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1. Divide 17. Ib 9 oz 0 dwts 2 gr by 7.

Ads. 2 lb 6 oz 8 dwts 14 gr. 2. Divide 17 lb 5 oz 2 dr 1 scr 4 gr by 12.

Ans. 1 lb 5 oz 3 dr 1 scr 12 gr. 3. Divide 178 cwt 3 qrs 14 lb by 53. Ans. 3 cwt 1 ar

14 lb. 4. Divide 144 mi 4 fur 2 po 1 yd 2 st 0 in by 39.

Ans. 3 mi 5 fur 26 po Oyds 2 ft 8 in. 5. Divide 534 yds 2 qrs 2 na by 47. Ans. 11 yds 1

qr 6. Divide 71 ac 1 ro 33 po by 51.

Ans. 1 ac 2 ro 3 po. 7. Divide 7 tu 0 hods 47 gal 7 pi by 65. Ads. 27 gal 7 pi. 8. Divide 387 la 9 qr by 72. Ans. 5 la 3 grs 7 bu. 9. Divide 206 mo 4 da by 26. Ans. 7 mo 3 we 5 ds.


2 na.

THE GOLDEN RULE, OR RULE OF THREE. The Rule of Three teaches bow to find a fourth proportional to three numbers given : for which reason it is so. etimes called the Rule of proportion. It is called the Rule of Three, because three terms or numbers are given, to find a fourth. And because of its great and extensive usefulness, it is often called the Golden Rule. This Rule is usually conside ed as of two kinds, uamely, Direct, and Inverse.

The Rule of Three Direct is that in which more requires more, or less requires less. As in this ; if 3 men dig 21 yards of trench in a certain time, how much will-6 men dig in the same time? Here more requires more, that is, 6 men, which are more than 3 men, will also perform more work in the same time. Or when it is tbus : if 6 men dig 42 yards, how much will 3 men dig in the same time ? Here then less requires less, or 3 men will perform proportionably less work than 6 men, in the same time. In both these cases then, the Rule, or the Proportion, is Direct; and the stating must be

thus, As 3:21 ::6 : 42,

or thus, As 6 : 42 : :3:21. But the Rule of Three Inverse, is when more requires less, or less requires more. As in this: if 3 men dig a certain quantity of trench in 14 hours, in how many hours will 6 men dig the like quantity ? Here it is evident that 6 men, being more than 3, will perform an equal quantity of work in less time or fewer hours. Or thus : if 6 men perform a certain quantity of work in 7 hours, in how many hours will 3 men perform the same? Here less requires more, for 3 men will take more hours than 6 to perform the same work. In both these cases then the Rule, or the Proportion, is Inverse ; and the stating must be

thus, As 6 : 14 : :3 7,

or thus, As 3 : 7::6:14. And in all these statings, the fourth term is found, by multiplying the 2d and 3d terms together, and dividing the product by the 1st term.

Of the three given numbers ; two of them contain the supposition, and the third a demand. And for stating and working questions of these kinds observe the following general Rule :


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STATE the question, by setting down in a straight line the three given numbers, in the following manner, viz. so that the 2d term be that number of supposition, which is of the same kind that the answer or fourth term is to be ; making the other number of supposition the 1st term, and the demanding number the 3d term, when the question is in direct proportion ; but contrariwise, the other number of suppositio, the 3d term, and the demanding number the 1st term, wben the question has inverse proportion.

Then, in both cases, multiply the zd and 3d terms together, and divide the product by the ist, which will give the answer, or 4th term sought, viz. of the same denomination as the second term.

Note, If the first and third terms consist of different denominations, reduce them both to the same : and if the second term be a compound number, it is mostly convenient to reduce it to the lowest denomination mentioned. If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last, denomination. Proceed in the same manner with all the remainders, till they be reduced to the lowest denomination which the second admits of, and the several quotients taken together will be the answer required.

Note also, The reason for the foregoing Rules will appear, when we come to treat of the nature of proportions.-Sometimes two or more statings are necessary, which may always be known from the nature of the question.


1. If 8 yards of Cloth cost 11 4s, what will 96 yards cost?

yds 18 yds is
As 8:14::96 : 14 8 the Answer


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Ex. 2. An engineer having raised 100 yards of a certaia work in 24 days with 5 men; how many men must he employ to finish a like quantity of work in 15 days ?

ds men ds men
As 15:5:: 24 : 8 Ans.


15) 120 (8 Answer.




3. What will 72 yards of cloth cost, at the rate of 9 yards for 5l 12s ?

Aus. 441 168. 4. A person's annual income being 1461 ; how much is that . per day?

Ans. 8s. 5. If 3 paces or common steps of a certain person be equal to 2 yards, how many yards will 160 of his paces make ?

Ans. 106 yds 2 ft. 6. What length must be cut off a board, that is 9 inches broad, io make a square foot, or as much as 12 inches in length and 12 ja breadth contains ?

Aos. 16 inches. . 7. If 750 men require 22500 rations of bread for a month; how many rations will a garrison of 1200 men require ?

Aus. 36000. 8. If 7 cwt I qr of sugar cost 261 10s 4d; what will be the price of 4 cwt 2 qrs ?

Ans. 1591-25. 9. The clothing of a regiment of foot of 750 men amounting to 28311 5s; what will the clothing of a body of 3600 men amount to ?

Ans. 132121 10s. 10. How many yards of matting, that is 3 ft broad, will cover a floor that is 27 feet long and 20 feet broad?

Ans. 60 yards. 11. What is the value of 6 bushels of coals, at the rate of 1l 14s 6d the chaldron ?

Ans. 58 9d. 12. If 6352 stones of 3 feet long complete a certain quantity of walling; how many stones of 2 feet long will raise a like quantity ?

Aos, 9528. 13. What must be given for a piece of silver weighing 73 lb 5 oz 15 dwts, at the rate of 5s 9d per ounce ?

Ans. 2531 10s 03 d. 14. A garrison of 536 men having provision for 12 months ; how long will those provisions last, if the garrison be increased to 1124 men ?

Ans. 174 days and

T127 15. What will be the tax upon 7631 158, at the rate of 3s 6d per pound sterling?

Ans. 1331 13s 1 d.


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16. A certain work being raised in 12 days, by working 4 hours each day; how long would it have been in raising by working 6 hours per day?

Ans. 8 days. 17. What quantity of corn can I buy for 90 goineas, at the rate of 6s the bushel ?

Ans. 39 ars 3 bu. 18. A person, failing io trade, owes in all 9771 ; at which time he has, in money, goods, and recoverable debts, 4201 68 34d; now supposing these tbings delivered to his creditors, how much will they get per pound ?

Ans. 8s 7 d. 19. A plain of a certain extent having supplied a body of 3000 borse with forage for 18 days; then how many days would the saine plain have supplied a body of 2000 horse ?

Ans. 27 days. 20. Suppose a gentleman's income is 600 guineas a year, and that he spends 258 6d per day, one day with another; how much will he have saved at the year's end ?

Ans. 1641 12s 6d. 21. What cost 30 pieces of lead, each weighing 1 cwt 121b, at the rate of 16s 4d the cwt ?

Ans. 271 %$ 6d. 22. The governor of a besieged place having provision for 54 days, at the rate of 1 lb of bread ; 'but being desirous to prolong the siege to 80 days, in expectation of succour, in that case what must the ration of bread be? Ans. 1 ' lb.

23. At half a guinea per week, how long can I be boarded for 20 pounds ?

Ans 38 wks. 24. How much will 75 chaldrons 7 bushels of coals come to, at the rate of 12 13s 6d per chaldron ?

Ans. 1251 193 0 d. 25. If the penny loaf weigh 8 ounces when the bushes of wheat costs 78 3d, what ought the penny loaf to weigh when the wheat is at 8s 4d ?

Ans. 6 oz 15,3 dr. 26. How much a year will 173 acres 2 roods 14 poles of land give, at the rate of 11 78 8d per acre ?

Ans. 2401 2s 7.2.d. 27. To how much amounts 73 pieces of lead, each weighing 1 cwt 3 qrs 7 lb, at 101 45 per fother of 191 cwt ?

Ans. 691 4s 2d 1419. 28. How many yards of stuff, of 3 qrs wide, will line a cloak that is 18 yards in length and 34 yards wide ?

Ans. 8 yds 0 qrs ? nl. 29. If 5 yards of cloth cost 14s 2d, what must be given for 9 pieces, containing each 21 yards 1 quarter ?

Ans. 271 18 1010. 30. If a gentleman's estate be worth 21071. 128 a year; what may he spend per day, to save 5001 in the year ?

Ans. 41 8s 14.

31. Wanting

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