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The work is then performed by two operations in the fingle rule of three direct; the answer to the first operation forming the second number in the second operation.

2. By one operation. Multiply the two numbers which stand one over the other in the supposition, together, for the first number; the two numbers in the demand for the third number; and the second number in the first question will be also the second number in the work. Then the answer is found by one operation in the fingle rule of three direct, as in the following example:

Example 1. If 1001. principal gain 51. interest in 1 year, what will 1401. gain in 9 months ?

The Que/tion stated.
If zool. gain sl.

What will 1401.
in 12 months,

gain in 9 months? In this example, the numbers 100, 5, and 12, belong to the supposition, and 140 and 9 is the demand; for the meaning of the work is, suppose 100l. gain 51. interest in 12 months, (then follows the demand) I demand to know how much 1401. will gain in 9 months at the same rate of interest?

Thus the questions are stated according to the foregoing directions: the sl. being the interest of the money (and of 'the fame nature with the sixth number or anfwer) must be the second number ; and the two other numbers in the fup. position 100 and 12 are placed one above the other, as are the two numbers 140 and 9 in the demand. It matters not which of these two numbers is uppermoft, provided that the numbers in each, which are of the same nature, occupy the corresponding places respectively: thus, in the supposition, the pounds principal is the uppermost number, so it is in the demand, and the number of months is undermost in both.

The question being thus stated, the work is wrought by two operations of the fingle rule of three direct. The three uppermost numbers are the numbers for the first operation, and the fourth number, or answer to these forms, the second

number

number for the second operation; the bottom number of the fupposition forms the first number, and the bottom number of the demand the third number; then the answer, or fourth number of this fecond operation, is the true answer to the question ; as in the following example, which is the foregoing one at large.

First Operation.
If 100?. gain 51. what will 1401.

5
1,00)7,0007

Second Operation.
If 12 months gain 77. what will 9 months ?

7
12)6305

60

3 20

Answer 36 55.

12)6015 The answer would have been the same if the number of months had been the uppermost numbers instead of the , pounds principal; in which case the first question would be, if 12 months give 5l. interest, what will 9 months give? and, the answer is 31. 155.; then the second question would be, if 1ool. gain 31. 155. what will 140l. gain ? and the answer is as before, 51.55.

By one Operation.
If rool. gain 51. what will 1401. gain ?
months

o months
1260

5 12,00)61.00

12

1200

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In this example, the work is stated as in the former ; the two first numbers in the supposition are multiplied together, and the two numbers in the demand are also multiplied together, then these two products are made, the one the first, and the other the third number, and the second number in the first question is also the second number. This rule is the most sure and practical method of proving the double rule of three direct, when wrought by two single ones.

The foregoing example worked both ways will be suffic' cient to inttruct the learner. I fhall, therefore, give a few questions, with their answers, omitting the operation.

Qu. 2. Suppose 468 men consume 175 quarters of wheat in 168 days, I demand how many quarters will serve 5612 men 58 days ? - Ans. 724 quarters, and 14834 of a quarter, or a little more than half a quarter.

Qit. 3. Suppose So acres of grass be mowed by Sinen in 14 days, I deinand how many acres 28 men will mow in 12 days ?--Anf. 240 acres.

Qu. 4. Suppose the wages of 12 men for 6 days amount to 71. 45. what are the wages of 25 njen for 40 days :- Ans. 100l.

Qu. 5. If 150l. principal put out to interest for 9 months be increased, principal and interest, to 1561. 155. I demand how much is that per cent. per annum?-Anf. 91.

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SECT. X.

OF THE DOUBLE RULE OF THREE INVERSE.

The double rule of three inverse is when there are five given numbers to find a fixth, in an inverted proportion.

Rule

Rule. . Place the numbers as directed in the last feétior. Multiply the lower number of the first place by the upper one of the third, and make the product the first number; next multiply the upper tern of the first place by the lower one of the third, for the third number : then if the inverse proportion be found in the three upper numbers, the answer is given by one operation in the rule of three direct; but if the inverse proportion be found in the lower numbers, the work is performed by the inverse rule (for of every sum in this rule one question is direct and the other inverse). : Example 1. If 1001. gain 51. interest in 12 months, what principal will gain 51. 55. in 9 months ? 5l.

100l.

months 12

57. gs. months 9

The pounds interest being reduced to fillings, and mul. tiplied by the number of months, the question will stand, and operation be performed as follows:

1260

100

900

100

9,00)1260,00

1401. for the answer, If the number of months had been made the upper terms, the upper proportion would then have been direct, and would have been required to have been worked by the dire& method. It would in that care stand thus :

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* The rule laid down in this section will be found quite general, and fufficient for working all queftions in the double rule of three both direct and inverse ; and is so obvious, as to require no demonstration. Nevertheless, in consequence of receiving advice from some teachers (not the most competent) of the mathematics, that I had not given the improved method of working this rule, I fall Rate this much-approved. rule, verbatim, from a well-known treatise, and Thew its fallacy.

AA z

66 Rulo

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24. 2. If 481. serve for the maintenance of 12 men 8 days, how long will 2881. serve for 4 men ?- Anf. 144 days.

Qu. 3. If, when a bushel of wheat costs 6s. &d..a penuy loaf weighs 6 ounces, how much will a loaf weigh that costs 10{d. when the wheat is 1os. the busheli- Anf. 42 ounces.

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" Rule 1.

Let the principal cause of loss or gain, interest or de creare, action or paffion, be put in the first place.

Let that which betokeneth time, distance, or place, and the like, be in the fecond place; and the remaining one in the third. 3. Place the other two terms under their like in the fuppofition.

If the blank falls under the third term, multiply the firft and second terms for a divisor, and the other three for a dividend; but,

5: If the blank falls under the firft or second term, multiply the third and fourth terms for a divisor, and the other three for the divi. dend;

and the quotient will be the answer.'
Proof. By two single rules of three.
Example 1.

If 14 horses eat 56 bushels of oats in 16 days, how many bushels will be fufficient for 20 horses for 24 days ?

By two fingle rules:

hor. bu. hor. bu. 1. As 14:56 :: 20: So da. bu. da.

bx. 2. As 16: 80 :: 24 : 120

Or, in one faring, worked thus : kor, da bu.

. 14: 16:56

56 X 20 X 24.

=120." 20:24:

WALKINGAME. That this rule is not founded on mathematical principles is evident from inspection; for by a different statement of the question (though exactly agreeable to the rules) a different answer will arise. Thus, how eafily might the learner, required to work this question by one ftate. meat, order the numbers as follows:

kor. bj. da.
14: 56:16

56 X 24X14

= 594 20::24

10 x 20 According to this fateinent the answer will be 584 bushels; whereas, the true answer is 1 20 bufrels. The question is, nevertheless, stated in this laft case agreeable to the rules,

14 x 16

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