1 distance B had run equaled the distance he was ahead of A; how far did B run? 6. Three men bought a horse; A paid $25, which was of what Band C paid; B paid as much as A and C; what did B and C pay? 7. The hour and minute hands of a watch are together at 12 o'clock; when will they next be together? ANALYSIS. Since the minute hand passes the hour hand 11 times in 12 hours, if both are at 12, the minute hand will pass the hour hand the first time in of 12 hours, or in 1 hours, equal to 5 minutes past 1 o'clock. Therefore, &c. 8. A gentleman, being asked the time, replied, "It is between 3 and 4 o'clock, and the hands are directly opposite." What was the time? 9. A fox is 40 leaps ahead of a hound, and takes 7 leaps to the hound's 4; but 3 of the hound's leaps equal 6 of the fox's; how many leaps will the fox take before being caught? 10. A wolf ran 80 rods to catch a sheep; & of the distance the sheep ran was equal to the distance between them when the chase commenced; what was the distance? 11. The sum off of A's and of B's money, being on interest 2 years 7 months and 6 days, at 5 per cent., amounted to $2260; of B's money was 14 times of A's; how much had each? 12. B's fortune is 11⁄2 times A's; the interest of of A's fortune and of B's for 5 years, at 6 per cent., is $600; what is the fortune of each ? 13. A drover paid $76 for calves and sheep, paying $3 apiece for calves and $2 for sheep; he sold of his calves and of his sheep for $23, and in so doing lost 8 per cent. on their cost; how many of each did he purchase? THE author desires to call the attention of teachers to the number and variety of chưnges that may be wrought upon the examples of the Sections from which the following have been selected, as well as upon many others of the same class, and which may appropriately be termed "Ringing the Changes." It is not recommended that these changes be applied to all the examples, but used as an occasional exercise, and limited according to the time and convenience of the teacher. It is recommended to teachers to exercise their classes in these "changes," not so much for the novelty of them, as for the enlivening effect produced upon the class, and especially for the peculiar facilities they afford for thorough mental discipline; and while, at first, to some they may appear severe and too mathematical, they will very soon become easy and comprehensible. To aid beginners, the blackboard may be used to advantage; and to illustrate, take the example which has been written out in full in Section 71, page 171-3: of 35 is & of how many thirds of 18? which may be written on the board thus: × 35 = × 3 × 18? Call upon scme pupil to read the question according to the first form, and to solve the question according to the form of analysis given; and the answer, if correct, placed over the 3, will fulfill the conditions of the question; substitute this number, and erase another, thus: of 35 is & of & of what number? Solve this as before, and so continue until all the numbers have been erased, and replaced by solution. When the class have become familiar with the exercise, the use of the blackboard should be dispensed with, and the pupil be required to "ring the changes" by simply hearing the first form of the question read. Each question, and its form of solution, should be given by the pupil, but he may be allowed to give results in the solution that have been previously determined by analysis, in order to shorten the work and avoid monotonous repetition. Members of the class should be called upon promiscuously, and encouraged to ask questions, and to detect errors either in the statement or solution of questions. The teacher should read an example once, slowly and distinctly, then call upon some one of the class who shall commence operations, and continue to give questions, forms, solutions, and conelusions, until signaled to sit, or another pupil detecting an error, raises the hand, when some other one should be called upon, who should resume where the last one left off; and so on. A teacher who has not practiced a class upon these changes, cannot conceive of the interest, close attention, and perfect enthusiasm that may be created, as well as the thorough mental discipline attained, by this exercise; and it is firmly believed that, upon trial, it will commend itself to the judgment and approval of every live teacher. 12 (169) 67. 1. 24 is of how many times 10? 24 is of as many times 10 as 10 is contained times in 5 times of 24; 4 of 24 is 6, and 5 times 6 is 30; 10 is contained in 30, 3 times. Therefore 24 is of 3 times 10. 2. 24 is of 3 times what number? 24 is of times of 5 times 4 of 24; 4 of 24 is 6, and 5 times 6 is 30; 3 of 30 is 10. Therefore 24 is of 3 times 10. 3. 24 is how many fifths of 3 times 10? 24 is as many fifths of 3 times 10 as of 3 times 10 is contained times in 24. 3 times 10 is 30, and of 30 is 6, and 6 is contained in 24, 4 times. Therefore 24 is of 3 times 10. 4. 24 is 4 times what part of 3 times 10? 24 is 4 times such a part of 3 times 10 as 3 times 10 is contained times in 4 of 24. 3 times 10 is 30, and 1 of 24 is 6; 30 is contained in 6,3 or times. Therefore 24 is 4 times, or, of 3 times 10. 5. What number is of 3 times 10? 4 4 times of 3 times 10. 3 times 10 is 30, of 30 is 6, and 4 times 6 is 24. Therefore 24 is of 3 times 10. 70. 1. 4 of 56 is of 3 times what number ? of 56 is of 3 times of 9 times of 56; of 56 is 8, and 4 times 8 is 32; and 9 times 4 is 36; 1 of 36 is 12. Therefore, of 56 is of 3 times 12. 2. 4 of 56 is of 4 times of 32 is 4, of how many times 12? of 56 is of as many times 12 as 12 is contained times in 9 times of 4 times of 563 of 56 is 8, and 4 times 8 is 32; 1 of 32 is 4, and 9 times 4 is 36; 12 is contained in 36,3 times. Therefore of 56 is of 3 times 12. 3. 4 of 56 is how many ninths of 3 times 12 ? 4 of 56 is as many ninths of 3 times 12 as of 3 times 12 is contained times in 4 times of 56. * of 56 is 8, and of 56 is 4 times 8; 4 times 8 is 32; 3 times 12 is 36, of 36 is 4, and 4 is contained in 32, 8 times. Therefore of 56 is of 3 times 12. 4. 4 of 56 is 8 times what part of 3 times 12? Such a part of 3 times 12 as 3 times 12 is contained times in of 4 times of 56.4 of 56 is 8, and 4 times 8 is 32; of 32 is 43; 3 times 12 is 36, and 36 is contained in 4, or times. Therefore of 56 is 8 times , or, of 3 times 12. 1 5. of 3 times 12 is of what number? of 3 times 12 is of 7 times 4 of 8 times 0. 3 times 12; of 36 is 4, and 8 times 4 is 32; of 32 is 8, and 7 times 8 is 56. Therefore of 3 times 12 is of 56. 6. of 3 times 12 is how many sevenths of 56? As many sevenths of 56 as 4 of 56 is contained times in 8 times of 3 times 12. 3 times 12 is 36, of 36 is 4, and 8 times 4 is 32; 4 of 56 is 8, and 8 is contained in 32, 4 times. Therefore of 3 times 12 is of 56. 7. of 3 times 12 is 4 times what part of 56 ? Such a part of 56 as 56 is contained times in of 8 times of 3 times 12; 3 times 12 is 36; 4 of 36 is 4, and 8 times 4 is 32; 14 of 32 is 8, and 56 is contained in 8, or times. Therefore of 3 times 12 is 4 times 1 , or, of 56. 71. 1. 4 of 35 is of how many thirds of 18? Expressed by signs on the blackboard thus : ×35×3×18? Of as many thirds of 18 as of 18 is contained times in 9 times of 4 times of 35; 4 of 35 is 5, and 4 times 5 is 20; of 20 is 4, and 9 times 4 is 36; of 18 is 6, and 6 is contained in 36, 6 times. Therefore of 6 thirds of 18. 2. # of 35 is of of what number ? ? of 35 is of 35; of 35 is 5, and 4 times 5 is 20; of 20 is 4, and 9 times 4 is 36; of 36 is 6, and 3 times 6 is 18. Therefore of 35 is of of 18. ×35×× of 35 is off of 3 times of 9 times of 4 times 3. of 35 is 4 of 6 times what part of 18? ×35××18? of 35 is of 6 times such a part of 18 as 18 is con tained times in of 9 times of 4 times of 35. of 35 is 5, and 4 times 5 is 20; 1 of 20 is 4, and 9 times 4 is 36; of 36 is 6, and 18 is contained in 6, or times. Therefore of 35 is of 6 times, or, of 18. 6 4. 4 of 35 is how many ninths of of 18? ×35×× 18? of 35 is as many ninths of of 18 as of 6 times of 35 is of 18 is contained times in 4 times of 35. 5, and 4 times 5 is 20; 1 of 18 is 6, and 6 times 6 is 36; 4 of 36 is 4, and 4 is contained in 20, 5 times. Therefore of 35 is of of 18. 5. 4 of 35 is 5 times what part of of 18 ? 4×35×× 18? of 35 is 5 times such a part of of 18 as 6 times of of 18 is contained times in of 4 times of 35. 35 is 5, and 4 times 5 is 20; 1 of 20 is 43 of 18 is 6, and 6 times 6, is 36; 36 is contained in 4,3 or times. Therefore of 35 is 5 times, or, off of 18. 6. of of 18 is of what number ? of of 18 is of 7 times 4 of 5 times of 6 times of 18; of 18 is 6, and 6 times 6 is 36; 1 of 36 is 4, and 5 times 4 is 20; 1 of 20 is 5, and 7 times 5 is 35. Therefore off of 18 is of 35. 7. of of 18 is how many sevenths of 35? ק×18× 35? of of 18 is as many sevenths of 35 as of 35 is contained times in 5 times of 6 times of 18; of 18 is 6, and 6 times 6 is 36; 1 of 36 is 4, and 5 times 4 |