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57. Dividing by 10 and its Powers. To divide by 10 is merely to find how many tens there are in the number, and this we can tell at a glance.

Thus in 1230 there are 123 tens and no units. In 2373 there are 237 tens and 3 units remainder; that is, 2373 ÷ 10 = 237%.

Therefore, to divide an integer by 10 cut off the right-hand figure of the dividend. The result is the quotient, and the right-hand figure expresses the remainder.

To divide an integer by any power of 10 cut off as many figures from the right of the dividend as there are zeros at the right of the divisor. The result is the quotient, and the right-hand figures express the remainder.

=

Thus 12,400 100 124; 43,203 ÷ 100 =

= 624; 736,1171000736,117.

4321; 624,000 ÷ 1000

58. Dividing by Multiples of Powers of 10. In this case we divide by the power of 10, as in § 57, and then by the number indicating the multiple.

200)42800 214 300)27407

For example, to divide 42,800 by 200 we cut off two zeros from divisor and dividend and then divide by 2. So to divide 27,407 by 300 we cut off two zeros from the divisor and two figures from the right of the dividend. We then divide by 3, and the quotient is 91, with remainder 1 in the hundreds' place, and to this we join the 07 cut off, obtaining the remainder 107.

91183

EXERCISE 19

Find the quotient and the remainder (if any):

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Find the quotient and the remainder (if any):

13. 325 30. 21. 20,300 100. 29. 98,207900.

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37. At $90 each, how many horses can I buy for $2160?

38. There being 60 minutes in an hour, how many hours are there in 1020 minutes?

39. There being 60 seconds in a minute, how many minutes are there in 1620 seconds?

40. At $30 each, how many harnesses must a manufacturer sell to receive $1110?

41. A farmer pays $6600 for some land at $40 an acre. How many acres does he buy?

42. A dealer sells some village lots for $10,200, at $600 a lot. How many lots does he sell?

43. An investor has $87,750 with which to buy some. bonds at $500 each. What is the greatest number he can buy, and how much will he have left?

44. An automobile dealer has $35,100 to invest in automobiles at $2000 each. What is the greatest number he can buy for this sum, and how much will he have left?

45. A ton is 2000 pounds. If a coal pocket contains 583,260 pounds of coal, how many tons does it contain, and how many pounds are left over?

EXERCISE 20

PROBLEMS OF THE FARM

1. A farmer sowed 60 acres of land, using 780 quarts of timothy seed. How much seed did he use per acre?

2. He also sowed 70 acres, using 1050 quarts of timothy and clover mixed. How much seed did he use per acre?

3. Another farmer sowed 90 acres, using 2520 quarts of timothy and redtop mixed equally. How many quarts of each kind did he use per acre?

4. A Maryland farmer experimented with Italian rye grass, using 3250 pounds to sow 50 acres. How many pounds did he use per acre?

5. A farmer found that 30 sheep ate 11,550 pounds of dry fodder in a year. What was the average amount per sheep?

6. He found that 20 horses consumed 162,400 pounds of hay and grain in a year. What was the average amount per horse?

7. A dealer found that 60 bushels of white clover seed weighed 3780 pounds. How much did it weigh per bushel?

8. He told a customer 1080 pounds would seed 90 acres. How much is that per acre? If he mixed other seed with the clover so as to use this amount to cover 4 acres, how much clover seed would he use per acre?

9. A farmer found that it took 980 pounds of red clover for 70 acres, and 1200 pounds of crimson clover for 80 acres. What was the amount of each used per acre?

10. How many pounds of Paris green will be needed to spray 300 trees, allowing 5 gallons to a tree, if every 150 gallons contains 1 pound of Paris green?

59. Long Division. When the divisor has two or more figures, except in such cases as are given in § 57 and § 58, it is customary to use a process known as long division.

We shall first consider the simplest case, that in which the units' figure is 1.

Required to divide 7182 by 21.

If all the work should be written, it would appear as in the first of these forms, but for practical purposes the second is used.

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First we will consider (1), the more complete form. We see that 21 is evidently not contained in 7, so there are no thousands in the quotient. The divisor is contained 3 times in 71, and since the 71 is hundreds, the 3 is written above, over the hundreds' place. Then 300 × 21 6300, and there remains 882 still to be divided; 21 is contained in 88 four times, and since the 88 is tens, the 4 of the quotient is written over the tens' place. Then 40 × 21 = 840, and there remains 42 to be divided; 21 is contained in 42 twice, and 2 is written over the units' place. The quotient is therefore 342. It will be seen that 6300, 840, and 42 make up 342 × 21, as should be the case.

Consider now the second form, which is always used in practice. Since at any time in this example we need only two figures of the dividend or two figures of such a partial dividend as 882, we may omit the zeros in 6300 and 840, and use only the 88 in the remainder 882, thus making the work shorter.

We may check our result by multiplying 342 by 21. Then 21 × 342 =7182, the dividend.

The pupil should check his work in enough examples to become thoroughly familiar with the process.

Required to divide 6447 by 21.

307 21 6447

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Here we have 64 ÷ 213, and since the 64 is hundreds, the 3 is written over the hundreds' place. Then 3 x 21 63, and when 63 is subtracted, there is a remainder 1. Bringing down one figure as before, we have 14 tens to be divided by 21. Since 21 is not contained in 14, we write 0 in the tens' place to show that there are no tens, and bring down the next figure 7. Then 147 ÷ 21=7, and 7 × 21= 147 exactly, so there is no remainder.

63
147
147

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