48. Divide the number 120 into two such parts, that the smaller may be contained in the greater 1 times. Ans. 48 and 72. 49. "I have a certain number in my mind," said A to B; "if I multiply it by 7, add 3 to the product, divide this by 2, and subtract 4 from the quotient, the remainder is 15." What is the number? Ans. 5.

50. What number is that, which, if you multiply it by 5, subtract 24 from the product, divide the remainder by 6, and add 13 to the quotient, will give the number itself? Ans. 54.

51. Two persons, A and B, engaged in trade, the capital of B being that of A; B gained, and A lost, 100 dollars; after which, if of what A had left, be subtracted from what B now has, the remainder will be 134 dollars; with what capital did each commence? Ans. A $786, B $524. 52. A man having spent 3 dollars more than of his money, had 7 dollars more than of it left; how many dollars had he at first? Ans. $75.

53. Two men, A and B, have the same annual income; A saves of his, but B spends 25 dollars per annum more than A, and at the end of 5 years finds he has saved 200 dollars; what is the annual income of each? Ans. $325.

54. In the composition of a quantity of gunpowder, of the whole, plus 10 pounds, was nitre; of the whole, plus 1 pound, was sulphur; and of the whole, minus 17 pounds, was charcoal; how many pounds of gunpowder were there? Ans. 69tb.

55. A person bought a chaise, horse, and harness, for 245 dollars; the horse cost 3 times as much as the harness, and the chaise cost 19 dollars less than 23 times as much as both horse and harness; what was the cost of each?

Ans. Harness $18, horse $54, chaise $173. 56. What two numbers are as 3 to 4, to each of which, if 4 be added, the sums will be to each other as 5 to 6? Ans. 6 and 8.

57. What two numbers are as 2 to 5, from each of which, if 2 be subtracted, the remainders will be to each other as 3 to 8? Ans. 20 and 50.

58. The ages of two brothers are now 25 and 30 years, so that their ages are as 5 to 6; in how many years will their ages be as 8 to 9?

How many years since their ages were as 1 to 2?

Ans. 15. A. 20 yrs.

59. A cistern has 3 pipes to fill it; by the first, it can be filled in 1 hours, by the second, in 3 hours, and by the third, in 5 hours; in what time can it be filled, by all three running at once? Ans. 48 min.

60. Find the time in which A, B, and C together, can perform a piece of work, which requires 7, 6, and 9 days respectively, when done singly. Ans. 2 days. 61. From a certain sum I took one third part, and put in its stead 50 dollars; next, from this sum I took the tenth part, and put in its stead 37 dollars; I then counted the money, and found I had 100 dollars; what was the original sum? Ans. $30.

62. A teacher spent of his yearly salary for board and lodging, of the remainder for clothes, and of what remained, for books, and still saved 120 dollars per annum; what was his salary? Ans. $375.

63. A laborer was engaged for a year, at 80 dollars and a suit of clothes; after he had served 7 months, he left, and received for his wages, the clothes and 35 dollars; what was the value of the suit of clothes? Ans. $28.

64. A man and his wife can drink a cask of wine in 6 days, and the man alone can drink it in 10 days; how many days will it last the woman?

Ans. 15.

65. A steamboat, that can run 15 miles per hour with the current, and 10 miles per hour against it, requires 25 hours to go from Cincinnati to Louisville, and return; what is the distance between those cities? Ans. 150 miles.

66. A and B engaged in a speculation; A with 240 dollars, and B with 96 dollars; A lost twice as much as B, and, upon settling their accounts, it appeared, that A had 3 times as much remaining as B; what did each lose? Ans. A $96, and B $48.

the whole, plus 25 gal

67. In a mixture of wine and water, lons, was wine, and of the whole, minus 5 gallons, was water; required the quantity of each in the mixture.

Ans. 85 galls. of wine, and 35 galls. of water. 68. It is required to divide the number 91 into 2 such parts, that the greater, being divided by their difference, the quotient will be 7. Ans. 49 and 42.

69. It is required to divide the number 72 into 4 such parts, that if the first be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, the difference, the product, and the quotient shall all be equal.

Ans. 14, 18, 8, and 32. Let the four parts be represented by x-2, x+2, 4x, and 22. 70. A merchant having cut 19 yards from each of 3 equal pieces of silk, and 17 from another of the same length, found, that the remnants taken together, measured 142 yards; what was the length of each piece? Ans. 54 yds.

71. Suppose, that for every 10 sheep a farmer keeps, he should plow an acre of land, and allow 1 acre of pasture for every 4 sheep; how many sheep can the person keep, who farms 161 acres? Ans. 460.

72. It is required to divide the number 34 into 2 such parts, that if 18 be subtracted from the greater, and the less be subtracted from 18, the first remainder shall be to the second as 2 to 3. Ans. 22 and 12.

73. A person was desirous of giving 3 cents a piece to some beggars, but found that he had not money enough in his pocket by 8 cents; he therefore gave each of them 2 cents, and then had 3 cents remaining; required the number of beggars. Ans. 11.

74. A father distributed a number of apples among his children, as follows: to the first he gave the whole number, less 8; to the second the remainder, diminished by 8; and in the same manner, with the third and fourth; after which, he had 20 apples remaining for the fifth; how many apples did he distribute?

Ans. 80. 75. A could reap a field in 20 days, but if B assisted him for 6 days, he could reap it in 16 days; in how many days could B reap it alone? Ans. 30 days.

76. There are two numbers in the proportion of to 3, which, being increased respectively, by 6 and 5, are in the proportion of to; required the numbers. Ans. 30 and 40.

77. When the price of a bushel of barley wanted but 3 cents to be to the price of a bushel of oats as 8 to 5, nine bushels of oats were received as an equivalent for 4 bushels of barley and 90 cents in money; what was the price of a bushel of each?

Ans. Oats 30 cts., and barley 45 cts.

78. Four places are situated in the order of the 4 letters, A, B, C, and D; the distance from A to D is 34 miles; the distance from A to B is to the distance from C to D, as 2 to 3; and the distance from A to B, added to the distance from C to D, is 3 times the distance from B to C. Required the respective distances. Ans. A to B 12, B to C 4, and C to D 18 miles. 79. The ingredients of a loaf of bread are rice, flour, and water, and the weight of the whole is 15 pounds; the weight of the rice increased by 5 pounds, is the weight of the flour; and the weight of the water is the weight of the flour and rice together; what is the weight of each?

Ans. Rice 2b, flour 101, and water 24.

SIMPLE EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES. ART. 157.-In order to find the value of any unknown quantity, it is evident, that we must obtain a single equation containing it, and known terms. Hence, when we have two or more equations, containing two or more unknown quantities, we must obtain from them a single equation containing only one unknown quantity. The method of doing this, is termed elimination, which may be briefly defined thus: Elimination is the process of deducing from two or more equations, containing two or more unknown quantities, a less number of equations containing one less unknown quantity. There are three methods of elimination.

1st. Elimination by substitution.

2d. Elimination by comparison.

3d. Elimination by addition and subtraction.

ELIMINATION BY SUBSTITUTION.

ART. 158.-Elimination by substitution, consists in finding the value of one of the unknown quantities in one of the equations, in terms of the other unknown quantity and known terms, and substituting this, instead of the quantity, in the other equation.

To explain this, suppose we have the following equations, in which it is required to find the value of x and y.

NOTE. The figures in the parentheses, are intended to number the equations for reference.

x+2y=17 (1.) 2x+3y=28 (2.).

By transposing 2y in the equation (1), we have x=17-2y. Substituting this value of x, instead of x in equation (2), we have 2(17-2y)+3y=28

Hence, when we have two equations, containing two unknown quantities, we have the following

RULE,

FOR ELIMINATION BY SUBSTITUTION.

Find an expression for the value of one of the unknown quantities in either equation, and substitute this value in place of the same unknown quantity in the other equation; there will thus be formed a new equation, containing only one unknown quantity.

y=3.

NOTE. In finding an expression for the value of one of the unknown quantities, let that be taken which is the least involved.

Find the values of the unknown quantities in each of the following equations.

1. x+5y=38. 3x+4y=37 2. 2x+4y-22. 5x+7y=46.

5x-3y=10.

3y

y=30.

=0.

Ans. x=21.

y=2.

7 8

2x 3y-26.

y=16.

X

-3=12.

y=12.

3

4

ELIMINATION BY COMPARISON.

ART. 159.-Elimination by comparison, consists in finding the value of the same unknown quantity in two different equations, and then placing these values equal to each other.

To illustrate this method, we will take the same equations which were used to explain elimination by substitution.

x+2y=17 (1.)

2x+3y=28 (2.)

By transposing 2y in equation (1), we have x=17-2y.
By transposing 3y in equation (2), and dividing by 2, we have

28-3y
2

Placing these values of x equal to each other,

The value of x may be found in a similar manner, by first finding the values of y, and placing them equal to each other. But, after having found the value of one of the unknown quantities, the value of the other may be found most readily by substitution, as in the preceding article. Thus, x=17-2y=17-12=5.

REVIEW.-157. What is necessary in order to find the value of any unknown quantity? When we have two equations, containing two unknown quantities, what is necessary, in order to find the value of one of them? What is elimination? How many methods of elimination are there? 158, In what does elimination by substitution consist? What is the rule for elimination by substitution? 159. In what does elimination by comparison consist?