Division-Decimals. 133. Find the quotient of 785.6 divided by .5. Operation .5)785.6'5 1571.3 Explanation. First place a separatrix (♥) after that figure in the dividend that is of the same denomination as the right hand figure of the divisor—in this case, after the figure 6. Then divide, writing the decimal point in the quotient when, in the process of division, the separatrix is reachedin this case, after the figure 1. It was required to find how many times 5 tenths are contained in 7856 tenths. 5 tenths are contained in 7856 tenths 1571 times. There are yet 15 hundredths to be divided. 5 tenths are contained in 15 tenths, 3 times; in 15 hundredths, 3 tenths of a time. NOTE.-By holding the thought for a moment upon that part of the dividend which corresponds in denomination to the divisor, the place of the decimal point becomes apparent at once. 5 apples are contained in 7856 apples, 1571 times. 134. Solve and explain the following problems with special reference to the placing of the decimal point : DENOMINATE NUMBERS. 138. Divide 46 rd. 12 ft. 8 in. by 4. Operation. 4)46 rd. 12ft. 8 in. 11 rd. 11 ft. 5 in. Explanation. This means, find 1 fourth of 46 rd. 12 ft. 8 in. One fourth of 46 rd. is 11 rd. with a re mainder* of 2 rd.; 2rd. equal 33 ft. 33 ft. plus 12ft. equal 45 ft. One fourth of 45 ft. equals 11 ft. with a remainder of 1 ft.; 1 ft. equals 12in.; 12 in. plus 8 in. equals 20 in. One fourth of 20 in. equals 5 in. One fourth of 46 rd. 12 ft. 8 in. equals 11 rd. 11 ft. 5 in. PROBLEM. The perimeter of a square garden is 46 rd. 12 ft. 8 in. How far across one side of it? 139. MISCELLANEOUS. Tell the meaning of each of the following, solve, explain, and state in the form of a problem the conditions that would give rise to each number process. 1. Multiply 64 rd. 14 ft. 6 in. by 8. 2. Divide 37 rd. 15 ft. 4 in. by 5. 3. Divide $675.36 by $48. 4. Divide $675.36 by 48. 5. Divide $675.36 by .48 6. Divide $675.36 by $4.8 7. Divide $675.36 by 4.8 8. Divide $675.36 by $.48 9. Multiply $356.54 by .36 10. Multiply $356.54 by 3.6 11. Multiply $356.54 by 36. 12. Multiply $275.56 by 2.25. 13. Multiply $275.56 by 24. 14. Can you multiply by a number of dollars? 15. Can you divide by a number of dollars? * The word remainder in this connection suggests incomplete division. When the division is complete there can be no remainder. 1. Observe that in the above examples we divide each term of the dividend by the divisor. 2. Prove Nos. 1 and 3, by (1) reducing each dividend to its simplest form, (2) dividing it so reduced, by the divisor, and (3) comparing the result with the quotient reduced to its simplest form. 3. Verify No. 2, by letting a = 3, and b = 5. 4. Verify No. 4, by letting a = 3, b = 5, and c = 7. 141. (6 χαχαχαχαχα) ÷ (2 ×a×a) = 6 a5 ÷ 2 a2 = 3 a3. Observe that to divide one algebraic term by another we must find the quotient of the co-efficients and the difference of the exponents. 6. Verify Problem 5, by letting a = 3, and b = 5. Algebraic Division. 143. PROBLEMS. 1. Divide 4 a3x + 8 a2x2 + 6 ax3 by 2 ax. 2. Multiply the quotient of problem 1, by 2 ax. 3. Verify problems 1 and 2, by letting a = 4. Divide 3 ab3 + 6a2b2 + 9 ab by 3 ab. 2, and x = 5. Multiply the quotient of problem 4, by 3 ab. b 3. 6. Verify problems 4 and 5, by letting a = 3, and 6 = 5. 7. Divide 2 x3y + x22 – xy3 by xy. 8. Multiply the quotient of problem 7, by xy. 9. Verify problems 7 and 8, by letting x = 2, and y = 3. 10. Divide 5 a3y2 - 2 a3y3 + ay by ay. 11. Multiply the quotient of problem 10, by a3y. 12. Verify problems 10 and 11, by letting a = 1, and y = 2. 14. Multiply the quotient of problem 13, by bx. 15. Verify problems 13 and 14, by letting b 3, and x = = 4. Observe that when the divisor is a positive number, each term of the quotient has the same sign as the term in the dividend from which it was derived. One half of +8 is + 4; one half of 6 is – 3. 16. 2x)4x-6x+8x3- 2x2 + 6x. 17. Verify by letting x = 2. |