4. Simple Division. SIMPLE DIVISION is the method of finding how many times one number is contained in another of the same denomination. Thus if it were required to know how many times 6 is contained in 18, the answer is 3 times. As in Multiplication, there are always two numbers given to find a third. The largest number, or number to be divided, is called the dividend. The number by which you divide, is called the divisor. The result of the operation, or number of times the divisor is contained in the dividend, is called the quotient. If there be any left after performing the operation, that excess is called the remainder. The remainder is always less than the divisor, and is of the same kind as the dividend. Rule.* 1. Having written the divisor at the left hand of the dividend, find * As Multiplication performs the office of many additions, so does Division perform the office of many subtractions. Hence Division is only an abridgment of Subtraction. Thus the result is the same, whether we divide 14 by 4 three times, or subtract 4 three times from it. Each shows that 14 contains 4 three times, and that 2 remains. + By the rule the dividend is resolved into parts, and there is found the number of times the divisor is contained in each of these parts, and the sum of these several quotients is the true quotient. This will appear plain from the following example and observations. Divisor. 25)4 5 6 8 Dividend. 100 Here the dividend is divided into 3 parts, the first is 4500, the second 60, and the third 8. The first part of the dividend employed is 45, but its true value is 4500, and instead of seeking how many times 25 in 45, we in fact seek how many times 25 in 4500, and the quotient, instead of 1, is 100, and the re80 mainder 2000. To this remainder add the second part, and the sum divided by 25, the quotient, instead of 8, is 80, and the remainder 60. Again, to this remainder bring down the other part 8, making it 68, and dividing it by 25, the quotient is 2, and 18 remainder. Now the sum of these several quotients, 182, is the proper quotient arising from dividing 4568 by 25, and 18 remainder, as in example second. Rem. 18 Quot. 182 From the preceding example and observations, we perceive the complete value of the first part of the dividend is 10, 100, or 1000, &c. times the value of which it is taken by the rule, according as there are 1, 2 or 3, &c. figures on the right hand, and also that the value of the quotient figure is the same number of times its simple value as the part the dividend employed. Hence to express the true value of any quotient figure, annex as many ciphers as there are places in the dividend, at the right hand of the part employed in obtaining that quotient figure; but as a figure is added to the quotient for each figure brought down in the dividend, at the last division the quotient is complete. The method of proof must be sufficiently obvious; for as the quotient is the. how many times it is contained in as many of the left hand figures of the dividend as will contain it once or more, and place the answer as the highest figure in the quotient. 2. Multiply the divisor by the quotient figure, and set the product under the part of the dividend used. 3. Subtract the product from the part of the dividend used. 4. Bring down the next figure in the dividend to the right of the remainder, and divide the sum as before. If this sum be less than the divisor, place a cipher in the quotient, and bring down another figure. Proof. Multiply the divisor by the quotient, and to the product add the remainder, if any, and if the sum equal the dividend, the work is right. Examples. Having written down the dividend and included it within the reversed parenthesis, with the divisor, (4) at the left hand, assume as many figures in the dividend as will contain it once Here it is necessary to assume the two first figures, (14) because 4 is not contained in the first figure, (1). But 4 is contained in 14 three times; therefore, write 3 for the first quotient figure, and multiply ing 4, the divisor, by it, place the product. (12) under 14 in the dividend. Subtract 12 from 14 and to the remainder, 2, bring down the next figure 7, making it 27. Again, how many times is 4 contained in 27; 6 times: place 6 in the quotient, multiply the divisor, 4. by it, and the product, 24, place under 27, and subtract, and the work is done. Thus we find that 4 is contained in 147, 36 times, and that 3 remains. number of times the dividend contains the divisor, the product of the quotient and divisor is evidently equal to the part of the dividend exhausted by dividing; and if there be a remainder, remainder, or part of the dividend which was not exhausted, it is plain that it must be added to the product of the divisor and quotient to obtain a sum equal to the dividend. There are several other methods of proving Division. The following is a very expeditious way of doing it by casting out the 9's. Cast the 9's out of the divisor and quotient, multiply the excesses and add the remainder to the product, if any. Cast the 9's from the sum, and also from the dividend, and if the two excesses agree, the work is right. This method is liable to the same inconvenience here as in the preceding rules. * When there is no remainder after division, the dividend is completely exhausted, and the quotient is the complete answer. But when there is a remainder, that would give a part of another unit in the quotient If the remainder be one fourth, one half, or three fourths as large as the divisor, one fourth, one half or three fourths of the divisor is contained in the dividend in addition to the figures already found in the quotient. Therefore, to express the true value of any remainder, write it after the quotient over a horizontal line, with the divisor under it. The quotient in the first example is expressed thus, 364. This is called a Vulgar Fraction, and 36 a mixed number. I. When the Divisor is a single digit, the operation may be performed mentally without setting down any figures except the quotient. This is called Short Division. 1 1. Divide 867 by 3. Examples. Here seek how many times 3 in 8, and finding it 2 times and 2 over, write 2 under the 8 for the first figure of the quotient, and place the 2 which remained before 6, making it 26. Then 289 Quot.seeking how many times 3 in 26, the answer is 8 times and 2 over; therefore write the 8 under 6, and place the 2 which remained before the 7, making it 27. Lastly, seek how many times 3 in 27, and the answer is 9, which write under the 7, and the work is done. II. To divide by any number with ciphers at the right hand. Rule. Cut off the ciphers from the divisor, and as many figures from the right hand of the dividend; making use of the remaining figures, divide as usual, and place the figures cut off from the dividend at the right hand of the remainder. III. When the divisor is a composite number. Rule. Divide continually by the component parts instead of the whole 1 divisor at once. * The first remainder, 1, is evidently a unit of the given dividend, but the second remainder, 6, is so many units of the second dividend, and a unit in the second dividend is plainly equal to 7 units in the first. Therefore to find the true remainder, multiply the last remainder by the last divisor but one, and add the preceding remainder, the sum will be the true remainder if there are but two divisors; but if more than two, multiply this sum by the next preceding divisor, and so on till you arrive at the first divisor. 1. The first settlement of New-England was commenced in 1620; how many years from that time to the declaration of Independence in 1776 P Ans. 156 years. 2. What number shall I multiply by 8, that the product may be 552? 3. There are two numbers, the greater is 27 times the less is a 9th part of 27; what is the greater ? 4. If 9000 men march in a column of 750 deep, how abreast? Ans. 69. the less, and Ans. 81. many march Ans. 12. 5. What is the sum of 16, added to the difference of twice five Ans. 36. and twenty, and twice twenty-five? 6. A was born when B was 26 years old; how old will A be when B is 47? Ans. 21. 7. The remainder of a division is 325, the quotient 467, and the divisor 43 more than the sum of both; what is the dividend? Ans. 390270. 8. If a man's income be 730 dollars a year, what is that per day? Ans. 2 dollars. 9. A gentleman left his son 725 dollars more than his daughter, whose fortune was 15 thousand, 15 hundred and 15 dollars; what was the amount of the whole estate ? Ans. 33755 dollars. |