Multiply the dividend by the same figure that the divisor is multiplied by, and then divide it.

To divide by a number which becomes 10, 100, or 1000, &c. by adding one of its aliquot parts to it, or by subtracting one of its aliquot parts from it, add the same aliquot part of the dividend to itself, or deduct it from itself, which was added to, or subtracted from the divisor, and then divide the new dividend by the new divisor.

The reason of these contractions are so apparent, that they require no explanation.

Those who are tolerably expert at performing division, by the foregoing methods, should endeavour to divide any number, by another, without putting down any more figures than the quotient and the several remainders, obtained by subtracting the different products of the divisor, by each of the quotient figures, as in the following example :

432)12345678(28577 Quotient.

3705

2496

3367

3438

414 Remainder.

This is sometimes called the Italian method, and differs only from the common method, in performing the necessary multiplications and subtractions mentally, instead of visibly.

Instead of bringing down each figure of the dividend, the arrangement would be neater to put down the figures only that remain, after subtracting the different products, and drawing a line through, or making a mark under each figure of the dividend, as it is used in the division.

The last example would then be arranged thus:

432)12345678(28577 Quotient.

3709604

24341
334

METHODS OF PROVING MULTIPLICATION AND

DIVISION.

DIVISION, when compared with MULTIPLICATION, will be found to be exactly opposite in its tendency; hence these two rules mutually prove each other.

1. To prove Multiplication-Divide the product by either factor, the quotient will be the same as the other factor, without any remainder, when both operations are rightly performed.

2. To prove Division-Multiply the quotient and the divisor together; to their product add the remainder, if any; and,if

the operations have been accurately performed, it will make up the dividend.

3. Divide the dividend by the quotient, after having subtracted the remainder, when there is any, and the quotient, thus obtained, will be the same as the divisor, when the operations are accurately performed.

4. Add the remainder, and all the lower lines of figures together, and, if the work is right, the sum will be like the dividend.

This method of proof depends on this principle, that the product of the divisor and quotient, with the remainder added, equals the dividend; now the numbers directed to be added, are the products of the divisor by the several quotient figures, toge ther with the remainder, which ought to make up the dividend, if the operation be rightly performed. This is, perhaps, the best method of proving Division.

3. Cast the 9's out of the divisor, and also out of the quotient, multiply the remainders together, and cast out the 9's from their product, if more than 9; cast the 9's also out of the remainder, and add these two last remainders together, and their sum, or the excess of their sum, above 9, will be equal to the remainder derived from the dividend, after casting out the 9's from it.

The method of proving operations by casting out the 9's depends on the following principles:

If several numbers be separately divided by any divisor, and the respective remainders always added to the next number, the sum of the quotient figures and the last remainder will be the same as that obtained by dividing the sum of these numbers. Thus 14, 17, 25, contain as many 6's, as many 7's, &c. when divided separately, as their sum does, and the remainders are also the same: hence addition may be proved by division.

It is from the correspondence of the remainders, that the proof, by casting out the 9's, is inferred.

If any figure, with ciphers annexed to it, (except 9,) he divided by 9, the quotient will consist of that figure only, repeated as many times as there are ciphers attached to the significant figure, with the same figure as a remainder.

Thus, 50 divided by 9, quotes 5 with a remainder of 5; and 500 divided by 9, quotes 55, with a remainder of 5.

The reason of this will easily be perceived; for every figure, with a cipher annexed, contains exactly as many tens as there are ciphers; it must therefore contain an equal number of 9's and a remainder of an equal number of units.

If any number be divided by 9, the remainder is equal to the remainder of the sum of the figures, composing that number, when divided by 9.