PROB. 49. It is required to divide the number 99 into five such parts, that the first may exceed the second by 3; be less than the third by 10; greater than the fourth by 9; and less than the fifth by 16. Ans. 17, 14, 27, 8, and 33. PROB. 50. Two persons began to play with equal sums of money; the first lost 14 shillings, the other won 24 shillings, and then the second had twice as many shillings as the first. What sum had each at first? Ans. 52 shillings.

PROB. 51. A Mercer having cut 19 yards from each of three equal pieces of silk, and 17 from another of the same length, found that the remnants together were 142 yards. What was the length of each piece? Ans. 54 yards.

PROB. 52. A Farmer had two flocks of sheep, each containing the same number. From one of these he sells 39, and from the other 93; and finds just twice as many remaining in one as in the other. How many did each flock originally con

tain?

Ans. 147.

PROB. 53. A Courier, who travels 60 miles a day, had been dispatched five days, when a second is sent to overtake him, in order to do which he must travel 75 miles a day. In what time will he overtake the former ? Ans. 20 days.

PROB. 54. A and B trade with equal stocks. In the first year A tripled his stock, and had $27 to spare; B doubled his stock, and had $153 to spare. Now the amount of both their gains was five times the stock of either. What was that ?

Ans. 90 dollars.

PROB. 55. A and B began to trade with equal sums of money. In the first year A gained 40 dollars, and B lost 40; but in the second A lost one-third of what he then had, and B gained a sum less by 40 dollars, than twice the sum that A had lost; when it appeared that B had twice as much money as A. What money did each begin with? Ans. 320 dollars.

PROB. 56. A and B being at play, severally cut packs of cards, so as to take off more than they left. Now it happened that A cut off twice as many as B left, and B cut off seven times as many as A left. How were the cards cut by each? Ans. A cut off 48, and B cut off 28 cards. PROB. 57. What two numbers are as 2 to 3; to each of which if 4 be added, the sums will be as 5 to 7 ?

Ans. 16 and 24.

PROB. 58. A sum of money was divided between two persons, A and B, so that the share of A was to that of B as 5 to 3; and exceeded five-ninths of the whole sum by 50 dollars. What was the share of each person?

Ans. 450, and 270 dollars.

PROB. 59. The joint stock of two partners, whose particular shares differed by 40 dollars, was to the share of the lesser as 14 to 5. Required the shares.

Ans. the shares are 90 and 50 dollars respectively. PROB. 60. A Bankrupt owed to two creditors 1400 dollars; the difference of the debts was to the greater as 4 to 9. What were the debts? Ans. 900, and 500 dollars. PROB. 61. Four places are situated in the order of the four letters A, B, C, D. The distance from A to D is 34 miles, the distance from A to B : distance from C to D:: 2 : 3, and onefourth of the distance from A to B added to half the distance from C to D, is three times the distance from B to C. What are the respective distances?

Ans. AB=12, BC=4, and CD=18 miles. PROB. 62. A General having lost a battle, found that he had only half his army plus 3600 men left, fit for action; one-eighth of his men plus 600 being wounded, and the rest, which were one-fifth of the whole army, either slain, taken prisoners, or missing. Of how many men did his army consist?

Ans. 24000. PROB. 63. It is required to divide the number 91 into two such parts that the greater being divided by their difference, the quotient may be 7. Ans. 49 and 42. PROB. 64. A person being asked the hour, answered that it was between five and six; and the hour and minute hands were together. What was the time?

Ans. 5 hours 27 minutes 16 seconds. PROB. 65. Divide the number 49 into two such parts, that the greater increased by 6 may be to the less diminished by 11 as 9 to 2. Ans. 30 and 19. PROB. 66. It is required to divide the number 34 into two such parts that the difference between the greater and 18, shall be to the difference between 18 and the less:: 2 : 3.

PROB. 67. What number is that to which if severally added, the first sum shall be to the second is to the third.

Ans. 22 and 12. 1, 5, and 13, be second, as the Ans. 3.

PROB. 68. It is required to divide the number 36 into three such parts, that one-half of the first, one-third of the second, and one-fourth of the third, shall be equal to each other.

Ans. 8, 12, and 16. PROB. 69. Divide the number 116 into four such parts, that if the first be increased by 5 the second diminished by 4, the third multiplied by 3, and the fourth divided by 2, the result in each case shall be the same. Ans. 22, 31, 9, and 54.

PROB. 70. A Shepherd, in time of war, was plundered by a party of soldiers who took of his flock, and of a sheep; another party took from him of what he had left, and 3 of a sheep; then a third party took of what now remained, and 1 of a sheep. After which he had but 25 sheep left. How many had he at first? Ans. 103.

FROB. 71. A Trader maintained himself for 3 years at the expense of 50l. a year; and in each of those years augmented that part of his stock which was not so expended by thereof. At the end of the third year his original stock was doubled. What was that stock? Ans. 7401.

PROB. 72. In a naval engagement, the number of ships taken was 7 more, and the number burnt two fewer, than the number sunk. Fifteen escaped, and the fleet consisted of 8 times the number sunk. Of how many did the fleet consist?

Ans. 32. PROB. 73. A cistern is filled in twenty minutes by three pipes, one of which conveys 10 gallons more, and the other 5 gallons less, than the third, per minute. The cistern holds 820 gallons. How much flows through each pipe in a minute? Ans. 22, 7, and 12 gallons.

PROB. 74. A sets out from a certain place, and travels at the rate of 7 miles in five hours; and eight hours afterwards B sets out from the same place, and travels the same road at the rate of five miles in three hours. How long, and how far, must A travel before he is overtaken by B?

Ans. 50 hours, and 70 miles. PROB. 75. There are two places, 154 miles distant, from which two persons set out at the same time to meet, one travelling at the rate of 3 miles in two hours, and the other at the rate of 5 miles in four hours. How long, and how far did each travel before they met?

Ans. 56 hours; and 84, and 70 miles.

134

CHAPTER V.

ON

SIMPLE EQUATIONS,

INVOLVING TWO OR MORE UNKNOWN QUANTITIES.

183. It has been observed (Art. 159), that an equation was the translation into algebraic language of two equivalent phrases comprised in the enunciation of a question; but this question may comprehend in it a greater number, and if they are well distinguished two by two, and independent of one another, they furnish a certain number of equations.

Thus, for example, let us propose to find two numbers, such that double the first added to the second, gives 24, and that five times the first, plus three times the second, make 65. We find here two phrases, which express the same thing in different terms; 1st, the double of an unknown number, plus another unknown number, then the equivalent 24; 2d, five times the first unknown number, plus three times the second, then the equivalent 65. The translation is easy, and it gives these two determinate equations

2x+y=24; 5x+3y=65.

:

When two or more equations, involving as many unknown quantities, are independent of one another, they are called determinate. But if for the second of these two conditions we had substituted this and such that six times the first number, plus three times the second, make 72; these two phrases express nothing more than the first two, since that we have only tripled two equal results; we should have but one translation, and consequently a single equation. It can therefore happen that we may have less equations than unknown quantities, and then the question is said to be indeterminate; because the number of conditions would be insufficient for the determination of the unknown quantities, as we shall see clearly illustrated in the following section.

§ I. ELIMINATION OF UNKNOWN QUANTITIES FROM ANY NUM

BER OF SIMPLE EQUATIONS.

184. Elimination is the method of exterminating all the unknown quantities, except one, from two, three, or more given

equations, in order to reduce them to a single, or final equation, which shall contain only the remaining unknown, and certain known quantities.

185. In order to simplify the calculations, by avoiding fractions, we shall here make use of literal equations, which will modify the process of elimination: And also, to avoid the inconvenience arising from the multitude of letters which must be employed in order to represent the given quantities, when the number of equations involving as many unknown quantities surpasses two, we shall represent by the same letter all the coefficients of the same unknown quantity; but we shall affect them with one or more accents, in order to distinguish them, according to the number of equations.

186. In the first place, any two simple equations, each involving the same two unknown quantities, may, in general, be written thus:

ax + by=c a'x+b'y=c'

(A), (B).

The coefficients of the unknown quantity x are represented both by a; those of y by b; but the accent, by which the letters of the second equation are affected, shows that we do not regard them as having the same value as their correspondents in the first. Thus a' is a quantity different from a, b'a quantity different from b.

187. We can readily see, by a few examples, how any two simple equations, each involving the same two unknown quan, tities, may be reduced to the above form.

Ex. 1. Let the two simple equations,

5x+3y-5=y-2x+7,

9x-2y+3x-7y+16,

be reduced to the form of equations (A) and (B),
By transposition, these equations become

5x+3y—y+2x=7+5,
9x-2y-x-7y=16-3;

by reduction, we shall have

7x+2y=12,
8x+5y=13;

equations which are reduced to the form of (A) and (B), and which may be expressed under the form of the same literal equations, by substituting a, b, and c, for 7, 2, and 12; and a', b', and c', for 8, 5, and 13.

Ex. 2. Let the two simple equations,

mx+6y-7-px-2y+3,
rx-9y+6=3y-3x+12,

be reduced to the form of equations (A) and (B).