(4) _Least common multiple._--The whole set of bonds involved in learning 'least common multiple' should be left out. In adding and subtracting fractions the pupil should _not_ find the least common multiple of their denominators but should find any common multiple that he can find quickly and correctly. No intelligent person would ever waste time in searching for the least common multiple of sixths, thirds, and halves except for the unfortunate traditions of an oversystematized arithmetic, but would think of their equivalents in sixths or twelfths or twenty-fourths or _any other convenient common multiple_. The process of finding the least common multiple is of such exceedingly rare application in science or business or life generally that the textbooks have to resort to purely fantastic problems to give drill in its use. (5) _Greatest common divisor._--The whole set of bonds involved in learning 'greatest common divisor' should also be left out. In reducing fractions to lowest terms the pupil should divide by anything that he sees that he can divide by, favoring large divisors, and continue doing so until he gets the fraction in terms suitable for the purpose in hand. The reader probably never has had occasion to compute a greatest common divisor since he left school. If he has computed any, the chances are that he would have saved time by solving the problem in some other way! The following problems are taken at random from those given by one of the best of the textbooks that make the attempt to apply the facts of Greatest Common Divisor and Least Common Multiple to problems.[6] Most of these problems are fantastic. The others are trivial, or are better solved by trial and adaptation. 1. A certain school consists of 132 pupils in the high school, 154 in the grammar, and 198 in the primary grades. If each group is divided into sections of the same number containing as many pupils as possible, how many pupils will there be in each section? 2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he desires to put into the least number of boxes of the same capacity, without mixing the two kinds of grain. Find how many bushels each box must hold. 3. Four bells toll at intervals of 3, 7, 12, and 14 seconds respectively, and begin to toll at the same instant. When will they next toll together? 4. A, B, C, and D start together, and travel the same way around an island which is 600 mi. in circuit. A goes 20 mi. per day, B 30, C 25, and D 40. How long must their journeying continue, in order that they may all come together again? 5. The periods of three planets which move uniformly in circular orbits round the sun, are respectively 200, 250, and 300 da. Supposing their positions relatively to each other and the sun to be given at any moment, determine how many da. must elapse before they again have exactly the same relative positions.