5. There is a certain number, which being subtracted from 22, and the remainder multiplied by the number, the product will be 117. What is the number? Ans. 13 or 9.

6. In a certain number of hours a man traveled 36 miles, but if he had traveled one mile more per hour, he would have taken 3 hours less than he did to perform his journey. How many miles did he travel per hour? Ans. 3 miles.

7. A person dies, leaving children and a fortune of $46,800, which, by the will, is to be divided equally among them; but it happens that immediately after the death of the father, two of the children also die; and if, in consequence of this, each remaining child receive $1950 more than he or she was entitled to by the will, how many children were there? Ans. 8 children.

S. A gentleman bought a number of pieces of cloth for 675 dollars, which he sold again at 48 dollars by the piece, and gained by the bargain as much as one piece cost him. the number of pieces?

What was

Ans. 15.

This problem produces one of the equations in (Art. 107.)

9. A merchant sends for a piece of goods and pays a certain sum for it, besides 4 per cent. for carriage; he sells it for $390, and thus gains as much per cent. on the cost and carriage as the 12th part of the purchase money amounted to. For how much did he buy it? Ans. $300..

10. Divide the number 60 into two such parts that their produce shall be 704. Ans. 44 and 16.

11. A merchant sold a piece of cloth for $39, and gained as much per cent. as it cost him. What did he pay for it? Ans. $30.

12.

A and B distributed 1200 dollars each, among a certain

How many

number of persons. A relieved 40 persons more than B, and B gave to each individual 5 dollars more than A. were relieved by A and B? Ans. 120 by A, and 80 by B.

This problem can be brought into a pure equation, in like manner as (Problem 1.)

13. A vintner sold 7 dozen of sherry and 12 dozen of claret for £50, and finds that he has sold 3 dozen more of sherry for £10 than he has of claret for £6. Required the price of each? Ans. Sherry, £2 per dozen; claret, £3.

14. A set out from C towards D, and traveled 7 miles a day. After he had gone 32 miles, B set out from D towards C, and went every day of the whole journey; and after he had traveled as many days as he went miles in a day, he met A. Required the distance from C to D.

Ans. 76 or 152 miles; both numbers will answer the condition.

15. A farmer received $24 for a certain quantity of wheat, and an equal sum at a price 25 cents less by the bushel for a quantity of barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of each ?

Ans. 32 bushels of wheat, and 48 of barley.

16. A and B hired a pasture, into which A put 4 horses, and B as many as cost him 18 shillings a week; afterwards B put in two additional horses, and found that he must pay 20 shillings a week. At what rate was the pasture hired?

Ans.

B had six horses in the pasture at first, and the price of the whole pasture was 30 shillings per week.

17. A mercer bought a piece of silk for £16 4s., and the number of shillings he paid per yard, was to the number of yards as 4 to 9. How many yards did he buy, and what was the price per yard. Ans. 27 yards, at 12 shillings per yard. 18. If a certain number be divided by the product of its two digits, the quotient will be 2, and if 27 be added to the number, the digits will be inverted. What is the number?

Ans. 36.

19. It is required to find three numbers, whose sum is 33, such that the difference of the first and second shall exceed the difference of the second and third by 6, and the sum of whose squares is 441. Ans. 4, 13, and 16.

20. Find those two numeral quantities whose sum, product, and sum of their squares, are all equal to each other.

Ans. No such numeral quantities exist. In a strictly algebraic sense, the quantities are

√3, and F√3.

21. What two numbers are those whose product is 24, and whose sum added to the sum of their squares is 62?

Ans. 4 and 6.

22. It is required to find two numbers, such that if their product be added to their sum it shall make 47, and if their sum be taken from the sum of their squares, the remainder shall be 62?

Ans. 7 and 5.

23. The sum of two numbers is 27, and the sum of their cubes 5103. What are their numbers? Ans. 12 and 15.

24. The sum of two numbers is 9, and the sum of their fourth powers 2417. What are the numbers? Ans. 7 and 2.

25. The product of two numbers multiplied by the sum of their squares, is 1248, and the difference of their squares is 20 What are the numbers? Ans. 6 and 4

Let x+y=the greater, and x-y-the less.

26. Two men are employed to do a piece of work, which they can finish in 12 days. In how many days could each do the work alone, provided it would take one 10 days longer than the other? Ans. 20 and 30 days.

27. The joint stock of two partners, A and B, was $1000. A's money was in trade 9 months, and B's 6 months; when they shared stock and gain, A received $1,140 and B $640. What was each man's stock?

Ans. A's stock was $600; B's $400.

28. A speculator from market, going out to buy cattle, met with four droves. In the second were 4 more than 4 times the square root of one half the number in the first. The third con

tained three times as many as the first and second. The fourth

was one half the number in the third and 10 more, and the whole

ーん

number in the four droves was 1121. How many were in each drove? Ans. 1st, 162; 2d, 40; 3d, 606; 4th, 313.

29. Divide the number 20 into two such parts, that the pro- y duct of their squares shall be 9216. Ans. 12 and 8.

30. Divide the number a into two such parts that the product of their squares shall be b.

31. Find two numbers, such that their product shall be equai to the difference of their squares, and the sum of their squares shall be equal to the difference of their cubes.

Ans. 5 and 4(5±√5).

SECTION V.

ARITHMETICAL PROGRESSION.

CHAPTER I.

A series of numbers or quantities, increasing or decreasing by the same difference, from term to term, is called arithmetical progression.

Thus, 2, 4, 6, 8, 10, 12, &c., is an increasing or ascending arithmetical series, having a common difference of 2; and 20, 17, 14, 11, 8, &c., is a decreasing series, having a common difference of 3.

(Art. 115.) We can more readily investigate the properties' of an arithmetical series from literal than from numeral terms. Thus let a represent the first term of a series, and d the common difference. Then

a,(a+d),(a+2d),(a+3d),(a+4d), &c., represent an ascend

ing series; and

a, (a—d), (a—2d), (a—3d),(a—4d), &c., represent a descendng series.

Observe that the coefficient of d, in any term is equal to the number of the preceding terms.

The first term exists without the common difference. All other terms consist of the first term and the common difference multiplied by one less than the number of terms from the first.

Wherever the series is supposed to terminate, is the last term, and if such term be designated by L, and the number of terms by n, the last term must be a+(n-1)d, or a―(n-1)d, according as the series may be ascending or descending, which we draw from inspection.

(Art. 116.) It is manifest that the sum of the terms will be the same, in whatever order they are written.

Take the series a, a+d, a+2d, a+3d, a+4d,

Inverted,

Sums will be

a+4d, a+3d, a+2d, a+ d, a

2a+4d, 2a+4d, 2a+4d, 2a+4d, 2a4d.

Here we discover the important property, that, in an arithmetical progression, the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes. Also, that twice the sum of any series is equal to the extremes, or first and last term repeated as many times as the series contains

terms.

Hence, if S represents the sum of a series, and n the number of terms, a the first term, and L the last term, we shall 2S=n(a+L)

have

The two equations (A) and (B) contain five quantities, a, d,