Art. 34.—From the foregoing we derive the following definitions : 1. Division is a concise method of performing many subtractions, or, the reverse of Multiplication. 2. Division consists in finding how many times one number contains another. As in Multiplication two numbers are required to perform the operation, so in Division. The number to be divided is called the Dividend; the number by which you divide is called the Divisor. The Dividend is to be regarded as the product of two factors, of which the Divisor is one, and the other is sought, which is the Quotient after division. The Divisor and Quotient multiplied together produce the Dividend. Thus, it appears that Division and Multiplication mutually prove each other. OBS.-All questions in Division may be performed by Subtraction; but all questions in Subtraction cannot be performed by Division.-When a number is to be divided into equal parts, the operation may be performed by Division. Art. 35.-When the Divisor is not greater than 12, the process of operation may be carried on in the mind, and the Quotient only be written down. This process is called SHORT DIVISION. 17. If 336 dollars be divided equally among 3 men, how many dollars will each receive? Illustration.—To subtract 3 from 336 as many times as would be necessary to give each man his share, would be long and tedious; but, by Short Division the operation becomes simple: 336 is 3 hundreds, 3 tens, and 6 units. It will be perceived, that if 300 be divided into 3 equal parts, one of these parts will be 100; and 3 tens divided in like manner become 1 ten; and 6 units divided by 3 become 2 units. The answer, then, is 1 hundred, 1 ten, and 2 units, or 112. 300 divided by 3 gives 100, 30 divided by 3 gives 10, and 6 divided by 3 gives 2: Then, 100+10+2=112 Ans. Operation. 300 3 100 30 3 10 2 336 3 112 Ans. QUESTIONS.-1. What is Division? 2. How many numbers are required to perform the operation? 3. What are they called? 4. How is the dividend to be regarded? 5. What are the two factors, which, multiplied together, will produce the dividend? 6. How is Division proved? By carrying on the process partly in the mind, the operation be made still shorter, thus: may 18. If 4 shares of bank stock cost 456 dollars, what will 1 share cost? 4)456 We first set down the number to be divided, or the dividend; at the left of this number, place the number by which we divide, or the divisor. Taking the first left-hand figure, or hundreds, we find how many times the divisor is contained in it. The number of times, or 1, we place directly under the divided figure. We next divide the tens of the dividend: 4 is contained in 5 once, and 1 ten over, which also must be divided. If this ten be added to the unit, it will make 1 ten and 6 units equal to 16 units: 4, then, is contained in 16, four times, which we place in the column of units; and in 456, 114 times. 19. Six brothers received a legacy of 1512 dollars. What was the share of each? 6)1512 252 Ans. In this question the divisor is not contained in the first lefthand figure of the dividend. We, therefore, take the next figure, 5, which with the 1 makes 15 hundred; 6 is contained in 1500, 200 times, and 30 tens, or 300 over. The 3 added to the 1 ten in the next column, is 31 tens; 6 is contained in 31, five times, or in 31 tens, 50 times and 1 ten over, which, with the two units, makes 12 units: this, divided by 6, is 2 units. The answer, then, is 200+50+2=252. From the foregoing, we derive the following RULE. Write the divisor at the left-hand of the dividend, with a line drawn between them. QUESTIONS.-7. How is the process partly carried on in Short Division? 8. When do you work by Short Division? 9. What is the method of procedure in the 18th question? Find how many times the divisor is contained in the first lefthand figure or figures of the dividend, setting the result directly under the divided figure or figures. The remainder, if there be any, carry to the next figure, calling it so many tens. Find how many times the divisor is contained in this dividend, and set it down as before; and so continue to do until the figures in the dividend are all divided. EXERCISES. Art. 36.-1. Paid 150 dollars for six tons of hay. How much was it a ton? 2. In a certain town there are average number in each family is 8. there? Ans. 25 dollars. 1280 inhabitants. The How many families are Ans. 160. 3. How many yards of cloth can be bought for 1155 dollars, at 7 dollars per yard? Ans. 165. 4. If a man labor one month for 12 dollars, how many months will he labor for 1008 dollars? and how many years, allowing 12 months to a year ? Ans. 84 months-7 years. Art. 37.—When the divisor is a composite number, and greater than 12. 1. If 15 horses consume 2550 bushels of oats in one year, how many will one horse consume? As Division Operation. 3)510 It will be seen that 15 is a composite number, produced by the multiplication of 5 and 3, thus: 5×3=15. is the reverse of Multiplication, it is evident that when the divisor is a composite number, we may divide, first, by one of the component parts, and that quotient by the other. For example: Suppose 30 apples to be divided equally between 15 place, we divide the whole number by 5. 5 boys, they would receive 6 apples each; but as there are 3 times 5 boys, they can have only one-third as many as 5 boys would have. 170 Ans. boys. In the first If there were only 2. How many days would it take a man to travel from Boston to New York, travelling at the rate of 30 miles a day, the distance being 240 miles? Ans. 8 days. QUESTIONS.-10. What is the rule for Short Division? 11. When the divisor is a composite number, how may you proceed? 12. What is the first step in the 1st example? Second step? miles Ans. 52 miles. 3. If a horse travel 2184 miles in 42 days, how many will he travel in one day? 4. If 112 barrels of flour cost 672 dollars, what will 1 barrel cost? Ans. 6 dollars. Art. 38.- When the divisor is not a composite number, and is greater than 12. 1. If 2624 bushels of corn be divided equally among 41 men, how many bushels will each receive? As 41 is greater than 12, and not a composite number, the operation must be performed by the whole divisor at once. This process is called LONG DIVISION. Operation. 41)2624(64 246 164 164 Setting down the numbers as before directed, and taking 41 for the divisor, we find that it is not contained in the 1st figure, nor in the 1st and 2d taken together, but it is contained in the first three figures, 6 times; that is, 41 is contained in 262 tens, 60 times and something over. To find what this remainder is, we find the product of 6 times 41, which we place under the three figures employed in the dividend, and, subtracting it therefrom, we find the remainder to be 16, which is 16 tens. We next bring down the 4 units of the dividend, and place them at the right hand of the 16 tens, which make 164 to be divided by 41, which is contained in it 4 times. The answer, then, is 64. By examining the work of the last question, it will be seen, that it is the same as Short Division, only that the operation is all set down, instead of being carried on in the mind. For example-divide 868 by 7, Long Division. Operation. 7)868(124 7 16 14 28 28 By Short Division, we say, 7 in 8, once, and 1 over; 7 in 16, twice, and 2 over; 7 in 28, 4 times, and no remainder. In Long Division, we say, 7 in 8 once, and place 1 for the first figure in the quotient; we then multiply 7 by this quotient figure, and place the result under the 8, and subtracting it we find the difference to be 1, to which we bring down the next figure for a new dividend, and proceed as before. From the preceding explanations is deduced the following RULE. Draw a Place the divisor at the left hand of the dividend. line at the right and left of the dividend, and take as many figures of the dividend as will contain the divisor one or more times. Place the number of times at the right hand of the dividend, for the first figure of the quotient. Multiply the divisor by this quotient figure, and place the result under the divided figures; find the difference between them, by subtraction, and to this difference bring down the next figure of the dividend, and divide as before; so continue to do until all the figures of the dividend are brought down. Should it be necessary to bring down more than one figure to contain the divisor, a cipher must be annexed to the quotient. OBS. The number of figures of the dividend we assume at any one step is a matter of convenience. EXERCISES. Art. 39.-2. A man raised 6996 bushels of potatoes on 33 How many did he raise per acre? acres. Ans. 212 bushels. 3. How many years in 32485 days, if 365 days make a year? Ans. 89 years. 4. A legacy of 15808 dollars was left to a certain number of men, giving them 832 dollars each. How many men were Ans. 19 men. there? 5. How many pounds in 11520 farthings, there being 960 farthings in one pound? Ans. 12 pounds. 6. How many hogsheads in 49896 pints, if 504 pints make one hogshead? Ans. 99 hogsheads. 7. There are 8 furlongs in one mile. How many miles in 123 furlongs? Operation. 15 Ans. QUESTIONS.-13. What is the difference between Long and Short Division? 14. Rule for Long Division? |