4. Three players after a game count their money; one had lost, the other two had gained each as much as he had brought to the play; after the second game, one of the players, who had gained before, lost, and the two others gained each a sum equal to what he had at the beginning of this second game; at the third game the player, who had gained till now, lost with each of the others a sum equal to that, which each possessed at the beginning of this last game; they then separated, each having $1,20; how much had they each, when they commenced playing? Ans. $1,95, $1,05 and $0,60 respectively. 5. Three laborers are employed in a certain work. A and B together can perform it in 8 days, A and C together in 9 days, and B and C together in 10 days. In how many days can each alone perform the same work? Ans. A in 143, B in 1723 and C in 237 days. 6. Three men, A, B, C, driving their sheep to market, says A to B and C, if each of you will give me 5 of your sheep, I shall have just half as many as both of you will have left. Says B to A and C, if each of you will give me 5 of yours I shall have just as many as both of you will have left. Says C to A and B, if each of you will give me 5 of yours I shall have just twice as many as both of you will have left. How many had each. Ans. 10, 20, and 30 respectively. 7. A cistern is furnished with three pipes, A, B and C. By the pipes A and B it can be filled in 12 minutes, by the pipes B and C in 20 minutes, and by A and C in 15 minutes. In what time will each fill the cistern alone, and in what time will it be filled if all three are open together? Ans. A will fill it in 20, B in 30, and C. in 60 minutes, and the three together in 10 minutes. 8. It is required to divide the number 72 into four such parts, that if the first part be increased by 5, the second part diminished by 5, the third part multiplied by 5, and the fourth part divided by 5, the sum, difference, product and quotient shall all be equal. Ans. The parts are 5, 15, 2, and 50. SECTION VII. NEGATIVE QUANTITIES. QUESTIONS PRODUCING NEGATIVE RESULTS. 70. The length of a certain field is eight rods and the breadth five rods, how much must be added to the length, that the field may contain 30 square rods? Let x = the quantity to be added, then by the question In this result 8, the quantity to be subtracted, is greater than that, from which it is required to be taken; the subtraction therefore cannot be performed. We may, however, decompose 8 into two parts 6 and 2, the successive subtraction of which will be equal to that of 8, and we shall have for 6-8 the equivalent expression 6-6-2, which is reduced to 0 2 or more simply — 2, the sign — being retained before the 2 to show that it remains to be subtracted. tive quantity, thus, -2, -3a, -5 a b, are negative quantities. Monomials with the sign either prefixed or understood are called positive quantities. Thus 2, 3 a, 5 a b, are positive quantities. Negative quantities, it will be perceived, differ in nothing from positive quantities except in their sign. They are derived from endeavoring to subtract a larger quantity from one that is smaller, and are to be regarded merely as positive quantities to be subtracted. 71. If it now be asked what is the sum of the monomials +a, b, c, the question, from what has been said, is reduced to this, what change will be produced in the quantity a, if the quantity b be subtracted from it and the quantity c be added to the remainder. Indicating the operations required to obtain the answer to the question thus proposed, the result will be In order then to add monomials affected with the signs + and it will be sufficient to write them one after the other with the signs with which they are affected, when they stand alone. 72. If we now add the quantities + b,b, the result b — b, it is evident, will be equal to zero. If then the expression b. b be added to a, it will not affect the value of a; and a+b-b will only be a different form of expression for the same quantity a. If it now be proposed to subtract + b from a, it will be sufficient, it is evident, to efface +b in the equivalent expression a+bb, and the result will be a- – b. Again, if it be required to subtract b from a, it will be sufficient to efface b in the same expression, and we shall have for the result a+b. Thus, to subtract a positive quantity is the same as to add an equal negative quantity, and to subtract a negative quantity is the same as to add an equal positive quantity. To subtract monomials therefore of whatever sign, we change the signs, and then proceed as in addition. 73. If we multiply bb by a the product must be a b — a b, because the multiplicand being equal to zero, the product must be zero. Since then the product of b by a is evidently a b, that of -b by a must be a b in order that the second term may destroy the first. For a similar reason the product of a by bb will be a b -ab; ty be multiplied by a positive, or a positive by a negative, the product will be negative. b by Whence if a negative quanti Again, if we multiply · a by bb, from what has been proved above, the product of-a by b will be a b, the product of a must therefore be +ab, in order that the result may be zero, as it should be, when the multiplier is zero. Whence, the product of a negative quantity by a negative quantity will be positive. 74. We arrive at the same conclusions by the definition of multiplication given in arithmetic, according to which to multiply one number by another, we form a number by means of the multiplicand in the same manner that the multiplier is formed by means of unity. Thus to multiply 5 by -3, we form a num ber by means of 5 in the same manner that 3 is formed by means of unity; but -3 is formed by the subtraction of three units, the product required will therefore be formed by the subtraction of three fives, by consequence it will be - 15. In like manner the product of 5 by -3 will be found by the subtraction of three minus fives', it will therefore be + 15. The rules for division follow necessarily from those for multiplication. We have therefore the same rules for the signs in the multiplication and division of isolated simple quantities, as are applied to these quantities, when they make a part of polynomials; and in general, monomials, when they are isolated, are combined in the same manner with respect to their signs, as when they make a part of polynomials. 75. From what has been said, it will be perceived, that the term addition does not in algebra, as in arithmetic, always imply augmentation. Thus, the sum of a and b is, strictly speaking, the difference between a and b; it will therefore be less than a. To distinguish this from an arithmetical sum, we call it an algebraic sum. Thus the polynomial 3ab5bccd-ef, considered as formed by uniting the quantities 3 ab,-5bc,+ cd, -ef, Its with their respective signs is called an algebraic sum. proper acceptation is the arithmetical difference between the sum of the units contained in the terms, which are additive, and the sum of those contained in the terms, which are subtractive. In like manner the term subtraction in algebra does not always imply diminution. Thusb subtracted from a gives a+b, which is greater than a. This result may, however, be called an algebraic difference, since it may be put under the form a — (—b). 76. Resuming now the question proposed, art. 70, we have for the answer x=-2. In order to interpret this negative result, we return to the equation of the question 40+5x=30. Here, the addition intended in the enunciation of the question being arithmetical, it is evidently absurd to require that some thing should be added to 40 in order to make 30, since 40 is already greater than 30. The negative result indicates therefore, that the question is arithmetically impossible, or in other words, that it cannot be solved in the exact sense of the enunciation. If, however, in the equation 40 +5 x = 30, we substitute - 2 for x, we have 40 10= 30, an equation which is exact. In order then that the result may be positive, or which is the same thing, that the question may be arithmetically possible, the enunciation should be modified, thus, The length of a certain field is eight rods, and its breadth five rods; how much must be subtracted from the length, that the field may contain 30 square rods? Putting x for the quantity to be subtracted, we have by this new enunciation 40-5 x = 30, from which we obtain x = = 2. 2. The length of a certain field is 11 rods and its breadth 7 rods; how much must be subtracted from the length, that the field may contain 98 square rods? Let x the quantity to be subtracted; then by the ques77-7x= = 98 tion whence x=- 3 To interpret this result, we return to the equation of the question. Here, as an arithmetical subtraction is intended in the enunciation, it is evidently absurd to require, that something should be subtracted from 77 to make 98, since 77 is already less than 98. The question therefore cannot be solved in the exact sense of the enunciation. If, however, instead of x in the equation of the question, we substitute - 3, we have 77 +21= = 98, an equation, which is exact. In order then that the result may be positive, the question should be modified, thus, The length of a certain field is 11 rods and the breadth 7 rods; how much must be added to the length in order that the field may contain 98 square rods? Resolving the question according to this new enunciation, we obtain x= 3. Let us take as a third example the following question. 3. A laborer wrought for a person 12 days and had his wife |