CASE II. 63. To divide by Long Division. 1. Let it be required to divide 34531 by 15. sands, 2 thousands times; we write the 2 thousands in the quotient. 15 X 2 thousands = 30 thousands, which, subtracted from 34 thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we have 45 hundreds. 15 in 45 hundreds, 3 hundreds times; we write the 3 hundreds in the quotient. 15 × 3 hundreds = 45 hundreds, which subtracted from 45 hundreds, leaves nothing. Adding the 3 tens, we have 3 tens. 15 in 3 tens, 0 tens times; we write 0 tens in the quotient. Adding to the three tens, which equal 30 units, the 1 units, we have 31 units. 15 in 31 units, 2 units times; we write the 2 units in the quotient. 15 X 2 units 30 units, which, subtracted from 31 units, leaves 1 unit as a remainder. Indicating the division of the 1 unit, we annex the fractional expression, unit, to the integral part of the quotient. Therefore, 34531 divided by 15 is equal to 2302. In practice, we may say: 15 in 34, 2 times; write 2 in the quotient; 15 × 2=30, which from 34 leaves 4. Bring down 5; 15 in 45, 3 times; write 3 in the quotient; 15 × 3 = : 45, which from 45 leaves 0. Bring down 3; 15 in 3, 0 times; write 0 in the quotient. Bring down 1, 15 in 31, 2 times; 15 X 2 = 30, which from 31 leaves 1. Answer, 2302 Explain the operation. When, as above, the work of dividing is mostly written out, the process is called LONG DIVISION. RULE. Write the divisor at the left of the dividend. Begin at the left, divide the number expressed by the fewest figures of the dividend that will contain the divisor, and write the quotient at the right of the dividend. Multiply the divisor by this quotient; subtract the product from the part of the dividend used, and to the remainder bring down the next figure of the dividend. Divide as before, till all the figures of the dividend have been used. If there be a final remainder, write it, with the divisor beneath, after the integral part of the quotient. PROOF. The same as in Short Division. Examples. 2. How many times is 24 contained in 7816? 4. 1618 by 9. 7. 86009 by 63. Ans. 13651. 8. 18570 by 34. 5. 15702 by 11. 6. 25620 by 12. Ans. 1427· 9. 5783 by 108. Ans. 53. When is the process called Long Division? Repeat the Rule. the Proof? What is 15. 13354÷ 17= how many? 18. 10064 110= how many? 31. Divide 6421284 by 642. Ans. 10002. 32. Divide 120345 by 3102. Ans. 383163. 33. Divide six million three hundred forty-six thousand two hundred and sixty-nine by one thousand two hundred and sixty-nine. Ans. 5001. 34. Divide two million nine hundred fifty-three thousand and seventy-nine by one thousand seven hundred and twentyeight. Ans. 17081988 64. When the divisor is 10, 100, 1000, etc., the quotient may be obtained, at once, by removing the decimal point in the divi‹ dend, as many places to the left as there are ciphers in the divisor For, since the value denoted by figures is multiplied by 10 by re moving the decimal point one place to the right, by 100 by removing it two places, etc. (Art. 54); and as division is the reverse of multiplication (Art. 60), removing the decimal point in the dividend one place to the left divides it by 10, two places divides it by 100, etc. REVIEW QUESTIONS. What is Multiplication? (47) Division? (57.) How may the quotient be obtained when the divisor is 10, 100, 1000, etc. ? The integral part of the quotient will be on the left of the decimal point, and the remainder will be the part on the right of the point. Thus, 124310=124.3 = 124, read, one hundred twentyfour units and three tenths; 1243 ÷ 100 = 12.43 = 12,43, read, twelve units and forty-three hundredths; 5004 ÷ 1000 5.0045100, read, five units and four thousandths, etc. That is, The first order at the right of the decimal point expresses tenths; the second, hundredths; the third, thousandths, etc. Divide 35. 1463 by 10. Ans. 146. 39. 60013 by 1000. 36. 6700 by 100. Ans. 60-80 37. 16301 by 100. Ans. 163o. 40. 33444 by 10000. 38. 85761 by 1000. 41. 80000 by 1000. Ans. 80. 65. When the divisor has any number of significant figures with ciphers on the right, the work may be abridged by cutting off the ciphers at the right of the divisor and an equal number of jigures from the right of the dividend, and then dividing the remaining part of the dividend by the remaining part of the divisor, and, if there be a remainder, prefixing it to the figures that were cut off from the dividend for the entire remainder. 42. Find how many times 1700 is contained in 39792. What will mark the parts of the quotient? What will the part on the right of the point be? How may the work be abridged when the divisor has any number of significant figures with ciphers on the right? In the operation, what is the remainder found by using the hundreds in dividing? What is taken to form the entire remainder? 1. James Brown bought a farm of 387 acres for 8514 dollars; how much did it cost him per acre? Ans. 22 dollars. 2. An army of 9870 men lost one-fifteenth of its whole number in storming a fort; how many were lost? Ans. 658 men. 3. If a field of 109 acres produces 3379 bushels of wheat, how much is the yield per acre? Ans. 31 bushels. 4. If a field yielding at the rate of 31 bushels to the acre produces 3379 bushels of wheat, how many acres are there in the field? 5. How long should 17 men subsist upon a supply of provisions which would suffice for one man 6205 days? 6. A teamster removed 31340 bricks at 20 loads; how many did he remove at a load? Ans. 1567 bricks. 7. If the valuation of a certain town of 7500 inhabitants is 2625000 dollars, what is the average to each individual ? Ans. 350 dollars. 8. How many years must a person labor to accumulate 23100 dollars, by saving 1100 dollars a year? 5* |