tributes twice as much as a and 207. more; and c as much as A and B together. What sum did each contribute? Ans. A 601. B 140l. and c 2001.

7. A person paid a bill of 100l. 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, 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 78. and the whole reckoning 8 guineas.

17. A jockey has two horses: and also two saddles, the one valued at 18. the other at 3. 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 6/. 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.

VOL. I.

Kk

2. Complete

express that time, and R must be taken the amount for that time also.

Note 2. When the compound interest, or amount, of any sum, is required for the parts of a year; it may be determined in the following manner :

1st, For any time which is some aliquot part of a year :— Find the amount of 17. for 1 year, as before; then that root of it which is denoted by the aliquot part, will be the amount of 1. This amount being multiplied by the principal sum, will produce the amount of the given sum as required.

2d, When the time is not an aliquot part of a year:Reduce the time into days, and take the 365th root of the amount of 17. for 1 year, which will give the amount of the same for 1 day. Then raise this amount to that power whose index is equal to the number of days, and it will be the amount for that time. Which amount being multiplied by the principal sum, will produce the amount of that sum as before. And in these calculations, the operation by logarithms will be very useful.

OF ANNUITIES.

ANNUITY is a term used for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c. payable from time to time, but mostly by annual payments.

Annuities are divided into those that are in Possession, and those in Reversion: the former meaning such as have commenced; and the latter such as will not begin till some particular event has happened, or till after some certain time has elapsed. ·

When an annuity is forborn for some years, or the payments not made for that time, the annuity is said to be in Arrears.

An annuity may also be for a certain number of years; or it may be without any limit, and then it is called a Perpetuity.

The Amount of an annuity, forborn for any number of years, is the sum arising from the addition of all the annuities for that number of years, together with the interest due upon each after it becomes due.

The

The Present Worth or Value of an annuity, is the price or sum which ought to be given for it, supposing it to be bought off, or paid all at once.

Let a =

the annuity, pension, or yearly rent;

n = the number of years forborn, or lent for ;
R= the amount of 17. for 1 year;

m = the amount of the annuity;

v = its value, or its present worth.

Now, 1 being the present value of the sum R, by proportion the present value of any other sum a, is thus found:

the present value of a due 1 year hence.

continued to n terms, will be the present value of all the n And the value of the perpetuity, is the sum

years' annuities.

of the series to infinity.

But this series, it is evident, is a geometrical progression,

having

I

R

both for its first term and common ratio, and the

number of its terms n; therefore the sum v of all the terms, or the present value of all the annual payments, will be 1 1 1

When the annuity is a perpetuity; n being infinite, ¤1 is