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THE 
 
 NORMAL 
 
 MENTAL ARITHMETIC 1 
 
 A 
 
 THOROUGH AND COMPLETE COURSE, 
 
 BY 
 
 ANALYSIS AND INDUCTION. 
 
 BY 
 
 EDWAKD BKOOKS, A.M., 
 
 PRINCIPAL AND PROFESSOR OP MATHEMATICS IN PENNSYLVANIA STATE 
 NORMAL SCHOOL, AND AUTHOR OF " NORMAL PRIMARY ARITHMETIC," 
 
 "NORMAL ELEMENTARY ARITHMETIC," " NORMAL WRITTEN 
 ARITHMETIC," "NORMAL GEOMETRY," ETC. 
 ? 
 OFTHE 
 
 UNIVERSITY &Kstb Cbifio* 
 
 WITHA 
 
 TREATISE ON MENTAL ALGEBRA 
 
 A*yiis and Induction are the golden keys which unloeK the various enmpies comt 
 nations ot numbcre,. " 
 
 PHILADELPHIA : 
 SOWER, POTTS & CO., 
 
 530 MARKET ST., AND 523 MINOR ST. 
 
OFFICE OF THE CONTROLLERS OF PUBLIC SCHOOLS, ") 
 
 FIRST DISTRICT OF PENNSYLVANIA. \. 
 
 PHILADELPHIA, March 29, 1869. j 
 
 At a meeting of the Controllers of Public Schools, First Dis- 
 trict of Pennsylvania, held at the Controllers' Chamber, Tues- 
 day, March 9, 1869, the following resolution was adopted:- 
 
 Resolved, That the Normal Series of Arithmetics, comprising 
 "Brooks's Normal Primary Arithmetic," "Brooks's Normal 
 Elementary Arithmetic," "Brooks's Normal Mental Arithme- 
 tic," and ''Brooks's Normal Written Arithmetic," by Edward 
 Brooks, Esq., be and the same are hereby placed on the list of 
 books to be used in the Public Schools of this District. 
 From the minutes : 
 
 H. W. HALLIWELL, Sec'y. 
 
 Entered, according to Act of Congress, in the year 1858, by 
 EDWARD BROOKS, 
 
 In the Clerk's Office of the District Court of the United States, in and for the 
 Eastern District of Pennsylvania. 
 
 Entered, according to Act of Congress, in the year 1863, by 
 
 EDWARD BROOKS, 
 
 In the Clerk's Office of the District Court of the United States, in and for th 
 Eastern District of Pennsylvania. 
 
 MEARS ft DUSENBERY, STEREOTYPERS. SHERMAN & CO., PRINTERS. 
 
 The Board of School Trustees for the STATE OP MARYLAND, 
 recommend Brooks's Normal Arithmetics and Fewsmith's Gram- 
 mars, for use in all the Public Schools of that state. 
 
 The Board of School Commissioners for the CITY OP BALTI- 
 MORE, have adopted Brooks's Normal Series and Fewsmith'i 
 Grammars, for exclusive use in the Public Schools of that city. 
 
10 
 
 J P. WICKER SH AM, A. M 
 
 PRINCIPAL OF THE LANCASTER COUNTY NORMAL SCHOOL, 
 
 A TOKEN OF HIGH PERSONAL "REGARD, 
 
 AND 
 . TRIBUTE OF ADMIRATION FOR HIS NOBLE EFFORTS 
 
 IN THE 
 GREAT CAUSE OF POPULAR EDUCATION, 
 
 )p0 Jittte fflotame 
 
 18 M081 SINCERELY INSCRIBED BY 
 
 THE AUTHUS, 
 
 165026 
 
PREFACE. 
 
 THE science of Arithmetic, as taught unui quite re- 
 cently, has been much less beneficial as an educational 
 agency than it should have been. Consisting mainly 
 of dictated methods of deriving results, without present- 
 ing the reasons for the various operations, it failed to 
 afford that high degree of mental discipline, which, when 
 properly taught, it is so well calculated to impart. But 
 a great change has been wrought in this respectn a new 
 era has dawned upon the world of science the royal road 
 to Mathematics has been so graded and strewn with the 
 flowers of reason and philosophy, that it is now full of 
 interest and profit to the youthful learner ; and one of 
 the most influential agents in this work has been the 
 system of Mental Arithmetic. 
 
 The present is a proud period in the history of popular 
 education. At no previous time has there been so deep 
 and universal an interest manifested in the advancement 
 of the cause ; and this, combined with the elevation of 
 the standard of qualification among our teachers, has 
 created a demand for a more philosophic, systematic, 
 
 (3) 
 
IV PREFACE. 
 
 and comprehensive text book on the subject of Mental 
 Arithmetic, than has hitherto been presented ; and, at 
 the solicitation of his friends, the author of the present 
 work has attempted to meet that demand ; how success- 
 fully, must be determined by those whose province it is to 
 decide. 
 
 In the prosecution of this object he has availed himself 
 of the experience of several years of actual instruction in 
 the science, and of the careful examination of what has 
 been previously written, for the purpose of ascertaining 
 wherein improvements might be made. He does not 
 assume that every problem, or class of problems, or the 
 principle of every solution, is original with himself; as 
 every intelligent teacher knows that certain problems and 
 solutions have stood the test of years, and no book would 
 be faultless without them. The object has been not to be 
 merely original, but to prepare a suitable and valuable 
 text book; he feels confident, however, that it will be 
 found to contain sufficient distinct and original features 
 to recommend it to the favourable consideration of the 
 friends of education. 
 
 The work, like others of the same class, is not designed 
 for the child's first book in the science of numbers, and, 
 therefore, the more elementary operations have not been 
 uselessly enlarged upon ; yet the arrangement is so syste- 
 matic, and the transition from the easy to the complex so 
 gradual, that even very young pupils can pursue it with 
 ease and advantage. 
 
 A system of Mental Arithmetic should be based upon 
 the principles of Analysis and Induction. Results should 
 be derived by analytic processes, and methods inferred 
 from these processes and this is the philosophy upon 
 
PREFACE V 
 
 which the present treatise is founded. Analysis and In- 
 duction are the golden keys which unlock the various 
 complex combinations of numbers the magic wands by 
 which the intricate and abstruse are unfolded in logical 
 simplicity. The great element of Analysis is comparison, 
 and the equation is the great Archimedean lever of com- 
 parison. It enters into every operation, from the simplest 
 combination of Arithmetic, to the most complicated prob- 
 lem of the transcendental Analysis. In Geometry the 
 axioms and definitions are the standards of comparison j 
 in Algebra we compare the unknown with the known to 
 determine its value ; and in Arithmetic we compare all 
 numbers, and the effects produced by a number of equal 
 causes, with the unit, or effect of the single cause. And 
 thus the science of Mathematics is evolved, comprising 
 a vast series of dependent truths, derived from successive 
 comparisons of the unknown with the known, the theo- 
 retic with the axiomatic, the complex with the simple. 
 
 In the science of numbers, this relation of the col- 
 lection to the unit is so evident, that it is intuitively 
 apprehended, and hence the simplicity of this elementary 
 process of Analysis. But as the pupil progresses in the 
 science, it will be perceived that different collections bear 
 certain relations to each other, and he should be taught 
 to discover and apply these new relations. To develope 
 this theory fully and completely has been the object of 
 the author. 
 
 Attention is also called to the arrangement and treat- 
 ment of Fractions. First is given the fractional word, 
 treating them like denominate numbers, then the nume- 
 rical fractional expression, and then, after extensive 
 exercise iu analytic processes, mechanical methods are 
 
VI PREFACE. 
 
 derived from these processes by Induction,. At the 
 close of Fractions, some of these methods are stated ID 
 the form of Propositions, and their truth substantiated 
 by demonstration, in a manner different from any which 
 the author has met with, and which he has employed in 
 teaching, with entire satisfaction. In nearly every sectioD 
 of the book will be found matter not previously introduced 
 into works of this kind, and much that has never before 
 been published. Questions upon definitions and principles 
 are given in various parts of the work, without answers to 
 them, the object being to awaken original thought with 
 the pupil, which is deemed far more valuable than the 
 most accurate definition. 
 
 But it is impossible, as well as unnecessary, to call the 
 attention to. all the different peculiarities of the work, 
 since intelligent teachers, upon examination, will see and 
 judge for themselves. The whole, with its merits and 
 demerits, is respectfully submitted to the public, with 
 the sincere wish that it may prove a valuable auxiliary 
 in the promotion of that noble science, which promises 
 BO much for the advancement of the great cause of popu- 
 lar education. 
 
 E. B. 
 
 LANCASTER COUNTY NORMAL SCHOOL, 
 April 7, 1858, 
 
SUGGESTIONS TO TEACHERS. 
 
 THE attention of teachers is respectfully solicited to the following Method 
 of Recitation. Some of them are preferable to others, but all may occasionally 
 bo used with advantage. 
 
 COMMON METHOD. By this method the problems are read by the teacher and 
 assigned promiscuously, the pupils not being permitted to use the book during 
 recitation, nor retain the conditions of the problems by means of pencil and 
 paper, as is sometimes done. The pupil selected by the teacher arises, re- 
 peats the problem, and gives the solution, at the close of which the mistake? 
 that may have been made should be corrected by the class or teacher. 
 
 SILENT METHOD. By this method the teacher reads a problem to the class, 
 and then the pupils silently solve it, indicating the eompletion of the solution 
 by the upraised hand. After the whole class, or nearly the whole class, have 
 finished the solution, the teacher calls upon some member, who arises, repeats 
 the problem, and gives the solution as in the form<?r muhod. 
 
 By this method the whole class must be exercised, upon every problem, thus 
 securing more discipline than by the preceding method. It, however, requires 
 more time than the first; hence, not so many problems can be solved at a reci- 
 tation. We prefer the first method for advanced pupils, and the second, at 
 least a portion of the time, with younger pupils. 
 
 CHANCE ASSIGNMENT. This method differs from the first only in the assign- 
 ment of the problems. The teacher marks the number of the lesson and the 
 number of the problem, upon small pieces of paper, which the pupils may take 
 out of a box passed around by the teacher or some member of the class. The 
 teacher then, after reading a problem, instead of calling upon a pupil, merely 
 gives the number of the problem, the person having the number arising, repeat- 
 ing, and solving it. By this method the teacher is relieved of all responsibility 
 with reference to hard and easy problems, and it is also believed that better 
 attention is secured with it. It is particularly adapted to reviews and public 
 examinations. 
 
 DOUBLE ASSIGNMENT. By this method the pupil who receives the problem 
 from the teacher, arises, repeats it, and then assigns it to some one else to 
 solve. It may be combined with cither the first or second methods. The 
 objects of this method are variety and interest. 
 
 METHOD BY PARTS. By this method different parts of the same problem are 
 solved by different pupils. The teacher reads the problem and assigns it to a 
 pupil, and after he has given a portion of the solution, another is called upon, 
 who takes up the solution at the point where the first stops ; the second is suo 
 oeeded in like manner by a third ; and so on, until the solution is completed. 
 The object of this method is to secure the attention of the whole class, which it 
 does very effectually. It is particularly suited TO a large class consisting of 
 young pupils. 
 
 UNNAMED METHOD. By this method the teacher reads and assigns several 
 problems to different members of the class, before requiring any solutions, after 
 which those \*bo have received problems are called upon in the order of assign- 
 
 (vii) 
 
Vlii SUGGESTIONS TO TEACHERS. 
 
 ment fin Iheir solutions. The advantages of this method are, first, the pupil, 
 having some time to think of the problem, is enabled to give the solution with 
 more promptness and accuracy, and secondly, the necessity of retaining th 
 numbers and their relations in the mind for several minutes, affords a good 
 discipline to the memory. 
 
 CHOOSING SIDES. This is a modification of the old spelling class method, and 
 Is one calculated to elicit a very great degree of interest. By it two pupils, ap 
 pointed by the teacher, select the others, thus forming two parties for a trial 
 of skill, as in a game of cricket or base ball. The problems may be assigned 
 alternately to the sides, by the teacher, by chance, by the leaders of the sides, 
 or in any other way that may be agreed upon by the teacher and class. 
 
 In regard to these methods, the first, second, and third are probably the best 
 for the usual recitations,, but the other methods can very profitably be employed 
 with younger classes, or, in fact, with any class, to relieve monotony and awaken 
 interest. With advanced pupils we prefer the first method, or the first com- 
 bined with the third. 
 
 ERRORS TO BE AVOIDED. 
 
 There is a large number of errors to which pupils in every section of the 
 country are liable, a few of which we will mention. We classify them as errors 
 of Position and errors of Expression. 
 
 ERRORS OF POSITION. Pupils are exceedingly liable to assume improper posi- 
 tions and awkward attitudes during recitation, such as leaning on the desk or 
 against the wall, putting the foot upon a seat, jamming the hands in the pockets, 
 particularly when the problem is bard, playing with a button, watch-chain, &c. 
 All of these laults should be carefully guarded against, for reasons so obvious 
 that they need not be mentioned. An erect and graceful carriage, aside from 
 its relation to health, is of advantage to every lady and gentleman. 
 
 ERRORS OF EXPRESSION. Under this head we include errors of Articulation, 
 Pronunciation. Grammar, &c. There is quite a large number of words which 
 pupils in their haste mispronounce, and also quite a large number of combina- 
 tions, which by a careless enunciation make ridiculous sense, or nonsense. We 
 will call the attention to a few of them, suggesting to the teacher to correct 
 these and others he may notice. 
 
 " And" is often called " an ;" "/or" is called "fur ;" " o/" is pronounced as 
 If the o was omitted ; words commencing with wh, as when, which, where, &c., 
 are pronounced as if spelled " wen," " wich," " were" &c. 
 
 "Gave him" is called "gavim;" "did he" is called "diddy;" "had he" is 
 called ' haddy ;" "give, him" is called "givim;" "give ?ier" is called "giver;" 
 " which is" is often changed into "witches;" and "how many" is frequently 
 transformed into "hominy" "How many did each earn" is often rendered 
 * hominy did e churn." 
 
 A very common error, and one exceedingly difficult to correct, is involved in 
 the following solution: "If 2, apples cost 6 cents, one apple will cost the i of 6 
 cents, which are 3 cents." Here " the" is superfluous, and " are" is ungram- 
 matical. 
 
 The following is a frequent error: "If one apple cost 3 cents, for 12 cents you 
 can buy as many apptes as 3 is contained in 12, which are 4 times" The objec 
 tions are, first. 3 is not contained any apples in 12; secondly, the result obtained 
 is times, when it should be apples, or a number which applies to both times and 
 apples. The solution should be, " You can ouy as many apples for 12 cents ca 3 
 is contained times in 12, which are 4." 
 
MENTAL ARITHMETIC. 
 
 SECTION I. 
 
 LESSON L 
 
 1. If I have 2 cents in one hand, and 1 cent in the 
 other, how many cents have I in both ?' 
 
 SOLUTION. If I have 2 cents in one hand, and 1 cent in the 
 other, in both hands I have 2 cents plus 1 cent, which are 3 cents. 
 Therefore, if I have 2 cents in one hand, and 1 cent in the other, 
 I have 3 cents in both. 
 
 2. John has 3 apples, and William has 2 \ how many 
 have they hoth ? 
 
 3. Fanny has 4 peaches, and her sister has 3 ; how 
 many have they both ? 
 
 - ,4- James is 3 years old, and Sarah is 6 ; what is the 
 sum of their ages ? 
 
 5. Joseph bought 6 peaches, and his brother gave him 
 4 ] how many had he then ? 
 
 6. I paid 10 cents for a slate, and 2 cents for a pencil j 
 i what did they both cost ? 
 
 7. A cap cost 3 dollars, and a coat, 12 dollars ; what 
 did they both cost ? 
 
 8. Peter had 8 cents, and found 10 more; how many 
 had he then? 
 
 9. Martin had 6 birds, and caught 9 more ; how many 
 had he then ? 
 
 (7) 
 
8 MENTAL ARITHMETIC. 
 
 10. A pig cost 5 dollars, and a sheep, 7 dollars ; what 
 did they both cost ? 
 
 11. If Cyrus had 9 cows, and bought 11 more, how 
 many would he then have ? 
 
 12 A saddle cost 8 dollars, and a harness, 13 dollars 5 
 how much did they both cost ? 
 
 13. Sally had 10 pins in her cushion, and put in 9 
 more ; how many were then in the cushion ? 
 
 14. If James rode 8 miles, and walked 13, how far did 
 e travel ? 
 
 15. Mustard has 10 sisters and 11 brothers ; how many 
 are there in the family ? 
 
 16. Taylor had 5 horses, and his brother sold him 13 \ 
 how many had he then ? 
 
 17. In a garden there are 9 plum trees, and 11 peach 
 trees ; how many are there of both ? 
 
 18. Required the cost of Hunter's coat, if the cloth 
 cost 12 dollars, and the making, 5 dollars. 
 
 19. Rose gave 13 cents to her brother, and 10 to hei 
 sister ; how many cents did she give away ? 
 
 20. Mary's mother gave her 11 apples, and her father 
 gave her 7 ; how many did they both give her? 
 
 21. If I have 12 pencils, and find 5 more, how many 
 pencils will I then have ? 
 
 22. Russell gave 3 cents for a top, 4 for a whip, and 
 5 for a book; what did they all cost? 
 
 23. Louisa had 5 peaches, her mother gave her 6, and 
 her sister gave her 7 ; how many had she then ? 
 
 24. Ruth bought 5 yards of silk for a hat, 7 for a 
 cloak, and 9 for a dress ; how many yards did she buy ? 
 
 25. The head of a fish is 6 inches long, the tail 8, and 
 the body 10 ; what is the length of the fish ? 
 
 26 Philo caught 12 fish, Milo 9, and Nero 6; how 
 many did they all catch ? 
 
 27. Howard shot 11 robins, Howell 12, and Hoyt 13; 
 how many did they all shoot ? 
 
 28. A man bought a robin for 10 cents, a jay for 20 
 
MENTAL ARITHMETIC. 9 
 
 cents, -and a blue bird for 30 cents; required the cost of 
 ail. 
 
 29. A sleigh cost 50 dollars, a horse 150 dollars, and 
 a whip 5 dollars ; what was the cost of all ? 
 
 30. A lady gave 6 cents for needles, 12 cents for 
 thread, and 10 cents for muslin; what was the cost of 
 all ? 
 
 31. Addie has 9 roses, Lizzie has 15, and Amy had 
 20 ; how many have they all ? 
 
 32. A man bought some rye for 25 dollars, and some 
 wheat for 26 dollars ; what did he pay for both ? 
 
 83. A merchant sold some rice for 15 dollars, some 
 sugar for 17 dollars, and some molasses for 22 dollars; 
 required the amount received. 
 
 34. How many are 4 and 8 ? 5 and 7 ? 3 and 9 ? 5 
 and 12 ? 8 and 17 ? 6 and 23? 7 and 15 ? 8 and 26? 
 
 10 and 28 ? 10 and 32 ? 
 
 35. How many are 5 and 21 ? 8 and 23 ? 9 and 31 ? 
 
 11 and 16 ? 3 and 19 ? 8 and 27 ? 10 and 27 ? 11 and 
 11? 12 and 12? 12 and 24? 
 
 36. How many are 6 and 16 ? 7 and 17 ? 8 and 18 ? 
 
 9 and 19 ? 10 and 20 ? 11 and 21 ? 12 and 22 ? 13 and 
 23? 20 and 30? 21 and 22 ? 
 
 37. How many are 2 and 12 ? 3 and 13 ? 4 and 14 ? 
 5 and 15 ? 6 and 26 ? 7 and 27 ? 8 and 28 ? 9 and 29 ? 
 
 10 and 30 ? 11 and 31 ? 
 
 38. How many are 2 and 17 ? 3 and 27 ? 4 and 36 ? 
 5 and 46 ? 12 and 41 ? 15 and 29? 24 and 37 ? 43 and 
 57? 38 and 65? 56 and 44? 
 
 A single thing is a unit. One or more units is a number. 
 Similar numbers are those in which the units are the same 
 Addition is the process of finding the sum of two or more 
 numbers. 
 
 The symbol, -)-, is the sign of addition. It is read plus, and 
 when placed between two numbers denotes that they are to be 
 added together. The symbol, , is the sign of subtraction 
 It is read minus, and when placed between two numbers denotes 
 that the second is to be subtracted from the first. 
 
10 MENTAL ARITHMETIC. 
 
 LESSON II. 
 
 1. If I have 4 cents, and give 2 of them away, how 
 many will I have remaining? 
 
 SOLUTION. If I have 4 cents, and give away 2 cents, I will 
 have remaining the difference between 4 cents and 2 cents, which 
 is 2 cents. Therefore, &c. 
 
 2. Hunter had 6 apples, and gave 2 of them away ; 
 how many had he remaining ? 
 
 3. Morgan having 9 peaches, gave his sister 3 of them : 
 how many had he left ? 
 
 4. Ada culled 10 roses, and gave Lydia 5 of them; 
 how many did she retain ? 
 
 5. A man bought 12 lemons, and sold 7 of them; how 
 many remained unsold? 
 
 6. Reuben finding 13 cents, spent 8 of them ; how 
 many had he remaining ? 
 
 7. A watch was bought for 20 dollars, and sold for 27 
 dollars ; what was the gain ? 
 
 8. Mary has 28 pins, and Susan 15; how many more 
 has Mary than Susan ? 
 
 9. A cow was bought for 27 dollars, and sold for 19 
 dollars; required the loss. 
 
 10. In a school of 40 pupils, only 29 are present; how 
 many are absent ? 
 
 11. A horse cost 125 dollars, and was sold for 140 
 dollars ; required the gain. 
 
 12. William said he found 43 marbles, and lost 13 of 
 them; how many remained? 
 
 13. A farmer having 27 cows, sold 18 of them ; how 
 many cows had he remaining ? 
 
 14. How many are 9 less 5? 11 less 7? 13 less 9? 15 
 less 11? 18 less 12? 
 
 15. How many are 16 less 8? 14 less 6? 12 loss 4? 
 10 less 2? 14 less 8? 
 
MENTAL ARITHMETIC. 11 
 
 16. Required the value of 12 8? 13 7? 14--6? 
 15 5? 16 4? 
 
 17. Inquired the value of 10 5? 126? 14 7? 
 1<5 8? 189? 
 
 18. Required the value of 11 7? 14 8? 179? 
 20 10? 23 11? 26 12? 
 
 19. Required the value of 6 2? 10 4? 146? 
 8 8? 22 10? 26 12? 
 
 20. What is the value of 6 + 84? 7 + 5 6? 
 5 + 10 12? 
 
 What is the value 
 
 21. Of 7 + 6 5? 28. Of 10 6 + 4? 
 
 22. Of 3 + 8 9? 29. Of 16 7 + 8? 
 
 23. Of5 + 6 7? 30. Of 18 6 + 3? 
 
 24. Of 7 + 8 9? 31. Of 15 8 + 2? 
 
 25. Qf9 6 + 3? 32. Of 16 7 + 8? 
 
 26. Of8 5 + 10? 33. Of 17 9 + 4? 
 97. Of 6 3 + 12? 34. Of 22 12 + 13? 
 
 " 35. Silas having 3 dollars, found 6 ; and then lost 4 ; 
 how many had he remaining ? 
 
 36. A boy having 12 apples, bought 6 more, and then 
 sold 8 ; how many had he left ? 
 
 37. A man sold a colt for 25 dollars, which is 3 
 dollars more than it cost; required the cost. 
 
 38. James had 31 cents, and Harry had 27; how 
 many had they both, and how many had James more than 
 Harry ? 
 
 39. John having 20 peaches, eat 4, and gave hi? 
 sister 6 ; how many had he remaining ? 
 
 40. Philo having 12 cents, spent 7, and then found 9; 
 how many had he then? 
 
 41 A man having a certain number of cows, bought 
 6, and sold 10, and then had none left; how many had 
 be at first ? 
 
 42. Morris having a certain number of books, bough! 
 10, and giving 30 to his sister, had none remaining; ho\v 
 many books bad he at first 9 
 
12 MENTAL ARITHMETIC. 
 
 43. Paxton had 16 peaches, lie gave 6 to James, and 
 
 7 to Henry ; how many had he remaining ? 
 
 44. Edwin lost 4 cents, and found 6, and then had 
 1 1 ; how many had he at first ? 
 
 45. A man sold 13 cows, then bought 10, and then 
 had 1 2 ; how many had he at first ? 
 
 46. Two boys commenced playing with 20 maibles 
 Dach, at the close of the game one had 16; how many 
 bad the other? 
 
 47. A merchant bought goods to the amount of 27 
 dollars; how must he sell them to gain 11 dollars? 
 
 48. Thomas had 11 cents, Susan gave him 12, and 
 Walton gave him enough to make his number 30 ; how 
 many did Walton give him ? 
 
 49. Mr. A gave 35 dollars for a case of goods, and 
 
 Iid 4 dollars for cartage ; how must he sell them to gain 
 dollars ? 
 
 50. How many are 4 plus 6 minus 5 ? 7 plus 8 minus 
 
 10 ? 8 plus 12 minus 9 ? 12 plus 13 minus 14 ? 16 plus 
 20 minus 30 ? 7 plus 15 minus 12 ? 
 
 51. How many are 7 and 20 minus 13 ? 9 and 13 
 oiinus 16 ? 13 and 15 minus 16 ? 18 and 19 minus 20 ? 
 16 and 17 minus 18 ?' 20 and 30 minus 40 ? 
 
 52. How many are 6 plus 8 minus 10 ? 7 plus 9 minus 
 
 11 ? 8 plus 10 minus 12 ? 9 plus 11 minus 13 ? 10 plus 
 
 12 minus 14 ? 15 plus 17 minus 19 ? 
 
 53. How many are 4 plus 7 minus 10 ? 5 plus 8 minus 
 11? 6 plus 9 minus 12? 7 plus 10 minus 13? 8 plus 
 11 minus 14 ? 9 plus 12 minus 15 ? 
 
 54. How many are 6 plus 10 minus 14 ? 7 plus 11 
 minus 15 ? 8 plus 12 minus 16? 9 plus 13 minus 17 ? 
 10 plus 14 minus 18 ? 11 plus 15 minus 19 ? 
 
 55. How many are 4 plus 24 minus 14? 6- plus 26 
 minus 16 ? 7 plus 27 minus 17 ? 5 plus 25 minus 15 ? 
 
 8 plus 28 minus 18 ? 9 plus 29 minus 19 ? 
 
 56. How many are 7 plus 37 minus 27? 10 plus iiO 
 minus 20 ? 11 plus 31 minus 21 ? 12 plus 32 minus 22 ? 
 
 13 plus 33 minus 23 ? 14 plus 34 minus 24 ? 
 
MENTAL ARITHMETIC. 18 
 
 57. How many are 4 and 4 minus 2 and 2 ? 6 and 6 
 minus 3 and 3 ? 12 and 12 minus 6 and 6 ? 23 and 28 
 minus 13 and 13 ? 27 and 27 minus 17 and 17 ? 
 
 58. How many are 2 and 20 taken from 3 and 30? 
 5 and 55 from 6 and 66 ? 4 and 44 from 5 and 55 ? 7 
 and 77 from 8 and 88 ? 9 and 99 from 12 and 112 ? 
 
 Subtraction is the taking of one number from another. The 
 larger number is called the minuend; the smaller, ^the subtra- 
 hend; the result, the difference or remainder. 
 
 LESSON III. 
 
 1. What cost 4 apples, at 2 cents apiece? 
 
 SOLUTION. If one apple cost 2 cents, 4 apples will cost 4 
 times 2 cents, which are 8 cents. Therefore, c. 
 
 2. What cost 5 oranges, at 3 cents apiece? 
 
 3. At 5 cents each, what will 4 melons cost? 
 
 4. What will 8 sheep cost, at 6 dollars apiece ? 
 
 5. At 3 dimes each, what will 12 turkeys cost? 
 
 6. What cost 7 cows, at 20 dollars each ? 
 
 7. What cost 12 horses, at 200 dollars apiece ? 
 
 8. At 30 cents apiece, what cost 11 knives ? 
 
 9. What cost 12 books, at 20 cents apiece ? 
 
 10. At 4 dimes each, what cost 9 ducks ? 
 
 11. How far will a man travel in 9 days, at the rate 
 of 12 miles a day? 
 
 12. A boy has 13 apples, and 5 times as many peaches ; 
 required the number of peaches, 
 
 13. Mary has 14 apples, and John has 7 times as many 
 how many has John ? 
 
 14. A has 13 dollars, and B has 9 times as many; 
 what number have they both ? 
 
 15. A farmer sold 4 horses, and then bought 6 times 
 as many ; how many did he buy ? 
 
 2* 
 
14 MENTAL ARITHMETIC. 
 
 16. How many dimes must be paid for 9 books, at the 
 rate of 14 dimes each ? 
 
 17. Amos saw 13 flocks of pigeons, with 20 pigeons in 
 each flock ; how many pigeons did he see ? 
 
 18. A merchant having 10 melons, sold 6, and then 
 bought 5 times as many as he sold; how many had he 
 then ? 
 
 19. B having 20 sheep, sold 12, and then bought 4 
 times as many as remained; how many did he then 
 have ? 
 
 20. A boy borrowed 6 cents from a friend, and then 
 earned 7 times as much as he borrowed; how many cents 
 had he then? 
 
 21. Mary found 10 pins, and then bought 8 times as 
 many as she found ; how many had she then ? 
 
 22. What will 7 yards of muslin cost, at 8 cents a 
 yard? 
 
 23. How much cost 11 yards of ribbon, at the rate of 
 11 cents a yard ? 
 
 24. If 3 men can do a piece ot work in 5 days, how 
 long wi! 1 it take 1 man to do it ? 
 
 , 25. If 5 men can do a piece of work in 20 days, how 
 long will it require 1 man to do it? 
 
 26. How far will a man travel in 9 hours, at the rate 
 of 3 miles an hour ? 
 
 27. In an orchard there are 11 rows of trees, and 20 
 trees in each row; how many trees are there in the 
 orchard ? 
 
 28. If 9 men mow a field of grass in 14 days, how 
 long will it take 1 man to mow it ? 
 
 29. Mason earned 10 dollars a week, and paid 3 dol- 
 lars a week for his board; how much would he save in 5 
 weeks ? 
 
 30 If 1 melon is worth 3 peaches, how many peaches 
 are 7 melons worth ? 
 
 31. Passmore earned 3 dollars a day, and paid 1 dol 
 lar for his board ; how much could he save in a week ? 
 
 32. Morton earned 30 dollars a month, he paid o del- 
 
MENTAL ARITHMETIC. 15 
 
 lars a week for board, and 1 dollar for other expenses : 
 how much could he save in a year ? 
 
 33. Thomas travels 5 miles an hour, and John, 3; 
 how much farther will Thomas travel in 12 hours than 
 John? 
 
 84. How many are 6 times 6, plus 6? 5 times 8, plus 
 8 ? 7 times 10, plus 10 ? 4 times 11, plus 11 ? 8 times 
 
 I, plus 4? 5 times 9, plus 9 ? 
 
 35. How many are 3 times 4, plus 5 ? 6 times 7, plus 
 
 8 ? 8 times 9, plus 10 ? 9 times 10, plus 11 ? 10 times 
 
 II, plus 12 ? 11 times 12, plus 13 ? 
 
 36. How many are 4 times 4, minus 5? 5 times 7, 
 minus 6? 7 times 9, minus 8? 8 times 10, minus 9? 
 
 9 times 12, minus 11 ? 11 times 12, minus 13 ? 
 
 37. How many are 7 and 7 times 8 ? 5 and 5 times 6 ? 
 8 and 8 times 4 ? 6 and 6 times 7 ? 9 and 9 times 11 ? 
 
 10 and 10 times 12? 
 
 38. How many are 4 and 5 and 4 times 5 ? 5 and 6 
 and 5 times 6 ? 6 and 7 and 6 times 7 ? 7 and 8 and 7 
 times 8 ? 8 and 9 and 8 times 9 ? 
 
 39. How many are 23 minus 2 times 3 ? 34 minus 3 
 times 4 ? 25 minus 2 times 5 ? 37 minus 3 times 7 ? 
 49 minus 4 times 9 ? , 
 
 40. B and C start from the same place, and travel in 
 opposite directions, B at the rate of 5, and C 4 miles an 
 hour ; how far are they apart in 6 hours ? 
 
 41. Two men start at the same place, and travel in 
 the same direction, the one 7, and the other 5 miles an 
 hour ; how far will they be apart in 10 hours ? 
 
 Multiplication is the process of taking one number as many 
 times as there are units in another. The number repeated u 
 the multiplicand; the number of times it is repeated is the 
 multiplier; the result is the product. 
 
 The symbol, X' ^ s tne s ^g n f multiplication; when placed 
 between two numbers it denotes that they are to be multiplied 
 together. The symbol, -5-, is the sign of division : when 
 placed between two numbers it denotes that the one before ii 
 is to be divided by the one following it. 
 
16 MENTAL ilUTHMETIC. 
 
 LESSON IV. 
 
 1. How many 2's are there in 6 ? 
 
 SOLUTION. In 6 there are 3 two's, because 3 times 2 are 6. 
 
 2. How many 2's in 8 ? in 10 ? in 12 ? in 16 ? 
 
 3. How many 3's in 6 ? in 9 ? in 15 ? in 18 ? 
 
 4. How many 4's in 8 ? in 16 ? in 24 ? in 28 ? 
 
 5. How many 5's in 15 ? in 20 ? in 30 ? in 45 ? 
 6 How many 6's in 18 ? in 30 ? in 40 ? in 48 ? 
 
 7. How many 7's in 21? in 35 ? in 28 ? in 42 ? 
 
 8. How many 8's in 32? in 40 ? in 56 ? in 72 ? 
 
 9. How many 9's in 36 ? in 45 ? in 63 ? in 81 ? 
 
 10. How many 10's in 40 ? in 50 ? in 70 ? in 90 ? 
 
 11. How many ll's in 44 ? in 33 ? in 66 ? i 88 ? 
 
 12. How many 12's in 48 ? in 72 ? in 96 i in 132 '< 
 
 13. 24 contains 3 how many times? 4 ? 6? 8? 12? 
 
 SOLUTION. 24 contains 3, eight times, smo-e 8 times 3 are 24 
 
 14. 30 contains 2 now many times? 3? 5? 6? 10? 
 
 15. 36 contains 3 how many times? 4? 6? 9? 12? 
 
 16. 40 contains 4 how many times ? 5 ? 8 ? 10 ? 20 ? 
 
 17. 60 contains 5 how many times? 6? 10? 12? 20? 
 IS. 80 contains 4 how many times? 8? 10? 20? 40? 
 
 19. 90 contains 3 how many times? 6? 9? 10? 301 
 
 20. 100 contains 4 how many times? 5? 10? 20? 25? 
 
 21. How many times 3 are 6 times 2 ? 
 22 How many times 5 are 4 times 10 ? 
 23. II o^ many times 7 are 3 times 14 ? 
 
 24 How many times 8 are 5 times 6, -f 2 t 
 
 2 a 4 times 7, -j- 2, are how many times 6 ? 
 
 20 6 times 8, 8, are how many times 10 ? 
 
 27. 7 times 9, 3, are how many times 12 ? 
 
 28. 8 times 7, -(- 4, are how many times 4 ? 
 
 29. 10 times 9, 6, are how many times 7 ? 
 BO 9 times 8, + 8, are how many times 8? 
 
MENTAL ARITHMETIC. 17 
 
 81. 5 times 12, 4, are how many times 1 . ? 
 32 4 times 11, -f- 6, are how many times 5 ? 
 
 33. 5 limes 6, -f- 10, 8, are how many times 8 ? 
 
 34. 7 times 8, 6, -f- 5, are how many times 5 ? 
 
 35. 8 times 9, -f- 8, 10, are how many times 10 ? 
 
 36. 9 times 9, -j~ 9, 2, are how many times 8 ? 
 
 37. 8, and 4, and 2, and 3, and 5, and 6, less 12, an 
 h >w many? 
 
 38. Five, and 6, and 4, and 7, and 8, and 10, less 20, 
 are how many ? 
 
 39. Four, and 7, and 6, and 5, and 3, and 2, less 17 , 
 are how many ? 
 
 40. Twelve, multiplied by 2, divided by 3, multiplied 
 by 4, divided by 8, plus 6, are how many? 
 
 41. Three times 8, increased by 6, divided by 3, 
 diminished by 5, are how many ? 
 
 42. How many times 5 times 6, increased by 10, is 6 
 times 10 ; increased by 20 ? 
 
 LESSON V, 
 
 1. At 3 cents each, how many melons can I buy fof 
 12 cents? 
 
 SOLUTION. If 1 melon cost 3 cents, for 12 cents I can buy as 
 many melons as 3 is contained times in 12, which are 4. There- 
 fore, &c 
 
 2 At 2 dimes apiece, how many books can I buy foi 
 3 d.mes? 
 
 3. At 4 cents a yard, how many yards of ribbon can I 
 buy for 16 cents ? 
 
 4. How many apples can I buy for 21 cents, at 3 cents 
 apiece ? 
 
 5. How many yards of ribbon, at 6 cents a yard, can 
 be bought for 42 cents? 
 
IS MENTAL ARITHMETIC. 
 
 6. A man gave 50 dollars for sheep, at the rate o * 
 doLars a head ; how many did he buy ? 
 
 7. If a man travel 4 miles an hour, how long will it 
 take him to travel 48 miles ? 
 
 8. A man gave 7 boys 56 cents; how much did euob 
 boy receive? 
 
 9 If 8 apples cost 24 cents, how much will one apple 
 cost? 
 
 10. A farmer received 88 dollars for sheep, at the rate 
 of 8 dollars each ; how many did he sell ? 
 
 11. How many kegs, of 9 gallons each, can be rilled 
 from a hogshead, containing 63 gallons of molasses? 
 
 12. How many days must a man worl^ to earn 44 dol- 
 lars, at the rate of 4 dollars a day ? 
 
 13. How many melons, at 7 cents apiece, may be 
 bought for 84 cents ? 
 
 14. How many oranges, at 3 cents each, may be had 
 for 5 lemons, worth 6 cents each ? 
 
 15. How many yards of lace, at 6 cents a yard, may 
 be bought for 3 yards of muslin, at 12 cents per yard ? 
 
 16. If a man digs 10 yards of ditch, for 8 dimes a 
 yard, how many bushels of rye, at 4 dimes a bushel, will 
 pay him ? 
 
 17. How many pounds of meat, at 6 cents a pound, 
 will cost as much as 9 yards of ribbon, at 8 cents a yard ? 
 
 18. How many boxes of wafers, worth 6 cents a box, 
 may be bought for 12 sheets of paper, worth 2 cents a 
 sheet? 
 
 19. How much wheat, at 6 dimes a bushel, may be 
 purchased for 12 bushels of corn, worth 5 dimes a bushel ? 
 
 20. How many knives, at 3 dimes each, can I buy foi 
 I dimes in money, and 7 boxes of raisins, at 4 diunis 
 such ? 
 
 21. If flour is worth 8 dollars per barrel, how man} 
 barrels can be bought for 3 dollars, and 7 barrels of fish, 
 at 11 dollars per barrel ? 
 
 22 A man gave 9 pencils, worth 5 cents each, for 3 
 packages of envelopes worth 11 cents each; what did he 
 lose? 
 
MENTAL ARITHMETIC. 19 
 
 23. How many lamps at 7 dimes each, can be bought 
 for 9 dimes in money, and 6 dozen eggs, at 2 dimes per 
 dozen ? 
 
 24. How many are 12 plus 6 divided by 6 ? 25 plus 5 
 divided by 5 ? 32 plus 8 divided by 8 ? 42 plus 7 divided 
 by 7? 45 plus 9 divided by 9? 
 
 25. How many are 16 and 4 divided by 4 ? 21 and 7 
 divided by Y? 44 and 11 divided by 11? 60 and 12 
 divided by 12? 96 and 8 divided by 8? 
 
 26. How many are 90 minus 9 divided by 9 ? SO 
 minus 8 divided by 8 ? 100 minus 4 divided by 4 ? 110 
 minus 10 divided by 10 ? 144 minus 12 divided by 12 ? 
 
 27. How many are 6 times 8 divided by 4 ? 7 times 
 9 divided by 3 ? 10 times 6 divided by 12 ? 9 times 9 
 divided by 3 : 12 times 10 divided by 5 ? 
 
 28. How many are 2 times 22 divided by 11 ? 4 times 
 14 divided by ? 3 times 15 divided by 5 ? 6 times 16 
 divided by 8 ? 8 times 18 divided by 12 ? 
 
 29. How many are 48 divided by 6, plus 8 ? 30 divided 
 by 5, plus 6 ? Si 8 divided by 7, plus 17 ? 50 divided by 
 5, plus 15 ? 35 divided by 7, plus 17 ? 84 divided by 12, 
 plus 22 ? 
 
 30. Twice a number, -f- 3 times the number, 4 times 
 the number, -)- 2 times the number, equals how many 
 times the number? 
 
 NOTE. With young pupils this and the following problems 
 should be read slowly, that the pupils may solve them as the 
 teacher reads. More advanced pupils should be required to 
 enunciate them after the teacher has read them, and then solve 
 them. 
 
 31. Three times a number, 2 times the number, -|- 
 4 times the number, 5 times the number, equals how 
 many times the number ? 
 
 32. Eour times Susan's age, -f- 3 times her age, 5 
 times her age, -f 2 times her age, 3 times her age, 
 equals 13 years ; how old is Susan ? 
 
 3o. Three times Henry's number of marbles, -f- twice 
 his number, 3 times his number, -f- 4 times his num- 
 ber,- -5 times his number, equals 15; how many has he ? 
 
20 MENTAL ARITHMETIC. 
 
 34 Eight times a number, -j- by 4,X by 6, -5- by 8, X 
 by 5, by 10, equals how many times the number ? 
 
 35. Five times a number, X by 4, -4- by 10, X by 6, 
 -f-by 4, X by 2, equals how many times the number? 
 
 36. Think of any number, multiply it by 6, divide by 
 8, multiply by 2, divide by 4, add 10, subtract tli 
 number thought of, divide by 5, and the quotient ie 
 what ? 
 
 '>! . Think of a number, multiply it by 10, divide it 
 by 5, multiply by 3, divide by 6, add 10, subtract the 
 number thought of, divide by 8, and the quotient i& 
 what ? 
 
 38. Three times Mary's age, multiplied by 6, divided 
 by 9, multiplied by 4, divided by 8, equals 16 years; 
 how old is Mary ? 
 
 39. Think of a number, multiply it by 5, multiply 
 that by 4, divide the product by 10, multiply by 6. di- 
 vide by 3, add 30, subtract 4 times the number, divide 
 by 5^ and name the quotient. 
 
 40. What is Division ? Name and define each term 
 employed. What kind of numbers may each term bo 
 respectively ? 
 
 SECTION II. 
 
 LESSON I, 
 
 1 . If I divide an apple into two equal parts, what is 
 'i>ae of these parts called? What are 2 of these parts 
 called ? 
 
 2 How many halves of an apple in one apple? 
 
 3 What is 1 half of 6 
 
 SOLUTION 1 half of 6 is 3, because 2 times 3 are 6 
 
. MENTAL ARITHMETIC. 21 
 
 4 Wnatislhalfof4? of 8? of 10? of 12 t 
 
 5 What is 1 half of 14 ? of 16 ? of 18 ? of 20 ? 
 
 6. What is 1 half of 22 ? of 26 ? of 28 ? of 32 ? 
 
 7. If 1 yard of cloth cost 16 cents, what will 1 half of 
 a yard cost ? 
 
 8 If 1 pound of sugar cost 10 cents, what will 1 half 
 of a pound cost ? 
 
 9. Paul having 20 apples, gave 1 half of them to bis 
 brother ; how many did he give away ? 
 
 10. Thompson bought 24 cows, and sold 1 half of 
 them ; how many did he retain ? 
 
 11. Phebe had 80 peaches, and gave 1 half of them 
 away ; how many remained ? 
 
 12. If I divide an apple into 3 equal parts, what is 
 one of these parts called ? 
 
 13. What are 2, and 3 of these parts called? 
 
 14. How many thirds in one apple ? 
 
 15. What is 1 third of 6? of 9 ? of 12? of 15? 
 
 16. What is 1 third of 21? of 24? of 30? of 36? ' 
 
 17. James had 30 cents, and lost 1 third of them; 
 how many did he lose ? 
 
 18. Henry had 39 pears, and Thomas had 1 third as 
 many how many had Thomas ? 
 
 19. Lucy had 21 pins, and gave Mary 1 third of them; 
 how many did Mary receive? 
 
 20. Matthew bought 24 oranges, and gave 1 third of 
 them to Stephen ; how many remained ? 
 
 21. A bought 42 cows, and sold 1 third of them to 
 B; how many had he remaining ? 
 
 22. What are 2 thirds of 9 ? 
 
 SOLUTION. 1 third of 9 is 3, and if 1 third of 9 is 3, 2 thirds 
 of 9 are 2 times 3, which are 6. Therefore, &c. 
 
 23. What are 2 thirds of 6? of 12? of 15? of IS? 
 
 24. What are 2 thirds of 24? of 30? of 27 ? of 33? 
 
 25. John had 21 cents, and gave 2 thirds of them to 
 Sarah ; how many cents did Sarah receive ? 
 
 3 
 
22 MENTAL ARITHMETIC. 
 
 26 Having 27 peaches, I sold 2 thirds of them ; ho\* 
 many did I sell ? 
 
 27 Henry gave his sister 2 thirds of 39 oranges; how 
 many did he retain ? 
 
 28. Hiram lost 2 thirds of 36 dollars; how much 
 money had he remaining ? 
 
 29. Chandler, having 30 pears, gave 1 third to Maiia ; 
 and 1 third to Jane; how many remained? 
 
 30. If I divide an apple into 4 equal parts, what are 
 1. 2, and 3 of these parts ? 
 
 31. How many fourths of an apple in a whole apple ? 
 32. What is 1 fourth of 4? of 8? of 20? of 32? 
 
 33. What is 1 fourth of 12? of 24? of 16? of 48? 
 
 34. What are 2 fourths of 24? of 16? of 28? of 36? 
 
 35. What are 2 fourths of 12? of 20? of 40? of 48? 
 
 36. What are 3 fourths of 20? of 24? of 12? of 16? 
 
 37. What are 3 fourths of 16? of 28? of 40? of 52? 
 
 38. Jacob having 44 pens, sold 2 fourths of them to 
 Joseph ; how many pens did Joseph receive ? 
 
 39. If a yard of cloth cost 8 dollars, what will 3 fourths 
 of a yard cost ? 
 
 40. A boy sold 2 fourths of 28 pears; how many had 
 he remaining? 
 
 41. Samson gave his brother 1 fourth, and his sistei 2 
 fourths of 28 oranges; how many did each receive? 
 
 42. Marion found 2 fourths of 28 cents, and then lost 
 3 fourths of 16 cents ; how many remained ? 
 
 43. A having 24 plums, gave 1 half of them to B, 
 and 1 third to C ; how many had he left ? 
 
 44. Harleigh is 24 years of age, a'nd Townsend is 3 
 fourths as old; how old is Townsend ? 
 
 45. A farmer had 36 sheep, of which A bought 1 
 half, and a dog kil-led 1 third ; how many remained ? 
 
 46. A merchant having 40 barrels of flour, sold 3 
 fourths of them, and then bought 1 third as many as he 
 sold ; how many had he then ? 
 
MENTAL ARITHMETIC. 23 
 
 LESSON II, 
 
 1. If you divide an orange into 5 equal parts, what 
 are 1 2, 3, and 4 of these parts called ? 
 
 2. How many fifths in one orange? 
 
 8. What is 1 fifth of 10 ? of 25 ? of 15 ? ot 30 ? 
 
 4. What are 2 fifths of 15 ? of 30 ? of 45 ? of 60 ? 
 
 5. What are 2 fifths of 65 ? of 35 ? of 40 ? of 50 ? 
 
 6. What are 3 fifths of 10 ? of 30 ? of 25 ? of 55 ? 
 
 7. What are 4 fifths of 20 ? of 50 ? of 60 ? of 100 ? 
 
 8. Mary has 15 oranges, and Rachael 2 fifths as many; 
 how many has Rachael ? 
 
 9. Susan's age is 25 years, and her sister is 4 fifths as 
 old ; how old is her sister ? 
 
 10. Rowland is 35 years of age, and his wife is 3 fifths 
 as old } how old is his wife ? 
 
 11. A horse cost 100 dollars, and a sleigh 3 fifths as 
 much ; required the cost of the sleigh. 
 
 12. A man having 40 sheep, lost 20, and found only 
 3 fifths of them ] how many remained ? 
 
 13. A man having 50 cows, sold 4 fifths of them, and 
 then bought 4 fifths as many as he sold"; how many had 
 he then ? 
 
 14. 4 fifths of 200 dollars, is 2 times what Emily's 
 watch cost; what was the cost of the 'watch? 
 
 15. If you divide a melon into 6 equal parts, what are 
 1, 2, 3, 4, and 5 of these parts called? 
 
 16 How many sixths are there in a single thing? 
 
 17. What are 2 sixths of 24? of 18? of 36? of 60? 
 
 18. What are 3 sixths of 12 ? of 42? of 30? of 66? 
 
 19. What are 4 sixths of 6 ? of 36 ? of 48 ? cf 54 ? 
 
 20. What are 5 sixths of 18? of 54? of 24? of 72? 
 
 21. What will 2 sixths of 12 yards of tape cost, at 3 
 cents a yard? 
 
 22. llaub having 48 pens, sold 3 sixths to Frescoln. 
 and 2 sixths tc Morgan ; how many did he sell tc both ? 
 
24 MENTAL ARITHMETIC. 
 
 23. Whai will 5 sixths of 36 yards of cloth cost, at 
 the rate of 2 dollars a yard ? 
 
 24. Warren had 12 marbles, and Oliver had 5 sixths 
 as many, lacking 4; how many had Oliver? 
 
 25. How many yards of cloth can you buy for' 20 dol- 
 lars, if 1 yard cost 2 fifths of 10 dollars? 
 
 26. Dana having 60 peaches, gave 2 sixths of them to 
 Barton, and 3 sixths to Benton; how many remained? 
 
 27. If 1 yard of linen cost 5 sixths of 36 cents, how 
 many yards can you buy for 3 fourths of 80 cents? 
 
 28. 2 thirds of 30 dollars, increased by 2 thirds of 60 
 dollars, is 10 dollars less than A's money; required A's 
 money. 
 
 29. Frazier having 40. pens, gave Brown 10, and Seal 
 2 sixths of the remainder ; how many had he left ? 
 
 30. If a melon be divided into 7 equal parts, what are 
 1, 2, 3, 4, 5, and 6 of these parts called? 
 
 31. How many sevenths are there in one ? 
 
 32. What is 1 seventh of 21 ? of 28 ? of 42 ? of 56 ? 
 
 33. What are 2 sevenths of 28 ? of 49 ? of 63 ? of 70 ? 
 
 34. What are 3 sevenths of 14 ? of 35 ? of 49 ? of 28 ? 
 
 35. What are 4 sevenths of 70 ? of 77 ? of 63 ? of 84 ? 
 
 36. What are 5 sevenths of 77 ? of 91 ? of 42 ? of 140 ? 
 
 37. What are 6 sevenths of 35 ? of 42 ? of 49 ? of 28 ? 
 
 38. 4 sevenths of 21, are how many times 3, 4, and 6? 
 
 39. 6 sevenths of 63, are how many times 3, 6, and 9? 
 
 40. A watch was bought for 70 dollars, and sold for 6 
 sevenths of its cost ; required the loss ? 
 
 41. If 1 half of 4 yards of cloth cost 10 dollars, what 
 will 1 fifth of 10 yards cost? 
 
 42. A gave 70 dollars for a watch, 3 sevenths as much 
 for a chain, and sold them both for 90 dollars ; required 
 tne loss. 
 
 43. 3 sevenths of 56 dollars is 6 dollars more than 1 
 third of a4oad of hay cost; what will 3 loads cost, at the 
 same rate ? 
 
 44. 4 sevenths of 42 dollars is 12 dollars less than 1 
 half of a building lot cost; required the cost of the lot 
 
MENTAL ARITHMETIC. 25 
 
 45. Richard had 360 dollars, 1 third of which he spent 
 for a horse, 1 fourth for a watch, and 1 sixth for a sleigh ; 
 how much had he remaining ? 
 
 46. Mr. A having 140 dollars, gave 3 sevenths of it to 
 the poor, and lost 3 fourths of the remainder; how much 
 then remained ? 
 
 LESSON III. 
 
 1. If anything be divided into 8 equal parts, what is 
 one of these parts called ? 
 
 2. What are 2, 3, 4, 5, 6, and 7 of these parts called, 
 and how many eighths in a unit ? 
 
 3. What is 1 eighth of 24? 48? 72? 88? 
 
 4. What are 2 eighths of 32? 40? 56? 72? 
 
 5. What are 3 eighths of 16? 64? 80? 96? 
 
 6. What are 5 eighths of 8 ? 24? 48? 64? 
 
 7. What is 1 half of 3 ? 5? 7? 9? 11? 
 
 8. What is 1 third of 5? 7? 8? 10? 13? 
 
 9. What are 2 thirds of 8? 14? 16? 17? 19? 
 
 10. What are 3 fifths of 9 ? 21? 17? 24? 31? 
 
 11. What are 5 eighths of 7? 10? 14? 18? 20? 
 
 12. 2 eighths of 24 are how many times 3 ? 
 
 13. 3 eighths of 40 are how many times 5 ? 
 
 14. 4 eighths of 80 are how many times 8 ? 
 
 15. 5 eighths of 56 are how many times 7 ? 
 
 16. 6 eighths of 64 are how many dmes 12 ? 
 
 17 7 eighths of 72 are how many times 3 ? 
 
 18 3 eighths of 32 are how many times 1 third of 12 ? 
 
 19. 6 eighths of 40 are how many times 1 fourth of 24 ? 
 
 20. 4 eighths of 4 are how many times 2 thirds of 18 ? 
 
 21 7 eighths of 96 are how many times 4 fifths of 10 ? 
 
 22 3 eighths of 56 are how many times 5 sixths of 42 ? 
 
 23. 5 sevenths of 28 are how many times 5 sevenths 
 of 14? 
 
 24. 2 thirds of 27 are how many times 3 fourths of 12 j 
 3* 
 
26 MENTAL ARITHMETIC 
 
 25. If a single thing be divided into 9 equal parts, 
 what are 1, 2, 8, 4, &c., of these parts called? 
 
 26. What are 2 ninths of 18? 27? 45? 36? 
 
 27. What are 3 ninths of 63? 72? 81? 27? 
 
 28. What are 4 ninths of 9 ? 36? 54? 81? 
 
 29. What are 5 ninths of 54 ? 72? 63? 27? 
 
 30. What are 6 ninths of 81 ? 18? 36? 90? 
 31 What are 7 ninths of 18 ? 99? 27? 108? 
 
 32. 3 times 6, and 2 thirds of 6, are how many? 
 
 33. 4 times 12, and 3 fourths of 12, are how many? 
 
 34. 5 times 10, and 3 fifths of 10, are how many? 
 
 35. 6 times 12, and 3 sixths of 12, are how many? 
 
 36. 5 times 7, and 4 sevenths of 7, are how many? 
 
 37. 9 times 8, and 5 eighths of 8, are how many ? 
 
 38. 2 times 18, and 7 ninths of 18* are how many? 
 
 39. 2 ninths of 18 are how many times 2 thirds of 3 ? 
 
 40. 5 ninths of 27 are how many times 5 sixths of 6 ? 
 
 41. 6 ninths of 54 are how many times 4 fifths of 15 ? 
 
 42. 3 ninths of 72 are how many times 2 eighths of 16 ? 
 
 43. 7 eighths of 24 are how many times 7 eighths 
 of 8? 
 
 44. 5 sixths of 48 are how many times 5 sevenths 
 of 14? 
 
 45. 4 sevenths of 56 are how many times 8 ninths 
 of 18? 
 
 46. A bought 15 horses, and sold 6 of them, and then 
 lacked 4 of having 20 ; how many had he at first ? 
 
 47. Hiram and Oliver had each 26 cents ; after Hiram 
 had given Oliver 10, and Oliver had given Hiram 6, how 
 many had each ? 
 
 48. A farmer having 48 bushels of oats, sold 4 sixths 
 of them to one man, and 1 fourth to another ; how many 
 bushels did he sell to each ? 
 
 49. A bought 60 cows, and sold 2 sixths of them to 
 B, and 3 times 2 tenths of the remainder to C j how 
 many then remained Jf 
 
MENTAL ARITHMETIC 27 
 
 LESSON IV, 
 
 1 Harry gave 1 third of an apple to his trother, and 
 2 thiids to his sister; how much did he give away? 
 
 2. Matthew gave 2 fifths of a peach to Elias, and 8 
 fifths to Morris; how much did he give to both ? 
 
 3. If I give 3 sevenths of a melon to Harry, and 1 
 sevenths to Harvey, how much do I give away? 
 
 4. Fanny eat 3 eighths of a quart of chestnuts yester- 
 day, and 4 eighths to-day; how many did she eat in all? 
 
 5. Ella gave 1 fourth of a melon to Phebe, 2 fourths 
 to Carrie, and 3 fourths to Kate ; how much did she give 
 away? 
 
 6. Philip gave 2 sixths of a dollar to Jane, 3 sixths to 
 Sarah, and 5 sixths to Eliza; how much did he give 
 away? 
 
 7. Willis lost 7 eighths of a dollar, and had 9 eighths 
 emaining j how much had he at first ? 
 
 8. Matthew lost 6 eighths of a dollar from one pocket, 
 and 7 eighths from the other, and had 5 eighths remain- 
 ing ; how much had he at first ? 
 
 9. Dora gave 3 ninths of a pound of raisins to Ella, 
 and 7 ninths to Daisy, and then had 3 ninths remaining; 
 how many had she at first ? 
 
 10 What is the sum of 1 fourth, 3 fourths, 5 fourths, 
 7 fourths, and 9 fourths ? 
 
 11. Jane had 7 eighths of a pound of candies, and gave 
 Maria 5 eighths ; how many eighths remained ? 
 
 12. Frank had 6 sevenths of a melon, and gave Abratfl 
 4 sevenths ; how much remained ? 
 
 13. Louisa having 10 eighths of a dollar, gave Lizzie 
 7 eighths of it ; how much remained ? 
 
 14. What is the difference between 5 sevenths, and the 
 sum of 4 sevenths and 6 sevenths ? 
 
 15 Sallie having 24 pears, gave Buela 2 eighths, 
 
28 MENTAL ARITHMETIC. 
 
 Amanda 3 eighths, and James 1 eighth of them ; hew 
 many remained ? 
 
 16. Rufus having 1 third of a quart of chestnuts, 
 bought 4 thirds of a quart, and then sold 1 quart ; what 
 part of a quart remained ? 
 
 17. Peter having 5 sixths of a bushel of apples, sold 
 sixths, and then bought 2 sixths of a bushel; how 
 
 many sixths had he then ? 
 
 18. What is the difference between the sum of 3 
 eighths and 7 eighths, and the sum of 4 eighths and 5 
 eighths ? 
 
 19. A bought 20 sheep, and sold 2 tenths of them to 
 B, 3 tenths to C, and 4 tenths to D ; how many sheep 
 remained ? 
 
 20. A lady having 36 yards of tape, sold 6 ninths of 
 it to one person, and 3 ninths to another; how much 
 had she then ? 
 
 21. Mariana had 3 fourths of a pint of nuts, Elva had 
 twice as many, and Ezra 3 times as many; how many 
 had they all ? 
 
 22. A bought 4 ninths of a bushel of wheat, and K 
 bought 3 times as much ; how much did B buy ? 
 
 23. If 1 yard of cloth cost 5 sixths of a dollar, what 
 will 6 yards cost ? 
 
 24. At 7 ninths of a dollar each, what will 9 turkeys 
 cost ? 
 
 25. What is the sum of 5 times 3 ninths, and 3 times 
 5 ninths ? 
 
 26. Mary having 11 fifths of a melon, gave 2 fifths 
 to Sarah, and twice as much to Sophia; how much re 
 luaiued? 
 
 27. How much greater is 7 times 2 sevenths, than 4 
 times 3 sevenths ? 
 
 28. Cornell gave 3 times 3 sixths of an apple to Gray, 
 and had 4 times as much remaining ; how much had he 
 at first? 
 
 29. How much is 5 times 3 fourths, minus 2 time? 3 
 fourths, plus 6 times 3 fourths, minus 7 times 3 fourths'/ 
 
MENTAL ARITHMETIC. 29 
 
 30. I gave A 6 tenths of a dollar, and I gave B 3 
 times as much, plus 2 tenths of a dollar ; how much did 
 I give both ? 
 
 31 What will 1 fifth of a yard of tape cost, at the rate 
 of 20 fourths cents a yard ? What will 3 fifths cost, at 
 the same rate ? 
 
 32. Johnston gave his brother 1 third of 6 eighths of * 
 part of walnuts ; what part of a quart did he receive ? 
 
 33. How much is 1 fourth of 8 sevenths? of 12 ninths? 
 of 16 tenths? of 24 twelfths? 
 
 34. Stanton having 2 thirds of a dollar, found 1 half 
 of 4 thirds of a dollar ; how many thirds had he then ? 
 
 35. Mr. A bought 7 tenths of a barrel of sugar, and 
 then sold 2 thirds of 6 tenths of a barrel ; how much re- 
 mained? 
 
 36. What is the difference between 5 times 3 sevenths, 
 and 1 fifth of 40 sevenths ? 
 
 37. Thornton having 4 times 2 sixths of a bushel of 
 corn, bought 3 fourths of 20 sixths of a bushel; how 
 much had he then ? 
 
 38. If 1 half of a yard of tape cost 2 tenths of a dime, 
 how many yards may be bought fur 3 fourths of 16 tenths 
 of a dim e ? 
 
 39. Martin sold 2 thirds of 6 sevenths of a peck of 
 beans, ard then had 3 fourths of 8 sevenths of a peck 
 remaining ; how many had he at first ? 
 
 40. Ferris lost 3 fourths of 8 ninths of a dollar, and 
 then, having found 3 ninths of a dollar, had 3 fourths of 
 8 ninths remaining ; how much had he at first ? 
 
 LESSON V. 
 
 1 What will 4 apples cost, if 3 apples cost 9 cents ? 
 
 SOLUTION. If 3 apples cost 9 cents, 1 apple will cost one 
 third of 9 cents, which is 3 cents; and if 1 apple cost 3 cents, 4 
 apples will cost 4 times 3 cenis, whish are 12 cents. There- 
 fore, &c. 
 
30 MENTAL ARITHMETIC. 
 
 2. What will 5 lemons cost, at the rate of 3 for 12 
 cents ? 
 
 3. If 3 pairs of shoes cost 6 dollars, how much will 5 
 pairs cost? 
 
 4. What will 9 candies cost, if 4 candies cost 3 cents? 
 
 5. If 4 peaches are worth 8 cents, what are 8 peaches 
 worth ? 
 
 6. What are 10 oranges worth, if 8 oranges cost 10 
 cents ? 
 
 7* If 7 pounds of meat cost 42 cents, what will 9 
 pounds cost? 
 
 8. What cost 7 sheep, at the rate of 5 for 30 dollars? 
 
 9. What cost 11 barrels of flour, at the rate of 5 barrels 
 for 30 dollars ? 
 
 10. If 12 boxes of figs cost 48 dollars, what will 7 
 boxes cost? 
 
 11. How much will 7 cows cost, if 3 cows are bought 
 for 60 dollars? 
 
 12. How far will a man travel in 12 days, at the rate 
 of 36 miles in 4 days? 
 
 13. How many tons of hay will a drover feed in 11 
 weeks, at the rate of 10 tons in 5 weeks ? 
 
 14. How much must be paid for the keeping of IS 
 horses, at the rate of 80 cents for 4 horses ? 
 
 15. Required the value of 5 ducks, at the rate of 120 
 cents for 3 ducks. 
 
 16. Mary gave 10 cents for apples, at the rate of 3 
 cents for 9 ; how many did she buy ? 
 
 17. Fanny paid 8 dollars for silk, at the rate of 5 dol- 
 tars for 15 yards; how many did she buy? 
 
 18. Wilkinson walked 7 hours, at the rate of 12 miles 
 in 4 hours ; how far did he travel ? 
 
 19. Robert gave 12 oranges for apples, at the rate of 
 3 oranges for 9 apples; how many apples did he get? 
 
 20. At the rate of 3 melons for 12 oranges, how many 
 oranges can be bought for 10 melons? 
 
 21. If 6 men can mow 12 acres of grass in a day, how 
 much can 8 men mow in the same time? 
 
MENTAL ARITHMETIC. 31 
 
 22. If 1C men can dig 30 rods of ditch in one day, 
 how much can 12 men do m the same time? 
 
 23. How long will it take 4 ladies to drink a box of 
 tea, if 6 ladies can drink it in 12 days? 
 
 24. If 5 boys can do a piece of work in 16 days, ho\v 
 long will it take 20 boys to do it? 
 
 25. In what time will 8 girls pick a bushel of berries, 
 if 4 girls can do it in 8 hours ? 
 
 26. How many men will be required to ouild a boat in 
 6 days, if 3 men can do it in 12 days? 
 
 27. How many men can do as much work in 4 days, 
 as 8 men can in 40 days? 
 
 28. If it require 10 men 8 days to build a wall, how 
 many men will be required to build it in 5 days ? 
 
 29. If 5 men build a boat in 20 days, how many men 
 will be required to do it in 1 fourth of the time ? 
 
 30. What cost 1 half of 12 yards of cloth, at the rate 
 of 12 dollars for 4 yards? 
 
 31. If 7 yards of cashmere cost 21 dollars, what will 2 
 thirds of 15 yards cost ? 
 
 32. What cost 3 fourths of 8 pounds of coffee, at the 
 rate of 10 pounds for 60 cents? 
 
 33. If 3 fifths of 10 yards of ribbon cost 30 cents, 
 what will 4 sixths of 12 yards cost? 
 
 34. Mary gave 7 apples for 21 chestnuts ; at this rate 
 how many chestnuts could she get for 8 apples? 
 
 35. If 8 lemons are worth 16 oranges, how many 
 oranges can you buy for 10 lemons ? 
 
 36 At the rate of 6 citrons for 18 melons, how many 
 melons may be purchased for 11 citrons ? 
 
 37. If 9 apples are worth 27 chestnuts, how many 
 chestnuts may be had for 12 apples? 
 
 38. If 3 sevenths of 14 bunches of grapes cost 24 
 cents, what are 2 fifths of 15 bunches worth? 
 
 39. Peter can walk 3 fourths of 8 miles while John 
 walks 12 ; how far can John go while Peter walks 7 miles? 
 
 40. If 8 quarts of molasses .cost 40 cents, what will 4 
 ixths of 24 qua-ts cost ? 
 
32 MENTAL ARITHMETIC. 
 
 41. What must I pay to ride 1 half of 14 miles, if it 
 cost me 20 dimes to ride 2 thirds of 15 miles? 
 
 42. I gave 8 yards of muslin for 6 gallons of molasses; 
 what did the molasses cost a gallon, if 4 yards of muslin 
 cost 48 cents ? 
 
 43. A gave 9 bushels of wheat for 3 barrels of flour ; 
 what was the wheat worth a bushel, if 8 barrels of flour 
 cost 72 dollars ? 
 
 44. B bought 7 yards of cloth for 21 dollars, and gave 
 4 yards for apples worth 2 dollars per barrel; how many 
 barrels of apples did he receive ? 
 
 45. C gave 7 grammars for 6 arithmetics ; how much 
 were the grammars worth each, if 5 arithmetics cost 35 
 dimes ? 
 
 46. If 7 apples cost 21 cents, how many apples must 
 be given for 9 peaches, bought at the rate of 3 for 12 
 cents ? 
 
 47. Reuben had 9 oranges worth 4 cents apiece, and 
 Jackson had 8 lemons worth 3 cents each, which he gave 
 to Reuben for a part of his oranges ; how many oranges 
 had Reuben remaining ? 
 
 LESSON VI. 
 
 1. What will one yard of tape cost, if 2 thirds of a 
 yard cost 4 cents ? 
 
 SOLUTION. If 2 thirds of a yard of tape cost 4 cents, 1 third 
 of a yard will cost 1 half of 4 cents, which is 2 cents, and 3 
 thirds, or one yard, will cost 3 times 2 cents, which are 6 cents. 
 Therefore, &c. 
 
 2. What will one box of soap cost, if 3 fourths of a 
 box cost 6 dollars ? 
 
 3. If 4 fifths of a box of tea cost 8 dollars, what will 
 one box cost ? 
 
 4. If 3 fifths of a yard of cloth cost 6 dollars, what 
 will one yard cost ? 
 
MENTAL ARITHMETIC. 33 
 
 5. What will 2 pounds of starch cost, if 5 sixths of a 
 pound cost 10 cents? 
 
 6. What cost 2 barrels of flour, at the rate of 4 dol- 
 lars for 4 sixths of a barrel ? 
 
 7. If 7 eighths of a keg of oysters cost 14 shillings, 
 what will be the cost of 3 kegs of oysters ? 
 
 8. What cost 3 yards of cloth, if 3 sevenths of a yard 
 cost 6 dollars ? 
 
 9. How far can A walk in 4 days, if in 5 sixths of a 
 day he can walk 20 miles ? 
 
 10. What cost 5 boxes of raisins, if 3 fifths of a box 
 cost 6 dollars ? 
 
 11. What is 5 times the distance to Lancaster, if 3 
 fourths of the distance is 3 miles ? 
 
 12. How much will 4 bushels of apples cost, if 5 tenths 
 of a bushel cost 50 cents? 
 
 13. What cost 8 barrels of flour, if 7 eighths of a barrel 
 cost 7 dollars? 
 
 14. Mary bought 9 pecks of beans, at the rate of 12 
 cents for 6 sevenths of a peck; required the cost. 
 
 15. How much will 5 tons of hay cost, if 10 dollars 
 will buy 5 sixths of a ton*? 
 
 16. What is the cost of 10 barrels of sugar, at the rate 
 of 20 dollars for 4 fifths of a barrel ? 
 
 17. Pelton bought 4 dozen eggs at the rate of 8 cents 
 for 2 thirds of a dozen ; how much did they cost ? 
 
 18. What is the value of 1 half of 6 bushels of peaches, 
 at the rate of 2 dollars for 2 thirds of a bushel ? 
 
 19. What cost 2 thirds of 9 yards of cloth, if 3 fourths 
 of twelve yards cost 27 dollars ? 
 
 20. If 4 fifths of 10 lemons cost 24 cents, what cost 3 
 fourths of 12 lemons ? 
 
 21. If 3 fourths of a barrel of flour cost 6 dollars, what 
 will 5 eighths of a barrel cost ? 
 
 22. 3 fourths of 40 dollars is 6 times what a farmer 
 gave for a plough ; required its cost. 
 
 23. A watch cost 30 dollars, and 4 fifths of its cost is 
 twice the cost of the chain ; what was the cost of the chain ?' 
 
84 MENTAL ARITHMETIC. 
 
 24 B's horse cost 200 dollars, and 3 fifths of its cost 
 lacks 80 dollars of being 4 times the cost of the sleigh j 
 required the cost of the sleigh. 
 
 25. A merchant having 20 barrels of flour, sold 8 
 fourths of it to A, and 3 fifths of the remainder to B ; 
 bow much remained ? 
 
 26. Elmina is 25 years old, and 4 fifths of her age iy 
 4 years less than twice Elmira's age; required Elmira's 
 age. 
 
 27. Think of a number, multiply it by 8, divide by 4, 
 multiply by 3, divide by 6, add 20, subtract the number 
 thought of, divide by 4, and name the result. 
 
 28. Think of a number, multiply by 12, divide by 3, 
 multiply by 2, divide by 8, add 12, subtract the number 
 thought of, divide by 4, and name the result. 
 
 29. Multiply 10 by 12, divide by 6, multiply by 5, 
 divide by 2, multiply by 4, divide by 10, and name the 
 result. 
 
 30. How many are 24 multiplied by 6, divided by 3, 
 multiplied by 8, divided by 4 ; multiplied by 2 ; divided 
 by 4 ; divided by 8 ? 
 
 LESSON VIL 
 
 1. How many thirds are there in 4 ? 
 
 SOLUTION. In 1 there are 3 thirds, and in 4 there are 4 times 
 8 thirds, which are 12 thirds. Therefore, c. 
 
 2. How many thirds in 2? 3? 5? 7? 8? 
 8. How many fourths in 3? 5? 6? 4? 7? 
 
 4. How many fifths in 5? 4? 3? 2? 8? 
 
 5. How many sixths in 3? 2? 5? 6? 4? 
 
 6. How many sevenths in 2? 5? 4? 7? 9? 
 
 7 How many eighths in 3? 6? 4? 5? 7? 
 
 8 How many ninths in 8? 4? 6? 3? 10? 
 
MENTAL ARITHMETIC, 35 
 
 * 9. How many thirds in 3 and 2 thirds? In 4 and 
 1 third? 
 
 10. How many fifths in 4 and 3 fifths? In 6 and 3 
 fifths? 
 
 11. How many fourths in 2 and 1 fourth? In 7 and 
 3 fourths ? 
 
 12. How many sevenths in 5 and 6 sevenths? In 3 
 and 4 sevenths? 
 
 13. How many ninths in 3 and 4 ninths? In 6 and 
 7 ninths? 
 
 14. How many sixths in 7 and 5 sixths? In 3 and 2 
 sixths ? 
 
 15. How many eighths in 5 and 3 eighths? In 5 and 
 7 eighths ? 
 
 16. If 5 yards of cloth cost 2 and 1 half dollars, wha* 
 will 6 yards cost? 
 
 17. If 4 pears are worth 2 and 2 third cents, what are 
 7 pears worth? 
 
 18. What cost 10 peaches, at the rate of 4 and 1 half 
 cents for 3 ? 
 
 19. If 11 ducks cost 4 and 2 fifths dollars, what will 
 12 ducks cost? 
 
 20. If 1 half of eight yards of cloth cost 3 and 1 fifth 
 dollars, what cost 3 fifths of 15 yards? 
 
 21. If 2 thirds of 9 apples cost 4 and 4 fifths cents, 
 what will 3 fourths of 12 apples cost? 
 
 22. If 4 fifths of ten pounds of sugar cost 5 and 1 third 
 dimes, what cost 5 sevenths of 14 pounds ? 
 
 23. What cost 4 fifths of 15 yards of muslin, at tho 
 rate of 7 and 1 half dimes for 3 fourths of 20 yards ? 
 
 24. How many whole ones in 6 thirds ? 
 
 SOLUTION. In one there are three thirds ; hence, in six thirds 
 there are as many ones as 3 is contained times in 6, which are 
 2. Therefore, in 6 thirds there are 2 ones. 
 
 25. How many ones in 6 halves? 9 thirds? 12 thirds? 
 
 26. How many ones in 12 fourths ? 20 fourths ? 8 
 fourths ? 
 
36 MENTAL ARITHMETIC. 
 
 27. How many ones in 10 fifths? 12 sixths? 14 
 sevenths ? 
 
 28. How many ones in 16 eighths? 21 sevenths? 24 
 eighths ? 
 
 29. How many ones in 18 ninths? 15 thirds? 25 
 fifths? 
 
 30. How many ones in 28 sevenths? 36 ninths? 24 
 fourths ? 
 
 31. How many ones in 15 thirds ? 20 tenths ? 33 elev- 
 enths ? 
 
 32. How many ones in 48 twelfths? 18 halves? 40 
 tenths ? 
 
 33. How many ones in 9 halves? 7 thirds? 11 fourths? 
 
 34. How many ones in 7 fifths? 16 fifths? 19 eighths ? 
 
 35. How many ones in 20 ninths? 17 sixths? 15 
 fourths ? 
 
 36. How many ones in 13 fifths? 11 thirds? 17 fifths ? 
 
 37. How many ones in 25 eighths ? 23 tenths ? 31 
 ninths ? 
 
 38. If 2 apples cost 6 fifths of a cent, what cost 5 
 apples ? 
 
 39. If 3 pens cost 9 eighths of a cent, what cost 8 
 pens? 
 
 40. If 5 pigs cost 10 sevenths of a dollar, what cost 
 7 pigs? 
 
 41. What cost 6 pairs of boots, if 4 pairs cost 12 fifths 
 dollars ? 
 
 42. What are 8 pies worth, if 3 pies are worth 15 
 fourths cents? 
 
 43. What cost 12 pine apples, if 3 cost 3 fourths of a 
 dollar ? 
 
 44. If 2 books cost 4 fifths of a dollar, what cost 10 
 books? 
 
 45. How much are 9 lamps worth, if 5 are worth 19 
 thirds dollars ? 
 
 46. Required the cost of 8 hats, if 6 cost 12 fourths 
 of a dollar. 
 
MENTAL ARITHMETIC. 37 
 
 47. If 3 peaches cost 9 eighths of a cent, what cost 8 
 peaches ? 
 
 43. How much are 4 mirrors worth, if 7 are worth 14 
 halves dollars? 
 
 49 What cost 3 halves of a yard of linen, if 5 yards 
 cost 10 ninths of a dollar ? 
 
 50. What cost 1 half of 8 yards of cloth, if 1 half of 
 6 yards cost 1 third of 27 fourths dollars ? 
 
 LESSON VIH. 
 
 1. 3 is 1 half of what number? 
 
 SOLUTION. If 1 half of some number is 3, two halves, or that 
 number, is 2 times 3, which are 6. Therefore, &c. 
 
 2. 4 is 1 third of what number? 
 
 3. 6 is 1 fourth of what number? 
 
 4. 5 is 1 sixth of what number? 
 
 5. 6 is 1 half of what number? 
 
 6. 8 is 1 seventh of what number? 
 
 7. 9 is 1 fifth of what number? 
 
 8. 7 is 1 ninth of what number? 
 
 9. 4 is 1 fifth of what number? 
 
 10. 5 is 1 seventh of what number? 
 
 11. 10 is 1 sixth of what number? 
 
 12. 9 is 1 third of what number? 
 
 13. 11 is 1 fourth of what number? 
 
 14. 15 is 1 third of what number? 
 
 , 15. A is 10 years old, which is 1 fifth of B's age ; 
 required B's age. 
 
 16. An apple cost 3 cents, which is 1 fourth of the 
 oost of a melon ] required the cost of the melon. 
 
 17 A sheep cost 6 dollars, which is 1 fifth of the 
 oost of a cow ; what was the cost of the cow ? 
 4* 
 
38 MENTAL ARITHMETIC. 
 
 18. John has 20 marbles, which is 1 third of Henry's 
 number; how many has Henry? 
 
 19. Mary's shawl cost 7 dollars, which is 1 fourth 
 of the cost of her dress ; required the cost of her dress. 
 
 20. Henry found 5 marbles, which is 1 third of 
 what he had ; how many had he then ? 
 
 21. 6 is 1 half of 3 times what number? 
 
 22. 5 is 1 fourth of 2 times what number? . 
 
 23. 8 is 1 third of 4 times what number? 
 
 24. 10 is 1 fourth of 8 times what number ? 
 
 25. 9 is 1 half of 6 times what number? 
 
 26. 7 is 1 sixth of 3 times what number? 
 
 27. 12 is 1 third of 9 times what number? 
 
 28. 4 is 1 ninth of 6 times what number ? 
 
 29. 11 is 1 sixth of 3 times what number? 
 
 30. Flora's cloak cost 10 dollars, which is 1 third of 
 6 times the cost of her dress ; required the cost of hei 
 dress. 
 
 31. A watch was bought for 20 dollars, which is 1 
 third of 4 times what the chain cost; required the cost 
 of the chain. 
 
 32. The head of a fish is 6 inches long, which is 1 
 fou th of 3 times the length of the body; what is the 
 length of the body ? 
 
 33. A. slate cost 20 cents, which is 1 fifth of 4 times 
 the cost of an arithmetic ; required the cost of them both. 
 
 34. A boy lost 15 cents, , which is 1 fourth of 5 times 
 the money he had remaining ; how much money had he 
 at first ? 
 
 35 Mary found 12 pins, which is 1 fifth of 3 times 
 what she then had; how many had she at first? 
 
 86. 8 is 1 third of 1 half of what number ? 
 
 37. 4 is 1 fifth of 1 third of what number ? 
 
 38 3 is 1 fourth of 1 fifth of what number? 
 
 39. 2 is 1 eighth of 1 fourth of what number? 
 
 40. 3 is 1 seventh of 1 sixth of what number ? 
 
 41. 4 is 1 tenth of 1 fifth of what number ? 
 42 5 is 1 half o^ 1 seventh of what number ? 
 
MENTAL ARITHMETIC. 39 
 
 43. 6 is 1 fifth of 1 third of what nurnbei f 
 
 44 Philip's vest cost 4 dollars, which is 1 lialf of 1 
 
 third of the cost of his coat; what was the cost of his 
 
 coat? 
 
 45. Martin is 4 years old, and his age is 1 third of 1 
 fourth of his father's age ; how old is his father ? 
 
 46. Ella's bonnet cost 3 dollars, which is 1 fifth of 1 
 half of the cost of her shawl ; required the cost of the 
 shawl 
 
 47. A paid 10 dollars for a saddle, which is 1 fifth 
 of 1 third of the cost of his horse ; required the cost of 
 the horse. 
 
 48. Mr. A walked 8 miles, which is 4 times 1 seventh 
 of the distance he rode ; how far did he travel ? 
 
 49. Benton's house cost 2000 dollars, which is 1 half 
 of 4 times the cost of his barn ; required the cost of both. 
 
 50. The head of a fish is 3 inches long, and the tail 5 
 inches, which is 1 half of 1 third of the length of the 
 body ; required the length of the fish. 
 
 LESSON IX. 
 
 1. 6 is 2 thirds of what number? 
 
 SOLUTION. If 2 thirds of some number is 6, 1 third of that 
 number is 1 half of 6, which is 3, and 3 thirds, or that 
 number, is 3 times 3, or 9. Therefore, &c. 
 
 2. 9 is 3 fourths of what number? 
 
 3. 6 is 2 thirds of what number? 
 
 4. 10 is 2 fifths of what number ? 
 5 12 is 4 sixths of what number ? 
 
 6. 10 is 5 sevenths of what number? 
 
 7. 8 is 4 ninths of what number ? 
 
 8. 9 is 3 fifths of what number? 
 
 9. 15 is 5 sixths of what number? 
 10. 10 is 5 eighths of what number? 
 
40 MENTAL ARITHMETIC. 
 
 11. 16 is 8 ninths of what number? 
 
 12. 14 is 7 thirds of what number ? 
 
 13. Frank is 12 years old, and his age is 3 fifths of 
 Fanny's age ; how old is Fanny ? 
 
 14. Augustus gave his brother 10 peaches, which is 
 2 thirds of all he had ; how many had he ? 
 
 15. Hampton lost 8 cents, which is 4 fifths of Lis 
 money ; how much had he remaining ? 
 
 16. A lady found 12 dollars, which is 4 sixths of 
 what money she then had ; how much had she at first ? 
 
 17. Lester sold a cow for 24 dollars, which is 6 fifths 
 of the cost of the cow ; required its cost. 
 
 18. A farmer sold a colt for 80 dollars, and thereby 
 gained 1 fifth of the cost of the colt ; required the cost, 
 
 19. Mr. M is 20 years of age, and 4 fifths of his age 
 is 8 ninths of his brother's age ; what is the age of his 
 brother ? 
 
 20. Frank lost 12 marbles, which is 2 fifths of what 
 he had at first ; how many remained ? 
 
 21. 10 is 1 half of 4 fifths of what number? 
 
 22. 12 is 1 third of 6 sevenths of what number? 
 
 23. 9 is 1 fourth of 4 fifths of what number ? 
 
 24. 16 is 2 fifths of 10 fourths of what number ? 
 
 25. 4 is 2 fifths of 5 sixths of what number ? 
 
 26. 6 is 3 fourths of 4 fifths of what number ? 
 
 27. 15 is 5 sixths of 6 sevenths of what number? 
 
 28. 14 is 7 fourths of 4 thirds of what number ? 
 
 29. 18 is 9 eighths of 4 sevenths of what numbei ? 
 
 30. 20 is 5 fourths of 8 thirds of what number? 
 
 81. Thomas sold a book for 40 cents, which is 4 
 fifths of 5 sixths of the cost j required the zost. 
 
 32 Smiley sold his horse for 140 dollars, which is 7 
 eighths of 4 thirds of its value ; required its value. 
 
 33. A' s hat cost 6 dollars, which is 3 fourths of 4 
 fifths of the cost of his vest ; required the cost of his vest. 
 
 34. 20 feet of a pole is in the water, which is 2 fifths 
 of 5 sevenths of the length in the air; what is the length 
 of the pole ? 
 
MENTAL ARITHMETIC. 41 
 
 35 A pole is 30 feet in the air, which is 3 fifths of 2 
 fourths of the length of the pole ; required the length in 
 the mud and water. 
 
 36. A cow cost 24 dollars, which is 6 tenths of 2 fifths 
 of the cost of the cow and a horse ; what was the cost of 
 the horse ? 
 
 37. A man sold his watch for 60 dollars, which is 5 
 fourths of 4 times what the chain cost, and the watch 
 cost 3 times as much as the chain ; what was the cost of 
 each ? 
 
 38. A's horse cost 200 dollars, and 4 fifths of the cost 
 f the horse is 8 times the cost of his harness } required 
 
 the cost of the harness. 
 
 39. A man has 24 geese, and 3 fourths uf the numbei 
 of geese equals 2 times the number of turkeys; how 
 many turkeys had he ? 
 
 40. A man sold his horse and sleigh for 200 dollars, 
 and 4 fifths of this is 8 times what his sleigh cost, and 
 the horse cost 10 times as much as the sleigh ; required 
 the cost of each. 
 
 LESSON X. 
 
 1. 6 are how many times 1? 2? 3? 
 
 2. 8 are how many times 1? 2? 4? 
 
 3. 12 are how many times 2? 3? 6? 
 
 4. 20 are how many times 4? 5? 10? 
 
 5. 32 are how many times 4? 8? 16? 
 
 6. 30 are how many times 5? 6? 10? 
 
 7. 72 are how many times 6? 8? 9? 
 8 80 are how many times 4? 8? 20? 
 9. What is the relation of 8 to 2? 
 
 REMARK 8 is 4 times 2. 
 
 10. What is the relation of 12 to 4? Of 15 to 5? 
 
 11 What is the relation of 16 to 8 ? Of 18 to 6? 
 
 12 What is the relation of 21 to 7 ? Of 24 to 8 ? 
 
42 MENTAL ARITHMETIC. 
 
 13 What is the relation of 32 to 4 ? Of 27 to 9 ? 
 
 14 What is the relation of 40 to 10? Of 45 to 5? 
 
 15. What is the relation of 28 to 7? Of 36 to 4? 
 
 16. If 4 yards of cloth cost 10 dollars, what will 8 
 yards cost ? 
 
 SOLUTION. If 4 yards of cloth cost 10 dollars, 8 yards, whicn 
 are 2 times 4 yards, will cost 2 times 10 dollars, or 20 dollars. 
 Therefore, &c. 
 
 17. If 3 bunches of grapes cost 8 cents, what will 6 
 bunches cost ? 
 
 18. If 6 combs cost 9 cents, what will 12 combs cost 
 at the same rate ? 
 
 19. If 7 peaches cost 8 cents, what will 21 peaches 
 cost? 
 
 20. If 5 pairs of shoes cost 9 dollars, what will 20 pains 
 cost? 
 
 21. If 4 pens cost 11 cents, what will 12 pens cost? 
 
 22. If 8 ducks cost 5 dollars, what will 24 ducks cost? 
 
 23. What cost 30 lamps, if 5 lamps cost 7 dollars ? 
 
 24. How much will 42 primers cost, at the rate of 6 
 primers for 20 cents ? 
 
 25. What cost 56 inkstands, if 7 inkstands cost 5 dol- 
 lars? 
 
 26. If 5 pitchers cost 3 dollars, what will 45 pitchers 
 cost? 
 
 27. How far will a man travel in 48 days, if he travel 
 30 miles in 4 days ? 
 
 28. If 6 men can build 10 rods of wall in a certain 
 tame, how many rods can 54 men build in the same time? 
 
 29. Hiram bought 6 pigs for 11 dollars; how many 
 could he have bought for 44 dollars ? 
 
 30. 7 men earn 12 dollars in 3 days; how much could 
 they earn in 27 days ? 
 
 31. If 5 peaches are worth one pear, how many pears 
 are 30 peaches worth ? 
 
 32. If 8 dollars will buy 5 gold pens, how many wiU 
 56 dollars buy ? 
 
MENTAL ARITHMETIC. 43 
 
 33. If 6 stands cost 2 thirds of 12 dollars, what will 
 30 stands cost? 
 
 34. 40 dollars is 2 thirds of what A gave for sheep, 
 at the rate of 10 dollars for 3 sheep; how many did he 
 purchase ? , 
 
 35. If 4 men can perform a piece of work in 18 days, 
 how long will it require 12 men to do it ? 
 
 36. If 6 men can build a boat in 10 fourths days, how 
 long will it take 3 men to build it ? 
 
 37. 15 dollars is 3 eighths of what A earns in 5 days j 
 how much will he earn in 15 days ? 
 
 38. 18 men are 3 fifths of the number required to mow 
 a field in 8 days ; how many men would be required to 
 mow it in 24 days ? 
 
 39. 20 dollars is 4 dollars more than 2 thirds of 4 
 times what B paid for a chain, and his watch cost 5 timee 
 as much as the chain ; required the cost of each ? 
 
 LESSON XL 
 
 1. 4 is what part of 8 ? 
 
 SOLUTION. 4 is 1 half of 8, since 2 times 4 are 8. 
 
 2. 3 and 6 are what parts of 12 ? 
 
 3. 4 and 8 are what parts of 16 ? 
 
 4. 3 and 6 are what parts of 24 ? 
 
 5. 7 and 3 are what parts of 21 ? 
 
 6. 4 and 9 are what parts of 36 ? 
 
 7. What is the relation of 2 to 6 ? Of 4 to 8 ? 
 
 8. What is the relation of 3 to 9 ? Of 5 to 10 ? 
 
 9. What is the relation of 3 to 12 ? Of 4 to 16 ? 
 10. What is the relation of 5 to 20 ? Of 6 to 36 ? 
 
 11 What is the relation of 7 to 42 ? Of 8 to 40 ? 
 
 12 What is the relation of 6 to 54 ? Of 9 to 27 ? 
 
44 MENTAL ARITHMETIC. 
 
 13. What is the relation of 10 to 40? Of 7 to 56? 
 
 14. What is the relation of 11 to 55? Of 12 to 48? 
 
 15. If 6 apples cost 10 cents, what wiL 3 apples cost? 
 
 REMARK. 3 apples, the half of 6, will cost one-half of 10 
 cents. 
 
 16. How much will 5 books cost, if 20 books cost 16 
 dollars? 
 
 17. What cost 3 knives, if 18 knives cost 24 dollars? 
 
 18. If 14 pencils cost 35 cents, what will 2 pencils 
 cost? 
 
 19. If 10 peaches are worth 12 oranges, how many 
 oranges are 5 peaches worth ? 
 
 20. How much will 4 apples cost, if 16 apples cost 24 
 cents ? 
 
 21. How much will 9 pigs cost, if 27 pigs cost 36 
 dollars ? 
 
 22. What cost 7 tons of hay, if 56 tons cost 96 
 eagles ? 
 
 23. If 42 sheep are sold for 108 dollars, what are 7 
 sheep sold for? 
 
 24. If 100 pens cost 50 cents, what, will 20 pens 
 cost ? 
 
 25. What cost 5 inkstands, if 15 inkstands cost 2 
 thirds of 18 dimes ? 
 
 26. If 5 eighths of 32 hens cost 10 dollars, what will 
 
 2 thirds of 6 hens cost? 
 
 27. If 3 fourths of 48 oranges cost 40 cents, what will 
 
 3 fourths of 1 2 oranges cost ? 
 
 28. If A walked 132 miles in 33 days, how far did he 
 walk in 3 days ? 
 
 29. If 8 times 6 cents is 48 cents, how much is 2 
 times 6 cents ? 
 
 30. Mary having 27 roses gave 1 third of them to 
 Sallie and 1 third to Annie 5 how many remained ? 
 
 31. A worked 5 weeks for 7 dollars a week, and re- 
 ceived in payment 12 bushels of wheat worth 1 dollar 
 and 1 half a bushel; how much remains due A? 
 
MENTAL ARITHMETIC. 45 
 
 32 A farmer gave 2 thirds of 15 bushels of rye, worth 
 
 dimes a bushel, for cloth worth 3 dollars a }ard ; how 
 many yards did he receive ? 
 
 33. Think of a number, multiply by 10, divide by 5, 
 maltiply by 3, divide by 6, add 30, subtract the original 
 number, divide by 10, add 7, and the result is what ? 
 
 3 . What is the value of 27 multiplied by 8, divided 
 by 4, multiplied by 6, divided by 3, multiplied by 9, 
 divided by 12 ? 
 
 LESSON XII. 
 
 1. What numbers multiplied together will produce 4? 
 8? 10? 16? 12? 18? 24? 
 
 2. What numbers multiplied together will produce 15? 
 21? 28? 35? 36? 39? 48? 
 
 3. What numbers multiplied together will produce 40? 
 42? 45? 49? 50? 51? 52? 
 
 Numbers which can be produced by multiplying together other 
 numbers, each of which is greater than a unit, are called com- 
 posite numbers. 
 
 Numbers which cannot be produced by multiplying together 
 two or more numbers, each of which is greater than a unit, are 
 called prime numbers. 
 
 4. Tell which of the following numbers are prime, and 
 which composite; 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 
 16, 17, 18, 19, 20, 21, 22, 23. 
 
 5. Name the prime and composite numbers in the fol- 
 lowing list; 24, 25, 26, 27, 28, 29, 30, 31 32, 33, 34. 
 85, 36, 37, 38, 39, 40, 41. 
 
 The numbers, which when multiplied together will produce d 
 composite number, are called factors of that number. 
 
 6 What are the factors of 12 1 20 ? 16 ? 33 ? 30 ? 24? 
 27? 18? 26? 32? 
 5 
 
 .iVERSlTY 
 
16 - MENTAL ARITHMETIC 
 
 7. What are the factors of 9? 10? 14? 34 f 36? 40? 
 
 48? 50? 56? 60? 63? 72? 
 
 When these factors are prime numbers, they are called prime 
 factors. 
 
 8. What are the prime factors of 4 ? 6 ? 9 ? 12 ? 15 ? 1 8 ? 
 
 9. What are the prime factors of 10? 20? 21? 22 v 
 24? 25? 
 
 10. What are the prime factors of 27? 28? 30? 32? 
 33? 35? 
 
 11. What are the prime factors of 44? 45? 46? 48? 
 49? 50? 
 
 12. What are the prime factors of 52? 54? 55? 56? 
 57? 60? 
 
 13. What are the prime factors of 64? 68? 70? 72? 
 75? 80? 
 
 14. What prime factors are common to 6 and 12 ? 
 
 15. What prime factors are common to 9 and 12 ? 
 
 16. What prime factors are common to 8 and 20 ? 
 
 17. What prime factors are common to JO and 20? 
 
 18. What prime factors are common to 12 and 18? 
 
 19. What prime factors are common to 16 and 24? 
 
 20. What prime factors are common to 14 and 42 ? 
 
 21. Is a number exactly divisible by any numbers ex- 
 cept its prime factors, or some product of them ? 
 
 22. Name the divisors of 4, 6, 8, 10, 12, 14, 16, and 20. 
 
 23. What divisors are common to 4 and 6? To 8 
 and 10? 
 
 24. What divisors are common to 6 and 9? To 9 
 and 18? 
 
 25. What divisors are common to 10 and 30? To * 
 and 24? 
 
 26. What divisors are common to 9 and 27? To 10 
 and 20? 
 
 27. What divisors are common to 16 and 24? To 12 
 and 18? 
 
 A divisor common to two or more numbers is called their 
 common divisor. 
 
MENTAL ARITHMETIC. 47 
 
 28 What is a common divisor of 8 and 24 ? Of 9 
 and 15? 
 
 29. What is a common divisor of 15 and 20? Of 18 
 and 80 ? 
 
 30. What is a common divisor of 16 and 32 ? Of 32 
 and 40? 
 
 The greatest divisor common to two or more numbers is called 
 their greatest common divisor. 
 
 31. What is the greatest common divisor of 4 and 8? 
 
 32. What is the greatest common divisor of 8 and 24? 
 
 33. What is the greatest common divisor of 3 aod 27? 
 34 What is the greatest common divisor of 16 and 24 ? 
 
 35, What is the greatest common divisor of 24 and 32 ? 
 
 A multiple of a number is any number that will contain it, a 
 vshole number of times, without a remainder. 
 
 36. What is a multiple of 4? Of 3? Of 5? Of 6? Of 8? 
 
 87. What is a multiple of 7? Of 9? Of 10? Of 
 
 of 20? 
 
 A multiple common to two or more numbers is called their 
 common multiple. 
 
 88. What is a common multiple of 2 and 3 ? Of 3 and 
 4? Of 4 and 5? Of 5 and 6? 
 
 39. What is a common multiple of 6 and 7? Of 4 
 and 6 '( Of 5 and 10 ? Of 9 and 12 ? 
 
 Tlu least multiple, common to two or more numbers, is called 
 their k-w common multiple. 
 
 40. What is the least common multiple of 4 and 6? 
 Of 6 and S ? Of 8 aud 10 ? Of 10 and 12 ? 
 
 41 What is the least common multiple of 8 and 12? 
 Of 9 and 6 > Of 9 and 12? Of 12 and 20? 
 
 If a number be multiplied by itself, the result is called the 
 square of the number; if the square be multiplied by the num- 
 ber, the result is the cube, it the cube be multiplied by the 
 number, the result is the fourth power, c. 
 
 42. What is the square, cube, and fourth power of Ijf 
 2? 3? 4? 5? 6? TV S? 9? 10? 11? 12? 
 
48 MENTAL ARITHMETIC. 
 
 The square root of a number is one of the two equal factors 
 which produce that number; the cube root, one of the throe 
 equal factors ; the fourth root, one of the four equal factors. 
 
 43. What is the-square root of 1? 4? 9? 16? 25? 
 49? 81? 36? 64? 
 
 44 What is the cube root of 1? 8? 27? 64? 125? 
 313? 729? 512? 
 
 45. What is the fourth root of 1? 16? 81 ? 256? 625? 
 .46. Define a prime number, composite number, factors, 
 prime factors, common divisor, least common divisor, com- 
 mon multiple, least common multiple, square, cube, and 
 fourth power; square, cube, and fourth roots. 
 
 SECTION III. 
 
 LESSON I. 
 
 The number of parts into which anything is divided, instead 
 of being expressed by a word, may be represented by a figure 
 beneath a line, thus: 
 
 U represents halves, 4- represents fourths, 
 
 5 represents thirds, g represents fifths. 
 
 The number of fractional parts taken may be represented by 
 a figure above the line, thus: 
 
 | represents 2 fourths, f represents 5 sevenths, 
 
 I represents 3 fifths, J represents 7 eighths, 
 
 | represents 4 sixths, | represents 4 ninths. 
 
 These expressions are called fractions. A fraction is that 
 definite part which a portion is of the whole. Custom, however, 
 has sanctioned the use of the word fraction to denote both the 
 definite part and the expression of it. The expression consists 
 of two figures with a line between them. 
 
 The figure be".ow the line is the denominator; it denotes the 
 number of equal parts into which the unit is divided. 
 
MENTAL ARITHMETIC. 49 
 
 The figure above the line is the numerator; it denotes he 
 number of equal par, 3 taken. 
 
 A proper fraction is one whose value is less than a unit ; ae 
 
 I.I- 
 
 An improper fraction is one whose value is equal to, cr greater 
 than a unit; as 4 7 
 
 A mixed number consists of a whole number and a traction j 
 as, 2i, 3|. 
 
 The reciprocal of a number is a unit divided by that number ; 
 thus the reciprocal of 4 is 1 ; of 7 is 1. 
 
 1. How many halves in 3^? 2|? 4^? 64? 
 
 2. How many thirds in 3|? 2|? 4| ? 5|? 
 
 3. How many fourths in 3| ? 4| ? 2| ? 7| ? 
 
 4. How many fifths in 11? 2|? 3|? 4|? 
 
 5. How many sixths in 21? 3|? 4|? 5|? 
 
 6. How many eighths in 2|? 61? 7|? 8|? 
 
 7. How many thirds in 5| ? 7|? 9|? 10|? 
 
 8. How many sevenths in 3^? 51? 4|? 2^? 
 
 9. How many fifths in 5|? 4J? 7|? 8|? 
 
 10. How many ninths in 2|? 3|? 7|? 6|? 
 
 11. How many tenths in 5 T 3 ? 7f<j? 3 T ? 6 T 7 ? 
 
 12. If one yard of tape cost 21 cents, how many third? 
 cents will 5 yards cost? 
 
 13. How many fourths of a dollar will 7 baskets of 
 peaches cost, at 2^ dollars a basket ? 
 
 14. What cost 3 barrels of flour, if 21 barrels cost 18 
 dollars ? 
 
 15. How far can Henry walk in 7 hours, if he can 
 walk ]2*uiiles in 2| hours? 
 
 16 If 3^ yards of muslin cost 44 cents, how much 
 will 10 yards cost ? 
 
 17. 2 \ bushels of apples cost ^ dollars, what will 2J 
 bnshels cost ? 
 
 18. How many ones in |? |? I*? |? 
 
 19. How many ones in !? | 9 u>? 12? 
 
 20. How many ones in f ? \ 4 ? y ? y ? 
 
 21. How many ones in |? U ? l f ? 21 ? 
 
 22 How many ones in 4 ? if ? ^ ? || ? 
 
 23 How man^ cnes in C 5 ? if" l g? y"? 
 6* 
 
50 MENTAL ARITHMETIC. 
 
 24. How many ones in y ? is ? u ? u> ? 
 25 How many ones in y* ? 2 / ? f*? 2^7? 
 
 26. How many ones in U ? 2_*? ||? i| ? 
 
 27. Reduce 3, 7f , 8-J, and 2^ to improper fractions. 
 
 28. Reduce y, 1 / ) i^ an( j 2.3 to mixed numbers. 
 
 29. Reduce 5|, 6|, 7f , and 8| to improper fractions. 
 
 30. Reduce ~ T 4 , 2 H 5 , ^ 6 , and f J to mixed numbers. 
 
 31. Reduce 2|, 3|, 4 T 5 Q, and 5 T <Y to improper fractions, 
 
 32. At | of a dime a pound, what will 9 pounds of 
 candies cost ? 
 
 83. If 1 pound of sugar cost 6| cents, what will 8 
 pounds cost? 
 
 SOL. 8 pounds will cost 8 times 6f cents. 8 times 6 is 48, 
 8 times f is ^ or 5J; 48 plus 5J equals 53J. 
 
 34. At 6 1 dimes a bushel, what will 12 bushels of 
 wheat cost ? 
 
 35. What cost 8 barrels of apples, at the rate of 3| 
 dollars a barrel ? 
 
 36. How much will a man earn in a week, at the rate 
 of 2 1 dollars a day? 
 
 37. At the rate of 18| cents a dozen, what will 3 
 dozen eggs cost? 
 
 38. What cost 5| boxes of butter, at the rate of 3| 
 dollars for 10 boxes ? 
 
 39. What cost 7 barrels of apples, at the rate of 9 
 dollars for 2i barrels ? 
 
 40. How many chestnuts must be given for 12 1 cents, 
 if 14 chestnuts cost 3i cents ? 
 
 41. What cost 5| pounds of beef, if 2 pounds cost 
 32 cents ? 
 
 42. How far can a man travel in 5| hours, if he can 
 walk 19 miles in 2-^ hours ? 
 
 43. A vessel sailed 23 miles in 4| hours; how far did 
 she sail in 2| hours ? 
 
 44. A kite arose 48 rods in 3| minutes; how far at 
 this rate will it ascend in 3^ minutes ? 
 
 45. What cost 50 pounds of meat, at the rate of 4f 
 pounds for 5| cents ? 
 
MENTAL ARITHMETIC. 5] 
 
 46. What cost 10 yards of ribbon, if 4J yards cost 6| 
 cents ? 
 
 47. If f of a pint of almonds cost |f of a shilling, 
 what cost 5 1 pints ? 
 
 48. If 4 of a ton of hay is worth 4| dollars, what cost 
 7 1 tons of hay ? 
 
 49. If a m.m walk 3 miles in 2^ hours, ho^w far will 
 he walk in ^ hours? 
 
 
 LESSON II 
 
 SOLUTION. In 1 there are 4, and iu ^ there are | of |, or | 
 Therefore, &c. 
 
 
 1. How many fourths in 
 
 SOLUTION. In 1 there are 4, t 
 jrefore, &c. 
 
 2. How many sixths in 2? ' 1^ f^ 1^ 
 
 3. How many eighths in <|? |? f? |? 
 
 4. How many tenths in ? ? f ? |? 
 
 5. How many twelfths in | ? ^ ? ? | ? 
 
 6. How many fourteenths in^? |? 4^ f ^ 
 
 7. How many fifteenths in | ? |? |? |? 
 
 8. How many sixteenths in |? |? ? |? 
 
 9. How many eighteenths in -| ? |? |? -|? 
 
 10. How many twentieths in |? |? y^? |? 
 
 11. Reduce |, |, and | to twelfths. 
 
 12. Reduce J, |, |, and T 9 Q to twentieths. 
 
 13. Reduce ^, 4. |, T % and ji to thirtieths. 
 
 14. Reduce |, |, T 7 -. | ? and ^ to fortieths. 
 
 When fractions have the same denominator they are sai'-l to 
 have a common denominator. 
 
 15. Reduce | and | to a common denominator 
 
 16. Reduce | and | to a common denominator. 
 
 17. Reduce ^ and i to a common denominator. 
 
 18. Reduce i and | to a common denominator. 
 
 19. Reduce | arid | to a common denominator. 
 
 20. Rsduce i and I to a common denominator. 
 
62 MENTAL ARITHMETIC. 
 
 21 Reduce | and f to a common denominator. 
 
 22. Reduce and | to a common denominator. 
 
 23. Reduce | and | to a common denominator. 
 
 24. Reduce T 6 ami | to a common denominator. 
 
 25. Reduce | and | to a common denominator. 
 
 26. Reduce f and T 3 ^ to a common denominator. 
 
 27 Rqduce i, |, and \ to a common denominator 
 
 28 Reduce |, |, and J to a common denominator. 
 29, Reduce ^, ^, and ^ to a common denominator. 
 SO. Reduce |, , and y 1 ^ to a common denominator. 
 
 The five following question are designed to illustrate the 
 manner in which the pupil may be led to derive, by induction, 
 an abbreviated method of obtaining the same results as those 
 given by the analysis. Similar questions occur in several lessons, 
 and claim particular attention. 
 
 31. Since | J, by what number may you multiply 
 both numerator and denominator of f to obtain | ? 
 
 32. Since f = T 9 2 , by what number may you multiply 
 both numerator and denominator of | to obtain T 9 2 ? 
 
 33. Does it change the value of a fraction to multiply 
 both numerator and denominator by the same number ? 
 
 34. By what must you multiply the numerator and 
 denominator of | to reduce it to tenths ? 
 
 35. By what must you multiply both numerator and 
 denominator of f to reduce it to twentieths ? 
 
 36. Reduce 
 
 38. Reduce f, , j 
 89. Reduce 1, %, | 
 40 Reduce 
 
 2' 3> 6 
 
 and T 8 to twentieths, 
 and T 7 to thirtieths, 
 and | to twelfths. 
 and ^ to eighteenths, 
 and I to sixteenths. 
 
 41. If 2^ yards of silk cost 20 dimes, what will 4 
 yards cost? 
 
 42. If 2 yards of tape cost 13 cents, what will 3 
 yards cost? 
 
 43. What cost 3 pounds of sugar, if 2\ pounds cost 
 25 cents ? 
 
 44. Twice 18 is | of a certain number; required the 
 number. 
 
MENTAL ARITHMETIC. 53 
 
 45. Mary lost 20 roses, which is | as many as she 
 then had; how many had she at first? 
 
 46. John found 60 cents, which is | of | of what he 
 then had; how much had he at first? 
 
 47. If | of | of a yard of lace cost 8 cents, what will 
 | of 9 yards cost ? 
 
 48. Henry gave his sister 20 cents, which is | of 
 what he had at first more than his sister, and ^ of what 
 she now has; how much had each at first? 
 
 LESSON III. 
 
 1. How many thirds are equal to | ? 
 
 SOLUTION. 1 is equal to |, therefore 1 of the number of 
 sixths equals the number of thirds ; 1 of 4 is 2. Therefore, &. 
 
 2. How many halves in 
 
 3. How many thirds in 
 
 4. How many fourths in 
 
 5. How many sixths in 
 
 6. How many eighths in 
 
 7. How many fifths in 
 
 8. How many sevenths in 
 
 9. How many ninths in 
 
 10. How many tenths in 
 
 11. Since J = |, by what may we divide both nume- 
 rator and denominator of | to produce | ? 
 
 12. Does dividing both numerator and denominator of 
 a fraction, by the same number, change its value ? 
 
 13. By what number must we divide both numerate! 
 and denominator of | to reduce it to fourths ? 
 
 14. By what must we divide both numerator and de- 
 nominator of -, 5 Q to reduce it to, halves? 
 
 15 Reduce f fl to fifths, and T 3 ^ to fourths. 
 
 16. Reduce ^ to halves, and 7 y 2 to fourths. 
 
 17. Reduce 9 j to fourths, and | to thirds. 
 
64 MENTAL ARITHMETIC 
 
 18. Reduce ^g to fifths, and ^ 4 - to sixths. 
 
 19. Reduce 2 9 T to sevenths, and ^| to ninths. 
 
 20. Reduce -J| to eighths, and |'jj to tenths. 
 
 21. Reduce || to tenths, and || to twelfths. 
 
 22. Reduce f f to ninths, and f f to elevenths. 
 
 When a fraction cannot be reduced to an equivalent one hav- 
 ing a less denominator, it is said to be in its lowest terms. 
 
 23. Reduce | and T 6 5 to their lowest terms. 
 
 24. Feduce {S and T 9 ^ to their lowest terms. 
 
 25. Reduce || and -f-% to their lowest terms. 
 26 Reduce J and || to their lowest terms. 
 
 27. Reduce || arid f | to their lowest terms. 
 
 28. Reduce |i and f | to their lowest terms. 
 
 29. Reduce || and f | to their lowest terms. 
 
 30. If 8 is I of some number, what is | of the same 
 number ? 
 
 31. If 6 is | of some number, what is | of 3 times the 
 same number? 
 
 32. If 8 is | of some number, what is | of 2 times the 
 same number? 
 
 33. Henry's horse cost 90 dollars, which is T 9 a of 5 
 times the cost of the sleigh; required the cost of the 
 sleigh. 
 
 34. 4 times 50 years is 10 years less than 10 times the 
 age of James j how old is he? 
 
 35. If 4 horses eat 2 tons of hay in 8 weeks, aow long 
 will it require 5 horses to eat the same ? 
 
 36 If 8 men can build a boat in 16 days, how long 
 will it require 32 men to build it ? 
 
 37. How many lemons will pay for 7 melons, if 6 
 ksinons are worth 4^ melons ? 
 
 38. If it require 8| yards of cloth to make 2 coats, 
 how much will be required to make 9 coats ? 
 
 39. 42 dollars is f of all the money A has, and B haa 
 3 times as much ; how much money has B ? 
 
 40. What number being multiplied by 6, divided by 3, 
 multiplied by 5, divided by 2, and 10 added, equals 30? 
 
 41 . What number being multiplied by 8 ; divided by 
 
MENTAL ARITHMETIC 55 
 
 4, multiplied by 3, divided by 2, and 10 subtracted, 
 equals 14 ? 
 
 42. A -gave B 48 cents, and | of this is 4 time?, as 
 many as he had remaining; how much had he at first ? 
 
 43. Amanda having 50 pins, lost | of them, and theu 
 found | as many as remained ; how many had she then ? 
 
 44. A watch cost $90, which is | of 10 times what 
 the chain cost; required the cost of both. 
 
 45. Mary gave Lilly 24 pins, which is f of what Lilly 
 already had, and | of what Mary had remaining; how 
 many had each at first? 
 
 LESSON IV. 
 
 1. What is the sum of \ and f ? 
 
 2. What is the sum of f and f ? 
 
 3. What is the sum of | and | ? 
 
 4. What is the sum of | and | ? 
 
 5. What is the difference between f and f ? 
 fi. What is the difference between | and | ? 
 
 7. What is the difference between | and |j ? 
 
 8. What is the difference between 2| and If 1 
 
 9. What is the difference between 3| and 2|i 
 
 10. How many fourths in -| and | ? 
 
 11. How many eighths in | and |? 
 
 12. How many tenths in \ and 1? 
 
 13. How many twelfths in | and I? 
 14 How many fifteenths in | and |? 
 
 15. How many sixteenths in | and g ? 
 
 16. How many eighteenths in | and | ? 
 
 What is the sum 
 
 17. Of ' and I? 21. Off and ? 
 
 18. Of l and I? 22. Of f andf? 
 
 19. Of 4 and if 23. Of | and |? 
 20 O/ \ and |? 24. Off and ? 
 
56 MENTAL ARITHMETIC. 
 
 25 'Of and ? 36. Of 2' and 34 ? 
 
 26 Of | and |? 37. Of 3j and 4J ? 
 
 27. Of | and I? 38. Of 2 and If? 
 
 28. Of and f ? 39. Of 34 and 2* ? 
 
 29. Of J, i, and |? 40. Of 6* and 5 ? 
 
 30. Of i, i, and ? 41. Of 4J and 5$? 
 
 31. Of |, |, and ? 42. Of 6 and 5| ? 
 
 32. Of | and |? 43. Of 7-3 and 81? 
 
 33. Off and T y 44. Of l|, 2f , and 3|? 
 
 34. Of | and |? 45. Of 24, 3}, and 4 b l ? 
 
 35. Of I and I? 46. Of 3|, 2j, and 4? 
 
 |? 
 llar; 
 
 47. Thomas having | of a dollar, found f of a dol 
 how much had he then ? 
 
 48. A man having | of a barrel of flour, bought | of 
 a barrel ; how much had he then ? 
 
 49. Mary having J of a dozen of pins, found | of a 
 dozen ; how many pins had she then ? 
 
 50. Peter having | of a certain sum of money, found 
 of the same sum, and then had $21 ; how much was 
 the sum ? 
 
 51. | of a certain number, increased by | of the same 
 number, equals 34; required the number. 
 
 52. Fanny's number of roses, increased by ^ and -| of 
 her number, equals 55 ; how many roses has she ? 
 
 53. | of A's money increased by its |, equals 69 doll 
 lars j how much money has A ? 
 
 54. B gave $24 for a watch, and | + f of this is 4 
 times what he paid for a chain ; required the cost of the 
 chain. 
 
 55. If } of James's age, increased by ^ and | of his 
 age, equals 57 years, what is James's age ? 
 
 Subtract 
 
 56 l from ^. 61. f from f 
 
 57. from \. 62. f from | 
 
 58. | from f . 63. | from f . 
 
 59. i from \ 64. | from f . 
 
 60. i from | 65. 4 from I. 
 
MENTAL ARITHMETIC. 57 
 
 66 ^ from |. 72. from f . 
 
 67 | from |. 73. | from |. 
 
 . 68 | from f . 74. 2^ from 3f 
 
 69 | from f. 75. 3 from 4-]-. 
 
 70. | from f 76. 21 from 3 
 
 71. { from . 77. 3 from 5|, 
 
 78 What number is that, which being diini Dished by 
 its f , equals 36 ? 
 
 79 Joshua's age, diminished by its |, equals 20 years j 
 how old is Joshua ? 
 
 80. A certain sum of money, diminished by its J and J, 
 equals 15 dollars; required the sum. 
 
 81. What number is that, which being increased by 
 its 1 and diminished by its |, equals 130 ? 
 
 82. A boy having 36 marbles, lost | of them, and then 
 found | as many as he had at first ; how many had he 
 remaining? 
 
 83. $40 is 4 times what A paid for a chain, and the 
 cost of the chain, increased by its 11 tenths, is { of the 
 cost of his watch; required the cost of the watch.. 
 
 84. Janson's age, diminished by its \ and 1, is 22 
 years, which is | of his uncle's age ; required the age 
 of each. 
 
 85. | of B's money, increased by the difference be 
 tween | and J of his money, equals 190 dollars \ required 
 his money. 
 
 What is the value 
 
 86. Of 1 + 4 |? 93. Of 2-| + 4^ 41? 
 
 87. Of i + l i? 94. Ofli + lJ 2|? 
 
 88. Of | + |i? 95. Of3| + 2| 51? 
 89 On + | |? 96. Of2i li + 3|? 
 90. Of l + l-i? 97. Of 4141 + 61? 
 91 Of ^ A + i? 98. Of6i 21 + 31? 
 92. Of | &'+!? 99. Of2| + 2i 2-Jj? 
 100. $60 is | of what B gave for a horse, and the cost 
 
 of the horse, increased by its 3 fifths, is 5 times what he 
 paid for a sleigh ; required the cost of the sleigh. 
 6 
 
58 MENTAL ARITHMETIC. 
 
 LESSON V. 
 
 1. A man spent | of his money for a horse, and then 
 had $60 remaining ; how much money had he at first ? 
 
 SOLUTION. After spending 1 of his money, there remained 
 3 1 = 2 of his money, which equals $60. If 2 O f his money 
 eqnals $60, | equals . of $60, which is $30, &c. 
 
 2. A lady, after giving away | of her money, had only 
 40 cents remaining; how much money had she at first ? 
 
 3. A farmer sold | of his cows, and then had 25 re- 
 maining ; how many had he at first ? 
 
 4. A boy lost 4 marbles and found 10, and then had | 
 as many as at first ; how many had he at first ? 
 
 5; Mary gave ^ of her money for silk, and J for satin, 
 and had 10 dollars remaining ; how much had she at first? 
 
 6. Philo lost $16 and found 6, and then had f as 
 much as he had at first; how much had he before his 
 loss? 
 
 7. | of the length of a pole is in the air, \ in the 
 water, and 10 feet in the ground; required the length of 
 the pole. 
 
 8. Henry's money, diminished by its J and J, equals 
 57 dollars ; how much money has he ? 
 
 9. Milo lost 24 cents, and then found | as many as h-e 
 lost, and then had only | of what he had at first ; how 
 many had he before his loss ? 
 
 10. Mr. A's money, increased by its ^ and , equals 
 34 dollars ; how much money has he ? 
 
 11. Says B to C, | of my age diminished by | of my 
 age, equals 12 years ; how old was he ? 
 
 12. Peter gave 10 cents for a pie, which is f of -J of 
 the cost of his supper; required the cost of his supper. 
 
 13. A lady being asked her age, replied that her 
 daughter's age is 8 years, which is | of ^ of her age; 
 required her age. 
 
MENTAL ARITHMETIC. 59 
 
 11. Frank, after spending | of his money, found that 
 f!6 was | of what he had remaining; how much money 
 had he at first ? 
 
 15. | of an army was killed, | taken prisoners, and 
 800 escaped ; of how many men did the army consist ? 
 
 16. When E was married he was 27 years old, and | 
 of his age was 4 years more than | of his wife's age; re- 
 quired the age of his wife. 
 
 17. A boy, after spending \ of his money for candies, 
 and | for peaches, found that 20 cents was of what re- 
 mained ; how much money had he ? 
 
 18. What number is that, which being increased by 
 its ^, and that sum diminished by | of the number, the 
 remainder is 50 ? 
 
 19. Mr. K's hat cost $6, which was $3 less than f of 
 the cost of his coat ; required the cost of the coat. 
 
 20. What number is that which being doubled, and 
 then diminished by its |, equals 60 ? 
 
 21. | of 30 cents is 5 cents less than | of what a bushel 
 of potatoes cost ; what will be the cost of 5 bushels at the 
 same rate ? 
 
 22. What number is that, which being tripled and 
 diminished by its |, the remainder will equal 70 ? 
 
 23. A merchant sold 1 of | of his stock in a, month; 
 how many thirds of j of his stock remained ? 
 
 24. Alsop having | of an apple, gave J of what he had 
 to Jane ; what part of % of an apple remained ? 
 
 25. A farmer having T 9 Q of a ton of hay, sold | of it ; 
 how much of it then remained ? 
 
 26. A boy having ^ of a bushel of berries, sold | of 
 them ; what part of a bushel had he remaining ? 
 
 27. A fishing rod is 15 feet long, and | of its length 
 lacks 3 feet of being ^ of the length of tie line ; requijed 
 the length of the line. 
 
 28. A tree is 60 feet high, which is | of ^ of the 
 length of its shadow, diminished by 20 feet ; required the 
 length of the shadow. 
 
 29. 10 dollars is | of what M paid for a cow, and the 
 
60 MENTAL ARITHMETIC. 
 
 cost of the eow is ^ of the cost of an ox ; required the 
 value of the ox. 
 
 30. A pen cost 16 cents, which is f of f of what the 
 pen and holder both cost ; required the cost of the holder 
 
 LESSON VL 
 
 1 How many are 4 times | ? 
 SOLUTION. 4 times Ji are 12, which equals 3 or 1 * . There- 
 
 88 2 /^ 
 
 fere, &c. 
 
 2. How many are 3 times | ? 3 times | ? 
 
 3. How many are 4 times | ? 5 times -fa ? 
 
 4. How many are 7 times r \ ? 3 times | ? 
 
 5. How many are 4 times f 6 ^ ? 6 times T 7 2 ? 
 
 How many are 
 
 6. 3 times f ? 14. 4 times f? 
 
 7. 8 times f ? 15. 8 times |? 
 
 8. 5 times |? 16. 5 times |? 
 
 9. 3 times |? 17. 4 times 2^? 
 10. 5 times |? 18. 6 times 3^? 
 11 6 times f ? 19. 8 times 5|? 
 
 12. 10 times & 20. 9 times 2f ? 
 
 13. 8 times T \? 21. 10 times 4|? 
 
 22. If you give to 8 ooys, each 2| apples, how many 
 apples will it require ? 
 
 23. How many dollars does that man give away, who 
 gives to each of 10 beggars | of a dollar ? 
 
 24. Mary gave to each of 12 poor children, | of a pity 
 and had 3 pies remaining; how many pies had she? 
 
 25. How many quarts of nuts will it require, to give 
 | of a quart to each of 18 boys, and have 2 quarts re- 
 maining ? 
 
 26. What will be the cost of 25 pounds of sugar, at 
 nhe rate of | of a dime a pound? 
 
MENTAL ARITHMETIC, 
 
 61 
 
 27. How much will 4| yards of muslin cost, at the 
 rate of 6 cents a yard ? 
 
 28. What will 15 books cost, at the rate of | of a dol- 
 lar apiece ? 
 
 29. How much will 9 inkstands cost, at the rate of 2 
 for | of a dollar ? 
 
 30. If 4 apples cost | of a dime what will 16 applee 
 cost ? 
 
 31 . What will 7 quarts of beans cost, if 4 quarts cost 
 ^ of a dollar? 
 
 32. If a man walk 3^ miles in 2 hours, how far wili 
 ho walk in 1 day of 10 hours ? 
 
 33. William lost 2J dollars, and then having found 4 
 times as much as he lost, had 12 dollars ; how much had 
 he at first ? 
 
 34. Mary found 4| shillings, and then losing J as much 
 as she found, had 6-J shillings remaining ; how many had 
 she at first ? 
 
 35. Since 2 times | == |, or |, how may the last result 
 be obtained by omitting the analysis ? 
 
 36. Since 3 times | = l g, or |, how may the last result 
 be obtained without the analysis ? 
 
 37. How then may a fraction be multiplied by a num- 
 ber which will divide the denominator ? 
 
 38. What is the effect of dividing the denominator of 
 a fraction by any number ? 
 
 How many are 
 
 39. 
 
 4 
 
 times 
 
 f 
 
 49. 
 
 10 times ^ 
 
 40. 
 
 6 
 
 times 
 
 I? 
 
 50, 
 
 8 times ||? 
 
 41 
 
 7 
 
 times 
 
 
 51. 
 
 6 times ||? 
 
 42 
 
 5 
 
 times 
 
 _9_? 
 
 52. 
 
 4 times ||? 
 
 43 
 
 8 
 
 times 
 
 jor 
 
 53 
 
 3 times 2|? 
 
 44. 
 
 8 
 
 times 
 
 3? 
 
 54 
 
 5 times 2|? 
 
 45. 
 
 9 
 
 times 
 
 {>? 
 
 55. 
 
 7 times Si? 
 
 ^6. 
 
 7 
 
 times 
 
 18? 
 
 56. 
 
 12 times 5 \ 
 
 47. 
 
 6 
 
 times 
 
 -! f 
 
 57. 
 
 10 times 8^ 
 
 48. 
 
 4 
 
 f i 
 
 times 
 
 & 
 
 58 
 
 11 times 2^ 
 
62. MENTAL ARITHMETIC, 
 
 59. 3 times 2f miles is twice the distance from the 
 Normal School to Lancaster; required the distance. 
 
 60. 5 times 4| miles is 3 times the distance from Paoli 
 to Oakland; required the distance. 
 
 ^61 A is 6| years old, and 8 times A's age is 5 times 
 B's age ; required the age of B. 
 
 62. Six times 2| miles is 7 ? of the distance from Oak- 
 land to Christiana ; what is the distance ? 
 
 63. 6| dollars is ^ of 4 times 3| times B's money; how 
 much money has B? 
 
 64. 1 times 3j miles is 5^ times the distance from 
 Columbia to Marietta; what is the distance? 
 
 65. James lost $5|, and then had 5 times $6/3 re- 
 maining; how much money had he at first? 
 
 66. Peter gave of his marbles to Samuel, and 2 times 
 J of them to Anson, and had 15 remaining; how many 
 had he at first ? 
 
 67. A having lost ^ of his hens, sold 3 times | of the 
 remainder, and then had 20 ; how many had he at first ? 
 
 68. 4 miles, the distance from M to B, being multi- 
 plied by 8, lacks 2 miles of being 2 times the distance 
 from M to W; required tH distance. 
 
 LESSON VII. 
 
 1. What is | of 4? 
 
 SOLUTION. J of 1 is , and of 4 is 4 times , which are j 
 Therefore ^ of 4 is ^ of one. 
 
 2 What is | of 5? {of 6? 
 
 3. Whatip lof 7? J of 9? 
 
 4. What is of 5? \ of 10? 
 
 5. What is 4 of 12? J of 20? 
 
 6. Whath of 24? J 3 of 32? 
 
 A 
 | 
 
 7. What is | of 6? $ of 10* 
 
MENTAL ARITHMETIC. 68 
 
 8. What is 4 of 3? f of 14? 
 
 9. What is | of 4? | of 15? 
 10 What is f of 2? f of 12? 
 
 11. What is | of 9? | of 16? 
 
 12. What is | of 6? |of20? 
 
 13. What is -^ of 5 ? T 2 of 18 ? 
 
 14 B has 5 dollars, and | of B's money equals ^ ct 
 A's; how much money has A? 
 
 15. James is 7 years old, and | of his age equals J of 
 Mary's age; how old is Mary? 
 
 16. One part of a pole is 9 feet long, which lacks 8 
 feet of being T 4 T of the length of the other part ; required 
 the length of the pole. 
 
 17. The distance from A to B is 11 miles, and f of 
 this is \^ of the distance from A to C ; required the dis- 
 tance to C. 
 
 18. A watch cost $21, and 4 of its cost is T 7 ^ of the 
 cost of the chain; required the cost of both. 
 
 19. A cask contains 5 gallons, and fy of its contents is 
 \ | of the contents of another cask ; required the contents 
 of the second cask 
 
 20. Charles has 9 apples, and | of his, -|- | of an apple, 
 equals f of Chester's; how many has Chester? 
 
 21. What is | off? 
 
 SOLUTION. 1 of is 2, and if 4 of & is 2, 5 of & are 2 times 
 
 388 38838 
 
 , which are i, or 1 Therefore, &e. 
 
 b 82 
 
 What is 
 22 | of 48? -30 | of 2?? 
 
 23. fof|? 31 Irfilf 
 
 24. jof j? 32. i of Sir 
 
 25. |ofA? 33. |ofl|? 
 
 26. I of li? 34. 2J times 1|? 
 
 27. f ofj|? 35. 24 times Si? 
 
 28. | of -j%? 36. 3| times 3 A? 
 
 29. f of2|?_ 37. 2| times 2f? 
 
 38. j of $3| is of what A gave for a bureau ; whal 
 was the cost of the bureau ? 
 
64 MENTAL ARITHMETIC. 
 
 39. | of $ii is ^j of the cost of a watch; now much 
 did the watch costf 
 
 40. B has 27 marbles, and f of -g 6 of B's number 
 equals | of C's ; how many has C ? 
 
 41. Henry's hat cost i of an eagle, which is f of f of 
 the cost of his coat ; required the cost of the coat. 
 
 42. Peter had | of a barrel of flour, and after selling 
 | of it, the remainder is T a ff of what Paxton had ; how 
 much had Paxton ? 
 
 43. If there are 48 chestnuts in a pint, how many 
 does each of two boys receive, if A receives | of a pint, 
 and B | as many as A ? 
 
 44. Mary bought | of a paper of needles, which is | 
 of what Sarah bought; how many did each purchase, pro- 
 vided there are 24 needles in a paper? 
 
 45. Frank has 18 apples, Francis | as many, and 
 Fanny | as many as Francis ; how many has each ? 
 
 46. A has 40 fruit trees, T 4 of which bear apples, ^ 
 of the remainder, pears, and the rest, peaches; how many 
 trees of each kind has he ? 
 
 LESSON VIII. 
 
 1. What is | off? 
 
 SOLUTION. J of is one of the three equal parts into which 
 J may be divided ; if each fourth is divided into three equal 
 parts, 4- fourths or the unit will be divided into 4 times 3, or 12 
 equal parts ; hence, each part is J 2 - of a unit. Therefore J of 
 
 i 1S i 1 *- 
 
 2. What is 4 of}? } of}? Jof }? 
 3 What is I of i ? i of j ? j of J ? 
 
 4. What is } of i ? I of | ? J of } ? 
 
 5. What is l of ^? i of I? -l of $? 
 f>. What is } of l ? I of 1 ? % of ? 
 
 7 What is i of T V ? of * ? i of ^ ? 
 
 8 What is 4 of T V ? ^ of T V ? J T of & ? 
 
MENTAL ARITHMETIC. 65 
 
 9. What is J of 4 ? j of J-j ? i of -^ ? 
 
 10. Mary having ^ of a pie, gave | of it *o Hannah j 
 what part of a pie did Hannah receive ? 
 
 11. Philo having i of an orange, gave | of it to Peter; 
 what part of an orange did Peter receive : 
 
 12. A had | of a dollar, and gave of it to B ; what 
 part of a lo/lar did B receive ? 
 
 13. Since | of \ equals yV, how may the same result 
 be obtained without the analysis ? 
 
 14. Since \ of equals ^Q, how may we obtain the 
 same result without the analysis ? 
 
 15. What is 4 of i ? | of J ? J of * ? I of 4 ? 
 
 16. What is | of J ? i of ? 4 of -j ? of $ ? 
 
 17. What is * of i? 4 of 4? $of i? T V f TV ? 
 
 NOTE. Problem 1st may also be solved thus: J equals ^, 
 and J of fy is y 1 ^, hence J of J is T ^. This is simpler than 
 the solution given, but since it does not show the reason why 
 J of = y 1 ^, we prefer the other solution. 
 
 18. What is the difference between A -of -*, and | of { ? 
 
 19. What is the value of | of ^, subtracted from 5 of | ? 
 
 20. How much greater is | of ^, than ^ of ? 
 
 21. A man owning of a farm, sold ^ of it to his 
 aeighbour ; what part of it did his neighbour receive ? 
 
 22. Susan bought of a cake, and gave Eliza | of it; 
 how much did Susan retain ? 
 
 .23. A man, having 4 of a share of bank stock, sold | 
 of it j how much of a share did he retain ? 
 
 24. Carlo, finding j of a pound of meat, let Towser eat 
 i of it ; what part of a pound did Carlo eat ? 
 
 25. 'What is A's age, supposing that -J of 10 years ; J 
 of -i of his age ? 
 
 26. How much money has Jacob, if \ of 13 dollars is 
 | of | of his money ? 
 
 27. A cow cost 17 dollars, and | of the cost of the cow 
 is | of -| of the cost of an ox ; required the cost of the ox 
 
 28. A pei cost 20 cents, and f of its cost is | of f of 
 the cost of the holder; required* the cost of the holder. 
 
MENTAL ARITHMETIC. 
 
 29. What is J of fj ? 
 
 SOLUTION. i of 1 is 1 , and if 1 of i is _^, 1 of 4 is 4 times 
 
 3 5 15 3 o 1 o ' 3 o 
 
 i which are _4^., Therefore JL of 4 4 . 
 
 } Sr J5 3515 
 
 30. What is | of | ? \ of | ? \ of f ? 
 
 31. What is \ of f ? J of f ? | of | ? 
 
 32. What is I of f ? i of | ? I of -^ ? 
 33 What is | of | ? f of f ? 4 O f f ? 
 34, What is I of | ? | of f ? f of f f 
 
 35 What is f of -f ? T 7 ^ of | ? f- of T 7 ^ ? 
 
 36 ^yhat is T 8 of ft ? | of T 6 5 ? f of fg ? 
 
 37. Since f of | = -, 8 ^, in what manner may we obtain 
 the same result by omitting the analysis ? 
 
 38. What is | of ? f of f ? | of -?- ? i of f ? 
 
 39. What is f of | ? | of | ? | of | ? f of T % ? 
 
 40. W T hat is | of | ? 4 of L ? f of ^ ? 7 of - 
 
 41. What is f of y ?| O f e ? | of j| ? ff of _ 
 
 42. Johnston, having | of a melon, gave | of it to 
 Martin ; how much of it remained ? 
 
 43. Having | of a bushel of apples, I gave | of them 
 to Westlake ; what part of a bushel remained ? 
 
 44. W r hat is the difference between | of | of a lemon, 
 and | of | of a lemon ? 
 
 45. How much money has Sarah, if | of 11 dollars is 
 | of her money ? 
 
 46. A can build a boat in 2| weeks, which is | of | 
 of the time in which B can build it; how long will' it take 
 B to build it? 
 
 47. A hat cost $5, and f of its cost is f of JJ of the 
 cost of a coat ; required the cost of the coat. 
 
 48 A man lost | of his money, and. then found as 
 much as J of the remainder; what part of his money did 
 he then have ? 
 
 49. A squirrel fell | of the distance from the top of a 
 tree tc the ground, and then ascended \ of the distance 
 he was from the ground ; what part of the whole dis- 
 tance was he from the ground? 
 
 50. Annie, having f of a pound of candies, shared 
 

 MENTAL ARITHMETIC. 67 
 
 t.hein equally with 5 of her schoolmates) what part of a 
 pound did each receive? 
 
 51. Richard, having J of f of a quart of chestnuts, 
 shared them equally with 4 of his playfellows ; what part 
 of a quart did each receive ? 
 
 52. Matthew lost | of his property, then selling \ of 
 the remainder, bought as much as 1 third of what then 
 remained ; what part of his property did he then have ? 
 
 53. A, having f of a dollar, gave i of it to B, and B 
 gave ^ of his to ; what part of a dollar did each then 
 have ? 
 
 54. A kite in the air fell ^ of the distance to the ground, 
 tli en arose | of the distance it was from the ground, and 
 then fell \ of the distance it arose; what part of the 
 whole distance was it from the ground ? 
 
 LESSON IX. 
 
 1. How many times is f contained in 4 ? 
 
 SOLUTION. 1 is contained in 4, 4 times; and if 1 is contained 
 in 4, 4 times, J is contained in 4, 3 times 4 times, which are 12 
 times, and 2 thirds is contained in 4 J of 12 times, or 6 times, 
 Therefore. &c. 
 
 2. How many times is | contained in 2 ? In 3 ? 
 
 3. How many times is | contained in 3 ? In 5 ? 
 
 4. How many times is | contained in 2 ? In 4 ? 
 
 5. How many times is | contained in 5 ? In 7 ? 
 
 6. How many times is | contained in 4 ? In 5 ? 
 
 7. How many times is | contained in 2 ? In 4 ? 
 
 8. How many times is | contained in 5 ? In 8 ? 
 
 9. How many times is -$$ contained in 7 ? In 9 ? 
 
 10. How many times is T 8 T contained in 2 ? In 4 ? 
 
 11. How many times is | contained in 4 ? In 12 ? 
 
 12. By what method may we derive the result^ obtained 
 above, without the analvsis? 
 
68 MENTAL ARITHMETIC. 
 
 13 If a yard of cloth cost | of a dollar, how manj 
 yards can you buy for 12 dollars / 
 
 14. If 2 quarts of nuts cost j of a dime, how many 
 quarts can be bought for 1 dollar: 
 
 15. How many yards of tape can be bought for 11 
 cents, if 3 yards cost 2^ cents ? 
 
 16. What cost 7 peaches, at the rate of 5 peaches for 
 6 1 cents? 
 
 17. If 5 pints of milk cost 12 cents, how many pints 
 can you purchase for 25 cents ? 
 
 18. If 2^ barrels of apples cost 15 dollars, how many 
 barrels can be bought for 8| dollars ? 
 
 19. If | of an apple cost j of a cent, what will of | 
 of an appk cost ? 
 
 20. What cost 2 oranges, at the rate of 4 oranges for 
 51 cents? 
 
 21. If 3| boxes of butter cost $5$, what will 10 
 boxes cost, at the same rate ? 
 
 22. 7^ dollars is ^ of 5 times what a watch-key cost, 
 and the chain cost 8 times as much ; required the cost of 
 the chain. 
 
 23. What cost 16 lemons, if 5-J lemons cost 12| cents? 
 
 24. What cost 17 yards of lace, if 4| yards cost 7^ 
 dimes ? 
 
 25. If 3 men can do a piece of work in 6f days, how 
 long will it take 12 men to do the same? 
 
 26. If it require 9 men to do a piece of work in 4| 
 days, how many men will be required to do it in 13 
 days ? 
 
 27. What cost 26 gallons of molasses, if 4| gallois 
 cost 2^ dollars? 
 
 28. How many times is f contained in |? 
 
 REMARK. Solution similar to that at the beginning 
 >son. 
 
 29. How many times is | contained in J ? In 
 
MENTAL ARITHMETIC. 69 
 
 30 How many times is | contained in f? In |? 
 
 81. How many times is | contained in |? In |? 
 
 32. How many times is | contained in | ? In | ? 
 
 33. How many times is | contained in f? In |? 
 
 34. How many times is | contained in | ? In -3$ ? 
 
 35. How many times is | contained in |? In |? 
 
 36. How many times is g contained in i? In ||F 
 
 37. How many times is ^ contained in 2 8 T ? ID i|? 
 
 38. How many times is | contained in T 7 g ? In ||? 
 
 39. How many times is | contained in J'/? 
 
 ANOTHER SOLUTION. 3 is equal to ^fl, and 1 is equal to 4 
 10 is contained as many times in 1 1 as 10 is contained in 12, 
 grhich is 1*2 or 6 times. Therefore, &c. 
 
 1 5 
 
 40. How many times is f contained in | ? In | ? 
 
 41. How many times is f contained in |? In ^? 
 
 42. How many times is | contained in |? In |? 
 
 43. How many times is g contained in |? In |? 
 
 44. How many times | is 4 ? | ? | ? 
 
 45. How many times | is |? f? T %? 
 
 46. How many times |is|? |? J? 
 
 47. How many times | is \ ? | ? | ? 
 
 48. How many times | is | ? |? T ^j? 
 
 49. From the results in either of the above solutions, 
 what method may be derived to divide fractions by omit- 
 ting the analysis ? 
 
 50. If a yard of muslin cost | of a dime, how many 
 yards can you purchase for | of a dime ? 
 
 51. If a quart of vinegar cost T 3 of a dollar, how many 
 quarts can you buy for 2^ dollars ? 
 
 52. How many pens, at ^ of a dime each, can be bought 
 for 10 sheets of paper, worth | of a cent each ? 
 
 53. A divided 14 apples equally among his compan- 
 ions, giving to each | of an apple ; required the number 
 of companions. 
 
 54. A lady distributed 29 dimes equally among some 
 poor children, giving to each 5| dimes; hew many chil- 
 dren were there ? 
 
70 MENTAL ARITHMETIC. 
 
 55. How many bushels of grain, at $| a bushel, can be 
 bought for 12 bushels of apples, worth $| a bushel ? 
 
 56. If 1 pound of butter cost $|, how many pounds 
 can be had for | of a barrel of flour, worth $6 per barrel ? 
 
 57. B bought 6 yards of ribbon, worth 5| cents a 
 yard; how many apples, worth 1| cents each, will be 
 required to pay for it ? 
 
 58. Harvey bought 8 bushels of potatoes, worth $| a 
 bushel, and paid for them with eggs, wortli $| a dozen ; 
 how many eggs did it take ? 
 
 LESSON X. 
 
 1. What part of 2 is ? 
 
 SOLUTION. 1 is of 2, and is J of | of 2, which is J of 2, 
 and | is 3 times , or f of 2. Therefore f is f of 2. 
 
 2. What part of 3 is f ? Of 2 is f ? 
 
 3. What part of 4 is ? Of 5 is 5 ? 
 
 4. What part of 4 is | ? Of 7 is g ? 
 
 5. What part of 9 is f ? Of 5 is I of |? 
 
 6. What part of 6 is f of f ? Of 7 is f of I ? 
 
 7. What part of 2 is f of f ? Of 5 is | of fg ? 
 
 8. What part of 6 is | of ? Of 8 is f of j ? 
 
 9. If a pole 10 feet long cast a shadow 3| feet, what 
 is the length of the shadow of a pole 8 feet long, at the 
 earne time of day? 
 
 10. If 7 apples are worth 5| peaches, how many apples 
 are 12 peaches worth ? 
 
 11. If at a certain time of day, a pole 9 feet long cast 
 a shadow 4| feet, what must be the length of a pole to 
 cast a shadow 54 feet long, at the same time of day ? 
 
 12. If 8 pipes fill a vessel in 2 hours, how many pipes 
 will be required to fill it in y 1 ^ of an hour ? 
 
 13. Required the length of the shadow of a pole, 16 
 
 
MENTAL ARITHMETIC. 71 
 
 feet long, at the same time that a pole 3| feet long casts 
 a shadow 7| feet in ength ? 
 
 14. Mr. B, having lost $10, found that only g of hia 
 money remained; how much money had he? 
 
 15. A merchant sold goods for I| of what they cost, 
 and thereby lost $24; what was the cost of his goods? 
 
 16. What part of f is f ? 
 
 SOLUTION. J is J of f, and | or one is 3 times j, or f of f . 
 Since one is f of f , is \ of f , which is j 3 ^ of f , and f is 4 times 
 f 0t which are |J or f of f. Therefore, f is f of f. 
 
 What part 
 
 17. Of | is ? 24. Of | is I? 
 
 18. Of | is | ? 25. Of | is | ? 
 
 19. Of I is I? 26. Of f is 2'? 
 
 20. Of f is | ? 27. Of f is | of ? 
 
 21. Of f is I? 28. Of J is I of I? 
 
 22. Of 1 is | ? 29. Of -fa is | of I ? 
 
 23. Of f is |? 30. Of \\ is | of 2|? 
 
 31. 16 is | of how many times 5? 
 
 32. 18 is | of how many times 8? 
 
 33. 25 is | of how many times 10? 
 
 34. 14 is y 7 ^ of how many times 4? 
 
 35. 27 is | of how many times 9 ? 
 
 36. 15 is | of how many times 3? 
 
 37. 28 is | of how many times 7 ? 
 
 38. 10 is T 5 2 of how many times 12? 
 
 39. 20 is | of how many times \ of 10? 
 
 40. 24 is | of how many times f of 12? 
 28 is ^ of how many times \ of 21 ? 
 30 is | of how many times J of 12 ? 
 18 is | of how many times ^ of 14 ? 
 40 is | of how many times | of 30? 
 36 is | of how many times | of 15? 
 60 is | of how many times | of 14 ? 
 
 4-8 ia 8 r\f hrinfr momr Hmoa 3 r\f I r\r 
 
 40. 
 41. 
 42. 
 43. 
 44. 
 45. 
 46. 
 47. 
 48. 
 
 49. 60 is | of how many times ^ ui ^v : 
 
 50. 80 is | of how many times of 30? 
 
 48 is | of how many times | of 16? 
 35 is | of how many times -| of 18? 
 60 is - of how many times | of 20 ? 
 
72 MENTAL ARITHMETIC. 
 
 51. T| of 16 is how many times ^ of 12? 
 
 52. | of 30 is how many times | of 10? 
 
 53. | of 40 is how many times f of 21 ? 
 
 54. | of 45 is how many times | of 15? 
 * of 42 is how many times ^ of 14 ? 
 
 of 48 is how many times -| of 16? 
 | of 56 is how many times T 6 T of 22 ? 
 | of 80 is how many times | of 25 ? 
 
 59. | of 72 is how many times | of 16? 
 
 60. | of 27 is how many times f of 12? 
 
 61. A's horse cost $200, and | of this is twice the cost 
 of his sleigh, and the sleigh cost 4 times as much as his 
 harness ; required the cost of each. 
 
 62. B's wedding-coat cost $40, and | of this is twice 
 the cost of his vest, and three times the cost of his hat ; 
 what was the cost of each and of all ? 
 
 63. Hannah's wedding-dress cost $50, and J of this is 
 4 times the cost of her bonnet, and | of the cost of her 
 cloak; required the cost of the bonnet and cloak, re- 
 spectively. 
 
 LESSON XI, 
 
 1. Myron, having f of a certain sum of money, found 
 | of the same sum, and then had $28 ; required the sum. 
 
 2. If | of John's age, increased by \ and \ of his age, 
 equals 34 years, what is his age? 
 
 3. What number is that, which being increased by its 
 , and diminished by its |, equals 35? 
 
 4. Henry's age, diminished by its \ and \, equals 15 
 years, and his age is f of his brother's age; required the 
 age of each. 
 
 5. Francis, after losing f of his money, found that $1'J 
 was | of what remained ; how much money had he ? 
 
 6. What is the sum of the fractions , |, , and \ ? 
 
MENTAL ARITHMETIC. 73 
 
 7. How many apples does that man give away, who 
 gives to 5 girls each | of an apple? 
 
 8. What is the cost of 7 pens, at the rate of | of a 
 cent apiece ? 
 
 9. B is 9| years old, and 4 times B's age is | of C's 
 ag'*; what is the age of C? 
 
 10. Peter gave ^ of his money to James, and 3 times 
 | of it to John, and then had $20 remaining; how much 
 money had Peter at first? 
 
 11. A chain cost $15, and of its cost is | of the cost 
 of a watch ; required the cost of the watch. 
 
 12. Required the value of | of |, | of i, i of i, and 
 
 t f t? 
 
 13. A cow cost $14, and -| of the cost of the cow is | 
 of the cost of a horse ; required the cost of the horse. 
 
 14. A, having $, gave ^ of it to B, and B gave \ of 
 his part to C ; what part did each then have ? 
 
 15. If a yard of cloth cost | of a dollar, how many 
 yards can be bought for 9 dollars ? 
 
 16. How many yards of muslin can be bought for $6, 
 if 2 yards cost | of a dollar ? 
 
 17. How many times | is |? How many times ^ is ^? 
 
 18. If ^ of an apple cost | of a cent, what will i of an 
 apple cost ? 
 
 19. What cost 30 yards of lace, if 3| yards cost | of a 
 dollar ? 
 
 20. If 3 men can do a piece of work in 3| days, how 
 long will it take 5 men to do it? 
 
 21. If 7 men can do a piece of work in 2^ days, how 
 long will it require 6 men to do it? 
 
 22. If a yard of muslin cost f of a dime, how much 
 can you buy for | of a dime? 
 
 23. What part of f is ? What part of 1 is 2-J ? 
 
 24. A shared 8 apples with his companions, giving to 
 each | of an apple ; required the number of companions. 
 
 25. How many bushels of corn, worth 5| shillings a 
 bushel, can be bought for 34 shillings? 
 
 7* 
 
74 MENTAL ARITHMETIC. 
 
 26 How many apples will pay for 10 peaches, if 5 
 apples are worth 8-i peaches ? 
 
 27. Mary shared 21 dimes with her schoolmates, 
 giving to each 2 dimes; how many schoolmates had 
 she? 
 
 28. B bought 4 yards of silk, worth $1| a yard, and 
 paid for it with cloth, worth $11 a yard; how many yards 
 of cloth did it take ? 
 
 29. The distance from Paoli to Christiana is 24 mites, 
 and | of this distance is | of the distance from Christiana 
 to Lancaster; what is the distance to Lancaster? 
 
 30. The distance from Columbia to Rockville is 30 
 miles, and | of this distance is | of the distance from 
 Columbia to Newport; required the distance to Newport. 
 
 31. The distance from Conewago to Duncannon is 3 
 miles, and | of this distance is ^ of the distance from 
 Conewago to Mexico ; what is the distance to Mexico ? 
 
 LESSON XII. 
 
 1. Cyrus, after spending | of his fortune, found that 
 $40 was | of what remained ; wht was his fortune ? 
 
 2. What 'is the length of a pole, the shadow of which 
 is 10 feet, at the same time that a pole 10 feet long casts 
 a shadow 2^ feet in length? 
 
 3. Martha is 35 years old, and | of her age is of 
 twice her son's age; how old is her son ? 
 
 4. Rachael had | of a peck of walnuts, which she 
 shared with 5 of her schoolmates ; what part of a peck 
 did each receive? 
 
 5. Henry, having | of a barrel of cider, sold his neigh* 
 bour J of it ; what part of a barrel remained ? 
 
 6. A farmer, having sold ^ of his Sheep, and 10 cows, 
 found 15 sheep and -J of his cows remained; how many 
 of eaah di 1 he own? 
 
MENTAL ARITHMETIC. 75 
 
 7. Take any number, multiply it by 4, divide by 2, 
 multiply by 6, divide by 3, then divide by twice the 
 number, and name the result. 
 
 8. A's money, being increased by its J, and theD 
 diminished by its ^, is $40; required his money. 
 
 9. | of B's money, increased by tke difference between 
 | and | of his money, is $55 ; what is his money ? 
 
 10. When B was married he was 25 years old, and | 
 of his age was 3 years more than | of his wife's age; 
 required the age of his wife. 
 
 11. What number is that, which being diminished by 
 its |, and the remainder increased by its |, equals 40? 
 
 12. Stephen lost 12 cents, then found ^ as much as he 
 lost, and then had j as much as he had at first ; how much 
 had he at first? 
 
 13. A fishing-rod is 16 feet long, and | of its length 
 lacks 2 feet of being | of the length of the line ; -required 
 the length of the line. 
 
 14. A watch cost $40, which is | of | of the cost of 
 the watch and chain ; required the cost of the chain. 
 
 15. Susan has 7 peaches, and | of. Susan's number, 
 minus | of a peach, is | of Elizabeth's number; how 
 many has Elizabeth ? 
 
 16. If there are 50 chestnuts in a pint, how many do 
 A and B receive respectively, if A has | of a pint, and B 
 | as many as A ? 
 
 17. A has 60 fruit trees, | of which bear peaches, 
 of the remainder, pears, and the remainder, apples ; how 
 many are there of each ? 
 
 18. A farmer had 40 sheep in one field, which was | 
 of the number in another field ; then | of the sheep in 
 each field jumped into the other; how many then in each 
 field? 
 
 19. Maria gave away some money, and then found 10 
 cents, which is ^ of what she then had, and \ of what 
 she a*, first had ; how much did she give away ? 
 
 20 A man lost | of all his money, and then won | as 
 
76 MENTAL ARITHMETIC. 
 
 much as lie lost, and then had $20 how much had he 
 at first ? 
 
 21. If to ^ of the cost of B's horse you add $80, the 
 sum will equal | of the cost; required the cost of his 
 horse. 
 
 22. A paid $80 for flour, and f of the number of dol- 
 lars, is 8 times the number of barrels purchased; what 
 was the price of the flour a barrel ? 
 
 23. Philo found 10 cents, and then lost | of what he 
 found, and then had j as much as he had at first; how 
 much had he at first? 
 
 24. The distance from Medway to Columbia is 42 
 miles, and ^ of this distance is | of the distance from 
 Medway to Rockville; required the distance. 
 
 25. A has 8 marbles, and B has 7, and 6 times what 
 they both have, is equal to the number that C has, in- 
 creased by 10 ; how many has G ? 
 
 26. Mary has 5 roses, Jane, 3 times as many, lack- 
 ing 5, and Susan has twice as many as both, increased 
 by 5 ; how many has each, and how many have they 
 all? 
 
 27. Frescoln had 20 apples, and Lucy gave him 10 
 more, he then gave his father 7, and his mother a certain 
 nrrnber, and had 13 remaining; how many did he give 
 his mother ? 
 
 28. A and B had each 30 apples; A gave B 10 of 
 his, and B gave A 6 of his, and then lost a certain 
 number, so that A had 12 more than B ; how many did 
 B lose ? 
 
 29. A and B had each 40 cents, A gave B 10 of 
 his, and B gave A twice as many of his, and then losing 
 a certain number, had twice as many as A gave him j 
 bow many did he lose ? 
 
 30. The distance from Oak-land to Parkesburg is 16 
 miles, and | of this distance is -* of the distance from 
 Parkesburg to Marietta, lacking 1 mile ; what is the 
 distance ? 
 
MENTAL ARITHMETIC. 77 
 
 31. The distance from Mount Joy to Harrisburg ia 
 25 miles, and | of this distance is f of | of the distance 
 from Harrisburg to McVeytown; what is the distance 
 to McVeytown ? 
 
 32. From Philadelphia to Lancaster the distance is 
 68 miles, and ^ of this, increased by 2 miles, equals \ 
 of the distance from Lancaster to Harrisburg, minus 1 
 mile ; required the distance to Harrisburg. 
 
 We have derived the principles of the following propositions 
 by induction from analytical processes ; we will now proceed to 
 establish their .truth by rigid demonstration. 
 
 PROP. 1. Multiplying the, numerator of a fraction by any num- 
 ber multiplies the value of the fraction by that number. 
 
 PROP. 2. Dividing the numerator of a fraction by any number 
 divides the value of the fraction by that number. 
 
 PROP. 3. Multiplying the denominator of a fraction by any 
 number divides the value of the fraction by that number. 
 
 PROP. 4. Dividing the denominator of a fraction by any number 
 multiplies the value of the fraction by that number. 
 
 PROP. 5. Multiplying both numerator and denominator of a 
 fraction by any number does not change the value of the fraction. 
 
 PROP. 6. Dividing both numerator and denominator of a frac- 
 tion by any number does not change the value of the fraction. 
 
 DEMONSTRATION OF THE FIRST. If we multiply the numera- 
 tor of a fraction by any number, as 6, the resulting fraction 
 will express 5 times as many parts each of the same size as 
 before; hence the value of the fraction is 5 times as great. 
 
 DEMONSTRATION OF THE THIRD. Since the denominator showa 
 the number of equal parts into which the unit is divided, if we 
 multiply the denominator by any number, as 4, the unit will be 
 divided into 4 times as many parts, hence each part will be \ 
 as large as before, and the same number of parts being taken 
 the value of the fraction will be \ as great. 
 
 DEMONSTRTAION OF THE FIFTH. Since multiplying the nume- 
 rator multiplies the value, and multiplying the denominator 
 divides the value of the fraction, multiplying both numerator 
 and denominator by the same number, multiplies and divides 
 the value by the same number, and hence does not change its 
 value. Therefore, c. 
 
 NOTE. The 2d is demonstrated very much like the 1st, the 
 4th like the 3d, the 6th like the 5th. 
 
78 MENTAL ARITHMETIC 
 
 SECTION IV. 
 
 LESSON I. 
 FEDERAL MONEY. 
 
 10 mills (m.) . , equal 1 cent, . , ct. 
 
 10 cents ... " 1 dime, j , d. 
 
 10 dimes ... " 1 dollar, . . c 9 
 
 10 dollars ... "1 eagle, ... E. 
 
 Federal money is the currency of the United StAtes. The 
 coins are of three kinds, gold, silver, and copper. The new cent 
 coin consists of 88 parts copper and 12 parts nickle. The gold 
 and silver coins contain JL alloy, excepting the 3 cent piece, 
 which is 1 alloy. The alloy of the silver coin consists of copper, 
 and of the gold coin, of equal parts of copper and silver. 
 
 Dollars and cents, written together, are separated by a point (.) , 
 thus $5.60 is read 5 dollars and 60 'cents. 
 
 1. How many mills in 4 cents? In 7 cents? In 8 
 dimes ? 
 
 2. How many cents in 5 dimes? In 6 dollars? In 3 
 eagles ? 
 
 3. How many dollars in 9 eagles ? In 40 dimes ? In 
 500 cents ? 
 
 4. How many eagles in 50 dollars ? In 300 dimes ? 
 In 7000 cents ? 
 
 5. If 5 apples cost 20 cents, how many apples can you 
 buy for $2 ? 
 
 6. How many sheep can you b~iy for 18 eagles, at the 
 rate of 3 sheep for 15 dollars ? 
 
 7. What part of 2 dollars is 5 cents, and what part of 
 3 dimes is 6 cents ? 
 
 8. What part of 4 eagles is 8 dimes, and what part of 
 5 cents is | of a dime ? 
 
MENTAL ARITHMETIC. 79 
 
 ENGLISH OR STERLING MONEY. 
 
 4 farthings (gr.) . . equal 1 penny, . d. 
 
 12 pence " 1 shilling, . s. 
 
 20 shillings .... :" 1 pound, . . 
 
 21 shillings .... "1 guinea. 
 
 1 = $4.84. 1 s. = $0.24. 1 d. = $0.02. 
 
 The symbols , s., d., and qr. are the initials of the Latin words 
 libra, solidus, denarius, and quadrans ; signifying, respectively 
 pound, shilling, penny, and quarter. 
 
 1. How many farthings in 2 pence? In 3? In 5? 
 In 6? In 8? In 10? ^ 
 
 2. How many pence in 8 farthings? In 2? In 20? 
 In 28? In 30'? 
 
 3. How many pence in 2 shillings? In 4? In 5? 
 In 6? In 10? 
 
 4. How many shillings in 24 pence? In 48? In 72? 
 In 96? In 150? 
 
 5. How many shillings in 3 pounds? In 5? In 6? 
 In 8? In 12? 
 
 6. How many pence in 3s. and 6d. ? In 1, 2s. and 4c?. ? 
 
 7. What part of 2 pence is 3 farthings, and what pait 
 of 3 shillings is 5 pence? 
 
 8. What part of 8 pence is f of a penny, and what 
 part of a guinea is | of a pound ? 
 
 LESSON II 
 
 TROY WEIGHT. 
 
 24 grains (#r.) . . equal 1 pennyweight, pw*,. 
 20 'pennyweights . " 1 ounce, . . oz. 
 12 ounces .... "1 pound, . . Ib 
 
 The term Troy is said to 'be derived from Troyes, the name of 
 a town in France, where the weight was first used in Europe. 
 The symbol (oz.) is from the Spanish word onza^ for ounce, and 
 (Ib.) from libra, a pound. 
 
80 MENTAL ARITHMETIC. 
 
 1. How many grains in 2 pwts. ? In 3 In 4 ? In 
 5? In 6? In 7? 
 
 2. How many pwts, in 48 grs. ? In 72 ? In 96? ID 
 120? In 240? 
 
 3 How ,many pwts. in 3 oz.? In 4? In 5? In 6? 
 la 7? In 10? 
 
 4. How many ounces in 5 pounds? In 7? In 10? 
 In 8? In 4? In 12? 
 
 5. If 10 pwts. of silver are worth 3 shillings, what is 
 the value of 5 Ibs. of silver ? 
 
 6. What is the value of 3 ounces of gold, if 3 grains 
 are worth 3 dollars ? 
 
 APOTHECARIES' WEIGHT. 
 
 20 grains (gr.~) . . equal 1 scruple, . . 9 
 
 3 scruples ... u 1 dram, . . . 3 
 
 8 drams .... "1 ounce, . '. . 3 
 
 12 ounces .... "1 pound, . . . Ib 
 
 This weight is used for mixing and retailing medicines. The 
 pound is the same as the pound Troy. 
 
 1. How many grs. in 3 9s? In 5 ^s? In 2 gs? In 
 
 ift? 
 
 2. How many scruples in 4 ^s? In 40 grs.? In 2 
 s and 3 s? 
 
 3. How many ounces in 3 Ibs? In 16 s? In 4 Iba 
 and 5 gs? 
 
 4. How many drams in 120 grs.? In 36 9s? In 3 
 pounds ? 
 
 5. Which is the heavier, an ounce of opium or an ounce 
 of silver ? 
 
 6. If 5 grs. of medicine cost 10 cents, what will 3 oz. 
 and 4 gs cost ? 
 
 7. Two-thirds of 9 scruples of a certain drug cost 18 
 cents, what will 3 fourths of 8 pounds cost ? 
 
MENTAL ARITHMETIC. 81 
 
 AVOIRDUPOIS WEIGHT. 
 
 16 drams (<#r.) . . equal 1 ounce, . . oz. 
 
 16 ounces .... u 1 pound, . . Ib. 
 
 25 pounds .... "1 quarter, . . qr. 
 
 4 quarters ... " 1 hundred weight, cwt. 
 
 20 hundred weight . " i ton, ... T. 
 
 The term Avoirdupois is derived from the French avoir du 
 poids, signifying to have weight. The pound consists of 7000 
 Troy grains. This weight is used for weighing almost all 
 articles except gold, silver, platina, and precious stones, which 
 are weighed by Troy Weight. 
 
 1. How many drams in 2 ounces? In 3 ? In 5? 
 In 10? 
 
 2. How many ounces in 3 pounds ? In 5 ? In 48 
 drams ? 
 
 3. How many quarters in 75 pounds? In 3200 oz. ? 
 In 5 cwt. ? 
 
 4. How many hundredweight in 36 quarters ? In 300 
 Ibs. ? In 6 tons ? 
 
 5. How many tons in 5 hundredweight? In 240 qrs. ? 
 In 600 Ibs. ? 
 
 6. What will 12 pounds of starch cost, if o ounces cost 
 20 cents ? 
 
 7. Wha,t will 2 cwt. of coffee cost, at the rate of 4 
 pounds for 60 cents ? 
 
 8. I gave 3 cwt. 2 qrs. of hay, worth $20 a ton, for 
 butter worth 25 cents a pound; how many pounds of 
 butter did I receive ? 
 
 9. Which is the heavier, a pound of gold or a pound 
 of lead ? An ounce of silver or an ounce of feathers ? 
 
 METRIC SYSTEM. In the Metric System of weights and 
 measures the unit of weight is the Gram. The gram = 15.44 
 Troy grs. ; it is used for light weights. The Kilogram (1000 
 grams), abbreviated into kilo, will be the ordinary weight of 
 commerce. The kilogram 2 Ibs. Avoirdupois, very nearly. 
 
82 MENTAL ARITHMETIC 
 
 LESSON III. 
 LONG MEASURE. 
 
 12 inches (in.) . . equal 1 foot, , . ft. 
 
 3 feet .... "1 yard, . , yd. 
 
 5i yards .... " 1 rod, . . . rd. 
 
 40~rods .... " 1 furlong, . . fur. 
 
 8 furlongs . - . . * " 1 mile, . . . m. 
 
 The yard is the standard unit of length. It is formed by 
 dividing a pendulum, which vibrates seconds in a vacuum, at 
 the level of the sa, in the latitude of London, into 391393 
 equal parts and taking 360000 of these parts. From this unit 
 all other measures and weights are derived. 
 
 1. How many inches in 3 feet? In 5 ? In 7 ? In 
 2 yds.? In 2 rods and 2 feet? 
 
 2. How many feet in 60 inches? In 96? In 108? 
 In 6 rds. ? In 4 rds. 7 yds. ? 
 
 3. How many yards in 45 feet? In 66? In 72 in.? 
 In 4 rds. ? In 1 fur. 4 rods ? 
 
 4. How many furlongs in 320 rods? In 440? In 
 660 feet ? In 9 miles ? 
 
 5. How many miles in* 104 furlongs? In 640? In 
 800? In 1760 yards? 
 
 6. Mary ran 60 rods, and 4 fifths of this distance 
 equals T 3 <j of the distance Henry ran ; how many furlongs 
 did Henry run ? 
 
 7. Three fourths of the length of a pole is 6 feet, which 
 is 2 sixths of the length of another pole ; how many yards 
 long is each pole ? 
 
 8. What part of 1 yard is 2 feet, and what part of 2 
 furlongs is 4 yards? 
 
 METRIC SYSTEM. In the Metric System the unit of length is 
 the Meter. It equals 39.37 inches, or very nearly 3 ft. 3 in. and 
 | of an inch. Long distances will be measured by the Kilometer 
 ( ] 000 meters), which equals f of a mile, very nearly. 
 
MENTAL ARITHMETIC. 83 
 
 CLOTH MEASURE, 
 
 2J inches . . . equal 1 nail, , . na. 
 4 nails .... " 1 quarter, . qr. 
 
 4 quarters ... " 1 yard, . . . yd. 
 
 8 quarters ..." 1 Ell Flemish, . E. Fl 
 
 5 quarters ... " 1 Ell English, . E. E. 
 
 6 quarters ... " 1 Ell French, . E. Fr 
 
 Cloth Measure is used in measuring cloth, lace, muslin, &c. 
 
 1. How many inches in 4 nails? In 5 quarters? In 
 
 3 ells Flemish? In 2 ells English? 
 
 2. How many nails in 18 inches? In 54 inches? In 
 
 4 E. E. and 3 qrs. ? In 5 E. Fr. and 2 qrs. ? 
 
 3. How many quarters in 24 nails? In 63 inches? 
 In 3 E. E., 4 E. Fl., and 2 qrs. ? In 52 yards ? 
 
 4. How many yards in 144 inches, 48 nails, and 32 
 qrs. ? In 7 E. Fr. and 2 qrs. ? 
 
 5. What is the difference between 8 E. Fr. and 4 E. 
 E.? Between 5 E. Fl. and 32 nails? 
 
 6. How many coats can be made from 9 E. E. and 3 
 qrs., if one coat require 3 yards ? 
 
 7. Four fifths of the number of yards in Mary's dres? 
 equals 8, which is 4 times | as many yards as her cloak 
 contains; how many yards more in the dress than cloak; 
 
 LESSON IV. 
 
 ALE OR BEER MEASURE. 
 
 2 pints (p.) . . equal 1 quart, . , . qt. 
 
 4 quarts ... " 1 gallon, . . gal. 
 
 36 gallons ... " 1 barrel, . . bar. 
 
 54 gallons ... "1 hogshead, . lihd. 
 
 Ale or Beer Measure is used in measuring ale, beer, &c. Tha 
 gallon consists of 282 cubic inches. 
 
84 MENTAL ARITHMETIC. 
 
 1. How many pints in 5 quarts? In 10 gallons? In 
 5 barrels ? In | of a hogshead ? 
 
 2. How many quarts in 46 pints? In 7 gallons? In 
 3 bar. 10 gals. ? In 2 hhd. 2 gals. 3 qts. ? 
 
 3. How many gallons in 64 pints and 20 quarts? In 
 | of a hhd. and | of a barrel? 
 
 4. What part of a hhd. is J of a barrel ? What is the 
 difference between 2 hhd. and 3 barrels ? 
 
 5. What part of 2 barrels is 9 gallons ? What cost a 
 gal. of aie ; at 5 cents a pint ? 
 
 LIQUID OR WINE MEASURE. 
 
 4 gills ((71.) . . . equal 1 pint, . . pt. 
 
 2 pints .... " 1 quart, . . qt. 
 
 4 quarts .... " 1 gallon, . . gal. 
 
 31 gallons ... "1 barrel, . . bar. 
 
 42 gallons ... "1 tierce, . . tier. 
 
 63 gallons ... "1 hogshead, . hhd. 
 
 2 hogsheads ... " 1 pipe, . . pi. 
 
 2 pipes .... " 1 tun, . . tun. 
 
 Wine Measure is used in measuring wine, and liquids gene- 
 rally. The gallon consists of 231 cubic inches. 
 
 1. How many gills in 6 pints? In 5 quarts? In 8 
 gallons ? In 2 barrels ? 
 
 2. How many pints in 24 gills? In 8 quarts? In 2 
 tierces ? In 1 hhd. and 7 gals. ? 
 
 3. How many quarts in 40 gills? In 26 pints? In 
 2 tuns and 3 hogsheads? 
 
 4. What part of of a hogshead is ^ of a tierce ? What 
 part of | of a hhd. is 7 quarts ? 
 
 5. What part of ^ of a gal., wine measure, is | of a 
 gal., beer measure ? 
 
 6 Which is the greater, a gallon of wine or a gallon of 
 beer? A hogshead of ale or molasses? 
 
 METRIC Si STEM. The unit of capacity is the Liter, equal to 
 l T 'g liquid quarts, very nearly. The Hectoliter (100 liters), 
 equal *o about 2| bus., will be used for measuring grain, etc. 
 
MENTAL ARITHMETIC. 85 
 
 DRY MEASURE. 
 
 2 pints (p.) . . equal 1 quart, . qt* 
 
 8 quarts .... " 1 peck, . pk. 
 
 4 pecks .... " 1 bushel, bu. 
 
 36 bushels .... "1 chaldron, . ch. 
 
 Dry Measure is used for measuring grain, fruit, coal, &c. 
 
 1. How many pints in 3 quarts ? In 2 pecks? In 2.| 
 bushels ? In one chaldron ? 
 
 2. How many quarts in 64 pecks? In 10 bushels? 
 In | of a chaldron ? In 24 pints ? 
 
 3. How many pecks in 5^ bushels and 3 pecks? In 
 96 pints ? In 56 quarts ? 
 
 4. How many bushels in 2 ch. 8 bu. ? In 192 quarts? 
 In 128 pints? In 78 pecks? 
 
 5. Which cost the more, and how much, 5 bu. 3 qts. 
 of salt, at 4 cents a quart, or 10 bu. 3 pks. of apples, at 
 50 cents per bushel ? 
 
 6. At ten cents a peck, how many bushels of pears can 
 be bought for 8 dollars ? 
 
 7. A grocer bought 7 bushels and 2 pecks of cherries 
 at the rate of 3 cts. a pint, and sold them for 2 cwt. 2 
 qrs. of sugar at 6 cts. per Ib. } how much did he gain ? 
 
 LESSON V. 
 
 TIME. 
 
 60 seconds (sec.) . . equal 1 minute, . . ra. 
 
 60 minutes ... " 1 hour, . . . hr. 
 
 24 hours .... "1 day, . . da. 
 
 7 days .... "1 week, . . . wJc. 
 
 4 weeks .... " 1 month, . . mo. 
 
 52 weeks .... " 1 year, . . . yr. 
 
 12 calendar months . " 1 year, . . . yr, 
 8* 
 
86 MENTAL ARITHMETIC. 
 
 1. How many seconds in 2 minutes? In 3 ? In 6? 
 In 8? In 10? 
 
 2. How many minutes in 3 hours? In 4 ? In 5? 
 In 120 seconds? 
 
 3. How many hours in 4 days? In 5? In 10? In 
 240 minutes? 
 
 4. How many days in 3 weeks ? In 9 ? In 12 ? In 
 96 hours? 
 
 5. How many weeks in 7 months ? In 11 ? In 28 
 days? In 168 hours? 
 
 6. How many months in 10 yearj ? In 24 weeks ? In 
 56 days? In 112 days? 
 
 7. How many years in 24 months? In 104 weeks? 
 In 364 days? 
 
 8. Mary is 4 years and 6 months old, and her brother 
 is 6 years and 4 months old ; how much younger is Mary 
 than her brother ? 
 
 9. How many days, of 10 hours, each, will he required 
 to make 40 hats, if 4 hats can be made in 7 hours ? 
 
 CHANGE OF CURRENCIES. 
 
 In New England, Virginia, Kentucky, and Ten- 
 nessee, 6*. =$1. 
 
 In New York, Ohio, and North Carolina, . . 8*'. = $1 
 
 In New Jersey, Pennsylvania, Delaware, and 
 
 Maryland, 7s.6e?. = $l. 
 
 In South Carolina and Georgia, 4s Sd. =$1. 
 
 In Canada and Nova Scotia, 5s. =$1. 
 
 1. Ten dollars, in New York, equal how many pounds? 
 
 SOLUTION. In $1 there are 8 shillings, and in 1 there arc 
 20 shilling One shilling is 1 of a pound, and 8 shillings, 
 or $1, are 8 times ^1 , which are ^ or |, and $10 are 10 times 
 |, wh ch are 4. ~ Therefore, $10, in New York, equals 4. 
 
 2. How many pounds in $20 in Ohio? In $60 in 
 North Carolina? 
 
 3. How many pounds in $30 of New England ? ID 
 $90, Kentucky ? In $1 00, Virginia ? 
 
MENTAL ARITHMETIC. 87 
 
 4. How manj pounds in $48, New Jersey I In $64, 
 Pennsylvania? In $72, Maryland? 
 
 5. How many pounds in $120, South Carolina? In 
 8240, Georgia ? In $56, Canada ? 
 
 6. How many dollars in 24, New York? In 26, 
 Ohio ? In 30, North Carolina ? 
 
 7. How many dollars in 27, Maine? In 54, Ver- 
 mont ? In 60, Massachusetts ? 
 
 8. How many dollars in 21, South Carolina? In 
 63, Georgia currency ? 
 
 9. What is the value of a shilling, in cents, in each 
 of the states mentioned in the table ? 
 
 REMARK. The class should now review, unless they are 
 entirely familiar with the preceding sections. 
 
 SECTION Y. 
 
 LESSON I. 
 
 Special attention is called to the adaptation of the remainder 
 of the book to elementary and advanced classes. Each lesson 
 commences simply enough for the most elementary pupils, and 
 near the close becomes sufficiently difficult for the most ad- 
 vanced. The more difficult problems are separated from the 
 easier by a line ; these are to be omitted by young pupils until 
 review, while the older or more advanced pupils take the whole 
 lesson. 
 
 1. If | of a yard of cloth cost J of a dollar, what will 
 | of a yard cost ? 
 
 2. If 8 horses eat a quantity of hay in 16 weeks, how 
 long would it last 32 horses? 
 
 3. If 5 men earn 30 dollars in a certain time, how 
 much will 8 men earn in j the time ? 
 
 4. If 6 persons spend 836 in 3 days, how much wiD 
 10 persons spend in 5 days ? 
 
88 MENTAL ARITHMETIC. 
 
 5. How long will 5 tons of hay last 8 horses, if 6 horses 
 eat it in 12 weeks ? 
 
 6. How long will 3 barrels of flour last 10 persons, if 
 4. persons eat 4 barrels in 40 weeks ? 
 
 7. If 7 men can earn 82 8 in 4 days, how many dollars 
 can 9 men earn in 6 days ? 
 
 8. How long will 6 men require to build 6 boats, if 7 
 men build 3 boats in 12 weeks ? 
 
 9. If 10 oxen eat 4 acres of grass in 6 days, in how 
 many days will 30 oxen eat .8 acres ? 
 
 10. If it require 4 men 7 days to perform a certain 
 piece of work, how many men can perform a piece, 3 
 times as large, in 6 days'/ 
 
 11. If it require 5 men 8 days to build 20 rods of 
 wall, how many men can, in 2 days, build | as much 
 wall? 
 
 12. How many men, in 10 days of 6 hours each, can 
 earn as much as 6 men in 20 days of 8 hours each ? 
 
 13. How many oxen will eat 5 tons of hay in 5 weeks, 
 if 12 oxen eat 4 tons in 4 weeks ? 
 
 14. If 3 horses, in J- of a month, eat | of a ton of hay, 
 how long will | of a ton last 5 horses ? 
 
 15. If 4 men can do a piece of work in 6 days, in what 
 time will it be completed if they receive the assistance of 
 5 men, when the work is half done ? 
 
 16. How many cents are 10 melons worth, if 4 melona 
 are worth 8 oranges, and 3 oranges are worth 9 cents ? 
 
 17. How many cents will 5 oranges cost, if 3 oranges 
 are worth 9 apples, and 4 apples are worth 8 cents ? 
 
 18. How many dollars will 10 sheep cost, if 5 sheep 
 are worth 2 cows,* and 4 cows are worth 180 ? 
 
 19. How many pigs can a man get for 2 cows, if 12 
 pigs are worth 3 sheep, and 12 sheep are worth 3 cows ? 
 
 20. How many oranges can you buy for 20 cents, of 4 
 oranges are worth 8 apples, and 4 apples are worth 8 
 cents? 
 
 21 How many hens can you purchase for $12, if 4 
 hens are worth 8 turkeys, and 3 turkeys are worth $6 ? 
 
MENTAL ARITHMETIC. 89 
 
 22. If G sheep are worth 2 cows, and 10 cows are 
 worth 5 horses, how many sheep can you buy for 3 
 horses ? 
 
 23. If a measure of flour make 5 four cent loaves, how 
 many 2 cent loaves will it make ? How many 5 cent 
 loaves will it make ? 
 
 24. If a certain sum of money buy 9 four cent oranges, 
 how many 6 cent oranges can you buy with the same 
 gum? ' 
 
 25. If a 5 cent loaf weigh 7 ounces, when flour is 
 worth 6 dollars a barrel, how much should it weigh when 
 flour is worth 7 dollars per barrel ? 
 
 26. If a 3 cent loaf weigh 9 ounces, when flour is 6 
 dollars a barrel, how much ought a 4 cent loaf to weigh 
 when flour is $S a barrel ? 
 
 27. A can do as much work in 2 days as B can in 4, 
 or C in 6 days ; in how many days can B do as much as 
 C can in 18 days ? 
 
 28. A can do as much in 6 days as B can in 2 days, 
 and B can do as much in 5 days as C can in 15 days; 
 in how many days can A do as much as C can in 4 days ? 
 
 29. A can do 3 times as much in a day as B, and B 
 can do twice as much as C ; in how many days can A do 
 as much as C can in 4 days ? 
 
 30. If 5 horses can eat a lot of grain in 12 days, in 
 what time will it be consumed, if 7 horses are added 
 when the grain is | eaten ? 
 
 31. If 8 boys can weed a garden in 5 hours, in what 
 time will the job be completed, provided 3 boys leave 
 when the work is half done ? 
 
 32. If 9 men build 10 rods of wall in 8 days, in what 
 time can 20 rods be built, if f of their number leave 
 when the work is \ part completed ? 
 
90 MENTAL ARITHMETIC. 
 
 LESSON II. 
 
 1. A gentleman gave 4 cents each to some poor child- 
 ren ; had he given them 7 cents each it would have taken 
 B6 cents more ; how many children were there ? 
 
 SOLUTION. By the second condition of the question fre gave 
 each child 7 4, which is 3 cents more than by the first, and 
 to them all, 36 cents more ; hence there were as many children 
 as 3 is contained times in 36, which are 12. Therefore, &c. 
 
 2. A teacher gave his pupils 2 questions each, and had 
 26 questions remaining; if he had given them 4 apiece 
 there would have been none remaining; required the 
 number of pupils and questions. 
 
 3. A father gave his sons 5 dollars each, and had 30 
 dollars remaining; had he given them 8 dollars each it 
 would have taken all his money ; required the number of 
 eons and amount of money. 
 
 4. Mary gave some beggars 6 cents each, and had 25 
 cents remaining; had she given them 8 cents each she 
 would have had 3 cents remaining ; how many beggars 
 were there ? 
 
 5. Edward bought a certain number of melons, at the 
 rate of 5 cents each; if he had paid 3 cents each they 
 would have cost 14 cents less ; how many melons did he 
 buy? 
 
 6. A lady, wishing to buy some ribbon, found if she 
 bought that at 10 cents a yard she would want 9 cents to 
 pay for it, but if she bought that at 7 cents ji yard she 
 would have 9 cents remaining; how much money had 
 ihe? 
 
 7. Morris and Robert have each a certain number of 
 peaches ; if Morris had 10 more he would have twice as 
 many as Robert, but if he had 30 more he would have 4 
 times as many as Robert; how many has each? 
 
 8. A drover bought a number of sheep at $3-J a head, 
 and found he lacked $6 of having money enough to pay 
 
MENTAL ARITHMETIC. 91 
 
 for them ; if he had paid $2 a head he would have had $9 
 remaining ; how much money had he ? 
 
 9. Sallie wishes to buy a silk dress; if she pays $5 a 
 yard she will lack $10 of having money enough to pay 
 for it, but if she pays $2.50 a yard she will have Sf) re- 
 maining; required her money, and the number of yards 
 in the dress ? 
 
 10. A certain number of oranges, at the rate of 3 for 
 12 cents, will cost 18 'cents more than the same number 
 of apples, at the rate of 4 for 8 cents; required the 
 number. 
 
 11. A certain number of peaches, at 3 for 10 cents, 
 will cost 20 cents more than the same number of pears, 
 at the rate of 3 for 5 cents ; required the number. 
 
 12. James and Henry have a certain number of 
 marbles ; if James had 8 more he would have 6 times 
 as many as Henry, but if 12 less he would have only 
 twice as many ; how many has each ? 
 
 13. A gentleman divided 28 apples between an equal 
 number of boys and girls, giving to each girl 3, and to 
 each boy 4 apples ; required the number of boys and girls. 
 
 14. A man bought an equal number of pigs and sheep 
 for $81, giving $4 each for the pigs, and $5 each for the 
 sheep ; how many of each did he buy ? 
 
 15. A boy expended 36 cents for an equal number of 
 melons and lemons, giving 4 cents each for the melons, 
 and 2 cents each for the lemons ; how many of each did 
 he purchase ? 
 
 16. A lady gave 60 cents to some poor children ; to 
 each boy she gave 2 cents, and to each girl 4 cents; how 
 many were there of each, provided there were 3 times as 
 many boys as girls? 
 
 17. Two boys had an equal sum of money; one bought 
 a certain number of oranges, at 4 cents each, and had 12 
 cents remaining the other bought twice as many apples. 
 
92 MENTAL ARITHMETIC. 
 
 for 3 cents each, and had 2 cents remaining; how much 
 money had each ? 
 
 18. A, B, and C, dig a ditch for $60 ; A receives $H. 
 B $2, and C $2-| a day ; how many days were they em- 
 ployed, and what did each receive ? 
 
 19. A and. B agree to perform a piece of work, A re* 
 ceiving $2, and B $3 a day ; A works twice as many days 
 as B, and they together receive 870 ; how many days did 
 each labour? 
 
 LESSON IIL 
 
 1. If to Harry's age its ^ be added, the sum will be 
 24 years ; what is his age ? 
 
 SOLUTION. By the condition of the problem, 2 O f Harry's age, 
 plus of his age, which is ^ of his age, equals" 24 years. If 3 
 of Harry's age equals 24 years, .1 of his age equals 1 of 24 years', 
 which is 8 years, and , or his age, equals 2 times 8, or 16 years. 
 Therefore, &c. 
 
 2. What number is that to which, if its ^ be *dded, 
 the sum will be 36 ? 
 
 3. Required the number, which being increased by 
 its I equals 40. 
 
 4. What number is that, which being increased by its 
 the sum will be 80 ? 
 
 5. What number is that, which being diminished by 
 its | the remainder will be 30 ? 
 
 6. Three times a certain number, increased by f of 
 itself, equals 22 ; required the number. 
 
 7. Reuben's age, being doubled and diminished by | 
 of his age, equals 50 years; how old is he? 
 
 8. Three and ^ times a number, minus f of the num- 
 ber, equals 34 ; what is the number ? 
 
 9. Two-fifths of a number being subtracted from of 
 the nurnbe~ equals 7 ; required the Dumber. 
 
MENTAL ARITHMETIC. 93 
 
 10 A boy being asked his age, replied, that his age t 
 increased by its | and -f , equalled 39 /ears; what was his 
 age I 
 
 11. What number is that, which being increased by 
 the difference between its | and i, equals 42 ? 
 
 12. If the height of a tree be increased by its f and 
 10 feet more, the sum will be twice the height; what is 
 the height of the tree ? 
 
 13. Jf twice the length of a pole be increased by its 3 
 and 2 feet more, the sum will equal 3 times the length 
 of the pole; required its length. 
 
 14. If 3 times Henry's age be increased by its ^, ^ 
 and 2 years more, the sum will equal 4 times his age, 
 what is his age ? 
 
 15. If the height of a steeple be increased by its J, 
 and that sum diminished by the difference between | and 
 J of the sum, it will equal J of its height, minus 13 feet; 
 required its height. 
 
 16. Two times a number, plus 6, equals 3 times the 
 same number, plus 2 ; what is the number ? 
 
 17. Three times a certain number, plus 8, equals 4 
 times the same number, plus 3 ; required the number. 
 
 18. ^ of a certain number, increased by 10, equals | 
 of the same number, plus 8 ; what is the number ? 
 
 19. Four times A's age, diminished by 10 years, equals 
 3 times his age, increased by 10 years ; what is his age ? 
 
 20. Two-thirds of Morton's apples, increased by 2, 
 equals | of his number, diminished by one ; how many 
 apples has he ? 
 
 21. Benton lost | of all his money, and then found -| 
 as much as he lost, and then had $120; how much money 
 had he at first? 
 
 22. Mary gave | of her money to the poor, and then 
 found | as much as she gave away, and then had $30 ; 
 how much had she at first? 
 
 23. William borrowed f of Emily's money, and after 
 spending | of it, returned the remainder, which was $20 ; 
 how much money had Emily? 
 
94 MENTAL ARITHMETIC. 
 
 24. A thief stole J of Harry's money, and before he 
 was caught spent f of it ; the remainder, which was $20 
 ess than he stole, was given back ; how much money had 
 
 Harry ? 
 
 25. Two times a certain number, -f- 10, equals 3 limes 
 the sum obtained by increasing the number by 2 ; what 
 is the number? i 
 
 26. Baldwin had stolen from him |>- of his money, arid 
 the thief was not caught until he had spent 4 of it ; the 
 remainder, which was $30 less than Baldwin had remain- 
 ing, was given back ; how much money had Baldwin '/ 
 
 LESSON IV. 
 
 1. William and Henry have 15 marbles; how many 
 has each, provided William has twice as many as Henry? 
 
 SOLUTION. By a condition of the problem, twice Henry's 
 number equals William's, which, added to Henry's number, 
 equals three times Henry's, which is what they both have, or 15 
 marbles. If 3 times Henry's number equals 15, once his num- 
 ber equals J of 15, which is 5, and twice his number, or Wil- 
 liam's, equals twice 5, or 10 marbles. Therefore, &c. 
 
 2. Robert has 3 times as many cents as Elia-s, and they 
 together have 24 ; how many has each ? 
 
 3. William has 4 times as many nuts as Oliver, and 
 they together have 20 pints; how many pints has each ? 
 
 4. Emma has 35 flowers, and 4 times the number of 
 roses equals the number of pinks ; how many has she of 
 each kind? 
 
 5. Divide the number 25 into two such parts, that 4 
 times one part shall equal the other. 
 
 6. A father and son earned in one week $12; how 
 much did each earn, if the father earned twice as much 
 as the son ? 
 
 7 . A pole, 36 feet in length, was broken into twc 
 
MENTAL ARITHMEIIC. 95 
 
 unequal p.eces, such that | of the longer piece equals 
 the shorter ; required the length of each piece. 
 
 8. In a certain school, consisting of 35 scholars, there 
 were | as many girls as boys ; how many boys and how 
 many girls in the school ? 
 
 9. The sum of two numbers equals 40, and * of the 
 greater equals the less ; required the numbers. 
 
 10. A man bought a horse and cow for $100, and the 
 how cost | as much as the horse ; required the cost of 
 each. 
 
 11. Twice the sum of two numbers is 30, and 3 times 
 the smaller equals twice the greater; what are the num- 
 ners ? 
 
 12. Two thirds of the number of dollars that A and B 
 have equals 40 ; how many has each, if 5 times A's num- 
 ber equals 7 times B's number ? 
 
 13. Three fourtks of 40 is | of the number of apples 
 and pears that Reuben has ; how many has he of each, 
 if 3 times the number of apples equals 7 times the num- 
 ber of pears ? 
 
 14. Divide 36 apples among three boys, so that the 
 second may have twice as many as the first, and the third 
 3 times as many as the first. 
 
 15. Divide 66 plums among Ella, Emma, and Ettie, 
 so that Ella shall have twice, and Emma three times as 
 many as Ettie. 
 
 16. A watch and chain cost 42 dollars ; what was the 
 cost of each, provided | of the cost of the watch equals 
 the cost of the chain? 
 
 17 A, B, and C, together, earned $70; A earned 
 twice as much as B, and B twice as much as C ; how 
 much did each earn ? 
 
 IS The sum of three numbers is 50 ; the second is 3 
 times the first, and the third is twice the second ; what 
 are the numbers ? 
 
 19. Harry and Thomas lost a purse of money contain- 
 ing $24, of which Harry owned ^ as much as Thomas; 
 how mucli did each lose ? 
 
96 MENTAL ARITHMETIC. 
 
 20. A turkey, duck, and hen, cost 66 dimes, the duck 
 cost twice as much as the hen, and the turkey 4 times as 
 much as the duck ; required the cost of each. 
 
 21. The difference between two numbers is 27, ami 
 the greater is 4 times the smaller; what are the num- 
 bers ? 
 
 22. Marie has 40 cherries more than Jane, arid 5 times 
 Jane's number equals Marie's; how many has each? 
 
 23. Two thirds of 30 is |- of the difference between two 
 numbers, and tho less is | of the greater; what are the 
 numbers ? 
 
 24. A man bought a horse, cow, and sheep for $105; 
 how much did he pay for each, provided the cow cost 4 
 times as much as the sheep, and the horse 4 times as 
 much as the cow? 
 
 25. A farmer has 102 animals, consisting of hogs, 
 eheep, and cows ; there are | as many sheep as bogs, and 
 | as many cows as hogs ; required the number of each. 
 
 26. Of a certain pole, whose parts are in the mud, air, 
 and water, | of the length in the air equals the length 
 in the water, and f of the length in the water equals the 
 length in the mud; required the length of each part, sup- 
 posing the part in the water to be 10 feet longer than the 
 part in the mud. 
 
 LESSON V 
 
 1 . A and B have 25 oranges ; how many has each, if 
 b has 5 more than A? 
 
 SOLUTION. By a condition of the problem, A's number -j- 5 
 oranges equals B's number, which, added to A's, is twice A'a 
 number -j- 5, which equals 25 oranges. If twice A's -|- 5 = ?-5, 
 twice A's =25 5, or 20 ; if twice A's = 20, once A's equals 
 | of 20, which is 10; and since B had 5 more than A ? 10 -|- , 
 or Jo, equals B's number. Therefore, &c. 
 
MENTAL ARITHMETIC. 97 
 
 2. Mary has 7 oranges more than William, and they 
 together have 27 ; how many has each ? 
 
 3. Stephen has 10 cents more than Martha, and they 
 together have 40 ; how many has each ? 
 
 4. The sum of two numbers is 31, and their difference 
 5 ; what are the numbers ? 
 
 5. Thomas and Reuben earned the same sum of money, 
 when Reuben found $9, and they then together had $45; 
 how much did each earn ? 
 
 6. Ella and Kate had each the same number of candies; 
 Ella eat 5 of hers, and they then together had 21 ; how 
 many had each at first ? 
 
 7. Divide the number 28 into two such parts, that one 
 part may be 6 less than the other. 
 
 8. Two boys found an equal number of cents; one lost 
 6, and the other 4, and they then together had 22 ; how 
 many did each find? 
 
 9. A and B had equal sums of money; A lost $5, and 
 B earned $7, and they then together had $36 ; how much 
 had each at first? 
 
 10. Daniel and Edwin had each the same number of 
 peaches; Daniel lost 6, and Edwin gave him 4, and they 
 then together had 14 ; how many had each then ? 
 
 11. Three times Harry's age, increased by 5 years, 
 equals Harvey's age, and the sum of their ages is 45 
 years; how old is each? 
 
 12. Divide the number 48 into two such parts, that 
 twice the first part, diminished by 6, shall equal the 
 second part ? 
 
 13. The sum of two numbers is 55, and the greater 
 equals 3 times the less, diminished by 5 ; required the 
 numbers. 
 
 14. A pole whose length was 48 feet, was broken Into 
 two unequal pieces, f of the longer part equalling the 
 shorter ; required the length of each piece ? 
 
 15. A tree whose length was 45 feet was broken into 
 two unequal parts, and | of the longer piece, plus 5 feet, 
 equals the shorter required the length of each piece, 
 
 9 * 
 
98 MENTAL ARITHMETIC. 
 
 16. A watch and chain cost $85, and T ^ of the cost 
 of the watch, plus .$7, equals the cost of the chain; re- 
 qui"ed the cost of each. 
 
 17. Francis has 9 cents more than ^ as many as Fan 
 nie, and they together have 42 ; how many cents has 
 each ? 
 
 18. A cow and horse cost $132 ; required the cost of 
 each, if the cow cost | as much as the horse, minus 8 
 dollars. 
 
 19. A man walked 110 miles in three days; he walked 
 5 miies further the second day than the first, and 10 
 miles further the third than the second ; how far did he 
 walk each day ? 
 
 20. A man bought a sleigh, horse, and harness, for 
 8152 ; for the sleigh he gave twice as much as for the 
 harness, plus $6, and for the horse 4 times as much as 
 for the harness, plus $6 ; what did he pay for each ? 
 
 21. A lady bought a hat, cloak, and shawl for $78; 
 what did she pay for each, supposing the cloak cost twice 
 as much as the hat, plus $4, and the shawl twice as much 
 as the cloak, lacking 4 dollars ? 
 
 22. One half of Mary's oranges equals Annie's, and ^ 
 of Annie's equals Emma's, and they together have 28 ; 
 how many has each ? 
 
 23. A earned f as much as B, and B earned | as much 
 as C, and they together earned $108; required the amount 
 earned by each. 
 
 24. In a certain field there are 42 animals, consisting 
 of horses, sheep, and cows; required the number of each, 
 provided of the number of sheep, -L 10, equals the 
 number of cows, and J of the number of sheep, -j- 1(\ 
 equals the number of horses. 
 
MENTAL ARITHMETIC. 99 
 
 SECTION VI. 
 
 LESSON I. 
 
 Percent., from th e Latin per, by, and centum, the hundred, mean? 
 by or on the hundred. Thus, 5 per cent, of a number of apples 
 is 5 apples of a hundred, 10 per cent, of a number of dollars is 
 10 dollars on a hundred, and so on, whatever be the denomina- 
 tion. 
 
 1. At a gain of 10 per cent., what part of the value 
 equals the gain ? 
 
 SOLUTION. A gain of 10 per cent, is a gain of 10 on 100. 
 If on 100 the gain is 10, on 1 it is J^ of 10, which is JH^, or 
 i . Therefore, at a gain of 10 per cent, J_ of the value equals 
 the gain. 
 
 2. At 2, 4, 5, or 8 per cent., what part of the cost 
 equals the gain ? 
 
 3. At a loss of 12, 14, 16, or 20 per cent., what part 
 of the value equals the loss 1 
 
 4. If I gain 25, 30, or 35 per cent, on an investment, 
 what part of the money invested equals the gain ? 
 
 5. A gains 50 per cent, on his capital; what part of 
 the capital equalled the gain ? 
 
 6. B gained at one time 60, at another time 70, and 
 at another time 80 per cent. ; what part of the capital 
 each time equalled the gain? 
 
 7 What part of the cost equals the gain at 8-1, 12.\, 
 16|. or 33| per cent ? 
 
 8. A man paid SI 50 for a horse, and sold it at a gain 
 of 30 per cent.; what was the gain? 
 
 SOLUTION. At a gain of 10 per cent. JjAc, or 1 , of the cost 
 equals the gain 1 of $150 is $15. Therefore, &c 
 
100 MENTAL ARITHMETIC. 
 
 9. A lady bought a shawl for $8, and sold it at a gain 
 of 25 per cent. ; required the gain. 
 
 10. A merchant sold 20 per cent, of 50 barrels of 
 flour ; how many barrels did he sell, and how many re- 
 mained ? 
 
 11. Henry sold a cow worth $40, at a loss of 5 per 
 cent. ; what did he receive for the cow ? 
 
 12. Samuel spent 20 per cent, of $50 for a watch, and 
 20 per cent, of the remainder for a chain ; how much had 
 he remaining? 
 
 13. How much is 5 times 4 per cent, of 400 barrels of 
 flour, and 6 times 5 per cent, of 300 barrels ? 
 
 14. Which is the greater, and how much ; 20 per cent, 
 of 50 apples, or 6 times 4 per cent, of 25 apples ? 
 
 15. Thomas having a horse which cost $120, sold it at a 
 gain of 25 per cent., and the buyer sold it at a loss of 20 
 per cent. ; what did the latter receive for it ? 
 
 16. A lady bought 6 yards of calico for 180 cents, and 
 sold it at a gain of 10 per cent. ; what was the gain on 
 each yard ? 
 
 17. 10 per cent, of $300 dollars is f of what Mary 
 paid for a shawl ; required the cost of the shawl. 
 
 18. A merchant purchased 10 barrels of flour for $50, 
 and sold them at a loss of 20 per cent. ; what did ne re- 
 ceive for each barrel ? 
 
 19. A man owning T 8 ff of a house, sold 25 per cent, of 
 it ; how much had he remaining ? 
 
 20. 8 per cent, of $200 is f of what A gave for a 
 watch ; he sold it so as to gain 20 per cent. ; for what did 
 he sell it ? 
 
 21. A and B together have $1600, of which A owns 
 | as much as B; A then obtains 20 per cent, of B's part \ 
 how much does each now possess ? 
 
 22. A owned 50 acres of land, and B owned three 
 times as much; A sold B 20 per cent, of his land, and 
 fchen bought 25 per cent, of B's; how much had each 
 after this operation? 
 
MENTAL ARITHMETIC. 101 
 
 LESSON II. 
 
 ] A man bought a watch for 20 dollars, and sold it 
 for $25 ; what was the gain per cent. ? 
 
 SOLUTION. If he bought it for $20, and sold it for $25, he 
 gained the difference between $25 and $20, which is $5. If 
 on $20 he gained $5, on $1 he gained ^ of 5, which is ^, or 
 $1, and on $100 he would gain 100 times 1, which are l|o, or 
 $25; hence the gain is 25 per cent. 
 
 2. A boy gave 25 cents for a knife, and sold it for 30 
 cents ; what did he gain per cent. ? 
 
 3. A lady bought a shawl for $5, and sold it for $8 ; 
 what was the gain per cent. ? 
 
 4. Thompson bought a boat for $20, and sold it for 
 816 ; what was the loss per cent. ? 
 
 5. Rose bought a dress for $4, and sold it for $6 ; what 
 was the gain per cent. ? 
 
 6. Edwin bought a horse for $150, and sold it for | of 
 the cost ; required the loss per cent. 
 
 7. Robert sold his horse for $150, which was | of 
 what he paid for it ; what per cent, did he lose ? 
 
 8. EHhu bought 10 cows for $200, and sold 8 of them 
 for what they all cost ; what was the gain per cent. ? 
 
 9. What per cent, of $25 is $5 ? Of 40 cows is 8 
 cows ? Of 60 apples is 6 apples ? 
 
 10. What per cent, of 16 is 4 ? Of 30 is 5 ? Of 200 
 is 8? Of 80 is 4? Of 96 is 12? 
 
 11. James having 50 marbles, sold 20 per cent, of 
 them ; what per cent, of the whole remained ? 
 
 12. A man bought 25 barrels of flour ; he lost 20 per 
 cent, of it, and sold 25 per cent, of the remainder; what 
 per cent, of the whole remained ? 
 
 13. What per cent, of \ is ? Of ^ is -^ ? Of \ is 
 I? Offis/,? Of|is T y Off1s|t 
 
 14. If a miller takes 8 quarts of every bushel he grinds 
 fur toll, what per cent, does he take for toll 1 
 
102 MENTAL ARITHMETIC. 
 
 15. | of $6 is twice what per cent, of of 50 dollars ? 
 
 16. 4 of $10 is I of what per cent, of i of 50 dollars 7 
 
 17. Two-thirds of 90 dimes is f of what Samuel paid 
 for 10 books; he sold them for 3 dimes apiece; required 
 the loss per cent. 
 
 18. A merchant bought 30 barrels of flour for $5 each, 
 and sold f of them at the rate of 3 barrels for $24, and 
 the rest for cost; required the gain per cent. 
 
 19. ^ of 10 per cent, is what per cent, of 20 per cent. ? 
 
 20. | of 8 per cent, is what per cent, of 30 per cent. ? 
 
 21. I of 15 per cent, is what per cent, of 4^ times 16 
 per cent. ? 
 
 22. Mary sold some silk for $12, and thereby cleared 
 i of this money ; what would she have lost per cent, if 
 she had sold it for 6 dollars ? 
 
 23. A man sold a cow for $25, and thereby cleared | 
 of this mon< y ; how much would he have gained per cent. 
 if he had sold it for $30 ? 
 
 24. Willis sold some books for $12, and thereby cleared 
 J of the cost ; what would he have lost per cent, by sell- 
 ing them for $8 ? 
 
 LESSON III. 
 
 1. Thomas sold his watch for $25, and thereby gained 
 25 per cent. ; what was the cost of the watch ? 
 
 SOLUTION. Tf he gained 25 per cent., then ._2p_, C r j, of the 
 cost equals the gain, which added to 4, the cost, is of the 
 cost, which equals $25. If | of the cost equals $25, | of the 
 cost equals 1 of $25, which is $5, and 4, or the cost, equals 4 
 times $5, which are $20. Therefore, &c. 
 
 2. Mary sold her shawl for $14, which was at a gain 
 of 40 per cent. ; required the cost of the shawl. 
 
 3. A farmer sold a cow for $23, and thereby gained 15 
 per cent. , required the value of the cow. 
 
MENTAL ARITHMETIC. 103 
 
 4. A student sold his library for $140, and thereby 
 lost 30 per cent. ; what was its value ? 
 
 5. By selling a hat for $8, Mary lost 20 per ceot. ; 
 what was the value of the hat ? 
 
 6. A dog was bought for 15, and sold at a gain of 20 
 per cent. ; for what was it sold ? 
 
 7. Mason gained 20 per cent, by selling cloth at $6 
 per yard ; how should he have sold it to gain 25 per cent. ? 
 
 8. If a merchant sells muslin at 39 cents a yard, and 
 thereby gains 30 per cent., how ought he to sell it to lose 
 40 per cent. ? 
 
 9. If by selling land at $75 an acre I gain 25 per cent., 
 how must I sell it to lose 40 per cent. ? 
 
 10. A boat was sold for $91, which was at a loss of 85 
 per cent. ; how ought it to have been sold to gain 40 per 
 cent, ? 
 
 11. Taylor lost 60 per cent, on a watch, by selling it for 
 840; what ought he to have received to gain 60 per cent. ? 
 
 12. Hinkston sold his horse and carriage for $240, and 
 thereby lost 4 per cent. ; what would he have gained per 
 cent, by selling it for $300 ? 
 
 1$. A wagon was sold for $90, which was 10 per cent, 
 less than its value ; what would have been the gain per 
 cent, if it had been sold for $120 ? 
 
 14. Mr. Bowman sold 2 books for $15 each ; on one he 
 gained 25 per cent., and on the other he lost 25 per cent. ; 
 how much did he lose by the transaction ? 
 
 15. A tailor sold 2 coats for $12 each; on one h 
 gained 20 per cent., and on the other he lost 20 per 
 cent. ; did he gain or lose by the sale, and how much ? 
 
 16. B bought a watch for $42, which was 40 per cent, 
 le?s than its value; he sold it for 30 per cent, more then 
 Us value ; what was the gain ? 
 
 1.7. A man sold 2 watches for $80 each; on one ho lost 
 20 per cent., and on the other he gained 25 per cent.; 
 how much was gained or lost by the transaction ? 
 
 18. A merchant sold a stove for $30, and thereby lost 
 25 per cent ; he then bought another for $30, and upon 
 
104 MENTAL ARITHMETIC. 
 
 it gained 25 per cent. ; what was gained or lost by the 
 transaction ? 
 
 19. Martha sold a painting, so that f of what she re 
 eeived for it equalled | of the cost ; did she gain or lose, 
 and how much per cent. ? 
 
 20. Ten-el sold his watch and chain for 8120, receiv- 
 ing 5 times as much for the watch as for the chain ; on 
 the watch he gained 25 per cent., and on the chain he 
 lost 20 per cent, j what was the gain ? 
 
 LESSON IV. 
 
 1. A man gained 25 per cent, by selling his watch for 
 $20 more than it cost ; required its cost. 
 
 2. A farmer gained 80 per cent, by selling a cow for 
 $9 more than she cost ; what did the cow cost ? 
 
 3. A, by selling his dog for $6 less than it cost, lost 
 15 per cent, on the sale j required the cost of the dog. 
 
 4; A hat was sold for 20 cents less than cost, which 
 was at a loss of 40 per cent. ; required the value of the 
 hat. 
 
 5. Four is 10 per cent., 5 is 20 per cent., and 6 is 25 
 per cent., of what number? 
 
 6. Eight is 40 per cent., 9 is 30 per cent., and 12 is 
 12 per cent., of what number ? 
 
 7. Thirty is 25 per cent, less, and 25 per cent, more, 
 fchan what numbers ? 
 
 8. A man gained $20 by selling a boat for 20 per cent 
 more than its value ; what would he have gained by sell- 
 ing it for 10 per cent, above its value ? 
 
 9. A piano was sold for $60 less than its value, which 
 was at a loss of 30 per cent. ; what would have been the 
 gain per cent, if it had been sold for $250 ? 
 
 10. $24 is 4 per cent, of the sum of A's and B's for- 
 tune ; how much money has each, provided A has twice 
 as much as B? 
 
MENTAL ARITHMETIC. 105 
 
 11. An agent receives $120 to purchase goods, after 
 deducting his commission, which is 20 per cect. on the 
 amount expended; required his commission. 
 
 REMARK. ^, his commission, -j- 4 what he expended, = | 
 of what he expended, which is $120. 
 
 12 A man receives 25 per cent, for purchasing goods, 
 how many dollars worth can he purchase with 200, re- 
 taining his commission ? 
 
 13. A receives $216 to buy goods, and is to retain 8 
 per cent, on the money expended ; required the amount 
 of money expended. 
 
 14. Frick received $2800 to invest in land, after de 
 ducting his commission, which is 12 per cent, on the 
 amount invested; required his commission. 
 
 15. How much grain must a farmer take to mill that 
 he may bring away the flour of 2 bushels, after the mil- 
 ler has taken 20 per cent, of all he took there ? 
 
 16. A's money if 25 per cent, more than B's ; then B's 
 money is how many per cent, less than A's ? 
 
 17. Morgan sold two horses for $180 each on om he 
 gained 25 per cent., and on the other he lost 10 pez 
 cent. ; did he gain or lose, and how much ? 
 
 18. A's shop is valued at $900 ; for what sum must he 
 have it insured, at 10 per cent., so that in case of Joss he 
 may receive both the value of the shop and premiu ia ? 
 
 19. At 5 per cent., what must be insured on a house 
 worth $1900, to include the premium in case of loss ? 
 
 20. At 8 per cent., what must $2300 worth of pro- 
 perty be insured for, so that the premium may be included 
 in case of loss ? 
 
 21. How many yards of cloth, at $4 a yard, must a 
 merchant buy, that by selling it at a profit of 20 per cent .. 
 he may gain $8 ? 
 
 22. A man receives $530 to purchase sheep and cows; 
 what sum will he expend for each, after deducting his 
 commission, which is 6 per cent, of the* money expended, 
 provided he expends 4 times as much for cows as sheep? 
 
 10 
 
106 MENTAL ARITHMETIC. 
 
 LESSON V 
 
 INTEREST. 
 
 Interest is money charged for the use of money or property. 
 ft is estimated at a certain rate per cent., per annum. 
 
 The Principal is the sum on which interest is computed. 
 
 The Amount is the sum of the principal and interest. 
 
 The Rate per cent, is the interest of 100 for one year. 
 
 In computing interest we shall consider 30 days to the month, 
 and 12 months to the year. 
 
 1. Reduce 2 years and 6 months to the fraction of a 
 year. 
 
 SOLUTION. In 1 yr. there are 12 months, hence 1 month is 
 _i of a year, and 6 months are 6 times J^, which are J^, or 1 
 of a year, which, added to 2 years, equals 2^, or 5 years. 
 
 2. Reduce each of the following to the fraction of a 
 year : 2 yr. 8 mo., 3 yr. 4 mo., 4 yr. 3 mo., 5 yr. 6 mo. 
 
 3. How many years in 3 yr. 9 mo. ? 7 yr. 2 mo. ? 4 
 yr. 10 mo.? 6 yr. 5 mo.? 6 yr. 8 mo.? 
 
 4. Reduce 3 years, 7 months, 15 days, to the fraction 
 of a year. 
 
 SOLUTION. There are 30 days in a month, hence 1 day is J^, 
 and 15 days are is O r 1 of a month, which, added to 7 mo., 
 equals 71, or y> months, &c. 
 
 5. How many years in 2 yr. 6 mo. ? 2 yr. 8 mo ? 3 
 yr. 4 mo. ? 4 yr. 3 mo. ? 5 yr. 6 mo. ? 
 
 6. How many years in 3 yr. 9 mo. ? 7 yr. 2 mo. ? 4 
 yr, 10 mo.? 6 yr. 5 mo.? 5 yr. 8 mo.? 
 
 7. How many years in 2 yr. 2 mo 2 da ? 3 yr. 3 mo. 
 da ? 2 yr. 4 mo. 5 da. ? 
 
 8. How many years in 4 yr. 7 mo. 6 da. ? 5 yr. 5 mo. 
 10 da.? 6yr. 2 mo. 12 da. 9 
 
 9. How many years in 7 yr. 3 mo. 18 da. ? 8 yr. 6 
 mo. 20 da.? 2 jr 1 mo. 6 da.? 
 
MENTAL ARITHMETIC. 107 
 
 10. At 5 per cent, for 4 years, what part of the princi- 
 pal equals the interest ? 
 
 SOLUTION. At 5 per cent. J g IJ of the principal equals the 
 iLterest for 1 year, and for 4 yrs., 4 times T g^, which are ^ g , 
 or i of the principal equals the interest. 
 
 11. At 10 per cent, for 5 years, or 20 per cent, for 2 
 years, what part of the principal equals the interest ? 
 
 12. At 8 per cent, for 5 years, or 9 per cent, for 10 
 years, what part of the principal equals the interest? 
 
 13. At 7 per cent, for 5 years, and 6 per cent, for 15 
 years, what part of the principal equals the interest? 
 
 14. At 6 per cent, for 2 yr. and 4 mo., what part of 
 the principal equals the interest ? 
 
 15. At 8 per cent, for 3 yr. and 3 mo., what part of 
 the principal equals the interest ? 
 
 16. At 6. per cent, for 5 yr. and 8 mo., what part of 
 the principal equals the interest ? 
 
 17. At 7 per cent, for 12 yr. and 6 mo., what part of 
 the principal equals the interest ? 
 
 18. At 8 per cent, for 1 yr. 4 rno. 15 da., what part 
 of the principal equals the interest ? 
 
 19. At 9 per cent, for 2 yr. 5 mo. 10 da., what part of 
 the principal equals the interest ? 
 
 20. At 5 per cent, for 3 yr. 7 mo. 6 da., what part of 
 the principal equals the interest ? 
 
 21. What is the interest of $60 for 6 years, at 5 per 
 cent. ? 
 
 22. What is the interest of $40 for 4 years, at 5 per 
 cent. ? 
 
 23. What is the interest of $30 for 5 years, at 4 per 
 eent. ? 
 
 24. What is the interest of $80 for 7 years, at 5 per 
 cent. ? 
 
 25. What is the interest of $75 for 8 years, at 6 per 
 cent. ? 
 
 26. What is the interest of $50 for 9 years, at S pel 
 cent ? 
 
108 MENTAL ARITHMETIC. 
 
 27. What is the interest of $250 for 6 years, at 4 per 
 cent. ? 
 
 28. What is the interest of $28 for 10 years, at 5 per 
 cent. ? 
 
 29. What is the interest of $400 for 11 years, at 5 per 
 cent. ? 
 
 30. What is the interest of $200 for 6^ years, at 6 per 
 oent.? 
 
 31. What is the interest of $300 for 5| years, at 9 per 
 cent. ? 
 
 32. What is the interest of $240 for 7^ years, at 9 per 
 cent. ? 
 
 What is the interest 
 
 33. Of $600 for 2 years, 3 months, at 8 per cent. ? 
 
 34. Of $300 for 4 years, 6 months, at 6 per cent. ? 
 
 35. Of $240 for 3 years, 9 months, at 8 per cent. ? 
 
 36. Of $225 for 6 years, 8 months, at 6 per cent.? 
 
 37. Of $500 for 5 years, 4 months, at 9 per cent. ? 
 
 38. Of $330 for 7 years, 6 months, at 4 per cent. ? 
 
 39. Of $222 for 8 years, 4 months, at 6 per cent. ? 
 
 40. Of $666 for 6 years, 3 months, at 8 per cent.? 
 
 41. Of $288 for 4 years, 2 months, at 12 per cent ? 
 
 42. Of $440 for 2 years, 1 month, at 12 per cent.? 
 
 43. Of $120 for 5 years, 10 months, at 12 per cent ? 
 
 44. Of $540 for 3 years, 7 months, 6 days, at 6 per 
 cent. ? 
 
 45. Of $300 for 5 years, 3 months, 18 days, at 10 per 
 cent. ? 
 
 46. Of $400 for 2 years, 3 months, 9 days, at 40 per 
 cent. ? 
 
 47. Of $500 for 2 years, 2 months, 12 days, at 5 per 
 cont.? ' 
 
 48. Of $600 for 1 year, 6 months, 12 days, at 15 per 
 cent. ? 
 
 49. Of $200 for 1 year, 6 months, 20 days, at 9 per 
 cent.? 
 
 50. Of $50C for 3 years, 8 months, 12 days, at LO per 
 cent. ? 
 
MENTAL ARITHMETIC. 109 
 
 LESSON VI. 
 
 1. What is the amount of $50 for 5 ytais, at 3 per 
 cent. ? 
 
 REMARK. if the principal equals the interest, which added 
 to 4, the principal, equals 1 of the principal, which equals the 
 amount. | of $50 = $70. 
 
 5 
 
 2. What is the amount of $250 for 4 years, at 5 pel 
 cent.? 
 
 3. What is the amount of $120 for 7 years, at 10 per 
 cent. ? 
 
 4. What is the amount of $400 for 5 years, at 7 per 
 cent. ? 
 
 5. What is the amount of $240 for 5 years, at 5 per 
 cent. ? 
 
 6. What is the amount of $200 for 2 years, 3 mo., at 
 8 per cent. ? 
 
 7. What is the amount of $600 for 7 years, 6 mo., at 
 6 per cent. ? 
 
 8. What is the amount of $300 for 3 years, 9 mo., at 
 
 8 per cent. ? 
 
 9. What is the amount of $320 for 8 years, 10 mo., at 
 3 per cent. ? 
 
 10. What is the amount of $400 for 7 years, 6 mo., at 
 6 per cent. ? 
 
 11. What is the amount of $360 for 8 years, 4 mo , at 
 
 9 per cent. ? 
 
 12. What is the amount of $100 for 2 years, 6 months, 
 20 days, at 9 per cent. ? 
 
 13. A and B wish to divide the amount of $500 for 8 
 years, at 5 per cent., so that A's part shal) be 6 times B's ; 
 required the share of each ? 
 
 14. The amount of $250 for 6 years, at 10 per cent., 
 is to be divided between C and D, so that C shall have % 
 times as much as D : what does each receive ? 
 
 10* 
 
110 MENTAL ARITHMETIC. 
 
 15. James and Henry have $1500 on interest for 4 
 years, at 10 per cent. ; what amount of interest will each 
 receive, provided James has twice as much as Henry ? 
 
 16. A's fortune is $200, which is I of-B's; what 
 interest will each receive on his money in 4 years, at 5 
 per cent. ? 
 
 17. C's money is $300, which is | of D's ; what is the 
 amount of each for 5 years, at 6 per cent. ? 
 
 18. A's money is $400, which is f of B's; how much 
 more interest will B receive than A, in 8 years, at 5 per 
 cent. ? 
 
 19. A, B, and C, together, have $1200, of which A 
 has twice, and B 3 times as much as C; what is the 
 interest of each for 5 years, at 6 per cent. ? 
 
 20. If the interest of $2500 for 4 years, at 10 per 
 cent., be divided into two parts, which are as 2 to 3, it 
 will respectively give | of B's, and ^ of A's money; how 
 much has each? 
 
 LESSON VII. 
 
 1. What principal will, in 6 years, at 5 per cent., give 
 $60 interest? 
 
 REMARK. We find JL O f the principal equals the interest, 
 which is $60. If _3^ of the principal equals $60, -^ equals J 
 of $60, which is $20, and JO, or the principal, equals 10 times 
 $20, which are $200. 
 
 2. What principal will, in 7 years, at 5 per cent., give 
 $21 interest? 
 
 3 What principal will, in 8 years, at 6 per cent., give 
 812 interest? 
 
 4. What principal will, in 3 years, at 8 per cent., give 
 $60 interest ? 
 
 5. What principal will, in 7 years, at 4 per cent., give 
 $70 interest ? 
 
 
MENTAL ARITHMETIC. Ill 
 
 6. What principal will, in 8 years, at 5 per cent., give 
 $60 interest ? 
 
 7. What principal will, in 3 years and 4 months, at 6 
 per cent., give $80 interest? 
 
 8. What principal will, in 7 years and 6 months, at 4 
 per cent., give $90 interest? 
 
 9 What principal will, in 6 years and 3 months, at 8 
 per cent., give $85 interest ? 
 
 10. A man pays $360 interest, at 6 per cent., annu- 
 ally, on money borrowed; what is the sum borrowed? 
 
 11. How much money must a person borrow, that he 
 must pay an annual interest of $150, at 5 per cent. ? 
 
 12. How much money has Howard on interest, sup- 
 posing he receives $320 for 5 years, 4 months, at 6 per 
 cent. ? 
 
 13. The interest of f of A's money for 6 years and 3 
 months, at 4 per coat., is $250; what is his money? 
 
 14. The interest of f of A's, and | of B's fortune, for 
 5 years, at 6 per cent., is $60, and $90, respectively, 
 required the fortune of each. 
 
 15. The interest of the sum of A's and B's fortune, 
 for 5 years, at 7 per cent., is $210; what is the fortune 
 of each, provided B. is worth twice as much as A? 
 
 16. Howard's money is 3 times HowelPs, and in 5 
 years, at 8 per cent., Howard receives $600 interest; 
 how much money has each ? 
 
 17. The interest on i of A's and | of B's fortune for 
 5 years, at 6 per cent., is $240; what is the fortune of 
 each, provided ^ of A's equals -| of B's? 
 
 18. The interest for 4 years, at 5 per cent., on tie 
 money Martin owes, is $40; and the interest for the 
 game time and rate per cent., on the money due him, 
 $70 ; how much more has he due than he owes ? 
 
 19. A's money is '4 times B's, and the sum of the 
 interest received by both for 3 years, at 8 per cent, is 
 $600 ; how much money has each ? 
 
 20. The interest on the money A paid for a farm, 
 house, and store, for 8 years, at 5 per cent, is equal to 
 
112 MENTAL ARITHMETIC. 
 
 $18,000; what was the cost of each, provided the farm 
 cost 3 times as much as the house, and the house twice 
 as much as the store ? 
 
 LESSON VIIL 
 
 The present worth of a debt, payable at some future time, 
 without interest, is such a sum as would, at a given rate pet 
 cent., amount to the debt at the time it becomes due. The pre- 
 sent worth may therefore be found in the same manner as the 
 principal, when we have given the amount, time, and rate per 
 cent. The Discount is the allowance made for the payment of 
 money before it is due. It equals the debt minus the present 
 worth. 
 
 1. What principal will, in 8 yeaps, at 5 per cent., 
 amount to $140 ? 
 
 SOLUTION. At 5 per cent., _|> of the principal equals the 
 interest for 1 year, and for 8 years, 8 times jj which are 40 
 or 2 of the principal equals the interest, which added to , the 
 principal, equals J of the principal, which equals the amount, 
 or $140, &c. 
 
 2. What principal will, in 7 years, at 6 per cent., 
 amount to $710 ? 
 
 3. What principal will, in 4 years, at 10 per cent., 
 amount to $420 ? 
 
 4. What principal will, in 6 years and 8 months, at 9 
 per cent., amount to $820? 
 
 5. What principal will, in 8 years and 9 months, at 8 
 per cent., amount to $510 ? 
 
 6. What principal will, in 5 years and 10 months, at 6 
 er cent., amount to $540 ? 
 
 7. What is the present worth of $60, due 4 years 
 hence, at 5 per cent. ? 
 
 8 What is the present worth of $52, due 5 years 
 hence, at 6 per cent. ? 
 
MENTAL ARITHMETIC. . 113 
 
 9. The amount of | of B's fortune, for 3 years and 4 
 months, at 6 per cent., is $600; what is his fortune? 
 
 10. The amount due on a note which had been on 
 interest for 3 years and 4 months, at 9 per cent., is $520 j 
 required the face of the note. 
 
 11. What is the present worth of $270, due 7 years 
 hence, at 5 per cent. ? 
 
 12. What is the present worth of $370, due 8 years 
 hence, at 6 per cent. ? 
 
 13. What is the discount of $580, due 9 years hence, 
 at 5 per cent. ? 
 
 14. What are the present worth and discount of $340, 
 due 10 years hence, at 7 per cent. ? 
 
 15. Required the discount of $700, due 5 years hence > 
 at 8 per cent. 
 
 16. Required the discount of $149, due 7 years hence, 
 at 7 per cent. 
 
 17. The sum of A's and B's money, being on interest 
 for 3 years and 9 months, at 8 per cent., amounts to 
 $2600 ; what is the money of each if A's is 3 times B's ? 
 
 18. A's money added to B's, being on interest for 5 
 years and 4 months, at 6 per cent., amounts to $660; 
 what sum has each if A's is 4 times B's t 
 
 19. A man wishes to place such a sum of money on 
 interest, at 6 per cent., that it will give an annual interest 
 of $360 for a poor sister; required the amount invested. 
 
 20. Four times A's money, added to 3 times B's, being 
 on interest for 4 years, at 10 per cent., amounts to $4200 ; 
 how much has each, if 3 times B's equals A's? 
 
 21. Two thirds of A's fortune, plus | of B's, being oo 
 interest for 6 years, at 5 per cent., amounts to $7800 , 
 what is the fortune of each, supposing | of A's equals | 
 ofBs? 
 
 22. | of the cost of Bowman's house, plus | of the cost 
 of his farm, being on interest for 5 years, at 8 per cent., 
 amounts to $2100 ; what is the cost of each, provided 
 the house ^ost ^ as much as the farm? 
 
 23 Two times the value of a horse, plus 8 timen the 
 
114 MENTAL ARITHMETIC. 
 
 value of a cow, which is \ of the value of the horse, in 8 
 years, at 5 per cent., gives $84 interest \ required the 
 value of each, 
 
 24. The money Henry paid for a horse, carriage, and 
 harness, in 10 years, at 5 per cent., would give such an 
 interest, that it on interest for the same time, and rate, 
 would amount to $270 ; how much did he pay for each, 
 if the horse cost twice as much as the carriage, and the 
 carriage 3 times as much as the harness ? 
 
 LESSON IX. 
 
 
 
 1. The interest of $200, for a certain time, at 5 pei 
 cent., is $60 ; required the time. 
 
 SOLUTION. At 5 per cent, for one year, -4*> or 1 , of the 
 principal equals the interest. ^ of $200 is $10. Iflt require 
 one year for $200 to gain $10/"to gain $1 it will require J^ of 
 a year, and to gain $60 it will require 60 times 1 of a year, 
 which are g or 6 years.* 
 
 2. In what time will $100, at 6 per cent., give $21 
 interest ? 
 
 3. In what time will $100, at 7 per cent., give $14 
 interest? 
 
 4. In what time will $200, at 5 per cent., give $40 
 interest ? 
 
 5. In what time- will $150, at 6 per cent., give $45 
 interest? 
 
 6. In what time will $100, at 8 per cent , give $32 
 interest? 
 
 7. In what time will $300, at 10 per cent., give $120 
 interest? 
 
 8. In what time will $200, at 8 per cent., give $48 
 interest? 
 
 * The latter part of this may be given thus : it will require as 
 many years as $10 is contained times in $60, which are 6. 
 
MENTAL ARITHMETIC, 115 
 
 9. In what time will $60, at 5 per cent., give 821 
 interest? 
 
 10. In what time will $25, at 6 per cent., give $9 
 interest? 
 
 11. In what time will $50, at 9 per cent., give $86 
 interest? 
 
 12. In what time will $50, at 4 per cent., amount to $62 ? 
 
 13. In what time will $150, at 5 per cent., amount to 
 8210? 
 
 14. In what time will $300, at 7 per cent., amount to 
 $510? 
 
 15. In what time will a principal gain twice itself, at 40 
 per cent. ? 
 
 REMARK. 2 of the principal = the interest in 1 year, and 
 twice the principal is U) ; hence it will require as many years 
 as - i s contained times in U>. 
 
 .-> o 
 
 16. In what time will a principal gain 3, 4, and 5 
 times itself, at 10 per cent ? 
 
 17. In what time will a principal double itself, at 5 
 percent.? At 6? 7? 8? 9? 10? 
 
 18. In what time will a principal double itself; at 10 
 percent.? At 12? 15? 20? 25? 50? 
 
 19. in what time will a principal treble itself, at 5 
 percent.? At 10? 20? 25? 40? 50? 
 
 20. In what time will a principal quadruple itself, at 
 5 per cent.? At 15? 30? 50? 60? 100? 
 
 21. The amount of a certain principal, for a certain 
 time, at 5 per cent., is $250, and the amount for the 
 same time at 8 per cent, is $280 ; required the principal 
 and time. 
 
 22. A certain sum of money, on interest, amounts, in 
 a certain time, at 6 per cent., to $310, and, at 10 per 
 cent., for the same time, to $350; required the time and 
 principal. 
 
MENTAL ARITHMETIC. 
 
 LESSON X. 
 
 1. At what per cent, will $60, in 5 years, give 82] 
 interest? 
 
 SOLUTION. For 5 years, at one per cent., _ , or i_ of the 
 principal equals the interest. * of $60 is $3. If $00 in 5 
 years, at 1 per cent., gains $3, to gain $1, it will require 
 of 1 per cent., and to gain $21 it will require 21 times JL, whic! 
 are ^1, or 7 per cent.* 
 
 2. At what per cent, will $40, in 5 years, give $20 
 interest ? 
 
 3. At what per cent, will $200, in 3 years, giv.e $36 
 interest ? 
 
 4. At what per cent, will $300, in 4 years, give $00 
 interest ? 
 
 5. At what per cent, will $80, in 5 years, give $32 
 interest ? 
 
 6. At what per cent, will $50, in 6 years, give $15 
 interest? 
 
 7. At what per cent, will $60, in 7 years, give $21 
 interest ? 
 
 8 At what per cent, will $50, in 8 years, give $22 
 interest ? 
 
 . 9. At what per cent, will $100, in 4 years, amount to 
 I] 20? 
 
 10. At what per cent, will $90, in 5 years, amount to 
 $117? 
 
 11. At what per cent, will $20, in 7 yr. 6 mo., amount 
 to $26? 
 
 12. At what per cent will $6, in 3 yr. 4 mo., amount 
 to $7? 
 
 * The latter part of this may be given thus: It will require 
 as many times 1 per cent, as $3 is contained times in $21, 
 which are 7. 
 
MENTAL ARITHMETIC. 117 
 
 13. At what per cent, will a given principal gain 3 
 times itself in 10 years ? 
 
 REMARK. JL of the principal equals the interest at 1 per 
 cent. ; hence it will require as many times 1 per cent, as 1 is 
 contained times in |g. 
 
 14. At what per cent, will a principal gain 2, 4, 5, 
 and 6 times itself in 30 years ? 
 
 15. At what per cent, will a principal double itself in 
 4 years? In 10? 12? 20? 
 
 16. At what per cent, will a principal double itself in 
 8 years? In 25? 331? 50? 
 
 17. At what per cent, will a principal treble itself in 
 10 years? In 20? 25? 40? 80? 100? 
 
 18. At what per cent, will a principal quadruple itself 
 in 10 years? In 15? 30? 60? 100? 150? 
 
 19. At what per cent, will a principal quintuple itself 
 in 4 years? In 20? 40? 80? 100? 200? 
 
 20. A gained $20 by selling an article for 20 per cent, 
 more than cost } required the cost and amount received 
 for it. 
 
 21. B lost 25 per cent, by selling a boat for 20 per 
 cent, of $150 ; required the value of the boat and the 
 amount received for it. 
 
 22. A man sold 2 cows for $50, gaining 25 per cent, 
 on the first, and losing 25 per cent, on the second; what 
 was the value of each, if he received | as much for the 
 second as for the first ? 
 
 23. The amount of a certain principal for 7 years, at 
 a certain per cent., is $540, and for 10 years, $600; re- 
 quired the principal and rate per cent. 
 
 24. The amount of a certain principal for 4 years, at 
 a certain per cent., is $420, and for 9 years, at the same 
 rate, $570 ; required the rate per cent, and principal. 
 
118 MENTAL ARITHMETIC. 
 
 SECTION VII. 
 
 LESSON I. 
 
 1. & *<? B hired a pasture for $36. A pastured 4 
 and B 5 cows ; how much should each pay ? 
 
 SOLUTION. If A pastured 4 cows and B 5, they both pastured 
 
 4 + 6, which are 9 cows. If the pasturage of 9 cows cost $36, 
 the pasturage of 1 cow will cost ^ of $36, which is $4. and 
 the pasturage of 4 cows, A's number, will cost 4 times $4, or 
 $16, and the pasturage of 5 cows, B's number will cost 5 times 
 $4, or $20. 
 
 2. Two boys bought 60 apples for 12 cents; one paid 
 
 5 cents, and the other 7 cents ; how many apples should 
 each receive ? 
 
 3. Rufus and William paid 20 cents for 40 peaches, 
 of which Rufus 'paid 9, and William, 11 cents; how many 
 peaches belong to each ? 
 
 4. Three men hired a horse for 20 days, at the rate of 
 $1 per day; the first used it 5, the second, 6, and the third, 
 
 9 days ; how much should each pay ? 
 
 5. A and B hired a pasture for $44; A puts in 12 
 oxen, and B 100 sheep; how much should each pay, 
 supposing an ox to eat as much as 10 sheep ? 
 
 6. Two farmers hire a pasture for $56 ; one turns in 
 
 10 cows, and the other 36 horses; how much should 
 each pay, provided a cow eats twice as much as a horse? 
 
 7. Three men, A, B, and C, bought 144 bushels of 
 peaches for $72, of which A paid J, B, f , and C, the re ' 
 mainder; how many bushels did each receive? 
 
 8. A and B engage to do a piece of work for $72 ; A 
 sends 6 men, and B, 15 boys; how much should each 
 receive, supposing 2 men to do as much as 3 boys ? 
 
 9 A and B agree to mow a field of grass for $54 ; A 
 
MENTAL ARITHMETIC. 119 
 
 sends 3 men 5 days, and B sends 4 n en 3 days ; how 
 much should A and B receive respectively ? 
 
 10. Two men hire a lot of pasture for $10 ; one turns 
 in 6 horses for 7 days, and the other, 7 horses for 4 
 days ; how much should each pay ? 
 
 11. A and B built a boat for $140; A sent 6 men 5 
 days, and B, 4 men 10 days; how much should A and B 
 receive respectively? 
 
 12. Two men gain in trade $440; A put in $25 for 4 
 nnonths, and B, $15 for 8 months; what is each man's 
 
 share of the gain ? 
 
 13. C and D build a wall for $120 ; C with 4 assistants 
 laboured 4 days, and D with 3 assistants laboured 5 days ; 
 how much do C and D receive respectively ? 
 
 14. A, B, and C build a boat for $80 ; A sent 3 men 
 4 days. B 5 men 2 days, C 3 men 6 days ; how much do 
 A, B, and C receive respectively ? 
 
 15. A and B plough a field for $76 ; A employed 12 
 horses, and B 18 oxen ; they completed it in 4 days ; 
 what was the value of the daily labour of each horse and 
 ox, supposing 3 horses do as much as 5 oxen ? 
 
 16. E and F engaged to reap a field of wheat for $54 ; 
 E sent 3 men 5 days, and F, 6 boys 4 days; how much 
 should each receive, if 1 man does as much as 2 boys ? 
 
 17. In a field of grass, which cost $24, M turned 16 
 horses for 3 weeks, and N, 25 cows for 4 weeks; how 
 much should each pay, if 4 horses eat as much as 5 cows? 
 
 18. R, S, and T hire a pasture for $63 ; R puts in 6 
 horses, S puts in 18 cows, and T, 48 sheep ; how mucL 
 should each pay, if a horse eat twice as much as a cow, 
 and a cow 4 times as much as a sheep ? 
 
120 MENTAL ARITHMETIC. 
 
 LESSON II. 
 
 1 . Divide 30 cents between A and B ; so that their 
 shares will be to each other as 4 to 6. 
 
 SOLUTION. Since the shares are to be to each other as 4 to 6, if 
 we divide 30 cents into 4 -f 6, which aro 10 equal parts, 4 uf 
 these parts, or j^, will be A's, and 6 of these parts, or ^ will 
 be B's number, &c. 
 
 2. Divide 45 apples between Thomas and Harry, so 
 that their shares may be to each other as 3 to 2. 
 
 3. Divide the number 50 into two parts, that shall be 
 to each other as 7 to 3. 
 
 4. In a school consisting of 45 pupils, there are 5 girls 
 for every 4 boys ; how many of each sex in the school ? 
 
 5. The sum of two numbers is 40, and the larger is to 
 the smaller as 5 to 3 ; required the numbers. 
 
 6. Divide 45 plums among three boys, so that their 
 shares may be in the proportion of 2, 8, and 4. 
 
 7. Two men bought a barrel of flour for $8, the first 
 paying $3, and the second $5; how .much of the flour 
 belongs to each ? 
 
 8. Three men bought 75 horses, and as often as the 
 first paid $4, the second paid $5, and the third, $6 ; how 
 many horses should each receive ? 
 
 9. Divide $44 between A and B, so that B shall have 
 $3^ as often as A $2. 
 
 10. The sum of 2 numbers is 50, and the first is to the 
 second as 4 to | ; what are the numbers ? 
 
 11. Divide the number 49 into two parts which are to 
 each other as | to |. 
 
 12. The sum of three numbers is 46; what is eacb 
 of the numbers, if they are to each other as ^, f , and f ? 
 
 13. Divide the number 50 into 3 parts which shall be 
 o each other as |, 1, and 2. 
 
 14. A, B, and C found a purse containing $1 00, which 
 
MENTAL ARITHMETIC. 121 
 
 fchey agree to divide in the proportion of |, |, and 1 ; 
 how much does each receive ? 
 
 15. A and B agree to pay $25 toward building a 
 church, which is to be situated 2 miles from A, and 8 
 miles from B; how much does each contribute, if the} 
 pay in proportion to the reciprocals of their distances ? 
 , 16. If $420 be divided into two parts, which are to 
 each other as 4 to |, it will respectively give | of A's, 
 and | of B's fortune ; required the fortune of each. 
 
 17. If the interest of $500 for 4 years, at 5 per cent., 
 be divided into two parts, to each other as 2 to 3, it will 
 respectively give f of A's, and | of B's fortune; required 
 the fortune of each. 
 
 18. A's fortune added to ^ of B's, being on interest 
 for 5 years at 6 per cent., equals $2600; what is the for- 
 tune of each, provided A's is to B's as 3 to 4 ? 
 
 19. | of A's fortune, plus \ of B's fortune, being on 
 interest for 6 years at 10 per cent., amounts to $800: 
 what is the fortune of each, if A's fortune is to B's as 9 
 to 8? 
 
 20. M's fortune, + 1 of N's, which is equal to of 
 M's, is $900, and if the sum of M's and N's be divfded 
 in the proportion of ^ to f , it will respectively give \ of 
 K's and | of T's fortune ; required the fortunes of each. 
 
 LESSON III, 
 
 1. A can do a piece of work in 4 days ; what part of 
 it can he do in one day ? 
 
 2. B can cut a cord of wood in | of a day; how much 
 can he cut in one day ? 
 
 3. A man can build | of a boat in a week; how long 
 will it require for him to build the whole boat? 
 
 11* 
 
122 MENTAL ARITHMETIC 
 
 4. A mason can build a wall in 2i days; what part of 
 it can he build in one day ? 
 
 5. If A and B can mow | of a field of grass in one day. 
 how long will it require to mow the whole field ? 
 
 6. A can do a piece of work in 3 days, and B in 6 
 days ; what part can each do in one day ? 
 
 7. If B can do a piece of work in 3 days, and C in 6 
 days, how much can they together do in one day ? 
 
 8. If B and C can do | of a piece of work in one day, 
 how long will it require to do the whole work ? 
 
 9. Fuller can eat a bushel of apples in 4 days, and 
 Brodhead in 6 days ; how many days would it last them 
 both? 
 
 10. A can dig a ditch in 5 days, and B in 6 days; in 
 what time will they do it working together? 
 
 11. C can make a chest in 4 days, and D in 7 days; 
 in what time can they make it working together ? 
 
 12. E can reap a field in 6 days, and F in 8 days; how 
 long will it take them both to reap it? 
 
 13. A can do a piece of work in 3 days, B in 4 days, 
 and G in 6 days; in what time can they together do it? 
 
 14. A cistern has two pipes, by the first of which it 
 may be filled in 12 hours, and by the second, in | of the 
 time; how long will both be in filling it? 
 
 15. A can make a book-case in 6 days, and A and B 
 can make it in 4 days; in what time can B make it 
 alone ? 
 
 16. A, B, and C can dig a ditch in 3 days. A can 
 dig it in 6 days, and B in 8 days ; in what time can 
 alone dig it ? 
 
 17. A pound of tea lasted a man and wife 3 months, 
 and the wife alone, 4 months ; how long would it last the 
 man alone ? 
 
 18. A, B, and C can mow a field in 4 days, A and B 
 IB 6 days, and B and C in 9 days ; how long will it take 
 each to mow it ? 
 
 19. A, B, and C can dig a ditch in 6 days, A and B 
 
MENTAL ARITHMETIC. 123 
 
 in 8 days, and B in 12 days ; how long will it take each 
 to do it ? 
 
 20. If 3 men, or 4 boys, can do a piece of work in 12 
 days, in what time can 3 men and 4 boys do it ? 
 
 21. If A can do a piece of work in | of a day, and B 
 in | of a day, how long will it take both to do it ? 
 
 22. C can cut a cord of wood in | of a day, and D tc 
 | of a day ; in what time can they together cut a cord ? 
 
 23. D can make a fence in 9 days, and I) and E in 6 
 days ; how long will it take E to make what remains after 
 D*has built of it? 
 
 24. Two men, or 3 boys, can plough an acre in | of a 
 day ; how long will it require 3 men and 2 boys to do it? 
 
 25. A can plough a field in J of a day, B in | of a day, 
 and C in | of a day ; how long will it take them togethei 
 to plough the field ? 
 
 26. A, B, and C can mow a field in 6 days, and A and 
 B in 9 days after the three had worked 2 days, C left ; 
 how long did it require A and B to finish it ? 
 
 27. Marie can make a dress in 6 days, Sallie in ^ of 
 the time, and Ewretta in | of the time ; in what time can 
 Marie and Sallie finish it, after the three have worked | 
 of a day ? 
 
 28. A can build a boat in | of a month, and B in | of 
 a month ; after A had wrought | of a month B joined 
 him ; how long was the boat building ? 
 
 29. Amos can plough 25 per cent, of a field in a month, 
 and Anson, 45 per cent. ; after they both had worked 2 
 weeks, how long would it require Amos to finish it? 
 
 30. A, B, and C can build a vessel in \ of a year, A 
 and C in ^ of a year, and C in | of a year; after they had 
 all laboured 1 month, A left ; in what time could B and 
 C finish it ? 
 
1 24 MENTAL ARITHMETIC . 
 
 LESSON IV. 
 
 1, A and B together have 34 apples, and f of 
 number equals | of B's number; how many has eacl 
 
 to I of B's, equals \i of B's, which equals 34 apples, &c. 
 
 2. Thomas and Walton together have $55, and f of 
 Thomas's money equals | of Walton's ; how much haa 
 each? 
 
 3. The sum of two numbers is 28, and | of the smaller 
 equals \ of the greater ; what are the numbers ? 
 
 4. Divide 46 oranges between Chester and Henry, so 
 that | of Chester's may equal f of Henry's number. 
 
 5. 4 of the number of apple-trees in an orchard equals 
 ^ of tne number of peach-trees, and in all there are 60 
 trees ; required the number of each. 
 
 6. A pole, whose length was 63 feet, was broken into 
 two parts, such that f of -the first part equals | of the 
 second ; required the length of each piece. 
 
 7. The sum of two numbers is 69 ; what is each of 
 the numbers, provided they are to each other as f to | ? 
 
 8. Walter bought a hat and coat for $26, and 2^ times 
 the cost of the hat equals f of the cost of the coat; required 
 the cost of each. 
 
 9. Says B to C, f of my age, + 6 years equals | of 
 yours, and the sum of our ages is 42 years ; required the 
 ago of each. 
 
 10. The difference between two numbers is 6, and | 
 of the first equals | of the second ; what are the num- 
 bers? 
 
 11. Fanny has 14 plums more than Sa Jie, and | of 
 Fanny's equal? | of Sallie's number; how many has 
 each? 
 
MENTAL ARITHMETIC. 125 
 
 12. of the difference between two numbers is 6, and 
 
 ;of the first number equals \ of the second; required 
 e numbers. 
 
 13. | of A's money equals $32, and | of B's money 
 equals | of A's; how much has each ? 
 
 14. Five sixths of the difference between A's and B's 
 fortune is $500, and f of A's equals | of B's fortune ; 
 what is the fortune of each ? 
 
 15. A and B can build J of a boat in a day, and twice 
 what A builds equals what B builds; how much can each 
 build in one day ? 
 
 16. Two boys can do a piece of work in 6 days, and 
 twice what A does equals what B does ; how long will it 
 take each to do it ? 
 
 17. Two pipes fill a cistern in 15 hours, and | of what 
 one pours in equals of what the other pours in ; how 
 long will it take each to fill it ? 
 
 18. John and Henry can mow 60 acres of grass in 6 
 weeks, and 4 of what John can mow in a day equals what 
 Henry can mow in a day; how long will v it take each to 
 mow it ? 
 
 19. Three men can build a boat in 6 days, and the 
 parts they each build are to each other as 1, 2, and 3 ; in 
 what time co ild each build it alone? 
 
 20. A horse and cow eat a quantity of hay in 3 weeks; 
 now long will it last each, provided the horse eat only | 
 as much as the cow ? 
 
 21. Three men, A, B, and C, can build a boat in 12 
 days, and their rates of working are as 4, -|, and |; how 
 long would it take each alone to build it ? 
 
 22. B can drink a keg of mead in 4 days; f of what 
 B drinks equals i of what A drinks, and \ of what 
 drinks. After the three had been drinking -f. of a day, 
 A and C drink the remainder; how long did it take 
 iheni ? 
 
MENTAL AR1TMMETJO. 
 
 LESSOR V. 
 
 1. A mat receives $3 a day for his labour, and forfeits 
 $1 each day he is idle, and at the expiration of 30 days 
 receives $50 ; how many days was he idle ? 
 
 SOLUTION. Had he laboured 30 days, he would have received 
 30 times $3, or $90; he therefore lost $90 $50, which is $40, 
 by his idleness. Every day he was idle he lost $3, his wages, 
 plus $1, his forfeit, which are $4. If in one day he lose $4, to 
 lose $1 it will require ^ of a day, and to lose $40, 40 times J, 
 which are 10 days, the number of days he was idle. 
 
 2. Warner receives $2.50 a day for his labour, and 
 pays 50 cents for his board, and at the expiration of 40 
 days he saves $50 ; how many days was he idle ? 
 
 3. A labourer agreed to work, on the condition, that 
 for every day he worked he should receive $2, and for 
 every day he was idle he should forfeit 50 cents ; how 
 many days did he labour, if at the end of 25 days he re- 
 ceived $30? 
 
 4. A boy agreed to carry 32 glasses to a certain store 
 for 5 cents apiece, on condition that for each one he broke 
 he should forfeit 10 cents; he received $1. How many 
 did he break ? 
 
 5. James engaged to labour on condition that for every 
 day he worked he should receive $1^, and for every day 
 he played he should pay $^ for his board ; at the expira- 
 tion of 30 days he received $35. How many days did be 
 work? 
 
 6. Eobeit agreed to carry 50 oranges to market for 
 a cent each, on condition that he should forfeit 2^ centa 
 for each one he eat ; he received 16 cents. How many 
 did he eat ? 
 
 7. The head of a fish is 10 inches long, the tail is as 
 long as the head, plus ' of the body, and the body is as 
 
MENTAL ARITHMETIC. 127 
 
 long as the head and tail both ; required the length of 
 the fish, 
 
 SOLUTION. Since the tail is as long as the head, + ^ f ^ e 
 body, i of the length of the body, -j- 10 inches, equals the length 
 of the tail, "which, added to the length of the head, equals of 
 the length of the body, -|- 20 inches, which, by the condition of 
 the problem, equals | of the length of the body. If | of the 
 length of the body equals 1 of the length of the body, -f 20, 
 2 ^, or of the length of the body, must equal 20 inches, &c. 
 
 8. The head of a fish is 8 inches long, the tail is as 
 long as the head, plus | of the body, and the body is as 
 long as the head and tail both ; required the length of the 
 fish. 
 
 9. The head of a perch is 4 inches long, the tail is as 
 long as the head, plus ^ of the body, and the body is as 
 long as the head and tail both; what is the length of the 
 perch ? 
 
 10. A has 6 cents, B has as many as A, plus | as many 
 as C, and C has as many as A and B both ; required the 
 number of cents possessed by each and all. 
 
 11. The tail of a pike weighs 3 ounces, the head 
 weighs as much as the tail, plus \ of the weight of the 
 body, and the body, twice as much as the head and tail; 
 required the weight of the fish. 
 
 12. A tree by falling was broken into 3 pieces; trie 
 bottom piece was 5 feet long, the top as long as the 
 bottom, plus 4- of the middle, and the middle 3 times as 
 long as the other two pieces ; what was the length of each 
 piece, and the tree ? 
 
 13. A agreed to labour a certain time for $60, on con- 
 dition, however, that for each day he was idle he should 
 forfeit $2 ; at the expiration of the time he received $30 : 
 how many days did he labour, supposing he receives $2 
 a day for his labour ? 
 
 14. The head of a trout weighs 2 pounds, the tai! 
 weighs 2 poun is more than the head, pJ as \ of the body, 
 
128 MENTAL ARITHMETIC. 
 
 and the body weighs as much as the head and tail to- 
 gether ; required the weight of the fish. 
 
 15. A man left $26,000 to his wife, son, and daughter, 
 on condition that if the daughter died before becoming 
 of age the widow should have | of the fortune, but if the 
 son died the widow should have | of it; required the 
 share of each if they all live. 
 
 REMARK. It will be readily seen that the widow has three 
 times as much as the daughter, and the son three times as much 
 as the widow, which is 9 times as much as the daughter. 
 
 16. A man wishing to erect some buildings, concluded 
 that if he built a store and house, the store should cost J 
 of his money, but if he built a house and barn, the barn 
 should cost J of his money ; what was the cost of each, 
 supposing he built all three, and his fortune was $77,000? 
 
 17. A gentleman receives $4 per day for his labour, and 
 pays $8 per week for his board ; at the expiration of 1 
 weeks he has saved $144 ; required the number of idle 
 and working days. 
 
 18. A man having a daughter in France, and a son in 
 Spain, willed, if the daughter returned, and not the son, 
 the wife should have | of the fortune, but if the son re- 
 turned, and not the daughter, he should have f of the 
 fortune ; they both returned, and it was found that the 
 son received $3000 more than the daughter ; required the 
 fortune and share of each. 
 
 19. A, B, and C, contemplating the purchase of a farm, 
 agreed if A and B bought it, A should pay | of the price, 
 but if B and C bought it, B should pay | of the price, 
 at length the three agreed to buy it together, when it 
 was found tha f! p^id $500 more than A ; what did thr 
 farm cot> and what did each pay ? 
 
MENTAL ARITHMETIC. 
 
 LESSON VI. 
 
 1. Henry is 35 years old, and Mary it 5, in howruanj 
 years will Henry be only 6 times as old LS Mary? 
 
 SOLUTION. At the required time, 6 times Mary's age will 
 equal Henry's age ; then 6 times Mary's age, which is Henry's 
 age, minus Mary's age, equals 5 times Mary's age, which equals 
 the difference between their ages, which is 35 5, or 30 years, 
 If 5 times Mary's equals 30 years, once her age equals 4. of 30 
 years, which is 6 years. Hence when Mary is 6 years old Henry 
 will be 6 times as old as she, but Mary is now 5, therefore in 
 6 5, or 1 year, Henry will be 6 times as old as Mary. 
 
 2. James is x 28 years old, and Ellen is 8 ; in how many 
 years will James be 3 times as old as Ellen ? 
 
 3. A is 30 years, and B is 6 years old; in how many 
 years will A be only 4 times as old as B ? 
 
 4. Eva is 6 years old, and her mother, 7 times as old; 
 in how many years will the mother be 5 times as old? 
 
 5. Morton is 10 years old, and Moses 30 ; how long 
 since Moses was 5 times as old as Morton ? 
 
 6. Jacob is twice as old as his son, who is 20 years cf 
 age ; how long since Jacob was 5 times as old as his sou t 
 
 7. Mary is | as old as her aunt, who is 40 years of 
 age ; how long since Mary was only ^ as old as her aunt I 
 
 8. Samuel is J5 years of age, which is | of Sarah's 
 age ; how long since Sarah was ^ as old as Samuel ? 
 
 9. Jason is 5 times as old as John, and the difference 
 of their ages is 20 years ; in how many years will Jasoo 
 be 3 times as old as John ? 
 
 10. Henry is 4 times as old as "William, and the sum 
 of their ages is 25 years; in how many years will Henrj 
 be but 3 times as old as William ? 
 
 11. | of A's age equals | of B's, and the difference 
 between their ages is 10 years; how long since A was 8 
 times as old as B? 
 
 12 
 
130 MENTAL ARITHMETIC. 
 
 12. A lady bought some ribbon at 5 cents a yard, and 
 as much more at the rate of 7 cents a yard, and sold it 
 all at 8 cents a yard ; how much was the gain on 20 
 yards ? 
 
 . 13. Albert bought some oranges at 2 cents each, and 
 as many more at 4 cents each, and sold them at the rate 
 of 2 for 8 cents, and gained 12 cents; required the num- 
 ber bought. 
 
 14. Witmer bought some tea at 4 shillings a pound, 
 and as much more at 8 shillings, and sold it all for 7 
 shillings a pound, and gained 40 shillings ; required the 
 number of pounds of each kind. 
 
 15. Bowman mixed some sugar worth 5 cents a pound, 
 with an equal quantity worth 9 cents a pound, and sold 
 the mixture for 10 cents a pound, and gained $6; required 
 the quantity of each kind. 
 
 16. Sophia bought a number of yards of silk at the 
 rate of 3 yards for $1, and as much more at the rate 
 of 4 yards for $1, and sold it all at the rate of 8 yarda 
 for $3, and thereby gained $5; how many yards did 
 she buy ? 
 
 17. A boy bought some apples at the rate of 4 for 
 1 cent, and as many more at the rate of 5 for 1 cent, 
 and sold them all at the rate of 10 for 2 cents, and 
 thereby lost 5 cents ; how many of each kind did he 
 buy? 
 
 18. \ of M's age equals | of N's, and 8 years is f of 
 fche difference of their ages; in how many years will | of 
 M's age equal \ of N's age ? 
 
 19. Two tenths of B's age equals of C's, and the 
 sum of their ages is 30 years; in how many years will B 
 be 3 times as old as C ? 
 
 20. | of A's age added to f of B's, which equals ^ of 
 A's, is 100 years; how long since A was 5 times as old 
 asB? 
 
 21 . | of Daniel's age equals J of David's, and the sum 
 
MENTAL ARITHMETIC. 131 
 
 of their ages is 68 years ; how long since of Daniel's 
 age equalled | of David's age ? 
 
 22. A boy bought some apples at 2 cents apiece, and 
 twice as many oranges at 4 cents apiece, and sold them 
 all at 3 cents each, and thereby lost $1 ; how many of 
 each kind did he buy ? 
 
 LESSON VII. 
 
 1. A's age equals 3 times B's, but in 10 years A's age 
 will be only twice B's; how old is each? 
 
 SOLUTION. By the first condition of the problem, 3 times B's 
 age equals A's age, and, in 10 years, 3 times B's age, now, -j- 10 
 years, will equal A's, and once B's, now, -j- 10 years, will equal 
 B's ; but A's age at that time is twice B's ; hence 2 times (B's, 
 now, -f 10) which is twice B's -f- 20= 3 times B's -f 10, there- 
 fore 3 B's, now, 2 B's, now, which is B's, at present time, 
 equals 20 10, or 10 years, and A's is 3 times 10, or 30 years. 
 Therefore, &c. 
 
 2. John is 5 times as old as Oliver, but in 8 years he 
 will be only 3 times as old ; what is the age of each ? 
 
 3. Mary is ^ as old as her aunt, but in 20 years she 
 will be ^ as old; what is the age of each ? 
 
 4. Henry is 1 as old as his father, but in 25 years he 
 will be | as old ; required the age of each. 
 
 5. Ten years ago, when I first met Mr. Morgan, I was 
 | as old as he, but now I am ^ as old as he is ; required 
 each of our ages. 
 
 6. Sixteen years ago, when Agnew married, he was 8 
 times as old as his wife, but now he is only twice as old ; 
 what is the age of each ? 
 
 7. A hare is 30 rods before a hound, and runs 3 rods 
 while the hound runs 6; how many rods must the hound 
 run to catch the hare ? 
 
 8. Stephen is 40 steps before James, and takes 5 
 steps to Ja-mes's 7 ; how many steps must James take to 
 catch Stephen, supposing their steps to be equal ? 
 
132 MENTAL ARITHMETIC. 
 
 9. A "hare takes 2 leaps while a hound takes 1, but 1 
 of the hound's leaps equals 4 of the hare's ; how much 
 does the hound gain on the hare in taking one leap ? 
 
 10. A hare is 30 leaps before a hound, and takes 4 
 leaps while the hound takes 2, but 2 of the hound's 
 equal 8 of the hare's ; how many leaps must the hound 
 take to catch the hare ? 
 
 11. A fox is 40 leaps before a hound, and takes 3 
 leaps while the hound takes 2. but 2 of the hound's 
 equal 4 of the fox's ; in how many leaps will the hound 
 catch the fox ? 
 
 12. A thief is 20 steps before an officer, and takes 6 
 steps while the officer takes 5, but 5 of the officer's equal 
 8 of the thief's ; how far will the thief run before he is 
 overtaken ? 
 
 13. A rabbit is 60 leaps before a hound, and takes 9 
 leaps while the hound takes 3, but 2 of the hound's 
 equal 7 of the rabbit's; how many leaps will the rabbit 
 take before being caught? 
 
 14. Twenty-five years ago, Willard was \ as old as his 
 uncle, but 5 years ago he was \ as old ; how old is each 
 at present ? 
 
 15. Four years ago B's house was four times as old as 
 his barn, but 2 years hence it will be only twice as old; 
 how long has each been built ? 
 
 16. Three years ago Emma's doll was only I- of the 
 age of herself, but 7 years hence it will be j of her age ; 
 required the age of each. 
 
 17 A is 10 steps before B, and takes 2 steps while B 
 takes 4, and 4 of A's steps equal (> of B's; how many 
 steps will each take before they are i ogether ? 
 
 18. B takes 30 steps to overtake C; how far was C 
 ahead of B when they started, provided B takes 2 steps 
 while C takes 3, and 2 of B's equal 5 of C's steps? 
 
 19. E takes 60 steps before he is overtaken by D; how 
 many steps does I) take to catch E, provided E takes 4 
 
MENTAL ARITHMETIC. 133 
 
 steps while D takes 3, and 5 of D's equal 8 of E's, and 
 how far ahead was E when they started ? 
 
 21. M and N are 60 rods apart and approach each 
 other; how far will each travel before they meet, provid- 
 ed M takes 3 steps while N takes 6, and 2 of M's equal 
 of N's steps ? 
 
 22. A and B are 150 of B's steps apart, and approach 
 each other ; how many steps will each take before they 
 are together, if 4 of A's steps equal 8 of B's, and B takep 
 9 steps while A takes 3 ? 
 
 LESSON VIII. 
 
 1. What is the time of day, provided ^ of the time 
 past midnight equals the time to noon ? 
 
 SOLUTION. By the condition of the problem, Jj. of the time 
 past midnight equals the time to noon, which, added to 2 of the 
 time past midnight, equals J| of the time past midnight, which 
 equals the time from midnight to noon, or 12 hours. If 3 of the 
 time past midnight equals 12 hours, 1 1 of 12 hours, which 
 is 4, and ^ = 2 times 4, or 8 hours, the time past midtight. 
 Hence, it is 8 o'clock in the morning. 
 
 2. What is the time of day, supposing | of the time 
 past midnight equals the time to noon ? 
 
 3. What is the time of day, if -| of the time past mid 
 night equals the time past noon ? 
 
 4. What is the time of day, if -| of the time past mid- 
 night equals the time past noon ? 
 
 5 What is the hour of day, supposing i, of the thne 
 past midnight equals the time to midnight again ? 
 
 6. What is the hour of day, provided | of the dme 
 past midnight equals the time to midnight again? 
 12* 
 
134 MENTAL ARITHMETIC. 
 
 7. Whau is the hour of day, if | of the time to noon 
 equals the time past midnight ? 
 
 8. What is the time of day, if U of the time to HOOD 
 equals the time to midnight ? 
 
 9. A person being asked the time of day, said. | of 
 the time past noon equals the time to midnight; vviuit 
 was the hour? 
 
 10. William recited his lesson when f of the time part 
 mon equalled the time past midnight; at what hour did 
 he recite? 
 
 11. A person being asked the time of day, said, | of 
 the time to midnight equals the time past midnight; what 
 was the time ? 
 
 12. A lady being asked the time of day, replied, that 
 of the time to midnight equalled ^ of the time past 
 noon ; what was the time ? 
 
 13. Required the hour of day, provided ^ of the time 
 to midnight equals f of the time to noon. 
 
 14. What time, after 12 o'clock, are the hour and 
 minute hands of a watch exactly together ? 
 
 REMARK. It will be seen that the minute hand gains on the 
 hour hand 11 spaces in going 12; hence, to gain 1 space, tho 
 distance they are apart at 1 o'clock, it must go _L of 12 
 spaces, which is 1? spaces, equal to 5JL minutes. 
 
 15. What time, after 2 o'clock, are the hour and 
 minute hands of a clock together ? 
 
 16. A man being asked the hour of the day, replied, 
 that it was between 4 and 5 o'clock, and that the hour 
 and minute hands >vere together; what was the time? 
 
 17. ^ of the time past 9 o'clock, A. M., equals | of the 
 time to midnight ; what is the time ? 
 
 18. A man being asked the hour of day, replied, that 
 \ of the time past 3 o'clock, equalled of the time to 
 midnight ; what was the hour ? 
 
 19. A has $20 in gold and silver, and for every $6 of 
 
 

 MENTAL ARITHMETIC. 185 
 
 gold he lias $4 of silver; how much gold must be added 
 that there may be $9 of gold for $3 of silver ? 
 
 REMARK. Since the gold is to the silver as 6 to 4, and there 
 are $20 in all, we find there are $12 of gold and $8 of silver. 
 After the addition, since 3 times the silver equals the gold, 3 
 times $8, or $24, is the gold, and $24 $12, or $12, equals 
 the amount added 
 
 20. A man has 40 pieces of money, consisting of copper 
 and silver, and for every 7 of copper there are 3 of silver ; 
 how many pieces of silver must be added, that for everj 
 1 of copper there may be two of silver ? 
 
 21. A drover has 100 animals, consisting of sheep and 
 cows, and he has 2 sheep for every 3 cows ; how many 
 sheep must he sell that he may have 2 sheep to 6 cows ? 
 
 22. There are 50 pupils in a certain school, consisting 
 of girls and boys, and there are 8 boys to 2 girls ; how 
 many boys must leave the school that there may be 6 
 boys to 2 girls ? 
 
 23. A man expends 60 cents for an equal number of 
 apples and pears, giving 3 cents each for his apples, and 
 2 cents each for his pears ; how many pear? must he sell, 
 that the remainder may be to his apples as 2 to 3 i 
 
 24. A man, being asked the hour of day, replied, that 
 | of\the time past 2 o'clock equalled | of the time to mid- 
 night ; what was the hour ? 
 
 25. What time, between 7 and 8 o'clock, are the hour 
 and minute hands of a watch exactly together ? 
 
 26. In how many minutes, after 4 o'clock, will the 
 hour and minute hands be 5 minutes apart? 
 
 27. A lady, being asked the hour of day, replied, that 
 | of the time past noon equalled | of the time to midnight, 
 minus | of an hour ; what was the time ? 
 
 ' 28. A tree 90 feet in length, by falling, was broken 
 into two parts, such that | of the shorter equalled -4 of 
 the longer; how much must be cut from the longer, so 
 ithat ^ of it may equal | of the other part ? 
 
136 MENTAL ARITHMETIC . 
 
 29. A boy bought 24 oranges and lemons, and | of the 
 number of oranges equals | of the number of lemons, 
 how many more oranges must be purchased, that oi the 
 number of oranges may equal | of the number of lemons? 
 
 LESSON IX 
 
 1. How far may a person ride in a coach, which goes 
 at the rate of 10 miles an hour, so that he may be gone 
 but 8 hours, provided he walks home at the rate of 6 
 miles an hour? 
 
 SOLUTION. If he walks 6 miles in 1 hour, to walk 1 mile it 
 will require ^ of an hour, and to walk 10 miles, the distance he 
 rides in an hour, it will require 10 times 4= 1 0; or 5, O f an 
 hour, hence 5 of the time he rides equals the time he walks, 
 which added to J, the time he rides, equals J of the time he 
 rides, which is 8 hours, &c. 
 
 2. How far may a person ride in a carriage, going at 
 the rate of 9 miles an hour, provided he is gone only 10 
 hours, and walks back at the rate of 6 miles an hour ? 
 
 3. How many miles may I sail in a steamboat, going 
 at the rate of 15 miles an hour, provided I am gone only 
 9 hours, and return at the rate of 12 miles an hour? 
 
 4. A steamboat whose rate of sailing in still water is 
 12 miles an hour, descends a river whose current is 4 
 miles an hour, and is gone 6 hours ; how far did it go ? 
 
 5. A boat whose rate of sailing is 10 miles an hour, 
 moves down a river whose current is 2 miles an hour j 
 how far may it go that it may be gone but 5 hours ? 
 
 6. Eight men hire a coach to ride to Lancaster, but by 
 taidng in 4 more persons, the expense of each is dimi- 
 nished by $| ; what do they pay for the coach ? 
 
 1 . Ten men hire a coach for a certain sum of money, 
 

 MENTAL ARITHMETIC 137 
 
 but taking in 5 more persons, the expense of each is 
 diminished Si ; what did the coach cost them '( 
 
 8. Twenty persons engage a pleasure boat for sailing, 
 but before they start, 12 of the company decline going, 
 by which the expense of each is increased 13 ; what did 
 they pay for the boat ? 
 
 9. A boy being asked his age, replied, 8 times the 
 square of my age equals 75 years ; how old was he ? 
 
 10. Albert being asked how many marbles he had, 
 answered, 4 of the square of the number equals 18 ; how 
 many marbles had he ? 
 
 11. Three fourths of the square of the number of 
 letters in a sentence equals 27; how many letters are 
 there in the sentence ? 
 
 12. The square of twice a number equals 256 ; what is 
 that number, and what is the square of ^ of the number? 
 
 13. If | of the number of trees in an orchard be 
 squared, the result will be 100 ; how many trees are there 
 in the orchard ? 
 
 14. The square of twice a number is 18 more than 
 twice the square of the number; what is the number? 
 
 15. Twice the square of a number is 8 more than 6 
 times the square of half the number; what is the number? 
 
 16. | of the square of a number is 36 more than | of 
 the square of half the number ; required the number. 
 
 17. 15 is 3 more than | of the cube of some number ; 
 what is that number ? 
 
 18. | of the cube of a number is 10 more than the 
 cube of | of the number ; what is the number ? 
 
 19. | of the square of twice a number is equal to J ol 
 the square of | of the number, diminished by 3 ; what is 
 the number ? 
 
 20. A lady being asked how many music pupils she 
 had, replied, f of the number multiplied by | of the 
 number is 9 more than the square of | the number; 
 how many had she ? 
 
138 MENTAL ARITHMETIC. 
 
 21. A company of 15 persons engage a dinner at a 
 hotel, but before paying the bill 5 of the company with 
 draw, by which each person's bill was augmented $ 
 what was the bill ? 
 
 LESSON X. 
 
 1. In what proportion must I mix tea worth 6 shilling^ 
 a pound, with that worth 11 shillings a pound, so that the 
 mixture may be worth 9 shillings a pound ? 
 
 SOLUTION. If I take 1 Ib. at 6.5., ana sell it for 95. , I will gain 
 9 6, or 3s., and to gain Is. I must take J of a pound; if I 
 take 1 Ib. at lls., and sell it for 9s., I lose 11 9, or 2s., and 
 to lose Is. what I gained on the first, I must take J of a pound; 
 hence I must take J of a pound at 6s. as often as J a pound at 
 lls., or, in whole numbers, 6 times J, which are 2 Ibs. at 6s., 
 as of* en as 6 times , which are 3 Ibs. at 11s. 
 
 2. In what proportion must I mix rice worth 7 cents 
 a pound, with that worth 12 cents, so that the mixture 
 may be worth 9 cents a pound ? 
 
 3. In what manner shall I mix sugar worth 5 cents a 
 pound, with that worth 12 cents a pound, so that the 
 mixture may be worth 8 cents a pound ? 
 
 . 4. Having two kinds of wine worth $2 and 89, respec- 
 tively, a gallon, how shall I mix them that the mixture 
 may be worth $4 a gallon ? 
 
 5. How many pounds of sugar worth 6 cents a pound 
 must I mix with 4 pounds at 11 cents a pound, so that 
 the mixture may be sold for 8 cents a pound ? 
 
 6. How many pounds of coffee worth 8 cents a pound 
 can I mix with 12 pounds worth 13 cents a pound, sc 
 that a pound of the mixture may be worth 11 cents ? 
 
 7. How many gallons of wine worth $2 a gallon must 
 be mixed with 9 gallons at $9 a gallon, so that the mixture 
 may be sold for $5 a gallon? 
 

 MENTAL ARITHMETIC. 139 
 
 8. How many pounds of tea worth 3 shillings a pound 
 must be mixed with 10 pounds worth 9 shillings a pound, 
 so that the mixture may be worth 5 shillings a pound ? 
 
 9. A grocer mixed 4 pounds of sugar worth 5 cents a 
 pound, with 6 pounds worth 10 cents a pound; what is 
 the yaJue of one pound of the mixture ? 
 
 10. In what proportion must I mix sugars worth 6, 7, 
 and 11 cents a pound, so that the mixture may be worth 
 9 cents a pound ? 
 
 11. A merchant mixed 5 gallons of wine worth 60 
 cents per gallon, with 5 gallons worth 80 cents a gallon ; 
 required the value of a gallon of the mixture. 
 
 12. In what proportion must I mix rice at 5, 6, and 
 12 cents a pound, so that one pound of the mixture may 
 be worth 8 cents ? 
 
 13. How shall teas worth 3, 4, 8, and 9 shillings per 
 pound be mixed, that the mixture may be sold for 6 
 shillings a pound ? 
 
 14. If a man travels 2 miles one day, 4 the next, 6 the 
 next, and so on, how many miles will he travel the sixth 
 day? 
 
 SOLUTION. Since the distance increases 2 miles each day 
 after the first, on the sixth day, or 5 days after the first, it will 
 increase 5 times 2, which are 10 miles, which, added to 2 miles, 
 .the distance he travels the first day, equals 12 miles, the "distance 
 he travels the sixth day. 
 
 15. If a man travels 2 miles the first day, and each 
 day travels 3 more than on the preceding day, how far 
 will he travel the tenth day ? 
 
 16. If a lady pays 4 cents for one yard of muslin, and 
 4 cents more for each yard than the preceding one, how 
 much will the eighth yard cost ? 
 
 17. If a person travels 3 miles the first day, and 13 the 
 sixth day, how much more did he travel each day than 
 on the preceding one ? 
 
 18. How many days did a person travel, supposing he 
 went 4 miles the first day, 14 the last day, and went 2 
 miles more each day than on the preceding one ? 
 
110 MENTAL ARITHMETIC. 
 
 19. If a person paid 5 cents for one yard of cloth, and 
 2 cents more for each succeeding one, how many yards 
 must be bought that the last yard may cost 25 cents ? 
 
 20. If a lady pays 4 cents for the first yard of muslin, 
 and 31 cents for the tenth yard, how much more did she 
 pay for each yard than for the preceding one ? 
 
 21. If Elmira bought 12 yards of muslin, paying 8 
 cents more for each yard than for the preceding, and 40 
 cents for the last yard ; how much did she pay for the 
 first yard ? 
 
 22. A merchant having sugar worth 4, 5, 10, and 11 
 cents a pound, wishes to make a mixture of 60 Ibs. worth 
 7 cents a pound ; how many pounds of each kind must he 
 take? 
 
 LESSON XI. 
 
 1. A lady bought two watches and a chain. The chain 
 and gold watch cost 4 times as much as the silver watch, 
 and the chain and silver watch cost twice as much as the 
 gold watch. What is the value of each, if the silver 
 watch is worth $30 ? 
 
 SOLUTION. By the first condition of the problem 4 times $80. 
 or $120, equals the cost of the gold watch and chain, which 
 added to $30, the cost of the silver watch, is $150, the cost of 
 the two watches and chain. By the second condition, twice the 
 cost of the gold watch equals the cost of the silver watch and 
 chain, which, added to the cost of the gold watch, equals 3 times 
 the cost of the gold watch, which equals the cost of all, or $150. 
 If 3 times the cost of the gold watch equals $150, &c 
 
 2. A farmer bought a horse, colt, and saddle. If the 
 liorse be saddled it will be worth 5 times as much as the 
 colt, but if the colt be saddled it will be worth ^ as much 
 as the horse. What is the value of the horse and the 
 saddle, supposing the colt to be worth $50 ? 
 
 3. Tte head of a fish is 10 inches long; 7 times the 
 
\ MENTAL ARITHMETIC. 141 
 
 length of the head equals the length of the body and tail, 
 and 3 times the length of the tail equals the length of 
 the head and body. Required the length of the tail 
 and body respectively. 
 
 4. A person has two silver cups and only one cover for 
 both. The first cup Weighs 12 ouilces. If the first cup 
 he covered it will weigh twice as much as the second, but 
 if the second cup be covered it will weigh 3 times as 
 much as the first. Required the weight of the second 
 cup and cover. 
 
 5. A farmer bought a cow for $30, which was | of 
 what he paid for a horse and sheep, and | of what he 
 paid for the horse and cow equals what the sheep cost ; 
 required the cost of the horse and sheep. 
 
 6. A has $20, which is ^ of what B and C have, and 
 C has ^ as much as A and B together; how much have 
 B and C respectively ? 
 
 7. A went to a store and borrowed as much money as 
 he had, and spent 4 cents ; he then went to another store 
 arid did the same, and then had 4 cents remaining; how 
 much money had he at first ? 
 
 8. A boy went to a store, borrowed as much money as 
 he had, and spent 8 cents; he then went to a second 
 store, borrowed as much as he had, and spent 12 cents, 
 and had no money remaining; how much money had he 
 at first ? 
 
 9. Rolland went to a store, borrowed as much money 
 as he had, and spent 8 cents ; he then went to a second 
 and third store and did the same, and had no money re- 
 maining; how much had he at first? 
 
 10. A gentleman has two silver cups and only one 
 cover for both. The first cup weighs 10 ounces, and if 
 the first cup be covered it will weigh 3 times as much as 
 the second, but if the second cup be covered it will weigh 
 5 times as much as the first ; required the weight of the 
 second cup and cover. 
 
 11. A expended 10 cents for apples, whicl was J of 
 he expended for peaches and oranges, and twice 
 
M2 MENIAL ARITHMETIC. 
 
 what he spent for oranges equals what he spent for applea 
 and peaches ; how many did he buy of each if the apples 
 cost 2, the peaches 3, and the oranges 4 cents each ? 
 
 12. Reynolds went to a hotel, borrowed as much money 
 as he had, and spent 2 cents; he then went to a second 
 and third hotel, did the same, and had 6 cents remaining ; 
 how much money had he at first ? 
 
 IB. William went to a store, borrowed 10 cents, and 
 then spent 8 cents, doing the same at a second and third 
 store, he found he had doubled his money; how much 
 money had he ? 
 
 14. A has $20, which is 1 of what B and have, and 
 twice B's money is equal to ^ of the sum of A's and C's 
 money ; how much money have B and C respectively ? 
 
 15. James went to a store, borrowed 10 cents, and then 
 spent 12 cents ; he did this at a second and third store, 
 and then had no money left ; how much money had he 
 at first ? 
 
 16. A man expended $5 for ducks, which was | of 
 what he paid for geese and turkeys, and twice what he 
 paid for geese equals what he paid for ducks and turkeys ; 
 how many of each kind did he buy, provided the ducks 
 cost $|, the geese $1, and the turkeys $3 each ? 
 
 LESSOR XII 
 
 1. Two men, A and B, in partnership gain $300. A 
 owns | of the stock, lacking $40, and gains $180; re- 
 quired the whole stock, and share of each. 
 
 SOLUTION. If A had owned 5 of the stock, his gain wotLtf 
 have been | of $300, or $200 ; but he gained only $180, there- 
 fore the $40 must gain the difference between $200 and $180, 
 which is $20. If $40 gain $20, to gain $1 it will require 
 ^ of $40, which is $2, and to gain $300 it will require 300 
 times $2 which are $600 
 
MENTAL ARITHMETIC. 143 
 
 2. A and B anter into partnership and gam $240. A 
 owns | of the stock, lacking $10, and his share of the 
 gain is $175 required the whole stock, and share of 
 each. 
 
 3. Two men in partnership gain $200. The first owns 
 | of the stock -f- $40, and his gain is $60; what is the 
 entire stock and share of each ? 
 
 4. A and B bought a lottery ticket with which they 
 drew a prize of $600. A paid | of the price of the ticket, 
 lacking $12, and his share of the prize was $340 ; what 
 did each pay for the ticket ? 
 
 5. A man and his son agree to mow a certain field for 
 $72. The son mowed of the whole -j- 10 acres, and 
 received $38; how many acres did each mow? 
 
 6. Two men pay $120 for the pasturage of some cattle. 
 The first turns in | of the whole number, -(- 20, and 
 pays $40 ; how many cattle does each turn in ? 
 
 7. A and B agree to mow a field of grass for $60; A 
 mows twice as much as B, lacking 8 acres, and receives 
 $24 ; how many acres does each mow ? 
 
 8. Two men engage to build a boat for $84. The first 
 labours -l as many days as the second, -f- 6 days, and re- 
 ceives $48 ; how many days does each labour ? 
 
 9. Two men receive the same sum for labour; but had 
 one received $10 more, and the other $6 less, one would 
 have received 5 times as much as the other; how much 
 does each receive? 
 
 10. A and B invested the same sum in speculation. A 
 lost $200, and B gained $1000, when it was found that 
 B had 4 times as much money as A ; what was the sum 
 invested ? 
 
 11. Said James to Isaac, our purses contain the same 
 sum of money, but if you give me $20, and I give you 
 $10, I shall have 3 times as much as you; how much 
 money had each ? 
 
 12. A person being asked the hour of the day, replied 
 that 2 hours ago the time past noon was | of the time to 
 midnight 2 hours hence ; required the time. 
 
144 MENTAL ARITHMETIC. 
 
 18. Said E to F toy age is 5 years more than yours, 
 but 4 years ago my age was ^ of what yours will be 4 
 years hence ; what was the age of each ? 
 
 14. A pol-3 whose length was 44 feet, was broken into 
 two unequal parts ; if the shorter be increased by 8 feet, 
 and the longer be diminished by 5 feet, the first will be ^ 
 of the length of the second ; required the length of each 
 part. 
 
 15. A staff, whose length is 33 feet, is in the air and 
 water ; and the length in the air, 2 feet, equals 4 times 
 the length in the water, -f- 6 feet ; required the length 
 in the air. 
 
 16. A person being asked the time of day, replied, 
 that of the time past midnight, 2 hours ago, equalled 
 | of the time to midnight, 3 hours and 20 minutes hence; 
 required the time. 
 
 17. A man having an equal number of cows in two 
 fields, sold | of the number from each, then 7 having 
 jumped from the first into the second, there were 3 
 times as many in the second as in the first ; required the 
 number in each field. 
 
 18. A had twice as many oranges as B, and then each 
 losing ^ and A giving B 8, they each have the same 
 number ; how many had each at first ? 
 
 19. C's fortune is twice D's, but after each had spent 
 | of his fortune, and C had given D $10, D had twice as 
 much money as ; required the fortune of each. 
 
 20. Two men, A and B, agree to dig a ditch for 850 ; 
 and | of what A digs, increased by 4 rods, equals f of 
 \vhat B digs, and B receives 830 ) how many rods did 
 eaoh dig ? 
 
MENTAL ARITHMETIC 145 
 
 LESSON XIII. 
 
 1 A lost 1 5 cents, and then found ^ as much as he 
 had remaining, and then had i as mucn as he had at 
 first, how much had he at first: 
 
 SOLUTION. After finding | as much as he had remaining, he 
 had | _|_ 1, which is | of what remained after losing 15 cents; 
 which, by the last condition of the problem, equals ^ of what 
 he had at first. If ^ of what he had at first equals | of what 
 remained, g, or what he had at first, equals 2 times J, or | of 
 what remained: then . of what remained, which is what he 
 had at first, minus J> of what remained, which is J> of what 
 remained, equals what he lost, or 15 cents, &c. 
 
 2. B lost $22, and then found i as much as he had 
 remaining, and then had ^ as much as he had at first; 
 how much had he at first ? 
 
 3. A, having a certain sum of money, found $20, and 
 then lost ^ of what he then had, and then had twice as 
 much as he had at first ; how much had he at first ? 
 
 4. A man having some money, borrowed 30 cents, and 
 then losing | of what he then had, found there remained 
 3 times as much as at first; how much had he at first? 
 
 5. A boy lost 44 cents, and then earning | as much 
 as remained, found he had | as much as at first; how 
 much had he at first ? 
 
 6. A man went to a store and spent 21 cents, and then 
 borrowing 1 of what he had remaining, had ^ as much 
 as he had at first; how much money had he at first? 
 
 7. A, at a game of chess, won $18, and then lost | of 
 what he then had, when, counting his money, he found ttt 
 had 24 times as much as at first; how much did he make 
 by playing ? 
 
 8. A and B lost 12 apples, and then bought | as many 
 as they lost, and then had | as many as they had at first ; 
 how many had they remaining? 
 
 13* 
 
146 MENTAL ARITHMETIC. 
 
 9. A, in a game of chance, lost | of his money and 
 then won 810, after which he had | as much as he had 
 at first ; how much did he lose at play ? 
 
 10. A and B lost 32 peaches, and then bought | as 
 many as remained, and then had j as many as at first; 
 how many had each at first, supposing their shares to be 
 as 2 to 3? 
 
 11. A boy being asked his age, replied, that if 11 
 years ago his age had been increased by its j, it would 
 then have been \ of what it now is ; required his age. 
 
 12. A lady being asked her age, said, that if her age 
 were increased by its i, the sum would equal 3 times her 
 age 12 years ago ; what was her age ? 
 
 13. A, B, and C, dine together; A furnishing 2 Ioaves 3 
 B 3 loaves, and C contributing 25 cents to be divided 
 between A and B ; required the share of each. 
 
 14. A boy being asked his age, replied, that if my age 
 in 3 years be diminished by its f , the remainder will be 
 | of my age now ; what was his age ? 
 
 15. A furnished 2 loaves for supper, and B 4, while 
 C contributed 20 cents to be divided between A and B; 
 how much of it should each receive ? 
 
 16. Two partners, A and B, lost $210, and the next 
 year gained ^ of what remained, which was J of the 
 original stock ; what was the stock of each ; if | of A's 
 equals | of B's stock ? 
 
 17. A furnished 6 eggs for a repast, and B 10 eggs, 
 while C contributed 16 cents to be divided between A 
 and B; how much shall each receive, provided A and B 
 eat the same number, and C, 4 more than each ? 
 
 18. A person being asked his age, said, that if my 
 age in 4 years be diminished by its f , the remainder will 
 equal ^ of my age 4 years ago ; what was his age ? 
 
 19. A, B, and C eat 14 peaches, of which A owned 5, 
 and B 9, and C contributed 24 cents; how much of the 
 money ought A and B each to receive, if B eats twice as 
 many as A, and C eats twice as many as B? 
 
 20 Two merchants having a certain number of yards 
 
MENTAL ARITHMETIC. 147 
 
 of cloth, bought 30 yards more, then scld I as many as 
 they had, and then had 3 times as many as at first ; how 
 many yards had each, if \ of A's number equals \ of B's? 
 
 LESSON XIV, 
 
 1. A man bought a certain number of sheep for $100, 
 when a dog having killed 8 of them, he sold | of the 
 remainder for cost, and received $20 ] how many did he 
 buy? 
 
 SOLUTION. If I of the remainder cost $20, J of the remain- 
 der cost 3 times $20, or $60 ; then, since they all cost $100, 
 the 8 must have cost $100 $60, or $40, &c. 
 
 2. A farmer bought a number of pigs for $80, when 5 
 of them having died, he sold | of the remainder for cost, 
 and received $40 ; how many did he buy ? 
 
 3. A lady purchased a number of yards of muslin for 
 $1.50, and after using 6 yards she sold f of the remain- 
 der for cost, and received 90 cents less than it all cost; 
 how many yards did she purchase ? 
 
 4. A bought a number of turkeys for $10, when, 
 having killed 10, he sold | of the remainder for cost, 
 receiving $8 less than the cost of all ; required the num- 
 ber purchased. 
 
 5. A bought a number of sheep for a certain sum, and 
 having lost 6 he sold | of the remainder for cost and 
 received $15, which was $35 less than they all sost; how 
 many did he buy ? 
 
 6. A farmer bought a number of hens for a certain 
 sum, and having killed 10, he sold | of the remainder 
 for cost, and received 48 dimes, which was 72 dimes less 
 than they all cost ; how many did he retain ? 
 
 7. A bought a p.umber of ducks for $16, and having 
 
148 MENTAL ARITHMETIC. 
 
 killed 12, he sold of the remainder, lacking 8, for 
 cost, and received $4; how many did he buy? 
 
 8. B bought a number of sheep for $30, and losing 2, 
 he sold I of the remainder, lacking 3. for cost, and received 
 621 less than all cost; required the number bought. 
 
 9. Henry bought a number of pigs for $48, and losing 
 9, he sold | of the remainder, minus 2, for cost, receiving 
 $32 less than all cost; required the number purchased. 
 
 10. A bought some calves for $80, and having lost 10, 
 he sold 4 more than f of the remainder for cost, and 
 received $32 less than all cost; required the number 
 purchased. 
 
 11. A dog killed j of A's sheep; now if he sells the 
 remainder for cost he will receive $60, but reserving 8 
 and selling J of the remainder for cost he will receive 
 $22 ; how many had he at first ? 
 
 12. A lost of his hens, and found if he sold | of the 
 remainder for cost he would receive 40 dimes, but if he 
 kept 15 and sold | of the remainder he would receive 20 
 dimes; how many did he have? 
 
 13. B lost | of his turkeys, and then finds by selling 
 | of the remainder for cost, he would receive $20, but 
 finding 6, and selling | of the number he then had, he 
 received $24 ; how many did he retain ? 
 
 14. A lost | of his hens; now if he finds 10 and sells 
 | of his number then for cost price he will receive 60 
 aimes, but if he loses 10 and sells | of the remainder for 
 cost he will receive 30 dimes; how many had he at first? 
 
 15. If 60 Ibs. of sea water contain 2 Ibs. of salt, how 
 much fresh water must be added to these 60 Ibs., so that 
 10 Ibs. of the new mixture may contain of a pound of 
 salt ? 
 
 16. Suppose that for every 4 cows a farmer has, he 
 should plough an acre of land, and allow one acre of pas- 
 ture for every 3 cows, how many cows could he keep OD 
 140 acres ? 
 
 17. B lost | of his sheep; now if he finds 5, and sell 
 
MENTAL ARITHMETIC; 149 
 
 | of what he then has for cost price, he wifl receive $18 5 
 but if he loses 5, and sells | of the remainder for cost 
 price, he will receive $6 ; how many had he at first ? 
 
 LESSON XV 
 
 1. A father divided $4400 between his two childre^ 
 A and B, whose ages were 11 and 16 years respectively, 
 in such a manner that the parts, at 5 per cent., simple 
 interest, would amount to equal sums when they became 
 of age ; what were the parts ? 
 
 SOLUTION. A's money was on interest 10 years and B's 5 
 years. For 10 years at 5 per cent., f of the principal equals 
 the amount, hence | of A's share equals A's amount ; and in the 
 same way we see that f of B's share equals B's amount. Now 
 the amounts are to be equal, hence f of A's share = f of B's 
 share, from which we find A's share = f of B's share ; then f 
 of B's + f of B's, or y of B's share = $4400, % of B's 
 $400, f of B's = $2400, and f of B's, or A's = .$2000. 
 
 2. A divided $5600 between his two sons, whose ages 
 were 11 and 15 years, in such a manner that the two 
 parts on interest, at five per cent., would amount to equal 
 sums when they became 21 years of age; required the 
 parts. 
 
 3. A gentleman dying divided $5100 among his 
 three sons, whose ages were 9, 11, and 17 respecti\ ely, 
 so that the different shares, being on interest at 5 per 
 cent., would amount to equa. sums when they became c 
 age y what were the shares ? 
 
 4. A widow divided $3700 among her sons, whoso 
 ages were respectively 14, 16, and 18 years, in such a 
 manner that the shares, being put on interest at 20 pei 
 
150 MENTAL ARITHMETIC. 
 
 cent., would amount to equal sums when they became 
 of age ; required the share of each. 
 
 5. A boy spends i of his money,-}- $^, then ^ of tne 
 remainder, -f &z an d then had $3 ; how much money 
 had he at first ? 
 
 6. C and D -together have 20 sheep, and | of C J s 
 number, -f- i of D's, equals of C ? s ; how many sheep 
 does each own ? 
 
 7. A man spent i of his money and $2 more, and then 
 spent $2 more than ^ of the remainder, and then had 
 $2 remaining ; required his money at first. 
 
 8. Sarah gave away | of her peaches, lacking ^ of a 
 peach, and then gave away -| of the remainder, lacking 
 | of a peach, and then had 5^ peaches remaining; how 
 many peaches had she at first ? 
 
 9. A fish caught in the Conowingo, weighs 8 pounds, 
 and | of the body, -f- | of the head and tail, weigh as 
 much as i of the body; required the weight of each part, 
 if the tail is | as heavy as the head. 
 
 10. A father left $5500 to his son and daughter, whose 
 ages are respectively 19 and 15 years, so that, being on 
 interest at 10 per cent., the son should receive twice as 
 much as the daughter, when they were 21 years of age; 
 what was the share of each ? 
 
 11. Harry gave of his money, lacking 3 cents, to 
 James, \ of the remainder, lacking 2 cents, to Willie, 
 and ^ of the remainder, lacking 1 cent, to Charles, and 
 then had 8 cents remaining ; what was Harry's money 
 before his gifts? 
 
 12. Divrde $290 between A and B, whose ages arc 
 respectively 15 and 19 years, in such a manner that 
 the parts, being placed on interest, at 10 per cent., shall 
 amount to such sums, at the time they are 21, that | of 
 A's shall be equal to f of B's money. 
 
 13. A lady, having two watches, bought a chain for 
 $20 If the chain be put on the silver watch their value 
 
MENTAL ARITHMETIC 151 
 
 will be | as much as the gold watch, but if it be pat on 
 the gold watch they will be worth 7 times as much as 
 the silver watch ; what was the value of each watch ? 
 
 14. Jordan gave i of his money, plus 4 cents, to 
 John, i of the remainder, plus 3 cents, to George, and 
 | of what now remained, plus 2 cents, to Jackson, and 
 found he had 4 as much as at first; how much money 
 had he at first ? 
 
 15. A person has two cups and a. cover which weighs 
 30 ounces. If the first cup be covered it will weigh 
 twice as much as the second, but if the second cup be 
 covered, it will weigh 3 times as much as the first ; what 
 is the weight of each cup ? 
 
 LESSON XVI. 
 
 1. A farmer bought a certain number of sheep for 
 $60 ; had he bought 5 more at $1 less each, they would 
 have cost him $75; how many sheep did he buy? 
 
 SOLUTION. By the conditions of the problem the 5 more, at 
 $1 less each, cost $75 $60, or $15, and one, at this rate, 
 cost i of $15, which is $3, which, increased by $1, equals $4, 
 the price of those purchased : hence, there were as many pur 
 chased as $4 is contained times in $60, which are 15. There- 
 fore, &e. 
 
 2. A farmer bought a certain number of cows for 
 $200 ; had he bought 2 more, at $2 less each, they would 
 have cost $216; how many did he buy? 
 
 3. A man bought a certain number of sheep for $80 : 
 and afterwards bought twice as many more, at $2 less 
 each, and paid for all $180 ; how many did he buy ? 
 
 4. Mr. A bought a certain number of turkeys for $5 ; 
 had he bought 3 times as many, -(- 4, for the same price, 
 they would have cost him $12 more; how many did he 
 purchase ? 
 
152 MENTAL ARITHMETIC. 
 
 5. A teacher bought a number of books for $8 ; had 
 he bought 4 times this number, diminished by 5, they 
 Would have cost $4 more ; how many did he buy ? 
 
 6. Willis sold 2 books for 60 cents each ; on one he 
 gained 20 per cent., and on the other he lost 40 per cent. ; 
 did he gain or lose by the sale, and how much ? 
 
 7. What part of 20 per cent, of 6 is 25 per cent 
 
 of a? 
 
 8. I gave 20 per cent, of my money to A, 25 pei 
 cent, of the remainder to B, and had $30 remaining ; how 
 much money had I at first ? 
 
 9. If A pays | of his salary for board, what per cent, 
 does he have left for other purposes ? 
 
 10. If I sell of a quantity of grain, and 4 of the re- 
 mainder is spoiled, what per cent, remains ? 
 
 11. Janson sold 20 per cent, of his apples to A, 25 
 per cent, of the remainder to B, and 33-J per cent, of this 
 last remainder to C, and had 20 barrels remaining ; how 
 many had he at first ? 
 
 12. A sold B a gun and gained 25 per cent., and B 
 sold it to C for $24 and gained 20 per cent. ; what did 
 A pay for it ? 
 
 13. Bought apples at 6 cents for 4, and sold them at 
 the rate of 6 for 4 cents ; what was the loss per cent. ? 
 
 14. B bought melons for 20 per cent, more than 10 
 cents each, and sold them for 20 per cent, less than 10 
 cents each ; required the loss per cent. ? 
 
 15. If a book is bought for | of its value, and sold for 
 20 per cent, more than its value, what is the gain per 
 cent. ? 
 
 16. If William has 25 Der cent, more money thau 
 Henry, how many per cent, has Henry less than William? 
 
 17. If a principal gain A uf itself in a year, how long 
 will it require to double itself? 
 
 18. At 25 per cent., in what time will a principal gain 
 | of itself ? | of itself ? 3 times itself? 
 
 19. A's and B's money on interest for 10 years at 5 
 
MENTAL ARITHMETIC 153 
 
 per cent, amounts to $900 ; how much Has fcsch, if A's 
 money equals 3 times B's money? 
 
 20. Henry received $224 to invest in property, after 
 retaining 12 per cent, on the amount invested; how much 
 did he invest? 
 
 21. If I gain $20 by selling an article for 20 per cent, 
 more than cost, required the cost and amount received 
 
 22 . A lost $60 by selling a horse for 30 per cent, less 
 than cost ; required the cost and amount received. 
 
 23. B bought goods 20 per cent, below par, and sold 
 them 20 per cent, above par ; supposing he gained $90, 
 what amount of goods did he buy ? 
 
 24. A man bought goods 25 per cent, below par, and 
 sold them 20 per cent, above par; how much did he 
 invest if he gained $270 ? 
 
 25. A merchant asked for cloth 20 per cent, more than 
 cost, but sold it for | of his asking price ; what was the 
 loss per cent. ? 
 
 26. B asked for flour 25 per cent, more than cost, but 
 sold it for 80 per cent, of the price asked; what did he 
 lose per cent. ? 
 
 27. A grocer asked for sugar 20 per cent, more than 
 cost, and sold it for 33 1 per cent, less than he asked for 
 it ; what was the loss per cent. ? 
 
 28. A man asked 10 per cent, less for an article than 
 cost, but sold it for 33| per cent, more than he asked for 
 it ; required the gain per cent. ? 
 
 29. What must I ask for hay, worth $10 a ton, that 
 after falling 20 per cent., I may gain 20 per cent, on the 
 value ? 
 
 30 What must I charge for flour, worth $5 a barrel, 
 that after falling 25 per cent, on my price, I may gain 20 
 per cent, on the cost? 
 
 31. What must I ask for cloth, worth $40, that after 
 falling 20 per cent, on my price, I may gain 30 per cent, 
 on the cost? 
 
 32 If my retail gain is 25 per cent., and my whole- 
 14 
 
154 ^ MENTAL ARITHMETIC. 
 
 sale gain is 20 per cent, of my retail less, what per cent 
 * I gain at wholesale ? 
 
 33. If I retail at a gain of 50 per cent., and sell at 
 wholesale for 25 per cent, less than at retail, what do I 
 gain per cent, at wholesale ? 
 
 34. If my gain at retail is 60 per cent., and my gain 
 at wholesale is 25 per cent, of my retail gain less, what , 
 is my gain per cent, at wholesale ? 
 
 35. If A's gain at wholesale is 20 per cent., and his 
 gain at retail is 25 per cent, of his wholesale more, what 
 does he gain per cent, at retail ? 
 
 36. If B's loss at wholesale was 10 per cent., and hi? 
 retail price was 33| per cent, more ; what was the gain 
 per cent, by retail ? 
 
 37. A lost 40 per cent, of his flour, and sold the re- 
 mainder at a gain of 50 per cent. ; did he gain or lose, 
 and how much per cent. ? 
 
 38. A barrel of molasses lost 20 per cent, by leakage, 
 and the remainder was rold at a gain of 40 per cent. ; re- 
 quired the gain per cent. 
 
 39. An article lost 25 per cent, by wastage, and the 
 remainder was sold for 20 per cent, above cost; what 
 per cent, was gained or lost ? 
 
 40. A drover lost 33| per cent, of his cattle, and sold 
 the remainder at a gain of 50 per cent. ; required the 
 gain or loss per cent. 
 
 41. The amount of B's fortune for 4 years, at 10 per 
 cent., is $200 more than its amount for 6 years at 5 per 
 cent. ; required the fortune. 
 
 42. The amount of C's fortune for 2 years, at 10 per 
 cent., is $400 less than its amount for 6 years at 5 per 
 cent. ; what is the fortune ? 
 
 43. The amount of M's money for 5 years, at 8 pei 
 cent., is $40 more than its amount for 4 years at 6 per 
 cent. ; required his money. 
 
 44. The amount of D's money for 2 years, at 5 per 
 cent., is $60 more than its interest for 9 years at 10 per 
 cent. ; what is his money ? 
 
MENTAL ARITHMETIC. 156 
 
 45. The amount of B's fortune for 5 ye&is, at 10 per 
 cent., is $330 mor< than the amount of C's for the same 
 time and rate per cent.; what is the fortune of each, 
 provided B's is twice C's? 
 
 LESSON XVII. 
 
 1. A has 3 times as many plums as B, and B has 
 twice as many as C ; how many has each, if A has 12 
 more than B and C together ? 
 
 2. A, B, and C can mow a field in 20 days, A and B 
 in 30 days, and B and C in 40 days ; after the three had 
 worked 5 days, A and C finished it; in what time was it 
 completed ? 
 
 3. A lady bought 10 yards of silk at the rate of $4 a 
 yard, but finding some of it damaged, for it she only 
 paid $1 a yard, and thus paid $28 ; how many yards were 
 damaged ? 
 
 4. A haviag his fortune on interest at 5 per cent., in 
 one year spends ^ of his income in travelling, | for edu- 
 cational purposes, and saves $100 ; what is his fortune? 
 
 5. A can do a piece of -work in 20 days, B and C iv 
 12 days, and if all work 6 days, C can complete it in 3 
 days; in what time could B and C each have done it? 
 
 6. A bought 12 cows for $360, and sold J of them for 
 what all cost ; what was the gain per cent. ? 
 
 7. B gained 50 per cent, in each of 3 years on what 
 he had at the beginning of the year, and then had gained 
 $190 ; what was his first capital? 
 
 8. A and B can do a piece of work in 15 days, and 
 after working 9 days they called in C, with whose assist- 
 ance they finished the work in 4 days; in what time 
 could C alone have done it ? 
 
 9. Edward and Ella have $900, and 20 per cent, of 
 
156 MENTAL ARITHMETIC. 
 
 Edward's money equals 25 per cent, of Ellas money j 
 how much money has each ? 
 
 10. A can do 15 per cent, of a piece of work in a day, 
 B 20 per cent., and C 25 per cent.; after B had wrought 
 3 days, and A 2 days, C joined them; in what time was 
 the work completed? 
 
 11. A man left $5000 to his wife, son, and daughter; 
 and if the daughter died before becoming of age, the 
 vridow should have | of the fortune, but if the son died, 
 she should have f of it ; required the shares of the son 
 and daughter if the widow dies ? 
 
 12. A boy bought some peaches at 4 cents each, and 
 3 times as many pears at 2 cents each, and sold them all 
 at 6 cents each, and thus gained 28 cents ; how many of 
 each did he buy? 
 
 13. Three fifths of A's age was his wife's age when 
 married, but in 40 years, i of his age equals hers ; what 
 was the age of each when married ? 
 
 14. C and D ran from the same point in the same 
 direction, and when D had run 40 rods, -^ of the distance 
 C had run equalled the distance he was ahead of D ; how 
 much did C, in running 40 rods, gain on D ? 
 
 15. What is the hour of day if | of the time past 9 
 A. M. equals f of the time to 11 o'clock p. M. ? 
 
 16. A boat whose rate of sailing is 5 miles an hour, 
 moves down a river whose current is 3 miles an hour; 
 how far may it go that it may be back in 10 hours ? 
 
 17. A and B, in partnership, gain $40 ; A owned | 
 of the stock, lacking $12, and B's share of the gain was 
 $10 ; required the whole stock and share of each. 
 
 18. A man went to a store and spent 20 cents, and 
 then losing | as much as remained, had -J as much as he 
 had at first, minus $1 ; how much had he at first? 
 
 19. A lost 20 per cent, of his peaches, and sold the 
 remainder for 20 per cent, more than they all cost ; what 
 did he gain per cent, on those sold ? 
 
 20. A drover lost 30 per cent, of his cattle, and sold 
 
MENTAL ARITHMETIC. 157 
 
 the remainder for 10 per cent, more thar tney all cost j 
 what did he gain per cent, on those sold ? 
 
 21. A digs | of a ditch in 8 days, and then calling ID 
 B, they together finish it in 9 days ; in what time could 
 B have done it alone ? 
 
 22. Bought hay for $8 a ton, but in getting it I lost 
 20 per cent, of it ; what did it really cost me a ton ? 
 
 23. I bought 8 apples for 16 cents, and lost | of them ; 
 what per cent, must I gain on the remainder, that I may 
 neither gain nor lose by the transaction ? 
 
 24. A bought 9 melons for 36 cents, but losing 33^ 
 per cent, of them, how must the remainder be sold to 
 gain 33| per cent, by the transaction ? 
 
 25. The amount of A's fortune for 3 years, at 1.0 per 
 cent., is $520 more than the amount of B's for 5 years, 
 at 6 per cent. ; required the fortune of each, supposing 
 A's to equal 3 times B's. 
 
 26. If | of a barrel of flour is sold for what i of it 
 cost, what is the gain per cent. ? 
 
 27. B lost 20 per cent, of his maroles; what must he 
 gain per cent, on the remainder, that he may gain 20 per 
 cent, on the whole ? 
 
 28. A barrel of molasses leaked away 20 per cent. ; 
 what per cent, must I gain on the remainder, that I may 
 gain 40 per cent, by the transaction ? 
 
 29. A sum of money at interest amounts, in 2 years, 
 to $240, and in 6 years to $320 ; required the sum and 
 rate per cent. 
 
 30. A man sold 2 horses for $250 ; on one he gamed 
 25 per cent., on the other he lost 20 per cent. ; did he 
 gain or lose, and how much, if he received for the second 
 | as much as for the first ? 
 
 31. The amount of a sum of money for 3 years is $230, 
 and the amount for 4 times as long, at ^ the same rate, 
 is $260 ; what are the sum and rates per cent. ? 
 
 32. A man sold 2 horses for $210 ; on one he gained 
 25 per cent, and on the other he lost 25 per cent. ; how 
 
 14* 
 
158 MENTAL ARITHMETIC. 
 
 much did he gain, supposing the second horse cost him f 
 as much as the first horse ? 
 
 33. A man sold a horse and carriage for $230; on the 
 horse he lost 20 per cent., and on the carriage he gained 
 25 per cent. ; did he gain or lose, and how much, if | of 
 what he paid for the horse equalled f of the cost of the 
 carriage ? 
 
 NOTE. For additional problems, see "METHODS OF 
 MENTAL ARITHMETIC, <fcc.," by the Author of this work; where may 
 be found a large collection, of an aqausing and interesting character, 
 ondar the head of Social Arithmetic. 
 
MENTAL ALGEBRA. 
 
 The transition from Arithmetic to Algebra being so 
 easy and natural, and a knowledge of the abbreviated 
 and general language of Algebra being so important, the 
 author is induced to present the following short treatise 
 upon the subject of Mental Algebra. 
 
 METHOD OF TREATMENT. The peculiarity of this 
 treatment is that Mental Arithmetic and Algebra are 
 taught in combination, like twin sisters going hand in 
 hand, one text-book answering for both subjects. A 
 solution is given to the first problem of a lesson, or class 
 of problems, which the pupils should be required to 
 apply to the other problems of that class. 
 
 METHOD OF SOLUTION. In Algebra we let some 
 symbol, as x, represent the number we wish to find, and 
 use this as we would the answer in proving a problem. 
 This will give an expression or equation from which 
 we may find the unknown number represented by the 
 symbol. 
 
 The following problem and solution will indicate the 
 method and spirit of the Algebraic treatment. The 
 pupil will observe how much shorter and simpler it .s 
 than the method of Arithmetic. 
 
 PROBLEM. Henry's number of apples increased by 3 
 his 'number equals 24; how many apples has he? 
 
 (159) 
 
160 MENTAL ALGEBRA. 
 
 ARITHMETICAL 
 SOLUTION. 
 
 Once Henry's number plus 3 times his number 
 
 equals 24. 
 
 Hence, 4 times Henry's number equals 24^, 
 _ And once his number equals of 24, which is 6 
 
 Now if we represent the expression " Henry's number 9 
 by some symbol as x, and use the sign, =, for the word 
 "equals" etc., the solution will be much shorter; thus: 
 
 ALGEBRAIC 
 SOLUTION. 
 
 NOTE. In Algebra 2x, 3x, &c., mean 2 times X, 3 times x, 
 
 112 
 etc., j of x, J of x, f of x are represented thus, -x, -x, -x, etc., 
 
 2 t o o 
 x x 2x 
 
 From this explanation the pupil will be prepared to 
 understand the solutions which follow. The solution 
 given to the first problem of' each class is to be applied 
 to the others of the same class. The class is indicated 
 by referring to the page and problem of the Mental 
 Arithmetic. 
 
 PAGE 37, PEOB. 1. 
 
 SOLUTION. Let x represent the Let x = the number, 
 
 number ; then J of x equals 3, and ^ en * _ ^ 
 x equals 2 times 3, or 6. Therefore 
 
 the number is 6. an d x = 6, Ans. 
 
 PAGE 58, PROS. 1. 
 
 SOL, Let x equal his mo- Let x = his money. 
 Qey ; then x minus f of x, or 
 | of x equals 60, and ^ of x 
 
 equals J of 60, or 30, and x an( j ?_ _ gQ 
 
 equals 6 times 30, or 150. 6 
 
 Therefore, etc. x = 15 Ans - 
 
MENTAL ALGEBRA. 161 
 
 PAGE 67, PROB. 1. 
 
 SOL. Let x equal the number of Let x the No. 
 
 times ; then, since the product of _ . 
 
 the divisor and quotient equals the c 3 X * == 4 
 
 dividend, we have fX z 4, hence 1 
 
 J X * = i 'rf 4, or 2, and x = 3 and - X * = 2 
 
 times 2, or 6. 
 
 x = 6. Ans. 
 
 PAGE 68, PROB. 28. 
 
 SOL. Let x = the number, then, since the product of the 
 divisor and quotient equals the dividend, we have f X x J> 
 hence J X x = f > an d x = f Therefore, etc. 
 
 PAGE 70, PROB. 1. 
 SOL. Let x equal the part of 2 Let x = the part 
 
 o 
 
 which equals J ; then 2 multiplied 2 X * = - 
 
 by x equals J, and x equals J of J, 3 
 
 which is |. ' X = 8' An8> 
 
 PAGE 71, PROB. 16. 
 
 SOL. Let x equal the part, then f times x equals f , hence 
 J of x equals f and x equals f . Therefore, etc. 
 
 PAGE 90, PROB. 1. 
 
 SOL. Let x equal the number Let x = No. of children, 
 
 of children ; then 4z equals what then, 4x = first sum, 
 
 they received by the first condi- and 7x = second sum. 
 
 tion, and 7x, what they received hence, 3z 36, 
 by the second condition ; hence x = 12. Ans. 
 
 7x 4z, or 3z, equals 36, and x 
 s= J of 36, or 12. Therefore, etc. 
 
 PAGE 94, PROB. 1. 
 
 SOL. Let x = Henry's num- Let x = Henry's number, 
 
 ber ; then 2x will equal Wil- then, 2x = William's number 
 
 liam's number, and 2x plus x, and Sx = 15. 
 
 or 3z, equals what both have, hence, x = 5, Henry's, 
 
 or 15. If 3z=rl5, x= J- of 15, and 2z = 10, William's, 
 which is 5, and 2z = 2 times 5, 
 or 10. Therefore, &c. 
 
162 MENTAL ALGEBRA. 
 
 PAGE 96, PROB. 1. 
 
 SOL. Let x = A's number ; Let x = A's number, 
 
 then will x -f- 5 equal B's number, then x -f- 5 = B's numoer. 
 
 and, adding, 2x -\- 5 will equal and 2z -f- 5 = 25. 
 
 what both have, which is 25. If hence, 2x = 20, 
 2z -}- 5 25, 2x must equal 25 x = 10 A's. 
 
 minus 5, which is 20, &c. x -}- 5 = 15 B's. 
 
 DEFINITIONS AND PRINCIPLES. 
 
 Having applied the above solutions to the different 
 lessons, the pupils will now be able to understand the 
 following definitions and principles : 
 
 An equation is an expression of the equality or two 
 equal quantities ; thus 2x -f- 4 = 10. 
 
 The quantity on the left of the sign of equality is 
 called the first member, the quantity on the right is 
 called the second member of the equation. 
 
 To transpose a quantity is to change it from one 
 member of an equation to the other. 
 
 We now present four simple principles, which will 
 enable the pupil to understand the following solutions. 
 
 PRIN. 1. If the same number be added to or subtracted 
 from both members of an equation, the resulting members 
 will be equal. 
 
 PRIN. 2. If both members of an equation be multiplied 
 or divided by the same number, the resulting members 
 will be equal. 
 
 PRIN. 3. In transposing a number from one member of 
 the equation to the other, we change the sign preceding it. 
 
 PROOF. Take the equation 3z -f 4 = 16. If 8x increased 
 by 4 equals 16, it is evident that So; = 16 4, hence in chang- 
 ing 4 from the first member to the second we change its sign. 
 Again, 
 
 From the equation 4x = 22 6, it is evident that 22 is 6 
 more than 4x, hence 4z -{- 6 = 22, therefore the 6 may be taken 
 from the second member to the first by changing the sign. 
 
 PRIN. 4. An equation may be cleared r f fractions by 
 multiplying both members by any number which is divi- 
 sible by all the denominators. 
 
MENTAL ALGEBRA. 163 
 
 PROOF. Take the equation --{---. If we multiply this 
 
 2> o 4 
 
 by 12 we have 6z -f- 4x = 30, which is an equation without 
 fractions. Therefore, &c. 
 
 NOTE. The symbol ( ) is called a parenthesis, and denotes 
 that the quantities enclosed in it are to be treated as a single 
 quantity ; thus (30 x) X 3 denotes that 30 z is to be mul- 
 tiplied by 3, and = 90 3z, while 30 x X 3 = 30 3s. 
 
 EXAMPLES FOR PRACTICE. 
 
 Given, 5z = 24 -f- 2z, to find Z. 
 
 Transposing, 5z 2x 24. 
 Subtracting, 3z = 24. 
 
 Hence, z 8. Ans. 
 
 2 5 
 
 Given, -- - = 4, to find x. 
 
 6 2i 
 
 Multiplying by 6, 4x 15 = 24. 
 Transposing, 4x = 24 -f 15 = 39. 
 
 Hence, x = 9J. 
 
 3. Findzin3z 7 = 8; In 4z = 18 2z; In Ix = 35 2x 
 
 ; In (8x 4) X 5 = 25. 
 
 PAGE 101, PROS. 1. 
 SOL. Let x = the gain per cent. 
 Then, JLx 20 = 25 -20 =5. 
 
 Hence, - = 5. 
 
 o 
 
 a; = 25, Ans. 
 
 PAGE 102, PROB. x. 
 SOL. Let x = the cost. 
 Then, + Jjc = 25 
 Or, f==26 
 
 And x = 20 Ans. 
 
 PAGE 104, PROB. 1. 
 SOL. Let a; = the cost. 
 Then, Jz = 20. 
 And zr=80. Ans. 
 
MENTAL ALGEBRA. 
 
 PAGE 118, PROS. 1. 
 
 SOL. Let x = what they pay for .me eow 
 Ihen, 4x = what A pays. 
 
 And 5x = what B pays. 
 
 Then,4*-f 5z:=36. 
 Or, 9ar = 36, 
 
 x~4. 
 
 4z=16, A's part. 
 bx ' = 20, B'spart. 
 
 PAGE 120, PROB. 1. 
 SOL. Let 4z = A's share. 
 And 6z = B's share. 
 
 Then, 4z -}- 6z := 30, 
 Or, lOa: 30. 
 
 z = 3. 
 
 4z=12, A's share. 
 6a; = 18, B's share. 
 
 PAGE 122, PROS. 9. 
 
 SOL. Let x = the time in which both can eat it. 
 Then, .- = what both can eat in one day. 
 
 | = what Fuller can eat in a day. 
 = what Brodhead can eat in a day 
 
 5x = 12. 
 2 = 2. Ans. 
 
 PAGE 126, PROS. 1. 
 
 SOL. Let z = number of idle days. 
 Then 30 x = number of working days 
 
 (30 x) X 3 = what he earned. 
 
 x = what he forfeited. 
 Hence, 6C -f x = (30 x) V 3. 
 Or 60 -fz=:90 8z. 
 
 4x = 40. 
 x = 10, Ans. 
 
MENTAL ALGEBRA. 
 
 165 
 
 + 8 = 24, the tail. 
 
 PAGE 127, PROB. 8. 
 OL. Let x = length of body. 
 
 Then, | + 8 = length of tail. 
 
 And 8 = length of head. 
 
 Hence, 2x = 16 -f- 
 
 And - = 16. 
 
 x = 32, the body. 
 Then, 
 
 32 -f 24 + 8 == 64, Ans. 
 
 PAGE 129, PROB. 1. 
 SOL. Let x = the time. 
 Then, 35 ^-j- x = H's age at that time. 
 
 And 5 '-}- x = M's age at that time. 
 
 Hence, (5 + x) X 6 = 35 -f x. 
 Or, 30 -{- 6 * 35 -f- x. 
 
 Hence, 5z = 5. 
 
 z = l, Ans. 
 
 PAGE 130, PROB. 16. 
 SOL. Let x = the number of each kind. 
 Then, 2x = cost of first kind, 
 And 4x = cost of second kind. 
 And Qx = cost of both kinds. 
 
 Since there were 2x in all and they were sold at the rate 
 of 4 cts. each, 
 
 8x = what they were sold for. 
 Then Sx 6x = 12, 
 2x = 12. 
 x = 6, Ans. 
 
 PAGE 131, PROB. 1. 
 SOL. 
 Let z = B's age. 
 
 3x = A's age. 
 
 x -\- 10 3E= B's age in 10 years. 
 
 Sx -j- 10 = A's age in 10 years. 
 
 f 10 = 2x -f20. 
 + 10 = 20. 
 
 x =a 10, B's age. 
 
 16 
 
 PAGE 133, PROB. 1. 
 SOL. 
 Let x = time past midnight. 
 
 x 
 
 - = time to noon. 
 
 A 
 
 X 
 
 x -f- ^ = 12 hours. 
 
MENTAL ALGEBRA. 
 
 PAGE 134, PROB. 15. 
 SOL. Let x = distance the minute hand goes. 
 
 x 
 Then, = distance the hour hand goes. 
 
 Andz - 10 the number of minute spaces they are 
 
 12 ~~ ' apart at 2 o'clock. 
 
 Ilz=rl20. 
 
 z = 10{, Ans. 
 
 PAGE 136, PROS. 1. 
 SOL. Let x = the distance. 
 
 
 Then, = time in going. 
 
 And - = time in returning. 
 
 
 - 
 
 And 6z-f 3z=240. 
 Sx = 240. 
 z = 30, Ana. 
 
 PAGE 136, PROS. 6. 
 
 SOL. Let x = what they pay for the coach 
 Then, - = what each of eight pay. 
 
 And = what each of twelve pay. 
 
 3* 2z = 18. 
 
 = 18 Ans. 
 
 PAGE 140, PROB. 1. 
 SOL. Let O = value of the gold 
 
 C = value of the chain. 
 Then, 2G = C +30. 
 
 And a+ C = 30X4 = 120. 
 
 Or, 67 = 120 (7. 
 
 Putting this in the 1st, 2G = 120 G + 30. 
 Transposing, 3 O = 150. 
 
 G = 50 Ans. 
 
 (7^120 50 = 70 Ans. 
 
MENTAL ALGEBRA. 167 
 
 PAGE 145, PROS. 1. 
 
 SOL. Let x = what he had at first 
 Then, x 15 = sum after losing 15 cents, 
 And |(z 15) = sum after finding J of remainder 
 
 Then, |(*-16) = g. 
 
 Or, f*-20 = |. 
 Hence, fa; = 20. 
 
 *=r4 
 6 
 
 3 = 24 Ans. 
 
 PAGE 150, PROS. 14. 
 
 SOL. Let Sib.Q value of the silver watch, 
 G= the value of the gold watch, 
 = the value of the chain. 
 Then, G=ZS+3C. 
 And 7S=G+C. Putting for G its value we have 
 
 7S=3S+ZC+ C. 
 Or, 4S=4C. 
 
 S = (7=: 20, Ans. 
 
 G= 40X3 = 120, Ans. 
 
 The following equations are a brief statement of the 
 rules for all the cases of Simple Interest. The rate is 
 expressed as a fraction; thus, 5 per cent, is used as 
 T fo or .05. 
 
 Let p = principal. (1) i =p X r X t. i 
 
 i = interest. i \*J T 
 
 ' ' 
 
 A = amount. (3) t = 
 
 EXPLANATION. The 1st denotes that the interest equals the 
 product of the principal, rate, and time. Thus the interest of 
 $40 for 5 yrs. at 6 per cent, equals $40 X lib X 5 = $ 12 
 The 2d denotes that the principal equals the interest divided by 
 the product of the rate and time, etc. 
 
168 MENTAL ALGEBRA. 
 
 To TEACHERS. The author presents this little treatise 
 upon Mental Algebra to his friends who have exhibited 
 such a kind appreciation of his former labors, with the 
 following remarks and suggestions : 
 
 The plan of combining Mental Arithmetic and Mental 
 Algebra in one book, is, so far as the author knows, 
 a new one, and is therefore a distinctive feature of this 
 work. Its advantages are supposed to be important, a 
 few of which will be briefly stated. 
 
 First, There is an economy in time and money ; 
 secondly, there is greater convenience in teaching; 
 thirdly, it is in accordance with the logical relation of 
 the subjects; fourthly, the arithmetical solution will 
 assist in understanding the algebraic, and the algebraic 
 will often throw light upon the arithmetical. 
 
 METHODS OF TEACHING. For those who desire to 
 try the plan here presented, developing the twin sisters 
 together, we present the following methods of instruc 
 tion : 
 
 1st. The same problem may be solved first arith- 
 metically and then algebraically, by the same or different 
 pupils. 
 
 2d. The algebraic solution may be deferred until 
 several or all of a class of problems have been solved 
 arithmetically. 
 
 3d. The book may first be completed arithmetically 
 and then reviewed algebraically. It is probable that 
 the second method will be preferred by most teachers. 
 
 If the pupils are carefully drilled upon the problems 
 under the several classes which we have solved, they 
 will be able to apply the algebraic method to other 
 problems not included in these classes. 
 
 THE END. 
 
OVERDUE. 
 
02390 
 
 
 Bfcy 
 
SCHOOL BOOKS 
 
 PUBLISHED BY 
 
 SOWER, POTTS & CO., 
 
 530 Market St, and 523 Minor St., Philada, 
 
 Brooks's Normal Primary Arith- 
 metic, 
 
 Brooks's Mental Arithmetic, 
 Brooks's Elementary Arithmetic, 
 Brooks's Written Arithmetic, 
 
 Broo!:s's Written Arithmetic, 
 
 
 
 Brooks's Key to Montal Arith- 
 metic, 
 
 Brooks's Key to Written Arith- 
 metic, 
 
 Brooks's Geometry and Trigono- 
 metry, 
 
 Roberts's United States History, 
 Fewsmith's English Grammar. 
 
 Peterson's Familiar Science, 
 no* 
 
 Peterson's Familiar Science, 
 
 18mo. 
 Wells's Familiar Science, 8vo. 
 
 Payson, Dunton & Scribner's 
 Series of Copy-Books, 
 
 Fairbanks's Book-Keeping, A 
 
 new work; best yet published. 
 
 Bouvier's Familiar Astronomy, 
 
 Complete. 8vo. 
 
 Bouvier's Familiar Astronomy, 
 
 Abridged. 8vo. 
 
 Sheppard's Text-Book of Con- 
 stitution of United States, 
 
 New Edition. 
 
 Sheppard's First Book of Con- 
 stitution of United States, 
 
 New Edition. 
 
 Pepper's Scientific Amusement. 
 Hillside's Geology, 
 Rhoads's Primary Arithmetic. 
 Raub's Normal Speller. 
 Raub's Primary Speller. 
 Pelton's Outline Maps, Muslin. 
 Pelton's Outline Maps, Paper. 
 Gilpin's Central Gold Region, 
 A new Music Book for Schools, 
 
 In Press.