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8. If $120 be divided between A and B so that A shall receive $20 more than B, how many dollars will each receive?

9. A sum of $2500 is divided between A and B. B receives $4 as often as A receives $1. How much does each receive?

10. Divide 75 into two parts, such that three times the first part shall be 15 greater than seven times the second.

11. Divide 190 into three parts so that the second shall be three times the first, and the third five times the second.

12. A father's age exceeds his son's by 18 years, and the sum of their ages is four times the son's age. What are their ages?

13. A man bought a horse, a carriage, and harness for $320. The horse cost five times as much as the harness, and the carriage cost twice as much as the horse. How much did each cost?

14. The deposits in a bank during three days amounted to $16,900. If the deposits each day after the first were one-third of the deposits of the preceding day, how many dollars were deposited each day?

15. A merchant, after selling one-third, one-fourth, and one-sixth of a piece of silk, has 15 yards left. How many yards were there in the piece? 16. If two trains start together and run in the same direction, one at the rate of 20 miles an hour, and the other at the rate of 30 miles an hour, after how many hours will they be 250 miles apart?

17. A teacher proposes 16 problems to a pupil. The latter is to receive 5 marks in his favor for each problem solved, and 3 marks against him for each problem not solved. If the number of marks in his favor exceed those against him by 32, how many problems will he have solved?

18. A merchant paid 30 cents a yard for a piece of cloth. He sold one-half for 35 cents a yard, one-third for 29 cents a yard, and the remainder for 32 cents a yard, gaining $ 18.15 by the transaction. How many yards did he buy?

19. Two men start from points 100 miles apart and travel toward each other, one at the rate of 15 miles an hour, and the other at the rate of 10 miles an hour. After how many hours will they meet, and how far will their point of meeting be from the starting point of the first?

20. A father is 32 years old, and his son is 8 years old. After how many years will the father's age be twice the son's?

21. Divide 130 into five parts so that each part shall be 12 greater than the next less part.

22. A, traveling at the rate of 20 miles a day, has four days' start of B, who travels at the rate of 25 miles a day in the same direction. After how many days will B overtake A ?

23. A sum of money is equally divided among four persons. If $60 more be divided equally among six persons, the shares will be the same as before.

How many dollars are divided?

24. Atmospheric air is a mixture of four parts of nitrogen with one of oxygen. How many cubic feet of oxygen are there in a room 10 yards long, 5 yards wide, and 12 feet high?

25. A merchant paid $6.15 in an equal number of dimes and five-cent pieces. How many coins of each kind did he pay?

26. A man has $4.75 in dimes and quarters, and he has 5 more quarters than dimes. How many coins of each kind has he?

27. A leaves a certain town P, traveling at the rate of 21 miles in 5 hours; B leaves the same town 3 hours later and travels in the same direction at the rate of 21 miles in 4 hours. After how many hours will B overtake A, and at what distance from P?

28. The circumference of the front and hind wheels of a wagon are 2 and 3 yards, respectively. What distance has the wagon moved when the front wheel has made 10 revolutions more than the hind wheel?

29. The sum of two numbers is 47, and their difference increased by 7 is equal to the less. What are the numbers ?

30. The sum of three consecutive even numbers exceeds the least by 42. What are the numbers?

31. The sum of the two digits of a number is 4. If the digits be interchanged, the resulting number will be equal to the original one. What is the number?

32. A father is three times as old as his son, and 10 years ago he was five times as old as his son. What is the present age of each?

33. One barrel contained 48 gallons, and another 88 quarts of wine. From the first twice as much wine was drawn as from the second; the first then contained three times as much wine as the second. How much wine was drawn from each ?

34. A child was born in November. On the 10th of December the number of days in its age was equal to the number of days from the 1st of November to the day of its birth, inclusive. What was the date of its birth?

35. A regiment moves from A to B, marching 20 miles a day. Two days later a second regiment leaves B for A, and marches 30 miles a day. At what distance from A do the regiments meet, A being 350 miles from B?

36. The sum of two digits of a number is 12. If the digits be interchanged, the resulting number exceeds the original one by three-fourths of the original number. What is the number?

37. Three boys, A, B, and C, have a number of have 44, B and C have 43, and A and C have 39. have they all, and how many marbles has each ?

marbles. A and B How many marbles

38. The tail of a fish is 4 inches long. Its head is as long as its tail and one-seventh of its body, and its body is as long as its head and onehalf of its tail. How long is the fish, and how long are its head and its

body?

39. A father divided his property equally among his sons. To the oldest son he gave $1000 and one-seventh of what remained; to the second son he gave $2000 and one-seventh of what was then left; to the third son he gave $3000 and one-seventh of the remainder; and so What was the amount of his property, and how many sons had he? 40. A man, wishing to give alms to several beggars, finds that in order to give 15 cents to each one, he must have 10 cents more than he has; but that if he were to give 12 cents to each one, he would have 14 cents left. How many beggars are there?

on.

41. A train runs from A to B at the rate of 30 miles an hour; and returning runs from B to A at the rate of 28 miles an hour. The time required to go from A to B and return is 15 hours, including 30 minutes' stop at B. How far is A from B?

42. A cistern has 3 taps. By the first it can be emptied in 80 minutes, by the second in 200 minutes, and by the third in 5 hours. After how many hours will the cistern be emptied, if all the taps be opened?

43. A cistern has 3 taps. By the first it can be filled in 6 hours, by the second in 8 hours, and by the third it can be emptied in 12 hours. In what time will it be filled if all the taps be opened?

44. An inlet pipe can fill a cistern in 3 hours, and an outlet pipe can empty it in 9 hours. After how many hours will the cistern be filled if both pipes be open half the time, and the outlet pipe be closed during the second half of the time?

45. In my right pocket I have as many dollars as I have cents in my left pocket. If I transfer $6.93 from my right pocket to my left, I shall have as many dollars in my left pocket as I shall have cents in my right. How much money have I in my left pocket ?

46. A servant is to receive $170 and a dress for one year's services. At the end of 7 months she leaves her place and receives $95 and the dress. What is the value of the dress?

47. A farmer found that his supply of feed for his cows would last only 14 weeks. He therefore sold 60 cows, and his supply then lasted 20 weeks. How many cows had he?

48. At 6 o'clock the hands of a clock are in a straight line. At what time between 7 and 8 o'clock will they be again in a straight line? At what time between 9 and 10 o'clock ?

49. A cistern has 3 pipes which can empty it in 6, 8, and 10 hours respectively. After all three pipes have been open for 2 hours they have discharged 94 gallons. What is the capacity of the cistern?

50. At what time between 10 and 11 o'clock are the minute-hand and the hour-hand of a clock at right angles to each other? Find two solutions. At what time between 12 and 1 o'clock ?

51. At what time between 3 and 4 o'clock will the minute-hand of a clock be 5 minute-divisions in advance of the hour-hand? At what time 17 minute-divisions?

A watch has the second-hand attached at the same point as the hourand the minute-hand :

52. At what time between 1 and 2 o'clock is the second-hand over the minute-hand? At what time between 8 and 9 o'clock ?

53. At what time between 11 and 12 o'clock does the second-hand of a watch bisect the angle between the hour- and the minute-hand? At what time between 4 and 5 o'clock ?

54. A woman sells an apple more than one-half of her apples. She next sells an apple more than one-half of the apples not yet sold, and then has 6 apples left. How many apples had she at first?

55. A steamer and a sailing vessel are both to sail from M to N. The steamer sails 40 miles every 3 hours, and the sailing vessel 24 miles in the same time. The sailing vessel has traveled 134 miles when the steamer sails, and arrives at N 5 hours later than the steamer. How long is the steamer in sailing from M to N, and how far is M from N?

56. A wall can be built by 20 workmen in 11 days, or by 30 other workmen in 7 days. If 22 of the first class work together with 21 of the second class, after how many days will the work be completed?

57. In a certain family each son has as many brothers as sisters, but each daughter has twice as many brothers as sisters. How many children are in the family?

58. A merchant's investment yields him yearly 33% profit. At the end of each year, after deducting $1000 for personal expenses, he adds the balance of his profits to his invested capital. At the end of three years his capital is twice his original investment. How much did he

invest?

59. I have in mind a number of six digits, the last one on the left being 1. If I bring this digit to the first place on the right, I shall obtain a number which is three times the number I have in mind. What is the number?

60. A dog caught sight of a hare at a distance of 50 dog's leaps. The dog makes 3 leaps while the hare makes 4 leaps, but the length of two dog's leaps is equal to the length of 3 hare's leaps. How many leaps will the hare make before the dog overtakes him?

CHAPTER VI.

TYPE-FORMS.

We shall in this chapter consider a number of products and quotients which are of frequent occurrence. They are called Type-Forms.

$ 1.

TYPE-FORMS IN MULTIPLICATION.

The Square of an Algebraic Expression.

1. By actual multiplication, we have

(a + b)2 = (a + b) (a + b) = a2 + 2 ab + b2.

That is, the square of the sum of two numbers is equal to the square of the first number, plus twice the product of the two numbers, plus the square of the second number.

E.g.,

(2x+5y)2= (2x)2 + 2 (2 x) (5 y) +(5 y)2
=4x2+20 xy + 25 y2.

2. By actual multiplication, we have

(a - b)2 = (a - b) (a — b) — a2 — 2 ab + b2.

That is, the square of the difference of two numbers is equal to the square of the first number, minus twice the product of the two numbers, plus the square of the second number.

E.g.,

(3x-7y)2= (3 x)2 − 2 (3 x) (7 y) + (7 y)2

= 9 x2 - 42 xy + 49 y2.

=

Observe that this type-form is equivalent to that of Art. 1, since a-ba + ( − b).

E.g.,

(3 x − 7 y)2 = (3 x)2 + 2 (3 x ) ( − 7 y) + ( − 7 y)2

=9x2-42xy + 49 y, as above.

The signs of all the terms of an expression which is to be squared may be changed without changing the result.

For,

(a — b)2 = [ − (b − a)]2 = (b − a)2.

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