156 USE OF LOGARITHMIC TABLES. 8. Calculate approximately the area and circumference of a circle inscribed in a square whose side is 359-5678 feet. 9. Find the volume of a prism on a triangular base the sides of which are 47.3, 38.9, 2.7 inches respectively, the height being 55.8 inches. 10. If V is the volume of a sphere and A the area of its surface, prove that 3 log A=2log 6+log +2 log V. Calculate the value of A, if V=796·325 cubic inches. 2. Find to six places of decimals the value of 87.327 x 784.55 x ⚫020868 61659 × 58.844 3. How many terms of the series 04, 08, 16, 32, 64, ... will amount to 41943 ? 4. A person borrowed £11000 for two months at 5 per cent. per annum. At the end of the time he was unable to pay the interest; it was, therefore, added to the principal and the debt was allowed to run for another two months. This was continually repeated, till at the end of 2 years the debt and interest were paid; how much did this amount to? 5. Looking due South from the top of a cliff which is 377 yards high, an observer sees two objects whose angles of depression are 27° 18′ 35′′ and 43° 12′ 14′′ respectively; find the distance between them. 6. A ship sailing due East is observed at 10 o'clock to be in a direction 15° 37' East of South; at 12.15 the ship is observed to be 57° 12′ East of South; if the ship sails 10 miles an hour, find the distance of the observer from each position of the ship. 7. Compute to the nearest square inch the area of a circle in which a chord 4 ft. in length subtends at the centre an angle 28° 36'. 8. The diagonal of a cube is 58.9342 inches; find the radius of a sphere whose surface is equal to that of the cube. Globe 8vo. Without Answers, 2s. With Answers, 2s. 6d. ALGEBRA FOR BEGINNERS By H. S. HALL, M.A. FORMERLY SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE AND S. R. KNIGHT, B.A., M.B., CH.B.. FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY All rights reserved EXTRACT FROM PREFACE THE present work has been undertaken in order to supply a demand for an easy introduction to our Elementary Algebra for Schools, and also to meet the wishes of those who, while approving of the order and treatment of the subject there laid down, have felt the want of a beginners' text-book in a cheaper form. It will be found that full provision has been made in the following pages for all the public examinations which do not require a knowledge of Algebra beyond Quadratic Equations. The Explanations are given in somewhat simpler language than in the Elementary Algebra, while the examples for practice are entirely new; thus the two books are supplementary to each other, and in many classes may be conveniently used together. Aberdeen Free Press-"To give the learner a sufficient knowledge of the ordinary algebraical processes without presenting him with too many of their difficulties requires a thorough knowledge of what is necessary for a beginner, and of what may be omitted until a more advanced stage. A very slight examination of the book shows that the authors possess this qualification in an eminent degree. The definitions and explanations of symbols are full and simple. An important feature of the work is the great number of problems given to show the practical applications of the science." Schoolmaster-"To teachers who have had experience of either the Elementary or the Higher Algebra it will only be necessary to say that this book is marked by the same qualities which have brought these works into such deserved repute. To those who are still in ignorance of these books, we can say that for clear, simple, and concise explanation, convenient order of subject-matter and copious land well graduated exercises, these books have, to say the least, no superiors. Quite early the student is introduced to easy problem work, which can only be looked upon as an advantage. The very numerous exercises are entirely new, so that the book might easily serve as a companion and supplement to the elementary work." Western Daily Press-"This little book seems to be a really valuable addition to the best text-books on elementary algebra. The authors wisely take simple equatoins much earlier than they used to be taken in the text-books of twenty years ago. There is nothing so interesting to the young student as to feel that he can work at something tangible like equations, and the drudgery of evolution and fractions can be taken later. Nothing is more admirable than the clearness and simplicity of statement adopted by the authors in illustrating the working of problems." Educational Times-" Algebra for Beginners is dealt with on the same lines as the earlier and somewhat more advanced book. The learner is introduced as soon as possible to the practical and more interesting side of the subject, such as equations and problems, while work which largely consists in the manipulation and simplification of elaborate expressions is postponed till later on. The examples for practice are copious, and have been newly composed for this particular book; and, as heretofore, the explanations are clear, concise, and simply expressed. Indeed, without hesitation, we pronounce this book the best of its size which we have seen. University Correspondent-"Those masters who have already adopted Messrs. Hall and Knight's Elementary Algebra in their schools will welcome this new work for the use of their junior classes. . The numerous exercises for the student are excellent in quality and entirely new. We can unhesitatingly recommend the book to the notice of both teachers and students." Scottish Leader-"Goes as far as quadratic equations, covering the ground on the same lines as the author's former books, but much more fully, and with a wealth of illustration and example that will commend the book to all masters of junior forms in schools." National Teacher "The exercises throughout are well graduated, and so selected as to make the difficult subject of Algebra interesting to the youthful student, Altogether it is an admirable text-book for beginners." Glasgow Herald-"Seems in all respects to deserve an equally favourable reception to that of its predecessor." Guardian-"It possesses the systematic arrangement and the lucidity which have gained so much praise for the works previously written by the same authors in collaboration," 62 ALGEBRA. [ОНАР. Example 3. A and B are playing for money: A begins with £p and B with q shillings. B wins £x; express by an equation the fact that A has now 3 times as much as B. What B has won A has lost; .. A has p-x pounds, that is 20(p − x) shillings, B has q shillings +x pounds, that is q+20x shillings. Thus the required equation is 20(p − x) = 3(q+20x). Example 4. A man travels a miles by coach and b miles by train; if the coach goes at the rate of 7 miles an hour, and the train at the rate of 25 miles per hour, how long does the journey take? The coach travels 7 miles in 1 hour; Example 5. How many men will be required to do in p hours what q men do in np hours? np hours is the time occupied by q men ; .. 1 hour........ that is, p hours.... Therefore the required number of men is qn. EXAMPLES X. b. 1. Write down three consecutive numbers of which a is the least. 2. Write down four consecutive numbers of which b is the greatest. 3. Write down five consecutive numbers of which c is the middle one. 4. What is the next odd number after 2n-1 ? 5. What is the even number next before 2n ? 6. Write down the product of three odd numbers of which the middle one is 2x+1. 7. How old is a man who will be x years old in 15 years? 8. How old was a man x years ago if his present age is n years? 9. In 2x years a man will be y years old, what is his present age? 110 ALGEBRA. (CHAP. 139. By suitably grouping together the terms, compound expressions can often be expressed as the difference of two squares, and so be resolved into factors. Example 1. Resolve into factors 9a2 - c2+4cx-4x2. 9ac4cx-4x2 = 9a2 - (c2 - 4cx+4x2) Example 2. Resolve into factors 2bd-a2 - c2 + b2 + d2+2ac. Here the terms 2bd and 2ac suggest the proper preliminary arrangement of the expression. Thus 2bd-a2 - c2+b2 + d2 + 2ac = b2 +2bd+d2 − a2+2ac - c2 = = b2+2bd+d2 - (a2 - 2ac + c2) = (b+d+a−c){b+d-a+c) 140. The following case is important. Example. Resolve into factors xa+x2y2+ya. xa+x2y2+ya = (x2 + 2x2y2+y1) − x2y2 141. Sometimes an expression may be resolved into more than two factors. Example 1. Resolve into factors 16a4-81b4. 16a4-81b4= (4a2+9b2)(4a2 - 962) = (4a2+9b2)(2a+3b)(2a − 3b). Example 2. Resolve into factors x6 - y6. 26 - y = (x2+y3)(x3 — y3) = (x + y)(x2 − xy + y2)(x − y)(x2+xy+y2). = Note. When an expression can be arranged either as the dif ference of two squares, or as the difference of two cubes, each of the methods explained in Arts. 136, 137 will be applicable. It will, however, be found simplest to first use the rule for resolving into factors the difference of two squares. |