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second stating ; and so on, as far as there are any numbers in the general stating, always making the fourth number, resulting from each simple stating, the second term of
So shall the last resulting number be the answer required.
2. By one operation.--Multiply together all the ternis in the first place, and also all the terms in the third place. Then multiply the latter product by the middle term, and divide the result by the former product, and the quotient will be the answer required.
Note 1. It is generally best to work by the latter method, viz. by one operation. And after the stating, and before the commencement of the operation, if one of the first terms, and either the middle term, or one of the last terms, can be exactly divided by one and the same number, let them be divided, and the quotients used instead of them
j which will much shorten the work. Note 2. The first and third terms of each line, if of different denominations, must be reduced to the same denomination.
1. How many men can complete a trench of 135 yards long in 8 days, provided 16 men can dig 54 yards in 6 days ?
GENERAL STATING. 54 yds. or 22
16 men ::
135 yds. or 57 8 days, or 45
6 days, or 3 )
2. If rool. in one year gain 51. interest, what will be the interest of 750l. for 7 years ?
Ans. 2621. IOS. 3. What principal will gain 2621. 1os. in 7 years, at 5l. per cent. per annum ?
Ans. 7501. 4. If a footman travel 130 miles in 3 days, when the days are 12 hours long; in how many days, of 10 hours each, may he travel 360 miles ?
Ans. 9 days. 5. If 120 bushels of corn can serve 14 horses 56 days ; how many days will 94 bushels serve 6 horses ?
Ans. 1021 days 6. If 702. 5dwts. of bread be bought at 4 d. when corn is at 4s. 2d. per bushel, what weight of it may be bought for is. 2d. when the price of the bushel is 5s. 6d. ?
Ans. ilb. 4oz. 327.dwt 7. If the carriage of 13cwt. Iqr. for 72 miles be 21. 105. 6d. what will be the carriage of 7cwt. 3qrs. for 112 miles ?
Ans. 21. 5$. IId. 175.4. 8. A wall, to be built to the height of 27 feet, was raised to the height of feet by 12 men in 6 days ; how many men must be employed to finish the wall in 4 days, at the same rate of working ?
Ans. 36 men. 9. If a regiment of soldiers, consisting of 939 men, can eat up 351 quarters of wheat in 7 months ; how many soldiers will eat up 1464 quarters in 5 months, at the rate ?
Io. If 248 men, in 5 days of 11 hours each, dig a trench 230 yards long, 3 wide and 2 deep ; in how many days of 9 hours long, will 24 men dig a trench of 420 yards long, 5 wide and 3 deep?
CONJOINED PROPORTION is when the coins, weights, or measures, of several countries are compared in the game question ; or it is the joining together of several ratios, and the inferring of the ratio of the first antecedent and the last consequent from the ratios of the several antecedents and their respective consequents.
NOTE 1. The solution of questions, under this rule, may frequently be much shortened by cancelling equal numbers, when in both the columns, or in the first column and third term, and abbreviating those that are commensarable, NOTE 2
The proof is by so many statements in the single rule of three as the nature of the question requires.
When it is required to find how many of the last kind of coin,
weight, or measure, mentioned in the question, are equal to a given number of the first.
1. Multiply continually together the antecedents for the first term, and the consequents for the second, and make the given number the third.
2. Then find the fourth term, or proportional, which will be the answer required.
EXAMPLES. 1. If olb. at Boston make glb. at Amsterdam ; golb. at Amsterdam, u2lb. 'at Thoulouse ; how many pounds at Thoulouse are equal to 50lb. at Boston ?
Or by abbreviation. 10: 9:: 50 10: 90: 112 10: II2 IO: 112 2:112 :: 1 56
56 the answer. 2. If 20 braces at Leghorn be equal to 10 vares at Lisbon ; 40 vares at Lisbon to 80 braces at Lucca ; how many braces at Lucca are equal to 100 braces at Leghorn ?
Ans. 100 braces,
CASE 11. When it is required to find how many of the first kind of coin,
weight, or measure, mentioned in the question, are equal to a given number of the last.
Proceed as in the first case, only make the product of the consequents the first term, and that of the antecedents, the second.
* In performing this example, the first abbreviation is obtained by dividing 90 and 9 by their common measure 9 ; the second by dịviding 10 and 50 by their common measure 10 ; the third by dividing 10 and 5 by their common measure 5 ; and the fourth, or answer, by dividing 2 and 112 by their common measure 2.
1. If 100lb. in America make 9515. Flemish 3 and 191b. Flemish, 25lb. at Bolognia'; how many pounds in America are equal to 50lb. at Bolognia ?
Or by abbreviation. 95 ; 100
5 : 100 5:4 25 19 :: 50 25 : 1 :: 50 1;1;:50 1:4:: 10 :
2. If 25lb. at Boston be 221b. at Nuremburg ; 881b. at Nuremburg, 92lþ. at Hamburg ; 46lh. at Hamburg, 491b. at Lyons ;. how many pounds at Boston are equal to 981b. at Lyons ?
Ans. ioolb. 3. If 6 braces at Leghorn make 3 ells English ; 5 ells English, 9 braces at Venice ; how many braces at Leghorn will make 45 braces at Venice?
Ans. 50 braces.
BARTER is the exchanging of one commodity for another, and directs traders so to proportion their goods, that neither party may sustain loss.