RULE. Multiply or divide the fraction proposed by the numbers which connect the different denominations in order, according as the value of the unit in the required fraction, is less or greater than that of the unit in the one which is given. For, let the proposed fraction be £, where the unit is one pound: then if it be required to find the corresponding fraction when the unit is one farthing, it is manifest from what has been said in the Reduction of compound quantities, that in order to retain the same absolute value, we must have 20 × 12 × 4 times as great a fraction as the original one: that is, 1 960 and the value of the unit in the latter fraction being th part of that in the former, the same absolute value is retained by taking 960 times as many parts in the latter, as in the former. Again, reversing the operation, we shall have the divisors 4, 12 and 20 being inverted, according to the rule laid down for the Division of Fractions. Ex. Let it be required to find what fraction of a crown, is equivalent to of a pound. According to the rule just given, we have and we know very well that of £1, or 5s., is equal to 1 crown, expressed fractionally by Examples for Practice. 4 (1) Reduce, and of a pound, to fractions of 64 a penny. 240 320 45 Answers: and 2 5 -1680-300 (2) Express of a shilling, of a penny and a farthing, as fractions of a pound. -x21 9 20 42 (3) Reduce of a guinea, and 3 of a half-guinea, to fractions of £1. 180=7 20 7 Answers: and 30 63 160 32 63 3 feet 7x36-252 342=1444 13421 108. 7 (5) Express 342 of a of a yard as the fraction of an 108 145 inch, and of an inch as that of a pole. 148798 (6) Find the fraction of a yard, which expresses of an ell of 5 quarters; and that of a day which is equivalent to 146 of a year of 365 days. 535-1925 25 (8) Required the fractions of £10., which are equivalent to of a guinea, of a shilling, and 15 of a farthing. 10 = 16 900044000-9000 92. The value of a compound quantity may be exhibited in the form of a fraction, whereof the unit is of a specified denomination. RULE. Reduce the proposed quantity to the lowest denomination contained in it, and also the proposed unit to the same denomination; then the fraction whose numerator and denominator are these results respectively, will be the one required. For, let it be required to represent 2qrs. 15lbs. as the fraction of 1 cwt: then we have and of the 112 pounds or equal parts into which 1 cwt. is supposed to be divided, 71 are here taken, so that according to Article (72), the fraction required will becwt. 93. By means of the two preceding rules, magnitudes of the same kind, consisting of fractions of simple or compound quantities, and connected by the operations of addition or subtraction, may be reduced to simple fractions of any given denomination. Ex. Find the fraction of £1., which is equivalent to the excess of of a guinea, above the sum of of a shilling and of 78. 6d. 4 3 of 7s. 6d. = 3 of 8 1 00100 1 = 6 44x2-440×12-2160 6 9 360 and therefore the required fraction will be £. 119 80 6 240' Examples for Practice. (1) Express 17s. 111⁄2d.; 19s. 10 d., and £1. 13s. 7 d. 4, as fractions of £1. 431 191 Answers: and 11293 =1613 480' 192 6720 76560 (2) What fraction is 2cwt. 1qr. 16lbs. of a ton; 2ft. 9in. of a pole, and 3ro. 25 po, of an acre? (3) Express 5 bush., 3pks., 1 gal., as the fraction of a quarter; and 2 wks., 5 days, 18 hrs., as the fraction of a year of 365 days. (4) Reduce of 2s. 4d. to the fraction of a half crown; and 9s. 10d. to the fraction of 13s. 2d. (5) Find the simple fraction of £1. which expresses the sum of of of 13s. 4d. and 2 of 5 of 10s. 6d. 94. If the species of the unit be given, the value of a fraction of it may be expressed by means of its known parts. RULE. Multiply the numerator of the fraction by the number of parts of the next inferior denomination which are equivalent in value to the unit, divide the product by the denominator, and the quotient is the required number of parts of that denomination: proceed in the same way with the remainder, if any, and the parts of the next denomination will be found: and repeat this process till the lowest denomination to which the unit is capable of being reduced, is obtained. have For, if the fraction proposed be of a yard, we and therefore the value of of a yard, expressed in the known parts of a yard, is 2ft. 6in., or 30 in. 95. The preceding articles enable us to find the value of the sum or difference of fractional parts of magnitudes, of the same kind. Ex. Required the sum and difference of of a pound, and of a guinea. S. d. S. d. £. S. d. therefore the sum = 13 4+9 4 = 1 and the difference 2 8: 0.4.0. The same results may also be obtained as follows: |