Examples. 1. Reduce 19 s. 51 d. to the fraction of a pound. Answer, 2. Reduce 19 s. 103 d. to the fraction of a pound. 311 Answer, 1 3. Reduce 12 s. 93 d. to the fraction of a pound. 191 192 Answer, 1. 4. What part of a cwt. is of an ounce Avoirdupois? Answer, 2048. 5. What part of a ton is 5 cwt. 2 qrs. 24 lb. ? Answer, 2. 6911 6. What part of a league is 2 m. 7 fur. 39 p. 4 yds. 2 ft. 2 in.? Answer 12 7. Reduce 3 R. 6 P. 15 s. yds. 51 s. ft. to the fraction of an acre. 835 Answer, 8. What is the ratio of 1 R. 23 P. 28 s. yds. 33 s. f. to 3 R. 39 P. 25 s. yds. 6 s. f.? Answer, 2. 316. Having seen (158) that a fraction may be considered as expressing the division of its numerator, as an integral number, by its denominator, we can, by performing this division, find the value of a fraction of the principal unit, or of a unit of any superior denomination, in terms of the inferior denominations. For example, to find the value of 1 of a pound, as this is the same as the nine hundred and sixtieth part of £791, we divide £791 by 960 in the following manner : 791 960) 15820 (16 s. 5 d. 960 96 6220 5760 460 12 960) 5520 (5 d. 4800 720 4 960) 2880 (3 qrs. or 4 d. **** It is evident that, when the given fraction is proper, we must reduce the numerator to the next lower denomination before it can contain the denominator; and, if the product is yet too small, to the next lower, &c. In the above example, therefore, we first reduce £791 to shillings, and dividing by 960, we have 16 s. for the quotient, with a remainder of 460 s.: this remainder, reduced to pence, is 5520 d.; and, dividing by 960, we have 5d. for the quotient, and a remainder of 720 d.: this remainder, reduced to farthings, is 2880 qrs. ; and, dividing by 960, we have 3 qrs., ord. without remainder. The value of £ is, therefore, 16 s. 53 d. This operation may also serve as an illustration of the method of dividing any compound number by a whole number. Or thus: 961848) 791 (16 s. 53 d. 48 311 288 18=4) 23 Here, as the divisor 960 is divisible by 20, we divide, and have 48, which we take for a divisor. Now 15820 = 791 X 20, and 96048 × 20. But, (165,) in dividing the dividend and divisor both by the same number, the quotient is not altered: wherefore, 791÷48 will give the same quotient as 15820960. In like manner 234 will give the same quotient as 5520960, because 5520=23 240 and 960 = 4 X 240. Therefore, in all divisions like the above, whenever the multiplier of the remainder will divide the divisor, we may so divide, which will very much shorten the operation. In the above example, the scholar may observe that, as 960 qrs. make £1, the numerator 791 represents a number of farthings, which may be reduced thus: 3. What is the value of £191 2048 Answer, 19 s. 103 d. of a cwt.? Answer, & oz. 4. What is the value of 1 Answer, 5 cwt. 2 qrs. 24 lbs. 6. What is the value of 6911 of a league? Answer, 2 m. 7 fur. 39 p. 4 yds. 2 ft. 2 in. 7. What is the value of 835 of an acre? 1056 Answer, 3 R. 6 P. 15 s. yds. 51 s. f. 8. What is the value of 1⁄2 of a mile? Answer, 5 fur. 28 p. 3 yds. 0 ft. 5 in. 317. We can, with equal facility, reduce the lower denominations of a compound number to a decimal fraction of the principal unit, and this again to the terms of the lower denominations. This is done in the same manner as in the preceding articles, except that we divide or multiply decimally. For example, to reduce 9 d. to the decimal fraction of a shilling, we first reduce 9 d. to the vulgar fraction of a shilling, which gives or s.; then, reducing to a decimal, we have,75 for the fraction required. Or, we divide 9 immediately by 12, thus: 12) 9,00 ,75 In the same manner units of any inferior denomination are reduced to the decimal of a higher denomination. To find the value, in pence, of ,75 of a shilling, we multiply by 12, thus: ,75 12 9,00 and we have 9 d. Each operation, therefore, proves the other. To reduce 5 s. 61 d. to the decimal of a pound, we operate thus: 2)1,0 12) 6,5000 d. 20) 5,5416 s. £,277083 Having found,5 the value of d., we prefix the 6d., and have 6,5 d. for the value of 6 d.; then, to have the value of 6,5 d. in the decimal of a shilling, we divide by 12, and have ,5416 s.; to which prefixing the 5 s. we have 5,5416 s. for the value of 5 s. 63 d. Lastly, to reduce 5,5416 s. to the decimal of a pound, we divide by 20, as in Art. 313; that is, we remove the comma, or suppose it to be removed, one place towards the left; and, dividing,55416 by 2, we have,277083 of a pound for the value of 5 s. 6 d., as was required. Again, we find the value of £,277083 in terms of the lower denominations, thus: 2,77083 2 5,54166 12 6,50000 4 2,0 To find the value of £,277083 in shillings, we remove the comma one place towards the right, and multiply by 2, which is the same as to multiply by 10, and again by 2-that is to say, by 20; after which, as there are 5 decimals in the number 2,77083, we separate as many in the product, and have 5,54166 s., which is the same as 5 s. and ,54166 s. Then, to find the value of ,54166 s. in pence, we multiply by 12, and have 6,5 d. Lastly, we multiply,5 d. by 4, and have 2 qrs., or d. for the product. Wherefore, collecting the different denominations we have thus found, we have 5 s. 6 d. for the required value. This last operation proves the preceding one. In a similar manner, we reduce the inferior terms of a compound number of any other species to the decimal of a higher denomination, and this decimal, again, to the inferior terms. Also, we may, in any case, first reduce the given compound number to a vulgar fraction of the principal unit, and then reduce this fraction to a decimal fraction. 17* Examples. 1. Reduce 19 s. 51 d. to the decimal of a pound. Answer, £,971875. 2. Reduce 19 s. 103 d. to the decimal of a pound. Answer, £,9947916. 3. Reduce 12 s. 93 d. to the decimal of a pound. Answer, £,640625. 4. Reduce oz. Avoirdupois to the decimal of a cwt. Answer, ,00048828125. 5. Reduce of an ell English to the decimal of a yard. Answer, ,89285714. 6. Reduce 2 qrs. 33 n. to the decimal of an ell English. Answer, ,571428. 7. Reduce 28 s. yds 24 s. f. to the decimal of a square pole. Answer,,935064. 8. Find the value of ,6697916 of a pound sterling. Answer, 13 s. 43 d. 9. Find the value of ,857142 of an ell English. Answer, 1 yd. 0 qrs. 14 n. 10. Find the value of ,06428571 of an acre. Answer, 10 P. 8 s. yds. 511 s. f. 11. Find the value of ,428571 of a league. Answer, 1 m. 2 fur. 11 p. 2 yds. 1 ft. 0 in. 12. Required the value of ,285714 of a ton? Answer, 5 cwt. 2 qrs. 24 lbs. 318. We have seen (256) that the decimal value of a fraction, the denominator of which is prime to 10, will be infinite; that is, (251,) it will be a repeating decimal. The decimal value of a proper fraction, therefore, having 7 for its denominator, will be infinite. Now, in dividing a series of nines by 7, we find (257) that the period will contain 6 places of decimals. Hence, in dividing a unit by 7, there must (251) be a different remainder at each division, till we reach the limit of the period: consequently the several remainders must be 1, 2, 3, 4, 5, 6. Wherefore, in dividing by 7 any of the numbers 1, 2, 3, 4, 5, 6, the same figures will always succeed each other, as soon as we find, for remainder, the figure first divided: |