86. RELATIONS OF NUMBERS. COMPOUND NUMBERS Reduced TO FRACTIONS AND DECIMALS. 1. What part of 4 is 1? What part is 2 3 4 5? What part of 5 is 3? 2? 4? 8? 2. What part of 15 is 1? 8? 7? 9? 13? 37? 26? 44? 3. What part of £1 is 1s.? 2s.? 69.? 18s.? 15s.? 4. How many pence are there in £1? 3d.? 9d.? 3s. 5d.? 4s. 6d.?! ld.? What part of a pound is 8s. 3d.? 13s. 6d.? 5. What part of a pound Troy is 1 oz. 3 oz.? 11 oz.? 1 dwt.? 3 oz. 5 dwt.? 6. What part of a bushel is 1 pk.? 3 pk.? 1 qt.? 3,qt.? 3 pk. 5 qt. 1 pt.? 7. What part of 75 gallons is 1 gal. 47 gal.? 1 qt.? 8 gal. 3 qt.? 8. How many farthings are there in £1? 9. What part of £1 is 1qr.? 1d. 3qr.? 8d. 3qr.? In £1 4s. 6d. ? 6d. 3qr.? 15s. 10. What part of 1 pound Troy is 1 gr.? 17 gr.? 4 oz. 16 dwt. 22 gr. ? 11. What part of 1 gallon are 3 qt. 1 pt. 3 gi.? 12, 13, 14. Reduce the last three answers to decimal expressions. (78, RULE.) ££.0010416; 2916. 15. Reduce 35 rods to the decimal of a mile. 320=64 35 = £.007 16. Reduce 34 rods 3 yards to the decimal of a mile. 17. Reduce 15 rods 5 yards 2 ft. 8 in. to the decimal of a mile. 18. What part of 1 week are 1 d. 7 h.? Express it decimally. Compound numbers may also be reduced to decimals of a higher denomination, by the following method. 19. Reduce 3 d. 15 h. 45 m. to the decimal of a week. 601 45 24 15.75 7 min. 3.65625 d. We first divide 45 min. by 60, to reduce them to the decimal of an hour, and annex it to the 15 hours; we then have 15.75 hours. Dividing 15.75 hours by 24, reduces them to the decimal of a day; which added to the 3 days, gives 3.65625 days. Dividing the days by 7, reduces them to decimals of a week. .52232+wk. The examples in question 9 may be performed in the same Perform the 10th, 11th, 15th, 16th, 17th and 18th questions by this method. 20. Reduce 1 R. 15 sq. rd. 24 sq. yd. to the decimal of an acre. 21. Reduce 45 d. 17 h. 35 min. 17 sec. to the decimal of a year. 22. What part of 5 d. 13 h. 5 min. are 4 d. 6 h. 5 sec.? 23. What part of 12 cwt. 3 qr. is 1 lb.? 1 cwt. 2 qr. 18 lb.? 24. What part of 5 m. 2 fur. 8 rd. are 15 rd. 4 yd. 2 ft.? 25. What part of $6.05 is $3? $4.10? $1.005? $0.06? $0.105 ? 26. What part of $37 is $4.15? $0.108? $8.415? $0.005? 87. TO REDUCE THE FRACTION OF ONE DENOMINATION TO THE FRACTION OF ANOTHER DENOMINATION. 1. Reduce of an acre to the fraction of a square rod. 2. Reduce of a penny to the fraction of a pound. 3. Reduce .07 of a yard to the fraction of an inch. 4. Reduce .3 of a rod to the fraction of a mile. TT NOTE 1. Specific rules for solving such problems as the above seem to be unnecessary; for to reduce of an acre to the fraction of a square rod is the same, in principle, as to reduce any number of acres to square rods. To do either, we multiply the acres or parts of an acre by 4, to reduce them to roods or parts of a rood; and these roods or parts of a rood by 40, to reduce them to square rods or parts of a square rod. Thus, of an acre are equal to A. X4f R., and if R. X 40-1440 sq. rd. So, .07 of a yard are equal to .07 yd. X 3= .21 ft. ; 21 ft. X 122.52 in. = NOTE 2. Again, to reduce pence or parts of a penny to pounds, we divide the pence or parts of a penny by 12, to reduce them to shillings or parts of a shilling; and these by 20, to reduce them to pounds or parts of a pound. Thus, d. 12s., and Tes. ÷ 20 = £3ʊ (79); or, id. X 12 X 28=£30. So, .3 of a rod÷ 40.075 of a fur., and .075 fur. 8.008375 miles. Therefore, If the fraction is to be reduced to a lower denomination, MULTIPLY it, and if to a higher, DIVIDE it, by the same numbers that you would use in reduction of whole numbers. (43.) 5. Reduce of a £ to the fraction of a farthing. Solution. X 20 X 12 × 1 =129qr. 6. Reduce .03 of a shilling to the fraction of a farthing. Solution. .03 X 12 × 4= 1.44qr. 7. Reduce to the fraction of a pint 18 of a peck. of a gallon. of a bushel. 8. Reduce .4 of a pint to the fraction of a gallon. 85 9. What part of a lb. are 15 of a cwt.? of a qr.? 10. What part of a £ are of a penny? Solution. d. X 12 X 20: = 11. What part of a lb. Troy are of a grain? of a dwt.? of an oz.? 12. Reduce of a rod to the fraction of a mile. 13. What part of an acre are of a sq. rod? 14. What part of a nail are .085 of a qr.? .17 of a yard? 15. What part of an inch are .08 of a yard? foot? II. 16. What part of a mile are inch? 17. What part of an acre is 18. What part of a year are 19. What part of a minute are .0003 of a year? week? .1084 of a of a foot? of an of a sq. yard? of a sq. ft.? of a day? of an hour? .04 of a .106 of .35 of a .35 of a 20. Reduce to the fraction of a mile 35 of a yard. a foot. 21. What part of an inch are .0006 of a mile? rod? 22. What part of 3 inches are .0006 of a mile? rod? 23.. What part of 5 miles are .35 of a yard? .106 of a foot? of an inch? 27. What part of 33 miles are of a foot? 88. FRACTIONAL COMPOUND NUMBERS REDUCED TO THEIR VALUES IN LOWER DENOMINATIONS. Specific rules for solving such questions as the above are unnecessary. Reduce the following fractions to their values in lower denominations: 3. £; s.; .3d. (4.) .67 lb. Troy; oz. Troy. 8. fur.; .35 rod; .871 yd; foot. 9. Reduce to cents and mills $3; 81; 10. Reduce to cents and mills $3 $20, §; $1; $17. 11. What is the value of .1456 of a £? .5716 of a shilling? .1847 of a square yard? . 12. What is the value of of a bushel? of a peck? II. 13. Reduce to its value in lower denominations of a mile. 14. Reduce of an acre; .3 of a rood; of a square rod. 15. Reduce to their values in lower denominations of a square mile; .5816 of an acre; of a year; .1084 of a mile; .30716 of a cord; .1516 of a cubic yard; of one deg.; of a cord. 89. ADDITION OF COMMON FRACTIONS. If fractions have a common denominator, they are added by adding their numerators and placing the sum over the common denominator. Thus, +⭑+&+Ã==1&• RULE. Having reduced compound fractions to simple ones, and all the fractions to their lowest terms, reduce them to their least common denominator, by Art. 84. Then add their numerators, and place the sum over the common denominator. If there are mixed numbers to be added, add the integers and the fractions separately. 1. What is the the sum of and? Of 41+2+2? Of 83 +5+? Of & + + 153 ? 2. What is the sum of +Of+? Solution. }=}; = 3. What is the sum of 4+33? Say, 4 and 3 are 7; }=}, ?=1; ++7=14, which added to 7 makes 84. 4. Find the sum of 143 +32; of 1 of 4+ of }. 5. Find the sum of and; of of 3+ of 8; of §+12. 6. Add 188152; 78 +38. 7. Add of a bushel + of a peck of a quart. Ans. 3 pk. 5 qt. 1 pt. NOTE. of a bushel 3 pecks; g of a peck: 5 quarts, &c. 8. Add of a week + § of a day + of an hour +1% of a minute. 9. What is the sum of £+s. + {d.? $3+§ ct. + Å m.? |