INTRODUCTION TO DIVISION.

MENTAL EXERCISES.

1. At 3 cents each, how many melons can I buy for 15 cents? SOLUTION.-If 1 melon costs 3 cents, for 15 cents I can buy as many melons as 3 is contained times in 15, which are 5.

2. How many yards of ribbon, at 8 cents a yard, can be bought for 56 cents?

3. A man gave 60 dollars for sheep, at the rate of 5 dollars a head' how many did he buy?

4. How many kegs, of 9 gallons each, can be filled from a hogshead containing 90 gallons of vinegar?

5. How many days must a man work, at the rate of $3 a day, to earn $45?

6. How many lemons at 6 cents apiece may be bought for 84 cents ?

7. How many are 15 plus 5, divided by 5? 18 plus 6, divided by 6? 40 plus 8, divided by 8? 35 plus 7, divided by 7?

8. How many are 3 times 8 divided by 4? 5 times 9 divided by 3? 6 times 10 divided by 12? 8 times 7 divided by 4?

9. How many are 3 times 33 divided by 11? 4 times 21 divided by 7? 3 times 25 divided by 5? 3 times 24 divided by 8?

10. How many pencils, worth 5 cents each, may be bought for 4 erasers worth 15 cents each?

11. How many barrels of flour, at $9 a barrel, can be bought for 15 barrels of apples at $3 a barrel ?

12. A woman carried to the store 6 dozen eggs at 25 cents a dozen; how many yards of calico, at 10 cents a yard, will pay her?

13. If mackerel is worth $12 a barrel, how many barrels can be bought for $6 in money and 6 barrels of pork at $15 a barrel?

14. The process of finding how often one number is contained in another is called division. The result is called the quotient.

15. The number divided is the dividend; the number we divide by is the divisor. The sign of division is÷, and is read, "divided by.' 16. The sign () signifies that the quantities included are to be subjected to the same operation.

17. Find the result of the following:

DIVISION.

82. Division is the process of finding the quotient of two numbers.

83. The Quotient of two numbers is a number which expresses how often one number is contained in another.

84. The Dividend is the number to be divided.

85. The Divisor is the number by which we divide. 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 line 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 27 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 divisor does not exceed twelve, the largest multiplier in the mul tiplication table.

OPERATION.

2)358

179

1. How many times is 2 contained in 358? SOLUTION. We write the divisor at the left of the dividend, with a curved line between them, draw a line beneath the dividend, and begin at the left to divide. 2 is contained in 3 hundreds 1 hundred times, and 1 hundred remaining; 1 hundred 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 dividend.

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, the 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 3 once and one remaining; 2 is contained in 15, 7 times, etc.

26. If a book cost 4 dollars, how many books, at the same rate, can you buy for 252 dollars?

SOLUTION.-If 1 book cost 4 dollars, for 252 dollars we can buy as many books as 4 dollars are contained times in 252 dollars, which are 63. Therefore, etc.

OPERATION.

4)252

63

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 rods apart, if one goes 5 and the other 7 rods in a minute? Ans. 2423 minutes.

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.

OPERATION.

23)5848(254 46

124

115

98

92

6

SOLUTION.-23 is not contained in 5 thousands any thousands times, hence there are no thousands in the quotient. 5 thousands and 8 hundreds are 58 hundreds; 23 is contained in 58 hundreds 2 hundreds times: 2 hundreds times 23 are 46 hundreds, which subtracted from 58 hundreds leave 12 hundreds 12 hundreds and 4 tens are 124 tens; 23 is contained in 124 tens 5 tens times: 5 tens times 23 are 115 tens, which subtracted from 124 tens leave 9 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 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 quotien

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: 1st. Write the number; 2d. Divide; 3d. Multiply; 4th. Subtract; 5th. Bring down.

II Pupils often have difficulty in finding the correct quotient figure : this difficulty can be greatly diminished by attention to the following suggestions.