Explanation. As there are no pecks or quarts expressed in the minuend, we take 1 bu. from the 17 bu. and reduce it to lower denomina. tions. 1 bu. = 3 pk. 8 qt. Therefore 17 bu. = 16 bu. 3 pk. 8 qt. The subtraction can now be performed as before. (a.) In examples involving fractional denominations, it will usually be more convenient to make all the reductions and changes in the minuend before beginning to subtract, as in the following: Third Example. What is the difference between 8 rd. 3 yd. 1 ft. 4 in. and 2 rd. 4 yd. 2 ft. 5 in.? WRITTEN WORK. 7 8 2 10 Minuend, changed in form. Explanation. Since there are more yards, feet, and inches expressed in the subtrahend than in the minuend, we will take 1 rd. from the 8 rd. and reduce it to lower denominations. 1 rd. = 5 yd. = 5 yd. 1 ft. 6 in., which added to the 3 yd. 1 ft. 4 in. equal 8 yd. 2 ft. 10 in. Therefore 8 rd. 3 yd. 1 ft. 4 in. · 2 rd. 4 yd. 2 ft. 5 in. = 7 rd. 8 yd. 2 ft. 10 in. — 2 rd. 4 yd. 2 ft. 5 in. NOTE. - Had not the 1 rod been reduced to yards, and the yard to feet and inches, before commencing the subtraction, the answer would have taken the form of 5 rd. 3 yd. 1 ft. 11 in., from which, by reducing the yard to feet and inches, we should get 5 rd. 3 yd. 2 ft. 17 in. = 5 rd. yd. 0 ft. 5 in. = the answer obtained directly by first method. 13. £17 6 s. 8 d. 1 qr. £13 17 s. 3 d. 2 qr.? 14. 25 cwt. 1 qr. 13 lb. 10 oz. 7 dr. 13 cwt. 2 qr. 15 lb 3 oz. 11 dr.? 16 gr.? 17. 4147 bu. 1 pk. 4 qt. 2878 bu. 2 pk. 7 qt. 1 pt.? 18. 49 w. 3 da. 19 h. 13 m. 45 sec. 18 w. 1 da. 22 h. 19. £236 11 s. 8 d. 3 qr.? 40 m. 53 sec.? £487 6 s. 0 d. 1 qr. 20. 27° 24' 47" 19° 37' 51"? 21. 38 m. 4 fur. 23 rd. 4 yd. 0 ft. 3 in. 32 m. 5 fur 28 rd. 5 yd. 1 ft. 5 in.? 22. 54 m. 6 fur. 3 rd. 8 in. 2 ft. 10 in.? 48 m. 3 fur. 3 rd. 2 yd. 23. 3 R. 14 sq. rd. 7 sq. yd. 2 sq. ft. 19 s sq. in. 11 sq. yd. 5 sq. ft. 138 sq. in.? 24. 15 rd. 5 yd. 2 ft. 11 in. 16 rd. 1 ft. 4 in. ? 23 sq. rd. 68. The Changed Minuend not usually written. (a.) The changed form of the minuend has been written in the preceding examples to insure that the nature of the reductions and changes shall be understood by the pupil. It is not, however, customary to write it. The full explanation is the same whether it is written or omitted; but when it is omitted, and every step of the process is understood and mastered, abbreviated explanations like the following may be adopted : WRITTEN WORK OF FIRST EXAMPLE Under 65. First Abbreviated Explanation. 6 hundredths cannot be taken from O hundredths, but 1 tenth = 10 hundredths, and 6 hundredths from 10 hundredths leave 4 hundredths. 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; 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. 5 10 16 18 Remainder. First Abbreviated Explanation. - 23 gr. 17 gr.; but 1 dwt. = 24 gr., and 17 gr. added = 18 gr. cannot be subtracted from are 41 gr. 23 gr. from 41 gr. 18 dwt. cannot be taken from 14 dwt.; but 1 oz. 20 dwt., and 14 dwt. added = 34 dwt. 18 dwt. from 34 dwt. = 16 dwt. 7 oz. cannot be taken from 5 oz.; but 1 lb. = 12 oz., and 5 oz. added are 17 oz. 7 oz. from 17 oz. 10 oz. 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 considering 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. = 10 oz. 8 lb. from 13 lb. leave 5 lb. (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 1 oz., or 20 dwt., = 16 dwt. Subtracting 5 of the 7 oz. leaves 2 oz., and 2 oz. from 1 lb., or 12 oz., (d.) Abbreviating still more, we have 14 from 18 4, and 4 dwt. from 1 oz., or 20 dwt, = 16 dwt. 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. 5. 45842.7 6. 2594.326? g. 2.78457? 100000000 4.000563? 10. 7000000 ..000007? 6243.984? 11. 847.96 47.96823? 7564.001756.4002? 12. 487.6307 48.76307? 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; |