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when the question is in direct proportion; but contrariwise, the other number of supposition the 3d term, and the demanding number the 1st term, when the question has inverse proportion *.
Then, in both cases, multiply the 2d and 3d terms together, and divide the product by the first, which will give the answer, or 4th term sought, viz. of the same denomination as the second term.
Note I. 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 II. 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: but in this case it falls under compound proportion, and may be more easily worked by the rule for that case.
Note III. When the first term is divisible by any number which also divides the second or third, we may so divide them, using the quotients instead of the original terms. This will often diminish the labour of the calculation considerably.
1. If 8 yards of cloth cost 17 4s what will 96 yards cost? yds £
S yds £ S Or, in accordance with note iii :— As 81 4:96: 14
yds £ S 4:12: 14 8
- yd £ s As 1: 1
£ 14 8
£ 14 8 Answer.
* If we adhere to the rigid geometrical principles of ratio, as Mr. Bonnycastle has done, we should put the term which is of the same kind with the answer in the third place instead of the second. It is not, however, with concrete but with abstract numbers that we work; and, hence, though the relations of things show us the relations of the numbers by which they are represented, still we may conceive a ratio between the numbers whilst the things themselves are dissimilar. Such a restriction was necessary in the geometry of the Greeks, but is not at all implied, and is therefore not necessary, in the arithmetic of symbols.
It should be added, that this rule is of very great European antiquity, and it has been universally given in this form: though the application of Jones's Rule (see Compound Proportion) is certainly more simple, and upon the whole more easily applied. As, however, the very form of stating the Rule of Three has been almost universally adopted in writing proportions, and it has acquired so strong a hold upon the language, habits, and practice of mankind, it has not been considered desirable to alter it here.
Ex. 2. An engineer having raised 100 yards of a certain work in 24 days with 5 men; how many men must he employ to finish a like quantity of work in 15 days?
3. What will 72 yards of cloth cost, at the rate of 9 yards for 57 12s?
Ans. 447 16s.
4. A person's annual income being 1467; 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, to make a square foot, or as much as 12 inches in length and 12 in breadth contains?
Ans. 16 inches. 7. If 750 men require 22500 rations of bread for a month, how many rations will a garrison of 1200 men require? Ans. 36000. 8. If 7 cwt 1 qr of sugar cost 267 10s 4d; what will be the price of 43 cwt 2 qrs? Ans. 159/ 2s.
9. The clothing of a regiment of foot of 750 men amounting to 2831l 5s; what will the clothing of a body of 3500 men amount to? Ans. 132127 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 six bushels of coals, at the rate of 1/ 14s 6d the chaldron ? Ans. 5s 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? Ans. 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 Oåd. 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. 15. What will be the tax upon 7631 15s, at the rate of 3s 6d per pound sterling? Ans. 133 13s ltd. 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 guineas, at the rate of 6s the bushel? Ans. 39 qrs 3 bushels.
18. A person, failing in trade, owes in all 9771; at which time he has, in money, goods, and recoverable debts, 4201 6s 34d; now supposing these things delivered to his creditors, how much will they get per pound? Ans. 8s 74d.
19. A plain of a certain extent having supplied a body of 3000 horse with forage for 18 days; then how many days would the same 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 25s 6d per day, one day with another; how much will he have saved at the year's end? Ans. 164/ 12s 6d.
21. What cost 30 pieces of lead, each weighing 1 cwt 12 lb, at the rate of 16s 4d the cwt? Ans. 271 2s 6d.
22. The governor of a besieged place having provision for 54 days, at the rate of 14lb 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? 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 17 13s 6d per chaldron ? Ans. 125/ 19s Old. 25. If the penny loaf weigh 8 ounces when the bushel of wheat cost 7s 3d, what ought the penny loaf to weigh when the wheat is at 8s 4d?
Ans. 6 oz 15 dr. 26. What rent will 173 acres 2 roods 14 poles of land yield, at the rate of 17 7s 8d per acre? Ans. 2401 28 7d. 27. To how much amount 73 pieces of lead each weighing 1 cwt 3 qrs 7 lb, at 101 48 per fother of 19 cwt? Ans. 691 48 2d 11q. 28. How many yards of stuff, of 3 qrs wide, will line a cloak that is 1 yards in length and 3 yards wide? Ans. 8 yds. O qrs. 23 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 1s 101d.
30. If a gentleman's estate be worth 21077 12s a year; what may he spend per day, to save 500l in the year ? Ans 41 88 1d.
31. Wanting just an acre of land cut off from a piece which is 131⁄2 poles in breadth, what length must the piece be? Ans. 11 po 4 yds 2 ft 0 in. 32. At 7s 94d per yard, what is the value of a piece of cloth containing 53 ells English 1 qr. Ans. 251 18s 1d. 33. If the carriage of 5 cwt 14 lb for 96 miles be 17 12s 6d; how far may have 3 cwt 1 qr carried for the same money? Ans. 151 m 3 fur 3 34. Bought a silver tankard, weighing 1 lb 7 oz 14 dwts; what did it cost me at 6s 4d the ounce?
Ans. 61 48 9fd.
35. What is the half year's rent of 547 acres of land, at 15s 6d the acre?
Ans. 2117 19s 3d.
36. A wall that is to be built to the height of 36 feet, was raised 9 feet high by 16 men in 6 days; then how many men must be employed to finish the wall in 4 days, at the same rate of working? Ans. 72 men,
37. What will be the charge of keeping 20 horses for a year, at the rate of 143d per day for each horse?
Ans. 4417 Os 10d. 38. If 18 ells of stuff that is yard wide, cost 39s 6d; what will 50 ells, of the same quality, cost, being yard wide? Ans. 71 6s 33 d. 39. How many yards of paper that is 30 inches wide, will hang a room that is 20 yards in circuit and 9 feet high.
Ans. 72 yards. 40. If a gentleman's estate be worth 3847 16s a year, and the land tax be assessed at 2s 94d per pound, what is his net annual income?
Ans. 3317 is 94d.
41. The circumference of the earth is about 25000 miles; at what rate per hour is a person at the middle of its surface carried round, one whole rotation being made in 23 hours 56 minutes? Ans. 1044 miles. 42. If a person drink 20 bottles of wine per month, when it cost 8s a gallon,
how many bottles per month may he drink, without increasing the expense,
when wine costs 10s the gallon?
Ans. 16 bottles.
43. What cost 43 qrs 5 bushels of corn, at 11 8s 6d the quarter?
Ans. 627 3s 32d.
44. How many yards of canvass that is ell wide will line 50 yards of say that
is 3 quarters wide?
Ans. 30 yards.
45. If an ounce of gold cost 4 guineas, what is the value of a grain?
Ans. 2 d.
46. If 3 cwt of tea cost 407 12s; at how much a pound must it be retailed, to gain 101 by the whole? Ans. 33365.
COMPOUND PROPORTION is a rule by means of which the student may resolve such questions as require two or more statings in simple proportion.
The general rule for questions of this kind may be exhibited in the following precepts, viz.
1. Set down the terms that express the conditions of the question in one line. 2. Under each conditional term, set its corresponding one, in another line, putting the letter Q in the (otherwise) blank place of the term required.
3. Multiply the effective terms of one line, and the objective terms of the other line, continually, and take the result for a dividend.
4. Multiply the remaining terms continually, and let the product be a divisor. 5. The quotient of this division will be Q, the term required *.
Note. By effective terms are here meant whatever necessarily and jointly produce any effect; as the cause and the time; length, breadth, and depth; buyer and his money; things carried, and their distance, &c. all necessarily inseparable in producing their several effects. In short, the causes of the effect. By objective terms, those which express the effect itself.
Thus, if the number of men, the time of the siege, and the daily rations, be the effective terms in producing the consumption of the quantity of food in the garrison; then, in reference to the same problem, the quantity of food constitutes the objective term.
In a question where a term is only understood, and not expressed, that term may always be expressed by unity.
A quotient is represented by the dividend put above a line, and the divisor put below it.
1. How many men can complete a trench of 135 yards long in 8 days, when 6 men can dig 54 yards of the same trench in 6 days?
Here 16 men and 6 days are the effective terms of the first line, and 135 yards e objective term of the other. Therefore, by the rule,
* This rule, which is as applicable to Simple as to Compound Proportion, was given, in 1706,
by W. Jones, Esq. F.R.S., the father of the late Sir W. Jones.
If a garrison of 3600 men have bread for 35 days, at 24 oz each day; how much a day may be allowed to 4800 men, each for 45 days, that the same quantity of bread may serve ?
AN EXAMPLE IN SIMPLE PROPORTION,
If 14 yards of cloth cost 217; how many yards may be bought for 737 10s?
2. If 1007 in one year gain 57 interest; what will be the interest of 750l for 7 years? Ans. 2621 10s. 3. If a family of 8 persons expend 2007 in 9 months; how much will serve a family of 18 people 12 months? Ans. 6001. 4. If 27s be the wages of 4 men for 7 days; what will be the wages of 14 men for 10 days?
Ans. 67 15s. 12 hours long; Ans. 98 days.
5. If a footman travel 130 miles in 3 days, when the days are in how many days, of 10 hours each, may he travel 360 miles? 6. If 120 bushels of corn can serve 14 horses 56 days; how many days will 94 bushels serve 6 horses? Ans. 1021 days.
7. If 3000 lbs of beef serve 340 men 15 days; how many lbs will serve 120 men for 25 days? Ans. 1764 lb 11 oz. 8. If a barrel of beer be sufficient to last a family of 8 persons 12 days; how many barrels will be drunk by 16 persons in the space of a year?
Ans. 60% barrels. 9. If 180 men, in 6 days, of 10 hours each, can dig a trench 200 yards long, 3 wide, and 2 deep; in how many days of 8 hours long, will 100 men dig a trench of 360 yards long, 4 wide, and 3 deep? Ans. 483 days.
OF VULGAR FRACTIONS.
A FRACTION, or broken number, is an expression of a part, or some parts, of something considered as a whole.
It is denoted by two numbers, placed one below the other, with a line between them :
The Denominator, or number placed below the line, shows how many equal parts the whole quantity is divided into; and it represents the Divisor in Division. And the Numerator, or number set above the line, shows how many of these parts are expressed by the fraction: being the remainder after division. Also, both these numbers are in general named the Terms of the Fraction.