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Art. 110. Ex. 1. How many barrels of flour, at 8 dollars per barrel, can you buy for 56 dollars ?
Analysis. Since flour is 8 dollars a barrel, it is obvious you can buy 1 barrel as often as 8 dollars are contained in 56 dollars ; and 8 dolls. are contained in 56 dolls. 7 times. Ans. 7 barrels.
Ex. 2. A man wished to divide 72 dollars equally among 9 bego gars :
dollars would each receive ? Solution.—Reasoning as before, each beggar would receive as many dollars as 9 is contained times in 72; and 9 is contained in 72, 8 times. Ans. 8 dollars.
Obs. The learner will at once perceive that the object in the first example, is to find how many times one number is contained in another; and that the object of the second, is to divide a given number into equal parts, but its solution consists in finding how many times one number is contained in another, and is the same in principle as that of the first.
111. The Process of finding how many times one number is contained in another, is called DIVISION.
The number to be divided, is called the dividend.
The number obtained by division, or the answer to the question, is called the quotient. It shows how many times the divisor is contained in the dividend. Hence, it may be said,
112. Division is finding a quotient, which multiplied into the divisor, will produce the dividend.
Note.—The term quotient is derived from the Latin word quoties, which sig. nifies how often, or how many times.
QUEST.-111. What is division? What is the number to be divided called ? The num. ber by which we divide? What is the number obtained called? What does the quotient show! 112. What then may division be said to be ?
113. The number which is sometimes left after division, is called the remainder. Thus, when we say 5 is contained in 38, 7 times, and 3 over, 5 is the divisor, 38 the dividend, 7 the quotient, and 3 the remainder.
Obs. 1. The remainder is of the same denomination as the dividend; for, it is a part of it.
2. The remainder is always less than the divisor; for, if it were equal to, or greater than the divisor, the divisor could be contained once more in the dividend.
114. It will be perceived that division is similar in principle to subtraction, and may be performed by it. For instance, to find how many times 7 is contained in 21, subtract 7 (the divisor) continually from 21 (the dividend), until the latter is exhausted; then counting these repeated subtractions, we shall have the true quotient. Thus, 7 from 21 leaves 14; 7 from 14 leaves 7; and 7 from 7 leaves 0. Now by counting, we find that 7 has been taken from 21, 3 times; consequently, 7 is contained in 21, 3 times. Hence,
Division is sometimes defined to be a short way of performing repeated subtractions of the same number.
OBs. 1. It will be observed that division is the reverse of multiplication. Multiplication is the repeated addition of the same number; division is the repeated subtraction of the same number. The product of the one answers to the dividend of the other; but the latter is always given, while the former is required.
2. When the dividend denotes things of one denomination only, the operation is called Simple Division.
Ex. 3. How many hats, at 2 dollars apiece, can be bought for 4862 dollars ? Operation.
We write the divisor on the left of the divi
dend with a curve line between them; then, 2) 4862
beginning at the left hand, proceed thus : 2 is Quot. 2431
contained in 4, 2 times. Now, since the 4 de
QUEST.-113. What is the number called which is sometimes left after division ? Obs. Of what denomination is the remainder? Why? Is the remainder greater or less than the divisor? Why? 114. To what rule is division similar in principle ?
Obs. Of what is division the reverse ? When the dividend denotes things of one denomination only, what is the operation called ?
notes thousands, the 2 must be thousands; we therefore write it in thousands' place, under the figure divided. 2 is contained in 8, 4 times; and as the 8 is hundreds, the 4 must also be hundreds; hence we write it in hundreds’ place, under the figure divided. 2 in 6, 3 times; the 6 being tens, the 3 must also be tens, and should be set in tens' place. 2 in 2, once; and since the 2 is units, the 1 is a unit, and must therefore be written in units' place. The answer is 2431 hats.
115. When the process of dividing is carried on in the mind, and the quotient only is written, as in the last example, the operation is called SHORT DIVISION.
116. The reason that each quotient figure is of the same order as the figure divided, may be shown in the following manner :
Having separated the dividend Analytic Solution. of the last example into the orders 4862=4000+800+60+2 of which it is composed, we per
2)4000+800+60+2 ceive that 2 is contained in 4000, 2000+400+30+1 2000 times; for 2 X 2000=4000,
Again, 2 is contained in 800, 400 times; for 2 X 400=800, &c. Ans. 2431.
Ex. 4. A man left an estate of 209635 dollars, to be divided equally among 4 children: how much did each receive ?
Since the divisor 4, is not contained in Operation. 2, the first figure of the dividend, we find 4)209635
times it is contained in the first Ans. 524088 dolls. two figures. Thus, 4 is contained in 20,
5 times; write the 5 under the 0. Again, 4 is contained in 9, 2 times and 1 over; set the 2 under the 9. Now, as we have i thousand over, we prefix it mentally to the 6 hundreds, making 16 hundreds; and 4 in 16, 4 times. Write the 4 under the 6. But 4 is not contained in 3, the next figure, we therefore put a cipher in the quotient, and prefix the 3 to the next figure of the dividend, as if it were a remainder. Then 4 in 35, 8 times and 3 over; place the 8 under the 5, and setting the remainder over the divisor thus &, place it on the right of the quotient,
Note. To prefix means to place before, or at the left hand.
117. When the divisor is not contained in any figure of the dividend, a cipher must always be placed in the quotient.
Obs. The reason for placing a cipher in the quotient, is to preserve the true local value of each figure of the quotient. (Art. 116.)
118. In order to render the division complete, it is obvious that the whole of the dividend must be divided. But when there is a remainder after dividing the last figure of the dividend, it must of necessity be smaller than the divisor, and cannot be divided by it. (Art. 113. Obs. 2.) We therefore represent the division by placing the remainder over the divisor, and annex it to the quotient. (Art. 25.)
Obs. 1. The learner will observe that in dividing we begin at the left hand, instead of the right, as in Addition, Subtraction, and Multiplication. The reason is, because there is frequen a remainder in dividing a higher order, which must necessarily be united with the next lower order, before the division can be performed.
2. The divisor is placed on the left of the dividend, and the quotient under it, merely for the sake of convenience. When division is represented by the sign : , the divisor is placed on the right of the dividend; and when represented in the form of a fraction, the divisor is placed under the dividend.
Ex. 5. At 15 dollars apiece, how many cows can be bought for 3525 dollars ?
Operation. Having written the divisor on the left of
Divisor. Divid. Quot. the dividend as before, we find that 15 is 15) 3525 (235 contained in 35, 2 times, and place the 2 on
30 the right of the dividend, with a curve line
52 between them. We next multiply the di
45 visor by this quotient figure, place the prod
75 uct under the figures divided, and subtract
75 it therefrom. We now bring down the next figure of the dividend, and placing it on the right of the remainder 5, we perceive that 15 is contained in 52, 3 times. Set the 3 on the right of the last quotient figure, multiply the divisor by it, and subtract the product from the figures divided as before. We then bring down the next, which is the last figure of the dividend, to the right of this remainder, and finding 15 is contained in 75, 5 times, we place the 5 in the quotient, multiply and subtract as before. The answer is 235 cows.
119. When the result of each step in the operation is written down, as in the last example, the process is called LONG Division. Long Division is the same in principle as Short Division. The only difference between them is, that in the former, the result of each step in the operation is written down, while in the latter, we carry on the process in the mind, and simply write the quotient.
Obs. 1. When the divisor contains but one figure, the operation by Short Division is the most expeditious, and therefore should always be practiced; but when the divisor contains two or more figures, it will generally be the most convenient to use Long Division.
2. To prevent mistakes, it is advisable to put a dot under each figure of the dividend, when it is brought down.
3. The French place the divisor on the right of the dividend, and the quotient below the divisor,* as seen in the following example.
Ex. 6. How many times is 72 contained in 5904 ?
5904 (72 divisor. The divisor is contained in 590, the
82 quotient. first three figures of the dividend, 8 144
times. Set the 8 under the divisor, 144
multiply, &c., as before.
Ex. 7. How many times is 435 contained in 262534 ?
Since the divisor is not contained 435)262534(603111 Ans. in the first three figures of the divi2610".
dend, we find how many times it is 1534
contained in the first four, the few1305
est that will contain it, and write 229 rem.
the 6 in the quotient; then multi
QUEST.-115. What is short division ? 119. What is long division ? What is the difference between them ?
* Eléments D'Arithmétique, par M. Bourdon. Also, Lacroix's Arithmetic, translated by Professor Farrar.