yard ! 16 21. How many cwt. of coffee in 17} bags, each bag containing 2cwt. Iqr. 7lb. ? Ans. 41cwt. Oqr. 5ļlb. 22. If 18yd. 1qr. of cloth cost $ 36.50, what is the price of 1 Ans. $ 2.00. 23. If $ 477.72 be equally divided between 9 men, what will be each man's share ? Ans. $ 53.08. 24. A man bought a barrel of flour for $ 5.37.5, 7gal. of molas. ses for $ 1.78, 9 gal. of vinegar for $1.1875, 1 gal. of wine for $ 1.125, 141b. of sugar for $ 1.275, and 5lb. of tea, for $ 2.625 ; what did the whole amount to ? Ans. $ 13.36.71. 25. A man purchased 3 loads of hay; the first contained 23 tons, the second 3} tons, and the third 11 tons; what was the value of the whole, at $ 17.625 a ton ? Aps. $ 128.88.213 26. At $13.625 per cwt., what cost Scwt. 2qr. 7lb. of sugar? Ans. $ 48.53.916. 27. At $ 125.75 per acre, what cost 37A. 3R. 35rds. ? Ans. $ 4774.57.06. 28. At $11.25 per cwt. what cost 17cwt. 2qr. 21lb. of rice ? Ans. $ 198.98.43. 29. What cost 74 bales of cotton, weighing 3.37cwt., at $ 9.374 per cwt. ? Ans. $ 244.85.16 30. What cost 7hhd. 49gal. of wine, at $ 97.625 per hhd. ? Ans. $759.30.53 31. What cost 7yd. Sqr. 3na. of cloth, at $ 4.75 per yard ? Ans. $ 37.70.3}. 32. What cost 27T. 15cwt. Iqr. 3 lb. of hemp, at $ 183.62 per ton ? Ans. $ 5098.03.7 33. What is the cost of constructing a railroad 17m. 3fur. 15rd., at $ 1725.87.5 per mile ? Ans. $ 30067.97.824 63 SECTION XXIII. EXCHANGE OF CURRENCIES. Previous to the year 1776, all accounts this country were kept in pounds, shillings, pence, and farthings; but owing to the depreciation of the currency, a dollar was estimated differently în different countries. In New England, Virginia, Kentucky, Tennessee, ? 65. Od. and Ohio, the dollar is valued at 8s. Od. 7s. 6d. South Carolina and Georgia, 4s. 80. Canada and Nova Scotia, 5s. Od. England and Newfoundland (sterling), 4s. 6d. } In order, therefore, to change any of the above currencies to federal money, the shillings, pence and farthings, if there be any, must first be reduced to decimals of a pound, and annexed to the pounds. We then adopt this general RULE. Divide the pounds by the value of a dollar in the given currency, EXPRESSED BY A FRACTION OF A POUND ; that is, to change the old New England currency to federal money, divide by to; because 6 shillings is io of a pound. To change the old currency of New York, &c., to federal money, divide by 1o; because 8 shillings is o of a pound. To change the old currency of Pennsylvania, &c., to federal money, divide by t; because 7 shillings and 6 pence is of a pound. To change the old currency of South Carolina and Georgia to federal money, divide by ; because 4 shillings and 8 pence is 3 of a pound. To change Canada and Nova Scotia currency to federal money, divide by ! ; because 5 shillings is of a pound. To change English (sterling) money to federal money, divide by &; because 4 shillings and 6 pence is of a pound. To change sterling money to lawful money, add { to the sterling, and the sum will be the lawful; because 4 shillings and 6 pence sterling is 6 shillings lawful; and, for the same reason, take from the lawful, and the remainder will be sterling. To reduce federal money to any of the above currencies, the federal money must be MULTIPLIED by the above fractions. EXAMPLES. 1. Change 18£. 4s. 6d. of the old New England currency to federal money. 18.225£.+ id=$60.75 Answer. In this example we reduce the 4 shillings and 6 pence to a decimal of a pound, which we find to be .225. This decimal we annex to the pounds, and multiply the 18.225£. by 10, and divide by 3, and it produces the ans er $ 60.75. The reason for this process has already been shown. 2. Change $ 60.75 to the old currency of New England. $ 60.75Xid=18.225= 18£. 5s. 60.= Answer. 'The decimal .225 is reduced to shillings and pence by Case IV. of Decimal Fractions. 3. Change 78£. 7s. 6d. of the old currency of New England to federal money. Ans. $ 261.25. 4. Change $ 261.25 to the old currency of New England. Ans. 78£. 7s. 6d. 5. Change 46£. 16s. 6d. of the old currency of New York to federal money: Ans. $ 117.064. 6. Change $117.064 to the old currency of New York. Ans. 46£. 16s. 6d. 7. Change 387£. of the old currency of New Jersey and Pennsylvania to federal money, Ans. $ 1032. 8. Change $ 1032 to the old currency of New Jersey and Pennsylvania. Ans. 387 £. 9. Change 12£. 12s. of the old currency of South Carolina and Georgia to federal money. Ans. $ 54. 10. Change $ 54 to the old currency of South Carolina and Georgia. Ans. 12£. 12s. 11. Change 128£. 18s. 6d. of Canada and Nova Scotia to federal money. Ans. $ 515.70. 12. Change $ 515.70 to Canada and Nova Scotia currency: Ans. 128£. 18s. 6d. 13. Change 162£. 18s. English money (sterling) to federal money. Ans. $724. 14. Change $724 to sterling money. Ans. 162£. 18s. 15. Change 347 £. sterling to lawful money. Ans. 462£. 135. 4d. 16. Change 462£. 13s. 4d. lawful to sterling. Ans. 347 £. SECTION XXIV. CIRCULATING DECIMALS.* DEFINITIONS. 1. Those decimals that are produced from Vulgar Fractions, whose denominators do not measure their numerators, and are distinguished by the continual repetition of the same figure or figures, are called infinite decimals. 2. The circulating figures, that is, those that continually repeat, are called repetends ; and, if the same figure only repeats, it is called a single repetend ; as .11111 or .5555, and is expressed by writing the circulating figure with a point over it ; thus, .11111, and is denoted by .i, and .5555 by .5. * As Circulating Decimals are not so much of a practical nature as many other rules, and as they are somewhat difficult in their operation, the student can omit them until he reviews arithmetic. a 3. If the same figures circulate alternately, it is called a compound repetend ; as .475475475, and is distinguished by putting a point over the first and last repeating figures; thus, .475475475 is written .475. 4. When other figures arise before those which circulate, it is called a mixed repetend ; as .1246, or .17835. 5. Similar repetends begin at the same place; as.3 and ..; or 5.123 and 3.478. 6. Dissimilar repetends begin at different places; as .986 and .4625. 7. Conterminous repetends end at the same place; as .631 and 465. 8. Similar and conterminous repetends begin and end at the same place ; as .1728 and .4981. REDUCTION OF CIRCULATING DECIMALS. CASE I. a To reduce a simple repetend to its equivalent vulgar fraction. If a unit with ciphers annexed to it be divided by 9 ad infinitum, the quotient will be 1 continually; that is, if be reduced to a decimal, it will produce the circulate .i ; and since .i is the decimal equivalent to ], , will be equivalent to }, .3 to $, and so on, till .9 is equal to or 1. Therefore every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and denominator 9. Again, go, or gšs, being reduced to decimals, makes .01010101, and .001001001 ad infinitum .oi and .boi ; that is šš=.oi, and šs=.001; consequently ýšc.02, and Bšs=.002 ; and, as the same will hold universally, we deduce the following RULE. Make the given decimal the numerator, and let the denominator be a number consisting of as many nines, as there are recurring places in the repetend. If there be integral figures in the circulate, as many ciphers must be annexed to the numerator, as the highest place of the repetend is distant from the decimal point. EXAMPLES. 123 1. Required the least vulgar fraction equal to .. and .123. 6== Ans. .i25== Ans. 2. Reduce .3 to its equivalent vulgar fraction. Ans. j. 3. Reduce 1.62 to its eq valent vulgar fraction. Ans. 134. 4. Reduce 769230 to its equivalent vulgar fraction. Ans. 13. CASE II. To reduce a-mixed repetend to its equivalent vulgar fraction, 1. What vulgar fraction is equivalent to .138 ? OPERATION. 13 .188=160 +50=+=*= Ans. As this is a mixed circulate, we divide it into its finite and circulating parts; thus .138=.18 the finite part, and .008 the repetend or circulating part; but .13= id; and .008 would be equal to s, if the circulation began immediately after the place of units; but, as it begins after the place of hundreds, it is , of 100= Therefore .138=16+=567+56=== Ans. Q. E. D. From the above demonstration we deduce the following 117 8 900 RULE. To as many nines as there are figures in the repetend, annex as many ciphers as there are finite places for a denominator; multiply the nines in the denominator by the finite part, and add the repeating decimal to the product for the numerator. If the repetend begins in some integral place, the finite value of the circulating, must be added to the finite part. 2. What is the least vulgar fraction equivalent to .53 ? Ans. is. 3. What is the least vulgar fraction equivalent to .5925 ? Ans. 4. What is the least vulgar fraction equivalent to .008497133? Ans. 9788 83 5. What is the finite number equivalent to 31.62 ? Ans. 3131 |