CASE 11. 327. To reduce a common denominate fraction to integers of lower denomination. 1. What is the value of of a pound Troy? SOLUTION.--There are 12 oz. in one pound, hence 12 times the number of pounds equals the number of ounces; 12 times & equals or 6 oz.: there are 20 pwt. in one oz., therefore 20 times the number of oz. equals the number of pwt.; 20 times equals 40, or 13} pwt., etc. SOLUTION 2d. -§ of a pound equals of 5 lb.; and of 5 lb. we find by dividing is 6 oz. 13 pwt. Sgr. OPERATION. X12=60-63 oz. OPERATION. lb. oz. pwt.gr. 9)5 0 0 0 6 13 8 Rule I. Reduce the fraction until we reach an integer and a fraction of a lower denomination, set aside the integer and reduce the fraction as before, and thus continue as far as necessary. Rule II.-Regard the numerator as so many units of the given denomination, and divide by the denominator. 328. To reduce a denominate decimal to integers if lower denominations. 1. Reduce .875 gal. to integers of lower denominations. SOLUTION.-There are 4 quarts in one gallon, there- OPERATION fore 4 times the number of gallons equals the number of quarts; 4 times .875 equals 3 qt. and .5 qt.; there are 2 pints in one quart, therefore 2 times the number of quarts equals the number of pints: 2 times .5 equals 1 pt. Therefore .875 gal. equals 3 qt. 1 pt. .875 4 3.500 2 1.000 Rule-Reduce the decimal until we reach an integer and a decimal of a lower denomination, set aside the integer, and reduce the decimal as before, and thus continue as far as necessary. WRITTEN EXERCISES. What is the value 7. Of .3218 of a ton of iron? Ans. 9 oz. 18 pwt. Ans. 3 yd. 211 ft. Ans. 58 P. 718 sq. yd. Ans. 20 gal. 2 qt. 13 pt. Ans. 3 pk. 79% qt. Ans. 6 cwt. 43 lb. 9.6 oz. 8. Of 2.1365 of a tun (4 hhd.) of wine? Ans. 2 tuns, 34 gal. 1 qt. 1 pt+. REDUCTION ASCENDING. CASE I. 329. To reduce a common denominate fraction to a common fraction of a higher denomination. 1. Reduce of a farthing to the fraction of a shilling. SOLUTION.-There are 4 farthings in a penny, therefore of the number of farthings equals the number of pence: there are 12 pence in 1 shilling, therefore of the number of pence equals the num OPERATION. 8. ber of shillings; hence far. equals ×× of a shilling. Rule.-Express the division by the required divisors, and reduce by cancellation. Reduce WRITTEN EXERCISES. 2. of a grain to the fraction of an ounce. 3. of a foot to the fraction of a mile. Ans. to. Ans. 17800. Ans.. Ans. 186 Ans. To boo Ans. 22100. 8. What part of a cord of wood is a pile containing 48 cubic feet? Ans. CASE II. 330. To reduce a compound number to a common fraction of a higher denomination. 1. Reduce 3 s. 6 d. 2 far. to the fraction of a pound. SOLUTION.-By reduction we find 3 s. 6 d. 2 far. equal to 170 far., and also £1=960 far.; one farthing is of a pound, and 170 far. equals 170 times 178, which reduced to its lowest terms, equals Therefore, etc. OPERATION. 3s. 6 d. 2 far.=170 far. £1960 far. 170=17, Ans. Rule.-Reduce the number to its lowest denomination, and write under it the number of units of this denomination which make a unit of the required denomination; and then reduce the resulting fraction to its lowest terms. WRITTEN EXERCISES. 2. Reduce 3 oz. 8 pwt. 12 gr. to pounds. 3. What part of 2 bushels is 1 bu. 4 qt. ? 7. Of 3 years is 3 wk. 6 da. 20 h.? Ans. 17. Ans. 16. Ans. Ans. 1. Ans. Ans. 1876. Ans. ; 2. 8. What part of 4 inches square is 4 square inches? What part of a 2 inch cube is 2 cu. in.? CASE III. 331. To reduce a compound number to a decimal of a higher denomination. 1. Reduce 1 bu. 2 pk. 4 qt. to the decimal of a bushel. of SOLUTION. There are 8qt. in 1 pk., hence the number of quarts equals the number of pecks; t of 4 equals .5, which, with 2 pk. equals 2.5 pk.; there are 4 pk. in a bushel, hence of the number of pecks equals the number of bushels; of 2.5 equals 625, which, with 1 bu., equals 1.625 bu.; hence 1 bu. 2 pk. 4 qt. equals 1.625 bu. OPERATION. 84 42.5 1.625 Ans. Rule.-I. Divide the lowest term by the number of units which equals one of the next higher, and annex the decimal quotient to the integer of the next higher denomination. II. Proceed in a similar manner until the whole is re duced to the required denomination. NOTE.-It may also be done by reducing to a common fraction, and the common fraction to a decimal. Reduce WRITTEN EXERCISES. 2. 5 oz. 10 pwt. 12 gr. to the decimal of a lb.. Ans. .4604166 lb.+. 3. 3 pk. 4 qt. 1 pt. to the decimal of a bu. Ans. .890625 bu. 4. 93 13 20 8 gr. to the decimal of a Hb. Ans. .76875 lb. 5. 286 rd. 1 yd. 33 in. to the decimal of a mile. Ans. .894375 mi. 6. What decimal part of 5 gal. is 1 qt. 3 gi.? Ans. .0725. 7. Reduce 295 rd. 3 yd. 2 ft. 8 in. to the decimal of a mile. Ans. .92408459 mi.+. 8. Reduce 91lb. 127 oz. to the decimal of a ton. Ans. .04590234375 ton. MENTAL EXERCISES. MISCELLANEOUS EXAMPLES. 1. How many seconds are there in the circumference of a carriage wheel? 2. A lady put up 50 quart cans of strawberries; how many pecks did she put up? 3. When the sun has seemed to pass over 4 signs, how many seconds has he appeared to move? 4. How many pages of an octavo volume can be printed on 3 quires of paper? 5. A hostler at a hotel stable fed out 48 half-peck measures of oats; how many bushels did he feed? 6. Sarah picked a peck of cherries, and sold them at 4 cents a pint; Low much did she receive for them? 7. A sailor took the sounding, and found the water 75 feet deep; how many fathoms was it? 8. My grandfather's age is "three-score years and ten;" how many years old is he? 9. A merchant buys dozen hand-saws, at $20 a dozen; how shall he sell them to gain 50 cents apiece? 10. I paid $3.20 for a bushel of huckleberries; how shall I sell them to gain 2 cents a quart? 11. A merchant bought dozen shovels, at $9.60 a dozen; how shall he retail them to gain 20 cents apiece? 12. What must I pay a gross for hair-pins, that I may sell them for 5 cents a dozen, and gain 2 cents a dozen? 13. How many cubic inches in a 2-inch cube? in a 4-inch cube? in a 6-inch cube? 14. In 6 francs how many centimes? How many in 8 francs? in 10 francs? 15. How many leap years in every century? At what time did the 19th century begin? 16. What part of a circumference is 90°? 60°? 180°? Number of degrees in a quadrant? 17. A grocer paid $6.40 for a bushel of cranberries; how shall he sell them a quart to gain 2 cents a pint? 18. A huckster woman sold one day a bushel of pea-nuts by the half-pint glass, at 5 cents a glass; how much did she receive? 19. A lady bought 2 pairs of vases, 3 sets of chairs, and a dozen knives and forks; how many individual articles did she buy? 20. If Henry Clay spoke 90 words in a minute, how many words would he speak in an address of one hour? 21. Bought paper collars at 40 cents a box, each box containing a dozen; how much will I gain on each collar by selling them for 5 cents apiece? WRITTEN EXERCISES. MISCELLANEOUS PROBLEMS. 1. If 8 barrels of flour cost £24 12 s. 8 d., what will 12 barrels cost at the same rate? Ans. £36 19 s. Ans. 166 cwt. 2. If 7 bales of goods weigh 20 cwt. 75 lb., what will 56 bales of the same size weigh? 3. What cost 8 cords of wood, at the rate of £11 5 d. for 5 cords? Ans. £16 12 s. 2 d. 4. What cost 8 yd. of cloth, at the rate of $4.50 per yard? Ans. $39.37. 5. Multiply 8 d. by 6.331, and from the product take 2s. 6 d. Ans. 1 s. 11 d. 6. Add .003 sq. yd., .04 sq. ft., and .0008 sq. in. Ans. 18.1808 sq. in 7. Reduce 1 shilling, 11 pence and 1.0384 farthings to the decimal of a guinea. Ans. .0923 guinea. |
