whence, the decimal will be .078125 + .875 = .953125 of a mile also, the same processes are comprised in the following more convenient and practical form: 40) 25.000 8)7.6250 00 which suggests the Rule. 953125 115. The value of a Decimal may be expressed by means of the known parts of its units. RULE. Multiply the decimal by the numbers which connect the successive denominations in order; and the integral parts of the products taken out, as they occur, will be the value required. For, to find the value of .655 of a day, we have that is, 15 hrs. 43 min. 12 sec. is the value required: and the following form amounts to the same thing, furnishing the Rule. days. .655 24 2620 1310 hrs. 1 5.720 60 min. 4 3. 200 60 sec. 12.000 Examples for Practice. (1) Express £.00375 as decimals of a shilling and a penny. Answers: .075s. and .9d. (2) What decimals of a pound are 8.4 of a penny and .4068 of a farthing? Answers: £.035 and £.00042375. (3) Reduce 2.15 lbs. to the decimal of a cwt, and 24 yards to the decimal of a mile. Answers: .01919 &c. and .0136 &c. (4) Reduce 7 oz. 4 dwts. to the decimal of 1 lb. and 2 qrs. 33 nls. to the decimal of an English ell of five quarters. Answers: .6 and .555 &c. (5) Reduce 12 hrs. 55min. 23 sec. to the decimal of a day and 5 days 12 hrs. 25 min. 37.92 sec. to the decimal of a week. Answers: .538461 &c. and .788257 &c. (6) Express 12s. 6d. and 15s. 9 d. as decimals of £1 and £4. 13s. 44d. as a decimal of £5. Answers: 628125, .790625 and .93375. (7) Reduce 1s. 3d. to the decimal of 10s., 5s. to the decimal of 13s. 4d. and 13s. 6d. to the decimal of 15s. 6d. Answers: .125, .375 and .875. (8) Find the values of .45 of £1., .16875 of £3. and 2.36875 of £6. Answers: 9s., 10s. 14d. and £14. 4s. 3d. (9) Required the values of £.5675, .375 cwt., .6875 yds. and 13.3375 acres. Answers: 11s. 4 d., 1 qr. 14 lbs., 2 qrs. 3 na. and 13 ac. 1 ro. 14 po. (10) What are the values of .203125 qrs. and .73625 bush. of corn? Answers: 1 bush. 2 pks. 1 gal. and 2 pks. 1 gal. 31 qts. (11) What is the value of .07 of £2. 10s, and of .0474609375 of £10. 13s. 4d.? Answers: 3s. 6d. and 10s. 14d. (12) Find the value of .5 shillings +.7 crowns +.125 pounds. Answer: 6s. 6d. (13) Reduce £24. 16s. 44d. and £167. 10s. 6d. f., to decimals of the same denomination, so as to find how often the former is contained in the latter. 116. DEF. In the conversion of a vulgar fraction into a decimal, if the division performed according to the rule laid down in Article (107) do not terminate, but the figures of the quotient continually recur in some certain order, the result is called a recurring or circulating decimal: the quantity repeated is styled its period and is termed a simple or a compound repetend according as it consists of one or more figures: and the extent of the period is denoted by means of dots placed over the first and last of the figures which compose it. . If the quotient comprise other figures besides those which are repeated, it is called a mixed circulating decimal, as it consists of a non-recurring and a recurring part. Ex. 1. Convert and into decimals. Proceeding according to the Rule, we have 3)1.0000 &c. . 3 3 3 3 &c. (3)4.00 0 0 0 &c. (9)1.3 3 3 3 3 3 &c. 1 481 48 &c. whence, = = .3333 &c. and = .148148 &c. : the former having the simple repetend 3 and the latter the compound repetend 148: and these repetends being denoted by 3 and 148 respectively, Ex. 2. What is the decimal corresponding to ? As in the preceding instances, we have whence, is equivalent to the mixed circulating decimal .13888 &c., the non-recurring part being 13 and the recurring part 8, and it is written 5 36 =.138. Conversely, every pure or mixed circulating decimal must be equal to, and expressible by, a vulgar fraction. 117. To find the vulgar fraction which shall be equivalent to a pure recurring decimal. Let the circulates be .666 &c. and .9696 &c., or .6 and .96: then if, for the sake of conciseness, we suppose the symbols x and y to represent their values, we shall have the following results from Article (106): x = .666 &c. 10 times x = 6.666 &c. y = .9696 &c. 100 times y = 96.9696 &c. whence, subtracting in each case, the former from the latter, we obtain These results may easily be verified, and from them we derive the following Rule. RULE. Make the repetend the numerator of a fraction whose denominator shall consist of as many nines as there are figures in the said repetend: and this reduced to its simplest terms will be the vulgar fraction required. 118. To find the vulgar fraction which shall represent the value of a mixed recurring decimal. Ex. To ascertain the vulgar fractions equivalent to 27 and .2457, we have, by abbreviating the forms, whence, subtracting the second line from the third in each case, we find RULE. Make the non-recurring and the recurring parts taken together, diminished by the non-recurring part taken alone, the numerator of a fraction whose denominator shall consist of as many nines as there are recurring figures, followed by as many ciphers as there are non-recurring figures; and this reduced to its lowest terms will be the vulgar fraction required. 119. It will hence appear that the arithmetical operations upon recurring decimals, may be correctly effected by means of the same operations performed upon their equivalent vulgar fractions. Ex. Let it be required to find the sum, difference, product and quotient, of the recurring decimals .6 and |