18. How much butter, at 18 cents a pound, must be given for 12 gal. 3 qt. of molasses, at 374 cents a gallon? Ans. 251 lb. 19. From a piece of cloth containing 97 yards, 9 coats, each containing 3 yd. were taken; required the value of the remainder at $4.37) a yard? Ans. $306.25. 20. A grocer lost from of a hogshead of molasses, of a gallon and of a quart. How much of the hogshead, expressed decimally, leaked out, and how much remained? Ans. 1.008 gal. leaked out, and 54.11 gal. remained. 21. What is the value of of a year of 365 days, and of a week? Ans. 296 da. 9 hr. 36 mi. 22. Bought 12 T. 12 cut. 2 qr. 20 lb. of iron, and sold 8 T. 4 cwt. 1 qr. 23 lb. of it; what is the value of of the remainder at 6 cents a pound. Ans. $223.94. 23. Bought a quantity of grain for $258.40, and sold of it to one man, Ty of it to another man, and used of it myself; what is the value of the remainder? Ans. $6.40. 24. A gentleman gave of his estate to his wife,& of the remainder to his son, and of what then remained to his daughter, and had remaining $630; required the value of the estate, and the amount each received ? Ans. $30240 the estate, $10080 W. received, $15120 S., and $4410 D. 25. Bought 42 tons of hay for $12.50 per ton; for of which I paid sheep, at $1.25 each; and for the remainder I paid butter, at 234 cents a pound. Required the number of sheep, and the number of pounds of butter it took to pay the debt? Ans. 280 sheep and 744 lb. DUODECIMALS. DUODECIMALS. 240. Duodecimals are denominate numbers, the denominations of which increase according to the scale of 12; or denominate fractions, whose denominators are 1, 12, 144, 1728, etc. Each foot is The Foot is the unit of the measure. divided into 12 equal parts called Primes, each prime into 12 equal parts called Seconds, etc. denotes fourths. etc. 12 make 1 ft. Duodecimals are employed in linear, superficial, or cubic measure NOTE-The marks, ', '', ''', '''', etc., are called indices. ADDITION AND SUBTRACTION OF DUODECIMALS, 241. Duodecimals are reduced, added, and subtracted the same as other denominate numbers. EXAMPLES. 1. Add 4 ft. 3' 5" 8", 6 ft. 7' 9" 1", 12 ft. 10′ 11′′ 8"", Ans. 31 ft. 8' 4" 9'"". 6 ft. 5' 7" 7'', and 1 ft. 4' 6" 9". 2. What is the sum of 17 ft. 8' 9" 3''', 22 ft. 3' 10", 8ft. 11' 7" 9'", 32 ft. 8' 7" 6''', and 39 ft. 6' 7" 3'"'? Ans. 121 ft. 3' 5" 9". 3. What is the sum of 43 ft. 9' 11" 5""" 10"'", 15 ft. 1' 2" 3'"" 5'"'", 28 ft. 6' 11" 8'"" 9'"'", 71 ft. 3' 8" 2'" 3"", and 47 ft. 7' 10" 3'' 8'"''? Ans. 206 ft. 5' 7" 11'' 11'"'". 4. 47 ft. 8' 9" 1'''-23 ft. 11' 3" 4"= ? Ans. 23 ft. 9' 5" 9"". 5. What is the sum, and what is the difference of 49 3' 2" 4'"" 9'"'" and 29 ft. 8' 7" 11'"'" 2'"'"? Ans. Sum 78 ft. 11' 10" 3''' 11'"'", difference 19 ft. 6' 6" 5'' 7'"'". 6. What is the sum of 6 ft. 3' 4" 9''' 11'"'", 37 ft. 9' 11" 7'", 46 ft. 8' 10", and 16 ft. 9' 3" 8''' 11''''? Ans. 107 ft. 7' 6" 1'' 10'"'". 7. 41 ft. 9"-27 ft. 1' 3" 4'' 9''''=? 8. 100 ft. 9''''-1 ft. 11′ 11′′ 11'"'= ? 9. What is the sum and difference of 100 ft. 10' 9" 8'"" and 91 ft. 11' 11' 11''' 11''''? 10. 65 ft. 5" 8''-(13 ft. 6' 7"+46 ft. 3'")=? 11. 14 ft. 7'+the half of 6 ft. 3' 9"=how much more or less than 20 ft. 6'? MULTIPLICATION OF DUODECIMALS. 242. Multiplication of Duodecimals is an abbreviated process of finding the measure of surfaces or solids. The product of two duodecimals expresses a Surface, the product of three expresses a Volume or Solid. 243. In Superficial measure 1 ft. denotes 1 square foot; 1' denotes of 1 sq.ft. 12 square inches; 1" denotes of of 1 sq.ft.=1 square inch, etc. 244. In Cubic measure 1 ft. denotes 1 cubic foot; 1' denotes of 1 cu.ft.=144 cubic inches; 1" denotes of of 1 cu.ft.=12 cubic inches, etc. Hence, a surface 1 inch wide and 12 inches long, makes 1' square measure, and a solid 1 inch thick, 12 inches long, and 12 inches wide, makes 1' cubic measure. 1. How many square feet in a floor 9 ft. 7' long and 7 ft 9' wide? OPERATION. 9ft. 7' 7ft. 2' 3" 67 ft. 1' ANALYSIS.-To find the area multiply the length by the width. 7'= ft. and 9= ft., and their product is fift.=63"-5'3". We write the 3", and add the 5' to the next product. 9×9 ft.ft.=81', and 5' added = 86' 7 ft. 2', which we write in the product. 7 ft. x 7'= ft.=49′=4 ft. 1'. Write the 1' 74 ft. 3' 3" and add the 4 ft. to the next product. 7 ft. × 9 ft=63 ft., and 4 ft. added 67 ft. Adding these partial products we have 74 ft. 3' 3". Rule-1. Place the multiplier under the multiplicand, so that units of the same order will stand in the same column. 2. Begin at the right to multiply, and make the index of each product equal to the sum of the indices of the factors. 3. Reduce each product to the next higher denomination, and add the partial products. NOTE.-1. In multiplication the product of feet and feet is square feet, of feet and primes is primes, of primes and primes is seconds, of primes and seconds is thirds, etc. 2. The multiplier is considered an abstract number. PRACTICAL QUESTIONS. 1. What is the area of a marble slab, the length of which is 9 ft. 8' 11", and width 3 ft. 7'? Ans. 34 ft. 10' 11" 5''. 2. How many square feet are contained in the floor of a room 40 ft. 10' long, 32 ft. 8' wide? Ans. 1333 ft. 10' 8". 3. How many square feet in 10 boards, each 18 ft. 10' long and 1 ft. 8' wide? Ans. 313 ft. 10′ 8′′. 4. How many square feet of boards will it take to inclose a piece of land 80 ft. 10 in. long, and 60 ft. 8 in. wide, with a close fence 7 ft. 6 in. high? Ans. 2122 ft. 6'. 5. How many square yards in a floor which is 48 ft. 6' long, and 36 ft. 10' wide? Ans. 1985 sq. yd. 6. What will the plastering of a room cost, at 18 cents a square yard, the length of which is 30 ft. 10 in., width 24 ft. 6 in., and height of ceiling 8 ft. 4'? Ans. $33.55. 7. In a building there are 32 windows; in each window 16 lights; and cach light is 1 ft. 10' by 11'. How many square feet of glass, in the 32 windows? Ans. 860 ft. 5' 4". 8. How many solid feet in a pile of wood 24 ft. 6 in. long, 6ft. 5' high, and 4 ft. C' wide? Ans. 707 ft. 5' 3". NOTE.-Multiply the length, height, and width together to find the solid contents. (213). 9. How many cubic feet in a stick of timber 32 ft. 9' long, 2 ft. 2' wide, and 2 ft. 8' thick? Ans. 189 ft. 2′ 8′′. 10. How many bricks, each 8 in. long, 4 in. wide, and 2 in. thick, are required to build a wall 144 ft. long, 6 ft. 6 in. high, and three bricks wide, no allowance being made for the mortar? Ans. 25272 bricks. 245. DIVISION OF DUODECIMALS. 1. Since the superficial contents of a surface is found by multiplying its length by its width; the superficial contents divided by either of these factors will give the other. 2. Since the solidity of a body is found by multiplying its three dimensions together; the solidity divided by the product of eithei two of its dimensions will give the other dimension. 3. Since the indices of the product are equal to the sum of the indices of the two factors, the indices of the quotient are equal to the indices of the dividend minus those of the divisor. 1. There are 8 ft. 5' 3" in the surface of a marble slab, the length of which is 3 ft. 9'; what is its width? ANALYSIS -3 ft. is con OPERATION, tained in 8 ft. 2 times. Mul- 3 ft. 9')8 ft. 5' 3" (2 ft. 3.' Ans tiplying the whole divisor by 2 ft. gives 7 ft. C' for the product, which we subtract from the corresponding denominations of the dividend, and ob 7 ft. 6' 11' 3" 11' 3" 0 tain 11' for a remainder, to which annex the next denomination of the dividend, and we have 11' 3". 3 ft. is contained in 11', 3 times The divisor being multiplied by this 3' give 11' 3", which being sub |