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4. If 14 horses can be kept so many days (the last answer), on 1 bu. of grain, how many days can they be kept on 201 or 87 bushels ?

The student should observe that the question in each case has been the same as that of the original problem, i. e. “How many days,” and that the denomination of each answer is days.

57. If 43 lb. of meat will last 25 men 7 days, how many days will 58} lb. last 13 men ?

58. If 9f acres of land will produce hay enough to keep 11 cows 44 months, how many acres will produce hay enough to last 8 cows 215 months ?

59. If 12 men can build a wall 314 rods long in 5f days, in how many days can 20 men build a wall 734 rods long?

60. How many days will it take a man to dig a trench 78;ft. long, 48 ft. wide, and 2 ft. deep, if it takes him 9 days to dig a trench 204 ft. long, 2r ft. wide, and if ft. deep?

61. If 10 men, by working 12 hours per day and 54 days in the week, earn $366% in 2 weeks, how many dollars will 15 men, working 9$ hours per day and 6 days per week, earn in 3} weeks?

62. If 18 bu. of corn are worth 24 bu, of oats, and 15 bu. of oats are worth 27 lb. of butter, how many pounds of butter are 12 bu. of corn worth?

SOLUTION. — As the answer is to be in pounds of butter, we write 27 (the number of pounds of butter mentioned in the question), above a line as a numerator, and take care to express the values throughout in the same denomination. Since 15 bu. of oats were worth 27 lb. of butter, 1 bu. of oats is worth 4 of 27 lb. of butter, expressed by making 15 a factor of the denominator, and 24 bu. of oats must be worth 24 times the last result, expressed by making 24 a factor of the numerator. But as this is also the value of 18 bu. of corn, 1 bu. of corn must be worth of this, expressed by making 18 a factor of the denominator, and 12 bu. of corn must be worth 12 times the last result, expressed by making 12 a factor of the numerator. This gives the following written work:

3 8 6

27 X 34 X H 1b. of butter, **

18 X 18

- = 245 lb. of butter. Alns.

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64. If 7 peaches are equal in value to 4 pears, and 9 pears to 24 apples, and 16 apples to 35 plums, and 8 plums to 60 cherries, how many cherries are equal in value to 56 peaches ?

65. If Edward can walk 6 miles while Arthur walks 4, and Arthur can walk 10 miles while Robert can drive 21, and Robert can drive 7 miles while Samuel can read 12 pages, and Samuel can read 18 pages while David can perform 4 problems, how many problems can David perform while Edward is walking 8 miles ?

66. When 16 sheep are equal in value to 5 cows, and 6 cows to 4 oxen, and 15 oxen to 8 horses, how many horses are equal in value to 36 sheep ?

67. Multiply .06 by .036, divide the result by .0012, and add .044 of 1.37 'to the quotient.

68. Divide 53.7 acres of land into house-lots each containing .375 of an acre.

69. A piece of work was to be completed in 40 days, but, after 50 men had been employed 30 days, only of it was done. How many men must be employed to complete the remainder of the work in the required time?

70. I gave 25.38 yards of cloth, worth $3.75 per yd., in exchange for 7.2 tons of hay. I sold .25 of the hay at an advance of $1.50 per ton on its cost, and the rest at the rate of $12.75 per ton. Did I gain or lose, and how much?

71. I owned a pile of wood 37.5 ft. long, 4 ft. wide, and 4.6 ft. high. From it I sold 1 cord 3} cord feet at $6.25 per cord, and the rest for enough to make up $35. For how much per cord did I sell the last lot?

109. Duodecimal Fractions.

(a.) DUODECIMAL FRACTIONS, or simply DUODECIMALS, are fractions whose denominator is 12, or some power of 12, and, instead of being written, is indicated by one or more marks or accents placed at the right of the numerator and a little above it.

Thus: 4' = t; 7" =11; 11"' =itto, etc.

(b.) In reading duodecimals, twelfths are usually called PRIMES; one-hundred-and-forty-fourths are called SECONDS, etc.

(c.) Duodecimals are employed in measuring lengths, surfaces, and solids ; so that the unit of measure is a foot of either Long, Square, or Cubic Measure, according to the nature of the thing measured.

(d.) Since a linear inch equals t of a linear foot, a square inch ita of a square foot, and a cubic inch of a cubic foot, it follows that in Long Measure the inch is represented by the prime, in Square Measure by the second, and in Cubic Measure by the third.

(e.) Duodecimals may be added, subtracted, multiplied, and divided, as other fractions are. In performing these operations, it is necessary to notice that a unit of any denomination equals 12 units of the next lower denomination, and it of a unit of the next higher denomination; also that 1' x 1' = t X = id or l"; that 1" x 1' = ida x t = hteo or 1", etc.

1. What are the contents of a floor 16 ft. 5' long and 11 ft. 8 wide ?

REASONING PROCESS. - A floor 16 ft. 5' long and 1 ft. wide must contain 16 sq. ft. 5'. Hence a floor 16 ft. 5' long and 11 ft. 8' wide must contain 118 times 16 sq. ft. 5'. The work may be written as below.

EXPLANATION. — Multiplying 16 5 by 8', we have 8' times 5' = 40" = 3' 4" (i. e. ex pe = M =

10 11' 4" Bet 14a). Writing the 4", we

180 71 have 8' times 16 ft. = 128 (i. e. & x 16 ft. = 448 ft.), 191 6' 44" = 191 sq. ft. 76 sq. in. and 3' added equal 131' = 10 ft. 11', which we write.

Now, multiplying by 11, we have 11 times 16 ft. 5' = 180 ft. 7', which, added to the product by 8', gives 191 ft. 6' 4"', which, being in Square Measure, the 4" = 4 sq. in., and the 6' = 6 times 12 sq. in. The result then equals 191 sq. ft. 76 sq. in.

2. What are the contents of a board 17 ft. 5' long and 2 ft. 1 wide ?

3. What are the contents of a platform 18 ft. 7 long and 9 fte 4' wide ?

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4. How many square feet and inches are there in the outer surface of a box 6 ft. 3' long, 5 ft. 5' wide, and 2 ft. 7/ high?

5. How many square feet and inches are there in the inner surface of the box described in the last problem, allowing that the boards composing it are 1 inch thick ?

6. How many feet of boards will it take to make the abovenamed box?

7. What are the solid contents of a block of granite 8 ft. 11 long, 3 ft. 5' wide, and 2 ft. 6' thick ?

8. How many cubic feet of earth would be removed in digging a ditch 83 ft. 8' long, 4 ft. 4wide, and 2 ft. 10% deep?

9. How much will a pile of wood 26 ft. 3' long, 4 ft. wide, and 5 ft. 8 high, cost at $.75 per cord foot ? '

10. How many square yards of carpeting will it take to cover a floor 17 ft. 3' long and 14 ft. 6' wide ?

11. How many yards of carpeting of a yard wide will it take to cover a floor 16 ft. 5' long and 14 ft. 9' wide ?

12. My room is 15 ft. 6' long, 14 ft. 10' wide, and 9 ft. 2 high. It contains 3 windows each 5 ft. 4' high and 2 ft. 11' wide: two doors each 6 ft. 11' high and 2 ft. 11' wide; a closet-door 6 ft. 11° high and 2 ft. 9' wide; and a mopboard 6' wide extending around the room. How many square yards of plastering are there in its top and walls ?

13. A man who owned a garden spot 127 ft. 10% long and 96 ft. 8 wide, built a heavy stone-wall around it 8 ft. 5/ high and 2 ft. 4' thick, leaving space for one gate 7 ft. 6' wide and as high as the wall, and for two smaller gates each 2 ft. 7' wide and 6 ft. 8' high. How many cubic feet did the wall contain ?

Note.- Observe what allowances are to be made for the thickness of the wall.

14. How many bricks each 8? long, 4' wide, and a thick will be required to build the four walls of a house 23 ft. 2 long, and 18 ft. 7' wide, the walls to be 21 ft. 3' high and 1 ft. thick, and allowance to be made for 1 door 7 ft. 10% high and 3 ft. 8' wide, for 1 door 7 ft. 10% high and 3 ft. 2' wide, and for 18 windows each 5 ft. l' high and 3 ft. 2 wide ?

SECTION XVI.

RATIO AND PROPORTION.

103. Ratios.

(a.) Ratio is the part which one number is of another, or the quotient of one number divided by another.

ILLUSTRATIONS.— The ratio of 3 to 4 is 4, or 14, because 4 equals of 3, or 14 times 3; or because 4 : 3 = = 13.

The ratio of 6 to 18 is 18, or 3, becauso 18 equals 4 of 6, = 3 times 6; or because 18 + 6 = 18 = 3.

The questions in 90 were really questions in Ratio. Note.—Some writers consider the ratio of 3 to 4 to be 3, and that of 6 to 18 to be , instead of being and 49, as in the above illustrations. The difference is not practically of much consequence, for the term ratio is almost invariably used in some such connection as the following: “9 has the same ratio to 12 that 6 has to 8,” which, by the first interpretation, means that ý = f, or f = ; and, by the second, i = f, or = }, both of which are manifestly true. The important thing is to apply the same interpretation to all ratios which are compared with each other.

(b.) A ratio can always be established between abstract numbers, but it can only exist between concrete numbers when they are of the same denomination.

ILLUSTRATION.–6 plums are no part of 3 peaches, and hence have no ratio to them.

(c.) Every ratio is a true fraction, and may be written and dealt with as such.

(a.) Ratios are usually expressed by writing one number after the other, and placing two dots between them.

ILLUSTRATION. - The ratio of 5 to 7 = 5:7, or 7.

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