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from the division of an inferior, and as the remainder resulting from the division of any superior order is (48) easily resolved into a number of units of any of the inferior orders, we always begin division at the left hand.
To divide a number consisting of several figures by any number not exceeding 12.
1. Take the smallest number of the left-hand figures of the dividend that will contain the divisor: that is, take as many as are contained in the divisor, or, when this is not sufficient, (121,) only one figure more.
2. Seek how often the divisor is contained in this part.
3. Place the number of times, which cannot exceed 9, in the quotient.
4. Multiply the divisor by the quotient figure. 5. Subtract the product, thus found, from the part divided.
6. To the right of the remainder annex the next figure of the principal dividend towards the right, and proceed to take all the steps, as before.
Having taken a number of the left-hand figures of the dividend sufficient to contain the divisor, we place the number of times that the divisor is contained in this part, in the quotient, under the unit figure of the part taken. Now this quotient figure, being (150) of the same order as the unit figure of the part divided, must have as many figures on the right of it as there are figures remaining in the dividend. For this reason, when any of the partial dividends is too small to contain the divisor, we must place a cipher in the quotient, annex one more figure to the partial dividend, (if any remain,) and proceed as before.
152. As the federal coin of the United States is constructed upon the same principle as ordinary numbers, ten units of each lower order being equal to one unit of the next higher, (see Table of Federal Money,) we shall take an example in this coin to illustrate the division of those numbers. Suppose we have 71436 mills to divide equally among
6 persons, we distinguish the different orders of Federal Money in this number, thus: the lowest order of units, expressed by the figure 6 is mills : each unit of the next order, on the left, being ten of those, is one cent; the figure 3 is, therefore, 3 cents. Again, each unit in the 4, being ten cents, is one dime: the 4 is, therefore, 4 dimes. The 1, being ten dimes, is one dollar. Lastly, as each unit in the 7 is ten dollars, or one eagle, the 7 is 7 eagles. To perform the operation, we
place the divisor 6 on the left of the dividend and draw a line underneath, thus :
Divisor 6) 71436 dividend.
11906 quotient. Then, beginning at the left, as 6 is contained in 7 once, we write 1 in the quotient, under 7. This 1 shows that each person will receive one eagle. Then, as there is one eagle remaining to be divided, we reduce this to dollars, the next order on the right, and connecting the amount, 10 dollars, with the 1, we have 11 dollars. We then say, 6 in 11, once and 5 over; we write 1 under 1, which shows that each person will receive one dollar. Then, as there are 5 dollars remaining to be divided, we reduce these to dimes, and, connecting the amount 50 with the 4 dimes on the right, we have 54 dimes. We then
say, 6 in 54 nine times, and write 9 under 4. This 9 shows that each person will receive 9 dimes. Then, as there is no remainder, we proceed to the next order; and, as 6 is not contained in the 3 cents, we say 6 in 3, no times and 3 over: we write 0 under 3, which 0 shows that there are no units of this order in the share of each person. We then reduce the 3 cents to mills; the amount 30, together with the 6 mills, makes 36 mills: we then say, 6 in 36, six times, and, writing 6 underneath, the operation is finished, each person's share being 11906 mills, or 1 eagle, 1 dollar, 9 dimes, and 6 mills : or (48) eleven dollars, ninety cents, and six mills. The latter is the usual method of reading.
153. As the remainder is always tens with regard to the next figure on the right, (48,) we have only to suppose it placed on the left of that figure, and read the number as if it stood alone.
11160900 : 12 = 930075
11160900 proof. (See Art. 141.) Here, as the two first figures of the dividend are not sufficient to contain the divisor, we take three, and say, 12 in 111, nine times and 3 over, writing 9 under the unit figure of the
We then suppose the remainder 3 to be placed on the left of the next figure 6, and say, 12 in 36, three times,
writing 3 under 6: we then say, 12 in 0, no times, writing a under 0; also 12 in 9, no times, and 9 over, writing 0 under 9; then, 12 in 90, seven times and 6 over; and, lastly, 12 in 60, five times, writing the quotient figure as before.
Examples. 1. 13506 ; 2
5. 292236 : 6 2. 25476 ;3
6. 608176 ; 7 3. 21868 : 4
7. 608176 4. 195425 ; 5
8. 842256 ;9= 9. Divide 47900160 by each of the numbers 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, and find the difference between the sum of the quotients and one trillion.
Answer. Nine hundred and ninety-nine billions, nine hundred and four millions, forty-five thousand, eight hundred and eighty-eight.
154. Division is the converse or contrary of multiplication; consequently, the one destroys the effect of the other. Hence, a number multiplied and divided by the same number is still the same.
Operation, 8 Thus : 8 X 7:7=8 or
For, (59 and 63,) 8X7=7+7+7+7+7+7+7+7: and it is evident that, from the sum of 8 sevens, we can subtract 7 eight times, which is, in substance, what we do in the division. Wherefore, the multiplication and division of a given number by one and the same number amounts to this, namely, the addition of the multiplier to itself as often as there is a unit in the multiplicand, and the subtraction of the multiplier from the sum the same number of times. Also, let us observe, that 8 X 7=7 is the same as 8:7 X 7. For, the first signifies the seventh part of 7 times 8, which is evidently 8; and the last 7 times the seventh part of 8, which is also evidently 8. The same reasoning will apply to all such cases : wherefore it is of no consequence, as to the result, which of the two operations is performed first. Lastly, as the number upon which we operate is not affected, both operations may be spared.
155. If we divide the product of two numbers by either of them, (60,) the quotient will be the oner. Also, by the above, a number multiplied and divided by the same number is still the same. Therefore, having found the product of 563 x 5
2815, if we divide this product by 5, we shall have 563 for the quotient, thus :
Here, as 2 is not sufficient, we say, 5 in 28, five times, (the double of the tens figure 2, plus 1, because 8 contains one more 5,) and 3 over. This 3 hundreds or 30 tens makes, with the 1 ten, 31 tens. We then say, 5 in 31, (doubling the 3,) six times, and 1 over. This 1 with the 5 makes 15. Then, 5 in 15, three times, and the work is done; the quotient being 563, as was proposed. Or thus : reading the binomial value, (49,) which is 281 tens and 5 units; we double the tens, for the number of times 5 is contained in them, which gives 562, to which adding 1, for the 5 units, we have 563, as before.
The student may observe, that the figures 1 and 3, which we carry in multiplying 563 by 5, are the same which we bring back again in dividing 2815 by 5.
156. We have seen (149) that the remainder of a division may be any number which is less than the divisor; and (146) that the division of this remaining part of the dividend is expressed by placing the divisor underneath it, with a line between : also, (145,) that this quotient, being less than a unit, is a fraction. The upper number, or dividend, is called numerator, because it shows the number of parts in the fraction; and the lower number, or divisor, is called denominator, because it determines the name of those parts. The quotient of any division, therefore, which has left a remainder, is rendered complete by placing the fraction, here described on the right of the unit
figure of that quotient. It is plain (60) that if we multiply a fraction by its denominator, the result or product will be the numerator : this is the same as to take away the denominator. Wherefore, to prove by multiplication, multiply the integral part of the quotient by the divisor, and to the product add the remainder, or numerator of the fraction, which will give the dividend when the work is right.
The student will easily understand this by considering, that the integral part of the quotient shows the number of times the
divisor has been subtraeted, and that when a fraction is multiplied by its denominator the result is the numerator. 5734999 = 7=8192854 7) 5734999
Note. A number which is com8192854
posed of a whole number and a frae7
tion is called a mixed number. Thus,
5, 6%, &c., are mixed numbers. 5734999 Here, writing as usual the numbers signified by the words in italics, we say, 7 in 57, eight times, and l over: 7 in 13, once, and 6 over: 7 in 64, nine times, and 1 over : 7 in 19, twice, and 5 over: 7 in 59, eight times, and 3 over : 7 in 39, five times, and 4 over, and placing this 4 over 7, on the right of the unit's figure 5, the quotient is complete. To prove the work, we multiply the quotient by 7, thus : 7 times foursevenths is 4: 7 times 5 is 35, and 4 is 39; nine and go 3, &c., as usual, and having reproduced the dividend, we consider the division correct. In the same manner the student may prove the following examples : 1. 76947385 = 2
6. 7924865731 :7= 2. 1062047953 ; 3
7. 9301068145 = 8= 3. 5799511287 = 4
8. 8991068357 ; 11= 4. 6397173134 ; 5
9. 1098106577 ;-9= 5. 1169387459 : 6
10. 11989275575 : 12 157. When the divisor consists of several figures, we seek how often its left-hand figure is contained in the left-hand figure or two left-hand figures of the partial dividend. Also, we increase by a unit the left-hand figure of the divisor when the next figure on the right of it is very large. Having found the quotient figure, we multiply the divisor by it, placing the product in order as we find it under the partial dividend ; and, having performed the subtraction, we bring down, to the right of the remainder, the next figure of the principal dividend; because, with regard to such figure, the remainder (46, 48, and 54) is always a number of tens. Having thus formed a new partial dividend, we proceed as before, and continue till the operation is finished. The student should attentively observe, that, if the remainder equals or exceeds the divisor, the quotient figure is too small. Also, if the product of the divisor and quotient figure exceeds the partial dividend, the quotient figure is too great.