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1. State the question, by placing the three conditional terms in such order that that number which is the cause of gain, loss, or action, may possess the first place ; that which denotes space of time, or distance of place, the second ; and that which is the gain, loss, or action, the third.
2. Place the other two terms, which move the question, under those of the same name.
3. Then, if the blank place, or term sought, fall under the third place, the proportion is direct, therefore, multiply the three last terms together, for a dividend, and the other two for a divisor; then the quotient will be the answer.
4. But if the blank fall under the first or second place, the proportion is inverse, wherefore multiply the first, second and last terms together, for a dividend, and the other two, for a divja sor; the quotient will be the answer.*
1. If a principal of $100 gain $6 interest in one year, what will $500 gain in 8 months Statement & Operation.
$P. M. INT. In this question the cause of $6 gain is $100; 100 : 12 : : 6 therefore, $100 possesses the first place in 500 : 8
the statement. The time, 12 months, is set
in the second place, as the rule directs; and 100
6 the gain, $6, in the third place. Then, the 12
8 other two terms being placed under those of
the same name, the blank falls under the third 1200
48 place: therefore the proportion is direct; 500 consequently, I multiply the three last terms,
6, 8, and 500, together, and take the product, 1200)2401000 24000, for the dividend ; and the product of
the two first terms, 100 and 12= 1200, is the $20 Ans. divisor. The dividend being divided by the
divisor, the quotient, $20, is the Ans. sought,
2. If $100 in 12 months will gain $6 interest; in what time will $750 gain $30 interest?
Ans. 8 months.
* When a question contaias more than five given terms it can always be solved by several statements in the Single Rule of Three, and so may any question in the Double Rule.
241. Repeal the rule ?
3. If 7 men can reap 84 acres of wheat in 24 days; how many men can reap 100 acres in 10 days ?
Ans. 20 men.
4. If 12 men can earn £8 8s. in 8 days; what will 21 men earn in 15 days?
Ans. £27 11s. 3d.
5. If a family of 9 persons spend 450 dollars in 5 months ; how much would be sufficient to maintain them 8 months, if 5 persons more were added to the family?
Ans. 1120 dolls.
6. If 2 mer in 6 days of 12 hours each, build 30 rods of wall, how many hours long is the day when 8 men build 64 rods in 4 days ?
REMARK.—In this question as there are seven giver terms or numbers, it cannot be solved by the rule given, but must be wrought by statements in the Single Rule of Three as observed in the note.
1. By separate statements. M. H. M. H. Explanation. In the 1st statement, as · As 8 : 12 :: 2 : 3 more men will require less time, the 3rd
H. D. H. term must be less than the let. In the 2d As 4 : 3 :: 6 : 41 statement, less days require more hours,
R. H. R. H. therefore, the 3rd term must be greater than As 30:47 :: 64 : 9•6 Aps. the 1st. In the 3rd statessent, more rods
require more men, therefore the 3rd term must be greater than the 1st; observing, always, that the 4th term or answer resulting from any statement, must be the 2d term in the next statement.
2. By reducing all the statements to one. 8m. h. 2m.
Explanation. The 1st and 3d 12
terms in the first three state30r. Seur.
ments are the same as those in the separate statements.
The 960 : 12 :: 768 : 9.6b. as before. middle term, by tbis method,
must always be of the same name as the answer required. All the given first terms being multiplied together, the product will be the 1st term of the last statemen:, and the product of all the third terms will be the 3d term. Then the product of the 2d and 3rd terms divided by the 1st, will give the
CONJOINED PROPORTION is when the coins, weights or measures of several countries are compared in the same question ; or, in other words, it is joining many proportions together, and by the relation which several antecedents have to their consequents, the proportion between the first antecedent and last consequent is discovered, as well as the proportion between the others in their several respects.
NOTE.—This rule may generally be abridged by cancelling equal quantities or numbers that happen to be the same in both columns ; and it may be proved by as many statings in the Single Rule of Three as the pature of the question may require.
When it is required to find how many of the first sort of coin, weight
er measure, mentioned in the question, are equal to a given quantity of the last.
RULE. Place the numbers alternately, that is, the antecedents at the left hand, and the consequents at the right, and write the last number on the left hand ; then multiply all the numbers in the left hand column continually together for a dividend ; and all the numbers in the right hand column for a divisor ; divide, and the quotient will be the answer.
1. If 12 lbs. of Boston be equal to 10 lbs. of Amsterdam, and 10 lbs. of Amsterdam be equal to 12 lbs. of Paris; how many pounds of Boston are equal to 80 lbs. of Paris ?
Antecedents. Consequents. 12 of Boston=10 of Amster. Then 12X10X80=9600 dividend ; apu 10 of Amster.=12 of Paris, 10X12–120 the divisor ; 80 of Paris ?
Therefore 9600-120=80 lbs. Apswer.
242; What is the meaning of Conjoined Proportion ?-243. When you wish to know how many of the first sort of coin, &c. mentioned in any question are equal to a giz, en quantity in the last, how do you proceed ? .
2. If 140 braces of Venice be equal to 150 braces of Leghorn, and 7 braces of Leghorn be equal to 4 American yards; how many Venitian braces are equal to 32 American yards?
Ans. 524 braces.
When it is required to find how many of the last sort of coin, weight,
or measure, mentioned in the question, are equal to a given quantity of the first.
Rule.- Place the numbers alternately, begiming at the left band, and write the last number on the right hand ; then multiply all the numbers in the right hand column continually together for a dividend, and all the numbers in the left hand column for a di. visor ; divide, and the quotient will be the answer.
1. If 12 lbs. of Boston be equal to 10 lbs. of Amsterdam, and 100
| ; 100 of Amsterdam=120 of Paris. and 12X100=1200 divisor ;
80 of Boston ? Then 96000-1200=80 lbs. Ans.
2. If 140 braces of Venice be equal to 150 braces of Leghorn, and 7 braces of Leghorn be equal to 4 American yards ; how many Amer ican yards are equal to 52 Venetian braces ?
Ans. 32 yards.
244. Horo, when you wish to ascertain how many of the last are equal to a certain quantity named in the first ?
PRACTICE is a contraction of the Rule of Three Direct, when the first term happens to be an unit or oae ; it has its name from its daily use among Merchants and tradesmen, being an easy and concise method of working most questions which occur in trade and business.
PROOF. By the Single Rule of Three, Compound Multiplication, or by varying the parts.
Before any advances are made in this rule, the learner should commit to memory the following
TABLES OF ALIQUOT, OR EVEN PARTS.
Aliquot, or even parts of Money. Fts. of a shill. of a £. Parts of a Pound. Parts of a Dollar. d. £
* Perhaps no method can be more simple and concise to find the value of goods in Federal Mamey, than the general rule of multiplying the price by the quantity, as given in Multiplication of Federal Money or recimals; therefore, the application of this rule to Federal Money is almost useless. Yet, as English merchants, trading with Americans, make out the invoices of their gorus in sterling money, an acquaintance with this excel. lont rule is necessary to every one, empioyed in mercantile pursuits.
What is Practice 3-246. Explain to me the use of the tables !---- 247. What parts of any quantity ?