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82. Division is the process of finding the quotient of two numbers.
83. The Quotient of two numbers is a number which Axpresses how often one number is contained in another.
8A. The Dividend is the number to be divided.
86. The Remainder is the number which is sometimes left after dividing.
87. The Sign of Division is -, and is read divided by. It denotes that the number preceding it is to be divided by the number following it.
NOTES.-1. The sign of division is a short line, in the Une of writing, with one dot above and another below the middle of it.
2. The symbol = was introduced by Dr. John Pell, an English mathematician, born in 1610.
3. Division is also indicated by writing the divisor beneath the dividend, with a straight line between them; or by writing the divisor at the left of the dividend, with a curved line between them; thus 27, and 9)27, mean 87 divided by 9.
PRINCIPLES. 1. The divisor and dividend are similar numbers.
2. The quotient is an abstract number; the remainder is similar to the dividend.
3. In dividing a number into equal parts, the dividend and quotient are similar, and the divisor is abstract.
METHODS OF DIVISION. 88. There are Two Methods of performing division, called Short Division and Long Division.
SHORT DIVISION. 89. Short Division is that method in which only the dividend, divisor, and quotient are written, the operation being performed mentally.
90. Short division is generally employed when the divi. sor does not exceed twelve, the largest multiplier in the maltiplication table
1. How many times is 2 contained in 358 ? SOLUTION.-We write the divisor at the left of the OPERATION, dividend, with a curved line between them, draw a
2)358 line beneath the dividend, and begin at the left to di
179 vide. 2 is contained in 3 hundreds 1 hundred times, and 1 hundred remaining; 1 bundred equals 10 tens, which with 5 tens are 15 tens. 2 is contained in 15 tens 7 tens times, with a remainder of 1 ten; 1 ten equals 10 units, which with 8 units equals 18 units: 2 is contained in 18 units 9 units times, and we have for the quotient, 179. Hence we have the following
Rule.-I. Write the divisor at the left of the dividend, with a curved line between them and a line beneath the divis dend.
II. Begin at the left, divide each term of the dividend by the divisor, and write the quotient beneath.
III. If there is a remainder after any division, regard it as prefixed to the next term of the dividend, and divide as before.
IV. If any partial dividend is less than the divisor, write a cipher in the quotient and prefix the dividend to the next term.
V. When there is a final remainder, annex it, with the divisor written beneath, to the integral part of the quotient.
Proof.-Multiply the quotient by the divisor, and add the remainder, if any, to the product; if the work is correct, tho result will equal the dividend.
NOTE.-In practice we need not name the denomination of the different partial dividends. Thus, in the above solution we say 2 is contained in 8. once and one remaining; 2 is contained in 15, 7 times, etc.
14. Divide 3248 by 7.
Ans. 464. 16. Divide 7185 by 7.
Ans. 10264 16. Divide 8097 by 7.
Ans. 11564 17. Divide 9136 by 8.
Ans. 1142. 18. Divide 72352 by 8.
Ans. 9044. 19. Divide 91672 by 8.
Ans. 11459. 20. Divide 23769 by 9.
Ans. 2641. 21. Divide 30564 by 9.
Ans. 3396. 22. Divide 9876504 by 7.
Ans. 14109294 23. Divide 3201567 by 6.
Ans. 5335943. 24. Divide 4187002 by 8.
Ans. 523375%. 25. Divide 83702507 by 9.
Ans. 93002785. 26. If a book cost 4 dollars, how many books, at the same rate, can you buy for 252 dollars ?
OPERATION. SOLUTION.—If 1 book cost 4 dollars, for 252 dollars
4)252 We can buy as many books as 4 dollars are contained
63 tinues in 252 dollars, which are 63. Therefore, etc.
27. There are 3 feet in one yard; how many yards in 291 feet?
Ans. 97 yards. 28. There are 8 quarts in one peck; how many pecks are there in 1728 quarts?
Ans. 216 pecks. 29. There are 7 days in one week; how many weeks in 364 days?
Ans. 52 weeks. 30. A man gave 324 dollars to some boys, giving 6 dollars to each; how many boys were there? Ans. 54.
31. If one sheep cost 9 dollars, how many sheep can you buy for 1935 dollars ?
Ans. 215 sheep. 32. If there are 12 pence in 1 shilling, how many shillings are there in 571,836 pence? Ans. 47,653 shillings.
33. If it require one sheet of paper to print 12 pages of a book, how many sheets will be required for a book of 504 pages?
Ans. 42 sheets. 34. How long will it take two boys, starting at the same place, and traveling in opposite directions, to be 29,076 roda apart, if one goes 5 and the other 7 rods in a minute?
Ane 2423 minntea
LONG DIVISION. 91. Long Division is the method of dividing when the work is written out in full. It is generally used when the divisor exceeds 12.
1. Divide 5848 by 23. SOLUTION.--23 is not contained in 5 thousands any thousands times, hence there are no thousands
23)5848(254 in the quotient. 5 thousands and 8 hundreds are 58
46 hundreds ; 23 is contained in 58 hundreds 2 hundreds times: 2 hundreds times 23 are 46 hundreds,
124 which subtracted from 58 hundreds leave 12 hun
115 dreds : 12 hundreds and 4 tens are 124 tens ; 23 is
98 contained in 124 tens 5 tens times : 5 tens times 23
92 are 115 tens, which subtracted from 124 tens leave 9
6 tens: 9 tens and 8 units are 98 units; 23 is contained in 98 units 4 times; 4 times 23 equals 92: subtracting there is a remainder of 6, which will not contain 23; hence the quotient is 2 hundreds, 5 tens, and 4 units, or 254, with a remainder of 6.
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 remain der 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.
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.
NOTES.-I. The pupils will notice that there are five operations : 18t. 'Vrite the number; 2d. Divide ; 3d. Multiply ; 4th. Subtract ; 5th. Bring doron
[ Puulls often have difficulty in finding the correct quotient figure: this difficulty can be greatly diminished by attention to the following sug. gestions :
1st. Notice how often the left-hand term of the divisor 18 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, 80 that the remainder can be anited to the number of units of the next lower order, giving a new partia Wridend. The sign + is used to denote a remainder.
1. Divide 840 by 24. 2. Divide 903 by 21 3. Divide 455 by 13. 4. Divide 4956 by 14. 5. Divide 3510 by 15. 6. Divide 5024 by 16. 7. Divide 4720 by 20. 8. Divide 5778 by 18. 9. Divide 8736 by 21. 10. Divide 9591 by 23. 11. Divide 4056 by 24. 12. Divide 5450 by 25. 13. Divide 9672 by 26. 14. Divide 7946 by 29. 15. Divide 9840 by 30. 16. Divide 11286 by 27. 17. Divide 17500 by 28. 18. Divide 13872 by 34. 19. Divide 24588 by 36 20. Divide 25536 by 38. 21. Divide 31629 by 39. 22. Divide 29725 by 41. 23. Divide 28896 by 43. 24. Divide 72945 by 45. 25. Divide 81098 by 46. 26. Divide 60466 by 49. 27. Divide 141050 by 50. 28. Divide 316160 by 52. 99. Divida 833224 hy 54
Ans. 35. Ans. 43. Ans. 35. Ans. 354. Ans. 234. Ans. 314. Ans. 236. Ans. 321. Ans. 416. Ans. 417 Ans. 169. Ans. 218. Ans. 372 Ans. 274. Ans. 328. Ans. 418. Ans. 625. Ans. 408. Ans. 683. Ans. 672. Ans. 811. Ans. 725. Ans. 672 Ans. 1621. Ans. 1763. Ans. 1234. Ans. 2821. Ans. 6080. Ano 6171.