5. Reduce 4 cwt. 23 qrs. to the decimal of a ton. Note. 232'6. 7. Reduce 38 gals. 3'52 qts. of beer, to the decimal of a hhd. 9. Reduce 1 qr. 2 n. to the decimal of a yard. 11. Reduce 17 h. 6 m. 43 sec. to the decimal of a day. 13. Reduce 21 s. 10 d. to the decimal of a guinea. 15. Reduce 3 cwt. 0 qr. 7 lbs. 8 oz. to the decimal of a ton. 6. What is the value of 2325 of a ton? 8. What is the value of "72 hhd. of beer? 10. What is the value of 375 of a yard? 12. What is the value of 713 of a day? 14. What is the value of 78125 of a guinea? 16. What is the value of '15334821 of a ton ? Let the pupil be required to reverse and prove the follow ing examples: 17. Reduce 4 rods to the decimal of an acre. 18. What is the value of " of a lb. of silver? 19. Reduce 18 hours, 15 m. 50'4 sec. to the decimal of › day. 20. What is the value of '67 of a league? 21. Reduce 10 s. 94 d. to the fraction of a pound. T76. There is a method of reducing shillings, pence and farthings to the decimal of a pound, by inspection, more simple and concise than the foregoing. The reasoning in relation to it is as follows: = of 20 s. is 2 s.; therefore every 2 s. is, or '1 £. Every shilling is 155, or '05 £. Pence are readily reduced to farthings. Every farthing is £. Had it so happened, that 1000 farthings, instead of 960, had made a pound, then every farthing would have been obʊ, or '001 £. But 960 increased by part of itself is 1000; consequently, 24 farthings are exactly 3, or '025 £., and 48 farthings are exactly 880, or '050 £. Wherefore, if the number of farthings, in the given pence and farthings, he more than 12,4 part will be more than ; therefore add 1 to them: if they be more than 36, 4 part will be more than 11; therefore add 2 to them: then call them so many thousandths, and the result will be correct within less than of robo of a pound. Thus, 17 s. 53 d. is reduced to the decimal of a pound as follows: 16 s. '8 £. and 1 s. — '05 £. Then, 5 d. = 23 farthings, which, increased by 1, (the number being more than 12, but not exceeding 36,) is ́024 £., and the whole is '874 £. the Ans. Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection,-Call every two shillings one tenth of a pound; every odd shilling, five hundredths; and the umber of farthings, in the given pence and farthings, so many housandths, adding one, if the number be more than twelve and not exceeding thirty-six, and two, if the number be more than 'hirty-six. 77. Reasoning as above, the result, or the three first figures in any decimal of a pound, may readily be reduced back to shillings, pence and farthings, by inspection. Double the first figure, or tenths, for shillings, and, if the second figure, or hundredths, be five, or more than five, reckon another shilling; then, after the five is deducted, call the figures in the second and third place so many farthings, abating one when they are above twelve, and two when above thirty-six, and the result will be the answer, sufficiently exact for all practical purposes. Thus, to find the value of '876 £. by inspection: '8 tenths of a pound '05 hundredths of a pound 026 thousandths, abating 1, 25 farthings '876 of a pound EXAMPLES FOR PRACTICE. 1. Find, by inspection, the decimal expressions of 9 s. 7 d., and 12 s. 03 d. Ans. 479 £., and ‘603 £. 2. Find, by inspection, the value of '523 £., and '694 £. Ans. 10 s. 5 d., and 13 s. 102 d. 3. Reduce to decimals, by inspection, the following sums, and fad their amount, viz.: 15 s. 3 d.; 8 s. 11 d. ; 10 s. 6 d.; 1 s. 84 d. ; 1⁄2 d., and 24 d. Amount, £1'833. 4. Find the value of 47 £. Mote. When the decimal has but two figures, after taking out the shillings, the remainder, to be reduced to thousandths, will require a cipher to be annexed to the right hand, or supposed to be so. Ans. 9 s. 44 d. 5. Value the following decimals, by inspection, and find their amount, viz.: 785 £.; '357 £.; '916 £.; '74 £., '5 £.; ‘25 £.; '09 £.; and '008 £. Ans. 3£. 12 s. 11 d. SUPPLEMENT TO DECIMAL FRACTIONS: QUESTIONS. 1. What are decimal fractions? 2. Whence is the term derived? 3. How do decimal differ from common fractions? 4. How are decimal fractions written? 5. How can the proper denominator to a decimal fraction be known, if it be not expressed? 6. How is the value of every figure. determined? 7. What does the first figure on the right hand of the decimal point signify? the second figure? third figure?fourth figure? 8. How do ciphers, placed at the right hand of decimals, affect their value? 9. Placed at the left hand, how do they affect their value? 10. How are decimals read? 11. How are decimal fractions, having different denominators, reduced to a common denominator? 12. What is a mixed number? 13. How may any whole number be reduced to decimal parts? 14. How can any mixed number be read together, and the whole expressed in the form of a common fraction? 15. "What is observed respecting the denominations in federal money? 16. What is the rule for addition and subtraction of decimals, particularly as respects placing the decimal point in the results? multiplication? division? 17. How is a common or vulgar fraction reduced to a decimal? 18. What is the rule for reducing a compound number to a decimal of the highest denomination contained in it? 19. What is the rule for finding the value of any given decimal of a higher denomination in terms of a lower? 20. What is the rule for reducing shillings, pence and farthings to the decimal of a pound, by inspection? 21. What is the reasoning in relation to this rule? 22. How may the three first figures of any decimal of a pound be reduced to ⚫shillings, pence and farthings, by inspection? EXERCISES. 1. A merchant had several remnants of cloth, measuring as follows, viz. : How many yards in the whole, and what would the whole come to at $3'67 per yard? Note. Reduce the common fractions to decimals. Do the same wherever they occur in the examples which follow. Ans. 36'475 yards. $133'863+, cost. 2. From a piece of cloth, containing 36 sold, at one time, 7 yds., and, at another how much of the cloth had he left? yds., a merchant time, 128 yds.; Ans. 16'7 yds. 1 cow 3. A farmer bought 7 yards of broadcloth for 86 £., a barrel of flour for 2 £., a cask of lime for 18 £., and 7 lbs. of rice for £.; he paid 1 ton of hay at 376 £., at 6., and the balance in pork at £. per lb.; how many were the pounds of pork? Note. In reducing the common fractions in this example, it will be sufficiently exact if the decimal be extended to three places. 4. At 124 cents per lb., what will 37 Ans. 1084 lb. lbs. of butter cost? Ans. $47184. 5. At $1737 per ton for hay, what will 11ğ tons cost? Ans. $201 925 6. The above erample reversed. At $201'92 for 11 tons of hay, what is that per ton? Ans. $1737. 7. If 5 of a ton of hay cost $9, what is that per ton? Consult ¶ 65. Ans. $20. 8. At 4 of a dollar a gallon, what will '25 of a gallon of molasses cost? Ans. $'1.. 9. At $9 per cwt., what will 7 cwt. 3 qrs. 16 lbs. of sugar cost? Note. Reduce the 3 qrs. 16 lbs. to the decimal of a cwt., extending the decimal in this, and the examples which follow, to four places. Ans. 71'035+. 10. At $69'875 for 5 cwt. 1 qr. 14 lbs. of raisins, what is that per cwt.? Ans. $13. 11. What will 2300 lbs. of hay come to at 7 mills per ib.r Ans. 16'10. 12. What will 7654 lbs. of coffee come to, at 18 cents per Ans. $13779. lb. ? N* 13. What will 12 gals. 3 qts. 1 pt. of gin cost, at 28 cents per quart? Note. Reduce the whole quantity to quarts and the deci mal of a quart. Ans. 14'42. 14. Bought 16 yds. 2 qrs. 3 na. of broadcloth for $ 100'125; what was that per yard? 15. At $1'92 per bushel, how bought for $72? 16. At $92'72 per ton, how chased for $60'268? Ans. $6. much wheat may be Ans. 1 peck 4 quarts much iron may be purAns. 13 cwt. 17. Bought a load of hay for of $16 per ton; what was the weight of the hay? $9'17, paying at the rate Ans. 11 cwt. 1 qr.23 lbs. 18. At $302'4 per tun, what will 1 hhd. 15 gals. 3 qts. of wine cost? Ans. $94'50. 19. The above reversed. At $94'50 for 1 hhd. 15 gals. 3 qts. of wine, what is that per tun? Ans. $302'4. Note. The following examples reciprocally prove each other, excepting when there are some fractional losses, as explained above, and even then the results will be sufficiently exact for all practical purposes. If, however, greater exactness be required, the decimals must be extended to a greater number of places. 20. At $1'80 for 31 qts. of wine, what is that per gal.? 22. If of a ton of potashes cost $60'45, what is that per ton? 21. At $2215 per gal., what cost 34 qts.? 23. At $96'72 per ton for pot-ashes, what will of a ton cost? 24. If '8 of a yard 25. If a yard of 26. At $25 per of cloth cost $2, cloth cost $2'5, yard, how much what is that per what will '8 of a cloth may be puryard? chased for $2? yard cost? 27. If 14 cwt. of 28. If a ton of pot-ashes cost 19£. pot-ashes cost 27. 5 s., what is that 10 s., what will 14 29. At 27 £.10 s. a ton for pot-ashes, what quantity may be bought for 19 £, 5 s.? Note. After the same manner let the pupil reverse and prove the following examples: |