16. What part of a piece of work can be performed in 30 days, if with the same force of men of it can be performed in 25 days?

17. If of the marked price of an article yield a profit of of its cost, what part of its cost would be gained by selling the article at of its marked price?

18. What principal will accrue $1.59 interest, if $820 will accrue $5.30 interest in the same time and at the same rate %?

19. A received a dividend of $420 on $8400 of stock. What would he have received if he had owned $9300 of the same stock?

20. In the preparation of a certain compound, 12 gills of water were mixed with gills of other ingredients. How many gills of other ingredients should be mixed with 42 gills of water?

21. How long a shadow will be cast by a vertical staff 11 feet high, if at the same moment a staff 6 feet high casts a shadow 4 feet long? 22. A pole 18 feet high casts a shadow 6 feet long. How high is a tree which, at the same time, casts a shadow 13 feet long?

23. Two houses cast shadows respectively 15 feet and 18 feet in length. If the height of the higher house is 27 feet, what is the height of the lower?

24. How many men can do the same work in 20 days that 45 men can do in 16 days?

25. A contractor agreed to build a house in 45 days, and estimated that 6 men would be sufficient to do the work. Should he desire to complete the job in 30 days, how many men must he employ?

26. A invested $5400 in a partnership, and B $3600; and the contract between them specified that each partner's gain or loss should be proportional to his investment. If the total gain is $1500, what should be each partner's share?

27. A merchant sells $3600 worth of goods and gains $648. At the same%% of profit, what amount of goods must he sell to gain $1620?

28. What should be the tax on a farm assessed at $32500, if a farm assessed at $7500 is taxed $18.21?

29. In what time can the whole of a piece of work be done, if of it can be done in 12 days?

30. A merchant sold an invoice of goods at 15% profit and gained $450. What % would he have gained by selling the invoice at a profit of $750?

31. A is given a start of 12 rods in a foot-race, and can run at the rate of 30 rods per minute. In what time will he be overtaken by his opponent, who can run at the rate of 33 rods per minute?

32. If, in Ex. 31, A had a start of 20 seconds, in what time would his opponent have overtaken him?

33. A grocer's pound weight weighs only 15 ounces, and his other weights in the same proportion. Out of what amount does he defraud his customers by using these weights in selling groceries for which he receives $320?

34. A merchant has a yard-stick which measures only 34 inches, and finds that by its use during a certain week he has defrauded his customers out of $23. What were his receipts during that week?

35. A borrowed $1200 from B, and kept it 1 year and 3 months. For what time should A lend $1500 to B to offset B's claim for interest? 36. If 15 men can plow a field in 12 days, in how many days can 18 men plow it?

37. An army of 12000 men has provisions for 14 days. How long will these provisions last if the army is reinforced by 9000 men?

38. If a man can make a journey in 18 days by traveling 12 hours each day, in what time can he make the same journey by traveling 10 hours each day?

39. If 3289 stones, each 2 feet in length, are required to build a certain wall, how many stones of the same width and depth, but 24 feet long, will be required to build the same wall?

40. A contractor could have executed a certain contract in 30 days by employing his full force of 24 men; but he detached a portion of his men to perform another piece of work, and with the remainder executed the contract in 40 days. How many men did he detach?

41. If A lend B $800 for 18 days, how long should B lend A $300 to offset his claim for interest?

42. If a vessel is provisioned for a crew of 30 men and a voyage of 168 days, how long a voyage can she make with the same provisions if the crew be increased to 35 men?

43. At what time between 7 and 8 o'clock will the hour and minute hands of a clock be together?

44. At what time between 4 and 5 o'clock will the hour and minute hands of a clock be together?

45. The two hands of a clock are together at 12 o'clock. When will they be next together?

46. At what time between 9 and 10 o'clock will the hour-hand be 6 minute-spaces ahead of the minute-hand?

COMPOUND PROPORTION.

When the required term in the second couplet of a proportion depends on the compound ratio of its first couplet, separately indicate the several simple ratios of the first couplet by taking each quantity mentioned in the question as a standard of comparison, making it the consequent of its couplet, if it is seen that the required fourth term must increase as the standard is increased, or decrease as the standard is decreased [direct proportion]; or making it the antecedent of its couplet if it is seen that the required fourth term must decrease as the standard is increased, or increase as the standard is decreased [inverse proportion].

EXAMPLES.

1. If 4 men can earn $84 in 7 days by working 10 hours a day, how many dollars can 6 men earn in 10 days by working 9 hours a day?

2. If $400 will accrue $5.60 interest in 84 days at 6% per annum, how much interest will $320 accrue in 96 days at 9% per annum?

3. A printer cut 30 pages of a certain book from a sheet of paper 18 inches wide and 25 inches long. How many pages of a book of the same size would a sheet yield which is 20 inches wide and 24 inches long?

4. A lumber dealer purchased two piles of wood. One pile was 24 feet long, 20 feet high, 16 feet wide, and contained 60 cords. The other pile was 32 feet long, 12 feet wide, and 18 feet high. proportion, the number of cords in the latter pile.

Find, by

5. An excavator contracted to dig two cellars at the same price per cubic yard, and obtained $120 for one which was 40 feet long, 30 feet wide, and 9 feet deep. The other cellar was 36 feet long, 27 feet wide, and 12 feet deep. How much was received for excavating the latter?

6. If 300 men can perform a certain work in 45 days by laboring 10 hours a day, how many hours a day will 600 men have to labor to perform the same work in 25 days?

7. If 20 men can earn $1260 in 15 days by working 12 hours a day, how many dollars can 17 men earn in 23 days by working 10 hours a day?

8. How many barrels of flour will supply a family of 9 persons for 20 weeks, if 3 barrels will supply a family of 8 persons for 15 weeks?

9. A livery-stable keeper's expenses for feeding 27 horses for 17 weeks were $91.80. What would be his expenses for feeding 23 horses for 13 weeks?

10. What should be the cost of pasturing 73 head of cattle for 7 weeks, if the cost of pasturing 95 head of cattle for 9 weeks is $513?

11. The freight on 12250 pounds of goods was $73.50 for 240 miles. A pro rata sum was charged on 28250 pounds of the same class of goods for 90 miles. How much was paid for freight for the shorter distance?

12. If $600 will accrue $9.60 interest at 6% per annum in 3 months and 6 days, at what rate per annum will $240 accrue $7.20 interest in 4 months?

13. If $840 will accrue $19.60 interest in 105 days at 8% per annum, in what time will $576 accrue $1.84 interest at 5%?

14. If, from 650 reams of paper, 4000 copies of a book containing 420 pages can be printed, how many reams will be required to print 7000 copies of a book containing 528 pages of the same size?

15. A contractor purchased a pile of stone 24 feet long, 12 feet high, and 9 feet wide for $120, and paid $300 for another pile of the same quality of stone which was 12 feet wide and 36 feet long. How high was the latter pile?

16. If 3 pipes of equal size will fill a cistern 5 feet long, 4 feet wide, and 9 feet deep in 3 hours and 20 minutes, in what time will 4 such pipes fill a cistern which is 3 feet long, 2 feet wide, and 5 feet deep?

17. If 250 men can dig a ditch 48 rods long, 24 feet wide, 3 feet deep, and 5 degrees of hardness, in 8 days of 10 hours each, in how many days of 8 hours each can 40 men dig a ditch 341 yards long, 3 feet wide, 4 feet deep, and 3 degrees of hardness?

18. If 7 men can plow a field 144 rods long and 105 rods wide in 9 days by using horses whose working gait is 5 miles per hour, in how many days can 5 men plow a field 216 rods long and 60 rods wide with horses whose gait is 4 miles per hour?

PARTITIVE PROPORTION.

INDUCTIVE EXERCISE.

1. If A and B together have $25 when A has $5 more than B, how much would A and B have together if A had the same sum as B? How much would each then have? If each has $10 when both have an equal sum, what would each have if A had $5 more than such equal sum?

2. A and B together have $30; how much has each if A has $6 more than B? If A has $8 more than B? If A has $12 more than B? If A has $20 more than B?

3. If A and B together have $25 when A has $5 less than B, how much would A and B have together if A had the same sum as B? How much would each then have? If each has $15 when both have an equal sum, what would each have if A had $5 less than such equal sum?

4. A and B together have $40; how much has each if A has $4 less than B? If A has $10 less than B? If A has $20 less than B?

[a] Take one of the parts as the standard of comparison, expressing the remaining parts in terms of that standard, and finding how many times the standard will denote the given value of all the parts.

EXAMPLES.

1. Divide 97 pounds of sugar into three parts so that the second part may weigh 15 pounds more than the first, and the third 8 pounds less than the second.

2. A and B started into business with a capital of $15000, of which A invested $1500 more than B. What was the investment of each?

3. The aggregate age of A, B, C, and D is 148 years: B is 6 years older than A; C is 16 years older than B; and D 14 years younger than C. What was the age of each?