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 gives 32, and whose sum, multiplied by the sum of their squares gives 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 order of 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. 18, 9, and 6.

27. What two numbers are those whose product is 24, and whose sum added to the sum of their squares gives 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 $1140 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

88. The greater of two numbers is a2 times the less, and the

b product of the two is b2. Find the numbers. Ans. and ab.

a'

39. A certain number is equal to 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? 120; that is, 4.5.6; or

Ans.

- 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 square 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 enclosure of six sides. Required the dimensions of the two lots. Ans. 62 rods and 16 rods, or 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 the form of a square, containing 36 square rods; B owns the adjacent lot on the same street, which is also a square, but greater than A's. Now if A should purchase all the front of B's lot, so as to extend the rear boundary line of his own through B's lot, parallel to the street, the two neighbors would possess equal quantities of land. Find the length of one side of B's lot. Ans. 6 (1 + √2) rods.

45. There are three numbers having the following relations to each other :-the sum of the squares of the first and second added to the first and second gives 32; the sum of the squares of the first and third added to the first and third gives 42; and the sum of the squares of the second and third added to the second and third gives 50. Required the quantities.

46. What is the edge of that cube which contains as many solid units as there are linear units in the diagonal through its opposite corners.

Ans. 3.

47. It is required to find two quantities such that their sum, their product, and the sum of their squares, shall be equal to each other. Ans. (3± √3), and ↓ (3 = √− 3).

48. Find two numbers whose sum, product, and the difference of whose squares, are equal to each other.

Ans. (3 ± √√5), and ‡ (1 ± √√5).

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

Ans. ±±√5, and ‡ (5 ± √√5).

SECTION VI.

PROPORTION, AND THE THEORY OF PERMUTATIONS AND COMBINATIONS.

PROPORTION.

315. Two quantities of the same kind may be compared, and their numerical relation determined, by finding how many times one contains the other. This mode of comparison gives rise to ratio and proportion.

316. The Ratio of two quantities is the quotient arising from dividing the first by the second.

There are two methods of indicating the ratio of two quantities. 1st. By writing the dividend before the divisor, with two dots between them; thus,

a: b

indicates the ratio of a to b, where a is the dividend and b the divisor.

2d. In the form of a fraction; thus, the ratio of a to b may be written

α

b

317. A Compound Ratio is the product of two or more ratios. Thus,

Simple ratios,

Compound ratio,

Sa: b
c: d

ac: bd.

318. The Duplicate Ratio of two quantities is the ratio of their squares.

319. The Triplicate Ratio of two quantities is the ratio of their cubes.

320. Proportion is an equality of ratios, both terms of each ratio being expressed. Thus, if two quantities, a and b, have the same ratio as two other quantities, c and d, the four