CASE II. 229. To reduce a compound number to a decimal. 1. Reduce 1bu. 2pk. 4qt. to the decimal of a bushel. SOLUTION. By reduction we find 1bu. 2pk. 4qt. equal to 52qt., and 1bu. equal to 32qt.; hence, 1bu. 2pk. 4qt. is, or of a bushel, which reduced to a decimal equals 1.625. OPERATION. 1bu. 2pk. 4qt. 52qt ==1.625, Ans. of the Since SOL. 2D.-Since there are 8qt. in 1pk., number of quarts equals the number of pecks. 4 equals .5, which with 2pk. equals 2.5pk. there are 4pk. in a bushel, of the number of pecks equals the number of bushels. of 2.5 equals .625bu. OPERATION. 814 4 2.5 .625, Ans. RULE.-Reduce as in the same case of common fractions, or divide as in reduction from a lower to a higher denomination. 2. Reduce 5oz. 10pwt. 12gr. to the decimal of a pound. Ans. .4604166+. 3. 3pk. 4qt. 1pt. to the decimal of a bushel. Ans. .890625. Ans. .76875. 4. 93 13 20 8gr. to the decimal of a lb. Ans. .894375. 6. What decimal part of 5gal. is 1qt. 3ggi.? 7. Reduce 7fur. 15rd. 3yd. 2ft. 8in. to the decimal of a mile. Ans. .92408459. ton. 8. Reduce 3qr. 16lb. 12oz. 14dr. to the decimal of a Ans. .04590234375. Ans. .5399-. 5lb. 12oz. AvoirduAns. 1.8229+. 230. MISCELLANEOUS PROBLEMS IN DENOMINATE NUMBERS. 1. If 8 barrels of flour cost 24£ 12s. 8d., what will 12 barrels cost at the same rate? Ans. 36£ 19s. 2. If 7 bales of goods weigh 20cwt. 3qr., what will 56 bales of the same size weigh? Ans. 166cwt. 3. What cost 8 cords of wood, at the rate of 11£ 5d. for 5₫ cords? Ans. 16£ 12s. 23d. 4. What cost Syd. 3qr. of cloth, at the rate of $1.50 per yard? Ans. $39.37. 5. How many cubic feet in 75.125 cords? How many cubic inches in the same? Ans. 9616cu. ft. 6. How many acres in a rectangular lot 65.5 rods long and 35.25 rods wide? Ans. 14A. 1R. 287P, 7. How much is the cost of 24cwt. 3qr. 12lb. of sugar, at $6.50 per hundred weight? Ans. $161.65. 8. A man bought 4hhd. 28gal. 3qt. of wine at $4.50 a gallon; what did it cost? Ans. $1263.375. 9. How many cubic feet in a box 83 feet long, 51 feet wide, and 3 feet in height? Ans. 140cu. ft. 10. An apothecary bought 16lb. 10oz. 8dr. of drugs, at the rate of $12.25 a pound; required the cost. Ans. $204.039. 11. How many acres in a rectangular field 35.5 chains long and 21.8 chains wide? Ans. 77.39 acres. 12. A druggist purchased 201b. 8oz. 12dr. of opium at 484 cents an ounce; what did it cost? $160.265+. 13. What is the area of a field 125rd. 12ft. long and 100rd. 15ft. in width? Ans. 79A. 1R. 7P. 63sq. ft. PROBLEMS FOR ADVANCED PUPILS OR REVIEW. 14. Required the contents of a box 16ft. 8in. long, 8ft. 10in. wide, and 6ft. 6in. high. 15. If 124£ 16s. 6d. are worth $599.16, are 136£ 10s. 6d. worth? 16. In what time will a man walk 120mi. 6fur. 20rd. 4ft., if he goes 12mi. in 3h. 20min.? Ans. 33h. 33min. 33sec.+. 17. The area of a rectangular field is 171A. 3R. 15P., and one side is 35.25 chains; required the length of the other side. Ans. 48.75. 18. If A travels 24mi. 4fur. 38rd. 4yd. in 6h. 30min., how far will he go in 9h. 45min. ? Ans. 36mi. 7fur. 18rd. lyd. 6in. 19. If B digs 363rd. 7yd. of ditch in 35wk. 5da., how long will it take to dig 910rd. 33yd., working 12h. a day, 6da. a week, and 4wk. a month? Ans. 22mo. 1wk. 31da. 20. What cost 321b. 8oz. 10pwt. of drugs, if 6lb. 6oz. 10pwt. cost $31.40? Ans. $157. 21. What cost 5cwt. 2qr. 15lb. of sugar, if .96 of a cwt. cost $7.50? Ans. $44.14. 22. What cost 10cwt. 3qr. 6lb. of hay, if 5cwt. 2qr. 5lb. cost 2£ 10s. 6d.? Ans. 4£ 18s. 4d.+: 23. What cost .75 of an ell English of cloth, at the rate of 4yd. 3qr. for 6£ 15s. 9d.? Ans. 1£ 6s. 9d. 2far.-+ 24. What cost 5.815625 of a pound Troy of drugs, if 13oz. 4dr. cost $76.35? Ans. $441.20—. 25. If a man walk 1931mi. 2fur. in 493da. 6h. 24min., how long will it take him to walk 965.625mi., walking 12h. a day? Ans. 246da. 9h. 12min. SECTION VII. CONTRACTIONS IN MULTIPLICATION AND DIVISION. CONTRACTIONS IN MULTIPLICATION. 231. Contractions in multiplication and division are abbreviated processes for multiplying and dividing. Many of these are very useful in facilitating the mechanical operations. CASE I. 232. When the multiplier is an aliquot part of 10, 100, 1000, etc. 1. To multiply any number by 21, we multiply the number by 10 and divide by 4. SOLUTION.-Suppose we wish to multiply 38 by 23. Since 2 equals of 10, 24 times a number equals of ten times the number; hence, 38 X 2 equals 38 X 10 divided by 4, which equals 95. M OPERATION. 4)380 95, Ans. 2. To multiply by 31, or 5, we annex a cipher to the multipli cand and divide respectively by 3 and 2. 3. To multiply by 121, 163, 25, 331, 50, we annex two ciphers and divide respectively by 8, 6, 4, 3, and 2. 4. To multiply by 125, 1663, 250, 3331, we annex three ciphers and divide respectively by 8, 6, 4, and 3. The following relations will enable us to multiply by other aliquot parts: of 100. 375 g of 1000. § of 100. 625 of 100. 6663 1. Multiply 956 by 121. SOLUTION. Since 12 of 100, 12 times 956 equals of 100 times 956, which equals of 9560011950. 2. 756 by 23. 3. 747 by 31. Ans. 1890. | 4. 895 by 121. Ans. 130900. 20. Multiply 67328 by 831; by 871; by 750; by 8331; by 875; by 750874; by 6258331. CASE II. 233. To multiply when all the digits of the multiplier are 9. 1. Multiply 856 by 999. SOLUTION.-999 times 856 equals (1000-1) times 856, which equals 1000 times 856, minus 856, which equals 856000856855144. Therefore, OPERATION. 856 999 856000 855144, Ans. RULE.-Annex to the number as many naughts as there are 9's in the multiplier, and subtract the number from the result. 234. To square a mixed number in which the fractional part is. The square of a number is the product of the number multiplied by itself. Thus, 3 x 3 or 9 is the square of 3. METHOD.-Multiply the whole number by the next larger whole number, and to the product add 1. DEM. This may be proved by squaring any such mixed number as 7. 7X71 (? + 1) (? + }), and multiplying this as in the margin, we have 7X7+1X7 +, which equals 8X7+, which proves the method. What is the square OPERATION. 7+季 7X7+7X1 7×7+1×7+1=8×7+& Ans. 901. | 5. Of 861? Ans. 74821. 1. Of 91? 4. Of 281? Ans. 8124.8. Of 9999†? Ans. 999900004. |