Division- Decimals. 133. Find the quotient of 785.6 divided by .5. Operation. .5)785.6 5 Explanation. First place a separatrix (ˇ) after that figure in 1571.3 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 at once. 5 apples are contained in 7856 apples, 1571 times. 5 tenths are contained in 7856 tenths, 1571 times. 134. Solve and explain the following problems with special reference to the placing of the decimal point: Division - United States Money. 135. Divide $754.65 by $.27. Operation. $.27)$754.65 (2795 54 214 189 256 243 135 135 Explanation. This means, find how many times 27 cents are contained in 75465 cents. 27 cents are contained in 75465 cents, 2795 times. PROBLEM. At 27¢ a bushel, how many bushels of oats can be bought for $754.65? As many bushels can be bought, as $.27 is contained times in $754.65. It is contained 2795 times. Therefore, 2795 bushels can be bought. 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. mainder* of 2 rd.; 2 rd. Explanation. This means, find 1 fourth of 46 rd. 12 ft. 8 in. One fourth of 46 rd. is 11 rd. with a reequal 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 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 21. 14. Can you multiply by a number of dollars? 15. Can you divide by a number of dollars? (P. 329.) * 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 ×a×a×a×a×a)÷(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 coefficients and the difference of the exponents. 2 a)6 a5b+4 a1b2 — 8 a3b3 + 10 a2b1 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. 4. Divide 3 ab3 + 6 a2b2 + 9 a3b by 3 ab. 5. Multiply the quotient of problem 4, by 3 ab. 6. Verify problems 4 and 5, by letting a = 3, and b = 5. 7. Divide 2x3y + 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 a2y3 + a2y by a2y. 11. Multiply the quotient of problem 10, by a2y. 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)4x5 - 6x4 + 8 x3 − 2 x2 + 6x. 17. Verify by letting x = 2. |