COMPOUND DIVISION.
For the operation of which the rule is : when the dividend only consists of differ ent denominations, divide the higher de nomination, and reduce the remainder to the next lower, taking in the given num ber of that denomination, and continue the division. When the divisor is not greater than 12, we proceed as before in short division.
Examples.
s. d. L. s.
5)84 .. 3 .. 9 11)976 .. 13 ..
Ans.16 16 9 Ans.88 15 91-8 lb. oz. &rots. cwt. qr. lb. oz.
8)994 .. 4.. 8 12)45 .. 2 .. 18 .. 8 124 .. 3 .. 11 3 .. 3 .. 6 .. 3 .. 5-4 When the divisor is greater than 12, the operation is performed by long divi sion.
E.rample.
L. s. d.
Divide 8467 .. 16 .. 8 by 659.
L. s. d.
659)8467 .. 16 .. 8(12 659 1877 1318 . 559 20 659)11196(16 659 4606 3954 .652 12 659)7832(11 7249 .583 4 659)2332(1 1977 .353 Ans. 12 .. 16 .. 111 tV, In connection with the rule of division, we may notice another kind of Reduc tion, so called, though improperly, as by it is meant to bring smaller denomina tions into larger : as pence into pounds, Dr drams into hundred weights, &c ; for which the rule is : divide by the parts of each denomination from that given to the highest sought : the remainders, if any, will be of the same name as the quantity from which they were reduced.
Kramples.
1. In 415684 farthings, how many pounds sterling.
4)415684 12)103921 2,0)866.0-1 Ans. /. 433 .. tt .. 1 2. Ilow many pounds troy are there in 67890 dwts.
2,0)6789,0 12)3394 —10 282 .. 10 .. 10 lb. oz. dints.
Ans. 282 .. 10 .. 10.
---- Before we conclude this article we may observe, that, in computations which re quire several steps, it is often immaterial what course we follow. Some methods may be preferable to others, in point of ease and brevity ; hut they all lead to the same conclusion. In addition or subtrac tion, we may take the articles in any order. When several numbers are to be multipli ed together, we may take the factors in any order, or we may arrange them into several classes ; find the product of each class, and then multiply the products to gether. When a number is to be divided
by several others, we may take the divi sors in any order, or we may multiply them into one another, and divide by the product ; or we may multiply them into several parcels, and divide by the pro ducts successively. Finally, when multi plication and division are both required, we may begin with either; and when both are repeatedly necessary, we may collect the multipliers into one product, and the divisors into another ; or we may collect them into parcels, or use them singly ; and that in any order. To begin with multiplication is generally the better mode, as this order preserves the account as clear as possible from fractions.
We have hitherto given the most ready and direct method of proving the forego ing examples, but there is another, which is very generally used, called "casting out the 9's," which depends on this prin ciple : That if any number be divided by 9, the remainder is equal to the remainder obtained, when that sum is divided by 9. For instance, if 87654 be divided by 9, there is a remainder of 3 ; and if 8, 7, 6, 5, 4, be added together, and the sum 30 be divided by 9, there will be likewise a remainder of 3.
l'o cast out the 9's of any number, add the figures, and when the sum is equal to or more than 9, pass by the 9, and pro ceed with the remainder: thus, in casting out the 9's of 56774 we say 5 and 7 are 12, 3 above 9 ; 3 and 7 are 10, 1 above 9, 1 and 7 are 8; 8 and 4 are 12, 3 above 9 : the last remainder is to be put down, and then proceed to the other lines, accord ing to the following rules.
To prove Addition. Cast out the 9's of the several articles, carrying the results to the following articles, and cast them out of the sum total; if the operations be correct, the two remainders, if any, will agree.
Example. 845 346 784 Sum 1975 Here, in casting out the 9's of the three lines to be added, I find a remainder of 4; there is also a remainder of 4 upon cast ing out the 9's of the sum.