27. A says to B and C, give me half of your money, and I shall have $55. B replies, if you two will give me one third of yours, I shall have $50. But C says to A and B, if I had one fifth of your money, I should have $50. Required the sum that each possessed.

Ans. A, $20; B, $30; C, $40.

28. A gentleman left a sum of money to be divided among his four sons, so that the share of the oldest was of the sum of the shares of the other three, the share of the second of the sum of the other three, and the share of the third of the sum of the other three; and it was found that the share of the oldest exceeded that of the youngest by $14. What was the whole sum, and what was the share of each person?

Ans. Whole sum, $120; oldest son's share, $40; second son's, $30; third son's, $24; youngest son's, $26.

171.

GENERAL SOLUTION OF PROBLEMS.

171. In the GENERAL SOLUTION of a problem, all the quantities are represented by letters, or the symbols of general values.

The unknown thus found in terms of the known quantities is a general expression, or formula, which can be used for the solution of any similar problem.

172. A problem is said to be generalized when letters. are used to represent its known quantities.

The algebraic solution of general problems discloses many interesting truths and useful practical rules, as may be seen from the consideration of the following, among

other

GENERAL PROBLEMS.

1. The sum of two numbers is a, and their difference is b; what are the two numbers ?

Hence, since a and b may have any value whatever, the values of x and y are general, and may be expressed as rules for the numerical calculations in any like case; thus,

To find two numbers, when their sum and difference are given, Add the sum and difference, and divide by 2, for the greater of the two numbers; subtract the difference from the sum, and divide by 2, for the less number.

2. A can do a piece of work in a days, which it requires b days for B to perform. In how many days can it be done if A and B work together?

SOLUTION.

Let x = the number of days required, and 1 = the entire work; then, in 1 day A can do

1

of the work, and

a

Ꮖ

α

B; therefore, in a days, they can do and of the

x

work. Hence,

To find the time required for two agencies conjointly to accomplish a certain result, when the times are given in which each separately can accomplish the same,Divide the product of the given times by their sum.

3. A cistern can be filled by three pipes; by the first in a hours, by the second in 6 hours, and by the third in c hours. In what time can it be filled by all the pipes running together?

Ans.

abc
ab + ac + b c

hours.

Here it will be seen that, when

three agencies are

employed, the required time is the product of the given times, divided by the sum of their products, taken two and

two.

4. In the last example, what will be the time required, if a= = 2, b = 5, and c = 10? Ans. 1 hours.

5. Three men, A, B, and C, enter into partnership for a certain time. Of the capital stock A furnishes m dollars; B, n dollars; and C, p dollars. They gain a dollars. What is each man's share of the gain?

To find each man's gain, when each man's stock and the whole gain are given, Multiply the whole gain by each man's stock, and divide the product by the whole stock.

Р

6. In the last example, if m

=

$ 300, n

=

$ 500,

$800, and a = $320, what is each man's share of

the gain?

Ans. A's share, $60; B's share, $100; C's share, $160.

7. Divide the number a into two parts which shall have to each other the ratio of m to n.

8. A courier left this place n days ago, and goes a miles each day. He is pursued by another, starting to-day and going 6 miles daily. How many days will the second require to overtake the first?

na

Ans.

b

days.

9. Required what principal at interest at r per cent. will amount to the sum a, in t years.

Ans.

100 a 100+ rt

10. A gentleman distributing some money among beggars, found that in order to give them a cents each, he should want b cents more; he therefore gave them c cents each, and he had d cents left. Required the number of beggars.

Aus.

b + d

a

prxx-a

10+2000

11. Required the number of years in which p dollars, at r per cent. interest, will amount to a dollars.

12. A banker has two kinds of money; it takes a pieces of the first to make a dollar, and b pieces of the second to If he is offered a dollar for c pieces,

make the same sum.
how many of each kind

Ans. First kind,

a (c — b)

13. In the last example, if a=10, b=20, and c=15, how many of each kind must he give?

Ans. First kind, 5; second kind, 10.

14. A mixture is made of a pounds of coffee at m cents a pound, pounds at n cents, and c pounds at p cents. Required the cost per pound of the mixture.

15. A, B, and C hire a pasture together for a dollars. puts in m horses for t months, B puts in n horses for t months, and C puts in p horses for t" months. What part of the expense should each pay ?

nťa

mt + nť + pt";
pt" a
mt+nt+pt”

DISCUSSION OF PROBLEMS

LEADING TO SIMPLE EQUATIONS.

173. The DISCUSSION of a problem, or of an equation, is the process of attributing any reasonable values and relations to the arbitrary quantities which enter the equation, and interpreting the results.

174. An ARBITRARY QUANTITY is one to which any reasonable value may be given at pleasure.