14-6= 8, is read 6 subtracted from 14, or 14 less 6, or 14 minus 6 is equal to 8. Observe that the signs + and affect the numbers which they stand immediately before, and no others. Thus 14 - 6 +8= 16; and 14 + 8--6 = 16; and 8-6 + 14 = 16; and, in fine, -6 + 8 + 14 = 16. In all these cases the 6 only is to be subtracted, and it is the same, whether it be first subtracted from one of the numbers, and then the rest be added, or whether all the others be added and that be subtracted at last. (x) (.) An inclined cross, or a point, is used to express multiplication ; thus, 5 X 3 =15, or 5.3 = 15. (3) A horizontal line, with a point above and'another below it, is used to express division. Thus 15 - 3=5, is read 15 divided by 3 is equal to 5. But division is more frequently expressed in the form of a fraction (Arith. Art. XVI. Part II.), the divisor being made the denominator, and the dividend the numerator. Thus 5, is read 15 divided by 3 is equal to 5, or one third of 15, is 5, or 15 contains 3, 5 times. Example. 6 x 9 + 15 — 3 = 7.8-4 + 14. This is read, 9 times 6 and 15 less 3 are equal to 8 times 7 less 16 divided by 4, and 14. To find the value of each side ; 9 times 6 are 54 and 15 are 69, less 3 are 66. Then 3 times 7 are 56, less 16 divided by 4, or 4 are 52, and 14 more are 66. In questions proposed for solution, it is always required to find one or more quantities which are unknown; these, when found, are the answer to the question. It will be found extremely useful to have signs to express these unknown quantities, because it will enable us to keep the object more steadily and distinctly in view. We shall also be able to represent certain operations upon them by the aid of signs, which will greatly assist us in arriving at the result. Algebraic signs are in fact nothing else than an abridgment of common language, by which a long process of reasoning is presented at once in a single view. The signs generally used to express the unknown quantities above mentioned are some of the last letters of the alphabet, as 2, y, z, &c. 1. 1. Two men, A and B, trade in company, and gain 267 dollars, of which B has twice as much as A. What is the share of each? In this example the unknown quantities are the particular shares of A and B. Let x represent the number of dollars in A's share, then 2 x will represent the number of dollars in B's share. Now these added together must make the number of dollars in both their shares, that is, 267 dollars. x + 2x = 267 Putting all the æ's together, 3x = 267 If 3 x are 267, 1 x is į of 267 in the same manner as if 3 oxen were worth $267, 1 ox would be worth of it. x = 89 = A's share. 2 x = 178 = B's share. 2. Four men, A, B, C, and D, found a purse of money containing $325, but not agreeing about the division of it, each took as much as he could get; A got a certain sum, B got 5 times as much ; C, 7 times as much ; and D, as much as B and C both. How many dollars did each get ? Let x represent the number of dollars thạt A got; then B got 5x, C7x, and D (5x + 7x) = 12 x. These, added together, must make $325, the whole number to be divided. 2 + 5x + 7 x + 12 x= 325 Putting all the x's together, 25 x = 325 X = 13 = A's share. 91 = C's 156 = D's Note. All examples of this kind in algebra admit of proof. In this case the work is proved by adding together the several shares. If they are equal to the whole sum, 325, the work is right. As the answers are not given in this work, it will be well for the learner always to prove his results. In the same manner perform the following examples. 3. Said A to B, my horse and saddle together are worth $130, but the horse is worth 9 times as much as the saddle. What is the value of each? 4. Three men, A, B, and C, trade in company, A puts in a certain sum, B puts in 3 times as much, and C puts in as much 12 x as A and B both; they gain $656. What is each man's of the gain? 5. A gentleman, meeting 4 poor persons, distributed cents among them, giving the second twice, the third times, and the fourth four times as much as the first. many cents did he give to each ? 6. A gentleman left 11000 crowns to be divided betw his widow, two sons, and three daughters. He intended the widow should receive twice the share of a son, and each son should receive twice the share of a daughter. quired the share of each. Lét x represent the share of a daughter, then 2x wall re sent the share of a son, &c. 7. Four gentlemen entered into a speculation, for wh they subscribed $4755, of which В paid 3 times as múch as and C paid as much as A and B, and D paid as much as Ba C. What did each pay? 8. A man bought some oxen, some cows, and some she for $1400 ; there were an equal number of each sort. F the oxen he gave $42 apiece, for the cows $20, and for t sheep $8 apiece. How many were there of each sort ? In this example the unknown quantity is the number of cac sort, but the number of each sort being the same, one chara ter will express it. Let x denote the number of each sort. Then x oxen, at $42 apiece, will come to 42 x dolls., and cows, at $20 apiece, will come to 20 x dolls., and x sheep, $8 apiece, will come to 8 x dolls. These added together mu make the whole price. 42 x + 20x + 8x=1400 Putting the x's together, 70x = 1400 Dividing by 70, X = 20 Ans. 20 of each sori 9. A man sold some calves and some sheep for $374, the calves at $5, and the sheep at $7 apiece; there were three times as many calves as sheep. How many were there of each ? Let x denote the number of sheep; then 3x will denote the number of calves. as A and B both; they gain $656. What is each man's share of the gain? 5. A gentleman, meeting 4 poor persons, distributed 60 cents among them, giving the second twice, the third three times, and the fourth four times as much as the first. How many cents did he give to each? 6. A gentleman left 11000 crowns to be divided between his widow, two sons, and three daughters. He intended that the widow should receive twice the share of a son, and that each son should receive twice the share of a daughter. Required the share of each. Lét x represent the share of a daughter, then 2 x will represent the share of a son, &c. 7. Four gentlemen entered into a speculation, for which they subscribed $4755, of which В paid 3 times as múch as A, and C paid as much as A and B, and D paid as much as B and C. What did each pay? 8. A man bought some oxen, some cows, and some sheep for $1400 ; there were an equal number of each sort. For the oxen he gave $42 apiece, for the cows $20, and for the sheep $8 apiece. How many were there of each sort ? In this example the unknown quantity is the number of cach sort, but the number of each sort being the same, one character will express it. Let x denote the number of each sort. Then x oxen, at $42 apiece, will come to 42 x dolls., and x cows, at $20 apiece, will come to 20 x dolls., and a sheep, at $8 apiece, will come to 8 x dolls. These added together must make the whole price. 42 x +20 x + 8x = 1400 Putting the x's together, 70 x = 1400 Dividing by 70, x = 20 Ans. 20 of each sort. 9. A man sold some calves and some sheep for $374, the calves at $5, and the sheep at $7 apiece; there were three times as many calves as sheep. How many were there of each ? Let x denote the number of sleep; then 3x will denote the number of calves. |