ITALIAN DIVISION. 5 5. Abbreviation in Division (the Italian method). A great saving of labour and time is effected by subtracting in all operations of Division as you proceed. This principle admits of very wide application and will be constantly used throughout this book. Its main applications are to Long Division (Simple and Compound), Division of Decimals, Greatest Common Measure, and Division-approximation. 1°. Long Division. Draw a line under the figures required for the first step in division. Multiply the divisor by the first figure of the quotient, subtracting as you proceed. Bring down the next figure and draw a line as before. Repeat the same process of multiplication and immediate subtraction with the second figure of the quotient and so on. Process for First Subtraction. (a) () (c) 4x5=20+1 3 =23 4x7=28+2=30+8 =38 4x8=32 +3=35+ | 4 =3 4x1= 4+3= 7+. Example. 79832041875. 4832 1329 rem. Thus practically the figure necessary to make up the figure above it is put down and the resulting tens-figure is added to the next multiplication. (a) gives the differences to be put down, (6) gives the figures to be carried, (c) shows the correctness of the subtraction, Similarly for any other line. 2o. Compound Division (Money, Weights, Measures), It is obvious that this principle may be applied and the above method used in all the actual dividing required in Compound Division of any sort. . Example 1. £891. 118. 44d. = 73. Example 2. 361 days 11 hrs. 36 mins. ;-81. £. 8. d. d. h. m. 73 | 891 11 41 | £12. 81 | 361 11 36 4 d. 161 37 15 24 20 899 11 h. 89 8 60 516 30 60 1800 180 18 30. Division of Decimals. This method is plainly applicable to Decimals and should always be used. 6 m. 22 s. Example. 8765.432 = 123.4. 1234 | 87654:32 | 71.03 1274 330 4. Greatest Common Measure. An application of this method renders the finding of G.C.M. surprisingly brief. Example. G.C.M, of 33495, 106260. 5 | 33495 | 106260 | 3 1155 | 6.C.M. 5. 4620 4620 5775. 1. 1155 4. ... G.C.M. is 1155. 1 MONEY DECIMALISATION. 7 EXAMPLES. 1. 865734968-3714. 2. 17345834651:87653. 3. 98534756385- 47351. 4. 876407653126:374582. 5. 93102356321: 7125. 6. 3567025831:871. 7, 9567435612-931524. 8. 51638451784;36195. 9. 16573481297: 81756. 10. 731256851-96145. 11. £891. 58. 6 d. = 732. 12. £73156. 118. 44d.--8154. 13. £7256. 98. 7 d. - 4239. 14. £631. 11s. 5 d. : 81. 15. £7316. 58. 31d. -- 654. 16. £351. 118. 9fd. = 716. 17. £95. 3s. 1fd. = 403. 18. £931. 178. 6td. :814. 19. £75312. 168. 8fd. -8137. 20. £516. 88. 9fd. :-64. 21. 8653:12805 ; 0625. 22. 73451:36:62:314. 23. 876.8314;609:315. 24. 7:31058; .00567. 25. •006785;61:532. 26. 83.00715:3•15814. 27. 1956210;83:514. 28. 00004561 ; 00375. 29. •00031 ; 00006. 30. 165•7312:-81.041. 31. G.C.M. of 86534, 9106. 32. G.C.M. of 75614, 89312. 69751, 6349. 34. 18905, 7300. 35. 743856, 9318. 36. 74165825, 93128625. 37. 94378, 612436. 38. 19500618, 8954. 39. 73256, 6303. 40. 856, 94732. 41. £75. 88. 94d. (in decimals) -- 19 to 3 places. 42. £861. 138. 4£d. -321 43. £7356. 98. 111d. • 709 44. £8340. 11s. 44d. ; 3131 45. £99. 10s. 4£d. : 834 46. £739. 98. 4£d. • 16 47. ' £810. 88. 7£d. ; 1000 48. £537561. 108. 11}d. • 851 49. £9007. 178. 5d. : 7014 50. £10000, Os. Od. • 347 33. 6. The Decimalisation of Money. The conversion of s. d. f. into decimals of a £ often greatly simplifies many calculations—more especially it renders of much greater service the Methods of Immediate Subtraction and Approximation, In general it is sufficient to get any sum of money in decimals of a £ correct to 3 places, but it must be remembered that these places must be correct in the final answer. Hence it is often necessary to find the exact decimal of a £. The methods given here will enable the student after a little practice to decimalise at sight. 1°. To convert £. s. d. f. into £ and decimals of a £. (1) To three places. Put down the tens-figure of the product as Ist dec. place. Reduce d. f. to farthings, adding 1 if pence are 6d. Add unit-figure of shillings-product to tens-figure of the farthings found and put down sum as 2nd dec. place. Put unit-figure of farthings as 3rd dec. place. Put unit-figure of this result (mentally) in the 4th place to see if its multiplication by 4 will have any effect on the carrying from the 3rd x 4 to get the true 4th place. Put down the unit-figure (thus corrected or not) as 4th place. Repeat this process with the 3rd and 4th to get the 5th, and then with the 4th and 5th to get the 6th and or over, So on. (Always carefully making the above correction when necessary.) Note 1. This correction can only occur when the last figure obtained (the figure on which the carrying depends) is 2 or 7. E.g. it can only occur in getting the 4th when the 3rd is 2 or 7, in getting the 6th when the 5th is 2 or 7. METHOD OF FIVE AND FOUR. 9 It does not always occur even then but it must be tried in these cases. If 2 or 7 does not occur at any stage this correction need not be considered. Note 2. This process brings 4999 ..... and therefore, practically, when 48 occurs or 49 after correction, change into 5 and end the decimal. Note 3. Observe—the 1st dec. place is left untouched. Note 4. If it is desirable to stop at any place and make allowance for the next it is easy to calculate it as above, and if it is found to be 5 or over add 1 to the place stopped at. Note 5. The operation of completing the Decimal given in (2) will be referred to as the method of Fours and the whole process of Deci. malisation as the method of Five and Four, Note 6. The student will very quickly know when the repeating figure is reached—the forms which always occur are 416, 83, 916, or 5 (without recurrence). Example 1. £65. 14s. 8 d.= £65•735. Mental process 5 x 14 = 70 735 Example 2. £17. 118. 3d.= £17.565. Mental process 5 x 11=55 565 Example 3. £86. 198. 17d. = 86-95'5/2018 3. Add 14 x 4=.955 4x5-3rd place) 4x5=20+0=20=.95520 (2 rejected: 0 to carry if no correction needed) [Here 4 x 2=8 :: put (mentally) 0 in 5th place and see its effect on 2x4. This effect is of course nothing as 4x0=0 :: no correction is needed.] 4x2=8+0=8=.955208 (0 carried from 4 x 0) 4x0=+3=3=.9552083 (3 carried from 4x8) 4x8=32+1=83=.95520833 3 rejected: 1 carried from 4x3) and so on, the 3 obviously repeating. |