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71. 1. How many hats, at $4 apiece, can be bought for $20 ?
SOLUTION. Twenty dollars will buy as many hats as 4 is contained times in 20, which is 5 times. Five hats, at $4 apiece, can be bought for $20.
2. A man gave $16 for 8 barrels of apples. What was the cost of each barrel?
3. If 1 cord of wood costs $3, how many cords can be bought for $15?
4. At $6 a barrel, how many barrels of flour can be bought for $24?
5. When flour is $5 a barrel, how many barrels can be bought for $30?
6. If a man can dig 7 rods of ditch in a day, how many days will it take him to dig 28 rods?
7. If an orchard contains 56 trees, and 7 trees in a row, how many rows are there?
8. I bought 6 barrels of flour for $42. What was the cost of 1 barrel ?
9. If a farmer divides 21 bushels of potatoes equally among 7 laborers, how many bushels will each receive?
10. How many oranges, at 3 cents each, can be bought for 27 cents?
11. A farmer paid $35 for sheep, at $5 apiece. How many did he buy?
72. Division is the process of finding how many times one number is contained in another, or of sepa rating a number into equal parts.
Division may also be regarded as a short method of performing several subtractions of a number.
73. The Dividend is the number to be divided, and the Divisor is the number by which to divide.
74. The Quotient is the result obtained by the process of division, and shows how many times the divisor is contained in the dividend.
1. When the dividend does not contain the divisor an exact number of times, the part of the dividend left is called the remainder, and it must be less than the divisor.
2. As the remainder is always a part of the dividend, it is always of the same name or kind.
3. When there is no remainder, the division is said to be exact.
75. The sign, ÷, placed between two numbers, denotes division, and shows that the number on the left is to be divided by the number on the right.
Thus, 20 ÷ 4 =
5, is read, 20 divided by 4 is equal to 5. Division is also indicated by writing the dividend above, and the divisor below, a short horizontal line. Thus, 12 = 4, shows that 12 divided by 3 equals 4.
76. In finding how many times one number is contained in another, the dividend and divisor are like numbers, and the quotient is an abstract number.
In finding one of the equal parts of a number the dividend and quotient are like numbers, and the divisor is an abstract number.
77. When the divisor consists of one figure. 1. How many times is 4 contained in 848?
Dividend. Divisor. 4)848 Quotient. 212
SOLUTION. After writing the divisor on the left of the dividend, with a line between them, we begin at the left hand and say: 4 is contained in 8 hundreds, 2 hundreds times, and we write 2 in hundreds' place in the quotient; 4 is contained in 4 tens 1 ten times; we write the 1 in tens' place in the quotient; 4 is contained in 8 units 2 units times; we write the 2 in units' place in the quotient, and the entire quotient is 212. 2. How many times is 4 contained in 2884?
SOLUTION. - As we cannot divide 2 thousands by 4, we take the 2 thousands and the 8 hundreds together. 4 is contained in 28 hundreds 7 hundreds times, which 721 we write in hundreds' place in the quotient; 4 is contained in 8 tens 2 tens times, which we write in tens' place in the quotient; and 4 is contained in 4 units 1 unit time, which we write in units' place, and we have the entire quotient, 721.
3. How many times is 6 contained in 1824?
OPERATION. SOLUTION. Beginning as in the last example, we 6)1824 say, 6 is contained in 18 hundreds 3 hundreds times, which we write in hundreds' place in the quotient; 304 6 is contained in 2 tens no times, and we write a cipher in tens' place in the quotient; taking the 2 tens and 4 units together, 6 is contained in 24 units 4 units times, which we write in units' place in the quotient, and we have 304 for the entire quotient.
4. How many times is 4 contained in 943 ?
SOLUTION. Here 4 is contained in 9 hundreds 2 hundreds times, and 1 hundred 235...3 Rem. over, which, united to the 4 tens, makes 14 tens; 4 in 14 tens, 3 tens times and 2 tens over, which, united to the 3 units, makes 23 units; 4 in 23 units 5 units times and 3 units over. The 3 which is left after performing the division, should be divided by 4; but we merely indicate the division by placing the divisor under the dividend, thus, . The entire quotient is written 235, which may be read, two hundred thirty-five and three divided by four, or, two hundred thirty-five and three fourths, or, two hundred thirty-five and a remainder of three.
Hence the following rule:
RULE. -I. Write the divisor at the left of the dividend, with a line between them.
II. Beginning at the left hand, divide each figure of the dividend by the divisor, and write the result under the dividend.
III. If there is a remainder after dividing any figure, regard it as prefixed to the figure of the next lower order in the dividend, and divide as before.
IV. Should any figure or part of the dividend be less than the divisor, write a cipher in the quotient, prefix the number to the figure of the next lower order in the dividend, and divide as before.
V. If there is a remainder after dividing the last figure, place it over the divisor at the right hand of the quotient.
PROOF. Multiply the quotient by the divisor, and to the product add the remainder, if any; if the result is equal to the dividend, the work is correct.
1. This method of proof depends on the fact that division is the reverse of multiplication. The dividend corresponds to the product, the divisor to one of the factors, and the quotient to the other.
2. In multiplication the two factors are given, to find the product; in division, the product and one of the factors are given, to find the other factor.
The sums of quotients and remainders of examples 17 to 22 are
23. Divide $47645 equally among 5 men. each receive?
24. There are seven days in one week. weeks are there in 17675 days?
Ans. 2525 weeks.
25. How many barrels of flour, at $6 a barrel, can be bought for $6756 ? Ans. 1126 barrels. 26. Twelve things make a dozen. How many dozen are there in 46216464?
27. How many barrels of flour 347560 bushels of wheat, if it takes one barrel?
Ans. 3851372 dozen. can be made from
5 bushels to make
Ans. 69512 barrels.
28. If there are 3240622 acres of land in 11 townships, how many acres are there in each township?
29. A man left his estate, worth $38470, to be shared equally by his wife and 4 children. What did each receive? Ans. $7694.
30. At $5 an acre, how many acres of land can be bought for $3875 ?
31. If a man walks 4 miles an hour, in how many hours will he walk 352 miles?
32. I paid $1792 for 7 horses.
What did each cost?
33. If 75000 bushels of grain are put into 8 bins of equal size, how many bushels does each bin contain ?