= 126roods, as before, by Compound Proportion.

SECOND EXAMPLE WORKED OUT. REDUCTION REQUIRED BEFORE STATING. If the cost of excavating a rectangular ditch, 95 ft. 3 in. long, 6 ft. 9 in. wide, and 6 ft. deep, be £5 19's. Oåd., what value of labour will be expended in digging another such trench, in similar ground, but 81 ft. 9 in. long, by 5 ft. 9 in. broad, and 4 ft. 3 in. deep?

Here, all the seven terms will require to be reduced before the work can be commenced.

It happens, in the present example, that all the six given terms of lineal measure can be reduced to quarter-feet.

Abbreviated Question; before reducing the terms.

ft. in.

£ S. d.

If 95 3 (long)... 6' 9' (wide)... 6 0 (deep)... 5 19.0

ft. in.

ft. in.
60.

EXERCISE CXX.

EXAMPLES IN COMPOUND

PROPORTION, OR DOUBLE

RULE OF THREE.

Work each example in two ways:

1. By Compound Proportion,

II. By Multiplication and Division, without reference to Ratio: explaining every line.

III. Resolve each question into the several Simple Proportion questions of which it is composed. See Pages 343 and 344.

A.

1. If £28 will pay for the board of 16 persons for 7 weeks, for how many will £39. pay for 4 weeks?

2. Calculating that 16 horses plough 128 acres in 8 days, in what time will 14 horses plough 56 acres?

3. How many loaves will suffice 19 people for 23 days, supposing that 34 persons, in 6 days, will consume 51 such loaves? 4. Granting that a man can walk 315 miles in 14 days of 9 hours each, in how many such days will he go 160 miles, allowing himself one hour per day for repose?

5. If 63 cwt. be carried 19 miles for 4's. 6.d., how far should 57. cwt. be conveyed for 3's.?

6. What number of loaves at 10 d. each, will serve 32 persons for 9 days, if 14 persons in 11 days require 55 loaves at 7.d.?

7. If 48 bushels of corn last 15 horses for 13 days, for what length of time will 32 bushels suffice for 5. horses?

8. If £14 be the cost of maintaining 8 boys for 7 weeks, what would 25 cost for 11 weeks?

9. How long could 300 soldiers be provisioned for £577 10's., if the keep of 114 for 19 days cost £379 1's.?

10. If £48 pay for the board and lodging of 33 people for 16. days, what sum will defray the like expenses of 15 persons for 77 days?

11. How many loaves at 7.d. will serve 14 persons for 11 days, if ́ 48 loaves at 15 d. suffice sixteen persons for 18 days?

12. If 39 horses can be maintained 28 days for £19 10's., how many can be supported 49 days for £14.?

13. In how many days will 32 horses plough 256 acres, working at the same rate as 7 horses which plough 28 acres in 4. days?

14. A man, walking per day 8 hours incessantly, travels 160 miles in 8 days. How much longer must he walk per day, at the same rate of speed, to complete 630 miles in 28 days? 15. What is the price, each, of 90 loaves, which will provide 16. boys with the same daily allowance for 9 days, as 77. loaves at 5 d. would furnish to 7 boys for 22 days?

16. How many workmen will mow 128 acres in 8 days, when 21. can mow 84 acres in 4 days?

17. Supposing a man could walk 210 miles in 7 days of 9. hours each, how many hours per day might he allow himself to

rest if he were required to walk, at the same rate, only 80miles in 4 such days?

18. For how many people will 218 lb. of beef serve for 11 days, supposing that 34 consume 102 lb. in 3 days?

19. How many people will consume twenty eighteenpenny loaves in 4 days, at the same rate as 77 consume 35 elevenpenny loaves in two days?

20. If 15 cwt. 3 qr. be carried 4 miles for 1's. ltd., how far should 14 cwt. 1. qr. be carried for eighteenpence?

21. If 19 men by excavating 50 yards of trenching in 12 days, earn 4's. per day, in what time ought 24 excavators to clear out 45 yards, each being paid 4's. 6 d. per day?

22. If £4 5 s. 6d. will maintain 6 people for 19 days, for what length of time will £52. 10's. pay the current expenses of 300 people?

23. Supposing that 110 loaves at 3d. be sufficient for 28 persons for 5 days, how long would 144 loaves at 5 d. last 64. people?

24 How much interest would £100 gain in 1 year, at the same rate that £103. 10's. gain £3° 0′s. 44d. in 7 months?

25. What would be the cost of digging a trench 29. ft. 3 in. long, 6 ft. 9 in. broad, and 2. ft. 3 in. deep, if another trench, in similar ground, 31. ft. 6 in. long, 7. ft. 3. in. broad, and 4· ft. 6 in. deep, cost £2. 10's. 9.d.?

26. If 16 men, by raising 5130 Tons of ore, in 57 days, earn 2's. 6 d. per day, what would be a fair daily wage for each of 20' men, who should raise 2250 Tons in 24 days?

27. What amount of Principal would gain £1. 18's. 9 d. Interest, in 4 months, if £27. 17's. 6 d. in 8 months, gain 18's. 7.d.? 28. If the keep of 19 horses, for 22 days, cost £5' 4's. 6d. when oats are at 19/ per quarter, what would it cost to maintain 35 horses for 15 days, when oats are 25/4 per quarter? 29 For what sum should 28 cwt. 2 qr. be carried 14 miles, if 31 cwt. 2 qr. be conveyed 19 miles for 6's. 9.d.?

30 What weight should be carried 42 miles for 4's. 6 d., if 10 cwt. 2 qr. be carried 85 miles for 10's. 1d.?

EXERCISE CXX. B.

1. If 15 men mow 48 acres in 4 days, how many acres will 30. men mow in 3 days?

2. Supposing 250 sheep can be pastured on 8 acres of land, for 3 weeks, how much land will be requisite for 420 sheep for 15 weeks?

3. If a trench 29 ft. 3. in. long, by 6. ft. 9 in. wide, and 2. ft. 3 in. deep, cost in labour £1 1's. 111d., what will another cost which is 31. ft. 6 in. long, 7 ft. 3. in. wide, and 4. ft. 6 in. deep?

4 The excavation of a trench 16 chains long, 4.5 ft. wide, and 4.5 ft. deep, cost £39 6.fl., how many chains in length of a similar trench 12.5 links wide, and 7.5 feet deep, should be dug for £105⋅ 17 s. 6.d.?

5. If cwt. be carried 54 miles for 8ğs., how far will 14 cwt. be transported for £13?

6. Supposing that £103 10's. gain £1· 18′s. 9 d. in 4 months, in what time will £27· 8·fl. 7·c. 5 m. gain 18's. 7.d.?

7. If 72 men complete a wall 450 ft. long, 10' ft. high, and 15 in. thick, in 9 days, and thus earn 5's. 6 d. per day, how much per day would each of 48 men deserve, who should build, in 7 days, a wall of similar material, but 320 ft. long, 9' ft. high, and 25 inches thick?

8. If 8 men earn 4's. 6 d. per day each, by completing in 9 days, a wall 72. ft. long, 12. ft. 6 in high, and 3 ft. 6 in. thick, what length of wall, 14 ft. high, and 4 ft. thick, should 15. men build, in 16 days, so as to earn 6's. 9 d. per day each? 9. If 19 men, by digging a trench 199 ft. 6 in. long, by 12 ft. wide, and 8 ft. 6 in. deep, in 14 days, each earn 5's. 11 d. per day, in what time should 21 men complete an excavation 91 ft. 7 in. long, 9. ft. wide, and 5' ft. 8 in. deep, to earn 3's. 3 d. per day, per man?

10. If 24 men excavate a trench 99 ft. 9 in. long, 12 feet wide, and 8 ft. 6 in. deep, in 42 days, thus earning each 5's. 114d. per day, how many men should complete a trench 274. ft. 9. in. long, 9. ft. wide, and 2. ft. 10 in. deep, in 5 days, to earn, per man, 3's. 31d. per day?

11. If £5. 1's. 6 d. be the value of a log 57 ft. 3 in. long, by 4 ft. 10 in. wide, and 4 ft. 10 in. thick, what is the length of a log of similar wood, which, being 6: ft. 9 in. wide, and 2' ft. 3 in. thick, costs £1. 1's. 11d.?

12. What length of embankment 9. ft. wide, and 24 ft. high, will cost £19 16's. in building, if 7 chains long, 1 pole wide, and 7. ft. 6 in. high, cost £105. 17·s. 6·d.?

13. If 27 men, by building a wall 42. ft. long, 9 ft. high, and 1.ft. 3 in. thick, in 4 days, of 10 hours each, earn 4's. 6 d. each, per day, how much per day would each of 15 men earn, by building in 6 days of 12 hours each, a similar wall, 19' ft. long, 7 ft. high, and 2. ft. 1 in. thick?

14. If a rectangular-sided block of stone 9. ft. 11 in. long, 2 ft. 2 in. wide, and 1ft. 9 in. thick, can be quarried, dressed, and raised to a height of 99 ft., by 6 men, in 2 days of 10. hours, each man earning 5's. 10'd. per day, what would be a proportionate payment per day for each of 8. men, for like preparation and raising to the height of 42. ft. in 4 days of 9 hours, a block 23 ft. 10 in. long, 2. ft. 10 in. wide, and 2. ft. 3 in. thick?

PERCENTAGES.*

The chief commercial use of percentages, or allowances upon the hundred, is to compute

1. Commission, which is a payment for disbursing, receiving, or remitting money for another person.

11. Brokerage, charge for buying or selling goods, shares, &c., for another person.

III. Interest, payment for the loan of money.

IV. Insurance Premium, previous payment for undertaking to replace loss by fire, shipwreck, &c., or to pay a certain sum at a person's death.

The sum on which any one of the above named percentages is charged or computed is called Principal.

Any percentage is, evidently, equal to as many onehundreth parts of the Principal as there are units in the Rate. Thus:

=

1. per cent. of P. =P X 180 (PRIN. IX.)÷P.÷100 ̊ (PR. XVII.) 2 per cent.=180 of P. =P X180

P.X2100.

&c.

To find any percentage of a given Principal. Multiply Principal by Rate and Divide by 100. Or, (PRINCIPLE XVII.) Multiply Toth part of Principal by the no. of units and parts of a unit in the Rate.

Now, the one-hundredth part of any number is found by removing its digits two places to the right. Whence the ordinary form of the rule is

Multiply by the Rate, and cut off the two right-hand figures of the Product.

Which would be more correctly expressed thus:

Multiply by the no. of units and parts of a unit in the Rate, and remove digits of Product two places to the right, or Decimal Point two places to the left.

FIRST EXAMPLE WORKED OUT. Required the Commission or Brokerage on £648 11's. 8 d. at £21 per cent. Example completed.