PROBLEM. 122. To divide one number by another. 1. Divide 7872 by 32. OPERATION. 32)7872(246 64 147 128 192 192 SOLUTION.-32 is not contained in 7 thousands any thousands times, hence there are no thousands in the quotient: 7 thousands and 8 hundreds are 78 hundreds; 32 is contained in 78 hundreds 2 hundreds times; 2 hundreds times 32 are 64 hundreds, which subtracted from 78 hundreds leave 14 hundreds: 14 hundreds and 7 tens are 147 tens; 32 is contained in 147 tens 4 tens times; 4 tens times 32 are 128 tens, which subtracted from 147 tens, leave 19 tens: 19 tens with 2 units are 192 units; 32 is contained in 192 units 6 units times; 6 units times 32 are 192 units. Hence the quotient is 246. Rule.-I. Draw curved lines at both sides of the dividend, and place the divisor at the left. II. Divide the number expressed by the fewest terms at the left that will contain the divisor, and place the quotient at the right. III. Multiply the divisor by this quotient, write the product under the partial dividend, subtract, and to the remainder annex the next term of the dividend. IV. Divide as before, and thus continue until all the terms of the dividend have been used. V. If any partial dividend will not contain the divisor, place a cipher in the quotient, annex the next term of the dividend, and proceed as before. VI. When there is a final remainder, annex it, with the divisor written beneath, to the integral part of the quotient. NOTE. When the divisor does not exceed 12, we usually draw a line under the dividend, and write the quotients beneath, doing the rest of the work mentally. This is called Short Division; the other method is called Long Division. Proof. Multiply the integral part of the quotient by the divisor, and add the remainder, if any, to the product; if the work is correct the result will be equal to the dividend. down. NOTES.-I. The pupils will notice that there are five operations: 1st, Write the numbers; 2d. Divide; 3d. Multiply; 4th. Subtract; 5th. Bring II. Pupils often have difficulty in finding the correct quotient figure; this ifficulty can be greatly diminished by attention to the following sugges tions: 1st. Notice how often the left hand term of the divisor is contained in the term or terms of the partial dividend, as far from the right hand term as the left hand term in the divisor is from the right hand term. 2d. If, when we multiply, the product is greater than the partial dividend, the quotient term must be diminished. 3d. If, when we subtract, the remainder is greater than the divisor, the quotient term must be increased. III. We commence at the left to divide, so that the remainder can be united to the number of units of the next lower order, giving a new partial dividend. The sign is used to denote a remainder. 15. 48384751874346 by 590778. 16. 5299770856733656 by 7904207. 17. 2016722783975663729 by 41927081. Ans. 81900057. Ans. 670500008. Ans. 48100720009. PRACTICAL PROBLEMS. 123. In Division there are two classes of practical problems: 1st. To find the number of equal parts of a number. 2d. To divide a number into equal parts. CASE I. 124. To find the number of equal parts of a number. 1. At 95 dollars each, how many oxen may be bought for 3040 dollars? SOLUTION.-If 95 dollars will buy one ox, 3040 dollars will buy as many oxen as 95 dollars are contained times in 3040 dollars, which are 32. Therefore, etc. 2. At 54 dollars a share, how many shares of bank stock can be purchased for 333234 dollars? Ans. 6171 shares. 3. If a student, on a pedestrian tour, walks 134 miles a week, how many weeks would it take him to walk 238788 miles? Ans. 1782 weeks. 4. If the construction of a railroad cost $116,188,800, how long was the road, provided it was built at the rate of $470,400 a mile ? Ans. 247 miles. 5. If a banker has a net gain of $7,420 annually, how long will it take him to pay for a farm of 175 acres at $212 per acre? Ans. 5 years. 6. A man traded 7425 acres of woodland, worth 48 dollars an acre, for farm land, worth 144 dollars an acre; how many acres did he receive? Ans. 2475 acres. 7. Suppose that A and B are 2376 miles apart and approach each other, A traveling 15 miles an hour and B 18 miles an hour; in how many hours will they meet? Ans 72 hours. 8. A man has $39,180 which he wishes to invest in land. He buys 246 acres at $145 an acre; how many acres can he buy with the balance of the money, at $130 an acre? Ans. 27 acres. 9. How many two-horse phaetons worth $245 apiece could be bought for the value of 348 horses worth $234 each, and $643 in money? Ans. 335 phaetons. 10. A man bought 140 acres of land for $10,500, and sold 95 acres at $125 an acre; at what price per acre must he sell the remainder to gain $5,425 ? Ans. $90. 11. A speculator wishes to trade land worth $95 an acre for 73 acres at $125 an acre, and gain $290 on this estimate of values by the exchange; how many acres will he exchange? Ans. 93 acres. CASE II. 125. To divide a number into equal parts. 1. Divide 456 into 6 equal parts. SOLUTION.-If we divide 456 into 6 equal parts, each part is of 456: of 45 tens is 7 tens and 3 tens remaining; 3 tens and 6 units equal 36; of 36 is 6; hence of 456 is 76, or 76 is one of the 6 equal parts of 456. 2. Divide 16512 into 12 equal parts. 3. Divide 42228 into 18 equal parts. 4. Divide 73260 into 37 equal parts. 5. Divide 52224 into 64 equal parts. OPERATION. 6)456 76 Ans. 1376. Ans. 2346. Ans. 816. 6. Messrs. Wilson and Co. purchased 24 shawls for $840; what did they cost apiece? SOLUTION.-If 24 shawls cost $840, one shawl will cost one twenty-fourth of $840, which, by division, we find is $35. Therefore, etc. OPERATION. 24)840(35 72 120 120 7. Messrs. Taylor & Brother bought 37 sets of furs for $6068; what did they pay per set? Ans. $164. 8. A young man shared a legacy of $24,780 with each of his 5 brothers, and another legacy of $14,300 with each of his 4 sisters; what sum did he receive? Ans. $6990. 9. I bought a farm of 136 acres for $8568, and sold 93 acres of it at $75 an acre, and the remainder for what it cost; how much did I gain by the bargain? Ans. $1116. 10. Miss Atherton bought 235 shares of Northern Central Railroad stock for $10,575, and sold a part off for $7448 at $56 a share; how many shares remained and what was the gain on those sold? Ans. Rem., 102 shares; gain, $1463. 11. Bought a farm for $35,380, and having made improvements valued at $3420, I sold one-half of it for $21,750 at $75 an acre; how many acres did I purchase, and at what price per acre? Ans. 580 acres; $61 an acre. 12. An army contractor bought some horses for $18,750, sold part of them for $4725 at $135 apiece, and lost $15 on each horse sold; and subsequently sold the remainder so as to gain $375 on the whole; at what rate were the remainder sold? Ans. $160 apiece. CONTRACTIONS IN DIVISION. 126. Contractions in Division are abbreviated forms of dividing. 127. Successive Division is the process of dividing one number by another, the quotient by a second divisor, etc. Successive division is the reverse of continued multiplication. PRINCIPLES. 1. The quotient of two numbers is equal to the quotient derived by the successive division of one of the numbers by the factors of the other. 2. The quotient derived by successive division is the same, in whatever order the divisors are taken. CASE I. 128. When the divisor is a composite number. 1. Divide 7875 by 35, using the factors 5 and 7. SOLUTION. Since 35 times a number equals 7 times 5 times the number, of a number equals of of the number; of 7875 is 1575, of 1575 is 225. OPERATION. 5)7875 7)1575 225 Rule. Divide the dividend by one factor of the divisor, the quotient by another factor, and thus continue for all the factors used; the last quotient will be the quotient required. Divide 129. The True Remainder in successive division being neither the last remainder nor the sum of all the remainders, it is necessary to explain the method of finding it. 1. Divide 2243 by 84, using the factors 3, 4, and 7. SOLUTION.-Dividing by 3 we find that 2243 equals 747 threes, and 2 remaining; dividing by 4 we find 747 threes equals 186 twelves and 3 threes, or 9 remaining; dividing by 7 we find that 186 twelves equals 26 eighty-fours and 4 twelves, or 48 remaining. Hence the true remainder is 2+9+48 59. = OPERATION. 3)2243 4)747-2= = 2018 26-4 twelves = 48 59 |