ELEMENTS OF ALGEBRA. SECTION I. EXPLANATION OF ALGEBRAIC SIGNS. 1. Let it be proposed to divide the number 56 into two such parts, that the greater may exceed the less by 12. To resolve this question, we remark that, 1o. The greater part is equal to the less added to 12. 2o. The greater part, added to the less part, is equal to 56. It follows therefore that, 3o. The less part, added to 12, added also to the less part, is equal to 56. But this language may be abridged, thus, 4o. Twice the less part, added to 12, is equal to 56, whence, 5o. Twice the less part is equal to 56 diminished by 12. Subtracting therefore 12 from 56, we have 6o. Twice the less part equal to 44; wherefore 70. Once the less part is equal to 44 divided by 2, or performing the division, we have 8°. Once the less part equal to 22. Adding 12 to 22 we have 34 for the greater part. The parts required therefore are 22 and 34. 2. In the process of reasoning required in the solution of the proposed question expressions, such as "added to,” “diminished by," "equal to," &c. are often repeated. These expressions refer to the operations, by which the numbers given in the question are connected among themselves, or to the relations, which they bear to each other. The reasoning therefore, which pertains to the solution of a question, it is evident, may be rendered much more concise, by representing each of these expressions by a convenient sign. It is agreed among mathematicians to represent the expression "added to" by the sign + read plus, the expression "diminished by" by the sign - read minus, the expression "multiplied by" by the sign X, that of "divided by" by the sign÷. Lastly the expression "equal to" is represented by the sign. 3. By means of the above signs the reasoning in the question proposed may be much abridged; still however we have frequent occasion to repeat the expression "the less part." The reasoning therefore may be still more abridged by representing this also by a sign. The less part is the unknown quantity sought directly by the reasoning pursued. It is agreed in general to represent the unknown quantity or quantities sought in a question by some one of the last letters of the alphabet, as, x, y, z. 4. Let us now resume the question proposed and employ in its solution the signs, which have been explained. Let us represent by x the less of the two parts required, we have then The multiplication of x by 2 may be expressed more concisely thus, 2. x, or still more concisely thus, 2 x. Division also is more commonly indicated by writing the number to be divided above a horizontal line, and the divisor beneath it in the form of a fraction; 14 divided by 2, for example, is indi 5. The question, which we have solved, is simple; it is sufficient, however, to show the aid, which may be derived from convenient signs in facilitating the reasonings, which pertain to the solution of a question. Indeed in abstruse and complicated questions, it would often be difficult, and some times absolutely impossible to conduct, without such aid, the reasonings required. 6. The signs which have been explained, together with those which will hereafter be introduced, are called Algebraic signs. It is from the use of these that the science of Algebra is derived. Let us now employ the signs already explained in the solution of some questions. 1. Three men, A, B and C trade in company and gain $405, of which B has twice as much as A, and C three times as much as B. Required the share of each. Let x represent the share of A, then 2 x will represent the share of B and 6 x the share of C. Then since the shares added together should be equal to the sum gained, we have 2. A fortress is garrisoned by 2600 men; and there are nine times as many infantry, and three times as many artillery as cavalry. How many are there of each? 3. From two towns, which are 187 miles distant, two trayellers set out at the same time, with an intention of meeting. One of them goes 8 miles, and the other 9 miles a day. In how many days will they meet? 4. In fencing the side of a field, whose length was 450 yards, two workmen were employed, one of whom fenced 9 yards and the other 6 yards per day. How many days did they work? 5. A gentleman meeting four poor persons distributed 5 shillings among them; to the second he gave twice, to the third thrice, and to the fourth four times as much as to the first. What did he give to each? 6. To divide the number 230 into three such parts, that the excess of the mean above the least may be 40, and the excess of the greatest above the mean may be 60. Let x represent the least part, then x + 40 will be the mean, and x + 40+60 will be the greatest part; we have therefore x+x+40+x+40 + 60: = 230 3x+140=230 3x=90 x=30 The parts will then be 30, 70 and 130 respectively. 7. A draper bought three pieces of cloth which together measured 159 yards. The second piece was 15 yds. longer than the first, and the third 24 yds. longer than the second. What was the length of each? 8. Three men, A, B and C made a joint stock; A puts in a certain sum, B puts in $115 more than A, and C puts in $235 more than B; the whole stock was $1753. What did each man put in ? 9. A cask which held 146 gallons was filled with a mixture of brandy, wine and water. In it there were 15 gallons of wine more than there were of brandy, and 25 gallons of water more than there were of wine. What quantity was there of each? 10. A gentleman buys 4 horses, for the second of which he gives £12 more than for the first, for the third £6 more than for the second, and for the fourth £2 more than for the third. The sum paid for all was £230. How much did each cost? 11. A man leaves by will his property, amounting, to $14000, to his wife, two sons and three daughters; each son is to receive twice as much as a daughter, and the wife as much as all the children together. What will each receive? 12. An express sets out to travel 240 miles in 4 days, but in consequence of the badness of the roads, he found he must go 5 miles the second day, 9 the third and 14 the fourth day less than the first. How many miles must he travel each day? 13. The sum of $300 was divided among 4 persons; the second received three times as much as the first, the third as much as the first and second, and the fourth as much as the second and third. What did each receive? 14. A sum of $1245 is to be divided among three persons, A, B and C. A is to receive $175 less than B, but C $320 more than B. What will each receive? 15. A silversmith has 3 pieces of metal. The second weighs 6 oz. more than twice the first, and the third 9 oz. less than three times the first. The weight of the whole being 52 oz., what is the weight of each? 16. A poor man had 4 children, the eldest of which could earn 7d. a week more than the second, the second 9d. more than the third, and the third 5d. less than the fourth. They together earned £2. 3s. How much could each earn a week? 17. To find a number such, that one half and one third of this number will be equal to 30. Let x represent the number sought, then one half of the 1 1 number will be represented by or and one third by x or 3'' 18. A farmer sold 96 loads of hay to two persons. To the first one half, and to the second one fourth of what his stack contained. How many loads were there in the stack? 19. A gentleman gave to three persons 98 pounds, the sec-. ond received twice the sum given to the first, and the third. one fifth of what the second had. What did each receive? 20. All the journeyings of a traveller taken together amount to 3040 miles; of which he travelled 3 times as much by water as on horseback, and 24 times as much on foot as by water.. How many miles did he travel in each of these three ways |