the third, $8000 for 9 months. They gained $12050 ; what ought each person to receive?

Although the stocks were in trade unequal times, they are reduced to the same time, in the following manner. The stock $3000 ought to gain as much in 6 months, as 6 times $3000, or $18000 in one month. The stock $4000 ought to gain in 5 months, as much as 5 times $4000, or $20,000, in one month. And the stock $8000 ought to produce in 9 months, as much as 9 times $8000, or $72000, in one month. Hence the question is reduced to this; three partners put into trade $18000, $20,000, and $72000, and gain $12050; what ought each one to receive ?

Proceeding as in the last example, we find 81971,81, $2190,901, and $7887,273, for their several shares.

REMARK ON THE PRECEDING RULE.

198. It may be useful here to examine a case, that might embarrass the learner. If it be required to divide the number 650 into three parts, of which the first shall be to the second :: 5 : 4, and the first to the third :: 7 3, we cannot apply the preceding rule without a preparation, which consists in making equal, in each given ratio, the parts which are proportional to one of the three parts sought. For example, that of the first; this is easily done by multiplying both terms of each ratio by the first term of the other ratio. Thus the two ratios 5: 4 and 7: 3, will have the same first term by multiplying both terms of the first by 7, and both terms of the second by 5; which does not change the value (170,) and gives the ratios 35: 28 and 35: 15. Hence the question becomes to divide the number 650 into three parts, which shall be to each other as the numbers 35, 28, and 15; which is easily done by means of the preceding rule.

If it be required to divide a number into four parts, of which the first is to the second :: 5 : 4, the first to the third 9: 5, and the first to the fourth: 7:3; these ratios may be reduced to have the same first term, by multiplying both terms of each by the products of the first terms of the other two. Thus, in this, example we change the three given ratios into these other

three, 315: 252, 315: 175, 315: 135; hence the question becomes to divide the given number into four parts, which shall be to each other as the numbers 315, 252, 175, and 135.

EXAMPLES FOR PRACTICE.

1. A and B buy goods, which cost £120; of which A pays £80 and B £40, and they gain £32: what part of the gain ought each to receive?

Ans. A 21 6s. 8d.: and B £10 13s. 4d. 2. Three merchants trading together, gained $800, A's stock was $1200, B's $4800, and C's $2000; what was each man's share of the gain ?

$120 A's share.

480 B's share. Ans. 200 C's share.

3. Three merchants trading together lost goods to the amount of 1920doll. A's stock in trade was 2880doll., B's 11520doll., C's 4800doll.; what share of the loss ought each man to sustain ?

Ans.

SDoll.

Doll.288 A's share. 1152 B's share.

480 C's share.

4. Three men are to share a legacy of $1500, of which B is to have, C, and D the remainder; but C relinquishes his share to B and D, to be divided according to their share in the whole. What part of the legacy will B and D resqectively receive?

Ans. B's part is $1000, and D's $500. 5. Divide the number 360 into 4 parts, that shall be to each other as the numbers 3, 4, 5, and 6.

6. A and B enter into partnership; A put in £45, and took of the gain; what did B put in ? Ans. £30. 7. A man dying left an estate of $12000, of which is to be given to his wife, to his eldest son, to his second son, and to his daughter. What share will each receive of the estate, divided in these proportions ?

The numerators of the fractions, when reduced to a common denominator, are 30, 20, 15, and 12, which show the proportions. Their sum 77 is the first term of the several proportions to be made in the Rule of Three. Hence, Ans.

The eldest son's share is $3116,8824.

The second son's

The daughter's

The wife's

Ans 66

2337,669.

1870,127.

4675,3236.

8. Two men A and B, traded in company. A put 350doll. for 8 months, and B 640doll. for 5 months ; they gained 250doll. What was the share of each? SDoll.116,66 A's share. { 133,33 B's share 9. B and D joined stocks of $500 each; they traded 2 years, and B took out one fifth of his stock, and then continued their trade 3 years longer, and gained $627; what is the share of each ?

Ans.

SB's share $993,4844. 333,51

D's

10. A family of 10 persons took a house for a year, for which they were to pay $500. At the end of 14 weeks they took in 14 new lodgers, and after 3 weeks 4 more; and for every 3 weeks during the remainder of the term they took in 4 more lodgers. What part of the rent for the whole time must one of each class payi P One of the first class must pay £39, 09

Ans.

66

2d

12,167

8,046

4,841

2,219

11. Three graziers hired a piece of land for £60 10s. A put in 5 sheep for 4 months, B put in 8 for 5 months, and C put in 9 for 6 months; how much must each pay of the rent? Ans. A 611 7s. 5d. 3125; B 20, 5s. 5d., and C 628 17s. 1d. 3103

235

12. Two men, K and I, were joint owners of a mill, in the building of which K laid out $150, and L $270. At the end of 7 months, K sold his share to L, and at the end of the year I sold the mill. The year's profit

of the mill was $260; on settlement, what was each K's share was $ 54,162 205,831

man's share?

L's

Ans.

OTHER RULES DEPENDING UPON PROPORTIONS.

199. The following rules are not only useful in themselves, but they also serve to explain the various applications of proportions.

SINGLE POSITION.

200. The first which we shall explain is called Single Position. It is used to solve questions which belong to the rules of Fellowship, from which it differs in this; that instead of taking parts proportional to those given in the enunciation of the question, it arbitrarily assumes one part to which the others are referred in conformity to the question. By this method calculation is rendered more easy.

EXAMPLE I.

Divide £640 between three persons, of which the second shall have four times as much as the first, and the third twice and as much as both the others. To represent the first part, we arbitrarily take the number 3, of which the part may be conveniently taken. The first part being 3, the second will be 12, and the third 35. The question then is to divide 640 into three parts, which shall be to each other as the three numbers 3, 12 and 35. This operation is performed by the rule in article (197).

The rule of Single Position is also used for solving such questions as are in some measure the reverse of those, which belong to the rule of Fellowship; for it teaches a method of descending from the sum of certain parts of a given number, to that number; as in the following example.

EXAMPLE II.

It is required to find a number, of which the,, and , shall make 808. We take a number of which the 1, and 2, may be conveniently founl. (This is easily done by multiplying the three denominators together.) This number is 105, of which we take the, which is

35, the, which is 21, and the 3, which is 45. On adding these three numbers, we find their sum is 101, which is composed of the parts of 105, in the same manner as 808 is composed of the parts of the number in question; then this number must have the same ratio to 808 as 105 has to 101. It will therefore be the fourth term of a proportion, which commences with these three;

101: 105: 808 :

The fourth term sought is 840, of which 808 contains the and 23/

DOUBLE POSITION.

201. The second rule, of which we shall speak, is called Double Position. It is used for solving questions, in which it is not required to divide the given number, but only a part of it, into parts proportional to given numbers. The following example will explain this rule and its application,

EXAMPLE III.

It is required to distribute $6954 among three persons, in such manner, that the second shall have as much as the first, and $54 more; and the third as much as both the others, and $78 more. If $54 and $78 were not in the question, it is plain, nothing would be required but to divide the given number into parts proportional to the numbers 1, I and 2; but as $54 must be deducted for the second person, and $54 plus $78 for the third, only a part of the given number can be divided into parts proportional to 1, 1 and 2. For finding this part, which is easily done in the present example, but in others may be accomplished with more difficulty, the method here given is to be followed.

Suppose for the first part, any convenient number, $1 for example; the second part will be $1, plus $54; that is $55; and the third will be $1 plus $55, plus $78; that is $134: the sum of these parts is $190. If it were only required to divide the given number into parts proportional to 1, 1 and 2; the first part being always sup