8. What number is that from which, if 5 be subtracted, two thirds of the remainder will be 40 ? Ans. 65. 9. At a certain election, 1296 persons voted, and the successful candidate had a majority of 120;; how many voted for each ? Ans. 708 for one, and 588 for the other. 10. A's age is double of B's, and B's is triple of c's, and the sum of all their ages is 140; what is the age of each? Ans. A's 84, B's 42, and c's 14 11. Two persons, A and B, lay out equal sums of money in trade; A gains 126l., and B loses 871., and A's money is now double of B's; what did each lay out? Ans. 3001. 12. A person bought a chaise, horse, and harness for 601.; the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness; what did he give for each? Ans. 131. 6s. 8d. for the horse, 61. 13s. 4d. for the harness, and 401. for the chaise. 13. A person was desirous of giving 3d. a piece to some beggars, but found he had not money enough in his pocket by 8d.. he therefore gave them each 2d., and had then 3d. remaining; required the number of beggars? Ans. 11. 14. A servant agreed to live with his master for 81. a year, and a livery, but was turned away at the end of seven months, and received only 21. 13s. 4d. and his livery;, what was its value ? Ans, 41. 16s. 15. A person left 5601. between his son and daughter, in such a manner, that for every half crown the son should have, the daughter was to have a shilling; what were their respective shares ? Ans. Son 4001., daughter 1601. 16. There is a certain number, consisting of two places of figures, which is equal to four times the sum of its digits ; and if 18 be added to it, the digits will be inverted; what is the number? Ans. 24. 17. Two persons, A and B, have both the same income; A saves a fifth of his yearly, but B, by spending 50l. per annum more than A, at the end of four years finds himself 1001. in debt; what was their income? Ans. 1251. 18. When a company at a tavern came to pay their reckoning, they found, that if there had been three persons more, they would have had a shilling apiece.less to pay, and if there had been two less, they would have had a shilling apiece more to pay; required the number of persons, and the quota of each. Ans. 12 persons, quota of each 5s. 19. A person at a tavern borrowed as much money as he had about him, and out of the whole spent 1s.; he then went to a second tavern, where he also borrowed as much as he had now about him, and out of the whole spent Is.; and going on, in this manner, to a third and fourth tavern, he found, after spending his shilling at the latter, that he had nothing left; how much money had he at first ? Ans. 11 d. 20. It is required to divide the number 75 into two such parts, that three times the greater shall exceed seven times the less by 15. Ans. 54 and 21. 21. In a mixture of British spirits and water, of the whole plus 25 gallons was spirits, and į part minus 5 gallons was water; how many gallons were there in each? Ans. 85 of wine, 35 of water, 22. A bill of 1201. was paid in guineas and moidores, and the number of pieces of both sorts that were used were just 100; how many were there of each, reckoning the guineas at 21s., and the moidores at 27s.? Ans. 50. 23. Two travellers set out at the same time from London and York, whose distance is 197 miles : one of them goes 14 miles a day, and the other 16: in what time will they meet ? Ans. 6 days 13 hours. 24. There is a fish whose tail weighs 91b., his head weighs as much as his tail and half his body, and his body weighs as much as his head and his tail; what is the whole weight of the fish ? Ans. 7216. 25. It is required to divide the number 10 into three such parts, that if the first be multiplied by 2, the second by 3, and the third by 4, the three products shall be all equal. Ans. 49, 375, 27 26. It is required to divide the number 36 into three such parts, that I the first, į of the second, and of the third, shall be all equal to each other. Ans. The parts are 8, 12, and 16. 27. A person has two horses, and a saddle, which of itself is worth 501.; now, if the saddle be put on the back of the first horse, it will make his value double that of the second, and if it be put on the back of the second, it will make his value triple that of the first; what is the value of each horse ? Ans. One 301, and the other 401. 28. If a give B 5s. of his money, B will have twice as much as the other has left; and if B give a 5s. of his money, A will have three times as much as the other has left: how much has each? Ans. A 13s. and B. 11s. 29. What two numbers are those whose difference, sum, and product, are to each other, as the numbers 2, 3, and 5 respectively? Ans. 10 and 2. 30. A person in play lost a fourth of his money, and then won back 3s., after which he lost a third of what he now had, and then won back 2s.; lastly, he lost a seventh of what he then had, and after this found he had but 12s. remaining; what had he at first? Ans. 20s. 31. A hare is 50 leaps before a greyhound, and takes 4 leaps to the greyhound's 3, but 2 of the greyhound's leaps are as much as 3 of the hare's; how many leaps must the greyhound take to catch the hare? Ans. 300. 32. It is required to divide the number 90 into four such parts, that if the first part be increased by 2, the second diminished by 2, the third multiplied by 2, and the fourth divided by 2, the sum, difference, product, and quotient, shall be all equal ? Ans. The parts are 18, 22, 10, and 40. 33. There are three numbers whose differences are equal, (that is, the second exceeds the first as much as the third exceeds the second), and the first is to the third as 5 to 7; also the sum of the three numbers is 324; what are those numbers ? Ans. 90, 108, and 126. 34. A man and his wife usually drank out a cask of beer in 12 days, but when the man was from home it lasted the woman 30 days; how many days would the man alone be in drinking it? Ans. 20 days. 35. A general, ranging his army in the form of a solid square, finds he has 284 men to spare, but increasing the side by one man, he wants 25 to fill up the square.; how many soldiers had he? Ans. 24000. 36. If A and B together can perform a piece of work in 8 days, A and c together in 9 days, and B and c in 10 days, how many days will it take each person to perform the same work alone. Ans. A 149 days, B 1721, and c 23.31. QUADRATIC EQUATIONS. A QUADRATIC EQUATION, as before observed, is that in which the unknown quantity is of two dimensions, or which rises to the second power, and is generally divided into simple or pure, and compound or adfected. A simple or pure quadratic equation, is that which contains only the square, or second powei, of the unknown quantity, as, 6 b axa = b, or 22 -; where x=V-. A compound or adfected quadratic equation, is that which contains both the first and second power of the unknown quantity, as, b ax? + bx = c, or 212 to 73 a !1 a a 2. C In which case it is to be observed, that every equation of this kind, having any real positive root, will fall under one or other of the three following forms: 1. *a* + ax = b... where x = +0). th +6). 4 3. 2012 b .. where x =+ - b). 2 Or, if the second and last terms be taken either positively or negatively, as they may happen to be, the general equation ax' + bx = Ic, or 22 §-x=+ which comprehends all the three cases above-mentioned, may be resolved by means of the following rule : RULE.—'Transpose all the terms that involve the unknown quantity to one side of the equation, and the known terms to 士 с a а * It may be observed, with respect to these forms, that In the case x2-+-ax-h=0, where ka tv (a2-+b), or -- ka -V (1a2+b), the first value of x must be positive, because v (102+) is greater than V$a2, or its equal sa; and its second value wilí evidently be negative, because each of the terms of which it is composed is negative. 2. In the case 22-ax-b=0, b = 0, where c= atv (la2+b) or la--V (ła2+b), the first value of x is manifestly positive, being the sum of two positive terms: and the second value will be negative, because v (1a27b) is greater than Vļa2, or its equal da. 3. In the case 22 — ax+b= 0, where x= where x=łatv (1a2-b), or la ka-V (1a2--b), both the values of x will be positive, when ļa2 is greater than Ò; for its first value is then evidently positive, being composed of two positive terms; and its second value will also be positive; because v (1a2-b) is less than Vļa2, or its equal ka. But if şa2, in this case, be less than b, the solution of the proposed equation is impossible; because the quantity ļa2-b, under the radical, is then negative; and consequently v (192—6) will be imaginary, or of no assignable value. 4. It may be also further observed, that there is a fourth case of the form 22-factb =0, where x=- katv (fa2-6), or x=-- a-V (ła2----b), the two values of 2 will be both negative, or both imaginary, according as ļa2 is greater or less than b; the imaginary roots, when they occur, being here of the forms -(a' tv-1) and (a-CV-1). From which it follows, that if all the terms of a quadratic equation, when brought to the lefthand side, be positive, its two roots will be both negative, or both imaginary; and conversely, if each of the roots be negative, or each imaginary, the signs of all the terms will be positive. So that, of all quadratic equations, which can have any real positive root, that of the third form, 22 - ax+b=0, is the only one, where the solution for certain numeral values of a and b will become impossible ܕ the other; observing to arrange them so that the term which contains the square of the unknown quantity may be positive, and stand first in the equation. Then, if this square has any coefficient prefixed to it, let all the rest of the terms be divided by it, and the equation will be brought to one of the three forms above-mentioned. In which case, the value of the unknown quantity x is always equal to half the coefficient, or multiplier of x, in the second term of the equation, taken with a contrary sign, together with the square root of the square of this number and the known quantity that forms the absolute or third term of the equation.* Note.—All equations, which have the index of the unknown quantity, in one of their terms, just double that of the other, are resolved like quadratics, by first finding the value of the square root of the first term, according to the method used in the above rille, and then taking such a root, or power of the result, as is denoted by the reduced index of the unknown quantity. Thus, if there be taken any general equation of this kind, as, 202m + axm we shall have, by taking the square root of .x2m, and observing the latter part of the rule, 22 Fax= * This rule, which is more commodious in its practical application than that usually given, is founded upon the same principle; being derived from the well-known property, that in any quadratic, Eb, if the square of half the coefficient a of the second term of the equation be added to each of its sides, so as to render it of the form + az +1a2 = } a2 b that side which contains the unknown quantity will then be a complete square; and, consequently, by extracting the root of each side, we shall have #ku=Iv (4a2 +b), or x= Fļa+v (şa2 #b), which is the same as the rule, taking a and b in for - as they may happen to be. It may here also be observed, that the ambiguous sign #, which denotes both + and --, is prefixed to the radical part of the value of x in every expression of this kind, because the square root of any positive quantity, as a2, is either tá or -a; for (+a) x (ta), or 6 x ( a) are each'=+a?: but the square root of a negative quantity, as - a2, is imaginary, or unassignable, there being no quantity, either positive or negative, that, when multiplied by itself, will give a negative product. To this we may also further add, that from the constant occurrence of the double sign before the radical part of the above expression, it necessarily follows, that every quadratic equation must have two roots; which are either both real, or both imaginary, according to the nature of the question. |