10. A person dying leaves half of his property to his wife, onesixth to each of two daughters, one-twelfth to a servant, and the remaining $600 to the poor: what was the amount of his property? Represent the amount of the property by x.

XC

2

+ ·+ +600-x the amount of the property,

3 12

which gives x=$7200.

11. A father leaves his property, amounting to $2500, to four sons, A, B, C and D. C is to have $360, B as much as C and D together, and A twice as much as B less $1000: how much does A, B and D receive? Ans. A $760, B $880, D $520.

12. An estate of $7500 is to be divided between a widow, two sons, and three daughters, so that each son shall receive twice as much as each daughter, and the widow herself $500 more than all

the children: what was her share, and what the share of each child? Widow's share $4000. Each son $1000.

Ans.

Each daughter $500.

13. A company of 180 persons consists of men, women, and children. The men are 8 more in number than the women, and the children 20 more than the men and women together: how many of each sort in the company?

Ans. 44 men, 36 women, 100 children.

14. A father divides $2000 among five sons, so that each elder should receive $40 more than his next younger brother: what is the share of the youngest? Ans. $320.

15. A purse of $2850 is to be divided among three persons, A, B, and C ; A's share is to be to B's as 6 to 11, and C is to have $300 more than A and B together: what is each one's share? Ans. A's $450, B's $825, C's $1575.

16. Two pedestrians start from the same point, the first steps twice as far as the second, but the second makes 5 steps while the first makes but one. At the end of a certain time they are 300 feet apart. Now, allowing each of the longer paces to be 3 feet, how far will each have travelled?

Ans. 1st, 200 feet; 2nd, 500.

17. Two carpenters, 24 journeymen and 8 apprentices, received at the end of a certain time $144. The carpenters received $1 per day, each journeyman half a dollar, and each apprentice 25 cents: how many days were they employed? Ans. 9 days.

18. A capitalist receives a yearly income of $2940: four-fifths of his money bears an interest of 4 per cent, and the remainder of 5 per cent: how much has he at interest? Ans. 70000.

19. A cistern containing 60 gallons of water has three unequal cocks for discharging it; the largest will empty it in one hour, the second in two hours, and the third in three: in what time will the cistern be emptied if they all run together? Ans. 32 min.

20. In a certain orchard are apple trees, peach trees, plum trees, 120 cherry trees, and 80 pear trees: how many trees in the orchard? Ans. 2400.

21. A farmer being asked how many sheep he had, answered that he had them in five fields; in the 1st he had 1, in the 2nd †, in the 3rd, in the 4th, and in the 5th 450: how many had he? Ans. 1200.

22. My horse and saddle together are worth $132, and the horse is worth ten times as much as the saddle: what is the value of the horse? Ans. $120. 23. The rent of an estate is this year 8 per cent greater than it was last. This year it is $1890: what was it last year?

Ans. $1750.

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

Ans. 65.

25. A post is in the mud, in the water, and ten feet above the water what is the whole length of the post? Ans. 24 feet. 26. After paying and of my money, I had 66 guincas left in my purse: how many guineas were in it at first?

Ans. 120.

27. A person was desirous of giving 3 pence apiece to some beggars, but found he had not money enough in his pocket by 8 pence he therefore gave them each 2 pence and had 3 pence remaining required the number of beggars.

Ans. 11.

28. A person in play lost of his money, and then won 3 shillings; after which he lost of what he then had; and this done, found that he had but 12 shillings remaining: what had he at first? Ans. 20s.

29. 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.

:

30. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings; he then borrows as much mo

ney as he had left, and going to another tavern, he there spends 2 shillings also; then borrowing again as much money as was left, he went to a third tavern, where likewise he spent two shillings and borrowed as much as he had left; and again spending 2 shillings at a fourth tavern, he then had nothing remaining. What had he at first? Ans. 3s. 9d.

Of Equations of the First Degree involving two or more

unknown quantities.

95. Although several of the questions hitherto resolved, contained in their enunciation more than one unknown quantity, we have resolved them by employing but one symbol. The reason of this is, that we have been able, from the conditions of the enunciation, to express easily the other unknown quantities by means of this symbol; but this is not the case in all problems containing more than one unknown quantity.

To ascertain how problems of this kind are resolved: first, take some of those which have been resolved by means of one unknown quantity.

1. Given the sum a, of two numbers, and their difference b, it is required to find these numbers.

Let x= the greater, and y the less number.

Then by the conditions

x+y=a.

x-y=b.

2x=a+b.

2y-a-b.

Each of these equations contains but one unknown quantity.

For a second example, let us also take a problem that has been already solved.

2. A person engaged a workman for 48 days. For each day that he labored he was to receive 24 cents, and for each day that he was idle he was to pay 12 cents for his board. At the end of the 48 days, the account was settled, when the laborer received 504 cents. Required the number of working days and the number of days he was idle.

It has already been shown that the two members of an equation can be multiplied by the same number, without destroying the equality; therefore the two members of the first equation may be multiplied by b, the co-efficient of y in the second, and we have

In like manner, multiplying the two members of the first equation by a, the co-efficient of x in the second, it becomes