QUEST. 6. A labourer engaged to serve for 30 days on these conditions: that for every day he worked, he was to receive 20d, but for every day he played, or was absent, he was to forfeit 10d. Now at the end of the time he had to receive just 20 shillings, or 240 pence. It is required to find how many days he worked, and how many he was idle?

Let be the days worked, and y the days idle. Then 20x is the pence earned, and 10y the forfeits ;

= 30,
10y= 240;

and 20x

The 1st mult. by 10, gives 10x + 10y=300;

These two added give

This div. by 30, gives

Hence

30x = 540;

x = 18, the days worked;
x= 12, the days idled.

QUEST. 7. Out of a cask of wine, which had leaked away, 30 gallons were drawn; and then, being gaged, it appeared to be half full; how much did it hold?

Let it be supposed to have held x gallons,

Then it would have leaked 1 gallons,

Conseq. there had been taken away +30 gallons.
30 by the question.

Hence =

Then mult. by 4, gives 2x = x + 120;

And transposing x, gives x = 120 the contents.

QUEST. 8. To divide 20 into two such parts, that 3 times the one part added to 5 times the other may make 76.

QUEST. 9. A market woman bought in a certain number of eggs at 2 a penny, and as many more at 3 a penny, and sold them all out again at the rate of 5 for two-pence, and by so doing, contrary to expectation, found she lost 3d.; what number of eggs had she?

Let x

number of eggs of each sort. Then will x = cost of the first sort, cost of the second sort ;

And

But

But 5 2
Hence x
Then by the question
Mult. by 2, gives
And mult. by 3, gives 5x

+ — fx = 3; x + 2x 3x =

6;

Also mult. by 5, gives x = 90, the number of eggs of each sort.

QUEST. 10. Two persons, A and B, engage at play. Before they begin, A has 80 guineas, and в has 60. After a certain number of games won and lost between them, a rises with three times as many guineas as B. Query, how many guineas

did a win of B?

Let x denote the number of guineas a won.
Then A rises with 80 + x,

1. To determine two numbers such, that their difference may be 4, and the difference of their squares 64.

Ans. 6 and 10.

2. To find two numbers with these conditions, viz. that half the first with a 3d part of the second may make 9, and that a 4th part of the first with a fifth part of the second may Ans. 8 and 15.

make 5.

3. To divide the number 20 into two such parts, that a 3d of the one part added to a fifth of the other, may make 6. Ans. 15 and 5.

4. To find three numbers such, that the sum of the 1st and 2d shall be 7, the sum of the 1st and 3d 8, and the sum of the 2d and 3d 9. Ans. 3, 4, 5.

5. A father, dying, bequeathed his fortune, which was 2800l. to his son and daughter, in this manner; that for every half crown the son might have, the daughter was to have a shilling. What then were their two shares ?

Ans. The son 2000l. and the daughter 800/

6. Three persons, A, B. c, make a joint contribution,

which in the whole amounts to 4001, of which sum в con

:

tributes

tributes twice as much as A and 201. more; and c as much as A and B together. What sum did each contribute?

Ans. A 601. B 140/. and c 2001.

7. A person paid a bill of 1007, with half guineas and crowns, using in all 202 pieces; how many pieces were there of each sort? Ans. 180 half guineas, and 22 crowns.

8. Says A to B, if you give me 10 guineas of your money, I shall then have twice as much as you will have left; but says B to A, give me 10 of your guineas, and then i shall have 3 times as many as you. How many had each ?

Ans. A 22, B 26.

9. A person goes to a tavern with a certain quantity of money in his pocket, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern, be there spend 2 shillings also; then borrowing again as much money as was left, he went to a third tavern, where likewise he spent 2 shillings; and thus repeating the same at a fourth tavern, he then had nothing remaining. What sum had he at first? Ans. 3s. 9d.

10. A man with his wife and child dine together at an inn. The landlord charged 1 shilling for the child; and for the woman he charged as much as for the child and as much as for the man; and for the man he charged as much as for the woman and child together. How much was that for each ?

Ans. The woman 20d. and the man 32d.

11. A cask, which held 60 gallons, was filled with a mixture of brandy, wine, and cyder, in this manner, viz. the cyder was 6 gallons more than the brandy, and the wine was as much as the cyder and of the brandy. How much was there of each. Ans. Brandy 15, cyder 21, wine 24.

12. A general, disposing his army into a square form, finds that he has 284 men more than a perfect square; but increas ing the side by 1 man, he then wants 25 men to be a complete square. Then how many men had he under his command ? Ans. 24000.

13. What number is that, to which if 3, 5, and 8, be severally added, the three sums shall be in geometrical progression? Ans. 1.

14 The stock of three traders amounted to 8601. the shares of the first and second exceeded that of the third

by

by 240; and the sum of the 2d and 3d exceeded the first by 360. What was the share of each ?

Ans. The 1st 200, the 2d 300, the 3d 260.

15. What two numbers are those, which, being in the ratio of 3 to 4, their product is equal to 12 times their sum? Ans. 21 and 28.

16. A certain company at a tavern, when they came to settle their reckoning, found that had there been 4 more in company, they might have paid a shilling a-piece less than they did; but that if there had been 3 fewer in company, they must have paid a shilling a-piece more than they did. What then was the number of persons in company, what each paid, and what was the whole reckoning?

Ans. 24 persons, each paid 7s, and the whole reckoning 8 guineas.

:

17. A jocky has two horses and also two saddles, the one valued at 18/ the other at 31. Now when he sets the better saddle on the 1st horse, and the worse on the 2d, it makes the first horse worth double the 2d but when he places the better saddle on the 2d horse, and the worse on the first, it makes the 2d horse worth three times the 1st. What then were the values of the two horses? Ans. The 1st 61. and the 2d 9l.

18. What two numbers are as 2 to 3, to each of which if 6 be added, the sums will be as 4 to 5 ? Ans. 6 and 9.

19. What are those two numbers, of which the greater is to the less as their sum is to 20, and as their difference is to 10 ? Ans. 15 and 45.

20. What two numbers are those, whose difference, sum, and product, are to each other, as the three numbers 2, 3, 5? Ans. 2 and 10.

21. To find three numbers in arithmetical progression, of which the first is to the third as 5 to 9, and the sum of all three is 63 ? Ans. 15, 21, 27.

22. It is required to divide the number 24 into two such parts, that the quotient of the greater part divided by the less, may be to the quotient of the less part divided by the greater, as 4 Ans. 16 and 8.

to 1.

23. A gentleman being asked the age of his two sons, answered, that if to the sum of their ages 18 be added, the result will be double the age of the elder'; but if 6 be

taken

taken from the difference of their ages, the remainder will be equal to the age of the younger. What then were their ages?

Ans. 30 and 12.

24. To find four numbers such, that the sum of the 1st, 2d, and 3d, shall be 13; the sum of the 1st, 2d, and 4th, 15; the sum of the 1st, 3d, and 4th, 18; and lastly the sum of the 2d, 3d, and 4th, 20. Ans. 2, 4, 7, 9. 25. To divide 48 into 4 such parts,' that the 1st increased by 3, the second diminished by 3, the third multiplied by 3, and the 4th divided by 3, may be all equal to each other. Ans. 6, 12, 3, 27.

QUADRATIC EQUATIONS.

QUADRATIC Equations are either simple or compound. A simple quadratic equation, is that which involves the square of the unknown quantity only. As ax2 = b. And the solution of such quadratics has been already given in simple equations.

A compound quadratic equation, is that which contains the square of the unknown quantity in one term, and the first power in another term. As ax + bx = c.

All compound quadratic equations, after being properly reduced, fall under the three following forms, to which they must always be reduced by preparing them for solution.

1. x2 + ax = b

2. x2 — ax =
3. x2

b

_b.

The general method of solving quadratic equations, is by what is called completing the square, which is as follows:

1. REDUCE the proposed equation to a proper simple form, as usual, such as the forms above; namely, by transposing all the terms which contain the unknown quantity to one side of the equation, and the known terms to the other; placing the square term first, and the single power second; dividing the equation by the co-efficient of the square or first term, if it has one, and changing the signs of all the terms, when that term happens to be negative, as that term must always be made positive before the solution. Then the proper solution is by completing the square as follows, 2. Complete,

viz.

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