EXAMPLES. Reduce 4s. 9 d. to the fraction of a £. sterling, The answer, therefore, is Here 49. 94d.=231 farthings, 26, or 3 in its lowest And 20s. 960 ditto. terms. That is, 960 times 4s. 94d.=£231; and 320 times 4s. 94d. equal to £77. From this case much entertaining information may be derived. I shall propose one line for the multiplication and the other for the division of the terms of the fraction.* Reduce 3s. 4 d. to the fraction of a £. sterling. Hence, by multiplying the terms, £=£=£+%%%%=£$5328=£ 3 5 7 9 8 8. And by dividing the terms 81 27 13 480 160 80 From the first line, we have 11 89 44 24 From the second line. 88 at we have In this manner you may proceed as far as you please. This is not all the use that may be made of this case of NOTE. * Most arithmeticians avoid having complex fractions in their operations, as if there was a difficulty in using them; but I would not on any account have the reader act upon this, for instead of difficulties they often serve to shorten the operation thus, for example; Suppose I have to multiply. 5 by 21; I know at once that it is 3 times 54, or that 7 174, is the answer. Whereas, if the above had been reduced C reduction; for, when we say that 3s. 4d. £., it shows that 480 yards at 3s. 4 d. may be bought for £81, and this is again equal to 44, which shows also that 1440 yards may be bought for £243. Now, if both the terms of the fraction be added together, we shall have 1440 +480 1920, which will cost £243+81=£324; again, is equivalent to either of the before mentioned fractions, viz., so and, which shews that if two equal fractions have their terms added together, the new fraction will be equal to either. 16 2 and therefore = 16+2 18 2 24+3 27 3 For example, we say and it must be observed that when the terms of equal fractions are added together, they are increased by one common ratio; for 16×11___18, 24 × 1 27 as above, and therefore the value of the fraction will not be altered. If the terms of equal fractions are substracted, the value 16-2 14 2, will be the same, for here the terms of 24-3 21 3 the fraction are decreased by one common ratio; for 16×3 14_2 24×7 21 3, terms of many equal fractions be added or substracted without altering the value of the fraction; and this, sometimes, is found to be useful in discovering whether a fraction be already in its lowest terms. as before, and in this manner may the This plan may also be adopted in the way of division; for example: How many yards of cloth at 7s. 3 d. may be bought for £130? Here 7s. 3 d.=£17. But the numerator of the above fraction is only 117, and if we add to this numerator its part we shall have the 130 which is wanting by the question; and thus, by proceeding with the denominator in the same manner, we have 117+17_130; and consequently, the answer is 355 320+30 355 yards. to a single fraction, it would have been thus 5 40 and 40×21=40×4=140=174; which is extremely tedious, and therefore ought to be avoided. oz. dwts. grs. Reduce 8 10 8 to the fraction of a lb. Troy Answer, $. Note. It is better when the lowest name can be reduced to the next greater to do so, thus, for oz. dwts. 8 10%, which reduce to thirds of dwts. Answer, f. The following rules being short and curious, if not new, may be useful. RULE I. Divide an integer into any number of fractional parts, the sum of these will be the fractional parts of the quantity. EXAMPLES. What part of a £. sterling is 4s. 9åd.? Here 4s. ÷ and 9d.=80 also } Then ++ when added = the frac£ tion required, namely, AGAIN: What part of an acre is 2r. 20p. ? Here 2 roods or Then, += the fraction = and 20 poles = required. What part of an ell English is 1 yard 2 nails? Answer, fo What part of a cwt. is 3qrs. 14lb. Answer, . Similar solutions of similar sums may be given by subtraction. RULE II. Take the difference between the fraction, and the difference will be the fraction required. What part of a yard is 2 feet 3 inches? MULTIPLICATION. Notwithstanding the simplicity of the operations of this rule, it is of great importance for the making of rules applicable to mental calculation. As this is a rule so well known to all arithmeticians, it is only necessary to treat upon such points as will be essentially useful to the mental calculator. 1st. Whenever a fraction is to be multiplied by a whole number, and the denominator is divisible by that number, it is best to divide it: for example, let be multiplied by 7. Here 35 can be divided by 7; and therefore the product will be; an easier operation when calculating mentally than 17×7 and then reducing them to their lowest terms. 35 2dly. When the denominator cannot be divided by the whole number, the terms of the fraction may be made greater or lesser, so as to make it divisible by the given number, and then divide as in the following examples. Let be multiplied by 21. Here the 18 cannot be divided by 21, but it is divisible by 3; and as 3 times 7 is 21, it follows that if their terms be multiplied by 7, the denominator will be divisible by 21; or +7=128; which being multiplied by 21 produces 185, or 173. It is thus fully demonstrated that the smaller the parts are in the denominator, the numerator continuing the same, the greater is the value of the fraction; and, contrariwise, the greater the parts in denominator, the numerator remaining the same, the less is the value of the fraction. Application of multiplication for the finding of arithmetical rules. To find a convenient rule for the interest of money at 5 per cent. for months.* Here let 1 represent the principal. Then 5 per cent. interest. oro of the principal for 1 year's And 1 month of a year. Also £1 240 pence. Therefore 1 X X X 21°=1. This shows that £1 put to interest at the above rate will be a penny per month, consequently the rule will be as follows. Multiply the principal by the given number of months, the product of the pounds considered as pence will be the answer. EXAMPLE. What is the interest of £29 3s. 4d. for 7 months? £29 3s. 4d. 7 178. Od. the answer. 204 3 4 204 pence, or Again -Suppose it be required to find the interest for days at 5 per cent. Then 1×× 365 × 240=3'65. This shows that the principal multiplied by the number NOTE. *This is introduced to shew the beauty of the science of fractions, and to what excellent purposes they may be applied, not only to this, but to many other branches of mental calculations, which may either have escaped my notice, or which I have not inserted in order to prevent this work from being unnecessarily large. Questions in interest may often be answered mentally; for example, What is the interest of £39 3s. 4d. for 5 years at 3 per cent. ? Here, the denominator being as many as there are shillings in a £; the numerator denotes the number of shillings on each pound of the principal; therefore, by converting the pounds in the principal to shillings, and multiplying by the numerator we have £1 19s. † ×3= £5 17s. 6d. answer. |