ANSWER. —3 lb. is the same part of 4 lb. that 3 is of 4, which is 3. 18. What part of 9 lb. 7 oz. 13 dwt. is 1 lb. 3 oz. 15 dwt. 171 gr.? = SOLUTION. - Reducing both quantities to the lowest denomination mentioned, i. e. to fourths of a grain, we have 9 lb. 7 oz. 13 dwt. 222048 of a grain; 1 lb. 3 oz. 15 dwt. 174 gr. 30309 of a grain, and 30800 is the same part of 22 2220 2048 that 30309 is of 222048, which is 30309 222048 Hence, 1 lb. 3 oz. 15 dwt. 174 gr. is 40198 of 9 lb. 7 oz. 13 dwt. or 1010 74016 (d.) The above solutions show that we may find what part one compound number is of another of similar denominations, by reducing both to the lowest denomination mentioned in either, and making the value of the number of which the fractional part is required the denominator, and the value of the other number the numerator. The fraction thus obtained should be reduced to its lowest terms. 34. Of 1 m. 6 fur. 43 rd. is 1 m. 2 fur. 19 rd.? 37. Of 1 T. is 13 cwt. 2 qr. 17 lb. 6 oz. 52 dr.? (f.) The following method is often better than the preceding, when we wish to find what part a compound number is of a unit of a higher denomination. 38. What part of 1 mile is 4 fur. 26 rd. 3 yd. 2 ft. ? 1ST SOLUTION.-Since 1 ft. which adding the 3 yd. gives 3 = of a yd., 2 ft. must equal yd. or 11 yd. rd. = of a yard, to of a rod,* Since 1 yd. = of a rod = of a rod, to which adding Since 1 rod of a fur., to which adding the 4 fur. gives of a mile, 4 fur. must equal 14 of = of a mile. Hence, 4 fur. 26 rd. 3 yd. 2 ft. of a mile. = ΤΣ 2ND SOLUTION.-There must be as many yards as feet, or, in 2 ft. yd. of a yard. But there must be as many rods as yards, or, yd., there must be of 33 rods of a rod. But there must be = many furlongs as rods, or, etc. This reduction is the opposite of that explained in 89. (g.) What part 39. Of £1 is 11s. 1 d. 14 far.? 40. Of 1 gal. is 2 qt. 1 pt. 14 gi.? 41. Of 1 wk. is 6 da.? 42. Of 1 cord is 6 cd. ft. 6 cu. ft. 691 cu. in.? * For 1 rod = 51 yd. = of 2 in 3 as 11 of a yd., and 1 yd. or 2 of a yd. is the same that 2 is of 11, which is 2. part of 11 * 43 Of 1 acre is 2 R. 36 sq. rd. 11 sq. yd.? 46. Of 1 degree is 47′ 463/? (h.) Should it be required to give the answers in a decimal form, it will only be necessary to reduce the vulgar fractions, obtained as above to decimal fractions. (i.) The process illustrated in the following solution may also be employed. 47. What part of 1 lb. is 3 dwt. 5.76 gr.? 20 SOLUTION. There must be as many pennyweights as grains, or, in 5.76 gr. there must be 4 of 5.76 dwt. = .24 of a dwt., to which adding the 3 dwt. gives 3.24 dwt. But there must be as many ounces as pennyweights, or, in 3.24 dwt., of 3.24 oz. = .162 oz., to which adding the 6 oz. gives 6.162 oz. But there must be as many pounds as ounces, or, in 6.162 oz., of 6.162 lb. = .5135 of a lb. Hence, 6 oz. 3 dwt. 5.76 gr. .5135 of a lb. 12 20 = NOTE.-The most convenient form of writing the work for the last solution, is to write the numbers to be reduced in a vertical column, the smallest denomination uppermost, as indicated below. (j.) What decimal part — 48. Of £1 is 9s. 2d. 1.12 far.? 49. Of 1 ton is 7 cwt. 14 lb. 6 oz. 6.4 dr.? 50. Of 1 week is 12 h. 46 min. 4.8 sec.? 51. Of 1 league is 2 rd. 3 yd. 1 ft. 8.6208 in.? 52. Of 1 rood is 7 sq. rd. 8 sq. yd. 8 sq. ft. 84.384 sq. in.? 53. Of 1 cord is 1256.32512 cu. in.? 54. Of 1 bu. is 1 pk. 7 qt. 1.6 gi.? 55. Of 1tb is 43 63 .96 gr.? 56. Of 1 mile is 2 ft. 7.68 in.? 57. Of 1 acre is 30 sq. rd. 4 sq. ft. 51.264 sq. in.? 91. To multiply by a Fraction. (a.) Multiplying a number by 1 gives the number itself for a product. Hence, multiplying a number by 2 of 1, or 3, must give of the number for a product; multiplying a number by .23 must give .23 of the number for a product; and, generally— (b.) To multiply a number by a fraction is merely to obtain such a part of it as the fraction indicates. (c.) Hence, to multiply by a fraction, we have only to multiply by the numerator of the fraction and divide by the denominator. 1. What is times 254? SOLUTION.-7 times 254 equals 7 of 254, which, found as explained in 86, is 2221. NOTE. The fraction in each answer should be reduced to its lowest terms. (d.) What is the product of 2. times 783? 3. times 4.86? 4. times 6.345? 5. times 868 ? 10. What is the product of SOLUTION.& multiplied by = 1 of 3, which is found by writing the fraction and then making 11 a factor of the numerator and 12 a factor of the denominator, as in the written work below. (e.) The above solutions show that to multiply vulgar fractions together is the same thing as to reduce compound fractions to simple ones, (See 87.) 5901 2329 1686 199.791 23.7 8.43 711 948 1896 199.791 = 1ST SOLUTION.-23.7 times 8.43 23.7 of 8.43, which, found by multiplying 8.43 by 237 and removing the point one place further to the left (86, 20th example) is 199.791. = 2D SOLUTION.-3.7 multiplied by 8.43 8.43* of 23.7, which, found by multiplying 23.7 by 8.43 and removing the point two places further to the left, is 199.791. NOTE. The slight difference in the above solutions of the last example results from the different reading of the sign of multiplication. The student should not be confined to either form, but should be prepared to use the one that is most convenient in the example he is considering. When mixed decimal numbers are to be multiplied together, the multiplier should in the solution be read as an improper fraction. The multiplicand may be read either as a mixed number or as an improper fraction. Thus, in the first of the above solutions, "23.7 of 8.4" should either be read 237 of 838, or 237 of 8.43; while in the second "8.4 of 3.7" should either be read of, or if of 31. |