MULTIPLICATION AND DIVISION OF SIMPLE QUANTITIES. §29. Were we to add a to itself four times, we should write the sum thus, a+a+a+a; which, when united, becomes 4a. Whence we see that any literal quantity is multiplied by a number, by putting that number before it as a co-efficient; as, 7 times x is 7x. 6 times a is 6a. The pupil may now multiply x by each of the numbers from 2 to 20. §30. If the quantity to be multiplied has already a coefficient, that co-efficient only is to be multiplied. Thus, 3x taken four times is 3x+3x+3x+3x, which when united =12x. The co-efficient is multiplied by 4. Thus, 4 times 3x=12x. The pupil may multiply 2x by each number from 2 to 12. He may then multiply 3x by the same numbers; and then 4x by the same. §31. It is evident that if 2 times 3x is 6x, then one half of 6x is 3x. Whence we learn that a quantity with a numeral co-efficient, may be divided, by merely dividing that co-efficient. Thus, 8x divided by 2=4x. 12x divided by 4=3x, &c. §32. In 1661, Rev. William Oughtred of England, published a work, in which he introduced the sign X to represent multiplication. Thus, 4x3=12, is read 4 multiplied by 3 equals 12; or, 4 into 3 equals 12. §33. In 1668, Mr. Brancker invented the sign for division. This sign is always put before the divisor; as 20÷4-5; read 20 divided by 4 equals 5; or, 20 by 4, equals 5. SIMPLE EQUATIONS. §34. The most general application of algebra, is that which investigates the values of unknown quantities by means of equations.. §35. An equation is an expression which declares one quantity to be equal to another quantity, by means of the sign being placed between them. = Thus, 5+3=8, is an equation, denoting that 5 with 3 added to it, equals 8. Also, 4-1-3, and 3+2—1=4, and 8—2—5+1, are equations, each denoting that the quantity on one side of the =, is equal to that on the other side. §36. The whole quantity on the left of is called the first member of the equation; and all on the right of = is called the last member of the equation. In order to be a member of the equation, it is of no importance whether the quantity is simple or compound. Thus, in the equation, x=4+a-b-18, x is the first member, and 4+a-b-18 is the last member. And in this case, the first member represents just as great a quantity as the last. §37. In order that an equation may be such that we can find the value of an unknown quantity by it, it must contain some quantity that is already known. And then, we find the value of the unknown quantity, by making that stand by itself on one side of =, and all the known quantities on the other side; taking care to change them in such a manner as not to destroy the equation. §38. The operation of managing an equation, so as to bring the unknown quantity to stand equal to a known quantity, is called solving or reducing the equation. EQUATIONS.-SECTION 1. Equations which are solved by merely uniting terms. In each of the following equations, the object is to find the value of x. Example 1. x+2x=45-15. Uniting terms, 3x=30. Now, as we have found that three x's 30, it is evident that one x will be one third of 30. Therefore, dividing by 3, x=10. 2. 8x-4x-x-7+26+51-15 Dividing by 3, x=23. 3. 10x-5x+4x=56+75+32-1 Uniting terms, 9x=162 4. x+2x+3x+4x=12+35+74-11 Uniting terms, 10x=110 8x-3x+2x=46+54+37-4. 5. Ans. x=19. 6. 4x-3x+4x=29-36+48+14. 7. 6x-8x+14x=12+36 +14+22. Ans. x 11. Ans. x=7. Ans. x=8. Ans. x=2. Ans. x=36. 8. 5x+4x+3x=49+14+22+11. 15. 17x-4x-3x-5x-x-57-32. Ans. x=5. Ans. x=9. 16. 14x-36x+29x+47x+x=504. PROBLEMS. §39. An algebraic problem is a proposition which requires the discovery or demonstration of some unknown truth. §40. In the solution of problems, the first thing to be done, is to make a statement of the conditions, in algebraic language, in the same manner as if the answer were already found, and you were required to see if it is right. In order to do this, it is customary to represent the unknown quantity by x, y, or some other final letters of the alphabet. §41. When the question has been fairly stated, it will be found that some condition has been represented in two ways; one having the unknown quantity in it, and the other having a known quantity. These two expressions must be put together, so as to form an equation. And then, by reducing the equation, the required result will be found. 1. The sum of $660 was subscribed for a certain purpose, by two persons, A and B; of which, B gave twice as much as A. What did each of them subscribe? 2. Three persons in partnership, put into the stock $4800; of which, A put in a certain sum, B twice as much, and C as much as A and B both. man put in ? What did each In all the succeeding problems, the learner should prove his answers. 3. A person told his friend that he gave 108 dollars for his horse and saddle; and that the horse cost 8 times as much as the saddle. What was the cost of each? It is advisable for the pupil while performing his sums, to write them on his slate in a manner similar to the three questions above; beginning the statement by making a the answer to the question. And in recitation, the whole of it is to be recited. 4. A father once said, that his age was six times that of |