4. A person has two horses, and a saddle worth £10, if the saddle be put on the first horse, his value becomes double that of the second; but if the saddle be put on the second horse, his value will not amount to that of the first horse by £13. Find the value of each horse.

5. There are two numbers such that the greater added to the less is 13; and if the less be taken from the greater, the remainder is nothing; find the numbers.

6. There is a certain number, to the sum of whose digits if you add 7, the result will be three times the left-hand digit; and if from the number itself you subtract 18, the digits will be inverted. Find the number.

7. A vessel containing 120 gallons is filled in 10 minutes by 2 spouts running successively, the one runs 14 gallons in a minute, and the other 9 gallons in a minute. For what time had each spout run?

8. A farmer has oxen worth £12. 10s. each, and sheep worth £2. 58. each; the number of oxen and sheep being 35, and their value £191. 10s.; find the number he had of them.

9. A person buys oxen at £13, and calves at £5 per head, and spends £144; if he had purchased as many oxen as he did actually purchase calves, and vice versa, he would have spent £288. How many of each did he buy?

10. Two clocks are together at 12, one loses two seconds while the other gains three, in twelve hours one is four minutes before the other; what is the time indicated by each clock?

11. Find two numbers such that if the first be added to 4 times the second, the sum is 29; and if the second be added to 6 times the first, the sum is 36.

12. A farmer wishing to purchase a number of sheep found that if they cost him £2. 2s. a head, he would not have money enough by £1. 8s. ; but if they cost him £2 a head, he would then have £2 more than he required; find the number of sheep, and the money which he had.

13. Divide 30 into two parts such that a certain multiple of the first part plus the second part is equal to 50, and the same multiple of the second part plus the first part is equal to 70.

14. Required two numbers such that three times the greater exceeds twice the less by 10, and twice the greater together with three times the less is 24.

15. Find a number consisting of two digits such that the digit in the unit's place is twice the other; if 6 be subtracted from double the number, the digits will be inverted.

16. A and B lay a wager of 10s. ; if A loses he will have twice as much, less 35s., as B will then have; but if B loses he will have ths of what A will then have; how much had each of them at first?

17. If A's money be increased by 36s., he will have 3 times as much as B; but if B's money be diminished by 5s., he will have half as much as A; find the sum possessed by each.

18. A man has to travel a certain distance; when he has travelled 40 miles, he increases his speed 2 miles per hour; if he had travelled with his increased speed during the whole of his journey, he would have arrived 40 minutes earlier; but if he had continued at his original speed, he would have arrived 20 minutes later; how far had he to travel?

19. A cask A contains 12 gallons of wine and 18 gallons of water; and another cask B contains 9 gallons of wine and 3 gallons of water; how many gallons must be drawn from each cask so as to produce by their mixture 7 gallons of wine and 7 gallons of water?

20. In a certain examination the difference between the number of those admitted to Honors and those sent down to the Poll is twenty times as great as the number rejected; the number admitted to Honors equals five times the number sent to the Poll, eight times the number rejected, and four over; and, lastly, if five of those rejected had been sent down to the Poll, then four times the number sent down would have exceeded half the number admitted to Honors by ten times the number rejected. Find the whole number of men examined.

21. Two plugs are opened in the bottom of a cistern containing 192 gallons of water; after 3 hours one of them becomes stopped, and the cistern is emptied by the other in 11 hours; had 6 hours occurred before the stoppage, it would have only required 6 hours more to empty it. How many gallons will each plug-hole discharge in an hour, supposing the discharge uniform?

22. What numbers are those whose difference is 20, and the quotient of the greater by the less is 3?

23. A boy spends his money in oranges. If he had received 4 more for his money, they would have averaged a half-penny each less; if 3 less, a half-penny each more. How much did he spend?

24. A vintner would mix wine at 10s. a gallon with another sort at 6s. a gallon, to make 100 gallons to be sold at 7s. a gallon; how much of each sort must he take?

25. A man invested 2s. 6d. in apples and pears, buying the apples at 4 a penny and the pears at 5 a penny; he sold half his apples and onethird of his pears for 13d., which was at the rate at which he bought them. How many did he buy of each sort?

26. There is a number consisting of two figures, and it is equal to 4 times the sum of its digits; if 18 be added to the number, the digits will be inverted. Find the number.

27. A certain sum of money is to be divided among a certain number of men; if there were 3 men less, each man would have £150 more; but if there were 6 men more, each man would have £120 less. Find the sum of money and the number of men.

28. A person has £27. 6s. in guineas and crown-pieces; out of which he pays a debt of £14. 178., and finds he has exactly as many guineas left as he has paid away crowns; and as many crowns as he has paid away guineas. How many of each had he at first?

Simultaneous Equations involving three unknown quantities.

108. If we have three independent equations, involving three unknown quantities, for instance,

ax+by+cz=m (1),

dx+ey+fz=n (2),

gx+hy+kz=p (3).

The values of x, y, and z may be determined thus:

Again,

adx+bdy+cdz=dm, multiplying equation (1) by d,
adx+aey+afz=an,

(2) by a ;

.. (bd-ae) y + (cd—af) z=dm—an, (4).

dgx + egy +fgz=gn, multiplying equation (2) by g,
dg+dhy+d=dp,

(3) by d;

.. (eg-dh) y+(fg-dk)z=gn-dp, (5).

from the equations (4) and (5) the values of y and z may be determined; and then by substituting their values so obtained in any of the given equations the value of a will be determined.

This method may be extended to equations containing four or more unknown quantities.

Ex. XLIX.

Example worked out.

4x-2y+5x=18 .......... (1)

x. 2x+4y-3x=22..... (2) find the values of x, y, and z.

6x+7y- z=63. (3)

om equation (1)

(2)

......

4x-2y+5x=18,

4x+8y-6x=44;

7. Find three numbers, such, that the first with the sum of the second and third shall be 120; the second with the difference of the third and first shall be 70; and the sum of the three numbers shall be 95.

QUADRATIC EQUATIONS.

109. QUADRATIC EQUATIONS are those into which the second power of the unknown quantity enters, with or without the first power.

If the second power of the unknown quantity alone enters, such equations are called PURE QUADRATIC EQUATIONS; thus, x2=49, and x2—a=6, are pure quadratic equations.

If the first power as well as the second power of the unknown quantity be involved, such equations are called ADFECTED QUADRATIC EQUATIONS; thus 2+3x=1, and ax2+ bx=c, are adfected quadratic equations.

Solution of Pure Quadratic Equations, and others easily reducible to

Simple Equations.

110. Pure quadratic equations are solved in the same manner, in every respect, as simple equations, except that, at the conclusion, the square root of each side of the equation has to be taken.

are prefixed to the root, because the square root

of a quantity may be either positive or negative.

Also the square root of 2 may itself be +x, or -x; but still x has only two values: thus, for instance, in the example

But in fact, equation (2) gives x = 8, that is, the values of x are +8 and -8 as given by equation (1), so that (1) and (2) only give two

values of x.

Ex. 1.

Ex. L.

Examples worked out.

3x2-4=71, find the values of x.

or 3x2=71+4=75,

or a2=35=25;

.. x=±5.