2. A company, dining at an inn, agreed to pay $3.50 for the entertainment; but before the bill was presented, two of the party left, in consequence of which each of the others had to pay 20 cents more than if all had been present. How many persons

dined?

Ans. 7.

3. 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.

4. It is required to divide the number 18 into two such parts, that the squares of these parts may be to each other as 25 to 16.

Ans. 10 and 8.

5. The difference of two numbers is 4, and their by the difference of their second powers, is 1600. numbers?

sum multiplied What are the Ans. 12 and 8.

6. What two numbers are those whose difference is to the less as 4 to 3, and whose product multiplied by the less is equal to 504? Ans. 14 and 6.

7. A man purchased a field, whose length was to its breadth as 8 to 5. The number of dollars paid per acre was equal to the number of rods in the length of the field; and the number of dollars given for the whole was equal to 13 times the number of rods round the field. Required the length and breadth of the field.

Ans. Length, 104 rods; breadth, 65 rods.

8. There is a stack of hay, whose length is to its breadth as 5 to 4, and whose height is to its breadth as 7 to 8. It is worth as many cents per cubic foot as it is feet in breadth; and the whole is worth at that rate 224 times as many cents as there are square feet on the bottom. Required the dimensions of the stack.

Ans. Length, 20 feet; breadth, 16 feet; height, 14 feet. 9. There is a number, to which if you add 7 and extract the square root of the sum, and to which if you add 16 and extract the square root of the sum, the sum of the two roots will be 9. What is the number? Ans. 9.

NOTE.-Represent the number by x2-7.

10. A and B together carried 100 eggs to market, and each received the same sum. If A had carried as many as B, he would

have received 18 pence for them; and if B had taken as many as A, he would have received 8 pence. How many had each? Ans. A 40, and B 60.

11. The sum of two numbers is 6, and the sum of their cubes is 72. What are the numbers? Ans. 4 and 2.

12. A man traveled 36 miles in a certain number of hours; if he had traveled one mile more per hour, he would have required 3 hours less to perform his journey. How many miles did he travel per hour?

13. The sum of two numbers is 100, the square roots is two; what are the numbers?

Ans. 3 miles.

difference of their

Ans. 36 and 64.

14. A gentleman bought a number of pieces of cloth for 675 dollars, which he sold again at 48 dollars a piece, and gained by the bargain as much as one piece cost him. What was the number of pieces ? Ans. 15.

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

Ans. $30.

16. A merchant sent for a piece of goods and paid a certain sum for it, besides 4 per cent. for carriage; he sold it for $390, and and thus gained 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.

17. From two towns, 396 miles apart, two persons, A and B, set out at the same time, and traveled toward each other; after as many days as are equal to the difference of miles they traveled per day, they met, when it appeared that A had traveled 216 miles. How many miles did each travel per day? Ans. A, 36; B, 30.

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

19. A vintner sells 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.

20. 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.

21. A farmer received $24 for a certain quantity of wheat, and an equal sum at a price 25 cents less per 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.

22. Two travelers, A and B, set out to meet each other, A leaving C at the same time that B left D. They traveled the direct road, and met 18 miles from the half-way point between C and D; and it appeared that A could have traveled B's distance in 15 days, and B could have traveled A's distance in 28 days. Required the distance between C and D. Ans. 252 miles.

23. Find two numbers, whose difference, multiplied by the difference of their squares, is 32, and whose sum, multiplied by the sum of their squares, is 272. Ans. 5 and 3.

24. A and B hired a pasture at a certain rate per week, agreeing that each should pay according to the number of animals he should have in the pasture. At first A put in 4 horses, and B as many as cost him 18 shillings a week; afterward B put in 2 additional horses, and found that he must pay 20 shillings a week. At what rate was the pasture hired? Ans. 30 shillings per week.

25. 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.

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

27. 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.

28. It is required to find two numbers, such that if their product be added to their sum, the result shall be 47; and if their sum be taken from the sum of their squares, the remainder shall be 62.

Ans. 7 and 5.

NOTE.-In many examples of two unknown quantities, giving rise to symmetrical equations, it will be found convenient to denote one of the unknown quantities by x+y, and the other by x—y.

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

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

31. 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.

32. 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. One in 20 days; the other in 30 days.

33. The joint stock of two partners 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, $600; B's, $400.

34. A speculator, 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 contained 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 number in the four droves was 1121. How many were in each drove ?

Ans. 1st, 162; 2d, 40; 3d, 606; 4th, 313.

35. Find two numbers, such that if the sum of their squares be subtracted from three times their product, 11 will remain; and if the difference of their squares be subtracted from twice their product, the remainder will be 14. Ans. 3 and 5.

36. Divide the number 20 into two such parts, that the product of their squares shall be 9216. Ans. 12 and 8.

37. Divide the number a into two such parts, that the product

39. A certain number is formed by the product of three consecutive numbers; and if it be divided by each of them in turn, the sum of the quotients will be 74. What is the number?

Ans.

{

120; that is, 4.5.6; or

—120; that is, (—4) · (—5) · (—6). 40. An engraving, whose length was twice its breadth was mounted on Bristol board, so as to have a margin 3 inches wide, and equal in area to the engraving, lacking 36 inches. Find the width of the engraving. Ans. 12 inches.

41. A man has two square lots of unequal dimensions, containing together 25 A. 100 P. If the lots were contiguous to each other, it would require 280 rods of fence to embrace them in a single inclosure of six sides. Required the dimensions of the two lots.

Ans. 62 rods and 16 rods, 50 rods and 40 rods.

42. A person has £1300, which he divides into two portions, and lends at different rates of interest. He finds that the incomes from the two portions are equal; but if the first portion had been lent at the second rate of interest it would have produced £36, and if the second portion had been lent at the first rate of interest it would have produced £49. Find the rates of interest.

Ans. 7 and 6 per cent.

43. A sets out from London to York, and B at the same time from York to London, both traveling uniformly. A reaches York 25 hours, and B reaches London 36 hours, after they have met on the road. Find in what time each has performed the journey.

Ans. A, 55 hours; B, 66 hours.

44. A owns a village lot, in square form, containing 36 square rods; B owns the adjacent lot on the same street, which is also a