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 60%. B 140l. and c 2001.

7. A person paid a bill of 100%. 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 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, 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 2 shillings; and thus repeating the same at a fourth tavern, he then had nothing remaining. What sum had he at first? Ans. 38. 9d.

10. A man with his wife and child dine together at an inn. The landlord charged I 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 increasing 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 860. the shares of the first and second exceeded that of the third

by

by 210; and the sum of the 2d and 3d exceeded the first by 260. 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 jockey has two horses: and also two saddles, the one valued at 187. 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 67. and the 2d 91.

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, Ans. 16 and 8.

as 4 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 first 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 ax2 + 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.

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, viz.

4. Compoundedly, a: a+ar:: b: b+br; or 2: 8:: 4:16. 5. Dividedly, a : ar — a : : b: br b 6. Mixed,ar+a: ara :: br+b: br

; or 2:4:: 4 : 8.

b; or 8:4:: 16:8.

7. Multiplication, ac : are: : be: bre; or 2.3: 6.3 :: 4:12.

a ar

8. Division, -:-::b: br; or 1: 3 :: 4:12.

1. Given the first term of a geometrical series 1, the ratio 2, and the number of terms 12; to find the sum of the series? X 211 = 1 x 2048, is the last term.

First,

2. Given the first term of a geometric series, the ratio I , and the number of terms 8; to find the sum of the series? = 3 × 15 = 3, is the last term.

First, × (4)

255

Then, ( × 1) ÷ (1 − 1) = (-7) ÷ ÷ = 28 × 7 =3, the sum required.

3. Required the sum of 12 terms of the series 1, 8, 9, 27, 31, &c. Ans. 265720.

177117

4. Required the sum of 12 terms of the series 1, 3, 77, , &c. Ans. 265720 5. Required the sum of 100 terms of the series 1, 2, 4, 8, 16, 32, &c. Ans. 1267650600228229401496703205375. See more of Geometrical Proportion in the Arithmetic.

SIMPLE EQUATIONS.

AN Equation is the expression of two equal quantities, with the sign of equality (=) placed between them. Thus, 10- 4 6 is an equation, denoting the equality of the quantities 10 .4 and 6.

Equations

Equations are either simple or compound. A Simple Equation, is that which contains only one power of the unknown quantity, without including different powers. Thus, x-α= b+c, or ax2 = b, is a simple equation, containing only one power of the unknown quantity x. But x2 2ax 6 is a compound one.

GENERAL RULE.

Reduction of Equations, is the finding the value of the unknown quantity. And this consists in disengaging that quantity from the known ones; or in ordering the equation so, that the unknown letter or quantity may stand alone on one side of the equation, or of the mark of equality, without a co-efficient: and all the rest, or the known quantities, on the other side.-In general, the unknown quantity is disengaged from the known ones, by performing always the reverse operations. So, if the known quantities are connected with it by + or addition, they must be subtracted; if by minus (-), or subtraction, they must be added; if by multiplication, we must divide by them; if by division, we must multiply; when it is in any power, we must extract the root; and when in any radical, we must raise it to the power. As in the following particular rules; which are founded on the general principle of performing equal operations on equal quantities; in which case it is evident that the results must still be equal, whether by equal additions, or subtractions, or multiplications, or divisions, or roots, or powers.

PARTICULAR RULE I.

WHEN known quantities are connected with the unknown by or; transpose them to the other side of the equation, and change their signs. Which is only adding or subtracting the same quantities on both sides, in order to get all the unknown terms on one side of the equation, and all the known ones on the other side*.

Thus,

Here it is earnestly recommended that the pupil be accustomed, at every line or step in the reduction of the equations, to name the particular operation to be performed on the equation in the last line, in order to produce the next form or state of the equation, in applying each of these rules, according as the particular form of the equation may require; applying them according to the

order