5. Divide 9876543210 by 8. 6. Divide 1357975313 by 9. Ans. 1234567901 7. Divide 3217684329765 by 17. Ans. 1892755488091 8. Divide 3211473 by 27. 9. Divide 1406373 by 108. Ans. 118943 I 2 Ans. 130211. 10. Divide 293839455936 by 8405. Ans. 34960078-340 11. Divide 4637064283 by 57606. Ans. 804963707. CONTRACTIONS. 1. To divide by any number with cyphers annexed. Cut off the cyphers from the divisor, and the same number of digits from the right hand of the dividend; then divide, making use of the remaining figures, as usual, and the quotient is the answer; and what remains, writ` ten before the figures cut off, is the true remainder. EXAMPLES. 1. Divide 310869017 by 7100. 71,00)3108690,17(4378433 the quotient, 284 * The reason of this contraction is easy to conceive: for the cutting off the same figures from each, is the same as dividing each of II. When the divisor is the product of two or more small num bers in the table. RULE.* Divide continually by those numbers, instead of the whole divisor at once: EXAMPLES. of them by 10, 100, 1000, &c. and it is evident, that as often as the whole divisor is contained in the whole dividend, so often must any part of the divisor be contained in a like part of the dividend. This method is only to avoid a needless repetition of cyphers, which would happen in the common way, as may be seen by working an example at large. * This follows from contraction the second in multiplication, of which it is only the converse; for the third part of the half of any thing is evidently the same as the sixth part of the whole; and so of any other number. I have omitted saying any thing, in the rule, about the method of finding the true remainder; for as the learner is fupposed, at present, to be unacquainted with the nature of frac tions, it would be improper to introduce them in this part of the work, especially as the integral quotient is sufficient to answer most of the purposes of practical division. However, as the quotient is incomplete without this remainder, and, in some computations, it is necessary it should be known, I shall here shew the manner of find ing it, without any assistance from fractions. RULE. Multiply the quotient by the divisor, and subtract the product from the dividend, and the result will be the true remainder. The truth of this is extremely obvious; for if the product of the divisor and quotient, added to the remainder, be equal to the dividend, their product taken from the dividend must leave the remainder. The E The rule which is most commonly made use of is this: RULE. Multiply the last remainder by the preceding divisor, or last but one, and to the product add the preceding remainder ; multiply this sum by the next preceding divisor, and to the product add the next preceding remainder; and so on, till you have gone through all the divisors and remainders to the first. III. To perform division more concisely than by the general rule. RULE.* Multiply the divisor by the quotient figures as before, and subtract each figure of the product as you produce it, always remembering to carry as many to the next figure as were borrowed before, EXAMPLES. 1. Divide 3104675846 by 833. 833)3104675846(3727101 the quotient. 6056 To explain this rule from the example, we may observe, that every unit of the first quotient may be looked upon as containing 9 of the units in the given dividend; consequently every unit, that remains, will contain the same; therefore this remainder must be multiplied by 9, in order to find the units it contains of the given dividend. Again, every unit in the next quotient will contain 4 of the preceding ones, or 36 of the first, that is, 9 times 4 ; therefore what remains must be multiplied by 36; or, which is the same thing, by 9 and 4 continually. Now this is the same as the rule; for instead of finding the remainders separately, they are reduced from the bottom upward, step by step, to one another, and the remaining units of the same class taken in as they occur. *The reason of this rule is the same as that of the general rule. |