33. There is a number consisting of two digits, which number divided by 5 gives a certain quotient and a remainder of 1, and the same number divided by 8 gives another quotient and a remainder of 1. Now the quotient obtained by dividing by 5 is twice the value of the digit in the tens' place, and the quotient obtained by dividing by 8 is equal to 5 times the digit in the units' place. What is the number?

Ans. 41.

34. The four classes in a certain college are to compete for four prizes, amounting in the aggregate to $119, and the prize money is to be raised by contribution, on the following conditions, namely: that the members of the class whose candidate obtains the 1st prize shall each pay one dollar, and the class whose candidate obtains the 2d prize shall pay the remainder. Now it is found that if a senior gets the 1st prize and a junior the 2d, each junior will pay of a dollar; if a junior gets the 1st prize and a sophomore the 2d, each sophomore will pay of a dollar; if a sophomore gets the 1st prize and a freshman the 2d, each freshman will pay of a dollar; and if a freshman gets the 1st prize and a senior the 2d, each senior will pay of a dollar. Of how many members does each class Freshman, 104; Sophomore, 93;

consist?

Ans.

35. Find four numbers, such that if 3 times the first be added to the second, 4 times the second be added to the third, 5 times the third be added to the fourth, and 6 times the fourth be added to the first, each sum shall be 359. Ans. 95, 74, 63, 44.

GENERAL SOLUTION OF PROBLEMS.

175. In the preceding problems, the given quantities have been expressed by numbers, and it has been required simply to determine the values of the unknown quantities from the numerical relations thus expressed.

If, however, the given quantities in any problem be represented by letters, the solution will give rise to a formula, showing not only the value of the unknown quantity, but indicating the precise ope

rations to be performed in order to obtain this value. This is called a general solution of the problem.

176. When any particular problem has been proposed, we may, by simply varying the numbers, form other problems of the same kind or class; and the solutions of all the problems of the class will require exactly the same operations. Hence,

177. The General Solution of a problem is the process of obtaining a formula which shall express, in known terms, the values of the unknown quantities in the given problem, or in any problem of its class.

178. An Arbitrary Quantity is one to which any value may be assigned at pleasure, in a general formula or equation.

179. For illustration, let the following questions be proposed : 1.—What number is that whose third part exceeds its fourth part by 6?

Instead of confining our attention to the particular numbers here given, we may first investigate the problem under a general form, as follows:

What number is that whose mth part exceeds its nth part by a ? Let x represent the number; then by the conditions,

Equation (3) is the formula which indicates the operations to be performed in solving all questions of this class.

If in this formula we put m = 3, n = 4, and a = 6, we shall

the number required by the particular question as at first proposed. 2.-What number is that whose fifth part exceeds its seventh part by 12?

To obtain the number by the formula, let m= a= 12; then

= 5, n = 7, and

x=

12×5×7

= 210, Ans.

EXAMPLES FOR PRACTICE.

1. Divide the number n into two such parts that the greater increased by a shall be equal to the less increased by b.

2. In the last example, what will be the two parts if n = 84,

4. My indebtedness to three persons, A, B, and C, amounts to a dollars; and I owe B n times the sum which I owe A, and C m times the sum which I owe A. What is my indebtedness to A?

5. In the last example, what is the sum due to A when a = $786, 2, and m = 3?

n =

Ans. $131.

6. A person engaged to work a days on these conditions: For each day he worked he was to receive b cents, and for each day he was idle he was to forfeit c cents; at the end of a days he received d cents. How many days was he idle?

7. My horse and saddle are together worth horse is worth n times the price of my saddle. of each?

α

Ans.

ab-d
I days.
b+c

a dollars, and my

What is the value

Ans. Saddle, ; Horse,
n+1

na

n+1

8. The rent of an estate is n per cent. greater this year than it was last. This year it is a dollars; what was it last year?

Ans.

100a 100+n

dollars.

9. A person after spending a dollars more than

of his income,

had remaining 6 dollars more than of it. Required his income.

10. A person after spending a dollars more than th of his income, had remaining b dollars more thanth of it. Required his income.

Ans.

mn(a+b)

mnmn

dollars.

11. If A can perform a certain piece of work in a days, and B can do the same in b days, and C the same in c days, in how many days can all together perform the work?

12. In the last example, what will be the time required, when a=6, b=8, and c= = 12? Ans. 23 days.

13. If from a times a certain number c be subtracted, the remainder will be equal to b times the number increased by d. Required the number.

Ans.

c+d

-b

14. A farmer would mix oats worth a cents a bushel with peas worth b cents a bushel, to form a mixture of c bushels worth d cents a bushel. How many bushels of each kind must he take?

15. There were a boys in one party, and b boys in another party, and each party had the same number of nuts. Each boy in the first party snatched m nuts from the second party, and ate them; then each boy in the second party snatched m nuts from the first party, and ate them. Each party then divided the nuts remaining to it equally among its members, when the boys in the two parties found that they had the same number of nuts apiece; how many nuts had each party at first? Ans. m(a+b).

16. Find four numbers, such that if a times the first be added to the second, b times the second be added to the third, c times the third be added to the fourth, and d times the fourth be added to the first, each sum shall be m.

17. A sent n pupils regularly to a certain school during a term of a days, and B sent m pupils regularly to another school for a term of b days. The two schools had the same number of pupils in attendance, and raised the same amount of money by rate-bill. There were c days' absence allowed for at the school to which A sent, and d days' absence at the school to which B sent; and A and B found that they had equal sums to pay. What was the number of pupils attending each school?

Ans.

18. Divide the number m into four parts, such shall be a times the first, the third a times the fourth a times the third.

Ans. 1st part,

bcm--adn

al(m-n)

that the second second, and the

m

a'+a+a+1

19. The sum of two numbers is s, and their difference is d. Required the numbers.

s+d

Ans. Greater, ; Less,
2

S -d

2

20. There are three numbers, such that the sum of the first and second is a, the sum of the first and third is b, and the sum of the second and third is c. What are the numbers?

21. There is a number consisting of two digits; the number is equal to a times the sum of its digits; and if c be number, the order of the digits will be reversed.

added to the Required the

22. Find what each of three persons, A, B, and C, is worth,knowing, 1st, that what A is worth added to 7 times what B and C are worth is equal to p; 2d, that what B is worth added to m times what A and C are worth is equal to q; 3d, that what C is worth added to n times what A and B are worth is equal to r.

We give here a solution of this example, partly to illustrate the method of simplifying algebraic formulas by the use of auxiliary quantities.