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Ex. 2. If the penny loaf weighs 4 ounces when flour is 4s. per peck, how much must it weigh when flour is 5s. 4d. per peck?

Ex. 3. A person lent me 240/. for 8 months: in return for his kindness, how much ought I to lend him for eighteen months?

Ex. 4. How many men must be employed to finish a canal in 12 days, which 5 could perform in six weeks, or 36 days?

Ex. 5. If 24 pioneers can make a trench in 12 days, what length of time would the same work employ 9 men?

Ex. 6. The floor of a chapel 96 feet in length and 70 feet in breadth, is to be covered with matting 2 feet six inches broad: how many yards will it require?

Ex. 7. If a person travel 12 hours a day, and finish his journey in three weeks: how long would the same journey take him, if he travelled only 9 hours a day at the same rate?

Ex. 8. If the town and garrison of Bhurtpoor, containing 22,400 persons, have provisions to last three weeks, how many inhabitants must Holkar send away, so as to make the same provisions last 7 weeks, which is as long as General Lake can carry on the siege ?*

Ex. 9. If a besieged garrison have 4 months provisions, at the rate of 18 ounces per man per day: how long will they be able to hold out, if each man is allowed only 12 ounces per day?

Ex. 10. If there are in a garrison provisions sufficient for 1500 men 10 weeks, which, on account of the rains, is seven weeks longer than the siege can last: how many soldiers may be brought in to defend the place for three weeks, without lessening the quantity of food to any individual? +

Ex. 11. If 9 plasterers can finish the inside of a chapel in 10 days: how long will it take 4 men, supposing the other 5 sent away to a new job?

Ex. 12. If 3 yards of broad cloth, 12 wide, will make a suit of clothes: how much will be necessary of cloth only wide?

NOTES.

that when flour is 4s. a peck, the loaf weighs 4 ounces; these are the first and second terms, and the question is, how much it will weigh when fiour is 5s. 4d. per peck. The answer, or unknown quantity, is weight. I therefore state it thus: 4s.: 4 oz. :: 5s. 4d. : 3 oz. the answer. To prove the truth of it I say, if I have 3 oz. of bread when flour is 5s. 4d. per peck, how much shall I have when it is 4s. per peck: thus 5s. 4d. : 3 oz.: 4s. and I find the answer is 4 ounces.

*The answer to this question is the number of people to be sent away; therefore, when I have found how many the provisions will support for 7 weeks, I subtract this number from that given, and the remainder shews what number are to be dismissed, which in this case will be found to be 12,800.

Having found the number of men that may be supported three weeks, subtract from that number the 1500 already in the garrison, and the remainder is the true answer.

Ex. 13. If 52 clerks in the Bank are sufficient to make up the books in a certain office in 15 days, how many clerks would be required to do the same work in 6 days?

Ex. 14. If the carriage of 15 cwt., for 60 miles, came to 7s. 9d: how far can I have carried 33 cwt. for the same sum?

Ex. 15. The apartment in which the late Duke of Gloucester lay in state previously to his funeral, was 50 feet long, 40 feet broad, and 24 feet in height: how many yards of black cloth, 1 yards wide, were used in covering the walls, and how much did it cost at 188. per yard.*

Ex. 16. If 12 inches in length, and 12 inches in breadth, make a square foot: what length of board, 8 inches broad, will be equal to the same measure?

Ex. 17. If 220 yards in length, and 22 in breadth, make an acre : what must be the breadth when the length is 121 yards?

Ex. 18. If 5 horses can be maintained when oats are 18s. per quarter: how many can be supported at the same cost, when they are 30 shillings per quarter?

Ex. 19. If 250l. gaín 127. 10s. at interest, in 12 months. what principal will gain an equal sum in 5 months?

Ex. 20. There are two rooms, in the floors of which there are an equál number of square feet; the length of the one is 50 feet, and its breadth is 42; but the breadth of the other is 48 feet: what is its length?

Ex. 21. The cock to a large water-tub will empty it in 36 minutes : how many such cocks will empty it in 4 minutes?

Ex. 22. The sides of a room are found to measure 138 feet in length, and the height is 14 feet 6 inches: how much paper, 2 feet 3 incl.es wide, will cover it; and what is the value of it at 9d. per yard?

Ex. 23. If 50 cows can be kept in a field 17 days: how long will the same pasture feed 70 cows?

Ex. 24. How many Venetian ducats, at 4s.4d. each, must I take in payment for 560 English crowns?

NOTE.

* As there are four sides to the room, add the length to the breadth, and multiply by 2, which gives the length of the sides: then say, as the height of the room is to the length of the sides found, so is the breadth of the cloth to the quantity used. The value of the cloth is found afterwards by the Rule of Three Direct,

THE DOUBLE RULE OF THREE.

The Double Rule of Three teaches, from five given numbers to find a sixth. Three of the numbers contain the suppositions, and the remaining two are terms of demand.

RULE (1.) Put the terms of supposition one above another in the first place, except that which is of the same nature with the term sought, which put in the second place.

(2.) Place the terms of demand one above another in the third place, in the same order as the terms of the supposition were put in the first place.

(3.) The first and third term in every row will be of the same nature, and must be reduced to one denomination; and the middle term must be brought to the lowest denonination mentioned.

(4.) Examine each stating separately, using the middle terin as common to both, in order to know if the proportion be direct or inverse. When it is direct mark the first term with an asterisk, and when it is inverse, mark the third term with an asterisk.

(5.) Multiply the numbers together which are marked for a divisor, and those which are not marked for a dividend, and the quotient will be the answer.

Ex. 1. If 12 persons spend £.160 in 4 months: how much will 32 persons expend in 8 months?

persons. £. persons.

*12: 160 :: 32

months.

* 4:

months.

::

or,

8

The terms of supposition are, that 12 persons spend, in 4 months, 160.; the 12 and 4 are therefore the first terms; and as the answer will be in money, the 160l. is the middle term. The terms of demand are, How much 32 persons will expend in 8 months; these are accordingly the third It is evident, from inspection, that both the statings in this example. are in direct proportion, because the fourth terms will be greater

12 X 4 160 :: 32 X 8

terms

than the second, that is, 32 persons will expend more than 12, and s months expenditure will be greater than 4 months. *

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Ex. 2. If a garrison of 600 men have provisions for 5 weeks, allowing each man 12 ounces per day: how many men can be maintained 10 weeks by the same quantity, if each man is limited to 8 ounces a day?

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Ex. 3. If 15 pecks of wheat will last a family of 9 persons 22 days: in how many days will six persons consume 20 pecks?

pecks. days. pecks. *15 : 22 ::

persons. 9 :

or,

9 X 22 X 20

15 X 6

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20

6*

In the first stating of this example, the proportion is direct, because a greater quanpersons. tity of wheat will last a greater number of days. In the second stating the proportion is inverse, because a smaller number of people will require more days to eat the same 44 days. quantity of wheat. The divisors are therefore 15 and 6, and the answer is 44 days.

Ex. 4. If 6 pioneers can dig a ditch 34 yards long in 10 days: how many yards may be dug by 20 men in 15 days?

Ex. 5. If 1050 soldiers consume 250 quarters of corn in 6 months : how many soldiers will 960 quarters serve 4 months?

NOTE.

* Examples in the Double Rule of Three may be worked by two statements in the Single method; and this will be a good method of proving the truth of the sums. Thus, in the foregoing example, I say,

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2d Statement, 4 : 426 13 4 :: 8 : 853 6

8

Thus the fourth term in the proportion, found by the first stating,

becomes the middle term of the second stating.

Ex. 6. If a cask of beer last 8 persons 14 days: how many casks will serve 2 persons 365 days?

Ex. 7. If 10 men in six weeks earn gol.: how many weeks must 15 men work to earn 150l.?

Ex. 8. Suppose I walk 66 miles in 4 days, of eight hours each day: how many days, of 14 hours each, shall I be in going from London to York, or 196 miles ?

Ex. 9. If 3 boats take 6000 herrings in 8 days: how long will 600 boats be in taking 20,000 barrels, each containing 700 herrings? *

Ex. 10. If, against a general mourning, 6 tailors can make 10 suits of clothes in 4 days: how many suits can 600 men make in the 7 days which occur before the mourning is wanted?

Ex. 11., If 12 mantua-makers can make 27 mourning dresses in 4 days: how many persons would be required to make 189 dresses in 8 days?

Ex. 12. If 3000 copies of a History of America, each containing 11 sheets, require 66 reams of paper: how much paper will 5000 take, if the work be extended to 12 sheets?

Ex. 13. As 12 inches in length, 12 in breadth, and 12 in thickness, make a solid foot: what length of plank, which is 7 inches broad and 3 inches thick, will make the same? +

Ex. 14. If 450 tiles, each 12 inches square, will pave my cellar: how many tiles must I have, if the tiles are 9 inches long and 8 broad?

Ex. 15. If the expense of 3 persons on a tour for five months be 1237.8s.: what will 2 persons spend in 9 months?

Ex. 16. If 12 ounces of wool make 24 yards of very fine cloth, 6 quarters wide: how much wool would be required to 150 yards, 4 quarters broad?

Ex. 17. If 300l. gain 9 per cent interest in a year, in what time will 900l. gain 1501.?

Ex. 18. If an iron bar 4 feet long, 3 inches broad, and 14 inch thick, weigh 36 lbs. how much will a bar weigh that is 6 feet long, 4 inches broad, and 2 inches thick?

:

NOTES.

* It is asserted, that this number of herrings have been caught in a single season in Loch Fyne, a salt lake communicating with the sea. The fact may be readily credited, when it is added, that at one single haul 12,000 makarel were drawn on shore at Sidmouch, a few years since. + The statements in this example are, broad,

long. broad. 12 : 12 :: 7*

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It is evident these are both inverse proportions, because 7 in breadth will require more length than 12 inches to make an equal surface. The same may be said of the 3 inches in thickness.

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