EXAMPLES.

1. What is the value of of an house, which is worth 3000X8 9000÷4-$2250. Ans.

$3000?

CASE VI.

(1) When the number of shares are unequal.

RULE.-(2) Divide the sum by the number of simple shares, the quotient will be the share of the first, which (3) multiply by so many as the second has more than the first, and thus continue till you have found all the shares.

PROOF.-(4) Add all the shares together, and if the sum is equal to the sum divided, the work is right.

Questions.-1. When is Case 6th in Compound Division used?-2. What is the first step in the rule?-3. How then do you proceed?-4. What is the method of proof?

EXAMPLES.

1. Divide $373·50 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.

2. Divide $1089.33 cents among 4 persons, and give the second, three times as much as the first; the third, four times as much as the second, and the fourth, five times as much as the third.

Ans. A's. $14.33†; B's. $42.99†; C's. $171.96f; and D's. $859.80 cts.; twenty five cents being lost in fractions.

CASE VII.

(1) When the shares are not equal, but increases by a certain ratio, as 1, 2, 3, 4, 5, §c.

RULE. (2) Divide the sum by the number of persons, the quotient is a mean, or middle share; (3) from the middle share subtract the ratio, the remainder is the next share that is less; (4) from the last found share subtract the ratio until you have found all the shares that are less; (5) 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. If the number of persons is an even number, as 4, 6, 8, &c. (7) divide as above, and from the quotient subtract half the ratio and the remainder is one share-and add half the ratio to the quotient, the sum is another share; these two shares are the two middle shares; (3) for the shares that are less continue to subtract the ratio, and for those that are larger, add the ratio, till you have found all the shares.

PROOF. (9) Add together the several shares found, and if the sum total is equal to the sum divided, the work is right.

Questions.-1.When is Case 7th in Compound Division used?-2. How do you find the mean or middle share?-3. How the next less?-4. How the other smaller shares?-5. How do you find the next share larger than the middle?-6. How the other larger shares?—7. If the number of persons are even, how do find the two mean or middle shares?-8. How do you you find those shares that are less, and those that are larger than the two middle shares?-9. What is the method of proof?

EXAMPLES.

1. Divide $600 among 5 persons, in such a manner that B. may have two more than A., and C. two more than B., &c.

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, $88.429; B's$89-424; C's $90-42; D's $91-42; E's $92-424; F's $93.42; G's $94-42 .

3. Divide $1600 among four persons, in such a manner that the second may have one more than the first, the third one more than the second, &c.

1600-00-4-400.00 the middle share.

DEFINITION.-Average Judgment, is (1) the mean, or middle judgment, of several persons, who are appointed to appraise any particular property.

RULE.-(2) Add together the several sums which the commodity is appraised at, for a dividend; and use the number of appraisers for a divisor; divide, and the quotient will be the mean, or middle judgment required.

Questions.-1. What is Average Judgment?-2. What is the general

rule?

EXAMPLES.

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 $12.30; and by D. at $13.40 cts. per acre?

$10.00

11.50

2. A. B. C. D. E. and F. were appointed to appraise a certain estate; they appraised it as follows, viz. A. at $8470; B. at $3650; C. at $3700; D. at $3500; E. at $3400; and F. at $3600; I demand the value of the estate.

Ans. $3553-33 cts. 33 m.

3. M. N. O. and P. appraised the ship Lucy as follows, viz. M. at $6700; N. at $9000; O. at $8750; and P. at $7380; what is the middle judgment?

Ans. $7957-5 d.

COMPOUND AVERAGE JUDGMENT.

Compound Average Judgment is (1) when the judgment of the referees is partly on one side of the equality and partly on the other.

RULE.

(2) Subtract one side from the other, and divide the remainder by the number of referees, and the quotient will be the average judgment.

Questions.-1. When is average judgment said to be compound?-2 Give the rule for its operation?

EXAMPLES.

1. A. and B. wishing to exchange horses and disagreeing as to the conditions, referred the matter to X. Y. and Z. by whose judgments they agreed to abide,which were as follows, viz.-X. said A. should pay B. $8, and Y. said A. should pay B. $6; but Z. said B. should pay A. $5. What is the average judgment? Ans. $3 B. receives.

2. E. and F. proposed to swap watches-agreed to refer it to A. B. C. and D. to say how they should exchange: A. marked that E. should have $4; B. said E. should have $5; C. said E. should have $2; but D. said F. should have $3,50; which receives the boot, and how much?

Ans. E. $1.87.5.

SINGLE RULE OF THREE DIRECT.

DEFINITION.—The Single Rule of Three Direct teachea, (1) by having three numbers given, to find a fourth, that shall have the same proportion to the third, as the secord has to first. (2) When more requires more, or less requirts less, the proportion is 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 (4) 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.-(5) 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. (6) 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 (7) multiply the second and third terms together, and divide the product by the first; the quotient will be (8) 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.

Questions.-1. What does the Single Rule of Three Direct teach?-2. When is proportion direct?-3. When does more require more?-4. When does less require less?-5. What is the first step in the Rule?-6. What the second step?-7. What the third?-8. What will the quotient be?-9. Of what denomiation will it be, and to what may it be reduced?

RULE OF THREE IN DECIMALS.

RULE.-State the question (1) as in the Rule of Three Direct; prepare the terms (2) by reducing the smaller denominations to the decimal of the highest; observing (5)that the Integer in the first and third terms are in the same denomination: (4) multiply and divide as in the Rule of Three Direct, and point off for decimals as is required in the rule of multiplication and division of decimals.

Questions.-1. How do you state the question in the Rule of Three in Decimals?-2. How do you prepare the terms?-3. What do vou observe? -4. How then do you proceed?

NOTE. As the currency of the United States is a decimal calculation, it becomes most necessary to calculate in that way; but I have done the questions in the Rule of Three by both methods, therefore one will prove the other.

*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.