sible by 8 or by 125, is itself divisible by 8 or by 125; for, as one thousand is divisible by 8 or by 125, any number of thousands is equally divisible. Every number terminated on the right by a cipher or by 5, is divisible by 5; because 10, or any number of 10s, is divisible by 5. Every number, the sum of whose significant figures is divisible by 3 or by 9, is itself divisible by 3 or by 9. Thus, the number 426,213 is divisible by 9, because, as 100,000 consists of 99,999 and 1, 100,000, divided by 9, will leave a remainder of 1, and 400,000, a remainder of 4. For a like reason, 20,000 will leave a remainder of 2, 6000, of 6, 200, of 2, &c.; that is, each of the figures in the above or any other number, if divided by 9, will leave a remainder equal to the simple value of the figure; and, if the sum of all the remainders is divisible by 9, of course, the whole number is divisible by 9. But the sum of the above remainders is divisible by 9; therefore, the number is divisible by 9. The same demonstration will answer for 3. Therefore, every number, the sum of whose significant figures is divisible by 9 or by 3, is itself divisible by 9 or by 3. And every number divisible by 9 is also divisible by 3, although every number divisible by 3 is not divisible by 9. Thus we can tell, almost by a glance, whether any number can be divided by 2, 3, 4, 5, 8, 9, 10, 25, or 125. T. What will be the remainder, if we divide 326 by 25? Tell by inspection. T. What will be the remainder, if any, of dividing 6892736 by 4? by 9? by 3? As soon as the teacher has fully explained to his class, on the blackboard, all that has been stated above, relative to fractions, and ascertained that it is thoroughly understood, the pupils should practise, on the slate, the reduction of fractions to their lowest terms, and the teacher proceed as follows, on the blackboard. An integer, or whole number, may be represented in a fractional form, by placing a unit under it, as its denomi nator. Thus, 6 or 8 may be written f, . An integer may also be represented in connexion with a fraction. Thus, contain the integer 2 and more. For, as g are equal to 1, 6-2; therefore, 23; or this result may be obtained, by actually performing the division indicated by . Thus, 8)19 Every fraction, then, whose numerator exceeds its denominator, contains a whole number, which may be obtained by performing the division indicated. Such fractions are called improper fractions; and, when the division is performed, (as above,) the quotient, if it consist of an integer and fraction, is called a mixed number. The improper fraction may evidently be reproduced, by multiplying the whole number by the divisor of the fraction, adding in the numerator, and replacing the denominator of the fraction. [Let the pupils now practise on the slate, the reduction of improper fractions to mixed numbers, and vice versa.] From the table given page 162, we see that a fraction can be multiplied two ways: namely, by multiplying its numerator, or by dividing its denominator; and that it can also be divided in two ways: namely, by dividing its numerator, or by multiplying its denominator. Hence it follows, that multiplication, alone, according as it is performed on the numerator or denominator, is sufficient, both for the multiplication and division of fractions. Thus, multiplied by 5, makes 15, and divided by 5, makes [Here give suitable examples, for practice, both in multiplication and division of fractions by whole numbers.] Multiplication and division of fractions by fractions, may be thus exemplified: Multiply 3 by 4. First, the 4 may be considered a whole number, and mutiplying by it, gives 2; but it is not 4, but the sev enth part of 4; therefore, our 12 is 7 times too much. Accordingly, if we divide by 7, by multiplying the denominator, we shall have 2, the true answer. Hence, the common rule, Multiply the numerators, for a new numerator, and the denominators, for a new denominator. Divide by 3. a 5 Dividing by 3 gives 3; but our divisor is not 3, but the fifth part of 3; therefore, we have divided by a number 5 times too large, and our is 5 times too small. Therefore, multiplying it by 5, gives 35, the correct answer. But, if we mark our two first factors a, and our two second factors b, it will appear, that, if we had reversed the divisor, we might have proceeded, as in multiplication. Hence, our common rule in division, Reverse the divisor, and proceed as in multiplication. These two demonstrations should be repeated on the blackboard, by every member of the class, till the subject has become perfectly familiar. The term fractions of fractions is sometimes given, to expressions like this: of, which signifies the product of such fractions. That the word of, here, represents multiplication, appears from expressions with which all are familiar. For instance; every one knows that of 6 is 2. Now, by multiplying, we have 2. Again, of 12 is 3; multiplying gives us 12=3. = The process of multiplying fractions may frequently be very much abbreviated. Thus, in the question of of of of % of of how much? = α f The whole affair consists merely in striking out factors and divisors that are equal, and may be thus demonstrated. It would be a waste of time to multiply marked b, and then to divide by 5, also marked b. The by 5, same reasoning allows us to strike out the two 8s, marked f, the two 9s, marked e; and, going on thus, we have, at length, nothing left but fo, which is the true answer. The same abbreviation may be practised in division, after we have reversed the divisor, and thus changed the division into multiplication. Many cases occur, in which abbreviations may be made, although the whole number may not, at once, be struck out, or even though all the factors may never disappear. No. 1 is thus abbreviated: divide a 2 and a 8 by 2; which gives a new denominator 4; then strike out the two denominators b 4 and b 4, and the numerator b 16. In No. 2, divide the two a s by 4, the two b s by 7, and the two cs by 4. [Here give examples, for practice, in multiplication and division of fractions by fractions, which should be examined by the teacher, to see if all the possible abbreviations have been made.] Many young students find it difficult to comprehend, why multiplying by a fraction should lessen, and dividing increase a number. But this difficulty may be removed, by explaining, that multiplying by a number is taking it as many times as there are units in the multiplier; that, consequently, when we multiply by 2, the product will be twice the multiplicand; when we multiply by 1, the product will be once the multiplicand; and when we multiply by, the product will be half the multiplicand, &c. In division, when the divisor is 5, the dividend is five times the quotient; consequently, the quotient is one fifth of the dividend. And if the divisor be a fraction,, for instance, the dividend can be but half the quotient, or the quotient double of the dividend. Let it be remarked here, once for all, that all the expla nations will appear somewhat abstruse and difficult, when merely given in words; but, when illustrated by numbers, on the blackboard, as these and what follow are intended to be, they will readily be understood by any intelligent child. Addition and Subtraction of Vulgar Fractions.—It will be recollected, that, in multiplication and division of integers, it is not necessary that the multiplicand and multiplier, or the divisor and dividend, should be of the same denomination; for 6 thousand could be multiplied or divided by 3 units, as readily as 6 units could. The same remark applies to fractions; for can be multiplied or divided by, just as easily as by. This, however, is not the case with addition and subtraction, either in integers or fractions. For 2 units cannot be added to 3 hundred, in any other manner, than by naming or writing them, one after the other, as they do not make 5 of any denomination; and, in like manner, and cannot be added, until they are changed into the same denomination. In subtraction, 6 units cannot be taken from 8ty, without first reducing one or more of the ty, into units. In fractions, also, cannot be taken from until their denomination is made the same, by changing the form of one or both. Thus it appears, that, although we can multiply or divide fractions, however they may differ in denomination; it is always necessary that they should have the same denominator before they can be added or subtracted; and that the same remark applies to integers. It becomes necessary, then, to point out the simplest mode of reducing fractions to the same denominator. Suppose, then, we wished to add or subtract to or from If each term of the first fraction be multiplied by 5, the denominator of the second; and each term of the second be multiplied by 3, the denominator of the first, the value of both will be unchanged, and they will have the same denominator 15, namely; 19 and 12. In this form, there is no difficulty, either in adding or subtracting them; and the process, which has been applied to the above fractions, may be applied to any other. In general, then, to reduce any two fractions to the same denominator, the two terms of each of them must be mul |