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ought to be greater than the second term, I mark the less extreme for a divisor; I multiply the two unmarked terms together, and divide the product by the marked term, the quotient 13.54166 &c. reduced to its proper terms is the answer.

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3 5. Bought s of a levant trader for 357 l. 5s. what sum will

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6. Paid 9.87 l. for 6.54 cwt. of stock-fish; what quantity can be had for 32.11. at that rate 2 Ans. 21cut. lgr. 2lb. 2297856.

7. If a lump of ore, weighing 15.253 lb., be valued at 3s. 9d. what is the cargo of a ship, carrying 180 tons of the same, worth 2 Ans. 4956l. 8s. 0d. . .939.

8. A piece of cloth was cut into two parts, one of which measured 5; English ells, and the other 8: Flemish ells; what is the value of the whole, at 8s. 4d.; per yard 2

239. COMPOUND PROPORTION IN DECIMALS.

RULE. Prepare the numbers (if they require it) as in the preceding rule, and work as in Compound Proportion in whole numbers".

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* This rule depends on the same principles with Compound Proportion in whole numbers.

CIRCULATING DECIMALS.

240. Circulating, repeating, or recurring decimals are those in which one or more of the figures continually recur, and may be carried on indefinitely; the figures that recur are called repetends. 241. A pure repetend is a decimal in which all the figures recur; as .222 &c. .012012 &c. .153153 &c. 242. A mixed repetend is a decimal in which some of the figures do, and some do not, recur; as .5333 &c. 341212 &c. .4.19375.375 &c. 243. A single repetend is that in which only one figure repeats, as .333 &c. and is denoted by a point placed over the circulating figure, as 3. 244. A compound repetend is that in which the same figures repeat alternately, as .1212 &c. .345345 &c. and is expressed by a point over the first and last repeating figure, as i2.345 &c. 245. Similar repetends are those which begin at equal distances from the decimal mark; thus 2357, .47 i, and 493857, are similar. 246. Dissimilar repetends are those which do not begin at equal distances from the decimal mark; thus 23iz3 and 453i are dissimilar. 247. Conterminous repetends are such as end at equal distances from the decimal mark; thus 332323 and 315315 are conterminous, as are 3451? and 82413. 248. Similar and conterminous repetends are such as begin at the same distance from the decimal mark, and end at the same distance; thus 785343434 and .000789789 are similar and conterminous, as are .12345 and 5432i.

REDUCTION OF CIRCULATING DECIMALS.

249. To reduce a pure repetend to its equivalent vulgar fraction. RULE I. Under the given repetend as a numerator write as many mines as the repetend has figures for a denominator. II. Reduce this fraction to its lowest terms, which will be the fraction required".

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Ans.

90909 - e e - 2 5 3. Reduce .6 and 45 to vulgar fractions. Ans. T3 and II" 4. Reduce 313 and #286 to equal vulgar fractions. Answer 71 7286 - and —, 333 99.99 5 37O

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3, &c. to + = 1; wherefore every single repetend is equal to a vulgar frac

tion, the numerator of which is the repeating figure, and the denominator 9. -- • - 3 -In the same manner + = .01 ; whence + = ,02, — = .03, &c. also 99 • a 99 99 999

- - 2 • .. 3 - = .001; whence 395 T 002, 995 = 003; and the same holds true universally.

Wherefore every pure repetend is equal to a vulgar fraction, the numerator of which consists of the repeating figures, and its denominator of as many nines as there are repeating figures; which was to be shewn,

250. When any part of the repetend is a whole number. Rule. Subjoin as many ciphers to the numerator as the highest place of the repetend is distant from the decimal mark". 6. Reduce ioi and i273 to fractions.

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7. Reduce 3.4%, is 24, and i234.8 to vulgar fractions. An

820 60800 41 16OOOO

333 T. *T2.3T.

8. Reduce i3, 21%, and 312.4 to equal vulgar fractions. 130 21700 d 2s1000.

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251. To reduce a mixed repetend to its equivalent vulgar fraction. Rule I. Prefix as many mines as there are places in the repetend, to as many ciphers as there are places in the finite part, for a denominator. II. From the given mixed repetend subtract the finite part for a numerator, and reduce the fraction to its lowest terms for the answer.

* This rule may be explained by example 6; where if we suppose ioi to be

wholly a decimal, its equivalent vulgar fraction will be lot, by the preceding 999

rule; but i.oi is ten times ini, whence the foregoing fraction multiplied by 101 1010

10, (thus 500 ° 10,) or , will be the value of i.oi. Again, if i278 9 - - - - 1278 be considered as a decimal, its equivalent vulgar fraction will be o ; but

iz73 is 100 times .i.278"; wherefore the vulgar fraction, equal to the former, 127800.

will be 100 times as great as that equal to the latter, that is, 12.78 = Toss '

which is the rule,

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