NOTE. Duodecimals are really common fractions, and can always be treated as such; but usually their denoin'nators are not expressed, and they are treated as compound numbers.

ADDITION AND SUBTRACTION OF DUODECIMALS.

228. We add and subtract duo decimals the same as other compound numbers.

EXAMPLES.

1. Add 13 ft. 4' 8", 10 ft. 6' 7", 145 ft. 9' 11".

Ans. 169 ft. 9′ 2′′.

2. Add 179 ft. 11' 4", 245 ft. 1' 4", 3 ft. 9' 9".

Ans. 428 ft. 10' 5".

3. From 25 ft. 6' 3" take 14 ft. 9' 8". Ans. 10 ft. 8' 7". 4. From a board 15 ft. 7' 6" in length, 3 ft. 8' 11" were sawed off; what was the length of the piece left?

Ans. 11 ft. 10' 7".

MULTIPLICATION OF DUOdecimals.

229. Length multiplied by breadth gives surface, and surface multiplied by thickness gives solid contents (198). 1. How many square feet in a board 11 feet 8 inches long and 2 feet 7 inches wide?

7'

OPERATION.

11 ft.

8'

2

6 ft.

23

30 ft. 1'

9' 8"
4'

ANALYSIS. We first multiply by the 7'. 7 twelfths times 8 twelfths equals 56 one hundred forty-fourths, which equals 4 twelfths and 8 one hundred forty-fourths. We write the 8 144ths-marked with two indices to the right, and add the 4 12ths to the next product. 7' times 11 equals 77', which added to 4' equals 81', equal to 6 feet and 9'. We write the 9' under the inches, or 12ths, and the 6 under the feet, or units. 2 times 8' equals 16', or 1 foot and 4′. We write the 4′ under the 9', and add the 1 foot to the next product. 2 times 11 feet are 22 feet, and 1 foot added make 23 feet, which we write under the 6 feet. Add

8"

How are duodecimals added and subtracted? Give analysis of example 1.

ing these partial products, and we have 30 ft. 1' and 8' for the entire product.

It will be seen from the above that the number of indices to every product of any two factors is equal to the sum of the indices of those factors; thus 7' X 8′ = 56′′ ; 4′′ X 5""20". Hence the

RULE. I. Write the several terms of the multiplier under the corresponding terms of the multiplicand.

II. Multiply each term of the multiplicand by each term of the multiplier, beginning with the lowest term in each, and call the product of any two denominations the denomination denoted by the sum of their indices, carrying 1 for every 12.

III. Add the partial products, carrying 1 for every 12; their sum will be the required answer.

EXAMPLES FOR PRACTICE.

2. How many square feet in a board 13 ft. 9' long and 11′ wide? Ans. 12 ft. 7' 3". 3. How many square feet in a stock of 4 boards, each 11 ft. 9' long and 1 ft. 3' wide? Ans. 58 ft. 9'.

4. How many square yards of plastering on the walls of a room 12 ft. 11' square, and 9 ft. 3′ high, allowing for two windows and one door, each 6 ft. 2' high and 2 ft. 4' wide?

Ans. 48 sq. yd. 2 ft. 9'. 5. How many solid feet in a mow of hay 30 ft. 4' long, 25 ft. 6′ wide, and 12 ft. 5' high? Ans. 9604 ft. 3' 6".

6. How many cords in a pile of wood 18 ft. 6' long, 12 ft. wide, and 5 ft. 6' high? Ans. 9 cords 69 ft. 7. How many cubic yards of earth must be removed in digging a cellar 36 ft. 10' long, 22 ft. 3′ wide, and 5 ft. 2' deep? Ans. 156 cu. yd. 22 ft. 3' 7".

8. What would it cost to plaster a wall 32 ft. 8' long and 9 ft. high, at 17 cents per square yard? Ans. $5.55. 9. How many yards of carpeting, 27' wide, will be required to cover a floor 48 ft. long and 33 ft. 9' wide?

Ans. 240 yards.

Give the rule.

DIVISION OF DUODECIMALS.

230. 1. A flagstone, 3 ft. 9' wide, has a surface of 20 ft

11' 3"; what is its length?

OPERATION.

3 ft. 9') 20 ft. 11' 3" (5 ft. 7'.

18 9'

2

2' 3"

2

2' 3"

ANALYSIS. We divide the surface by the width to obtain the length. The divisor is something more than 3 ft., and to obtain the first quotient figure, we consider how many times 3 ft. and something more is contained in nearly 21 ft. (20 ft. 11'); we estimate it to be 5 times, and multipiying the divisor by this quotient figure, we have 18 ft. 9', which, subtracted from 20 ft. 11', leaves 2 ft. 2', to which we bring down 3', the last term of the divi dend. We next seek how many times the divisor is contained in this remainder, and find by trial the quotient 7'; multiplying the divisor by this figure, we obtain 2 ft. 2′ 3′′, and there is no remainder. Hence the

RULE. I. Write the divisor on the left hand of the dividend, as in simple numbers.

II. Find the first term of the quotient either by dividing the first term of the dividend by the first term of the divisor, or by dividing the first two terms of the dividend by the first two terms of the divisor; multiply the divisor by this term of the quotient, subtract the product from the corresponding terms of the dividend, and to the remainder bring down another term of the dividend.

III. Proceed in like manner till there is no remainder, or till a quotient has been obtained sufficiently exact.

EXAMPLES FOR PRACTICE.

2. Divide 44 ft. 5' 4 by 16 ft. 8'

3. The square contents of a walk arc length is 40 ft. 11' 4"; what is the width?

Ans. 2 ft. 8'. 184 ft. 3', and the

Ans. 4 ft. 6'.

4. A blanket whose square contents are 14 ft. 6', is to be lined with cloth 2 ft. 7' wide; how much in length will be required?

Give analysis of example 1. Rule.

5. A block of granite contains 64 ft. 2' 5"; its widch is 2 ft. 6', and its thickness 3 ft. 7'; what is its length?

NOTE. Since the solid contents are the product of the three dimen sions, we divide the solid contents by any two dimensions or by their product, to obtain the other dimension.

Ans. 7 ft. 2'.

PROMISCUOUS EXAMPLES.

1. In 115200 grains Troy, how many pounds? -2. In 365 da. 5 h. 48 min. 46 sec., how many second

Ans. 31556926.

3. A man wishes to ship 1560 bushels of potatoes in bar rels containing 3 bu. 1 pk. each; how many barrels will be required?

4. Reduce 295218 inches to miles.

Ans. 480.

15. Reduce 456575 grains to pounds, apothecaries' weight.

Ans. 79 lb 33 13 1 9 15 gr.

6. How many sheets in 3 reams of paper?

7. What is the value of 4 piles of wood, each 20 ft. long, 6 ft. wide, and 10 ft. high, at $3.25 cord? per Ans. $121.87.

Ans. 112.

8. How many bottles, each holding 1 qt. 1 gi., can be filled from a barrel of cider? 9. At $26.40 per sq. rd. for land, what will be the cost of a village lot 8 rd. long, and 44 rd. wide? Ans. $980.10. 10. Divide 259 A. 1 R. 10 P. of land into 36 equal lots. Ans. 7 A. 321 P.

11. How many times can a box holding 4 bu. 3 pk. 2 qt. be filled from 336 bu. 3 pk. 4 qt.? Ans. 70.

12. What is the value of .875 of a gallon?

13. What part of a mile is 2 fur. 36 rd. 2 yd.? Ans. • 14. What part of 2 days is 13 h. 26 min. 24 sec.?

15. From 26 A. 2 R. of land, 5 A. 3 R. were sold; what

part of the whole piece remained unsold?"

Ans.

16. What is the difference between of a pound sterling and 51 pence? Ans. 11 s. 6 d. 17. What is the sum of of a yard, 4 of a foot, and 4 of an inch? Ans. 7 inches.

18. Reduce 3 cwt. 1qr. 7 lb. of coal to the decimal of a long ton. Ans. .165625. 19. Benjamin Franklin was born Jan. 18, 1706, and George Washington Feb. 22, 1732; how much older was Franklin than Washington? Ans. 26 yr. 1 mo. 4 da. 20. The longitude of Boston is 71° 4′ west, and that of Chicago 87° 30′ west; when it is 12 M. at Boston, what is the time in Chicago? Ans. 10 h. 54 min. 16 sec. A. M.

21. If the difference of time between New York and New Orleans be 1 h. 4 sec., what is the difference in longitude? Ans. 15° 1'.

22 Add of a mile, of a furlong, and of a rod together. Ans. 5 für 33 rd. 8 ft. 3 in. 23. If a bushel of barley cost $.80, what will 20 bu. 3 pk. 6 qt. cost? Ans. $16.75. 24. What is the value of .875 of a gross? Ans. 10 doz. 25. How many acres in a field 563 rods long, and 24.6 rods wide? Ans. 8 A. 2 R. 29.9 P. 26. How many perches of masonry in the wall of a cellar which is 20 feet square on the inside, 8 feet high, and 11⁄2 feet in thickness? Ans. 44.6+

27. A, B, and C rent a farm, and agree to work it upon shares; they raise 640 bu. 3 pk. of grain, which they divide. as follows: one fourth is given for the rent; of the remainder A takes 13 bu. more than one third, after which B takes one half of the remainder less 7 bushels, and C has what is left; how much is C's share? Ans 161 bu. 3 pk. 6 qt.

28. What is the value in Troy weight of 13 lb. 8 oz. 11.4 dr. avoirdupois weight? Ans. 16 lb. 5 oz. 10 pwt. 11.7+gr. 29. If 154 bu. 1 pk. & qt. cost $173.74, how much will 1.5 bushels cost? Ans. $1.687+. 30. What is the value of .0125 of a ton? Ans. 25 lbs. 31. What fraction of 3 bushels is of 2 bu. 3 pk.?

Ans. 77

32. How many wine gallons in a water tank 4 feet long,

8 feet wide, and 1 ft. 8 in. deep?

Ans. 174.