Here it must be observed that in laying out £66 13s. 4d., there is a loss of £10 13s. 4d.; but if £100 worth of goods had been bought and sold at the same rate as above, there would actually have been a loss of £16. Sold goods for £127%, and lost after the rate of 35 per cent., what did they cost? Answer, £1961. Sold apples for 2s. 6d., and lost 30 per cent., what did they cost? Answer, 3s. 64d. When the loss per cent. is an aliquot part of 100, questions in this rule may very well be answered mentally by the following very simple method. Suppose 25 to be the loss per cent. ; now there are four twenty-fives in 100; and if the selling price be multiplied by 4, and divided by 3 (4-1), the answer will immediately be obtained. If 12 per cent. loss, (or the 8th part of 100,) multiply by 8, and divide by 7 (8-1), and so on in like manner for any other aliquot part of 100. Because the loss when deducted from unity, the numerator will always be 1 less than the denominator; which being inverted becomes a multiplier. EXAMPLES. Sold Spanish wool for £140, and lost, by so doing, 121 per cent., what did it cost? Sold cloth for £25, and lost 25 per cent. by so doing, what did it cost? Answer, £33 6s. 8d. Sold a quantity of bitter almonds for £125, and by so doing lost 10 per cent., what did they cost? Answer, £138 17s. 9. PURCHASING OF STOCKS. Rule. Multiply the stock to be transferred by the fractional part of 100, and it will give the answer at once. Note. When the rate per cent. is such a part of 100, as is not in the table, for instance, £58 6s. 8d., we can take any two or more of the fractional parts of 100, and add them together: thus, 50 being half of 100, and £8 6s. 8d. the , and, as +; we find that £58 6s. 8d. is the of 100. Or, by subtraction. £66 13s. 4d., which is ; and £8 6s. 8d. is of 100. It may here be observed that the fractional part of £1 may be made the fractional part of 100, by merely adding two ciphers to the denominator; thus, 19s. 84d. is of a £., (which may be seen by the table, page 10,) it therefore follows that it is 43 of £100. What must be given for £750 16s. 24d. in the 3 per cent. Annuities when £663 will buy £100 ? *Here it may be remarked that the author chose, in the first place, to multiply by 6, (in preference to 11,) because, SQUARE ROOT.* The chief object in giving the extraction of the square root a place here is to make it as agreeable, instructive, and pleasant, as possible. When numbers are to be evolved, the reader should always cast his eye upon the two last right hand figures, merely to observe if the sum end with 25, 50, 75, or two ciphers; because, in that case, they are divisible by the square No, 25 : and, with the aid of the table, page 72, the root may more easily be found, thus: I want to know the square root of 6725. NOTE CONTINUED. by dividing this product by 9, he has immediately the of £750 16s. 24d.; for is: there is certainly in this a little ingenuity. Suppose it were required to multiply any sum, for instance, by 703; we should begin with 7 first, rather than the 10, because the product that results by multiplying by 7, being divided by 8, will give the value of the . Whenever the reader, even after finishing his operation, finds that he could have taken an easier method, he should decidedly adopt it; for this will undoubtedly, in the end, make him truly an arithmetical proficient. EXAMPLES. What is the purchase of £680 16s. Bank Stock, at 98 per cent. ? Answer, £669 9s. 03d. Required the purchase of £700 South Sea Stock at 87 Answer, £612 10s. per cent. * It may appear to many somewhat remarkable that the cube root of an exact cube, consisting of two figures, may be instantly found; and yet the means are not applicable in finding the square root of an exact square to the same Now this number is not to be found in the table: but by dividing it by 25 we have it in the manner following. 6725 269 Now, the square root of 269 by the table is 16'4012194; this multiplied by 5 gives 82.0060970, the square root required. NOTE CONTINUED. number of places; the reason is, that the square of each of the nine digits ends at equal distances from the centre the same, with the exception of the centre figure 5, as the following diagram will show. Roots 1 2 3 4 5 6 7 8 9 Squares 1 4 9 16 25 36 49 64 81 From this it is evident that to obtain the root of any square number, some little consideration is required, for example, What is the square root of 1156 ? Here we see that 3 must be the first or left hand figure in the root; but, as to the remaining figure, it leaves us in doubt whether it should be a 4 or a 6, since the square of both will end with the 6; such then must therefore be left to the reader's ingenuity for obtaining the correct root; it is however 34. But the further we recede from the centre figure 5 the root may be more easily found. EXAMPLE. What is the square root of 1936 ? To extract the square root of a number of figures, it is best to divide them (if they are divisible) in succession by other numbers than 25, such as have already been mentioned; those numbers probably may be brought so low that the square root of them may be obtained by the aid of the table. NOTE CONTINUED. AGAIN. What is the square root of 7225? Here in a moment the answer may be seen to be 85; for 8 is the nearest square to 72; and it ends with the centre figure 5, there not being any other of the nine digits, when squared, that will end with a 5. * The reason of this root ending differently from the other is, that the square root of 78 being made in the table an unit more in the last right hand decimal, the square root of 78 approximates to 8,8317609 nearer than 8,8317608; and this difference, though trifling in itself, being multiplied by 5, makes the difference in the two roots, which is not material: had the latter been multiplied by 5, it would not have been so near the truth. |