said to be known, only because the known values in each particular example may always be substituted for them. What is called the general solution of a problem in algebra, usually involves less labour than a particular solution, because, in the former, the operations being with letters instead of figures, are for the most part only indicated not actually performed. As another example of a general problem, we shall take that of which examples (8) and (9) are only particular cases. (14) If n agents A, B, C, &c. can each produce the same effect in the times a, b, c, &c. respectively, in what time will the effect be produced if they all act together? Suppose x to be the time, then since in one hour or one day, &c. according as a, b, c, &c. may represent hours or days, &c. the agents separately can do the parts 1 1 1 -" --> α с &c. of the whole, in x hours or days, &c. they can do together - +~+ &c. But as α с these several parts together make up the whole 1, we must •. (bc &c.)x+(ac &c.)x+(ab &c.)x=abc &c.; ab &c. +ac &c. + bc &c.; or, in words, the time of producing the effect by their united operation will be found by dividing the product of the times which they take separately by the sum of all the different products which arise from multiplying together all the times, omitting one.* Thus, in the case of three agents, the general soluabc tion will be x= ab+ac+bc In the particular instance of example (8), a=3, b=4, and c=6 hours; and substituting these particular values in the formula, we have 3x4x6 * Or the time will be expressed by a fraction of which the numerator is the product of all the separate times, and the denominator the sum of the results which arise from dividing the numerator by each of the times in succession. Instead of repeating the "&c." in the algebraic expression of this fraction, when the number of agents is arbitrary, dots may be employed: thus, In the case of two agents, the general solution or formula will be x= ab a+b ; and in the particular instance of Example (9) where a 12, and b=15, the solution is = (1) Find a number the double of which, diminished by 3, may be equal to the number itself, increased by 8. (2) What number is that the double of which exceeds the third part of it by 30? (3) What number is that of which the third part exceeds the fourth part by 6? (4) From two places, 204 miles apart, two persons set out at the same time to meet each other: one travels 16 miles a day and the other 18: in how many days will they meet? (5) Two persons, 153 miles apart, set out to meet each other: one goes 4 miles while the other goes 31: what part of the distance will each have travelled when they meet? (6) What number is that three-fourths of which, increased by 2, may be equal to five-sixths of it diminished by 2? (7) It is required to divide £110 among three persons, so that the first may receive £8 more than the second, and the second £10 more than the third. (8) Divide £700 among three persons, so that the first may receive double of what the second receives, and the second double of what the third receives. (9) Divide £143 among three persons, A, B, and C, so that A may have twice as much as B, and B three times as much as C. (10) It is required to divide a line 37 feet long into three parts, so that the first part may be 3 feet less than the second, and the second 5 feet more than the third. (11) It is required to find two numbers whose difference is 6, such that if one-third the less be added to one-fifth the greater, the sum shall be equal to one-third the greater diminished by one-fifth the less. (12) A and B perform a piece of work in 6 days which A alone can do in 9 days: in how many days can B alone do it? (13) A detachment of soldiers march from a certain place at the rate of 2 miles an hour, and two hours afterwards another detachment marching at the rate of 3 miles an hour is sent to overtake them: how far must they march to do this? (14) A gamester at play lost of his money, and then won 10s.: he afterwards lost of what he then had and won 3s. making his money just 3 guineas: how much had he at first? (15) A garrison of one thousand men was victualled for 25 days, but after nine days 250 men were removed: how long will the provisions last those who remain? (16) A person has three debtors, A, B, and C, whose debts he only so far remembers that A's and B's amounted to £60, A's and C's to £80, and B's and C's to £92: what was the exact debt of each? (17) Suppose that for every 10 sheep a farmer keeps he should plough an acre of land, and allow one acre of pasture for every 4 sheep: how many sheep may he keep who farms 700 acres? (18) A person has two sorts of spirits, one worth £1 a gallon and the other worth 12s.; and he wishes to make from these a gallon worth 14s.: how much of each must he take? (19) Two persons, A and B, have the same annual income: A lays by the fifth part of his; but B, by spending £80 a year more than A, at the end of 4 years finds himself £220 in debt: what is the annual income and expenditure of each? (20) Divide the number 116 into four parts, such that if the first be increased by 5, the second diminished by 4, the third multiplied by 3, and the fourth divided by 2, all the results may be equal. (21) A person relieving a poor family gives to the father half what he has in his pocket and one penny more: to the mother half what he has left and twopence more; and upon giving half the remainder and threepence more to the children, finds that all his money is distributed. How much did he give? (22) A and B can do a piece of work in 6 days, A and C in 8 days, and B can do twice as much as C in a given time: in what time can B and C together finish the work? (23) Divide the number a into two parts, such that the square of one part, increased by b, may be the same as the square of the other part diminished by b. (24) Gunpowder is to be conveyed into a garrison in full casks: more than 2 cwt. cannot be carried in each. Now there are two sorts of powder: a cask full of the inferior sort weighs 182 lbs.; a cask full of the superior sort weighs 230 lbs.: it is required to find how much of each sort must be mixed so that casks full of the best possible quality may be sent. (25) It is required to generalise the preceding question: a being the weight of a cask of the inferior powder, b the weight of a cask of the superior, and c the weight of the cask-full to be carried. (26) A merchant has spirits at a shillings a gallon and at b shillings a gallon, and he wishes to make a mixture of d gallons that shall be worth c shillings a gallon: how much of each must he take? (27) A trader supported himself for 3 years at the expense of £50 a year; and in each of those years increased his unexpended money by one-third of it. At the end of the third year he found that his original stock of money was doubled: how much was it? (28) Two workmen, A and B, were engaged to work for a certain number of days at different wages. At the end of the time A, who had been idle 4 days, received £3 15s.; and B, who had been idle 7 days, received £2 8s. But if A's idle days had been B's, and B's A's, they would have received exactly alike. For how many days were they engaged, how many did each work, and what had each per day? (29) Each of two traders, at the end of every year, finds the capital he had at the beginning of the year doubled: each expends £100 a year for his support; and at the end of 3 years, one finds that he has twice as much as he began with, and the other that he has only half as much: what sum did each commence with? (30) Two labourers who can separately mow a field in 4 days and 7 days, are joined on the second day by a third labourer, who alone could mow the field in 10 days. After helping them for a certain time the third labourer goes away, at which time four-fifths of the field have been mowed. How many days has the work, up to this point, been in hand? (31) At twelve o'clock, the hour and minute hands are exactly together: the minute hand then takes the lead: when does it next arrive at the hour hand? (32) At what time between 5 and 6 o'clock are the hour and minute hands of a watch exactly together? (33) A and B can do a piece of work in 3 days, A and C in 5 days, and B and C in 6 days: in what time can they finish it if they all work together? (34) It is required to generalise the preceding question, a representing the time occupied by A and B together, b the time occupied by A and C together, and c the time occupied by B and C together. (35) The conditions being as in the last question, to find the time in which A, B, C can perform the work separately. (36) A man and his wife can drink a cask of beer in 15 days; but after drinking together 6 days, the woman alone drank the remainder in 30 days. In what time could either alone drink the whole? (37) A fox pursued by a greyhound has a start of 60 leaps: he makes 9 leaps while the greyhound makes 6, but three leaps of the greyhound are equal to 7 leaps of the fox. How many leaps must the greyhound make to overtake the fox? (38) The sum of three numbers is 70, and the numbers themselves are such that the second, divided by the first, gives 2 for quotient and 1 for remainder, and the third divided by the second gives 3 for quotient and 3 for remainder: what are the numbers? (39) Three causes or agents, A, B, C, produce the effects a, b, c, in the times d, e, f, when they act separately: in what time will they produce the effect E if they act in conjunction? (40) The hands of a clock start together at 12 o'clock: state how often they are together during the next 12 hours, and how the different times of their conjunction may be found. 54. The learner who has carefully gone over what has preceded, must now have acquired a tolerably distinct idea of the nature of algebraic operations and of the great utility of the symbols employed in them in facilitating arithmetical inquiries. He has perceived, when a question is to be solved algebraically, that as soon as the conditions of it are translated into the symbolical language of the science, that he may dismiss from his mind all concern about the particular question in which his equation has originated, and confine himself to the single object of extricating therefrom the unknown quantity, and of thus discovering the value which must be put for it so as to satisfy the algebraical condition, that is, the equation of the problem. We have noticed above (page 122) the distinction between a particular and a generul solution; but, in strictness, no algebraical solution is a particular solution in the same restricted sense that the question solved is a particular question. It is pretty obvious that we may always modify a question in various ways without disturbing the algebraic expression of its conditions: the algebraic statement of these is always so general as to comprehend every conceivable modification of the question which does not |