the number of Irish members. What are the respective numbers? Ans. 496, 105, and 53. H. Four persons engaged in a speculation requiring an outlay of £2400. Of that sum, A contributed twice as much as B, and £10 more; C contributed £20 less than A; and D only two-thirds of C. What sums did they separately contribute? 9. Two boys, who made their living by selling nuts, commenced the week with the same sum; but, when they met on the Saturday night, the one found that, after paying for his maintenance, he had gained a half-crown; and the other that he had lost 1s. 6d. The consequence was, that the former had now three times as much as the latter. What did they begin the week with? Ans. 3s. 6d. each. I. Two fields were purchased at £43 and £28 per acre respectively. Their united area was 12 acres, and their united price £411. What were the sizes of the fields separately? 10. A gentleman, after travelling 12 hours without stopping, found that, if he had travelled 3 miles an hour faster, he would have accomplished the journey in two hours less of time. At what rate did he travel? Ans. 15 miles per hour. J. Two friends, living at Walton and Middleton, 24 miles apart, agreed to meet on an angling excursion between the two places. The one from Walton set out an hour after the other, but, having no encumbrance, got on at the rate of 4 miles an hour,—while his friend, having undertaken to bring fishing-tackle and provisions for both, could only proceed at three-fourths of that speed. At what point on the road did they meet? K. A man departs on a journey, walking at the uniform rate of 3 miles an hour; and, two hours later, another sets out after him, riding, at 7 miles an hour. At what distance on the road will the latter overtake the former? 11. If you divide a certain number by 9, and add together divisor, dividend, and quotient, their sum shall be 59. The number is required. Ans. 45. L. Divide the number 100 into four such parts that the first may be equal to half the second, but greater than the third by 6, and less than the fourth by 1. 12. Find three numbers, such that their sums, taken two by two, shall be 11, 12, and 13. Ans. 5, 6, and 7. M. A man and his wife were married at the respective ages of 40 and 20 years. How old will the man be when his wife's age becomes three-fourths of his own? 13. A certain reading-room is maintained at the joint expense of the subscribers. At the end of the first year they find they have each £1, 4s. to pay; but one of them remarked correctly that, if there had been seven more of them, the expenditure would not have been increased, and they would have got off at the rate of a sovereign a piece. Compute the number of members. Ans. 35. N. Half the difference of two numbers added to half the sum is 93.5, and half the difference taken from half the sum is 77-4. What are the two numbers? 14. What are the two numbers, whose sum is 133, and quotient 18? Ans. 126 and 7. O. Find two numbers whose difference shall be 1, and the difference of their squares 19. P. Divide 100 into two such parts that the difference of their squares may be 200. Q. The difference of two numbers is 40, and the difference of their square roots 2. What are the numbers? 15. A lady, on her birth-day, presented all the wives and children of her cottagers with donations of money. She gave the women a sovereign each, and the children a half-crown. Having done so, she found that she had bestowed 200 donations, and expended £58, 5s. What were the respective numbers of women and children? Ans. 38 and 162. R. Two servants lived together 40 years in the service of the same master, and at the same wages,-viz., £12 per annum. The one spent 10 shillings less every year than the other, and found at the end of the 40 years, that she had saved twice as much as her fellow-servant, and halfa-year's wages besides. What did they severally save in the year? 16. A broker bought two cabinets at a sale, suspecting that a £10 note was hid in one of the two. If in the one, he thought, it should be worth twice the other; but if in the other, it ought to be three times the value of the first. What values did he put upon the cabinets without the note? Ans. £6 and £8. S. Two foolish young gentlemen began to play against each other with equal sums of money. After playing 12 hours, and incurring 15 shillings each of expenses, the one remained with twice as much money as the other, having gained from him £15. What had each when they began? 17. A lady's ring cost 7 guineas; her brooch cost as much as her ring and half her bracelets, and the bracelets cost as much as the ring and brooch together. What was the cost of the whole? Ans. 56 guineas. T. On examining a meteorological table for a particular locality, it was found that the whole fall of rain for the year was 22.8 inches,-that the quantity which fell in the first and last quarters together was just equal to that which fell in the second and third,-that the fall in the first quarter was two-thirds of that in the fourth, and that the third exceeded the second by 2 inches. What were the depths of fall in the respective quarters? 18. A tea-dealer considers that 2 lb. of his best green tea is worth three of his best black. He mixes them together in equal proportions, and sells the mixture at 6s. 3d. per pound. At what rate would he sell the black and green separately? Ans. At 5s. and 7s. 6d. U. A lady, going to purchase a carpet, found, that if she took the best quality in the shop, at 8 shillings the yard, it would take £1 more than the money she brought with her. She therefore contented herself with the second, at 7s. 6d., and found that she had then £2 left. What did the carpet cost? 19. There are three numbers such, that the first, with half the sum of the other two, makes 46; the second, with one-third of the other two, makes 30; 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? with? 20. Two cousins, John and George, go to school at the beginning of the year with 5 guineas a-piece of pocketmoney. 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 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 hear ing 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? 21. 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 powderflask 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? 22. 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? 23. An itinerant orange-vender bought a quantity of oranges for sale, at the rate of 5 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 orange-vender, 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. Z. Two brothers went from their house to the neighbouring town, taking different roads. The two roads were of equal lengths, but the one was over level ground and the other over a hill,-so that the elder brother, who travelled along the level road, went at the uniform rate of 3 miles an hour; while the younger could advance at only two thirds of that rate for the first half of the way, thinking to make up for it by doubling his speed the other half. It turned out, however, although he did double his speed, that he was ten minutes longer in reaching his destination. What was the distance of the house from the town? CHAPTER XV. PROPORTION. DEFINITIONS. 1. When the first of four quantities contains the second as often as the third contains the fourth, the four quan tities are said to be Proportional. Thus, since 16 12 = we 3 then abc: d. NOTE. When we speak of the first containing the second as often as the third contains the fourth, we mean that the quotients are equal, whether they be integers, fractions, or mixed numbers, or even if they be roots or other expressions not convertible at all into definite numbers. Thus, Also 2: 20 :: 3:45, 3 : 4 : : 9 12, because 3 9 4 12 although we cannot express either 2÷√20 or 3 ÷ √ 45 exactly, in numbers, at all. 2. When four quantities are proportional, the first is Isaid to have to the second the same Ratio which the third has to the fourth. A proportion, therefore, expressing the equality of two ratios, is sometimes written thus, |