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and the third, with one-fourth of the first and second, amounts to 29. Calculate the numbers. Ans. 30,

14, and 18.

V. A boy bought 90 apples and pears for 2s. 4d., having got 3 apples for a penny, and 7 pears for twopence. How many had he of each kind?

53. Two cousins, John and George, go to school at the beginning of the year with 5 guineas a-piece of pocket money. At the end of the year John finds that he has spent twice as much as George, and 5 shillings more, and that he has remaining 5 shillings less than half of what George still possesses. How much does each return home with? Ans. £1 10s. and £3 10s.

W. Mr. Johnson, Mr. Thomson, and Mr. Wilson, being one day in company, were requested to contribute to a charitable purpose. Mr. Thomson said, that if Mr. Johnson subscribed more than £50, he would give, for his subscription, three times as much as all that Mr. Johnson's exceeded that sum; while Mr. Wilson offered to table four times as much as Mr. Thomson's exceeded £50. On hearing this, Mr. Johnson, resolving to strain the liberality of his friends, went to the utmost extent of his ability; and Mrs. Ellison, who requested their contributions, had the pleasure of departing with £730. What was Mr. Johnson's subscription?

54. A gentleman left £8400 to be divided among his four nephews in certain portions named, and directed his house to be sold by auction. As the four nephews were returning together from the sale, Edward observed, that if his uncle had left him twice as much as he did, he could have bought the house with the sum. "He left me enough," said Richard, "to buy it twice over if I were disposed." "It would have taken Harry's and mine put together to purchase it," replied Alfred. "I could have paid for it with my own share," exclaimed Harry, "if Dick had

been generous enough to add to it the third part of his. What was the house sold for? Ans. £2400.

X. A young sportsman returning unsuccessful from a day's shooting, and meeting his friend with six brace of black game, offered him his gun, shot-belt, and powder flask for 5 brace; or the gun and belt for 4 brace; or the belt and powder flask for 3 brace; or the belt, flask, and 6 guineas for the whole. What values did he put upon the different articles and the game?

55. A son, asking his father how old he was, received the following reply:-"Seven years ago I was four times as old as you; but seven years hence, if you and I live, my age will then be only double yours." It is required, from this information, to satisfy the son's curiosity. Ans. The father's age was 35.

Y. On making up the roll of an army after a battle, it was found that the number of effective men was only 714 more than half the number before the battle. Of the remainder, the wounded were twice as many as the slain, and the prisoners equal to one-third of all that were left for immediate service, while the number of wounded exceeded the number of prisoners by 677. What was the original strength of the army?

56. An itinerant orange-vendor bought a quantity of oranges for sale, at the rate of five for twopence. He then arranged the good and the bad in two separate baskets, containing equal numbers, and sold the one basketful at 3 a penny and the other at 2 a penny. In selling them he met another orangevender, who laughed at his simplicity, and said he would have no profit upon them; but, when he had sold the whole, he found he had gained sixpence. Please to calculate, from these data, how many oranges he bought and sold. Ans. 30 dozen.

ALGEBRA.

QUADRATIC EQUATIONS.

In simple equations we deal only with single or simple powers of the unknown quantities x, y, z, &c. Sometimes, however, we have problems and equations. involving higher powers of x and y, as the square x2, y2; and the cube, as a3, y3; we are then said to deal with quadratic and cubic equations respectively. The same thing is sometimes expressed by saying that quadratic equations involve two dimensions, x and x2, y and y2.

2

Thus in the equation a2+6x=27 we have a quadratic.

Now in the square of +y, which equals x2 + 2xy + y2, we note that the whole quantity x2+2xy + y2, of which the square root +y can be obtained, consists of the square of the square of y+twice the product of x and y.

If, therefore, the quantity a2+2xy alone were given, we could turn this into a square quantity, or one of which the root could be obtained, by the addition of y2.

But this quantity y is the square of half the number which is multiplied into a in 2xy, or y2 = (2)2.

If we call the number 2y which is multiplied into a to make the second term of the quantity the coefficient of x, we get a rule for turning a quantity not square into a square quantity.

Thus in the given example, a2+6x=27, the coefficient of a in 6x=6, and half of this =3, and the square of the half co-efficient =32=9.

If 9 therefore be added to each side of the equation we have equality as before,

2+6x+()2=27+32=36.

But now the quantity on the left hand side of the equation is a square quantity, and therefore the root of it can be obtained. This root is the sum of the roots of the first and last term, x+3.

:.x+3=√36.

But √36=6 or−6.

.x+3=6 or -6.

And a 3 or -9.

The quadratic therefore when solved gives us two possible answers.

The rule for the squaring of a quadratic equation may therefore be thus expressed:

To each side of the equation add the square of half the co-efficient of x, and take the square root. Similarly the root of

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