We can arrive at the same result by a shorter method, called Division. DIVISION teaches the manner of finding how many times a less number is contained in a greater. It is a short method of subtraction. The less number is called the divisor. The greater number is called the dividend. The number expressing how many times the dividend contains the divisor, is called the quotient. If there is a number left, it is called the remainder, which is always less than the divisor. There are three signs used to denote division. They are the following. 18÷4 expresses that 18 is to be divided by 4. 18 4 expresses that 18 is to be divided by 4. When the last sign is used, a curved line is also drawn on the right of the dividend to separate it from the quotient, which is generally set down on the right. Q. When Charles divides 12 apples equally, among his four brothers, how many does he give to each? How many times does 12 contain 4? In dividing 28 apples equally among 8 boys, how many does each receive? How many remain? Which number is the dividend? Which the divisor? Which the quotient? Which the remainder? What does division teach? What is the less number called? What is the greater called? What is the answer called? What is the number called which is left? Is this number greater or less than the divisor? How many signs are there in division? Make them? § 29. Let the following table be committed to memory. It is read 2 in 2, 1 time; 2 in 4, 2 times, &c. 5 in 5 1 time 5 in 10 2 times 5 in 15 3 times 5 in 20 4 times 5 in 25 5 times 5 in 30 6 times 5 in 35 7 times 5 in 40 8 times 5 in 45 9 times 6 in 6 1 time 6 in 12 2 times 6 in 18 3 times 6 in 24 4 times 6 in 30 5 times 6 in 36 6 times 6 in 42 7 times 6 in 48 8 times 6 in 54 9 times 7 in 7 1 time 7 in 14 2 times 7 in 21 3 times 7 in 28 4 times 7 in 35 5 times 7 in 42 6 times 7 in 497 times 7 in 56 8 times 7 in 63 9 times 8 in 8 1 time 8 in 16 2 times 8 in 24 3 times 8 in 32 4 times 8 in 40 5 times 8 in 48 6 times 8 in 567 times 8 in 64 8 times 8 in 72 9 times 9 in 9 1 time 9 in 18 2 times 9 in 27 3 times 9 in 36 4 times 9 in 45 5 times 9 in 54 6 times 9 in 63 7 times 9 in 72 8 times 9 in 81 9 times 10 in 10 1 time 10 in 20 2 times 10 in 30 3 times 10 in 40 4 times 10 in 50 5 times 10 in 60 6 times 10 in 70 7 times 10 in 80 8 times 10 in 90 9 times 1. Divide 86 by 2. Place the divisor on the left of the dividend, draw a curved line between them, and a straight line under the dividend. = = = = = = OPERATION. Divisor Dividend. )86 43 quotient. Now, there are 8 tens and 6 units to be divided by 2. We say, 2 in 8, 4 times, which being 4 tens we write the 4 under the tens. We then say, 2 in 6, 3 times, which are three units, and must be written under the 6. The quotient therefore, is 4 tens and 3 units, or 43. Q. When you divide 8 tens by 2, is the quotient tens or units? When 6 units are divided by 2, what is the quotient? 2. Divide 729 by 3. OPERATION. 3)729 243 In this example there are 7 hundreds 2 tens, and 9 units, all to be divided by 3. Now, we say 3 in 7, 2 times and 1 over. Set down the 2, which is hundreds, under the 7. But of the 7 hundreds there is 1 hundred or 10 tens not yet divided. We put the 10 tens with the 2 tens, making 12 tens, and then say, 3 in 12, 4 times, and write the 4 in the quotient, in the ten's place; then say 3 in 9, 3 times. The quotient therefore, is 243. Q. When the 7 hundreds are divided by 3, of what denomination is the quotient? To how many tens is the undivided hundred equal? When the 12 tens are divided by 3, what is the quotient? When the 9 units are divided by 3, what is the quotient? 3. Divide 729 by 9. In this example, we say, 9 in 7 we cannot, but 9 in 72, 8 times, which are 8 tens: then, 9 in 9, 1 time. The quotient is therefore 81. 4. Divide 8040 by 8. OPERATION. 9)729 81 OPERATION. 8)8040 1005 In this example, we say 8 in 8, 1 time, and set 1 in the quotient. We then say 8 in 0, 0 times, and set the 0 in the quotient: then say, 8 in 4, 0 times, and set the 0 in the quotient: then say, 8 in 40, 5 times, and set the 5 in the quotient. The true quotient is therefore 1005. § 30. It may be remarked that any number contains 1 as many times as there are units in the number, or that if any number be divided by 1, the quotient will be equal to the number itself. Q. How many times will any number contain 1? If any number be divided by 1, what is the quotient? CASE I. § 31. Short Division, or when the divisor does not exceed 12. RULE. I. Set down the divisor on the left of the dividend, draw a curved line between them, and a straight line under the dividend. II. Find how often the divisor is contained in the left hand figure or figures of the dividend, and place the figure so found under the straight line, for the first figure of the quotient. III. If there is no remainder, divide the next figure of the dividend for the next figure of the quotient. But when there is a remainder consider it as tens, to which add the next figure of the dividend, regarded as units, and divide this sum for the next figure of the quotient, and do the same for each of the figures of the dividend. IV. When any of the figures or sums that are to be divided, is less than the divisor, set down 0 in the quotient, and to such number regarded as tens, add the next figure of the dividend considered as units, and divide the sum for the next figure of the quotient. EXAMPLES. OPERATION. 5)36458 7291-3 remain. 1. Let it be required to divide 36458 by 5. In this example, we find the quotient to be 7291 and a remainder 3. This 3 ought in fact to be divided by the divisor 5, but the division cannot be effected, since 3 does not contain 5. The division must then be expressed by placing 5 under the 3, thus, 3. The true quotient, therefore, is 72913, which is read, seven thousand two hundred and ninety-one, and three divided by five. Therefore, Where there is a remainder after the division, it may be written after the quotient, and the divisor placed under it. Q. What is short division? How do you set down the numbers to be divided? How do you divide? Repeat the rule. If there is a remainder after division, how may it be written? (1.) 9486312 EXAMPLES. (2.) (3.) 15840087 § 32. Long Division, or when the divisor contains several figures. RULE. I. Set down the divisor on the left of the dividend, draw a curved line between them, and also a curved line on the right of the dividend. II. Note the fewest figures of the dividend, counted from the left hand, that will contain the divisor; find how often they contain it, and set the figure in the quotient. III. Multiply the whole divisor by this figure; set the product under the first figures of the dividend, and subtract it from them. IV. To the remainder annex the next figure of the dividend, then find how often the divisor is contained in this new number, and set the figure in the quotient. V. Multiply the whole divisor by the last figure of the quotient, and subtract the product from the last number containing the divisor. To the remainder annex the next figure of the dividend, and find the figures of the quotient in the same way, till all the figures of the dividend are brought down. NOTE 1. There are four numbers in division. First, the dividend; second, the divisor; third, the quotient; fourth, the remainder. NOTE 2. There are five operations in division. First, to write down the numbers; second, find how many times: third, multiply; fourth, subtract; fifth, bring down." Q. How do you set down the numbers for division? do next? What do you do next? What is the next many numbers are there in division? What are they? operations are there in division? Name them. What do you step? How How many |