Again, "If the share of three boys is five apples, how many will be sufficient for nine?"
Our pupil will attempt to proceed as in the former question, and will begin by endeavouring to find out the share of one of the three boys; but this is not quite so easy; he will see that each is to have one apple, and part of another; but it will cost him some pains to determine exactly how much. When at length he finds that one and two-thirds is the share of one boy, before he can answer the question, he must multiply one and two-thirds by nine, which is an operation _in fractions_, a rule of which he at present knows nothing. But if he begins by multiplying the second, instead of dividing it previously by the first number, he will avoid the embarrassment occasioned by fractional parts, and will easily solve the question.
3 : 5 : 9 : 15 Multiply 5 by 9 -- it makes 45
which product 45, divided by 3, gives 15.
Here our pupil perceives, that if a given number, 12, for instance, is to be divided by one number, and multiplied by another, _it will come to the same thing_, whether he begins by dividing the given number, or by multiplying it.
12 divided by 4 is 3, which multiplied by 6 is 18; And 12 multiplied by 6 is 72, which divided by 4 is 18.
We recommend it to preceptors not to fatigue the memories of their young pupils with sums which are difficult only from the number of figures which they require, but rather to give examples _in practice_, where aliquot parts are to be considered, and where their ingenuity may be employed without exhausting their patience. A variety of arithmetical questions occur in common conversation, and from common incidents; these should be made a subject of inquiry, and our pupils, amongst others, should try their skill: in short, whatever can be taught in conversation, is clear gain in instruction.
We should observe, that every explanation upon these subjects should be recurred to from time to time, perhaps every two or three months; as there are no circumstances in the business of every day, which recall abstract speculations to the minds of children; and the pupil who understands them to-day, may, without any deficiency of memory, forget them entirely in a few weeks. Indeed, the perception of the chain of reasoning, which connects demonstration, is what makes it truly advantageous in education. Whoever has occasion, in the business of life, to make use of the rule of three, may learn it effectually in a month as well as in ten years; but the habit of reasoning cannot be acquired late in life without _unusual_ labour, and uncommon fortitude.