and 5 oranges, for 28 cents; and at another, 5 peaches and 6 oranges, for 39 cents; required the cost of each kind of fruit.

Ans. Apples 1 cent, pears 2, peaches, 3, oranges 4 cts., each. 8. A and B together possess only as much money as C; B and C together, have 6 times as much as A; and B has 680 dollars less than A and C together; how much has each?

Ans. A $200, B $360, and C $840.

9. A, B, and C together, have 1820 dollars; if B give A 200 dollars, then A will have 160 dollars more than B; but if B receive 70 dollars from C, they will both have the same sum; how much has each? Ans. A $400, B $640, and C $780. 10. Three persons, A, B, and C, compare their money; A says to B, "Give me 700 dollars, and I shall have twice as much as you will have left." B says to C, "Give me 1400 dollars, and I shall have three times as much as you will have left." And C says to A, "Give me 420 dollars, and then I shall have five times as much as you will have left." How much has each?

Ans. A $980, B $1540, and C $2380. 11. A certain number is expressed by three figures, and the sum of the figures is 11; the figure in the place of units, is double that in the place of hundreds; and if 297 be added to the number, its figures will be inverted; required the number.

Ans. 326.

12. Three persons, A, B, and C, together, have 2000 dollars; if A gives B 200 dollars, then B will have 100 dollars more than C; but, if B gives A 100 dollars, then B will have only as much as C; required the sum possessed by each.

Ans. A $500, B $700, and C $800. 13. There are three numbers whose sum is 83; if, from the first and second you subtract 7, the remainders are as 5 to 3; but if from the second and third, you subtract 3, the remainders are to each other as 11 to 9; required the numbers. A. 37, 25, 21. 14. Divide 180 dollars between three persons, A, B, and C, so that twice A's share plus 80 dollars, three times B's share, plus 40 dollars, and four times C's share plus 20 dollars, may be all equal to each other. Ans. A $70, B $60, and C $50. 15. There are three numbers whose sum is 78; of the first is to of the second, as 1 to 2; also, of the second is to of the third, as 2 to 3; what are the numbers? Ans. 9, 24, and 45.

of

of the

16. A, B, and C, have a sum of money; A's share exceeds the shares of B and C, by 30 dollars; B's share exceeds of the shares of A and C, by 30 dollars; and C's share exceeds shares of A and B, by 30 dollars; what is the share of each? Ans. A's $150, B's $120, and C's $90.

17. If A and B can perform a certain work in 12 days, A and C in 15 days, and B and C in 20 days, in what time could each do it alone? Ans. A 20, B 30, and C 60 days.

18. A number, expressed by three figures, when divided by the sum of the figures plus 9, gives a quotient 19; also, the middle figure is equal to half the sum of the first and third; and, if 198 be added to the number, we obtain a number with the same figures in an inverted order; what is the number? Ans. 456. 19. A farmer mixes barley at 28 cents, with rye at 36, and wheat at 48 cents per bushel, so that the whole is 100 bushels, and worth 40 cents per bushel. Had he put twice as much rye, and 10 bushels more of wheat, the whole would have been worth exactly the same per bushel; how much of each kind was there? Ans. Barley 28, rye 20, and wheat 52 bushels.

20. A, B, and C, in a hunting excursion, killed 96 birds, which they wish to share equally; in order to do this, A, who has the most, gives to B and C as many as they already had; next, B gives to A and C as many as they had after the first division; and lastly, C gives to A and B as many as they both had after the second division; it was then found, that each had the same number; how many had each at first? Ans. A 52, B 28, and C 16.

CHAPTER V.

SUPPLEMENT TO EQUATIONS OF THE FIRST DEGREE.

GENERALIZATION.

ART. 164.-EQUATIONS are termed literal, when the known quantities are represented, either entirely or partly, by letters. Quantities represented by letters, are termed general values-because, by giving particular values to the letters, the solution of one problem, furnishes a general solution to all others of the same kind. The answer to a problem, when the known quantities are represented by letters, is termed a formula; and a formula, expressed in ordinary language, furnishes a rule.

By the application of Algebra to the solution of general questions, a great number of useful and interesting truths and rules may be established. We shall now proceed to illustrate this subject, by a few examples.

ART. 165.-1. Let it be required to find a number, which being divided by 3, and by 5, the sum of the quotients will be 16.

2. Again, let it be required to find another number, which being divided by 4, and by 7, the sum of the quotients will be 11.

By proceeding, as in the preceding question, we find the number to be 28.

Instead, however, of solving every example of the same kind separately, we may give a general solution, that will embrace all the particular questions. Thus:

3. Let it be required to find a number, which being divided by two given numbers, a and b, the sum of the quotients may be equal to another given number, c.

The answer to this question is termed a formula; it shows, that the required number is equal to the continued product of a, b, and c, divided by the sum of a and b. Or, it may be expressed in ordinary language, thus: Multiply together the three given numbers, and divide the product by the sum of the divisors; the result will be the required number.

The pupil may test the accuracy of this rule, by solving the following examples, and verifying the results.

4. Find a number, which being divided by 3, and by 7, the sum of the quotients may be 20. Ans. 42.

5. Find a number, which being divided by and, the sum of the quotients may be 1. Ans..

ART. 166.-1. The sum of 500 dollars is to be divided between two persons, A and B, so that A may have 50 dollars less than B. Ans. A $225, B $275.

To make this question general, let it be stated as follows:

REVIEW.-164. When are equations termed literal? When are quan tities termed general? When is the answer to a problem termed a formula? What is a formula called, when expressed in ordinary language? 165. Example 3. What is the answer to this question, expressed in ordinary language?

2. To divide a given number, a, into two such parts, that their difference shall be b. Or thus:

The sum of two numbers is a, and their difference b; required the numbers.

Let x= the greater number, and y= the less.

This formula, when expressed in ordinary language, gives the

RULE,

FOR FINDING TWO QUANTITIES, WHEN THEIR SUM AND DIFFERENCE

ARE GIVEN.

To find the greater, add half the difference to half the sum. find the less, subtract half the difference from half the sum.

To

Let the learner test the accuracy of the rule, by finding two numbers, such that their sum shall be equal to the first number in each of the following examples, and their difference equal to the second.

3. Sum 200, difference 50. 4. Sum 100, difference 25. 5. Sum 15, difference 10.

Ans. 125, 75, Ans. 621, 371. Ans. 12, 2

Ans. 31, 28.

6. Sum 5, difference 3. ART. 167.-1. A can perform a certain piece of work in 3 days, and B in 4 days; in what time can they both together do it? Ans. 15 days.

To make this question general, let it be stated thus: 2. A can perform a certain piece of work in m days, and B can do it in n days; in how many days can they both together do it? Let x= the number of days in which they can both do it.

1

Then ==the part of the work which both can do in one day.

х

Also, if A can do the work in m days, he can do

1

one day. And, if B can do the work in n days, he can do part

n

This result, expressed in ordinary language, gives the following

RULE.

Divide the product of the numbers expressing the time in which each can perform the work by their sum; the quotient will be the time in which they can jointly perform it.

The question can be made more general, by expressing it thus: An agent, A, can produce a certain effect, e, in a time, t; another agent, B, can produce the same effect, in a time, t'; in what time can they both do it? Both the result and the rule would be the same as that already given.

The following examples will illustrate the rule.

3. A cistern is filled by one pipe in 6, and by another in 9 hours; in what time will it be filled by both together?

A. 3 hrs.

4. One man can drink a keg of cider in 5 days, and another in 7 days; in what time can both together drink it? A. 21 dys. ART. 168.-Let it be required to find a rule for dividing the gain or loss in a partnership, or, as it is generally termed, fellowship. First, take a particular question.

1. A, B, and C, engage in trade, and put in stock in the following proportions: A put in 3 dollars, as often as B put in 4, and as often as C put in 5 dollars. Their gains amounted to 60 dollars; required the share of each, the gains being divided in proportion to the stock put in.

Let 3x

A's share of the gain, then 4x= B's, and 5x=- C's. (See Example 24, page 126.)

2. To make this question general, suppose A puts in m dollars, as often as B puts in n dollars, and as often as C puts in r dollars; and that they gain c dollars. To find the share of each.

REVIEW.-166. By what rule do you find two quantities, when their sum and difference are given? 167. When the times are given, in which each of two agents can produce a certain effect, how is the time found in which they can jointly produce it?