money is said to be loaned at 6 per cent. per annum, or 6 per hundred per year. If 7 are paid for the use of 100, the money is loaned at 7 per cent. per annum, &c. of When money is loaned at 6 per cent. per annum, for one year, the interest is of the principal; when at 7, the principal are equal to the interest. 1. If x represent the per cent., Ꮖ 100 of the principal or sum lent, will represent the interest for one year; 2x 100 cipal, will represent the interest for two years, &c. of the prin 2. What is the interest of 250 for 2 years, at 6 per cent.? What is the amount? What is the interest of a dollars for 2 years at 6 per cent.? 12 a Ans. 100 3. If a 720, what would be the answer to the preceding question? 4. What is the interest of a dollars for 3 years, at r per cent.? 3 ra Ans. 100 5. If a 50, and r = 10, what would be the answer to the preceding question? 6. What will represent the interest of p dollars for t years, at r per cent.? prt Ans. 100 7. What will represent the amount in the preceding ques tion? prt Ans. p + 100 = 500, t 3 and r = 6, what will be the answer to the preceding questions, 6 and 7? 9. A man loaned a sum of money for t years, at r per cent., and found that the interest wanted but c dollars of being equal to the principal. What will represent the principal ? If c = 12, t = 5 and r = 10, what will be the answer to the preceding question? 10. A man has a sum, p, which he wishes to lend at r per cent. to amount to d. What will represent the time that he must lend it? 11. If r= :6, p =200 and d = 224, what will be the answer to the preceding question? 12. A man wishes a sum p, which he has to lend, to amount to a, in t years. At what rate per cent. must he lend it? 13. If p=500, t = 3 and a = 590, what will the answer to the preceding question be? 14. A man wishes to lend a sum sufficient to amount to a, in t years, at r per cent. What will represent the sum which he must lend? 15. A man loaned a sum for t years, at r per cent. He recollects that interest was c dollars; but has forgotten the principal. Can you tell him what it was, or how to find it? SECTION XXXVI. Generalization. 1. The sum of two numbers is 2 a, and their product d. What are the numbers? · 2 a x + a2 = a2. d This is an affected equation, a2 is added that we may obtain the square root of the first member. Extracting the root, we have The at the right of the parenthesis, shows that the root of the quantity within the parenthesis is to be taken; and the double sign shows that the root may be either + or Supposing it +, we have (1) x = + (a2 — d)3 + a Supposing it (2) we have X= - (a2 — d)3 + a 2.- If 2 a= 10, d=24, what are the values of x? 3. What is the rule for finding two numbers, when their sum and their product are given? RULE.-Subtract the product from the square of half the sum; and extract the root of the remainder. Add half the sum to this root, and we have the greater of the numbers, Subtract this root from half the sum, and we have the less number. 4. The difference of two numbers is 2 a, and their product d. What are the numbers? 5. How do you find two numbers, when you know their difference and their product? 6. The sum of two numbers is 2 a, and the sum of their second powers is b. What are the numbers? 7. How do you find two numbers, when you know their sum, and the sum of their second powers? 8. The difference of two numbers is 2 a, and the sum of their second powers is b. What are the numbers? 9. How do you find two numbers, when you know their difference and the sum of their second powers? 10. The sum of two numbers is 2 a, and the difference of their second powers is b. What are the numbers? 11. How do you find two numbers, when you know their sum, and the difference of their second powers? 12. The sum of the second powers of two numbers is a, and the difference of their second powers is b. What will represent the numbers? 13. How do you find two numbers, when you know the sum and difference of their second powers? 14. A man bought a number of sheep for b dollars. If he had bought a number c sheep more for the same money, they would have come to him d dollars a-piece cheaper. What will represent the number of sheep he bought? SECTION XXXVII. Extraction of the Second Root. It is frequently necessary to find the roots of large numbers, whose roots cannot be immediately seen. This can often be done by re-solving the number into two or more factors. For the root of the product of two or more numbers is equal to the product of the roots of those numbers. It is required to find the second root of 196. 4 X 49 = 196. Hence the second root of 196 is equal to the root of 4 multiplied by the root of 49. The root of 4 is 2, and the root of 49 is 7; consequently, the root of 196 = 2 × 7 = 14. If the root of any number be required, find by trial, whether it can be divided by any number, the root of which can be immediately seen; such as, 4, 9, 16, 25, 49, 81, &c. The divisor will be one factor, and the quotient will be the other. Multiply the root of the divisor by the root of the quotient, and the product will be the root required. Find the roots of the following numbers: (1) 324 (3) 256 484 676 (5) 400 8 3136 5184 (6) 1764 When the root of a number cannot be found by the preceding method, we must resort to the process explained in the following example: Find the root of 1444. It can be immediately seen that the root is between 30 and 40, or 3 tens and 4 tens; for 30 X 30, or 3 tens X 3 tens, = 900, and 40 X 40, or 4 tens X 4 tens 1600. Hence 3 is the number of tens in the root. Then 30+x= the root of 1444. Hence 30+x multiplied by 30+x=1444: but 30+x multiplied by 30+x, is evidently equal to 30 times 30+x, added to x times 30+x. on the next page. See the multiplication performed a 2 30+x 900 + 30 x 30 x + x2 900+2, 30 x + x2 = 1444 Subtract 900 from both members, and we have 2. 30 x + x2= 544 We may now find x by trial. Assume x any number, and if twice 30 multiplied by the number assumed, added to the second power of the number assumed 544, the assumed number is the correct value of x, Or, what amounts to the same thing, as may be seen by inspecting the following equations 2. 30 x + x2=(2.30+x)x = 544, add the number to twice 30 or 60, and multiply the sum by the number. If the result is 544, the assumed number is the correct value of x. = Then (608) 8 68 X 8544 Hence 8 is the value of x; and 30+8 or 38 is the root required. Which is the greater, 2. 30 x or x2? We might sometimes make a number of trials before obtaining the exact value of x; but observe 1st., that x cannot And 2nd., by inspecting the equation be more than 9. 2.30 x + x2= 544 it will be seen that 2.30 x is nearly equal to 544; as the 2 makes a comparatively small part of the number. Hence divide 544 by 2.30, or 60, and it will give the value of x nearly. If 2. 30 x 544, we should obtain x exactly, by dividing 544 by 2.30, or 60; but as 544 is somewhat greater than 2. 30 x, this division will give something larger than x. The whole number, rejecting the remainder, may be the true value of x. It is evident that the true number cannot be greater than the whole number given by this division, nor can it be much less; so that the number of trials is, by this division, reduced to very few.. Sometimes the number has no exact root; we can then find only the root nearly, or the approximate root in whole num bers; unless we carry the process farther, as will be explained hereafter. We shall exhibit the operation of extracting the root, omit ting the x. |