Let be the time in which they could finish it if all worked together, then by Prob. 8.

Prob. 16. What two numbers are those whose difference is 7, and sum 33?

Ans. 13 and 20

Prob. 17. To divide the number 75 into two such parts, that three times the greater may exceed 7 times the less by 15.

Prob. 18. In a mixture of wine and cyder, wine, and part minus 5 gallons was cyder; each?

Ans. 24 and 21.

of the whole plus 25 gallons was how many gallons were there of Ans. 85 of wine, and 35 of cyder.

Prob. 19. A bill of 120%. was paid in guineas and moidores, and the number of pieces of both sorts that were used was just 100; how many were there of each? Ans. 50 of each.

Prob. 20. Two travellers set out at the same time from London and York, whose distance is 150 miles; one of them goes 8 miles a day, and the other 7; in what time will they meet? Ans. In 10 days.

Prob. 21. At a certain election 375 persons voted, and the candidate chosen had a majority of 91; how many voted for each?

Ans. 233 for one, and 142 for the other.

Prob. 22. What number is that from which, if 5 be subtracted, of the remainder will be 40?

Ans. 65.

P

Prob. 23. A post is in the mud, in the water, and 10 feet above the water; what is its whole length?

Ans. 24 feet.

Prob. 24. There is a fish whose tail weighs 9lb. 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. 72lb.

Prob. 25. After paying away and of my money, I had 66 guineas left in my purse; what was in it at first? Ans. 120 guineas.

Prob. 26. 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.

Prob. 27. Two persons, A and B, lay out equal sums of money in trade; A gains £126, and B loses £87, and A's money is now double of B's; what did each lay out?

Ans. £300.

Prob. 28. A person bought a chaise, horse, and harness, for £60, 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. £13 6s. 8d. for the horse, £6 13s. 4d. for the harness, and £40 for the chaise.

Prob. 29. Two persons, A and B, have both the same income: A saves of his yearly, but B, by spending £50 per annum more than A, at the end of 4 years finds himself £100 in debt; what is their income?

Ans. £125.

Prob. 30. A person has two horses, and a saddle worth £50: now, if the saddle be put on the back of the first horse, it will make his value double that of the second; but 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 £30, and the other £40.

Prob. 31. To divide the number 36 into three such parts, that of the first, of the second, and of the third, may be all equal to each other?

Ans. The parts are 8, 12, and 16.

Prob. 32. A footman agreed to serve his master for £8 a year, and a livery ; but was turned away at the end of 7 months, and received only £2 13s. 4d. and his livery; what was its value? Ans. £4 16s.

Prob. 33. A person was desirous of giving 3d. a-piece to some beggars, but found that 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.

Prob. 34. A person in play lost of his money, and then won 3s.; after which, he lost of what he then had, and then won 2s.; lastly, he lost of what he then had; and, this done, found he had but 12s. remaining; what had be at first? Ans. 20s.

Prob. 35. To divide the number 90 into 4 such parts, that if the first 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 to each other? Ans. The parts are 18, 22, 10, and 40, respectively.

Prob. 36. The hour and minute hand of a clock are exactly together at 12 o'clock; when are they next together? Ans. 1 hour 55 minutes.

Prob. 37. There is an island 73 miles in circumference, and three footmen all start together to travel the same way about it: A goes 5 miles a day, B 8, and C 10; when will they all come together again? Ans. 73 days.

Prob. 38. How much foreign brandy at 8s. per gallon, and British spirits at 3s. per gallon, must be mixed together, so that in selling the compound at 9s. per gallon, the distiller may clear 30 per cent. ?

Ans. 51 gallons of brandy, and 14 of spirits.

Prob. 39. 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.

Prob. 40. 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 143 days, B 1743, and C 23.

Prob. 41. A book is printed in such a manner, that each page contains a certain number of lines, and each line a certain number of letters. If each page were required to contain 3 lines more, and each line 4 letters more, the number of letters in a page would be greater by 224 than before; but if each page were required to contain 2 lines less, and each line 3 letters less, the number of letters in a page would be less by 145 than before. Required the number of lines in each page, and the number of letters in each line?

Ans. 29 lines, 32 letters.

Prob. 42. Five gamblers, A, B, C, D, E, throw dice, upon the condition that he who has the lowest throw shall give all the rest the sum they have already. Each gamester loses in turn, commencing with A, and at the end of the fifth game, all have the same sum, viz. £32. How much had each at first?

Ans. A £81, B £41, C £21, D £11, E £6.

Prob. 43. To divide a number a into two parts, which shall have to each other the ratio of m to n.

Prob. 44. To divide a number a into three parts, which shall be to each

Prob. 45. A banker has two kinds of change; there must be a pieces of the first to make a crown, and b pieces of the second to make the same: now, a person wishes to have c pieces for a crown. How many pieces of each kind

must the banker give him?

Prob. 46. A sportsman promises to pay a friend a shillings for each shot he misses, upon condition that he is to receive b shillings for each shot he hits. After n shots, it may happen that the two friends are quits, or that the first owes the second c shillings, or the contrary. Required a formula which shall comprehend all the three cases, and which shall give x the number of shots missed. b n + c

Ans. x =

a+b

In the first case, co, in the second case we must take the positive sign, in the third case the negative sign.

Prob. 47. If one of two numbers be multiplied by m, and the other by n, the sum of the products is p; but if the first be multiplied by m', and the second by n', the sum of the products is p'. Required the two numbers.

Prob. 48. An ingot of metal which weighs n pounds, loses p pounds when weighed in water. This ingot is itself composed of two other metals, which we may call M and M'; now, n pounds of M loses q pounds when weighed in water, and n pounds of M' loses r pounds when weighed in water. of each metal does the original ingot contain?

How much

n (r− p)
Ans.
Ч

pounds of M,

n ( p − q)

pounds of M'.

-q

REMARKS UPON EQUATIONS OF THE FIRST DEGREE.

152. Algebraic formulæ can offer no distinct ideas to the mind, unless they represent a succession of numerical operations which can be actually performed. Thus, the quantity b—a, when considered by itself alone, can only signify an absurdity when ab. It will be proper for us, therefore, to review the preceding calculations, since they sometimes present this difficulty.

Every equation of the first degree may be reduced to one which has all its signs positive, such as,

Subtracting cx+b from each member, we then have,

.. (1) *

.(2)

* We can always change the negative terms of an equation into others which are positive, since we can always add any quantity to both members.

This being premised, three different cases present themselves,

1o. db, and, ac.

2o. One of these conditions only may hold good.

3o. bd, and, ca.

In the first case, the value of x in equation (2), resolves the problem without giving rise to any embarrassment; in the second and third cases, it does not at first appear what signification we ought to attach to the value of x, and it is this that we propose to examine.

In the second case, one of the subtractions, d—b, ac, is impossible : for example, let b⇒d and ac, it is manifest that the proposed equation (1) is absurd, since the two terms ax and b of the first member are respectively greater than the two terms c x and d of the second. Hence, when we encounter a difficulty of this nature, we may be assured that the proposed problem is absurd, since the equation is merely a faithful expression of its conditions in algebraic language.

In the third case, we suppose b>d, and c>a; here both subtractions are impossible: but let us observe, that in order to solve equation (1), we subtracted from each member the quantity cab, an operation manifestly impossible, since each member cx+b. This calculation being erroneous, let us subtract a x + d from each member, we then have,

This value of x, when compared with equation (2), differs from it in this only, that the signs of both terms of the fraction have been changed, and the solution is no longer obscure. We perceive, that when we meet with this third case, it points out to us, that instead of transposing all the terms involving the unknown quantity, to the first member of the equation, we ought to place them in the second; and that it is unnecessary, in order to correct this error, to recommence the calculation,—it is sufficient to change the signs of both numerator and denominator.

One of the principal advantages which algebra holds out, is to enable us to obtain formulæ which shall include every variety of the same problem, whatever the numbers may be which it involves. We shall here attain this object by establishing a convention, to perform upon isolated negative quantities the same operations as if they were accompanied by other magnitudes. For example, if we had an expression, m+d—b, and b⇒d, we might express it, 712- -(b—d); if m does not exist, by our convention, we still write, d—b · (b—d), when b⇒d.

=

The value of x, in the second case, becomes x ——

b.

-

d

a C'

and we conclude, from what has been said above, that every negative solution denotes an absurdity

in the condition of the question proposed.

-

In order to divide the polynomial · a1 + 3a2 b2 + &c. by · a + b2 + &c. we divide the first term a by-a2, and we know (Art. 18.) that the quotient a2 has the sign+. We may say the same of the isolated negative quantities,