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NOTE:---In assessing taxes, the law allows the assessor to assess a certain sum, over and aboye the necessary sum to be raised ; or else the tax would not be large enough when there are remainders, which is almost always the case.

INVERSE PROPORTION. DEFINITION.-Inverse proportion teaches by having three numbers given to find a fourth, that shall have the same proportion to the third, as the first has to the second, when more requires less, the proportion is Inverse ; when less requires more, the proportion is also Inverse ; more requires less, when the third term is larger than the first, and the sense of the question requires the fourth to be less than the second ; less requires more when the third term is smaller than the first, and the sense of the question requires the fourth to be greater than the second. RULE.- Make that number which asks the

question the third term; that, which is of the same name, the first term ; and the other remaining number must be the middle term ; multiply the first and second terms together, and divide the product by the third ; the quotient will be the answer required. *

Examples. 1. If 8 men can build a tower in 12 days ; I demand how long it would take them, if 4 more were added to their number?

men. days.
As 8 : 12 : : 8

men.

inen.

8

12)96(8 da. Ans.

* I have followed the same method in Inverse proportion as I did in direct proportion; that is, I have done the sums in decimals and then the other way.

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NOTE:-In this example it is evident that 12 men would do the work quicker than 8, therefore more requires less; more men, require less time. The same question done the other way.

Ans. 8 days. 2. There is a piece of land 16.5 rods long ; I demand the width to make an acre, allowing 20 links to a rod.

Ans. 9•696 rds. The same question done the other way.

Ans. 9. rds. 137 lin. 3. A ship at sea is provided with bread and water sufficient to supply 20 men 9 months, 30 days to a month; but finding a wreck in which were 3 men and 3 boys, to whom they were willing to communicate relief: I demand the time they all can draw their allowance, allowing 2 boys equal to a man.

Ans. 7•346 mo.

The same question done the other way.

Ans. 7 months 10+ days. 4. There is a piece of land that is 10 rds. 10 links in width; I demand the length to make an acre, allowing 20 links to a rod.

Ans. 15.238 rds.

The same question done the other way.

Ans. 15 rds. 4f links. 5. How much in length of that which is 5 rods wide will make an acre

?

Ans. 32 rds. The same question done the other way.

Ans. 32 rds. 6. How much carpeting, that is 1.5 yds. wide will cover a floor that is 7.5 yds. long, and 5 yds. wide!

Ans. 25 yds. The same question done the other way.

Ans. 25. yds.

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DEFINITION.-The Double Rule of Three teaches to solve such questions as require two statings in the Single Rule of Three ; five numbers are given to find a sixth.

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Rule.-State the question by placing that number, which is the principal cause of gain or loss, in the first place ; that number which represents time, distance, &c. (that the first is gaining, or losing) in the second place ; that number which represents the gain or loss in the third place ; then place the other two numbers under those of the same name ; and if the term sought, or the blank place, fall under the third term; then multiply the three last terms together for a dividend, and the other two, for a divisor ; but if the blank place fall under the first, or second term, then multiply the first, second and last terms together for a dividend, and the other two for a divisor.

NOTE.The sixth term or answer always will come of the same name, and of the same denomination, of the term directly over the blank place.

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To solve questions that belong in the DOUBI.E RULE of THREE, by trvo statements in the SINGLE

RULE of THREE, RULE. Make the number which is the principal cause of gain, or loss, the first term; the gain, or loss the second ; the number, which is the demand of the question the third, the answer to this first statement will show what the third térm gained, or lost, in a time that the first term was gaining or losing, (which time is always mentioned in the question) there fore say in the second stating, as the space of time, or distance, &c. mentioned in the question, is to the answer of the first statement, so isthe required time or distance, &c. to the answer

Note.--I have done the questions in the Double Rule of Three, first by stating the five numbers at önce; and then by two statings in the Rule of Three Direct; the learner may follow my examples or not, as may best suit his inclination.

Examples. 1. If 200 dollars in 12 months will gain 12 dollars ; I demand how many 50 dollars will gain in six months ?

dolls.

dolls.
200 : 12 :: 12
50 :

6 : :
Three last terms 50 X 6X12_3600. dividend.
Two first terms 200x12=2400. divisor.

3600+2400=$1.50 Ans.
The same question done by-two statements.

Dolls. dolls, dolls dolls.
As 200

12
50

3

mo,

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:

Months,
As 12

dolls.

3

Months, coll. cts.

6 . 1.50 Ans.

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2. If 4-men spend 60 dollars, going 300 miles ; I demand what is sufficient for the expense of : 20 men, and one boy 700 milcs, allowing the boy, one half a man's expense?

Ans. $717.50 The same question done by two statements.

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3. If 3 men can build 360 rods of wall in 24 days ; I demand how many rods 7 men would buildl in 27 days ?

Ans. 945 rds,

The same question done by two statements,
rods

rods.
As 3
: 360

7 : 840

men.

men.

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:.

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men.

men.

:

4. If 3 men in 24 days can build 360 rods of wall.; I demand the number of men necessary to build 945 rods in 27 days?

Ans. 7 men. Tke same question done by two statements.

Ans. 7 men. 5. If 50 men can build a bridge in 144 days ; I demand the number of men necessary to build a {ike bridge in 720 days?

Ans. 10 men, The same question done by two statements, Days.

bridge. days. bridges. As 144 1. : ; 720 : 5 bridges,

bridge. As 5. 50 ::

1 10 Ans. 6. A man lent a friend 600 dollars, for 6 months ; for which he received 9 dollars interest: I demand the sum that will gain the same interest in 2 months?

Aas. $1800. The same question done by two statements. dolls,

dolls. As 6 : 9 :: 2 3 dolls. dolls.

dolls. As 3 :. 600 :: 9 1800 Ans. 7.A. received of B. 9 dollars, for the loan of 600 dollars six months ; now B, wishes to hire of A. 1800 dollars until the loan should amount to the game sumy; how long may he keep it?

Ans. 2 months.. The same question done by two statements.

dolls. interest dolls, interest.
As 600 9 :: 1800

27
dolls.
As 27 :

mo.

mo,

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dolls.

:

:

6 ; :

9 2 Ans.

mo..

dolls.

mo.

.:

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