90. One Number a Fractional Part of Another. 1. What part of 3 is 1 I? ANSWER. — 1 is } of 3, because times 3. times 1 = 3, or because 1 taken three (a.) What part — 4 Of 9 is 1 ? of 12. (b.) What part — 6. Of 7 is 4 ? 8. Of 10 is 9? 7. Of 10 is 6? 9. Of 9 is 10? 6 10. What part of 4 lb. is 3 lb. ? 15. Of 1% is ? 16. Of .64 is .48? 14. Of is a ? 17. Of .5 is .3? 18. What part of 9 lb. 7 oz. 13 dwt. is 1 lb. 3 oz. 15 dwt. 174 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. = of a grain; 1 lb. 3 oz. 15 dwt. 174 gr. 30309 109 of a grain, and 30 000 įs the same part of 2220 48 that 30309 is of 222048, which is 30,309 or 48/98 Hence, 1 lb. 3 oz. 15 dwt. 174 gr. is 481 8 of 9 lb. 7 oz. 13 dwt. (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. 2 2 2 0 48 2220489 (e.) What part- 26. Of 171% is 1214? 21. Of 4 qt. 1 pt. is 3 qt.? 27. Of 44 is 3 ? 22. Of 3 da. is 1 wk.? 28. Of is 1 ? 23. Of £7 is 78.? 29. Of 6 is .06 ? 24. Of 3 qt. is 1 pk.? 30. Of .08 is 1? 25. Of 2 qr. 23 lb. is 2 qr. 22 lb.! 31. Of 1.4 is 6.2? 32. Of £1 13s. 3d. 1 far, is 8d. 2 far.? 33. Of 3 bu. 2 pk. 54 qt. is 1 bu. ? 34. Of 1 m. 6 fur. 42 rd. is 1 m. 2 fur. 19 rd. ? 35. Of 1 wk. is 5 da. 3 h. 15 min. 37.3 sec. ? 36. Of 1 da. 13 h. 19 min. 24.7 sec, is 1 wk.? 37. Of 1 T. is 13 cwt. 2 qr. 17 lb. 6 oz. 53 dr. ? (f.) The following method is often better than the preceding, when we wish to find what part & 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. ? ist SOLUTION. – Since 1 ft. of a yd., 2 ft. must equal şof a yard, to which adding the 3 yd. gives 3; yd. or } yd. Since 1 yd. = î of a rod, * of a yard must equal şof ů of a rod of rod, to which adding the 26 rd. gives 26; rd. Bord. Since 1 rod = io of a furlong ord. must equal 30 of z' of a fur. of a fur., to which adding the 4 fur. gives 4; fur. = fur. Since 1 fur. of a mile, ļ4 fur. must equal of of a mile Te of a mile. Hence, 4 fur. 26 rd. 3 yd. 2 ft. of a mile. 2nd Solution. There must be į as many yards as feet, or, in 2 ft. of 2 yd. = 's of a yard. But there must be i as many rods as yards, or, in 3 yd., there must be i of 3j rods = of a rod. But there must be to as many furlongs as rods, or, etc. NOTE. - This reduction is the opposite of that explained in 89. 12 * For 1 rod = 51 = 51 yd. of a yd., and 1 yd. or of a yd. is the same part of ų that 2 is of 11, which is i. 43 Of 1 acre is 2 R. 36 sq. rd. 11 sq. yd. ? (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. 1 (i.) The process illustrated in the following solution may also be employed. 47. What part of 1 lb. is 3 dwt. 5.76 gr.? SOLUTION. - There must be as many pennyweights as grains, or, in 5.76 gr. there must be at 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., z 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., it of 6.162 lb. = .5135 of a lb. Hence, 6 oz. 3 dwt. 5.76 gr. = .5135 of a lb. NOTE. — The most convenient form of writing the work for the last solation, is to write the numbers to be reduced in a vertical column, the smallest denomination uppermost, as indicated below. 24.)5.76 gr. 3 dwt. 5.76 gr. .5135 lb. 6 oz. 3 dwt. 5.76 gr. (j.) What decimal part- 91. To multiply by a Fraction. (a.) Multiplying a number by 1 gives the number itself for a product. Hence, multiplying a number by 1 of 1, or ], must give h 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 į times 254 ? SOLUTION.— times 254 equals 1 of 254, which, found as explained in 86, 2224 NOTE. — The fraction in each answer should be reduced to its lowest is terms. 8 22 (d.) What is the product of 2. & times 783 ? 6. 4 times 7.92? 3. times 4.86 ? 7. ig times 6.07 ? 4. } times 6.345 ? 8. f times 4.158? 5. $ times 868 ? 9. times 943 ? 10. What is the product of f x ft? SOLUTION.— multiplied by 11 = 1t of , 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. 2 3 11. What is the product of 34 X 2 X 18? Solution. — 34 x 25 x 15 = } X Š XY. But 7 multiplied by f = $ of ], and this product multiplied by 4 = 4 of f of 7. Hence the following written work : 2 7 X 8 X 12 34 X 2 X 1 = = 16. % X 9 X 7 (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.) Find the following products: 12. & x ? 20. Š X X ? 13. Į x ? 21. $ x 11 x 1%? 14. f xf? 22. s x x 5? 15. _ x $ 23. 3} x 14 x ? 16. f x? 24. X X 214? 17. 3} x 1? 25. 4} X X 7%? 18. 4 Xb4? 26. 81 X 5 X 2} ? 27. 274 x 63 x 4}? 92 Co multiply by a Decimal Fraction. 2.43 23.7 1. What is 23.7 X 8.43 ? 0901 2529 1686 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. 199.791 23.7 2D SOLUTION. — 3.7 multiplied by 8.43 = 8.43 * of 8.43 23.7, which, found by multiplying 23.7 by 8.43 and 711 removing the point two places further to the left, is 948 199.791. 1896 Note. — The slight difference in the above solutions 199.791 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 of 172, or 237 of 8-40%; while in the second “8.4 of 3.7” should either be read 18 of ij, or if of 3 jo. |