. commencing at the left column with x, which expresses the number of the kind wanted, and stating opposite in the right column the number of the equivalent which is to be answered, then recommencing with the number of the kind last-mentioned, and to the right its equivalent. Continue, if the example should require it, in the same manner, by keeping up the chain or link between the columns until there appears at the end of the right column the number of the same kind as x, stated at the commencement, and in which the answer will be found. The above, though a short example, will show at once the stating or linking, and the working which is performed in After having drawn a line below, multiply 12 by 3; 3 × 12 = 36, and divide by 4, which gives the answer, equal to x, in £. But, as the Chain Rule admits cancelling or shortening of numbers in the two columns, whenever it is practicable, so as to lessen the working, after the line has been drawn, compare the respective numbers of each column, and cancel or shorten them by a suitable divisor. The above example will, therefore, stand thus: : 4 and 12 being divisable by 4, they have been cancelled by crossing them, and 3 put instead of 12; the right column is then multiplied; 3 times 3 are 9, and as there is none at the left for a divisor, the answer is £9, being the same kind as x. If, by way of illustration, the same example is to be proved, the principle of stating, &c., remains unaltered : 4 and 12 have been cancelled by saying 4 in 4, 1, cross the 4; 4 in 12, is 3, cross 12 and put down 3; 3 in 3 is one, cancel the 3; 3 in 9 is 3, cancel the 9; the remaining 3 is the answer in x. The cancelling or shortening may also be effected differently, 3 in 9 is 3, and in 12, 4, leaving 3 on the right, and 4 on the left; cancel 4 and 4, the answer is 3 as x or £. The supposed divisor is 3 for 9; and cancelling 9 and 3, the 4 multiplied by 3, gives the answer in x or yards. 3 and 9 have been cancelled and shortened by 3, and again 3 and 12 by 3, which is the answer in yards as x. In calculations of a complicated nature, the linking together of numbers is so beautiful, that the task of working the question at once becomes easy, which is different from the ordinary mode of stating and working arithmetical questions. In the ordinary mode separate statings are often required, and, besides the many figures to be put down, more have to be calculated mentally, in order partially to lessen the labour, not to mention the additional trouble which is occasioned by the calculation of fractions. By the Chain, many statings of an example in proportion can be combined and worked as one, and the final answer obtained without circuitous processes, often by the mere cancelling of figures, and without recourse being had either to mental calculations, or the assuming, as in Practice, of amounts different from what are in the example. The figures stated are all that is necessary for correct calculation. |