tive quantities have? ad+bd ac-bc-ad+bd 14. What sign would the continued product of three nega Ans. the sign minus. Why? 15. What sign would the continued product of four negative quantities have? 16. What sign would the continued number of negative numbers have? Ans. +. Why? product of any odd Why? Ans. 17. What sign would the continued product of any even number of negative numbers have? Ans. +. 18. What signs would the second, third, fourth, fifth, sixth, &c. powers of a have? SECTION XXXII. The Use of Parentheses. It is sometimes necessary to represent an arithmetical operation performed with an entire algebraic expression, consisting of several terms. We then make use of parentheses; thus, if we wished to signify that a number represented by a+b is multiplied by a, we write it thus, (a+b) a, or a (a+b.) In like manner, if we wished to signify that the difference between b and c, or b. -c is to be multiplied by a, we write it a (b-c.) If we wished to signify that the sum of a and b is to be multiplied by the sum of a and b, we write it (a+b) × (a + b,) or (a+b).(a+b,) or (a+b)(a+b,) or (a + b)2. (a+b) signifies the second power of the sum of a and b, (a+b) the third power of the sum of a and b, (a+b) the 1 fourth power of the sum of a and b, (a+b) the second root of the sum of a and b, (a+b) the third root of the sum of a and b, &c. If a = 2, b = 3, c = 4, and d=5, what numbers will the following formulæ represent? (1) (2) (3) (5) (10) (11) (12) (b + a) a (a+b) (b− a) b (a+b) (b—a) (a + b) (b+c) (c + d) (ba—a) × (cd+b) Ans. 10. (ab+bc)× (cd+ad) (c d—a d)2 SECTION XXXIII. Equivalent Expressions. There are different arithmetical operations which give the same result; and then the algebraic formula expressing these different operations, are equivalent. Five and 6 may be added together, and the sum multiplied by 7. The result is the same as if the two numbers were multiplied by 7, separately, and the sum of their products taken; thus, 7 (5 + 6) = 7 × 11 = 7 × 5 +7 x 6. Hence, c(a + b) = ac+cb. If 2 be subtracted from 8, and the remainder be multiplied by 6, the result will be the same as if 8 and 2 were multiplied by 6, separately, and the difference of their products taken. Thus 6(8-2), or 6 × 6 = 6 × 8—6 × 2 = 36 a (b—c) = ab a c Hence (a+b) signifies that a and b are added together, and the sum multiplied by itself; but this is evidently a times a+b added to b times a + b, a+b a+b a2 + ab a times a + b +ab+b2= b times a + b a2+2ab+b2=a+b times a+b Hence (a+b)2 = a2 + 2 a b + b2 If a 5 and b = 6; how would you find the number represented by the first member of this equation? How would you find the number represented by the second member? According to what has been observed c (a+b) is equivalent to ac+bc, and a (b—c) is equivalent to a b -ac. Find the equivalent of the following formula, and verify the result. Supposing a = 5, b = 6, c = 7, d = 8. (b-a)d Ans. bd-ad. = (6-5) 8 Proof. (b-a) d =6X8-5 X 8 = 48-40: 8. 1 X 88; and bd-ad Substitute any given numbers for the letters, in each of the preceding formulæ and its equivalent, and observe if the results are identical. Find the equivalent of the following: 1. Two men built a rods of fence. One built 5 rods per day, and the other 6. they worked? Let What represents the number of days x= the number of days, For finding the answer to the preceding question. RULE.-Divide the whole number of rods built, by 11. If they had built 121 rods, what would have been the num'ber of days? If 132, what? If 187, what? 2. Two men built a rods of fence. One built b, and the other c rods per day. What will represent the number of days they worked? For finding the answer to the preceding question. RULE. Add together the number of rods each built per day, and divide the whole number of rods by that sum. 3. A man bought three kinds of wine; one at a, one at b, and one at c shillings per gallon, and of each kind an equal quantity. The whole came to d shillings. What will represent the number of gallons of each kind? d Ans. a + b + c State the rule for finding the answer to the preceding quesand to all questions of the same kind. tion; If a= 2, b = = 3, c=4; what would be the answer to the preceding question? 4. The sum of two numbers is a, and their difference b; what will represent the numbers? b α Ans. + the greater. 2 2 What is the rule for finding two numbers, when we know their sum and difference? 5. The sum of three numbers is s; the second is a, and third b more than the first. What will represent the numbers? What is the rule for finding two numbers, when we know the difference between the first and third, the difference between the first and second; and also the sum of the three numbers? 6. A man meeting a company of beggars, gave each of them a cents a-piece, and had b cents left. If he had wished to give them c cents a-piece, he would have wanted d cents for that purpose. What will represent the number of beggars? Ans. b + d Ca State the rule for answering the preceding question, and others like it. 7. One man can do a piece of work in a hours, and another in b hours. In what time will both working together do it? Give the rule. Ans. ab a+b 8. Divide the number a into two such parts, that if one be divided by b, and the other by c, the quotients will be equal. Find the formula and the rule. SECTION XXXV. On Interest. În pecuniary transactions, a sum of money loaned out, is called the principal. The sum paid for the use of money borrowed, is called interest. The principal and interest together, are called the amount. The borrower and lender agree what shall be the interest of 100, for one year, and from that, the interest of any sum for any time, may be calculated. When 6 dollars are paid for the use of 100 for a year, the G |