Division-Decimals. 133. Find the quotient of 785.65 divided by .5. Operation. .5)785.6'5 1571.3 Explanation. First place a séparatrix (v) 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 reached— in 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. 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: (a) Find the sum of the twelve quotients. Division-Denominate Numbers. 138. Divide 46 rd. 12 ft. 8 in. by 4. Operation. 4)46 rd. 12 ft. 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.; 2 rd. equal 33 ft.; 33 ft. plus 12 ft. equal 45 ft. One fourth of 45 ft. equals 11 ft. with a remainder of 1 ft.; 1 ft. equals 12 in.; 12 in. plus 8 in. equals 20 in. One fourth of 20 in. equals 5 in. One fourth of 46 rà. 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. Can you multiply by a number of dollars? 13. Can you divide by a number of dollars? 1. 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. 2. Verify No. 2 by letting a = 3, and b = 5. 3. Verify No. 4 by letting a = 3, b = 5, and c = 7. 141. (6 ×a×a×a×a×a) ÷ (2 × a × a) = 6a3 ÷ 2a2 = 3a3. Observe that to divide one algebraic term by another we must find the quotient of the coefficients and the difference of the exponents. 6. Verify problem 5 by letting a 3 and b = 5. = Algebraic Division. 143. PROBLEMS. 1. Divide 4a3x + 8a2x2 + 6ax3 by 2ax. 2. Multiply the quotient of problem 1 by 2ax. 3. Verify problems 1 and 2 by letting a = 2 and x = 3. 4. Divide 3ab3 + 6a2b2 + 9a3b by 3ab. 5. Multiply the quotient of problem 4 by 3ab. 6. Verify problems 4 and 5 by letting a = 3 and b = 5. 7. Divide 2x3y + x2y2 – xy3 by xy. 8. Multiply the quotient of problem 7 by xy. 9. Verify problems 7 and 8 by letting x = 2 and y 10. Divide 5a3y2 — 2a2y3 + a3y1 by a2y. 11. Multiply the quotient of problem 10 by a'y. = 3. 12. Verify problems 10 and 11 by letting a = 1 and y = 2. 13. Divide 36*x + b3x2 - 3b2x23 by bx. 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 is derived. One half of +8 is + 4; one half of – 6 is – 3. 16. 2x)4x-6x+8x3- 2x2+6x. |