The Federal Arithmetic; Or, A Compendium of the Most Useful Rules of that Science

CASE 1.

RULE 1. First, seek how many times the divisor is contained in a competent number of the first left hand figures of the dividend, which figures must at least be equal to the divisor, and place the figure in the quotient.

2. Multiply the divisor by this quotient figure; set the product under the left hand figures of the dividend which you divided, and subtract the product therefrom.

3. To the right hand of the remainder, bring down the next figure of the dividend, and inquire how many times the divisor is contained in that number, which, when found, place in the quot ent; but if the number be less than the divisor, place a cypher in the quotient, and bring down another figure of the dividend; then proceed as before, till all the figures in the dividend áre brought down.

PROOF.

Multiply the quotient by the divisor, and if there be a remainder, add it to the product; then, if the sum be equal to the dividend, the work is right.

In this example, I first inquire how many times 36, the divisor, can be had in 153, the three first figures of the dividend, for it cannot be had in 15, and find it to be 4; I place 4 in the quotient, and multiply the divisor 36, by it, and place the product 144, under the number divided 153; then I subtract, and to the remainder 9, bring down 2, the next figure of the dividend, and inquire how many times 36 there are in the number 92, which being found to be 2, I place the figure in the quotient, and proceed in the same manner, till all the figures are brought down. Then to prove the work, I multiply the divisor and quotient together, and to the product add the remainder 27; which sum being equal to the dividend, proves the work to be right.

Note. The remainder after division, is always the numerator to a vulgar fraction, whose denominator is the divisor, and belongs to the quotient; thus, the quotient to the first example is 425527

CASE 2.

When there are cyphers at the right hand of the divisor, cut them off; also cut off the same number of figures at the right hand of the dividend; then divide with the other figures, as before, and place the figures cut off in the dividend, at the right hand of the remainder, for the true remainder.

41600)4900603

123|0)98765(

To divide by 10, 100, 1000, &c. cut off so many figures at the right hand of the dividend, as there are cyphers in the divisor; then the figures at the left hand will be the quotient, and the others the remainder.

When the divisor does not exceed 12, first seek how often the divisor can be had in the first figure; or figures, of the dividend; which, when found, place in the quotient; then, mentally, multiply the divisor by it, and subtract the product from the divided figures of the dividend ; join the next figure of the dividend to the remainder, and inquire how many times the divisor can be had in that number, which place in the quotient, and proceed as before, till all the figures in the dividend are divided.

Note.-The quotient may be set underneath the dividend.