DIVISION. 59. 1. How many 1's are there in 1? How many in 2? In 3? In 4? In 5? 2. How many 2's are there in 2? 2 in 2 how many times? 2 in 4 how many times? 2 in 6 how many times? In 8? 3. How many 3's in 6? 3 in 6 how many times? 3 in 9? 3 in 12? 3 in 15? 3 in 18? QUESTIONS. 1. If 12 apples be equally divided among 4 boys, how many will each have ? ANALYSIS. Since 12 apples are to be divided equally among 4 boys, one boy will have as many apples as 4 is contained times in 12, which is 3. 2. If 24 peaches be equally divided among 6 boys, how many will each have? How many times is 6 contained in 24? 3. A man has 32 miles to walk, and can travel 4 miles an hour, how many hours will it take him? 4. How many yards of cloth, at 3 dollars a yard, can you buy for 24 dollars? ANALYSIS.-Since the cloth is 3 dollars a yard, you can buy as many yards as 3 is contained times in 24, which is 8: therefore, you can buy 8 yards. 5. How many oranges at 6 cents apiece can you buy for 42 cents? 6. How many pine-apples at 12 cents apiece can you buy for 132 cents? 7. A farmer pays 28 dollars for 7 sheep: how much is that apiece? ANALYSIS. Since 7 sheep cost 28 dollars, one sheep will cost as many dollars as 7 is contained times in 28, which is 4; therefore, each sheep will cost 4 dollars. 8. If 12 yards of muslin cost 96 cents, how much does 1 yard cost? 9. How many lead pencils could you buy for 42 cents, if they cost 6 cents apiece? 10. How many oranges could you buy for 72 cents, if they cost 6 cents apiece? 11. A trader wishes to pack 64 hats in boxes, and can put but 8 hats in a box: how many boxes does he want? 12. If a man can build 7 rods of fence in a day, how long will it take him to build 77 rods? 13. If a man pays 56 dollars for seven yards of cloth, how much is that a yard? 14. Twelve men receive 108 dollars for doing a piece of work how much does each one receive? 15. A merchant has 144 dollars with which he is going to buy cloth at 12 dollars a yard; how many yards can he purchase? 16. James is to learn forty-two verses of Scripture in a week how many must he learn each day? 17. How many times is 4 contained in 50, and how many over? PRINCIPLES AND EXAMPLES. 60. 1. Let it be required to divide 86 by 2. Set down the number to be divided and write the other number on the left, drawing a curved line between them. Now there are 8 tens and 6 units to be divided by 2. We say, 2 in 8, 4 times, which being tens, we write it in the tens' place. We then say, 2 in 6, 3 times, which being units, are written in the units' place. The result, which is called a quotient, is therefore, 4 tens and 3 units, or 43. 2. Let it be required to divide 729 by 3. OPERATION. Divisor. Dividend. 43 quotie't. OPERATION. 3)729 243 ANALYSIS.-We say, 3 in 7, 2 times and 1 over. Set down the 2, which are 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 of the quotient in the tens' place; then say, 3 in 9, 3 times. The quotient, therefore, is 243. 3. Let it be required to divide 466 by 8. ANALYSIS.-We first divide the 46 tens by 8, giving a quotient of 5 tens, and 6 tens over. These 6 tens are equal to 60 units, to which we add the 6 in the units' place. We then say, 8 in 66, 8 times and 2 over; hence, the quotient is 58, and 2 over, which we call a remainder. This remainder is written after the last quotient figure, and the 8 raced under it; the quotient is read, 58 and 2 divided by 8. OPERATION. 8)466 58-2 remain. 58 quotient. 50. Ex. 1.-When you divide 8 tens by 2, is the unit of the quotient tens or units? When 6 units are divided by 2, what is the unit? ANALYSIS. In the first example 86 is divided into 2 equal parts, and the quotient 43 is one of the parts. If one of the equal parts be multiplied by the number of parts 2, the product will be 86, the number divided. In the third example 466 is divided into 8 equal parts, and two units remain that are not divided. If one of the equal parts 58, be multiplied by the number of parts, 8, and the remainder 2 be added to the product, the result will be equal to 466, the number divided. 61. DIVISION is the operation of dividing a number into two equal parts; or, of finding how many times one number contains another. The first number, or number by which we divide, is called the divisor. The second number, or number to be divided, is called the dividend. The third number, or result, is called the quotient The quotient shows how many times the dividend contains the divisor. If anything is left after division, it is called a remainder. 62. There are three parts in every division, and sometimes four 1st, the dividend; 2d, the divisor; 3d, the quotient; and 4th, the remainder. : There are three signs used to denote division; they are the following: 18÷4 expresses that 18 is to be divided by 4. 18 expresses that 18 is to be divided by 4. When the last sign is used, if the divisor does not exceed 12, we draw a line beneath, and set the quotient under it. If the divisor exceeds 12, we draw a curved line on the right of the dividend, and set the quotient at the right. 2. When the seven hundreds are divided by 3, what is the unit of the quotient? To how many tens is the undivided hundred equal? When the 12 tens are divided by 3, what is the unit of the quotient? When the 9 units are divided by 3, what is the quotient? Read the 3. How is the division of the remainder expressed? quotient. If there be a remainder after division, how must it be written? 61. What is division? What is the number to be divided called? What is the number called by which we divide? What is the answer called? What is the number called which is left? 62. How many parts are there in division? Name them. many signs are there in division? Make and name them? How SHORT DIVISION. 63. SHORT DIVISION is the operation of dividing when the work is performed mentally, and the results only written down. It is limited to the cases in which the divisors do not exceed 12. Let it be required to divide 30456 by 8. ANALYSIS-We first say, 8 in 3 we cannot. Then, 8 in 30, 3 times and 6 over; then 8 in 64, 8 times; then 8 in 5, 0 times; then, 8 in 56, 7 times; hence, OPERATION. 8)30456 3807 RULE I.-Write the divisor on the left of the dividend. Beginning at the left, divide each figure of the dividend by the divisor, and set each quotient figure under its dividend II.-If there is a remainder, after any division, annex to it the next figure of the dividend, and divide as before.. III. If any dividend is less than the divisor, write for the quotient figure and annex the next figure of the dividend, for a new dividend. IV. If there is a remainder, after dividing the last figure, set the divisor under it, and annex the result to the quotient. PROOF.-Multiply the divisor by the quotient, and to the product add the remainder, when there is one; if the work is right the result will be equal to the dividend. |