First Abbreviated Explanation. O hundredths, but 1 tenth hundredths leave 4 hundredths. = 6 hundredths cannot be taken from 10 hundredths, and 6 hundredths from 10 8 tenths cannot be taken from 6 tenths, but one unit equal 10 tenths, and 6 tenths added 16 tenths. 8 tenths from 16 tenths: ― 8 tenths. 5 units cannot be taken from 1 unit, but 1 ten 10 units, and 1 unit added = 11 units. 5 units from 11 units leave 6 units. 3 tens from 5 tens leave 2 tens. Hence the answer is 26.84. Second Abbreviated Explanation. 6 hundredths from 10 hundredths leave 4 hundredths; 8 tenths from 16 tenths leave 8 tenths; 5 units from 11 units leave 6 units; 3 tens from 5 tens leave 2 tens. This gives 26.84 for an answer, as before. (b.) All explanations should finally be dropped, and only results named, thus:-4 hundredths, 8 tenths, 6 units, 2 tens; giving for the answer 26.84, as before. WRITTEN WORK OF FIRST EXAMPLE UNder 66. 17 gr.; - Subtrahend. Remainder. 23 gr. cannot be subtracted from but 1 dwt. = 24 gr., and 17 gr. added are 41 gr. 23 gr. from 41 gr. Hence the remainder is 5 lb. 10 oz. 16 dwt. 18 gr. 69. Subtrahend Figure may be increased instead of diminishing Minuend. It is obvious that the result would be the same, if, instead of consid ering the minuend figure of a denomination from which a reduction has been made to be one less, we should consider the corresponding subtrahend figure to be one greater. In the former case, we subtract 1 (on account of the reduced unit) before subtracting the subtrahend figure while in the latter we add 1 to the subtrahend figure, and subtract both together. Thus, in subtracting the tenths of the first example, we may subtract 1 tenth from the 7 tenths before subtracting the 8 tenths, or we may add the 1 tenth to the 8 tenths, and subtract both at once. Many always subtract by the last method. One method is as convenient as the other, but the one to which we are most accustomed will seem the easiest. 70. Another, and usually shorter, Method of subtracting Compound Numbers. (a.) When in compound subtraction reductions are necessary, and the changed minuend is not written, one of the following methods is often, if not usually, easier than that hitherto taken. First. Subtract from the value of the reduced unit, and add the remainder to the minuend figure; or, Second. Subtract as many as possible from the written minuend figure, and the rest from the value of the reduced unit. (b.) Applying the first of the above methods to the example just considered gives the following work : — Subtracting 17 of the 23 gr. from the 17 gr. leaves 6 gr. to be taken from 24 gr., (the value of the. reduced unit.) 6 gr. from 24 gr. = 18 gr. Subtracting 14 of the 18 dwt. from the 14 dwt. leaves 4 dwt. to be subtracted from 20 dwt., (the value of the reduced unit.) 4 dwt. from 20 dwt. = 16 dwt. Subtracting 5 of the 7 oz. from the 5 oz. leaves 2 oz. to be taken from 12 oz., (the value of the reduced unit.) 2 oz. from 12 oz. 8 lb. from 13 lb. leave 5 lb. 10 oz. (c.) In practice, the above method may be abbreviated thus: Subtracting 17 of the 23 gr. leaves 6 gr., and 6 gr. from 1 dwt., or 24 gr., 18 gr. Subtracting 14 of the 18 dwt. leaves 4 dwt., and 4 dwt. from 1oz., or 20 dwt., = 16 dwt. Subtracting 5 of the 7 oz. leaves 2 oz., and 2 oz. from 1 lb., or 12 oz. = 10 oz. 8 lb. from 13 lb. = 5 lb: (d.) Abbreviating still more, we have 17 from 236, and 6 gr. from 1 dwt., or 24 gr., = 18 gr. 14 from 18= 4, and 4 dwt. from 1 oz., or 20 dwt., = 16 dwt. 5 from 7: = 2, and 2 oz. from 1 lb., or 12 oz., = 10 oz. 3 lb. from 13 lb. == 6 lb., giving, as before, 5 lb. 10 oz. 16 dwt. 18 gr. (e.) Adopting the second method, we have 23 gr. from 1 dwt., or 24 gr., = 1 gr., which added to the 17 gr. gives 18 gr. 18 dwt. from i oz., or 20 dwt., = 2 dwt., which added to the 14 dwt. gives 16 dwt. 7 oz. from 1 lb., or 12 oz., = 5 oz., which added to the 5 oz. gives 10 oz. 8 lb. from 13 lb. = 5 lb., giving, as before, 5 lb. 10 oz. 16 dwt. 18 gr. NOTE.- Practice will make the student so familiar with all these methods, that he will be able to see at once which is best adapted to the case he is considering. 13. A land company bought 8479 acres of wild land, and sold 3896 acres of it. How many did they have left? Reasoning Process. - If they bought 8479 acres, and sold 3896 acres of it, they must have left the difference between 8479 acres and 3896 acres, to find which we subtract 3896 from 8479. 14. A ship is valued at $27648, and its cargo at $49325. How much more is the cargo worth than the ship? Reasoning Process. If the ship is worth $27648, and the cargo is worth $49325, the cargo must be worth as many dollars more than the ship as there are in the difference between $49325 and $27648, to find which the latter must be subtracted from the former. 15. Census returns show that the United States contained 3929827 inhabitants in the year 1790; 5305941 in 1800; 7239814 in 1810; 9638191 in 1820; 12866020 in 1830; 17069453 in 1840; and 23263488 in 1850. How many more did they contain in 1800 than in 1790? 16. How many more in 1810 than in 1800? 3 oz. 9 dr.? 18 bu. 3 pk. 3 qt.? 47 yd. 3 qr. 2 na.? 15 lb. 7 oz. 3 dr. 2 cwt. 3 qr. 24 lb. 27. 37 gal. 2 qt. 1 pt. 2 gi. 12 gal. 3 qt. 1 pt. 3 gi. 4 lb. 7 oz. 5 dwt. 13 gr.? 5 lb 113 43 2 14 gr.? 2 qr. 3 na. ? £28 14 s. 7 d. 2 qr. ? 32. 187 T. 3 qr. 13 lb. 13 dr.? 67 T. 17 cwt. 1 qr. 18 lb. 5 oz. 33. 27 sq. rd. 5 sq. ft. 17 sq. in. 7 sq. ft. 53 sq. in.? 34. 6 fur. 8 rd. 3 yd. 1 ft. 1 in. 11 in. ? 35. 18 yd. 1 na. 36. 6 m. 1 ft. 37. 18 m. 1 fur. 8 rd. 4 yd. 2 ft. 14 yd. 2 qr. 3 na. ? 5 m. 7 fur. 39 rd. 5 yd. 1 ft. 2 in. ? 17 m. 7 fur. 39 rd. 5 yd. 1 ft. 5 in. ? 38. 231 A. 19 sq. rd. 197 A. 3 R. 27 sq. rd. 15 sq. yd. 40. 307 T. 8 cwt. 2 qr. 23 lb. 8 oz. 12 dr. 431 yd. 2 qr. 3 na. 2 in.? 27 m. 7 fur. 39 rd. 5 yd. 2 ft. 3 in.? 43. Bought 7 T. 14 cwt. sold 3 T. 7 cwt. 3 qr. 26 lb. 1 qr. 19 lb. of hay, from which I How much had I left? · 44. A goldsmith bought 7 lb. 7 oz. of gold. How much will he have left after manufacturing and selling 3 lb. 5 gr. of it? 45. A trader sold 9 yd. containing 27 yd. 1qr. 1na. piece? qr. 2 na. of cloth from a piece How much was left in the 46. A man set on foot to travel from Boston to Springfield, the distance being 98 miles. The first day he travelled 28 m. 7 fur. 19 rd., the second 24 m. 6 fur. 28 rd., the third 29 m. 4 fur. 36 rd. How far was he from Springfield at the end of the third day? 47. A man undertook to dig a ditch for a certain price per rod. On completing it he demanded payment for a ditch 37 rd. 0 ft. 3 in. long. His employer, doubting his honesty, measured it, and found it to be but 36 rd. 5 yd. 1 ft. 9 in. long. A dispute arising between them, they called in Mr. Jenks to settle it, agreeing to abide by his decision. He measured the ditch, and found its length to be 36 rd. 16 ft. 9 in. What was the difference in their measurements? 72. Subtraction from Left to Right. We can begin at the left to subtract as well as at the right, if we are only careful to reserve one for reduction from each denomination in the minuend when it is required by the lower denominations. This reduction will be necessary when the figures in the subtrahend at the right of the denomination considered are greater than the corresponding ones of the minuend. Example. How many are 508.935 249.748? 508.935 = Minuend. 259.187 Remainder. Explanation. Reserving 1 hundred from the 5 hundreds, we have, 2 hundreds from 4 hundreds 2 hundreds. Reducing the 1 hundred reserved to tens, and reserving I ten for further reduction, we have, 4 |