WRITTEN WORE. 49. ILLUSTRATIVE EXAMPLE II. How many barrels of flour at $ 8 a barrel can I buy for $ 2597 ? Explanation. — Here, after dividing, we have a remainder of $5: hence, 324 barrels can be 8) 2597 — 5 Rem. bought and $5 remain unexpended, which 324 may be expressed as in the margin. Ans. 324 barrels; $5 remain. The work may be proved by finding the product of the quotient and divisor (Art. 92, Note III.) and adding the remainder. Thus, 324 x 8 + 5 = 2597. 5. How many weeks are there in 585 days ? in 730 days ? 1st Ans. 83 weeks; 4 days remaini. 6. How many 8 quart cans can be filled with 1865 quarts of milk? with 2587 quarts ? 1st Ans. 233 cans; 1 quart remains. 7. How many years of 12 months each are there in 200 months ? 8. There are in an orchard 1608 trees, 12 in a row. How many rows of trees are there? 9. At 11 cents a yard, how many yards of cloth can I buy for 5972 cents ? 10. At 9 cents apiece, how many oranges can be bought for 29415 cents ? 100. ILLUSTRATIVE EXAMPLE III. If 8 men buy 9675 acres of land which they are to divide equally among themselves, what is each man's share? Ecplanation. - Each one will have 1 eighth of 9675 acres. 8) 9675 acres. We divide, briefly, thus : One eighth of 9 thousand is 1 thousand, and Ans. 12093 acres. 1 thousand (equal to 10 hundreds) remains. One eighth of 16 hundreds is 2 hundreds ; of 7 tens, O tens and 7 tens (equal to 70 units) remain. One eighth of 75 units is 9 units, with a remainder of 3 units yet to be divided. If 1 eighth of each of the 3 acres is taken, we shall have 3 eighths of This we express as in the margin, and have 12093 acres for the entire quotient. WRITTEN WORK. an acre. 11. What is the price of 1 hat if 6 hats cost 375 cents ? it 12 cost 2700 cents ? 1st Ans. 622 cents. 12. How far must a man travel each day to go 1761 miles in 4 days ? in 9 days ? 1st Ans. 4404 miles. 13. Mr. Stewart promises to sell me 5 rods of land for $ 1578. What is his price per rod ? 14. At $8 a thousand, how many thousands of bricks can be bought for $ 3287 ? 15. A man left by his will $ 45267 to be divided equally among his 6 children. What should each child receive ? 16. Eight times a certain number equals 324787. What is that number? 17. How many 9's are there in 10000 ? 18. To what number is 368,57 equal ? 19. To what number is 1493 6 7. equal ? 20. How many are 10101019 = 7? 21. How many are 98306572 = 5 ? 22. Divide 864024 by 7. 24. Divide 369801 by 9. 23. Divide 164408 by 8. 25. Divide 120087 by 11. 12 101. Division of Decimals. WRITTEN WORK. ILLUSTRATIVE EXAMPLE IV. What is 1 twelfth of 109.92 ? Esplanation. - Briefly thus: 1 twelfth of 109 is 9, and 1 remains; of 19 tenths is 1 tenth, and 7 tenths 12) 109.92 remain; of 72 hundredths is 6 hundredths. Ans. 9.16. 9.16 In the example above it will be seen that we have hundredths in the quotient as there are hundredths in the dividend. In dividing a decimal by a whole number, the quotient is of the same denomination as the dividend. In dividing a decimal by a whole number, fix the decimal point in the quotient as soon as you reach the decimal point in the dividend. 26. What is 1 fifth of 86.4055 ? 27. What is 1 eighth of 94076.8 ? (28.) $234.54 = 9 = ? (29.) $ 907.34 + 7 = ? WRITTEN WORK. 102. To Divide, carrying the Division to Decimals. ILLUSTRATIVE EXAMPLE V. Find 1 eighth of 9675 acres. Explanation. We divide as in Illustrative Ex ample III., until we come to the remainder, 3 acres. 8) 9675.000 This we change to 30 tenths. One eighth of 30 1209.375 tenths is 3 tenths, and 6 tenths remain, which are equal to 60 hundredths. One eighth of 60 hundredths is 7 hundredths, and 4 hundredths remain, which are equal to 40 thousandths. One eighth of 40 thousandths is 5 thousandths. The entire quotient is 1209.375 acres. Perform Examples 11 to 16 in Article 100, carrying the division to decimals. 103. Where the divisor is not greater than 12, it is customary to divide as shown above without expressing all the operations. Such a process is short division. For other examples in short division, see pages 61 and 63. LONG DIVISION. WRITTEN WORK. 104. ILLUSTRATIVE EXAMPLE VI. Divide 33075 by 82. Explanation. — We write the dividend and divisor as in the margin, and draw a curved 82) 33075 (403;} line at the right of the expression for the divi328 dend. 275 Since the divisor 82 is a larger number than 246 3 or than 33, we first divide 330 hundreds by 82. Now 330 divided by 82 will give about the 29 same quotient as 33 divided by 8,* which is 4. The first term of the quotient is then 4 hundreds, which we express by writing a figure 4 at the right of the curved line. Multiplying 82 by 4 hundreds, and subtracting the product, we find 2 hundreds remain ; uniting with these 2 hundreds the 7 tens of the dividend, we have 27 tens. Dividing the 27 tens by 82, we have no tens in the quotient ; so we write a zero to show that there are no tens in the quotient, and unite with the 27 tens the 5 units of the dividend, making 275 units. * So we make 8 our trial divisor. Dividing the 275 units by 82, using 8 for a trial divisor, we have 3 units in the quotient, which we write. Multiplying and subtracting as before, 29 units remain. Dividing each of the 29 units by 82, we have 32, which we write with the units, and have for the entire quotient 403 3. 105. When the divisor is larger than 12, it is usually convenient to express in full, as above, the work of dividing. The process is then called long division. WRITTEN WORK. To Divide, carrying the Division to Decimals. 106. ILLUSTRATIVE EXAMPLE VII. Divide 33075 by 82. Explanation. — We divide as in the last 82) 33075 (403.35... illustrative example until we reach the re328 mainder, 29 units. We now put a decimal point in the expression for the quotient, 275 and, changing the remainder to 290 tenths, 246 divide as before; and so we keep on dividing 290 Tenths. as far as desirable, or until there is no re246 mainder. In this example we stop dividing 440 Hundredths. at hundredths, and indicate that the divis ion is incomplete by writing a few dots. 410 107. Give answers to the following 30 examples as in Art. 104, or with the quotient carried to thousandths, as the teacher may direct : * 30. Divide 4684 by 31. 34. Divide 12157 by 23. 31. Divide 9632 by 43. 35. Divide 24898 by 72. 32. Divide 5872 by 54. 36. Divide 36872 by 84. 33. Divide 6748 by 62. 37. Divide 36072 by 91. 108. ILLUSTRATIVE EXAMPLE VIII. Divide 1849 by 192. Explanation. — As 192 is nearly 200, 1849 192) 1849 (91%} divided by 192 will give about the same quo1728 tient as 1800 divided by 200, or as 18 divided 121 by 2. We then make 2 our trial divisor. WRITTEN WORK. * The answers in the Key are given in both forms. 38. Divide 26832 by 96. 40. Divide 232848 by 56. 39. Divide 97684 by 79. 41. Divide 682345 by 88. 109. From the preceding examples we derive the following Rule for Division. 1. Write the dividend ; at the left draw a curved line : and at the left of this line write the divisor. 2. Divide the highest term or terms of the dividend by the divisor. 3. Express the result for the first term of the quotient at the right in long division, beneath in short division. 4. Multiply the divisor by this term. 5. Take the product thus obtained from the part of the dividend used. 6. Unite the next term of the dividend with the remainder for a new partial dividend ; divide, multiply, and subtract as before; and so continue till all the terms of the dividend are used. * 7. Express the division of the final remainder, should here be any, in the fractional form. (Or Change the remainder to tenths, hundredths, thousandths, etc., and continue the division as far as desirable.) Proof. Find the product of the quotient and divisor, and add to it the remainder, if there is one. The result ought to equal the dividend. 42. How many are 36247 = 189 ? 43. How many are 53004 - 398 ? 44. How many are 932480 - 287 ? 45. How many are 750010 - 677 ? * If at any time the divisor is not contained in a partial dividend, write a zero for the next figure of the quotient, and unite with the partial dividend the next term of the given dividend. |