CXLIV.] RACES. 391 16. A and B run a race; A starts at the rate of 400 yards a minute, and at the end of each minute diminishes his pace by 20 yards a minute. B increases his pace at the same rate, and catches A at the end of six minutes; at what pace did B start? 17. A and B run a race. A starts at the rate of 400 yards a minute, but in each successive minute increases his pace by a yard per minute; B diminishes his pace at the same rate, and is overtaken by A in 4 minutes; what was B's pace at first? 18. In a bicycle race of 5 miles A gives 5 400 yards and C 750 yards start. If their respective speeds be uniform, and as 25: 24: 23, in what order will they reach the goal? 19. A gives B 49 yards start in a race of one mile, and is beaten by 20 yards; how many yards ought A to have given B? Since A is beaten by 20 yards the distance he has run - = 1760 yards - 20 yards = 1740 yards. Now the distance B has run = 1760 yards - 49 yards = 1711 yards. .. A runs 1740 yards whilst B runs 1711 yards; Hence the start that A can give B = (1760 – 1730) = 29 yards. 20. A gives B 4 yards start in a race of 100 yards and is beaten by 1 yard; he gives C a start of 7 yards in a race of the same distance and wins by 1 yard: how much start could B give C in a race of 800 yards in order to just make a dead heat, the rates of running in each case being supposed to be always the same? In the first race B runs a distance of 100-4, that is, 96 yards. .. A runs 99 yards whilst B runs 96 yards; In the second race C has only 93 yards to run, and he is beaten by 1 yard, so that he runs 92 yards only; . A runs 100 yards whilst C runs 92 yards; 23 that is, 300 yard; 759 yard whilst C runs yard; 1 yard B runs 800 yards whilst Cruns 759 yards; 21. A can give B 5 yards and 6 yards in a 100 yards' race; if B give C one yard start in a 100 yards' race, which would win? 22. At a game of billiards A can give B 10 points in a game of 100, and he can give C 15 points in the same number; how many points can B give C in a game of 200? 23. At a game of billiards A can give B 15 points in 50, and he can give C 32 in 80; how many can B give C in a game of 70? FOREIGN EXCHANGES. 411 461. Foreign Exchanges. Suppose that an English merchant, A, sells machinery to a Frenchman, B, in Paris, for which he charges the latter 1000 francs. One way of settling the debt would be for B to send A 1000 francs. But this method of settling the debt would be inconvenient, both because of the expense of actually sending cash, and also because the amount of coin required in the course of trade would be considerably greater than the amount which is actually in existence. Suppose, now, that a second Frenchman, C, sells to a second Englishman, D, silk for which he charges D £40 (=1000 francs). If, then, D were to pay 1000 francs (or its equivalent in English money, say £40) to A, and B were to pay 1000 francs to C, the two transactions would be completed, and each of the four persons, A, B, C, and D, would have that to which he is entitled. The transaction would then stand as follows: The two sales would then have been completed without any transference of money across the Channel. 462. In actual practice it would be inconvenient, and perhaps impossible, for B to find a person, C, who required from London exactly the same sum of money which B`owed to London. This inconvenience would be overcome if the business were performed through a banking house that berbranches both in London and in Paris. In the above case A would draw a bill on B for (=£40, say), and B would accept it. every kind Crown 8vo. 4s. 6d. Key. Crown 8vo. IOS. 6d. ARITHMETIC IN THEORY AND PRACTICE BY J. BROOKSMITH, M.A., LL.B. London B give 23. At a MACMILLAN AND CO., LIMITED give C 32 in 80; .. YORK: THE MACMILLAN COMPANY 380 "In the following pages I have endeavoured to reason out in a clear and accurate manner the leading propositions of the science of Arithmetic, and to illustrate and apply those propositions in practice. In the practical part of the subject I have advanced somewhat beyond the majority of preceding writers; particularly in Division, in Greatest Common Measure, in Cube Root, in the chapters on Decimal Money and the Metric System, and more especially in the application of Decimals to Percentages and cognate subjects. So long as the mania for neat answers continues to exist, so long will Decimals fail to take their legitimate place in the class-room, and be relegated to the office and the counting-house. The chapter on Weights and Measures and the Metric System is longer than usual, but not I hope uninteresting. In the earlier part of the work I have used the method of Reduction to the Unit, but I am far from advising an exclusive adherence to that method; when the student has gained a clear and firm grasp of ratio, it would be unwise of him to neglect the powerful instrument that has come into his possession." PRESS OPINIONS School Board Chronicle-"We most heartily welcome every good book expounding the theory of arithmetic, and Mr. Brooksmith's is one of the best of them. The master of the arithmetic class and the clever boy trying to excel in study will hail it as a friend." Scotsman-"Arithmetic in Theory and Practice will make a good arithmetician of whoever has brains and perseverance enough to get through it. A clever young man who wishes to go thoroughly into the theory and practice of arithmetic, and has no prospect of studying algebra, could not get a better book." Literary Churchman-"This strikes us as a valuable manual of arithmetic of the modern, i.e. scientific kind. Indeed, pace the many other writers of excellent school books, this really appears to us the best we have seen." British Quarterly Review-" This is an essentially practical book, leading the student forward to commercial book-keeping, land mensuration, and percentages, and providing very definite help to candidates for almost every kind of competitive examination." § 133. DIVISION OF FRACTIONS. 81 hence (108) we have this Rule,-Divide in the usual way, and to the integral quotient add the fraction whose numerator is the remainder, and denominator the divisor. COR. Ex. I. 29 Since 29÷8= we shall use either notation indifferently. 8 Divide 868 by 37. which when multiplied by shall give the product; hence (128) we the same result as before; that is, to divide a number by multiply the number by ; but is the reciprocal of; hence to divide a number by a fraction we have this Rule: Multiply the number by the reciprocal of the Divisor. |