Ans. is. For is from each apple makes is in the whole. If we take 9 twelfths from each of the two divided ap. ples, we shall have in the whole. Now this is not of one apple, for nothing has more than 12 twelfths. Whenever therefore we find an improper fraction, we know that more than one unit has been divided. What part of 13 apples is 3 apples and of an apple ! Ans. It is a fourth of 13, because 4 times 3 and make 13. What part of 5 is 1 ? is 2 ? is 3 ? is 4 ? is 6 ? In the last question we reason thus: If 1 is one fifth of 5,6 must be 6 times as much, or of 5. What part of 8 is 1 ? is 4 ? is ñ ? is 9? If you had 19 pears, and divided them equally among 6 persons, how much would you give to each? What part of 19 is 3 and z? When one number is placed over another, it signifies that the upper number is divided by the lower. Thus, signifies that the 3 is divided by 4. For a fourth of three things is 3 fourths, and is signifies either 3 fourths of one thing, or a fourth of 3 things. If you wish to divide 3 dollars into 5 equal parts, what would it be necessary to do, before you could divide them? Ans. Change them to dimes. wished to divide 4 dimes into 10 equal parts, what would it be necessary to do before you could divide them? What would be the answer ? How can 3 dollars be divided so as to give ten of the class, each an equal part ? Ans. Change the dollars to dimes, and then dividing them into ten equal parts, there will be 3 dimes for each of the ten. Divide $1,2 so as to give 6 scholars, each an equal part. Divide $2,4 so as to give 8 scholars, each an equal part. Divide 1 dime equally between two scholars. If 3 dimes 9 cents are divided by 6, what is the quotient, and what the remainder ? If 5 dimes 6 cents are divided by 7, what are the quotient and remainder? If 4 dimes 7 cents are divided by 6, what'are the quotient and remainder ? In the above sums, it will be seen that when one order of the dividend will not contain the divisor once, it is reduced, and added to the next lower order, and then divided. Thus when 4 dimes, 6 cents were to be divided by 6, the 4 dimes were changed to cents, and added to the 6 cents, and then divided. It will also be seen, that the quotient and the remainder are always of the same order as the dividend. Thus if 4 dimes 7 cents are divided by 6, the 4 dimes are reduced, and added to the cents, and the quotient is 7 cents, and the remainder is 5 cents. Thus, also, if 17 thousands are divided by 5, the quo. tient is 3, and 2 remainder. The 3 is 3 thousands, and the 2, is 2 thousands. If the order of the dividend were millions, the quotient and remainder would also be millions. If the order were tens the quotient and remainder would also be tens. If we divide 8 tens by 3, the quotient is 2 tens, and the remainder 2 tens. When the dividend has several orders, we divide each order separately, beginning with the highest orders. This is called Short Division. If there is any remainder, after the division of each or. der, 'it is changed to the next lower order, added to it, and ther vided For example. Let 9358 be divided by 4. What is the method of dividing when one order of the dividend will not contain the divisor once ? Of what order are the quotient and remainder ? We first divide the 9 thousands by 4, add the remaio. der to the 3 hundreds and divide that. Then divide the tens and units. Place them thus : 4)9358 23397 The 9 thousands is first divided. In 9 units there would be 2 fours, and 1 remainder. But as this is 9 thousands, the quotient and remainder must be the same order as the dividend, and the 2, is 2 thousand fours, and is set under the 9 in the thousands order. The remainder also is 1 thousand, and is changed to hundreds and added to the 3, making it 13 hundred. This is then divided by 4. Thé quotient is 3 hundreds, which is put under that order, and the 1 hundred that remains, is changed to tens and added to the 5 tens, making 15 tens. This is divided by 4, and the quotient is 3 tens, which is set in that order. 3 tens remain, which, changed to units and added to the 8, make 38 units. This is divided by 4, and the quotient is 9 units, which is put in that order. 2 units remain, which are divided by the 4 thus 4. 9358, then, contains 4, 2 thousands of times, 3 hundreds of times, 3 tens of times, and 9 units of times. The 2 left over, is of another time. Let the pupil, in performing each operation on the slate, explain it thus : Note to TEACHERS.-Let such questions as those be. low be asked on several sums, till the pupil fully under. stands them. 7)2496 3564 7 is contained in 24 units 3 times, in 24 hundreds, 3 hundred times, which are set in the order of hundreds. 3 hundred are left over, which, changed and added to the 9 tens, make 39 tens. 7 is contained in 39 tens, 5 tens of times, which are set In the first example, what is divided first? Of what order is the first quotient figure, and why? What is done with the remainder ? Explain the remainder of the sum in the same way. in the order of tens. 4 tens are left over, which, changed and added to 6, make 46 units. Divide 46 units by 7, and the answer is 6 units, which are set in that order, and 4 remain, which have the 7 set under them, to show that they are divided by 7. RULE FOR SHORT DIVISION. Divide the highest order, and set the quotient under it. If any remains, reduce and add it to the next lower order, and divide as before. If the number in any order, is less than the divisor, place a cipher under it in the quotient; then reduce and add it to the next lower order, and divide as before. If any remains when the lowest order is divided, place the divisor under it as a fraction. EXAMPLES. 4329 66 7892 6 4 6548 66 5 3456 6 5 3621 66 6 78926 6 46386 7 1234 « 7 296396 8 5678 6 8 36964 56 9 91234 « 9 24697 6 10 56789 4 10 36941 " 11 12345 " 1263 - 12 67891 66 12 When both the divisor and dividend, have several orders, another method is taken, called Long Division. Let 6492 be divided by 15. In performing the operation described below, we set the figures thus : Dividend. 60 6 6 11 49 42 12 We first take as many of the highest orders as would if units, contain the divisor once, and not more than 9 times. In this case we take 64 hundreds. Now we can. not very easily find exactly how many times the 15 is con. tained in 64 hundreds. But we can find how many hun. dreds of times it is contained thus. As 15 would be con. tained 4 units of times, in 64 units, it is contained 4 hun. dreds of times, in 64 hundreds. Which 400 is to be set in the quotient, (omitting the ciphers.) As we have found that the dividend contains 15, 4 hundreds of times, we subtract 4 hundred times 15 from the dividend, to find how often 15 is contained in what remains. 400 times 15 is 60 hundreds (6000) which, subtracted from the 64 hundreds, leaves 4 hundreds. This. 4 hundreds changed to tens, and the 9 tens of the dividend put with it, make 49 tens. We now find how many tens of times the 15 is contained in the 49 tens, thus: as 15 would be contained 3 units of times in 49 units, it is contained 3 tens of times in 49 tens, which 3 tens is set in the quotient. We now subtract 3 tens of 15 (or 45 tens) from the 49 tens, and 4 tens remain. These are changed to units and have the 2 units of the dividend put with them, making 42 units. 15 is contained in 42 units 2 units of times, which is set in the quotient. Twice 15 from 42 units, leave 12, which is of another 15. The 15 then, is contained in the dividend 4 hundreds of times, 3 tens of times, 2 units of times, and is of another time, or 432 times, and is of another time. Again, divide 6998 by 24. To do it we first find how many hundreds of times the dividend contains the divisor, and subtract these hundreds; second, how many tens of times, and subtract these tens; third, how many units of times, and subtract these units; and fourth, what remains has the divisor set under it. What is the rule for Short Division? When is Long Division performed? How many of the highest orders are first taken? Do we find exactly how many times the divisor is contained? What do we find, and how do we reason in order to find it? What is the first quotient, and what is omitted in setting it down? After we have found how many hundred times the divisor is contained, what is done next and for what purpose ? What is done with the 4 hundred that remain ? |