then a xw: the part done by A in x days

a

and since the sum of these three parts must be equal to the whole work, therefore

multiplying by abc, bcx + acx + abx = alc

(ab+ac+bc) x = abc

abc

x= abac + bc

In the 11th example, a = 5, b = 6, c = 8,

1. What number is that which, being added to its fifth

part, gives 24 for the sum? Ans. 20. 2. Required a number which exceeds its fourth part by Ans. 36.

27.

3. There is a number which, being added to 7, gives a sum equal to 5 times its fourth part: what is that number? Ans. 28.

4. There is a number which exceeds its sixth part as much as 26 exceeds the fourth part of the same number: required the number. Ans. 24.

5. The sum of two numbers is 20, and their difference is 8: what are these numbers? Ans. 14 and 6.

6. At an election, the number of votes given for two candidates was 256; the successful candidate had a majority by 50 votes: how many voted for each of the candidates? Ans. 153 and 103.

7. Divide a line 24 inches long, so that the greater may exceed the smaller part by 10 inches. Ans. 17 and 7.

8. Divide a line of 42 feet in length into two parts, so that the one shall be double the other. Ans. 14 and 28. 9. Divide the number 40 into 3 parts, so that the second may be double of the first, and the third triple the first? Ans. 5, 10, and 15.

10. Divide a line of 60 inches in length into 3 parts, such that the second may be 3 times the first, and the third double the second. Ans. 6, 18, and 36.

11. An uncle divided among three nephews £385. To the first he gave four-fifths of the sum given to the second, and the third received a fifth part of the second's share more than the second: how much did each receive?

Ans. 100, 125, and 160. 12. The sum of two numbers is 60, and the less is to the greater as 5 to 7: what are these numbers?

Ans. 25 and 35. 13. The sum of £154 was paid in half sovereigns, half crowns, and groats, and an equal number of each of these coins was used: what was the number? Ans. 240. 14. The sum of two numbers is 10, and their product is equal to the excess of 60 above the square of the greater: required the numbers? Ans. 4 and 6. 15. Divide the number 20 into two parts, so that 5 times the greater may exceed 6 times the less by 12.

Ans. 12 and 8. 16. A cistern was found to be three-fourths full of water, and after running out 220 gallons, it was then found to be one-fifth full: how many gallons could it hold? Ans. 400. 17. A cistern supplied by 3 pipes can be filled by the first in 10 minutes, by the second in 12, and by the third in 15 minutes: in what time would it be filled by the three running at the same time? Ans. 4 minutes.

18. A cistern can be filled by one pipe in 16 minutes, and emptied by another in 20 minutes: supposing it at first empty, in what time would it be filled when both pipes are running? Ans. 80 minutes. 19. A person has a certain number of shillings in each hand, and if he takes 8 from the left, and puts them in his

right hand, he would then have 4 times as many in his right hand as in his left; but at first he had 5 more in his right hand than in his left: how many had he in each at first? Ans. 15 and 20. 20. The hour and minute hands of a watch are observed to coincide between 4 and 5 o'clock; how many minutes is it after 4?

7

Ans. 21 minutes 43 seconds.
Π

21. Two persons depart at, the same time from London and Edinburgh, and travel till they meet; the one goes 20 miles a-day and the other 30: in how many days will they meet, the distance being 400 miles? Ans. 8 days.

22. Two persons A and B depart from the same place to go in the same direction; B travels at the rate of two miles an hour and A 3 miles, but B has the start of A by 5 hours: in how many hours will A overtake B?

Ans. 10 hours.

23. Two persons A and B depart at the same time from the same place, to travel in the same direction round an island 36 miles in circumference. A travels 3 miles an hour, and B 24: after how many hours will they come together? Ans. 72 hours. 24. Divide £4400 among three persons, so that the first may have three-fifths of the second's share, and the second three-fourths of the third's share.

Ans. £900, £1500, and £2000. 25. A and B engage in trade with the same capital; A gains 160 pounds, and B loses 190, and A's money is now 8 times B's: with how much money did they begin?

Ans. £240. 26. Divide a number a into two parts, so that 6 times the greater shall exceed c times the less by d.

ab- d

ac + d

The less

the greater

b+c

b + c

ON NEGATIVE SOLUTIONS OF SIMPLE EQUATIONS.

(251.) Sometimes the value of an unknown quantity in an equation is found to be negative; and it will be necessary to consider the meaning of such a solution. It would appear at first to be absurd, and, in its literal sense, it is so; but a proper interpretation may be easily determined, by means of the conventional use of the negative sign (44.) To illustrate this circumstance, the following example may be taken :

(252.) Let it be required to find a number to which if a given number c be added, the sum shall be a given number a

If

the required number,

then + c = a

So long as a exceeds c, the value of x is positive; but let a be less than c, then a becomes negative; thus,

.8=2;

.4.

-=6

let a 10, c = 8, then x 10 let a = 5, c = 9, then x = 5 In this second case, a has a negative value, which implies, that, instead of adding c to x, x must be taken from c to give a; and hence the question, in its literal sense, is impossible with these values of a and c, or generally when ca. This sign, therefore, apprises us of the necessity of modifying the condition of the question, in order to adapt it to the case of ac. For such a case, the condition, to be literally fulfilled, must be expressed thus :

Required a number such that if it be subtracted from a given number c, the remainder shall be another given number a.

This question will be solved for the case of a = = 5, c = 9, 4, for then 9-4-5; and the general equation will be

if

Instead, however, of making two distinct questions for the

two cases of ca and ca, one question and one solution will serve for both, by considering that the subtracting of a positive quantity is equivalent to adding a negative one (56), or that, in this case, the subtracting of ⇒ 4 in this question is the same as adding x=- -4 in the preceding one when ca; therefore, with this understanding, the solution of the former question

x = c a

comprehends also the solution of the latter. Thus, when a=5, c = 9, and x = a—c=—

- 4,

if c be added to a, by the ordinary rule of algebraical addition, the result is

(253.) In order to explain more fully the meaning of negative solutions, let the equation

ax + d = cx + b

be given. From this equation,

x=

b-d

ac

Several cases requiring some consideration will occur according to the relative values of the given quantities a, b, c, d. I. Let b―d and ac, then a is positive.

II. If while one of the quantities (bd) and (ac) is positive, the other be negative, then the value of x is negative, and the equation becomes

ax+d=

cx + b

1st, When bd and ac, (bd) is negative, and (ac) positive, and

In order that the value of a may be positive, the terms in the given equation containing x are therefore in this case to be made negative, which is just subtracting ax and ca; but subtracting these is just adding, by the rule of addition, -ax and ca, or a and c multiplied by —x

gative, because bd is so, since bd. But this is just