28. If a tree 20 ft. high cast a shadow 30 ft. long, how long will be the shadow of a tree 50 ft. high?

29. How long will it take a steam ship to sail round the globe, allowing it to be 25000 miles in circumference, if she sails at the rate of 3000 miles in 12 days?

30. How many acres of land can a man buy for $840, if he pays at the rate of $56 for every 7 acres?

31. How much will 85 cwt. of iron cost, at the rate of $91 for 13 cwt.?

32. At the rate of $45 for 6 cwt. of beef, how much can be bought for $980?

33. If 9 ounces of silver will make 4 tea spoons, how many spoons will 25 pounds of silver make?

34. If 15 tons of wool are worth $90000, how much is 5 cwt. worth?

35. If 5 yds. of cloth is worth $271, how much is 501 yds. worth?

36. If 60 men can build a house in 90 days, how long will it take 15 men to build it?

37. A bankrupt owes $25000, and his property is worth $20000: how much can he pay on a dollar?

38. At 7s. 6d. per week, how long can a man board. for £24, 10s.?

39. What cost 94 tons of coal, if 141 tons cost £85? 40. What cost 291 yds. of cambric, if 13 yds. cost £8, 6s. 31d.?

41. What cost 3 lbs. of raisins, at £6,7s.6d. per 100 lbs.? 42. If 20 sheep cost £37, 12s., what will 311 cost? 43. At 7s. 6d. per ounce, what is the value of a silver pitcher weighing 9 oz. 13 pwt. 8 grs.?

44. If 405 yards of linen cost £69, 7s. 6d., what will 243 yards cost?

45. A can saw a cord of wood in 6 hours, and B in 9 hours how long will it take both together to saw a cord? 46. A cistern has 3 cocks, the first of which will empty it in 10 min.; the second in 15 min.; and the third in 30 min. how long will it take all of them together to empty it?

47. A man and a boy together can mow an acre in 4 hours; the man can mow it alone in 6 hours; how long will it take the boy to mow it?

COMPOUND PROPORTION.

329. Compound Proportion is an equality between a compound ratio and a simple one. (Arts. 308, 310.) Thus, 8: 4 And 6:35

123, is a compound proportion.

That is, 8 x 6:4x3:: 12:3; for, 8x6x3=4×3×12.

OBS. Compound Proportion is often called the Double Rule of Three.

Ex. 1. If 4 men can earn $24 in 6 days, how much can 10 men earn in 8 days.

Suggestion. When stated in the form of a compound proportion, the question will stand thus:

4m.: 8m. 6d. : 10d.

}

:: $24 to the answer required. That is, "the product of the antecedents, 4×6, has the same ratio to the product of the consequents, 8X10, as $24 has to the answer."

Operation. 24x8x10=1920,

and 4×6=24.

We divide the product of all the numbers standing in the 2d and 3d places of the proportion,

Now 1920-24-80. by the product of those standing Ans. 80 dollars. in the first place.

Note.-1. The learner will observe, that it is not the ratio of 4 to 8 alone, nor that of 6 to 10, which is equal to the ratio of 24 to the answer, as it is sometimes stated; but it is the ratio compounded of 4 to 8 and 6 to 10 which is equal to the ratio of 24 to the anThus 4X6: 8X10 :: 24: 80, the answer.

swer.

For, 4×6×80=8X10X24. (Art. 324.)

2. A compound proportion, when stated as above, is read, “ the ratio of 4 into 6 is to 8 into 10 as 24 to the answer."

2. If 5 men can mow 20 acres of grass in 4 days, working 10 hours per day, how much can 8 men mow in 5 days, working 12 hours per day?

Operation.

Acres.

5m. : 8m.

State the question, then multiply and divide as be

10hr. 12hr.

fore.

8×5×12×20=9600; and 5×4×10=200. Now 9600

200 48. Ans. 48 acres.

QUEST.-329. What is compound proportion? Obs. What is compound proportion sometimes called?

330. From the foregoing illustrations we derive the following general

RULE FOR COMPOUND PROPORTION.

I. Place that number which is of the same kind as the answer required for the third term.

II. Then take the other numbers in pairs, or two of a kind, and arrange them as in simple proportion. (Art. 327.) III. Finally, multiply together all the second and third terms, divide the result by the product of the first terms, and the quotient will be the fourth term or answer required.

PROOF.-Multiply the answer into all of the first terms or antecedents of the first couplets, and if the product is equal to the continued product of all the second and third terms multiplied together, the work is right. (Art. 324.) It may also be proved by analysis. (Art. 295.)

OBS. 1. Among the given numbers there is but one which is of the same kind as the answer. This is sometimes called the odd term, and is always to be placed for the third term.

2. Questions in Compound Proportion may be solved by Analysis ; also by Simple Proportion, by making two or more separate statements. (Arts. 302. N. 327.)

3. If 8 men can clear 30 acres of land in 63 days, working 10 hours a day, how many acres can 10 men clear in 72 days, working 12 hours a day?

8×63×10:10 × 72 × 12: : 30 to the answer.
But the prod. 10 × 72 × 12 × 30
Divided by

8 × 63 × 10

the Ans. (Art. 330.)

QUEST.-230. What is the rule for compound proportion? How are questions in compound proportion proved? Obs. Among the given numbers, how many are of the same kind as the answer? Can questions in compound proportion be solved by simple proportion? How?

Now by canceling equal factors, (Art. 140,) we have

331. After stating the question according to the rule above, if the antecedents or first terms have factors common to the consequents or second terms, or to the third term, they should be canceled before performing the multiplication and division.

Note. Instead of placing points between the first and second terms, that is, between the antecedents and consequents of the left hand couplets of the proportion as above, it is sometimes more convenient to put a perpendicular line between them, as in division of fractions. (Art. 140.) This will bring all the terms whose product is to be the dividend on the right of the line, and those whose product is to form the divisor, on the left. In this case the third term should be placed below the second terms, with the sign of proportion (:) before it, to show its origin.

4. If a man can walk 192 miles in 4 days, traveling 12 hours a day, how far can he go in 24 days, traveling 8 hours a day?

That is, the product of the antecedents, 4x 12, has the same ratio to the product of the consequents, 24 × 8, as 192 has to the answer required.

Ans. 192×2×2=768 miles.

5. If 8 men can make 9 rods of wall in 12 days, how many men will it require to make 36 rods in 4 days?

6. If 5 men make 240 pair of shoes in 24 days, how many men will it require to make 300 pair in 15 days? 7. If 60 lbs. of meat will supply 8 men 15 days, how long will 72 lbs. last 24 men?

8. If 12 men can reap 80 acres of wheat in 6 days, how long will it take 25 men to reap 200 acres?

9. If 18 horses eat 128 bushels of oats in 32 days, how many bushels will 12 horses eat in 64 days?

10. If 8 men can build a wall 20 ft. long, 6 ft. high,

QUEST.-331. When the antecedents have factors common to the consequents, what should be done with them?

and 4 ft. thick, in 12 days, how long will it take 24 men to build one 200 ft. long, 8 ft. high, and 6 ft. thick?

11. If 8 men reap 36 acres in 9 days, working 9 hours per day, how many men will it take to reap 48 acres in 12 days, working 12 hours per day?

12. If $100 gain $6 in 12 months, how long will it take $400 to gain $18?

13. If $200 gain $12 in 12 months, what will $400 gain in 9 months?

14. If 8 men spend £32 in 13 weeks, how much will 24 men spend in 52 weeks?

15. If 6 men can dig a drain 20 rods long, 6 feet deep, and 4 feet wide, in 16 days, working 9 hours each day, how many days will it take 24 men to dig a drain 200 rods long, 8 ft. deep, and 6 ft. wide, working 8 hours per day?

SECTION XIII.

DUODECIMALS.

332. Duodecimals are a species of compound numbers, the denominations of which increase and decrease uniformly in a twelvefold ratio. Its denominations are feet, inches or primes, seconds, thirds, fourths, fifths, &c.

Note. The term duodecimal is derived from the Latin numeral duodecim, which signifies twelve.

1"1 of 1 in. or 12 of 12 of 1 ft-1 of 1 ft.

1""=12 of 1", or 12 of 12 of 12 of 1 ft.=172 of 1 ft.

QUEST.-332. What are duodecimals? What are its denominations? What is the meaning of the term duodecimal? Repeat the Table. Obs. What are the accents called, which are used to distinguish the different denominations?