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CASE IV.

To divide by fractional parts. RULE.-To divide by fractional parts, is the same as multiplying by them ; (see case 5th, compound multiplication.)

Examples. 1. What is the value of of an house, which is worth 83000?

3000X39000;4=$2250. Ans.

RE the de

10

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When the number of shares are unequal. RULE. Divide the sum by the number of simple shares, the quotient will be the share of the first, which multiply by so many as the second has more than the first, and thus continue till you have found all the shares.

PROOF.--Add all the shares together, and if the sum is equal to the sum divided, the work is right.

Examples.
1. Divide 8373.50 cts. among A. B. and C, in such a
manner, that B. may have twice as much as A. and C.
twice as much as B.

A. has 1 simple share.
2

shares.
shares.

B.' is
C. «

7 number of simple shares.

$373.50 = 75$53•35, A's. A's, $53.35 X 2 =106.7047, B's. B's, $106.704, X 2 =21304027, C's.

PROOF..$373•50 cts. 2. Divide $1089.33 cents among persons,

and give the second, three times as much as the first; the third, four times as much as the second, and the fourth 5 times as much as the third.

Ans. A's $14.337; B's $42.997; C's $171.967; and D's $859.80 cts. ; twenty five cts. being lost in fractions.

CASE VI.

je as julti

.

When the shares are not equal, but increase by a certain

ratio, a: 1, 2, 3, 4, 5, &c. Rule.-Divide the sum by the number of persons, the quotient is a mean, or middle share ; from the middle share subtract the ratio, the remainder is the next share that is less; from the last found share subtract the ratio until you have found all the shares that are less; to the middle share add the ratio, the sum is the next share that is larger ; to the last found share add the ratio till you have found all the shares that are larger,

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Examples. 1. Divide $600 among 5 persons, and as often as A. has 2, give B, 4, C. 6, D. 8, and E. 10 dolls,

1-16 A's share,

1-18 B's
8600-5 120 C's

Ans.
122 D's
124 E's

a. .
PROOF-9600.
2. Divide $640 among 7 persons in such a manner,
that the second shall have one more than the first, the
third one more than the second, &c.

Ans. A's share; $88.424; B's $89.42%, ; C's $90.426 ; D's, $91.428 ; E's,$92.424 ; F's,$93.42 ; G's$94.424.

Divide $1600 among four persons, in such a manner, that as often as the first has one, give the second 2, the third 3, the fourth 4 dolls,

1600.00 -45400.00 the middle share. dol.

dol, cts. 399.50 - 100 cts. ratio =398.50 A's share. 400.00 50 cts. half ratio 399.50 B's share 400.00 + 50 .cts. half ratio 400-50 C's share. 400:50 + 100 cts. ratio

B's

and

-sti.

und

401.50 D's share.

57:

$1600.00

Proof.?

AVERAGE JUDGMENT. DEFINITION.--Average Judgment, is the mean, or middle Judgment, of several persons, who are appointed to appraise any particular property.

RULE. _Add together the several sums which the commodity is appraised at, for a dividend; and the number of appraisers for a divisor; divide, and the quotient will be the mean, or middle Judgment required.

Eramples. 1. What is the value of a piece of land, which is valued by A. at $10; by B. at $11.50 ; by C. at 812.30 ; and by D. at $13:40 cts. per acre ?

A. 1. $10.00

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B. 1.
C. 1.
D. 1.

11.50
12.30
13:40

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Ans. $11.80 cts. 2. A. B. C. D. E. and F. were appointed to appraise a certain estate ; they appraised it as follows, viz. A. at $3470; B. at $3650 ; C. at $3700; D. at $3500; E. at 83400; and F. at 83600; I demand the value of the estate,

Ans. $3553•33 cts. 3ş m. 3. M. N. O. and P. appraised the ship Lucy as follows, viz. M. at $6700; N. at $9000; O. at 88750; and P. at 87380; what is the middle judge ment?

Ans. $795705 dimes.

SINGLE RULE OF THREE DIRECT.

DEFINITION.—The Single Rule of Three Direct teaches, by having three numbers given, to find a fourth, that shall have the same proportion to the third, as the second has to the first. If more re

quire more, the proportion is direcť; if less require less, the proportion is also direct ; more requiring more, is when the third term is greater than the first, and the sense of the question requires that the fourth term should be greater than the second ; less requiring less, is when the third term is less than the first, and the sense of the question requires that the fourth term should be less than the second.

Rule.--State the question, or arrange the three given numbers in such order, that the one which asks the question may stand in the third place ;* that number which is of the same name with the third, must possess the first place ; the remaining number (which is always of the same name with the number required) must possess the middle place. Reduce the first and third terms, or numbers, into the same denomination; and reduce the middle number, or term, into the lowest denomination mentioned; then multiply the second and third terms together, and divide the product by the first ; the quotient will be the answer, or fourth term sought; and always will be of the same denomination as the middle term was in when it was multiplied with the third term ; and may be reduced to any other denomination required.

Rule of Three in Decimals, Rule.-State the question as in the Rule of Three Direct; prepare the terms by reducing the smaller denominations to the decimal of the highest ; observing that the Integer in the first and third terms are in the same denomination ; multiply and divide as in thee Rule of Three Direct, and point off for decimals as is required in the rule of multiplication and division of decimals.

* The third term always asks a question and is generally preceded by some such words as, what will ? How much? How far? How long ? How soon? What is ? Where will, &c. ?

62 SINGLE RULE OF THREE DIRECT, &C.

Note.--As the currency of the United States is a decimal calculation, the decimal mode of computing becomes much the easiest, and for the benefit of the learner, I have done the questions in the Rule of Three by both methods, therefore one serves as a proof to the other.

Examples. 1. If ten sheep are worth $22.22; what are 19 of the kind worth? Sheep D. cts,

sheep. As 10 : 22•22 : : 19

19

19998
2222

10)422:18

42.218=$42.21 cts. 8 m. Ans. NOTE.--In the first question 19 sheep is the number that asks the question, and is placed in the third place ; 10 being of the same name, viz. sheep, it is placed in the first place, and the remaining number, viz. $22.22 cents, is the number left and is used for the middle term; and is of the same name with the number sought, viz. money. The same question done by direct proportion, Sheep

sheep. As 10 : 22.22 : : 19

100

D. cts.

2222

19

19998
2222

1042218

100) 422118 cts.

$42.2176 Ans.

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