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RULE OF THREE INVERSE ;
OR,

BECIPROCAL PROPORTION.

The RULE of THREE INVERSE teaches, by having three numbers given, to find a fourth, which shall have the same ratio to the second, as the first has to the third.

RULE.—State and reduce the terms as in the Rule of Three Direct; then multiply the first and second terms together, and divide the product by the third, the quotient will be the fourth term, or answer.

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1. If 6 men can do a piece of work in 18 days, in what time can 12 men do it !

m. d. In. -
6 : 18 : : 12 In this example, the third term is greater than
6 the first, yet it is evident, that the question re-
- quires the fourth term or answer to be less than
12)108 the second; therefore, this question bekongs to
- the Rule of Three linverse.
9 days Ans.

2. If a man perform a journey in 15 days, when the day is 12 hours long, in how many days will he do it, when the day is but 10 hours? Ans. 18 days.

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The Double RULE,0F THREE, or Compound PRoportion, is the method of resolving such questions as require two or more operations by Simple Proportion.

It always consists of an odd number of terms given, as five, seven, &c. These terms are distinguished into terms of supposition, and terms of demand ; the number of the former always exceeding that of the latter by one, which is of the same name or kind with the answer or term required.

238. How do you proceed in the Rule of Three Inverse 3—239. What is Compound Proportion, or the Double Rule of Three ?—240. How is it distinguished from Sim. ple Proportion ?

RULE.

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 divisor; the quotient will be the answer.”

EXAMPLES. 1. If a principal of $100 gain $6 interest in one year, what will

$ 500 gain in 8 months 7

Statement & operation.

$ P. $ Int. $ P.

100 : 6 : : 500 In this question, the answer required is

12 8 interest, therefore, $6 must be the middle - — term. As $500 will gain more interest in 1200 4000 the same time than $100 ; $500 must be

6 placed on the right for the third term ; and $100 on the left for the first term. And as 1200)24000 the same sum will gain more interest in 12

-- months than in 8 months, the 8 must be pla$20 Ans. ced under the third term, and the 12 under the first term. The operation is obvious on inspecting it. *

*

2. If $100 in 12 months will gain $6 interest; in what time will $750 gain $30 interest ? * , - Ans. 8 months. , ‘. === , ;" *The reason of this rule for stating, and of the method of operation, is evident from the nature of Simple Proportion; for every line in this case is a particular statement in that rule. If, then, all the separate dividends be collected into one dividend, and all the divisors into one divisor, their quotient must be the answer.

241. Repeat the rule 2.

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.

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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.

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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 nature of the question may require.

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CASE I.

*. When it is required to find how many of the first sort of coin, weight, or measure, mentioned in the question, are equal to a given quantity of the last.

* - RULE.—Place the numbers alternately, that is, the antece

dents 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.

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 given quantity in the last, how do you proceed 3

EXAMPLES.

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 7

Antecedents. Consequents. 12 of Boston=10 of Amster. , Then 12×10x80=9600 dividend; 10 of Amster.–12 of Paris. and 10× 12=120 the divisor; 80 of Paris 7 Therefore 9600-120-80 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 Venitian braces are equal to 32 American yards ' Ans. 52; braces.

CASE II.

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, beginning at the left hand, 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 divisor; divide, and the quotient will be the answer.

EXAMPLES.

1. If 12 lbs. of Boston be equal to 10 lbs. of Amsterdam, and 100 lbs. of Amsterdam be equal to 120 lbs. of Paris; how many pounds of Paris are equal to 80 lbs. of Boston ? Antecedents. Consequents. 12 of Boston=10 of Amsterdam. || 10×120x80=96000 dividend; 100 of Amsterdam=120 of Paris. and 12× 100–1200 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 American yards are equal to 521, Venetian braces ! Ans. 32 yards.

244. How, when you wish to ascertain how many of the last are equal to a certain quantity named in the first 3

PRACTICE.}

PRActice is a contraction of the Rule of Three Direct, when the first term happens to be an unit or one ; 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

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* Perhaps no method can be more simple and concise to find the value of goods in Federal Money, than the general rule of multiplying the price by the quantity, as given in Multiplication of Federal Money or Decimals; 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 goods in sterling money, an acquaintance with this excellent rule is necessary to every one, employed in mercantile pursuits.

245. What is Practice 3–246. Explain to me the use of the tables 7–247. What are aliquot parts of any quantity ?

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