LIBRARY OF THE UNIVERSITY OF CALIFORNIA. OR Received Accession No. (9 7j~ J / . Class No. ROBINSON'S MATHEMATICAL SERIES. KEY PROGRESSIVE HIGHER AEITHMETIC. FOE TEACHEKS AND PRIVATE LEARNERS. NEW YORK: IVISON, PHINNEY, BLAKEMAN & CO. CHICAGO: S. C. GRIGGS & Co. 1866. R O B I N S O N'S The 'most COMPLETE, most PRACTICAL, and most SCIENTIFIC SEE ES of MATHEMATICAL TEXT-BOOKS ever issued in this country. I. Robinson's Progressive Table Book. ..... II. Robinson's Progressive Primary Arithmetic, - III. Robinson's Progressive Intellectual Arithmetic, - IV. Robinson's Rudiments of Written Arithmetic, - V. Hobiiison's Progressive Practical Arithmetic, - VI. Robinson's Key to Practical Arithmetic. ..... vii. Robinson's Progresses Si^ii^r arithmetic, - vm. Robinson's Key to Higher Arithmetic, - IX. Robinson's New .Elementary Algebra, - X. Robinson's Key to Elementary Algebra, - XI. Robinson's University A Igebra, - - - XII. Robinson's Key to University Algebra, ..... XIII. Robinson's New University Algebra, ..... XIV. Robinson's Key to ]Mew University Algebra, - ... AV . Robinson's New Geometry and Trigonometry, - XVI "rZoblnson's Surveying and navigation, ..... XVII. Hobinson's Analyt. Geometry and Conic Sections, XVIII. Robinson's Differen. and Int. Calculus, (in preparation ,)- XIX. Robinson's Elementary Astronomy, ...... XX Kobinson'3 University Astronomy. ...... XXI. Robinson's Mathematical Operations, XXIi. Robinson's Key to Geometry and Trigonometry, Conic Sections and Analytical Geometry, - - - . Entered, according to Act of Congress, in the year 1SfiA w DANIEL W. FISH & J, H. FRENCH, an* I ngnin i:i the year l>63, by DANIEL W. FISH. A.M., In the Clerk's Office of the District Court ,>f the TnHetl States for tbe Na them District of tlio New York. PROMINENT CHARACTERISTICS OF ROBINSON'S MATHEMATICAL SKRIES, The books of this Series, although many of them have so recently been published, have been recommended and adopted by hundreds of the most critical and su^vssful teachers, for the following reasons : 1. For the philosophical and scientific arrangement of the subjects. 2. For the conciseness of the rules and the brevity and accuracy of the definitions. 3. For the rigid and logical, yet full and comprehen- sive Analysis. 4. For the new, original, and improved methods of operations, not found in most other works of the kind. 5. For the very largo mnitbe.r and variety of practical examples practical, because adapted to the ordinary transactions of business life. 6. For their typographical execution, substantial bind" ing, and general attractiveness. 7. For the easy gradation and progressiveness, not only in the several books that compose the series, but in the arrangement and treatment of the subjects of each book. 8. For their adaptation to the various grades of schol- arship in all our schools. 9. For the general unity of their plan, and the clear- ness and perspicuity of their style. 10. For, scientific accuracy, combined with practical utility, throughout the whole. KEY. ADDITION. (67, page 25.) Ex. 3 An*. 1982738. Ex. 5. An*. 3189. Ex. 7. An*. 415184. Ex. 13. An*. 16977. Ex. 17. An*. 6076510. Ex. 4. Ans. 435058. Ex. 6. An*. 289142. Ex. 12. An*. 3001623. Ex. 16. An*. 1881. Ex. 20. An*. 21184000 Ex. 22. Ans. $1924950. ADDING TWO OR MORE COLUMNS AT ONE OPERATION. (68, page 28.) Ex. 5. Number of churches, 35887; " " persons accommodated, 13847902; Value of church property, $85774659. Ex. 6. Pounds of butter, 312625306; cheese, 105735893; " " wool, 52516961; Bushels of wheat, 100485844. SUBTEACTION. (75, page 31.) Ex. 5. Ans. 174333815. Ex. 6. Ans. 2361650877. Ex. 7. An*. 86602389426. Ex. 8. Ans. 9000989311. (25- 31) [5] 6 SIMPLE NUMBERS. Ex. 10. An*. 86 years. Ex. 13. Ans. $44656513 Ex. 15. Ans. 2121108 square miles; 316636286 population, Ex. 20. Ans. 2657043. TWO OB MORE SUBTRAHENDS. (76, page 34.) Ex. 5. 4568 Ex. 6. 4756+575+1404-84=5555 1320 1200 275 750 320 96 2653 Ans. 3509 Ans. Ex. 7. $15760 Ex. 8. $75860 2175 45640 3794 25175 4587 $5045 An? $5204 Ans. Ex. 9. 20000 Ex. 10. 398470 11000 157548 7000 143429 Ans. 2000 square miles. 97493 Ans. Ex. 11. $61307088 Ex. 12. $5760+$3575=$9335 52889800 2746 234000 4632 $8233288 Ans. $1957 Ans. (31-34) Ex. 13. 643J66 MULTIPLICATION. 7 Ex. 14. $8186793 65038 114624 Ans. 463504 Ex. 15. $12722470 7821556 424497 2355016 Ans. $2121401 5700314 904299 $1582180 Am. MULTIPLICATION. (85, page 38.) Ex. 7. Ans. 43506216. Ex. 16.^. 24500. Ex. 20. Ans. $909000. Ex. 8. Ans. 48288058. Ex. 19. Ans. $31647000 POWERS OF NUMBERS (91, page 39.) Ex. 1. 72X72=5184. Ex.2. 12X12X12X12X12=248832. Ex. 3. 25X25X25=15625. Ex.4. 7X7X7X7X7X7X7=823543. Ex. 5. 19X19X19X19=130321. Ex. 6. 3X3X3X3X3X3=729 (34-40) 8 SIMPLE NUMBERS. Ex.7. Ans. 9 5 =59049; 11 3 =1331; 18 2 =g24; 244140625; 786 2 =617796; 94^=689869781056; 100 4 =100000000; 17 3 =4913; 251 5 =996250626251. Ex.8. 8 3 =512; 15*=225; and 512x225=115200, Ans. Ex. 9. 25 2 =625; 3 4 =81; and 625x81=50625, Ans. Ex. 10. 7 3 X200=68600; 4 4 xll*=30976; and 68600 30976=37624, Ans. CONTRACTIONS IN MULTIPLICATION. (98, page 42.) Ex. 1. 736X6X4=17664, Ans. Ex. 2. 538X8X7=30128, Ans. Ex. 3. 27865X7X3X4, or 27865X7X12,=2340660,^/U Ex.4. 7856X4X4X3X3, or 7856X12X12= 1131264, Ans. Ex. 5. $185X8X7=$10360, Ans. Ex. 6. 17740872X8X12=1703123712 cubic feet, Ans. (99.) - Ex. 3. Ans. 50000 dollars. Ex. 4. Ans. 100000000000 (100, page 43.) Ex. 3. Ans. 10350000. Ex. 5. Ans. 192128000 (102, page 41 ) Ex. 1. 5784 Ex. 2. 3785 246 721 34704 26495 138816 79485 1422864 Ans. 2728985 Ans. (40-44) MULTIPLICATION. Ex. 3. 472856 54918 4255704 8511408 25534224 25968305808 Ans. Ex. 5. 573042 24816 4584336 9168672 13753008 14220610272 Ans. Ex. 7. 43725652 5187914 393530868 787061736 306079564 612159128 218628260 226847922169928 Ans. Ex. 9. 2703605 4249784 18925235 132476645 113551410 227102820 11489737271320 Ans. Ex. 4. 43785 7153 131355 656775 306495 313194105 Ex. 6. 78563721 127369 707073489 2828293956 2121220467 78563721 10006582580049 An* A Ex. 8. 3578426785 64532164 14313707140 57254828560 114509657120 17892133925 229019314240 230923624151612740 An* Ex. 10. 9462108 16824 75696864 227090592 151393728 159190504992 Ans. (44, 45) 10 SIMPLE NUMBERS. EXAMPLES COMBINING THE PRECEDING RULES. (Page 45.) Ex. 1. #28 X 175=44900; $37X320=$11840 5 $4900+ $11840=$16740, Ans. Ex. 2. $1200 ($364+$275+$150+$187)=$224; and $224X5=81120, Ans. Ex. 3. 29+32=61; 61X17=1037 miles, Ans. Ex.4. 834X127=84318; $47X97=$4559; and $4318+ $4559=18877, cost; 127+97=224; $40x224=88960, sold for ; $8960 $8877=$83, profit, Ans. Ex.5. 77+56=133; 675 133 = 542, multiplicand. 3X 156=468; 21428=186; 468 186=282, multiplier. 542X282=152844, Ans. Ex.6. 37+50=87; 87X6=522; 98+522=620, multipli- plicand. 6450=14; 14x5=70; 7010=60, multi- plier. 620X60=37200, Ans. Ex. 7. 14X25=350; 9x36 = 324; 350324 + 4324= 4350, multiplicand. 280112=168; 376 + 42 = 418; 418X4=1672; 168+1672=1840, multiplier. 4350X 1840=8004000, Ans. Ex. 8. $2751X29967= $82439217 $5030x23905=$120242150 $37802933 Ans. Ex. 9. 1449075X203=294162225 acres cultivated; 1922890880294162225=1628728655 acres, Ans Ex. 10. $2258+$105=$2363, valuation per farm; $2363x1449075=13424164225, Ans. Ex.11. 2 4 X5 5 =50000; 7 3 =343; 50000343=49657, Ans. (45, 46) M ULTIPLICATION. i 1 V Ex.12. 15=3375; 3 2 X2 5 =288; 208^=43264; 9x2 4 =144. 3375+43264=46639 ; 288+144=432 ; 46639 432=46207, Ans. Ex. 13. 4+27+256+3125+46656=50068, Ans. Ex. 14. 1200000X400=480000000 pounds, Ans. Ex. 15. $2450, value of house ; $2450X6 $500=14200 r farm; $2450X2= 4900, stock; s. $21550, total value. Ex.16. 1500 X $7=$10500 ; 800x*10=$8000; 700 X$6=$4200; $8000 + $4200=$12200; $12200 $1050C=$1700, Ans. Ex. 17. ($450+$780+$1250+$2275)X3=$14265,^t*. Ex. 18. $115X35000=$4025000, Ans. Ex. 19. $485X2500 =$1212500 $1450X10 == 14500 $1250X25 = 31250 $1258250 Ans. Ex. 20. 1401944X$20=$28038880, value of double eagles; 62990 X $10= 629900, eagles; 154555 X $5= 772775, half eagles; 22059 X $3= 66177, " " $3 pieces. Ans. $29507732, total value. DIVISION. (Ullage 49.) Ex. 1. Am. 78972. Ex. 2. Ans. 121562. Ex. 3. Ans. 152329. ^x. 4. ^ns. 6086847. (46-49) 12 SIMPLE NUMBERS. Ex. 9. Am. 7198. Ex. 10. Ans. 7071. Ex. 11. Ans. 15607. Ex. 12. Ans. 48340 2 f 2 . Ex. 13. Ans. 1253974? |. Ex. 14. Ans. 5479f| jf. Ex. 15. Ans. 2084768|f ff . Ex. 16. Ans. 24781. Ex. 17. Ans. 5851fff. Ex. 18. Ans. 591862f{}f Ex 19. Ans. 15395919^f fii. Ex. 20. Ans. Ex. 21. $147675^365^404f J | .4ns. Ex. 22. $30732518--556= $55274i|| Ans. Ex. 23. $5572470-v-287=$19416^ Ans. Ex. 24. $8186793--27977=4292^f of ABBREVIATED LONG DIVISION. (112 page 51.) Ex. 1. 204)77112(378 Ans. 159 163 Ex.2. 72)65664(912 Ans. 8 14 Ex. 3. 209)7913576(37864 Ans. 164 180 133 83 Ex. 4. 698)6636584(9508 Ans. 354 55 Ex. 5. 8903)4024156(452 Ans. 4625 . 1780 (49-51) DIVISION. Ex. 6. 6791)760592(112 Am. 814 1358 Ex. 7. 25203)101443929(4025^^ Am. 631 12786 1854 Ex. 8. 269181)1246038849(4629 An*. 169314 78062 242262 Ex. 9. 56240)2318922(41if |>f Ant. 6932 13082 Ex. 10. 17300)1454900(84^^% An*. 7090 1700 CONTRACTIONS IN DIVISION. (121 page 57.) Ex.1. 3(435 Ex.2. 7)4256 5)145 8)608 29 An*. 76 Am Ex. -3. 9)17856 Ex.4. 2)15288 8)1984 3)7644 248 Am. 7)2548 364 Ana. (51-57) 14 SIMPLE NUMBERS. Ex. 5, 8)972552 Ex. 6. 9)526050 7)121569 7)58450 3)17367 2)8350 5789 Ans. 4175 Am Ex. 7. 7)612360 5)87480 3)17496 5832 Ans. Ex.8. 3)553 5)184 1 Quotient, 36 - - 4x3=12 13, remainder. Ex.9. 3)10183 5)3394 - 1 7)678 - - - 4X3=12 Quotient, 96 - 6X^X3=90 103, remainder. Ex. 10. 2)10197 3)5098 1 4)1699 1X2= 2 5)424 - - -3X3X2=18 Quotient, 84 - 4x4x3x2=96 ^ 117, remainder. (57) DIVISION. 15 Ex. 11. 3)29792 8)9930 2 6)1241 2X3= 6 Quotient, 206 - - -5x8X3 =120 128, remainder Ex. 12. 4)73522 6)18380 - 2 7)3063 2X4= 8 Quotient, 437 - - - 4x6x4= 96 106, remainder Ex. 13. 3)63844 5)21281 1 9)4256 .... 1X3= 3 Quotient, 472 . - - 8x5X3=120 124, remainder. Ex. 14. 2)386639 3)193319 1 4)64439 2X2= 4 5)16109 - .-- 3XBX2= 18 6)3221 - - 4X4X3X2= 96 Quotient, 536 5X^X4X3X2=600 719, remainder (57) 16 SIMPLE NUMBERS. Ex. 15. 4)734514 6)183628 2 7)30604 4X4= 16 Quotient, 4372 18, remainder. Ex. 16. 9)636388 9)70709 7 9)7856 5X9= 45 Quotient, 872 8x9X9=648 700, remainder, Ex. 17. 5)4619 5)923 4 5)184 3X5= 15 Quotient, 36 4x5x5=100 119, remainder, Ex. 18. 3)116423 7)38807 - - 2 7)5543 6X3= 18 8)791 - 6X7X3= 126 9)98 7X7X7X3=1029 Quotient, 10 - - - 8x8x7X7x3=9408 10583, remainder. (57) DIVISION. 17 Ex. 19. 5)79500 5)15900 5)3180 . 7)636 7)90 6X5X5X5= 750 Quotient, 12 - - - - 6x7X5x5x5=5250 6000, remainder. (122, page 58.) Ex. 2. AM. 79-&V Ex. 4. Ans. 230 T V$ft. (123.) Ex. 2. Ans. 27f3. Ex. 6. Ans. 8206|f$j|. Ex. 7. ^ws. 3005. EXAMPLES COMBINING THE PRECEDING RULES. (Page 59.) Ex.1. $4X25=$100; $3x36=$108; $100+$108= $208 ; 2088=26, Ans. Ex. 2. $10x12=8192; $13xl7=$221; $192+$221= $413, cost; $18 X (12+17) = $522; $522 $413= $109, Ans. Ex. 3. $2X300+$750=$1350, value of produce t $3X120+ $90= $450, stock; $900--25=$36, AM. (57-59) 18 SIMPLE NUMBERS. Ex.4. 450+(24 12) X 5=510; (90-^-6)+ (8 X 11) 18=30 ; 510-r-30=17, Am. Ex.5. 648 x (3^X23)319 = 5184; 2910-f-15=194 ; 5184 194=4990, dividend; 4375-^-175=25 ; 25x4 2 + 3 2 =409 ; 2863 ~ 409 = 7, divisor. Hence, 4990-*- 7=712f , Ans. Ex. 6. 42X34=1428; 107100-^-1428=75, Ans. Ex. 7. Reversing the fifth operation, 12x24=288; reversing the fourth operation, 288-7-6=48 ; reversing the third operation, 48 +(28 16)=60; reversing the second operation, 60 (7 2 +l)=10; reversing the first operation, 10x45=450, Ans. Ex.8. $60 $42=$18; $36x50=81800; 1800-^18=100 months, Ans. Ex. 9. 251104-j-472=532, Ans. Ex. 10. 30422=9253764, Ans. Ex. 11. 453x307+109=139180, Ans. Ex. 12. $4+$7=$ll ; $1276--$11=116, number of each kind; 116x2=232, whole number purchased; $7 $4=$3; $3XH6=$348, difference in cost. Ex. 13. $950+$7500=$8450; $13982686-=-$8450=1654, and a remainder i of $6386. Ex. 14. 854x4300000-5-860=3870000 tons, Ans. Ex. 15. ^3191876~400=57979f flf. Hence, by this est* mate, 57979 persons died. Ex. 16. 508464-f-10593=48, Ans. (59, 60) PKOBLEMS. 19 Jlx 11, $7680-r-$64=120, number sold; $960-f-120=$8, gained per head; $64 $8=$56, cost per head ; $9800-^456=175, number bought. Ex.18. $95X6+$1200=$1770; $1770-:-30=$59, Am. Ex. 19. 36X16=576, number of days' work required; 576-r-24=24, number of days 24 men will require. Ex. 20. $1650H-275=$6, cost per barrel; ($9 $6) X 186=$558, gain. Ex. 21. 840-5-(5+10+15)=28, of each kind; hence, 28X 3=84, whole number. Ex. 22. $965 ($5X160) = $165; 165--3 = 55 tons, un- sold; 160+55=215 tons bought. Ex. 23. $3825-s-$85=45, number sold; $7560-v-($85-f-$5)=84, whole number of horses; ($7560+$945) $3825=$4680, to be realized on the remainder. Hence, $4680-K84 45)=$120, Am. Ex. 24. $22360+$1742=$24102, total cost; $15480--$18=860; 860x2=1720, No. acres; $22360~-1720=$13, original cost per acre. PROBLEMS IN SIMPLE INTEGRAL NUMBERS. (127, page 62.) The following are the general? solutions : Prob 1. Add the several numbers. Prob 2. Subtract the sum of the given numbers from the sum of all. Prob. 3. Add the parts. Prob. 4. Subtract the sum of the given parts from the whole (60-63) 20 SIMPLE NUMBERS. Prob. 5, Subtract the less from the greater. Prob. 6. Subtract the difference from the greater. Prob. 7. Add the difference to the less. Prob. 8. Subtract the subtrahend from the minuend. Prob. 9. Subtract the remainder from the minuend. Prob. 10. Add the subtrahend and remainder. Prob. 11. Multiply the numbers together. Prob. 12. Divide the product by the given factor. Prob. 13. Divide the continued product by the product of the given factors. .Prob. 14. Multiply the factors together in continued multi- plication. Prob. 15. Multiply the multiplicand by the multiplier. Prob. 16. Divide the product by the multiplicand. Prob. 17. Divide the product by the multiplier. Prob. 18. Divide each number by the other. Prob. 19. Divide the dividend by the divisor. Prob. 20. Multiply the divisor and quotient together. Prob. 21. Divide the dividend by the quotient. Prob. 22. Multiply the divisor by the quotient, and to the product add the remainder. Prob. 23. Subtract the remainder from the dividend, and di- vide the result by the quotient. Prob. 24. Multiply the final quotient and the several divisors together. Prob. 25. Divide the first dividend by the continued product: of the final quotient into all the given divisors. Prob. 26. Divide the dividend by the several divisors sue- cessively. Prob. 27. Add together the numbers comprising each set, and subtract the less sum from the greater. (63, 64) FACTORING. 21 Piob 28. Multiply together the factors comprising each set, and add the several products. Prob. 29. Multiply together the factors comprising each set, and then add the products and given numbers. Prob. 30. Multiply together the factors comprising each set, and subtract the less product from the greater. Prob. 31. Add the product of the given set or sets of fac- tors and the given number or numbers. Prob. 32. Subtract the sum of the products of the set or sets of factors which form the less number from the sum of the products of the set or sets of fac- tors which form the greater number. PROPERTIES OF NUMBERS. FACTORING. (142, page 71.) Ex. 1. Ans. 2, 5, 5, 43. Ex. 2. Ans. 3, 5, 163. Ex. 3. Ans. 2, 2, 3, 3, 5, 5, 7. Ex. 4. Ans. 2, 2, 2, 2, 2, 2, 2, 2 ,2, 2, 3, 7. Ex, 5. Ans. 2, 7, 13, 13. Ex. 6. 2, 2, 2, 5, 5, 5. Ex. 7. Ans. 5, 5, 5, 5, 5, 5, 5, 5. Ex. 8. Ans. 3, 3, 3, 7, 11, 13, 37. (144, page 74.) Ex. 2. Am. 2, 3, 7, 17, 17, 29. Ex. 3. Ans. 13, 17, 31 Ex, 4. Ans. 17, 19, 29. Ex. 5. Ans. 2, 11, 19, 487. Ex. 6. Ans. 7, 83, 103. Ex. 7. Ans. 97, 103. *f Ex. 8. Ans. 3, 5, 59, 139. Ex. 9. Ans. 3, 5, 7, 47, 181. (64-74) 22 PROPERTIES OF NUMBERS. Ex. 10. Am. 2, 2, 2, 2, 2, 2, 41, 149. Ex. 11. Ans. 7, 11, 37, 79. Ex. 12. Ans. 2, 5, 13, 17, 37. Ex. 13. Ans. 13, 17, 29. Ex. 14. Ans. 2, 2, 2, 3, 17, 19, 23. Ex. 15. Ans. 2, 3, 5, 7, 19, 179. (145, page 76.) Ex. 1. 120=1X2X2X2X3X5 1, 2, 4, 8 Combinations of 1 and 2. 3, 6, 12, 24 1, 2, and 3. 5, 10, 20, 40 ) "123 and 5 15, 30< 60, 120 J 1,4*,M Ans. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 Ex, 2. 84=1X2X2X3X7 1, 2, 4 Combinations of 1 and 2. 3, 6, 12 1,2, and 3. 7, 14, 28 } u u - o o fln j 7 21, 42, 84} 1, Z,d,and7. . 4tw. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. Ex. 3. 100=1X2X2X5X5 1, 2, 4 Combinations of 1 and 2. 5 > 10 > 20 l 1 2 and 5 25, 50, 100 } 1*3, and ^rcs. 1, 2, 4, 5, 10, 20, 25, 50, 100. Ex. 4. 420=1X2X2X3X5X7 1, 2, 4 Combinations of 1 and 2. 3, 6, 12 1,2, and 3. if; $ 6?} " "1,2, 3, and 5. 7, 14, 28) 21 > 42 > 84 I 1 2 3 5 and 7 35, 70, 140 f *' ^ 6 > *> and 7 ' 105, 210, 420 J . 1 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, " {30, 35, 42, 60, 70, 84, 105, 140, 210, 420. (74-76) GREATEST COMMON DIVISOR. 23 Ex. 5. 1050=1X2X5X5X3X7 1, 5, 25 Combinations of 1 and 5. 2, 10, 50 " 1, 5, and 2. 8 > 16 > 75 1 "152 and 3 6, 30, 150 j I, 0, ^, ai 7, 35, 175 1 14, 70, 350 I K 21* 105* 525 i ' > ; ; a 42^ 210 ? , 1050 J ( 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42, I 50, 70, 75, 105, 150, 175, 210, 350, .525, 1050. GREATEST COMMON DIVISOR. (149, page 77.) Ex. 2. 2X3=6, An*. Ex. 5. 6x7=42, Am Ex.8. 3X3X7=63, ^ras. Ex.9. 91, Ans. Ex. 11. 4X3X7=84, Ans. (150, page 81.) Ex. 4. Ans. 11. Ex. 5. ^is. 1. Ex. 7. -4ns. 17. Ex. 8. Ans. 337 Ex. 10. In the operation under this rule, the quotient figure may always be so taken that the product shall be either greater, or less, than the dividend; in either case, the new divisor will be the difference between the dividend and product. It will always be found advantageous to use that quotient figure which will give the least number for a new divisor. In the first operation below, the second quotient figure is 1, and the next divisor is 413690 ; in the second operation, the second quotient figure is 2, which gives 178593" for the next divisor, and abbreviates the subsequent work (76-81) PROPERTIES OF NUMBERS. FIRST OPERATION. L005973 4 4616175 4023892 592283 1 592283 413690 357186 1 2 3 413690 178593 169512 56504 54486 6 9081 2018 4 8072 2018 2 1009 SECOND OPERATION. 4616175 4023892 1005973 1184566 4 2 178593 8 169512 3 9081 6 8072 4 Am. 1009 2 592283 535779 56504 54486 2018 2018 Ex. 13. An*. 47. Ex. 15. In order that the bins may be equal, the number of bushels contained in one bin must be a common divisor of the two quantities. And in order that the number of bins may be the least possible, each must contain the greatest com- mon divisor of the two quantities. Ans. 91 bushels. Ex. 16. The pannels, to be of uniform length, must be a common measure or divisor of the three sides ; and to be of the greatest possible length, they must be the greatest com- mon divisor of the three sides. Ans. 11 feet. Ex. 17. The price to be paid by each is the greatest com- mon divisor of the three sums, $620, $1116, aud 81488, which is $124. Hence, B can purchase $620n-$124=5 ; C can purchase $1116 H- $124 = 9; and D can purchase $1488-:-$124=12. Ex. 18. The greatest common divisor of 14599 feet and 10361 feet is 13 feet, the length of 1 joint in the fence, (14599+10361)X2=49920 feet, the entire length of the (81) LEAST COMMON MULTIPLE. 25 fence. 49920 H- 13 = 3840, the number of joints in the fence; and 3840x7=26880, the number of rails, Ans. LEAST COMMON MULTIPLE. (155, page 83.) Ex. 1. 2X2X3X11X7X5=4620, Ans. Ex.2. 7X3X3X2X2X11X5=13860, Ans. Ex. 3. 5X3X2X2X2=120, Ans. Ex. 4. 7X5X3X2X2X2X2=1680, Ans. Ex. 5. 7X5X5X3=525, Ans. Ex. 6. 19X3X7X2=798, Ans. Ex. 7. 2X2X2X2X2X2X3X3X5=2880, Ans. Ex. 1. 5, 3, 2 2, 2, 3, 7, 5 (156, page 85.) 15.. 18.. 21.. 24.. 35.. 36.. 42.. 50.. 60 3.. 7.. 4.. 7.. 6.. 7.. 5.. 2 5X3X2X2X2X3X7X5=12600, Ans. , Ex. 2. 2, 2, 3 2,3,5 6. .8. .10. .15. .18. .20. .24 2.. 5.. 5.. 3.. 5.. 2 Ex. 3, 3, 5, 2 3, 5, 7, 2 2X2X3X2X3X5=360, 9. .15. .25. .35. .45. .100 3.. 5.. 7.. 3.. 10 3X5X2X3X5X7X2=6300, Ans. Ex. 4. 3, 3, 2 2, 2, 5, 3 18.. 27.. 36.. 40 3.. 2. .20 3X3X2X2X2X5X3=1080, (83-85) 26 P. Ex. 5. 3, 3 2,13 ROPEKTIES 12. .26. .52 2. .13. .26 2X2X3X13=156, Am. Ex. 6. 2, 2, 17 8,9 32.. 34.. 36 8.. 9 2X2X17X8X9=4896, Ans. NOTE. When numbers are prime to each other, as 8 and 9 in the above operation, their product will be their least common multiple. Ex. 7. 2, 2, 3 2,3,3 8. .12. .18. .24. .27. .36 2.. 3.. 2.. 9.. 3 2X2X2X3X3X3=216, Ans. Ex. 2,11 2,3,5 22. .33. .44. .55. .66 3.. 2.. 5.. 3 2X11X2X3X^=660, Ans. NOTE. The first three numbers in Ex. 7 and the first two ia Ex. 8, above, are factors of remaining numbers in the exam- ples respectively. They might, therefore, have been omitted in the operations. Ex.9. 2, 2, 3 64.. 84. .96.. 216 2, 2, 2, 2 16.. 7. . 8.. 18 3,3,7 7 9 2X2X2X2X2X2X3X3X3X7=12098, Ans. Ex. 10. The number of rods that will furnish whole days' work to each one, must be some common multiple of 14, 25, 8 and 20 ; and the least number of rods that will furnish (85) CANCELLATION. 27 whole days' work to each of the men, must therefore be the least common multiple of 14, 25, 8 and 20. Ans. 1400 rods. Ex. 11. The least common multiple of the prices, $4, $21, $49, and $72, which is $3528, Ans. Ex. 12. When all the men work together, they will dig 4 -f- 8 +6=1 8 rods per day. The ditch must therefore be the least common multiple of 4 rods, 8 rods, 6 rods, and 18 rods, which is 72 rods, Ans. Ex. 13. The least common multiple of 11 feet and 15 feet is 15 X 11=165 feet, the distance the carriage must move to bring the rivets up together. Hence, 165x575= 94875 feet, the entire distance traveled ; and 94875 feet-4- 5280=17 miles 5115 feet, Ans. CANCELLATION. (159, page 87.) K f> Ex. 2. -=80, Ans. Jfrv/ i oiv 9f'V # V 3f Af /\/Lfc> /\/> /\ y> XN/' O Ex. 3 -=32, ^Irw. 61 Ex.4, 61, Ans. (86-88) 28 PROPERTIES OF NUMBERS. 71 11 <^X190X^ -=14839, . Ex.5. Ex.6. Ex.7. Ex.9. =403, Ans. 9, 16 13 13, Ex. 10. 84+56=140 cents. 240 32 cents, 240, Ans. ~\ cost of 2 yards of the Ex. 11. 75X2+90=240 cents, [-first kind, and 1 yard j of the second. 11 yards of the second kind ; 11X2=22 " " first " Ex.12. =60 cents, Am. (88) NOTATION AND NUMERATION. 29 FRACTIONS. NOTATION AND NUMERATION. (169, page 90.) Ex. 3. Ans. |f. Ex. 4. Ans. 7 ^. Ex. 5. Ans. |f. Ex. 6. Ans. f|f Ex. 7. Ans. 25 H <>. Ex. 8. Ans. T1 /y> 52 . Ex. 9. Ans. T WzjoW Ex. 10. Four ninths; seven twelfths; seventeen thirty eighths; forty-five one hundredths; seventy -two three hundred seventy-fifths; forty-eight one thousand ninths; eighty-four seven thousand eight hundred sixty-thirds ; four hundred fifty- six five hundred thirty-sevenths. Ex.11. Twenty fourths; eighty-seven thirtieths ; ninety- five one hundredths; forty-eight twelfths; seventy-five four hundred thirty-sevenths ; one hundred seventy-five halves ; four hundred thirty-six fiftieths ; seven hundred sixty-six four thousand eight hundred seventy-ninths. Ex. 12. Four hundred sixty-seven nine hundred thirty- sixths; five hundred thirty-six two hundred forty-eighths ; ten thousand seventy-fifths; seventy-five ten thousandths; five thousand seven three thousand sevenths. Ex. 13. One hundred fifty Jive hundred thirty-sevenths ; four hundred thirty-six nine hundred seventy seconds ; thir- teen thousand seven hundred eighty-five forty-seven thousand nine hundred fifty-sixths ; one hundred fifty thousand seven- ty -two four hundred seventy-five thousandths ; one hundred thousand one two hundred thousand seconds. (90, 91) 30 FRACTIONS. SEDUCTION. (177, page 92.) Ex.6. ffl=ft,Ans. Ex.7. if|-||, Ans. Ex.8. &&=&, An*. Ex.9. f ff =Hf , ^- Ex. 13 m%=ll A ' Ex - 15 ^- im (178, page 93.) Ex.2. ? -afWI Ex.3. 51-7-17=3; Ex.4. 78-^-13=6; If =| f , ^TIS. A=W. Ex. 5. 3000--3758 ; Ex. 6. 8-*-4=2 ; 3%-3 4 9 6 0. ^' 7| = V = Ex.7. 16 2 \=16i=y; 176-^-4=44; \ 5 ^ 2 T 8 ^, Ex.8. 363--llr=33; Ex.9, 42V7--- ^6; ^=* ^ =* %** , Ans. (92-96) REDUCTION. 81 page 96.) , Ex.2. i,f=Jf, If, An*. Ex.6, i, >, &=&, A 3 T? 3 Vn ^ (184, page 98.) Ex. 1. 2X2X2X^=40, least common denominator; f ? A=Jfc l? ^*- Ex. 2. 3x2X2=12, least common denominator; I. f > t=A. A H> Ans - Ex. 3. 5x3X2X2=60, least common denominator, t? T 7 2> Ti=ife l> f & ^ w * Ex. 4. 2 X 2 X 2 X 3 X 3=72, least common denominator ; 2 g 3 48 64 27 ^? V) 8 72? 72^ 75^ Ex.5 A= |;^ ==A . 7,2 2,3 7. .12. .42 6.. 3 7X2X2X^=84, least common denominator. 3 5 13 _ 36 35 26 An* T 12? 42 - 84> 84? 54? ^ nS ' Ex. 6. 13X3X2=78, least common denominator; 2 4 25 4 _ 52 24 75 8 3> T3? 2^' 3~5> - 7H 78"? T8? 78"? Ex.7. 5 X 3 X 2 X 2 X 2=120, least common denominator ; V? T 7 5> A? |J=*4, i 5 ^? TTF? AV ^- Ex. 8. ||=i|; 7X2X2X2X3=168, least com. denom.,- 20 9 17 _ 160 27 102 27? ^6? 28" - T5f ? T68"? T5F? Ex.9. f=|; /A=A; il=A; 2,2,3 2,2,2 8.. 24.. 32 2.. 2.. 8 3X2 5 =96, least common denominator; 6? " 5 7 _ 60 20 21 > 23? 32 3 " (9698) 32 FRACTIONS, EX. 10. V=ft; iW=tt; iiS=^i; 13x17=221, least common denominator; A, K, 2 6 2 T= 3 T> ill, EX. 11. ^=3*,; ff?=II; tttf=H; 53X23=1219, least common denominator; Ex. 12. 7X^X270, least common denominator; V 9 . Ex.13. T 9 ^-ilx 3 !l; f4H=fl; fHi=5=i 97X59X31X2=354826, least com. denom.; T 9 8 3 ^ fi Ill-lf Ilil, 4n Ex. 14. 23X3 3 X5X7=7560, least com. denom.; 7? T5^ T^> 2TJ "35^? 4 O 1 ^ 5400 jg930 1008 2240 1944 3213 Avt* 7^560? 7^560? 7560? 7560? 7560? ^Se'O* - a71 *' Ex. 15. 13XTX2X2=364, least com. denom.; 435715 _ 208 84 65 49 30 7? T3? 28? 55? T82 - 3^4? 3^4^ 354? 354"? 304? Ex.16. ^=A; 11=4; 5X3X7=105, least com. denom.; ttfc ADDITION. (186, page 100.) Ex. 1. Ex.2. Ex. 3. ^^iJLLL = 4fi == 2, Am. Ex. 4. 7+8+2+5+4 =26 28|, (98-100) ADDITION. 33 Ex. 5. 37+12+13 =62 64|, Ans. Ex. 6. EX. 7. Ex. 8. i+f+ T > s =u 4 JJ>3=ff, _ EX. 9. T S 3 +If + 5 7,=25f|21= , Ex. 10. ^+|+i^+||=283 5 t-a^ = =2 g Y=,2f|, Am. Ex. 11. 3+^+12+11+14= 189+196+1^4+207+3091^^6=4^, Ans. Ex. 12. 3+4+2= 9 Ex. 13. 16+24=40 Ex. 14. 1+2+3+4+5=15 1844, Ex. 15. 4+8+2 =14 Ex. 16. Ex. 17. Ex.18 .9,9 + 44 + - , > Ex. 19. 4+^+A+|= |+ ^ = \^. ^m^, Ans (100) 34 FBACTIONS. Ex. 20. 41+105+300+241+472=1159 i + + I + I + i = 2 ! 1161f Ex. 21. 4+2+ 1 + 2 + 5 +7+4+6=31 Ex. 22. 36+42+39+51=168 169|| pounds, Ex. 23. 4+ 3 =7 Ex. 24. ^+^=11-^=4 Ex. 25. ^+ 1+1+^=^=1 of a dollar, Ex. 26. 46+64+76=186 187|| yards; $127+$226+$312=$ $666{|, received for the whole. SUBTRACTION. (188, page 102.) Ex.1. T?f = ft, An,. Ex.2. ^p=^=^ Am. Ex, 3. JLf^L=ff=4, Ans. (100102) SUBTRACTION. 35 rv A UX. 4. Ex. 5. Ex. 6. EV 7 9 Ifi _ 8132 _ 49 _ 7 &X - T3 B3 -- T2g- T25 T5' V-a- 8 14 1 9 _ 7 S 7 _ 13 _ 1 JliX. 0. -33 - gs - - T 9 5 -- T5 - T g, Ex. 9. Ex. 10. ^-^^i^U-^f^il, Ans. Ex. 11. Ex, 12. Ex. 13. 16| Ex. 14. 36^ * Ex. 15. 25 T ^=25|J Ex. 16. 75 14i|=14| 4f 10$=10|, Ans. 704, Ex. 17. 18f=18 T 4 g Ex. 18. 26 3 7 ? =26 T 3 3 % 12 T 7 F , ^4s. $fo=U, An * Ex. 19. 2841=28,% Ex. 20. 78^=78^ 32| =32jf 45{|=45|, Am. Ex 21. 364 7- l ' 3 =19&, ^w. Ex 22. 97| 184=79|| , Ans. Ex 23. 126^+240|=366||; 560|-366|f =198|J, Am (102) Kf> FRACTIONS. Bx. 24, -g + T^ + ig 7';? > II ?! f!> Ex. 25. | f =f f , ^Lw. Ex. 26. 81 14g=16f gallons, ^Irw. Ex. 27. $140| $ 775 1 456 $597^ > bought for; $1291ff, sold for; $1291f f 6597^= MULTIPLICATION. (193, page 105.) Ex. 1. |xf=!=2f , Am. Ex.2. Ex. 3. % 5. YXA=15;- fX'A=l= 2 i5 if^X 1=630 j Ex.6. J}X8= 1?v 7 lls/20 - 4. 16\v51 - S nx. I. 75X33 -g, T7AT5 5- V v 8 7 s/ I 7 _ 17. ^IA. O. 75^42 72 > Ex.9. i,8x\ 5 =10, ^ms. Ex. 10. f xffi=|=2J, Ex. 11. 1 R T Ex. 12. | Ex. 13. Ex.14. ii Ex. 15. ^ Ex. 16. T \ X f T X V X V X fj,= tf =lg, Ex. 17. ' Ex.18. (102-106) MULTIPLICATION. 37 Ex.19. 7f 2f=5 5 fcj |+J=1 T 2 F ; Ex. 20. xf XlHb -4W Ex. 21. | Ex.22. | Ex.23. Ex. 24. iXf Xf Xf Xf Xf XiXf X T 9 t5=T\ Ex.25. W=f?&l; WI=55?!J ll llxMXfl^ff xmtti=fti> Ans - Ex.26. i|5|iXf0if EX. 27. 4^!llx!^i Ex. 28. $3| X7=$25|, Ex. 29. Ex. 30. 4XlOX6=25f, Am. Ex. 31. 10^x^x^X^111^=2134^1 Ex.32. $9fXX|=44, Ans. Ex. 33. T 9 g Xlf=4 T %> ^ ws - Ex. 34. | Ex. 35. 156f Xf XiX!=47, Ans. Ex.36. Ex. 37. l|X6x8 T 9 o=94i|, Ans. Ex. 38. fXl21f=104f ; |x48f=36i; 104f +36^ 75=65 T % ; 150^ 65 T 9 ij=84|, multiplicand; X 3=4^, multiplier; Ex. 39. JxiXf== Ex. 40. | (|X|)=2 5 4, A 's stare; I X I (I X| X |)=T 5 g> B ' 8 share J ^Xf XS (!X!X!X|)=4 5 5 , C's share; |X|X|X|= 2 5 4, D's share. (106, 107) 38 FRACTIONS. Ex.41. 2iXl=A; |X41X(|) 2 -15; (3f) 3 -(3f) 3 =8511; DIVISION. (195, page 108.) Ex.2 j?Xi= T 2 T5 ilfX^gfs- Ex. 3. i T <>XI=35, .4ns. Ex.4. YX|= 1 | a =37i; fxV 8 =V= 7 i- Ex.5. VX-A= 36 i^- Ex - 6 - if Xf=|, Ex.7. i$Xf=fj HXY=fi=l{f; VXA=I- Ex.8. Ex.9. Ex. 11. X T 5 T X V X ^=1, Am. EX. 12. ^xAx|x^=V ! /.==i Ex. 13. iX 3 5 6 X T 3 (5Xil=|f=l T 7 T , An*. Ex. 14. VXJXT\X4=$> ^ ws - Ex. 15. L3 x V 9 X f X 35 X T 8 3= 2 5 ; Ex. 16. T 6 3 xHXfXf Ex. 17. MIX W^il Ex. 18. VxWx x B VX|f Ex. 19. Ex.22. - = | X M X 1 T 8 X T 2 T=2, Am. lOf Ex. 23. 7+3|=10f ; - ^8 H 5 X j| = . 2S== 7i, Am. 4 ^ 17 (107-110) DIVISION. 39 1_ Ex. 25. $ =& -1-orrjVXf Xj=i ^- IXI fxf Ex. 26. 6-5 T < s =if ; - -=f X f X|f=M, AM. If Ex. 27. Ex. 28. 16*41=X1=^=171, Am. Ex. 29. $30-s-$=$V ) Xf=$ J i fi = Ex. 30. *A-*-(iXi)=AXf X|=^=2| pints, Ex. 31. $2J T V= Ex. 32. Ex.33. Ex.34. 26^-r-|=i|ix|= 6 3 8 = Ex. 35. 27-^2!=YxA= 2 3 4 3 3 Ex. 36. 148|-!-16|i==i|Ax^=^=H; Am. Ex. 37. IXfXf Xi=|=Ui, An,. Ex.38. 28 7|=20|; 20^x|=12i; 720 12{|= 707 T 3 g 5 A-s-=H; 40i + li=41|j 41| X (i) 4 =2l > 271 T | T , Ex. 39. Ex. 40 iXT^Xyxf X 3 4 7 X 3 'VX^XT 8 TXf X|XA (110) 40 . FRACTIONS. GREATEST COMMON DIVISOR OP FRACTIONS. (198, page 112.) Ex. 1. The greatest common divisor of 7, 14, and 28 is 7 the least common multiple of 9, 27, and 45 is 135 ; Ans. T f 5. Ex.2. 3*,l$,f!=V>VM*; the greatest common divisor of 16, 12, and 24 is 4 ; the least common multiple of 5, 7, and 35 is 35 ; Ans. sV Ex.3. 4,2f,2f,^=f,^i_ 2? _ 2 _. greatest common divisor of 4, 20, 12, and 2 is 2 ; least common multiple of 1, 9, 5, and 45 is 45 ; Ans. ft. Ex. 4. 109, 1224=&ffi., |p; greatest common divisor of 546 and 858 is 78 ; least common multiple of 5 and 7 is 35 ; if =2^, Ans. Ex. 5. The measure will be the greatest common divisor of 18| feet and 57^ feet, which is 2-^ feet, Ans. Ex. & The greatest common divisor of 134| gallons, 128 1 gallons, and 115| gallons, is 6 T 5 2 gallons, the capac- ity of the casks required. Hence, 134|-j-6 T 5 2 ^=21, number of casks for the first kind $ 128i-r-6 T 5 5=20 ; second 115i-s-6/ 3 =18, third 59, (112) PROMISCUOUS EXAMPLES. 41 LEAST COMMON MULTIPLE OF FRACTIONS. (2O1, page 113.) Ex. 1. The least common multiple of 2, 7, 14, and 8 is 56 ; the greatest common divisor of 5, 10, 15, and 25 is 5 ; Ex. 2. The least common multiple of 7, 35, and 49 is 245 ; the greatest common divisor of 24, 36, and 60 is 12 ; \V=20 T ^, Ans. Ex. 3. 2||, Ift, $h=&, y, rfft ; least common multiple of 72, 112, and 63 is 1008 ; greatest common divisor of 25, 75 and 100 is 25 ; ^F=40^, Ans. Ex. 4. Least common multiple of 1, 2, 3, 4, 5, 6, 7, 8, 9, is 2520 ; greatest common divisor of 2, 3, 4, 5, 6, 7, 8, 9, 10, is 1. Ans. 2520. Ex. 5. The train must move a distance equal to the least common multiple of 15 T 5 g feet and 9| feet, which is 459| feet, Ans. PROMISCUOUS EXAMPLES (Page 114.) Ex. 1. ?Xf=f ; 135-9=15; |= T %, Ans. Ex. 2. 48--4^12; 48-j-6=r8 ; 488=6; 4812=4; Ex. 3. 11, f , 2, ^, | of |, | of i=|, f , f , /o 3X5X3X2X7=630, Ans. (113, 114) 42 FRACTIONS. Ex. 8. I f II- A number diminished by }| of itself will leave a remainder of 1 ||=|f of itself; hence, 141-5-|f=283^ Am. Ex. 9. ^-f-f+^jo s . i_i f a =T r7_ of his money left . hence, 8119-4-^^=8840, Am. Ex. 10. $42 X ,VX^X 8 8 7 -I311, Am. Ex. 11. Since the less is f of the greater, their difference is Jf f =2 of the greater ; hence, 25-& -i- f SU-^, the greater number; 89 T \ 25 T 7 g=63|, the less. Ex. 12. The two shares together must be |-|-|=y times the greater share; hence, 12000-r- 1 / $1125, the great- er share; $2000 $1125=1875, the less share. Ex 13. fx|X|= 10, Am. Ex. 14. Ex.15. Ex. 16. y X I bushels of corn that can be bought for $15 ; V 5 XIX|X|=18 bushels of barley, Ans. (114, 115) PROMISCUOUS EXAMPLES. 43 Ex.17. Ex. 18. y X|= the number of yards of cloth 1 yard wide; VX|X|= 34 yards, Ans. Ex 19 f XS=2 of the foundery sold for $2570f ; henee, $2570f X2=$5141, Ans. Ex. 20, $10000x|XfXV 2 Xf XIXV^^OOO, vessel; $10000+$12000:=$22000, Ans. Ex. 21. The second son had | of f = T 6 3 ; the third son had 1 (f +T 6 2) 5 5 4 > tne difference between the shares of the first and second is T 5 2 f =2*4 ; hence, $500 _^i_*=$1200 Ex. 24. I of the whole-{-12| acres=lst and 2d sons' shares; 3 a ( ( u _^_i2i ^3d son's share ; I of the whole -j- 24^ " =the whole ; therefore, 24 acres must be | of the whole; 24. JX 4=98 acres, the whole; and 98Xf+12|=49 acres, Ans. Ex. 25. $|X|XJX|=*|, Am. Ex. 26. $3500 $>740=:$2760, his money before gaining; hence, $2760-r-|=$4600, money invested ; and $4600 X =$1840, lost, .Ans. Ex. 27. | of f=4, Ans. Ex. 28. Since A can do | of the work in 1 day, and B can do 1 of it in 1 day, they can both do '-f-i^^ of the work in 1 day; and if ^ be done in 1 day, ^, or the whole work, will require 2 T 4 =z3| days, Au*. j OTEp The time required to perform any piece of wo k will always be the reciprocal of that iractioii of the work pu jormed in 1 unit of time. (115,116) 44 FRACTIONS. Ex. 29. % 5 Xf Xi=13 barrels, sold at $4 per barrel; 13+5=18 barrels, Am. Ex. 30. The number will be the least common multiple of f, f , f and |, which is 60, Ans. Ex. 31. According to the note above, A will travel round the island, and be again at the point of starting; once in | days, B once in ^ days, and C once in .y days ; and the least common multiple of | days, L 7 days, and. y days, is --|^=178^ days, Ans. Ex. 32. The sum of the distances traveled by the two men is 64J miles, and the difference of theee distances is 5^ miles. Hence, by Problem 33, page 64, of the Arithmetic, we have 64f + 5!=76;Jj 70^-5-2=351 miles, the greater journey; 64| 5^=59| ; 59-*-2=2'9f miles, the less journey. Ex.33. l^+|=li; li-*-2=f, the greater; Ilk 1=&> T 7 ^2=^, the less. Ex. 34. 1 T Ve {Jl, the swm of B's and O's shares. And since 7 7 ^ is the difference of B's and C's shares, we ^ve iil+T 7 H-=f ; *-s-2=A, B^s share; iH-A=if; if^-2=J|, C's share. Ex. 35. The reciprocal of | is | ; reversing the fourth operation, | X 2 6 ^=i% > reversing the third operation, T 3 Q-j-|= y 7 ^ ; reversing the second operation, T 7 a J= T 8 ; ; reversing the first operation, T 8 ^x$yf =$|, Ans. Ex 36. ?! is a quotient and If a divisor, and ] 3 6 xf= y\ 6 , the dividend ; \\ 5 is a product and 5| a multiplier, and j 3 4 5 -:- 2 5 7 ff> kh e multiplicand; f| is a re- mainder and | a subtrahend, and f|-(-|=L 2 J, the (116) NOTATION AND NUMERATION. 45 minuend ; ^ is the sum of two numbers and 1| is one of them, and ^g 1 J=f |> the required num- ber, Ans. Ex.37. fxfxixfxi-W; W+V=W; DECIMAL FRACTIONS. NOTATION AND NUMERATION. (21O, page 120.) Ex. 4. Ans. .496 Ex. 6. Ans. .0325 Ex. 6. Ans. .000001 Ex. 7. ^ks. .0000074 Ex. 8. Ans. .437549 Ex. 9. Ans. .3040010 Ex. 10. Ans. .00000024 Ex. 11. Ans. .08645 Ex. 12. Ans. .495705048 Ex. 13. Ans. .0000099009 Ex. 14 Ans. .04735901 Ex. 15. Ans. .000000000001 Ex. 16 Ans. .1001001001001 Ex. 17 Ans. .000841563436 Ex. 18. Ans. .000000000000000009 Ex. 19. Ans .3 Ex. 20. Ans. .105 Ex. 21. Ans. .0011 Ex. 22. Ans. .00085 Ex. 23. Ans. .100004 Ex. 24. Ans. .0000704 Ex. 25. Ans. 46.4 Ex. 26. Ans. 205.65 Ex. 27. Ans. 60.00036 Ex. 28. Ans. 705.000000005 Ex. 29. Ans. 300.10001001 Ex. 30. Ans. 52.000000000005 Ex. 31. Twenty-four hundredths. Ex. 32. Seventy-five thousandths. Ex. 33. Five hundred three thousandths. Ex. 34. Seven hundred twenty-five hundred-thousandths. Ex. 35. Forty million four hundred-millionths. (116-120) 46 DECIMALS. Ex. 36. Two hundred fifty- six ten-millionths. Ex. 37. Ten thousand seventy-five ten-millionths. Ex. 38. Eight, and twenty-five hundredths. Ex. 39. Seventy-five, and three hundred sixty-eight thous andths. Ex. 40. Forty- two, and six hundred thirty-seven ten-thous- andths. Ex. 41. Eight, and seventy-four ten thousandths. 1 Ex. 42. Thirty, and four thousand seventy-five ten-thous- andths. Ex. 43. Twenty-six, and five hundred-thousandths. Ex. 44. One hundred, and one hundred-millionth. Ex.1. .1800000 .4560000 .0075000 .0000010 .0500000 .3789000 .5943786 .0010000 Ex.1. Ex.3. REDUCTION. (211 ? page 121.) Ex. 2. .012000000000 .000185000000 .000000936000 .000000000007 Ex. 3. 57.300000 900.000000 4.755500 100.000001 (212, page 122.) Ex.2. Ex.4. (120-122) =-'^ Ans. REDUCTION. 47 Ex.6. T ft = T | 57 l Ex. 10. _JUflj{}=4, 66| Ex. 11. =f $$=, 100 Ex. 12. - =i{HJiJ==j, 1000 24f Ex. 13. - -=- 3 ^ 5 = T 1000 984| Ex. 14. - =iM=H Ex. 15. 7 T %=7|, Ex. 16. 24 T V 5 ^24|J, ^n,. Ex. 17. f $**=! Ex. 18. =, ^. Ex. 19. 2i = (314, page 123.) Ex. 3. Am .875 Ex. 4. ^s. .56 Ex. 5. Ans. .8125 Ex. 9. Ans. .001796875 Ex. 11. Ans. .60625 Ex. 15. Ans. 32.714286- Ex. 16. Ans. .245 Ex. 17. Ans. 5.783125 Ex. 20. Ans. .30007 022-124) 48 DECIMALS. ADDITION. (316, page 125.) Ex.1. .375 Ex. 2. 5.3756 .24 85.473 .536 9.2 .78567 46.37859 .4637 .57439 45.248377 191.675567, Ans. 2.97476, Ans. Ex.3. .5 Ex.4. .4675 .37 .325125 .489 .1616 .6372 .47856 .02524 .2754375 1.2296625, Ans 2.50000=2.5, Ans. Ex.5. 4.65 Ex. 6. 4.3785 7.322 5.3784125 2.6487875 2.66666+ 5.42857+ 12.4872 19.9992000, AM. 24.9609+, Ans. Ex.7. .137 Ex. 8. .0102 .435 .13426 .836 ,000567 .937 .000003 .496 .24007 2.841, Ans. (125) .3851, Ant. SUBTRACTION. 4Q 34.72 Ex. 10. f f=.24743+ 48.44 &=.17224+ 15217 ^=.24666+ 95.36 TT i 3F 56.18 .66691 db = Ans. 386.87 rods. .6669+, Ans. Ex. 11. 16^=16.316 Ex. 12. .45 15^=15.118 .0275 18if=18.484 .009125 14 ? \=14.155+ .000304 64.07+, Ans. .486929, Ans. Ex. 13. 1 dec. unit of the first order=.l " " " second " =.005 \ " " " third =.00033333333+ \ " fourth ==.000025 | fifth =.000002 \ " " sixth =.00000016666+ 4 , seventh =.00000001428+ Ans. .1053605143 SUBTRACTION. (217, page 126.) Ex. 4. 37.456 Ex. 5. 1.0066 24.367 .15 13.089, Ans. .8566, Am. Ex 6. 1000.000 Ex. 7. 36.75 .001 22.48 999.999, Ans. 14.27, Ans. (125, 126) 5 60 DECIMALS. Ex. 8. .56875 Ex. 9. 7.33333+ .55992 5.5625 .00883,^/is. 1.7708+, Am Ex. 10. ff i=.99398i9+ Ex. 11. 1. .000000000001 .0491725ih, ^ns. Ans. .999999999999 Ex. 12. 57436.00 Ex. 13. f4j| 1^4.400243+ If J|f= .227260+ 536.74 1756.19 4.17298+,^*. 3678.47 9572.15 7536.59 4785.94 Ans. 29569.92 acres. MULTIPLICATION. (219, page 127.) Ex. 2. Ans. .10464 Ex. 5. Ans. 9.3654 Ex. 8. Ans. 104.976 Ex. 9. Ans. 17.019 Ex. 11. Ans. 360. Ex. 12. Ans. 1. Ex. 13. Ans. 57600. Ex. 15. Ans. 15.15 Ex, 18. Ans. 4.626 Ex. 20. Ans. 168.48x27.375 =4612.14 pounds, Ans. Ex. 21. 2.8X36=100.8 bushels of oats for 36 bushels of corn; 100.8+48=148.8, Ans. (126-128) MULTIPLICATION. 51 CONTRACTED MULTIPLICATION. (222, page 131.) Ex.2. 36.275 Ex. 3. .24367 763.4 57.63 1451 731 109 146 22 17 2 1 158.4, Ans. Ex. 4. 4256.785 46500. 548.07, Ans. 8.95, Ans. Ex. 5. 357.84327 608700.1 21284 357 8433 * 2554 25049 170 2863 21 24.008 -f- Ans. AlJi 360.6366, Ans. Ex. 6. 400.756 Ex.7. 432.5672 85763.1 666660.1 40076 432 567 12023 25954 2405 2595 280 260 20 26 3 3 461.405:,<4n. (131) 52 DECIMALS. Ex. 8 48.4367 Ex. 9. 7.04424=7^ +31531.2 =2/7 +94658.3 =3|J| 96873 21133 4844 5635 1453 352 242 42 5 3 1 1 103.418db, Ans. 27.166:4:, ^w Ex 10. 142.8373+ Ex. 11. 35.8756 53025.2 S 8833.8 28567 28700 7142 1076 286 108 4 29 1 3 Ans. 360.00 degrees. Ans. 299.16ih pounds. Ex. 12. ^ 478.7862 Ex. 13. 6377397.6 " 65390.1 643126000. 47879 382644 4308 12755 144 638 24 191 3 25 A 4 Itui 69R &8-I- vnrrlfl Equatorial radius, 3962.57 miles. (131, 132) DIVISION. 63 6356078.96 643126000. 381365 12712 636 191 25 4 Polar radius, 3949.33 miles. DIVISION. '(224, page 133.) Ex. 4. Am. .2 Ex. 7. Am. .8666+ Ex. 9. Am. .00666+ Ex. 10. Am. .0075 Ex. 11. Am. .0000436 Ex. 12. Am. .8333+ Ex. 13. Ans. .6455 Ex. 14. Am. 6.165 Ex. 15. 16.2-f-2.7=6, Am. Ex. 16. 674-r-36.34=18.547+days, Ans. Ex. 17. 5280^-14.25=370.5+, Am. CONTRACTED DIVISION. (226, page 135.) Contracted decimal division is most readily performed by the method of inverting the quotient, described in Note 2 f page 135, of the Arithmetic. (132-135^ 54 DECIMALS. Ex. 1. 4.3267)27.3782 823.6 25960 Ans. 6.328 1418 1298 120 87 38 34 Ex.3. 75.430)8.47326 33211. 75430 An*. .11233 9302 7543 1759 1509 250 226 24 23 Ex.2. 1.003675)487.24 64.584 401 47 Ans. 485.46 8577 8029 548 502 46 40 6 Ex. 4. .075637).8487564 122.11 75637 9238 Ans. 11.221: 7564 1674 1513 161 151 10 In the following operations, Abbreviated Long Division is combined with decimal contraction, (see Arithmetic, 1 IS, page 50). 756.3452)8972.436 9268.11 140898 65263 4756 Ex.5. 1.436666+)478.325 Ex.6. 249.233 Ans. 332.942dz 47325 4225 1352 59 2 (135, 136) Ans. 11.8629 218 67 CIRCULATING DECIMALS. 65 Ex. 7. 1.007633)1.000000 Ex. 8. 44.736546).95372843 524299. 93130 77813120. 589975 2443 142610 Ans. ,992425 428 Am. .02131877+: 8400 25 3926 5 347 34 Ex. 9. 5737)4273.0 8 8447. 2571 276 Ans. ,7448 47 1 CIRCULATING DECIMALS. REDUCTION. (238, page 139.) Ex. 1. .45=f= T 5 T , Ans. Ex. 2. .66=f f=f, Am. Ex. 3. .279=111=^, Ans. Ex. 4. .423=|f f= T 4 T \, Ans. Ex. 5. .923076=fff^f=il, Am. Ex. 6. .9512i=ff ff=f f, Am. Ex. 7. 4.72=4|f =4 T 8 T , Am. CV ft 9 9Q7- _ 9'297 _ 911 _ 85 Ana HiX. O. L.LO( - ^~$^~Q - ^37 - 3T? ^- ns - Ex. 9. 2.97=2.972=2||f=2ff=i3V>, ^TW. Ex. 10. 15.0=15.015=15^=15^1^, Am. (239, page 141.) Ex. 1. .57=fr=$=H, Ex. 2. .048=^==^%=^ (136-141) 56 DECIMALS. Ex. 3. Ex. 4. .6590==%^t=|ff=i|, Ans. Ex. 5. Ex.6. .1004=^04-1 oo^ V ___i TV L, AnSf Ex. 7. Ex. 8. Ex. 9. Ex. 10. Ex. 11. 2.029268 ^ (24O, page 142.) Ex. 1. .43 ^.43333333 .57 =-.57575757 .4567^.45675675 .5037^.50373737 Ex. 2. .578 =.57888 Ex. 3. 1.34 =1.3413413 .37 =.37373 4.56 =4.5645645 .2485=.24855 .341= .3414141 .04 =.04040 Ex. 4. .56T4=.5674567456745 .34 =.3444444444444 .247 =.2472472472472 .67 =.6767676767676 Ex. 5. 1.24 =1.24124124124124124 .0578 = .05785785785785785 .4 = .44444444444444444 .4732147= .47321473214732147 Ex. 6. .7 =.7777777777777 .4567 =.4567777777777 .24 =.2424242424242 .346789=.3467894678946 (141, 142) CIRCULATING DECIMALS. 57 Ex. 7. .8 .36 .4857 .34567 =.36363636363636363636 =.48574857485748574857 =.34567345673456734567 .2784678943=.27846789432784678943 ADDITION AND SUBTRACTION. (242, page 143.) Ex, 1. 2.4444444 .3232323 .5675675 7.0565656 4.3777777 14.7695877, Am. Ex. 3. .7854854 .5959595 .1895258, Ans. Ex.5. .55 .32 .12 .99=1, Ans. Ex. 2. .4787878787878 .3213213213213 .7856485648564 .3222222222222 .5555555555555 .4326432643264 2.8961788070698, Am. Ex. 4. 57.0587 27.3131 29.7455, Am. Ex. 6. .4387 .8633 .2111 .3554 1.8686=1.86, Ans. (142, 143) 58 < DECIMALS. Ex. 7. 3.6537537 3.1351351 Ex. 8. .43243 2.5646464 .25000 .5353535 .18243, Ans. 9.8888888=9.8, An*. Ex. 9. 7.24574 Ex. 10. .99000 2.63463 .43343 4.6lili=4.6i, An*. Ans. .55656 Ex. 11. 4.638638 Ex. 12. :44 8.318318 .23 .016016 .545454 .2i=i=^, Ans. .454545 13.972972=13.972=13f!=13ff, An*. MULTIPLICATION AND DIVISION. (244, page 144.) Ex.1. 8.4=fS; 72=31; ^X?f=W=2.472, Ans. Ex.2. Ex. 3. 154=411; .2=|j if Ex. 4. 4.5724=- 4 ^W ; -fej ; - 4 ^WXf-fff?=5.8793, An*. Ex. 5. 4.37=W ; -27=^ ; WX 1^=1.182, An*. Ex. 6. 56.6=^; ^X T i 7 =-i?f =.41362530, Ans. Ex. 7. M||HXH=fi =-7857142 ; Ex. 8. iUlMX||= T AT (143, 144) UNITED STATES MONEY. 59 Ex.9. 3.456= 3fc^H3i=1.471037, Am. Ex, 10. 9.17045 V/AV > 3.36=^V; Ex. 11. |Xf?=A 6 5^=- 1395775941230486685032, UNITED STATES MONEY. NOTATION AND NUMERATION. (35O, page 146.) Ex. 2. ^rw. $4.07 Ex. 3. Ans. $10.04 Ex. 4. .4^8. $16.004 Ex. 5. Ans. $.31}, or $.315 Ex. 7. ^lw*. $1000.011 Ex. 8. Ans. $32.584 Ex. 9. Ans. $.06i, or $.0625 Ex. 10 Tweuty-one dollars eighteen cents; one hundred sixty-four dollars five cents; seven dollars ninety cents ; ten dollars one cent ; two hundred one dol- lars twenty cents one mill ; five dollars thirty-seven and one-half cents; eighty-one and one-fourth cents; fifteen dollars eight and one-third cents ; ninety -six dollars five mills. OPERATIONS IN UNITED STATES MONEY. (252, page 147.) Ex. 1. $3475.50 Ex. 2. $4.62} 310.20 1.75 1287.375 .87} 207.125 1.00 .62} $5280.20, Ans. $8.87}, Ans. (144-147) 60 DECIMALS. Ex 3. $390.375 $150.000 Ex. 4. $3800 175.84 190.87} 62.50 8 2.035, Ans. ^.87}, Am Ex. 5. $50.000 Ex. 6. $ .375x150=456.25 3.875X4 = 15.50 $10.75 5.50 $71.75 2.375 .875 $ .06}X84= 5.25 .625 ,62}X25= 15.62} 5.87}X2 = 11.75 $29.875, Ans. $39.12}, An* Ex. 7. 831.25X126.25=83945.31* 33.75X138.25:= 4665.931 $8611.25 $8611.25 $6726=$1885.25, Ans. Ex. 8. $.80 X28}X40 =8 912. .11JX29 X300= 1000.50 3 4 87}x36iX20 = 2809.37} l2= 213.50 811000 $4935.37i=$6064.62}, Ans. Ex. 9. $2189.25--139=$15.75, Ans. Ex. 10. $44.748--396x$.113, Am. Ex.11. $4.50X10.75 $48.375; $48.375-=-7.74=$6.25, Ans. Ex. 12. 84885.80-^-8287.40=17, Ans. (147, 148) UNITED STATES MONEY. Gi Ex. 13 $3.75+$2.875=$6.625, cost of a calf and a sheep; $265-46.625=40, Am. Ex. 14. I 128 Ex. 15. 9632 1730.75 41.25 2.5 23, Am. 125 112.20 136, Ans. Ex. 16. $2475.36 $1936.40 = 1538.96, the amount his money has diminished since the beginning of the month, which, by the conditions, must be f f =f of what he had at the beginning of the month. Hence, $538.96-i-f=$1347.40, Am. Ex. 17. $3200 $138X12=41544, saves yearly; $1544x8=$12352, Am. Ex.18. $45.75 Xl20=$5490; $5490 $1026=44464 ; $4464-f-120=$37.20, Am. Ex. 19. $6.25X425=$2656.25; $3088.25 $2656.25=$432 ; $432^-$4.50=96 ; 425+96=521 barrels, Ans. Ex. 20. $6315.12-r-36=$175.42, wages for 1 engineer; $21927.50--$175.42=125, Ans. Ex. 21. $2538+$750 $1378.56=$1909.44, Am. Ex. 22. $1.875 X-22=$41.2?; $41.25 $25.75=$15.50 ; $1116-r-$15.50=72 months; 75-^-12=6 years, Am. Ex. 23. ($453.75--27.5)+$3.625=$20.125, Ans. Ex. 24. $.95-f-$1.37+$.73=$3.05, cost of 1 bushel of each kind; $7G.15-:-$3.05=23 bushels of each kind; 23x3=69 bushels, Ans. Ex.25. 375.5-^-2 = 187.75; $1032.625 --187.75 = $5.50, jrofit per acre; $22.25+ $5.$0=$27.75, Am (148, 149) 62 DECIMALS. Ex. 26. 811. 374=8 V Ex. 27. 846.75=84* $91 4 2 51 161 187 32 | 2093 2 I 11 $65.40f , Ans. 5.5 hundred pounds. Ex. 28. $62.50 (83.25X12|)=821.875; 821.875-*-8.125=175 pounds, Ans. Ex.29. $10.04--16=$.6275; $17.50--20=8>875; $.875 $.6275 = $.2475; $.2475 X 320 = $79.20, Ans. PROBLEMS INVOLVING THE RELATION OF PRICE, COST, AND QUANTITY. , page 151.) Ex. 1. 81.32X187=8246.84, Ans. Ex. 2. $ T 3 gX70|x5=466.09f, Ans. Ex. 3. 8501.875-5-365=81.375, Ans. Ex. 4. 818.48-^-8.105=176, Ans. ( Ex. 5. $17.75-5-2=$8.875 j $8.875 X .1625 X 140=8201.906}, Ex. 6. $16.50X32.40=$534.60 ; Ans. Ex. 7. $66.44X842.75^155992.31, ^,s Ex. 8. $10660.125-i-325.5=$32.75, Ans. Ex.9. $1.94x8.40=816.296; 812.50x1.262=815.775; $16.296+$15^775=$32.071, Ans. Ex. 10. 837.6875 -j-81. 50=25.125=25 J bushels, Ans. (149-151) UNITED STATES MONEY. Dx. 11. $3.875-h-2=$1.9375; $1.9375X2.172=$4.208|, Am. Ex.12. $.37=$f; $|X|X^f fi =*106.87i, Ex. 13. $81.25--32.5=$2.50, An*. Ex. 14. $9.375X24.240=$227.25, An*. Ex. 15. $4234-5-$5f =752$, An*. Ex.16. $20.25 X .972=119.683 . 2.875X15.75 = 45.281^ 7.50 X 8.756= 65.670 $130.634|, An*. Ex. 17. $4.625 Xl0.46=$8.37f, An*. Ex.18. $4.70-4-2=$2.35 ; $5.25-f-2=$2.62 $2.35 X 5.840=113.724' 2.62^X4.376= 11.487 $25.211, An*. Ex. 19. $2-f-$2.50=| 7 ards ; Ans - Ex. 20. $5.75X37=$212.75, An*. Ex. 21. $.96X38.40=$36.864, Ans. Ex. 22. $2.875 X27|X9=$715.87^, Ans. Ex. 23. $80.745-v-$.42=192| pounds, An*. Ex. 24. $15.50 X .327=$ 5.0685 1.625X6.72 = 10.92 4.25 xl-108= 4.709 $20.6975, An*. Ex. 25. $15|--$|=18, An*. Ex. 26. $5X18.962=194.81, Ans. Ex. 27. $27.90-^-15.5=$1.80, An*. Ex. 28. $125.38 X27.86=$3493.0868, An*. Ex. 29. $13.125-r-.7-^$18.75, Ans. (152 i 64 DECIMALS. Ex.30. $12.75-r-2=$6.375; $15.50-s-2=$7.75 j $6.375 X- 720 =$4.59 $7.75 X.912 = 7.068 $11.658, Ans, LEDGER ACCOUNTS. (258, page 15.3 ) Ex. 1. Ans. $3434.80 Ex. 2. Ans. $7222.55 Ex. 3. Ans. $73785.18 "Ex. 4. Ans. $750026.82 ACCOUNTS AND BILLS. (267, page 155.) Ex. 1. Ex. 2. $2.85 X 10 =$28.50 $1.2B Xl25=$156.25 1.12^X16 = 18.00 1.75 X275= 481.25 .14 X?2 = 10.08 1.121X180= 202.50 .16^X42 == 6.93 .87^X210= 183.75 .40 X12 = 4.80 .84 X$0 = 67.20 .56 X24= 13.72 .90 X95 = 85.50 1.06 Xl75= 185.50 Ans. $82.03 30.50 X8 = 244.00 35.75 X3 = 107.25 .10^X958= 100.59 .37^X40 == 15.00 Ans. $1828.79 (152-155) ACCOUNTS AND BILLS. 65 Ex. 3. $27 50 X40=$1100.00 Ex.5 19.20 X25= 480.00 48.10 X16= 769.60 17.75 X12= 213.00 26.30 X20= 526.00 31.85 X15= 477.75 3.87^X36= 139.50 4.12^x42= 173.25 2.90 X25= 72.50 * Ans. ^ $}5951.60, Ex.4. $6.25 X 150=4937.50 7.16 X275=1969.00 5.87^170= 998.75 1.62^X326= 529.75 .82 X214= 175.48 .91 X300= 273.00 1.06 X500= 530.00 $64.30 X24 10.25 Xl5 7.78 X 7 8.45 X25 16.12^X14 5.90 X27 Dr. =$1543.20 $17.60 = 153.75 = 54.46 = 211.25 = 225.75 = 159.30 $5413.48, Ans. Or. .09^X1840= 174.80 $2522.51 (156, 157) = 156.26 9.37^X42= 393.75 1000.00 3.10 X75= 232.50 .87^X36= 31.50 $2166.00 2522.51 Balance, $356.51 66 DECIMALS. Dr. Ex. 6. $ .23 X96 =$206.08 .09^X872 = 80.66 54.77 68.82 13.50 21.00 ,16X81 = 1.40 X15 = 12X963J= 120.42 $565.25 Or. 12.25 X61=$137.25 .22 X70= 15.40 .87^X56= 49.00 .68|X31= 21.31 $222.96 565.25 Note to Bal. ; 8342.29 Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. Ex. 6. Ex. 7. Ex. 8. Ex.9. PROMISCUOUS EXAMPLES. 84.875 Xl2f=$61.54+, Ans. $33.75-4-$.375=90; 90-^-2^=36, Ans. 36X36=1296; $97.20-f-1296=$.075, Ans. $5.3.5-r-.625=$8.56, Ans. .0| =.033| .00|=.008| Or, iS rfu=A=- .025 , Ans. 814 2 ^x26^f=814.495x26.46875 =21558.66+, Ans> $75X5=375; $68x12=8816; 5+12=17; $375 + 8816 + $118 = $1309 ; $1309--17 = $77, Ans. $.625 X -8=4-50, Ans. $.87^ + $.18| + $.10| =$1.165; $27.96 -r- $1.165=24, Ans. (157, 158) PROMISCUOUS EXAMPLES. 67 Ex. 10. 13543.47--365.25=37.08 miles in 1 day; 37.08X1=32.445 miles, Am. Ex.11. $5.12;|X100=$512.50 $6.50 X 75=4487.50 1.06^X250= 265.62^ 1.37X250= 343.75 221.874 $831.25 $1000.00 8831.25=8168.75, to be realized on the remaining 25 barrels of flour; hence 3.75--25=$6.75, Ans. Ex. 12. 4580.289-4-114.45=40.02 bushels from 1 acre; 120.06-5-40.02=3 acres, Ans. Ex. 13. .017226+-r-.030625=.5625ib, Ans. Ex. 14. 13.5--.0225=600, Ans. Ex. 15. 8.5-4-5=1.7 rods, his daily work; 59.58.5=51 ; 51-^-1.7=30 days, Ans. Ex. 16. .375X28.5=810.68|; 12520--2000=6.26 tons; .75 X4.53= 3.391 $14.085-r-6.26=$2.25, Ans. Ex.17. 1826+1478+1921=5225; $8.80-5-2=; $4.40X5.225=$22.99 $.09x31=82.79 5.25X2.81 = 14.751 4.50x6^=29.25 $37.74^ $32.04=85.70 J, Ans. Ex.18. $122.50-v-35=$3.50; $3.50X29=$101.50, Ans. Ex.19. $.56|X1200=$675 168.675 $843.675 $.60X375.5= 225.30 1200375.5=824.5; 8618.375-5-824.5=8.75, Ans. (158, 159) Ex. 20. 1680 DECIMALS. $2.856 Ex. 21. 8 2000 .125 $127 25.42 $3.40, Ans. $3228.34, Ans. Ex. 22. ($100X150) $3900=411100, cost of whole; $11100-;-150 $74, cost per acre; $11100 $2250 $8850, sold for; $8850150 $59, sold for, per acre. Ex. 23. $14.375X212.5 $3054.68|, cost; 1.75 X 2125 $3718.75, avails; $664.06|, Ans. Ex. 24. $545-r-10 $54.50, cost of 1 acre; $17712.50--$54.50 325, Ans. Ex.25. 224.56x7XiXi=196.49,^7is. Ex. 26. $169.8125 $39.1875 $130.625; $130.625-^-104.5 $1.25, Ans. Ex. 27. $6.975-r-.93 $7.50, Ans. Ex.28. $4000X.375X-12=$180, Ans. flf 2^ Ex.29. - XfXi=3XlX|X$X|Xi [4| 2i J =^=.15, Ans. Ex.30. $.331X375 $125; 8125-5-7.5= $16.66|, Ans. Ex. 31. 1-|-. 84 1.84 times the sum invested; 1.84x2=3.68 times the sum invested, Ans. Ex.32. T %=.3; f .6; 1 (.3+.6)= .1. Now, if he pur- chases .3 of a bushel of barley, .6 of a bushel of wheat, and .1 of a bushel of oats, he will have 1 bushel of grain, worth $ .625X-3=$ .1875 1.875X-6= 1.125 1= .0375 $1.35 (159, 160) CONTINUED FRACTIONS. 09 And for $54, he can purchase as many bushels as $1.35 is contained times in $54. Therefore, |54-f-81.35=40, Am. Ex.33. 191X27=522 yards; $4.311 X522=$2251.12 881.871 9.621 $2642.62i-f-522-j-$5.06|, An*. Ex. 34. $1.18f 1356 $.41 736 1.12 870 .31 528 $ .061 x 486=$30.37i > $.10x208=$20.80 ; $30.37+820.80+13.62=$64.80, entire loss; $235.87| $64.80=$171.07|, gained, Arts. Ex..35. j? i+4J|= |; | -*-2=A, greater; Ex. 36. 1 + != y times his original capital, end of 1st year ; is x |= If " " " " " 2d ||Xl|=Vo 3 " " " " " 3d a $28585.70-- ^=$17991, his original capital; $28585.70 $17991=$10594.70 ; gain, Ans. CONTINUED FBACTIONS. (71, page 162.) The division may be performed in the same manner as in . finding the greatest common divisor. (See loO, Higher Arithmetic.) (160-162) 70 CONTINUED FRACTIONS. 1240 1042 Es 5 2 2 1 1 1 2 2 10 Ex 516! 447 .. 2. 6721 1 Ans. 6200 5+1 521 2+1 198 125 396 2+1 125 1+1 73 52 73 1+1 52 1+1 21 20 42 2+1 10 2+1 10 10 .3. 501 1 -A.71S 223874 207459 1 2 3 4 4 1 2 3 16 3 748 2+1 69153 3+1 16415 13972 65660 4+1 3493 4+1 2443 2100 2443 . 1+1 1050 2+1 343 336 1029 3+1 21 16+1 21 8 7 (162) REDUCTION. 71 Ex.4. 121 Av 1 29 4 116 ' 4+1 25 5 5 5+1 4 1 4 1+1 4 4 1 4 (273, page 163.) Ex. 1. Terms of continued fraction, ^, -J, approximate values, , f, ^, T %\, Ex. 2. Terms of continued fraction, |, |, 1, approximate values, J, T %, ^ 5 T , ^, B 8 Ex. 3. Terms of continued fraction, ^, A, | ; approximate values, 4; 3%? fVy; ^ ws - Ex. 4. Terms of continued fraction, *, , ^, | approximate values, , y\? A? 7%% T 4 5 Ex. 5. Terms of continued fraction, y> 3, }> -g j approximate values, 1, f , T 9 g, ||, Ans. COMPOUND NUMBERS REDUCTION DESCENDING. (367, page 194.) Ex. 2, 133x20+6 s.=2666 s. ; 2666 S.X12+8 d.=32000 d. ; 32001 d.x4=128000 far., Ans. (162-194) 72 COMPOUND NUMBERS. Ex. 3. 100 mi. X 63360=6336000 in., An*. NOTE. When the number given for reduction contains but one denomination, the scale of relation may be taken from the table of Unit Equivalents, and the answer obtained by a single operation. Ex. 4. 1| mi.x4=6 mi., length of fence; 6 mi. X 320=1920 rd., Ans. Ex. 5. 8X3X3=72 cu. ft., solid contents of the block; 175 Ib.X 72=12600 lb.; 12600 lb.-^-100=126cwt.; 126 cwt.-r-20=6 T. 6 cwt., Ans. Ex. 6. 1 hhd.=63 gal.; $.28x63=$17.64, Ans. Ex. 7. 1548 bu. 1 pk.=6193 pk. ; 2 bu. 3. pk.=ll pk. ; 6193-*-ll=563, Ans. Ex. 8. $3.75xlO=$37.50; 10 bu.=640 pt.; $.06|X 640=$40.00; $40 $37.50=$2.50, Ans. Ex.9. 90X60+17'=5417'; 5417' X 60+40" =325060," Ans. Ex. 10. The 18th century embraced the time from the com- mencement of A. D. 1701 to A. I}. 1800 inclusive, and 1800 was not leap year; hence, 100 yr.X365 +24 da.=36524 da., Ans. Ex. 11. lgreat-gross=1728 units ;$.06|X 1728=$108, Ans. Ex. 12. 4 bales 4 bundles 1 ream 10 quires=990 quires; 24-r-8=3 vol. per quire; 990x3=2970 vol., Ans. Ex. 13. 18 yr.x365+24 da.=6594 da.; 6594 da. X 24=158256 h.; 158256 h.x60=9495360 min., Ans. Ex. 14. 481 sov.X240=115440 d., Ans. Ex. 15. $7f =$7.375; $7.375x1000=7375 mills, Ans. Ex. 16. 3 P. X 130=390 gal.; 390 gal. X 4=1560 qt., An*. (194) REDUCTION. 73 Ex. 17. 37 ellsX5+l qr.=186 qr.; 186 qr.-^-4=46 yd. 2 qr., Ans. Ex, 18. 6 10s. 10d.=1570d.; $.02^X1570=$31.66+, Ans. Ex.19. 60.Xl6+14fg=110fg; 883 3X60+45^=53025^, Ans. Ex. 20. 1 T. 1 P. 1 hhd.=7 hhd. ; 7 hhd.X 52^=367^ gal, Ans. Ex. 21. 126 12 s. 6 d.=30390 d.; 30390 d.-7-60=$506, Ans Ex. 22. 1 hhd.=504 pt.; 2 qt.+l qt.+l pt.==7 pt.; Ex. 23. 2 ft. 9 in. = 33 in.; 63360^-33 = 1920 steps in 1 mile; 1920x95=182400, Ans. Ex. 24. $lf Xl2=$21, cost; 12 bbl.Xl26=1512 qt.; $.06X1512=190.72; $90.72 $21=$69.72, Ans Ex. 25. 75 A. X 10+4 sq. ch.=754 sq. ch.; 754 sq. ch.X 16+18 P.=12082 P.; 12082 P.X625+118 sq. 1=7551368 sq. 1., Ans. Ex. 26. 4 in. X 16=64 in., Ans. Ex. 27. 150 leaguesX3Xl-15=517.5 miles, Ans. Ex. 28. (50 A. 14 A.) X 160=5760 sq. rd., Ans. Ex. 29. 36 Ib. 8 oz.=8800 pwt.; $1.042X8800=89169.60, Ans. Ex.30 9 cwt. 42 lb.=9421b.; 942 Ib.X 8=7536 Ib.; 7536^-48=157, Ans. Ex 31. $1|X12= $15, cost; 12 bbl.X280=3360 Ib.; $.0075X3360=$25.20; $25.20 $15=110.20, Ans. (194, 195) 7 74 COMPOUND NUMBERS. Ex. 32. 1 Ib. 10 g^22g=10560 gr.=1056 doses; $2.25X22=$49.50; $.1 $132 $49.50=882.50, An*. (368, page 196.) Ex.1. Ex.2. Ex.3. Ex.4. - 3 1 2 gal.XfXfXf = E*. 5. Ex.6. Ex.7. Ex.8. Ex. 9. f X T 4 T rd. X V = V 2 J d -> Ex. 10. t % wk.Xf Xf=8j da., Ans. EX. 11. (369, page 197.) Ex.1. T 9 o yd.X3=2 T 7 o ft.; ^ ft.xl2 = 8f in.; Ans. 2 ft. 8| in. Ex. 2. | mo.X30=24 da. ? Ans. Ex. 3. || T.X20=18j^ cwt.; |4 cwt.XlOO=96f Ib.; | lb.Xl6=14 oz.,; Ans. 18 cwt. 96 Ib. 14 oz. Ex. 4. f T.X20=11^ cwt.; cwt.x4=| qr.; | qr.X28 = 12| Ib.; | lb.Xl6=7^ oz.; JL?is. 11 cwt. 12 Ib. 7-g oz. Ex.5. ?X41 (195197) REDUCTION. 75 Ex. 6 & A.X4=2 T ^ R; ^ iJsq. yd.x9= 5ff sq. ft.; || sq. ft.Xl44=127 T 5 y sq. in.; .4w. 2 E. 6 P. 4 sq. yd. 5 sq. ft. 127^ sq. in. Ex. 7. I mi.x8=3s fur.; f fur.x40=174 rd.; 2i 21 4rd.Xl6=2 ft.; ft.Xl2=4f in; .4n*. 3 fur. 17 rd. 2 ft. 4| in. Ex. 8. 4 great-gross Xl2=6f gross; f gross X 12=1 Of doz.; doz. X 12=3 1 units; ^tws. 6 gross 10 doz. 3| units. Ex. 9. T 9 g great ciro^X 360x60=12150 mi. ; Am. Ex. 10. V Cd.X|=2A Cd.; ^ Cd.x8=5 T % cd. ft.; T ^ cd. ft. X 16=9 1 cu. ft.; Ans. 2 Cd. 5 cd. ft. 9f cu. ft. Ex. 11. 438 mi.Xi=262| mi.; | mi.x8=6f fur.; f fur.x40=16 rd.; Ans. 262 mi. 6 fur. 16 rd. Ex.12. f| f 3X8=3/2 f 3; & f 3X60=35 TIL; Ans. 3 f ^ 35 T^ Ex. 13. | S.X30 12f; $ x60=51f'; f X60=25f"; ^iw. 12 51' 25f". Ex. 14. T ^ hhd.X63=33l| gal; if gal.x4=3 T ^ qt.; JL qt.X2=! T 5 3 pt.; A P^X4=1 T 7 3 gi.; ^Ins. 33 gal. 3 qt. 1 pt. 1 T ^ gi. (370, page 198.) Ex. 1. .645 da.X24=15.48 h.; .48 h.x60=28.8 min. ; .8 min.x60=48 sec.; Ans. 15 h. 28 min. 48 sec. Ex, 2 .765 Ib.x 12=9.18 oz. ; .18 oz.X 20=3.6 pwt.; .6 pwt.x24=14.4 gr. Ans. 9 oz. 3 pwt. 14.4 gr. (197, 198) 76 COMPOUND NUMBERS. Ex. 3. .6625 mi. X? =5.3 fur.; .3 fur.x40=12 rd.; Ans. 5 fur. 12 rd. Ex.4. ,8469X60=50.814'; .814'X60=48.84;" Ans. 50' 48.84". Ex. 5. .875 hhd.X 63=55.125 gal.; .125 gal.x8=l pt.; Ans. 55 gal. 1 pt. Ex. 6. .85251X20=17.0502 s.; .0502 s.Xl2=.6024d.; .6024 d.x4=2.4096 far.; Ans. 17 s. 2.4+far. Ex. 7. .715 X 60=42.9'; .9'X60=54"; Ans. 42' 54". Ex. 8. .88125 A.X4=3.525 K; .525 R.X40=21 P.; Ans. 7 A. 3 R. 21 P. Ex. 9. .625 fath.x6=3.75 ft.=3|%, Ans. Ex. 10. .375625 bbl.X200=75.1251b.; .125 Ib.X 16=2 oz.; Ans. 75 Ib. 2 oz. Ex. 11. .1150390625 Cong.x8=.9203125 0.; 8 f 5X60=48 nt; Ans. 14 f 5f 5 48 i^. Ex. 12. .61 tunX2=1.22 P.; .22 P.X2=.44 hhd.; .44 hhd.X 63=27.72 gal.; .72 gal.x4=2.88 qt.; .88 qt.X2=1.76 pt.; .76 pt.x4=3.04 gi.; Ans. 1 P. 27 gal. 2 qt. 1 pt. 3.04 gi. REDUCTION ASCENDING. (371, page 200.) Ex. 1. 1913551 dr.-f-16=119596 oz. 15 dr.; 119596 oz.-- 16=7474 Ib. 12 oz. ; 7474 lb.-f-2000=3 T. 1474 Ib. ; Ans. 3 T. 14 cwt. 74 Ib. 12 oz. 15 dr. (198-200) REDUCTION. 77 Ex. 2. 97920 gr.--20=4896 sc. ; 4896 sc.-^3^1632 dr.; 1632 dr.-i-8=204 oz. ; 204 oz.-T-12=17 lb., Am. Ex. 3. 1000000 in --12=83333 ft. 4 in.; 83333 fW3=27777 yd. 2 ft.; 27777 yd.-r-5,|=5050 rd. 2 yd.; 5050 rd.--40=126 fur. 10 rd.; 126 fur.--8=15 mi. 6 fur.; Ans. 15 mi. 6 fur. 10 rd. 2 yd. 2 ft. 4 in, Ex 4. 120X56=6720 sq. rd.; 6720 sq. rd-^160=42 A., Ans. Ex. 5. 60 X 15 X 10=9000 cu. ft. ; 9000 cu. ft.-4-16=562 cd. ft. 8 cu. ft.; 562 cd. ft.-*-8=70 Cd. 2 cd. ft. ; Ans. 70 Cd. 2 cd. ft. 8 cu. ft. Ex. 6. 28 ft. 6 in.=342 in. ; 6 ft.=72 in. ; 342--72=4f fath., Ans. Or, 28 ft. 6 in.=28! f M 28-s-6=4f fath., Ans. Ex. 7. 30876 gi.-s-4=7719 pt. ; 7719 p t.^-2=3859 qt. 1 pt.; 3859 qt.^-4^964 gal. 3 qt. ; 964 gal.-=-63=15 hhd. 19 gal.; Ans. 15 hhd. 19 gal. 3 qt. 1 pt. Ex. 8. 27072 qt. -^8=3384 pk.; 3384 p k.-^-4=846 bu., Ans. Ex. 9. 254-^-2=127 gi.; 127 gi.-?-4=31 pt. 3 gi.; 31 pt.-i-2==15 qt. 1 pt.; 15 qt.-^4=3 gal. 3 qt.; Ans. 3 gal. 3 qt. 1 pt. 3 gi Ex. 10. 1234567 far.-f-4=308641 d. 3 far.; 308641 d.-5-12 =25720 s. Id.; 25720 s.-^-20=1286; Ans. 1286 1 d. 3 far. Ex. 11. One half crown=2 s. 6 d.=30 d.; 2468H-30=82-r half crowns, Ans (200) 78 COMPOUND NUMBERS. Ex. 12. $88.350-v-$.186=475 francs, Am. Ex. 13. 622080 cu. in.--1728=360 cu. ft.; 360 cu. ft.--40=9 T., Am. Ex. 14. 84621 TTL --60 = 1410 f 3 21 r^ ; 1410 5 -5-8 = 176 f I 2 f 3 ; 176 f 2-5-16=11 0. ; 11 0.--8=1 Cong. 3 0.; Am. 1 Cong. 3 0. 2 f 3 21 TV Ex. 15. 135000000-r-1728=78125 great-gross, Am. Ex.16. 1020300" -5-60 = 17005'; 17005' -f- 60=283 25'; 283-^30=9 S. 13; Am. 9 S. 13 25'. Ex. 17. 411405 sec.-,- 60=6856 min. 45 sec. ; 6856 min.-s- 60=114 h. 16 min. ; 114 h.--24=4 da. 18 h; Am. 4 da. 18 h. 16 min. 45 sec. Ex. 18. 412'--60=6 52', Am. Ex.19. 360X60 = 21600'; 21600' X 20 = 432000 min. of time; 432000 min. -^60 = 7200 h. ; 7200 h.-s-24 300 da., Am. Ex.20. 120X144=17280; 17280-^20=864, Am. Ex. 21. 180X69.16=12448.8 mi., Am. Ex. 22. 45 min.-)-25 min. =70 min. gained each day; 36 yr.X365+9 da.=13149 da.; 13149 da. X 70=920430 min. ; 920430 min.-f-60=15340 h. 30 min. ; 15340 h.--24=639 da. 4 h. ; Am. 639 da. 4 h. 30 min Ex. 23. 20X4=80 qt. bought.; 20 X 282=5640 cu. in.; 5640-^-57|=97f4 qt. sold; 97^4 qt. 80 qt.=17f 4 qt. gained, Am. Ex. 24. 1500 bu.X 35=52500 lb.; 52500 lb.-~28=1875 bu., Am. Ex. 25. 120 lea X^X 1.15=414 mi., Ans. (200, 201) REDUCTION. 79 Ex. 26. 1 bbl. 1 gal. 2 qt.=33 gal. ; 33 gal. X 231=7623 ou. in.; 7623 cu. in.-r-282=27 15 3 3 : beer gal., Ans. Ex. 27. 150 bu.X 2150.4=322560 cu. in.; 322560-^-2218.2=145.415+ Imp. bushels, Ans. Ex. 28. 68 ft. 8 in.=68 ft.; 68f X33=2266 sq. ft.; 2266 sq. ffc.-5-100=22f-g squares, Ans. Ex. 29. 4 ft.=48 in.; 3 ft =36 in.; 1 ft. 6 in.=18 in.; 48X36X18=31104 cu. in., Ans. Ex. 30. 120X56=6720 P.; 6720 P.-*-160=42 A., Ans. Ex. 31. 856 dr.x3X20=21360 gr.; 21360 gr.-f-24=890 pwt.; 890 pwt.-=-20=44 oz. 10 pwt.; 44 oz.-^-12z=3 lb. 8 oz.; Ans. 3 lb. 8 oz. 10 pwt. Ex. 32. 175T.x2240=3920001b.; 392000--2000=196T., short ton weight; $3.75 X 175 = $656.25, cost; $4.50X196=J882, sold for; $882 $656.25=4225.75 gain, Ans. Ex. 33. 73750-^-1.25=59000 sq. ft.; 59000 sq. ft.-5-272|=216 P. 194 sq. ft; 216 P. -^160=1 A. 56 P.; Ans. I A. 56 P. 194 sq. ft Ex. 34. 2492 lb.-^-56=44.5 bushels of corn; 2175 lb.-f-60= 36.25 bushels of wheat; $.60X44.5=826.70; $1.20x36.25=843.50; $26.70+$43.50=$70.20, Ans. Ex. 39. 72 75 Ex. 40. 42 4 I 175 90 5 6 76 3 | 76 5 ; Ans. Ans. $25.33^ (201-203) 80 Ex. 41. COMPOUND NUMBERS. 21 2 126 Ex. 42. M y x f-* ) =$52.50j rated at in Vermont ; * $52.50+$7.50=$60, to be 24 lb., Ans. sold for in New Jersey ; (372, page 203.) Ex.1. * s.X^=^> A. Ex. 2. f pwt.X 3 1 T X T 1 3=?iff lb -> ^TW Ex. 3. | lb. Ex.4. fX Ex.5. pt. Ex. 6. ^I^XlXf pt.XiXi=| pk-= of 2 pk., Ex.7. |X|X T \cd. ft.x|=- Ex. 8. T %X T 4 ?X V P.X T |o Ex. 9. |X V fur. X 1=11 mi.; and || mi. is ^ of T \j of ||XfX T 2 nd-=12f mi., -4rw. Or, IX y XlXfXV 2 mi.=12| mi., Ex. 10. |Xf X^F cu. ft.X^=it Ex. 11. \f Xf X V cu - ft -X T i8- Ex. 12. f in.X^V^g 1 ^ E - E -? Ans - (373 9 page 205.) Ex. 1. 2 K. 20. P.^100 P. ; 1 A.=160 P.; i oo A _ 5 A An* T50 A 8 **^ -o-WS. Ex. 2. 6 fur. 26 rd. 3 yd. 2 ft.=4400 ft. ; 1 mi.=5280 ft.; Ex. 3 18 s. 5 d. 2 T 2 3 far.= U_5JLP f ar; lz=960 d. ; T=4|,^ (203-205) REDUCTION. 81 Ex. 4. 7 I 7 3 2 9 14 gr.=3834 gr. ; 21 ft. = 120960 gr.; Ex. 5. 4 da. 16 h. 30 min. = 6750mm; 3 wk.= 30240 min.; Ex. 6. bu. T 7 g bu., Ans. Ex. 7. 28 gal. 2 qt. = 114 qt. ; 1 hhd.=252 qt. ; 252 qt. 114 qt. = 138 qt.; ^||=f |, Ans. Ex. 8. 4 bundles 6 quires 16 sheets:=4000 sheets ; 1 bale=4800 sheets; |f8bales=| of a bale, Ans. Ex.9. Ex. 10. $ = =$6^=$6.30, Ans. Ex. 11. 3 0. 3 f g 1 f 3 36 nt =24576 nt ; 1 Cong.=61440 nt ; ||||^ Cong.=| Cong., Ans. Ex. 12. 36 cu. ft. 864 cu. in.=63072 cu. in.; 1 T.=50 cu. ft. =86400 cu. in.; Ex 1. 60 60 24 7 (374, 48.0 sec. page 206.) Ex. 2. 60 60 30 46.44" 46.80 min. 27.774' 9.78 h. 3.4629 5.4075 da. An*. .11543 S. ^Ins. .7725 wk. (205, 206) 82 COMPOUND NUMBERS. Ex.3. 40 4 11.52 P. Ex. . 3.25 I 7. 16 10 23040 4. 24 19.2 gr. 20 16.8 pwt 3r.288 E. 4 2.84 oz. Ans. Ex. 5. 12 3 5.5 40 .322 A. 11.04 in. Ex.6 Ans. .71 5-f-12=.27083 ft. 12.00 P. 1.92 ft. Ex. 2.64 yd. 4.75 sq. ch. 28.48 rd. 126.475 A. Ans. .712 fur. Ans. .0054893+Tj, Ex. 8. 3.75 ft.H-6=.625 fath., Ex. 9. .45 pk.-=-4=r.ll25 bu. ; .1125--1.25=.09, Ans. Ex. 10. 3 A. 2 E.=560 P.; IE. 11.52 P.=51.52 P.; 51.52-*-560=.092, Ans. Ex.11. Ex. 12. 63 36.00 HI Ex. 13. 2 4 252 1.0 pt. 8 5.6 f % 3.5 qt. Ans .7 f 50.875 gal. Ans. .20188+T. ADDITION. (377, page 208.) Ex. 3. Ans. 3 ini. 2 fur. 27 rd. 16 ft. Ex. 4. Ans. 1017 A. 2 E. 36 P. 15 sq. yd. 5 sq. ft. 72 sq. in. Ex. 7. Ans. 15 Cd. 4 cd. ft. 4 ou. ft. (206208) ADDITION. 83 fix, 8. 1| hhd. =1 hhd. 42 gal. 42 gal. 3 qt. 1| pt.= 42 3 qt. 1 pt. 1 gi. | gal. = 3 " 1 2 qt. | pt. = 2 " " 3 " 1.75 pt. = 1 3 . *. 2 hhd. 23 gal. 2 qt. pt. 3 gi. Ex. 9. 145| A =145 A. 3 E. 20 P. 7 " 2 " 29| " 1 3 " 16 " A.= 3 " 13 156 A. R. 39| P., Ans. Ex; 10. 31 bu. 2pk. 10| bu. =10 3 " 4 qt. 5 bu. 6 qt. = 5 6 1 pt. 14 bu. 2.75 pk.=14 " 2 6 62 bu. 1 pk. 5 qt. If pt., Am. Ex. 11. 42 yr. 7 mo. =42 yr. 7 mo. 15 da. 10 yr. 3 wk. 5 da. =10 " " 26 9| mo. 9 22 12 h. 1 wk. 16 h. 40 min.= 7 " 16 " 40 min. mo. = 25 3| da. = 3 19 12 min. s. 53 yr. 7 mo. 9 da. 23 h. 52 min. Ex. 13. 1 gross 1\ doz.= 304 8 1| = 453 | great-gross =1296 6| doz. = 75 4 doz. 7 units = 55 (208, 209) 84 COMPOUND NUMBERS. Ex. 15. 3| Pcli. 18 cu. ft.=4 Pch. 1 en. ft. 864 cu.in. 84.6 cu. ft. =3 10 " 604.8 Pch. = 20 " 1080 " f o cu. ft. = 1280 Ans. 8 Pch. 9 cu. ft. 804.8 cu ID Ex. 16, $ 3.75 25.50 12.875 2.40 2.5475 $47.0725, Ans. Ex. 18. 42.4 bu. 2866 lb.= 49.414- 36| bu. = 36.75 " 39 bu. 29 lb.= 39.5 " $.60X168.063=4100.84-, Ans. Ex. 19. 1.125 T. If T.=1.4 2500 lbs.=1.25 $8X3.775=$30.20, Ans. Ex.20 140|cu. yd. =140.8 cu. yd. 24.875 " 46 ou. yd. 20| cu. ft.= 46.75 212.425 cu. yd. remove* , $.18X212.425=$38.24 , cost. (209) SUBTKACTION. 85 SUBTRACTION. - (379, page 211.) Ex. 3. Am. 2 hhd. 54 gal. If qt. Ex.4. 45 yr. 1 mo. 3 wk. da. 17 h.=45 yr. 1 mo. 21 da. 17.5 h. 10 9 1 " 22 " 6.8 "=10 " 9 " 29 ' 6.8 h. Ans. 34 yr. 3 mo. 22 da, 10.7 h. Ex. 6. Ans. 12 cwt. 85 Ib. 6 oz. Ex. 7. 2 wk. 3| da.=2 wk. 3 da. 20 h. .659 wk. = 4 " 14 42 min. 431 S eo. Ans. 1 wk. 6 da. 5 h. 17 min. 16| sec. Ex. 8. | hhd.=32.90625 gal. .90625 32 gal., Ex. 9. | of 3| A.=l A. 1 K. 20 P. 3 " 12.56 " 2 E. 7.44 P., Ans. Ex. 10. 10 Ib. 8 oz. 8 pwt. D>-= 18 " 10 Ib. 7 oz. 10 pwt., Ex. 11. 36 Cd. 4 cd. ft. 10 " 6 " 12 cu. ft. 25 Cd. 5 cd. ft. 4 cu. ft., Ans. Ex. 12. 5 bbl.=5 bbl. 15 gal. 3 qt. 4hlid.=l " 4 2 4 bbl. 11 gal. 1 qt., (211) 86 COMPOUND NUMBERS. tfk.=4 da. 16 h. da.= 15 " 10 min. 30 sec. Ex. 13. | wk.=4 da. 16 h. 4 da. h. 49 min. 30 sec., Ans. i ' Ex. 14. | gross =7^ doz.; 7^ doz. | doz. =6 1 doz., Ans. Ex. 15. I mi.=6 fur. 8 rd. 4 yd. 2 ft. 8 in. rd.= 5 " 9 " 6 fur. 7 rd. 4 yd. 1 ft. 11 in. ; Or, 6 fur. 7 rd. 5 yd. ft. 5 in., Ans. Ex. 17. f pk.=6 qt.; .0625 bu.=2 qt. ; 6 qt. 2 qt.=4 qt., Ans. Ex. 18. of 365| da. = 28 wk. 6 da. 22 h. of 5 wk.= 4 " 1 4 33 wk. 1 da. 2 h. 49| min.= 33 wk. 1 da. 1 h. lOf min., Kx. 19. f of 3| mi.+174 rd. = l mi. 1 fur.; 1 mi. 1 fur. 5| fur. 3| fur., Am. Ex. 20. 15 bbl. 3.25 gal. =15 bbl. 3 gal. 1 qt. 14 24 3.54 " 9^ gal. 1.46 qt. ; Or, 9 gal. 3.46 qt., Ant. Ex. 21. 1457 lb.+1578 lb.+1420 lb.=4455 Ib. =92 bu. 39 Ib.; 200 bu. 92 bu. 39 Ib. = 107 bu. 9 Ib., Ans. Ex. 22. 50 A. 136.4 P. 48 123.3 200 A. 99 A. 99.7 P. =100 A. 60.3 P. = 100.376875 A.; $35x100.376875 =$3513.19+, Am. (211, 212) SUBTRACTION. 87 Ex. 23. 58x37x6:.rl2876cu.ft.=:476cu.yd.24cu.a; 476 cu. yd. 24 cu. ft. 471 " 16 972 cu. in. Ans. 5 cu. yd. 7 cu. ft. 756 cu. in. Ex.24. I lb.= 9oz.l2pwt. | oz. =12pwt. 4|oz.= 4" 16 "16gr. fpwt.= " 21 gr. 31|pwt.= 1 " 11 8gr. 1 Ib. 4 oz. pwt. gr. 11 3 1 Ib, 3 oz. 8 pwt. 21 gr., Ans. Ex. 25. 5 T 7 2 A. =5 A. 2 R. 13 P. 10 T ^ sq. yd. A.=4 " " 26 " 20 " 11 pwt. 3gr, R.= 30 P.= 18 a 9 A. 3 R. 30 P. 18 sq. yd. 4 25 " 12 ft 5 A. 3 K. 5 P. 6 sq. yd., Ans. (380, page 213.) ,, Ex. 1. 1783 yr. 1 mo. 20 da. Ex. 2. 1732 yr. 2 mo. 22 da. 1775 " 4 "19 " 1620 "12 22 " s. 7 yr. 9 mo. 1 da. Ill yr. 2 mo., Ans. Ex. 3. 1860 yr. 7 mo. 4 da. Ex. 4. 1861 yr. 6 mo. 3 da. 1607 " 5 " 23 " 1859 " 1 " 30 " Ans. 253 yr. 1 mo. 11 da. Ans. 2 yr. 4 mo. 3 da. (212, 213) 88 COMPOUND NUMBERS. Ex. 6 From July 4, 1855, to July 4, I860, is 366 da. +365 da.+365 da.+365 da.+366 da.=1827 da.; From July 4, I860, to Dec. 12, 1860, is 27 da. +31 da.+30 da.+31 da.+30 da.+12 da.=161 da.; from 16 minutes past 10 o'clock A. M., to 22 minutes before 8 o'clock p. M., is 9 h. 22 min.; hence 1827 da.+161 da.+9 h. 22 min. =1988 da. 9 h. 22 min., Ans. Ex. 7. 1862 yr. 1 mo. 1 da. 4 h. 55 min. 24 sec. 1860 " 4 " 21 12 40 " 25 1 yr. 8 mo. 9 da. 16 h. 14 min. 59 sec. As the full year is a common year, and the full months commence with May, the 1 yr. 8 mo. 9 da.=365 da. +31 da. +30 da.+31 da.+31 da.+30 da.+31 da.+30 da.+31 da. +9 da.=619 da.; hence, 619 da. 16 h. 14 min. 59 sec., Ans. Ex. 8. 274=23; 242 da.+23 da.=265 da., Ans. MULTIPLICATION. (38, page 215.) Ex. 3. Ans. 44 A. 3 E. 2 P. 9 sq. yd. 6 sq. ft. Ex. 4. Ans. 131 Cd. 5 cd. ft. 12 cu. ft. bu. pk. qt. pt. Ib. oz. pwt. Ex. 5. 34 3 6 1 Ex. 6. 4 10 18.7 2 9 69 3 5 44 2 8.3 7 3 489 1 3, Ana, 132 7 4.9, An* (214, 215) MULTIPLICATION. 89 lb. 3 Z 9 gr. gal. qt. pt. gi. Ex. 7. 9 3 2 13 Ex. 8. 5 2 1 3.25 7 12 5630 11 68 213 5 8 27 7 7 2 15, Am. 549 3 , Ans, A. R. P. sq. yd. Ex. 9. 78 3 15 15 1182 2 32 13^ product by 15 52 2 10 10 | 1235 1 2 23|, cu. yd. cu. ft. .cu. in. Ex.10. 9 10 1424 75 5 1024 product by 8; 9 676 23 576 product by 8x9=72 9 10 1424 " 1 Ans. 686 7 272 " 73 Ex. 11. 27 lb. 2 oz. 17 pwt. 12 gr. ; product by 10 163 5 "^ 5 " , 10X6=60 Subtract 2 8 13 18 , " 1 160 lb. 8 oz. 11 pwt. 6 gr., 601=59 Ex. 12. 22 'yd. ft. 11.5 in., product by 5 111 " 1 " 9.5 " , " " 5x5=25 Ans. 557 2 11.5 , " 5x5x5=125 (215, 216) 90 COMPOUND NUMBERS. Ex. 13. 1 qt. 2 gi.=l| qt.; 1| qt.X 144=180 qt.=45 gal., Ans. Ex. 14. 5 Cong. 5 0. 15 f I 2 f 3 30 nt, prod, by 6 Ans. 22 " 7 " 13 " 2 , 6x4=24 Ex. 15. 14 Hid. 46 gal. 1 qt. pt. 2.4 gi., prod, by 4 58 " 59 1 " " 1.6 " , " " 4x4=16 3 43 " " " 2.6 " , " 1 S. 62 hhd. 39 gal. 1 qt. 1 pt. .2 gi., 16X1=17 Ex. 16. 9 T. 13 cwt. 1 qr. 10.5 lb.=9.6671875 T.; 9.6671875 T. X 1.7=16.43421875 T. =16 T. 8 cwt. 2 qr. 20.65 lb., Ans. Ex. 17. 2 hhd. 23 gal. 2 qt. 1 pt.^2.375 hlid.j 2.375 Wid.x4.8=11.4 hhd. =11 hhd. 25 gal. 1 pt. 2.4 gi., Ans. Ex. 18. 9 oz. 13 pwt. 8 gr.xl2=9 lb. 8 oz.=9f lb., whole weight; $212.38 X9=$2053.00|, Ex. 19. 27 gal. 3 qt. 1 pt.=27.875 gal.; 27.875 gal.x5=139.375 gal.; $1.375X139.375 = $191.64+, Ans. Ex. 20. 37 bu. 3 pk. 5 qt.=37.90625 bu.; 37.90625 bu.x5=189.53+ bu.; $.65X189.53=$123.20 , Ans. %'at DIVISION. (383 9 page 218.) Ex. 6. Ans. 21 bu. 1 pk. 5 qt. 1 pi. yd. ft. in. Ex. 7. 3) 336 4 3| (216-218) DIVISION. 91 7)112 1 5f 16 2| ; Ans. Ex. 9. Ans. 10 cu. yd. 3 cu. ft. 428.15 cu. in. mi. fur. rd. yd. Ex.10. 9)1986 3 20 1 12) 220 5 28 5 18 3 5 4-^, Ans. sq. mi. A. R. P. Ex.11. 12 1 30 2 45)24 3 20 341 1 16f , Ans. Ex. 12. Ans. 1 da. 12|i h. cu. yd. cu. ft. cu. in. Ex. 13. 33|=-Mp; 3794 20 709f 3 Ex. 14. 13|^ 00) 11384 7 400 113 22 1300, An*. Ib. I 3 9 gr. 121 3 2 1 4 4 11)485 1 1 l 16 5)44 1 1 1 16 3962 S^Ans. (218) 92 COMPOUND NUMBERS. Ex. 15. 28 51' 27.765' ; :=28.8577125 ; 28.8577125-r-2.754=:10.478472 ; 10.478472=10 28' 42", Ans. yd. ft. in. da. h. min. Ex 16. 202 1 6| Ex. 17. 10) 1950 15 15f 5 3) 1012 1 9| 10)195 1 31f Ans. 19 12 9 337 1 7|, Ans. Ex. 18. 4 sq. mi.-5-124=82 A. 2 R. 12f f P., Ans. Ex. 19. 48^X24X6^=7566 cu. ft.=280 cu. yd. 6 cu. ft.; 48)280 cu. yd. 6 cu. ft. 5 cu. yd. 22 cu. ft. 1080 cu. in., Ans. Ex. 20. 2 bu. 3 pk. 6 qt. 94 qt.; 356 bu. 3 pk. 5 qt. =11421 qt.; 11421-^94=121^ Ans. LONGITUDE AND TIME. (385, page 219.) Ex.1, 9h. 8 " 7 min. 4 sec. Ex. 2. 1 h. 11 min. 56 sec. 15 52 min. 56 sec. 15 Take 17 59' From 89 2' 13 14' 0" din 7 . Ion. 77 V 71 90 15', Ans. (218, 219) LONGITUDE AND TIME. 93 Ex. 3. 9 h. 13 min. 20 sec. A. M., time at the easterly place ; 2 h. 30 min. A. M., " " westerly " 6 h. 43 min. 20 sec., difference of time ; 6 h. 43 min. 20 sec.Xl5=100 50', diff. Ion.; 100 50' 18 28'=82 22' west, An*. Ex. 4. 6 h. 8 min. 28 sec.Xl5=92 7.', Ans. Ex. 5 1 h. 18 min. 16 sec. X 15=19 34'; 940 44' 19 34'=75 10' west, Ans. Ex. 6. 2 h. 58 min. 23| sec. X 15=44 35' 50", diff. in Ion. ; 77 51'+44 35' 50"=122 26' 50" west, Ans. Ex. 7. 2 h. 33 min. 53i| sec. X 15 38 28' 29", diff. in Ion.; 71 12' 15"+38 28' 29" =109 40' 44" west, Ans. Ex. 8. 5 h. 40 min. 20 sec.Xl5=85 5' west, Ans. Ex. 9. 12 h. 5 h. 51 min. 41f sec.=6 h. 8 min. 18| sec., diff. time ;. 6 h. 8 min. 18| sec. X 15=92 4' 36", diff. in Ion.; 92 4' 36" 18 3' 30" =74 V 6" west, Ans. Ex. 10. 8 h. 53 min. 47 sec. 3.56sec.X24= 1 " 25.44 " f true time Take 8 h. 52 min. 21.56 sec. ( atN. York; From 10 4 36.80 " ship's time; 1 h. 12 min. 15.24 sec., diff. time; 1 h. 12 min. 15.24 sec. X 15=18 3' 48.6", Ans. (386, page 221.) Ex. 1. 84 24' 77 l'=7 23', diff. in longitude; 7 23'-f-15=29 min. 32 sec., diff. in time, Ans. (220, 221) 94 COMPOUND NUMBERS. Ex. 2. 113 14' 2 20' 110 54', diff. in longitude; 110 54'-^-15=7 h. 23 min. 36 sec., An*. Ex. 3. 78 55'+20 30'=99 25', diff. in Ion. ; 990 25' 15=6 h. 37 min. 40 sec., Ans. Ex. 4. 74 1' 63 36'=10 25', diff. in Ion.; 10 25'-=-15=41 min. 40 sec., diff. in time; 4 h. 30 mm. p. M. 41 min. 40 sec. =3 h. 48 min. 20 sec. p. M., Ans. Ex. 5. 71 7'+5' 2"=71 12' 2", diff. in Ion.; 71 12' 2"--15z= 4 h. 44 min. 48 T \.seo., diff. in time ; 12 h. M. 4 h. 44 min. 48f^ sec. =7 h. 15 min. lljf sec. A. M., Ans. Ex.6. 118+120=:238 , diff. in Ion. reckoned west from Pekin ; 360 238 =122, " "Sacramento; 238-7-15z=15 h. 52 min., Sacramento earlier than Pekin ; or 122-7-15=: 8 h. 8 min. later " Ex. 7. 35 32'+76 37'=112 9', diff. in Ion.; 112 9'-f-15=7 h. 28 min. 36 sec., diff. in time; 6 h. 40 min. A. M.-J-7 h. 28 min. 36 sec. 2 h. 8 min. 36 sec. p. M., Ans. Ex. 8. 6 h. p. M. 7 h. 28 min. 36 sec. =10 h. 31 min. 24 sec. A. M., Ans. NOTE. We always subtract the difference of time from the time at the easterly place to obtain the time at the westerly place, adding 12 h. to the minuend if necessary to make the subtrac- tion possible; and we always add the difference of time to the westerly place to obtain the time at the easterly place, rejecting 12 h. when the amount exceeds this time. And whenever 12 h. R,re borrowed n subtracting, or rejected in adding, the time changes from A. M. to p. M., or from p. M. to A. M. (221) PKOMISCUOtJS EXAMPLES. 95 Ex. 9. 94 46' 34" 72 35' 45"=22 10' 49", diff. in lon.j 22 10' 49"-5-15=l h. 28 min. 43 T % sec., diff k time ; 6 h. 20 min. A. M/ 1 h. 28 min. 43 T 4 ^ sec. =4 h. 51 min. 16{4 sec. A. M., Ans. Ex. 10. 28 49'+93 5'=121 54', diff. in Ion.; 121 54'-=-15=8 h. 7 min. 36 sec., diff. in time; 3 h. p. M.+ 8 h. 7 min. 36 sec. = 11 h. 7 min. 36 sec. p. M., Ans. .Ex. 11. 12 h. M. 8 h. 7 min. 36 sec. =3 h. 52 min. 24 sec. p. M., Ans. Ex. 12. 88 1' 29"+5' 2"=88 6' 31", diff. in Ion.; 88 6' 31"-5-15=5 h. 52 min. 26-^ sec., diff. in time ; 12 h. M.+5 h. 52 min. 26^ sec. =5 h. 52 min. 26yL p. M., Ans. PROMISCUOUS EXAMPLES IK COMPOUND NUMBERS. (Pag- 222.) Ex. 1. Ans. 55799 gr. Ex. 2. 3 cwt. 12 lb. = .156 T.; $15.50 X .156=82.418, Aw Ex. 3. 27 yd. 2 qr.=110 qr. ; 110 qr.-r-5=22 ells, Ans. Ex. 4. f>18.945-^-$4.84=:3.914256 =3 18 s. 3 d. 1.6-fqr., Ans. Ex. 5. Ans. 130413645 gee. Ex. 6. 24 sheetsX8X2=384 pages, Ans. Ex. 7. 1 s. 6 d.=18*d.; 5 6 s. 6 d.=1278 d.; 1278-^18=71 yd., Ans. Ex. 8. Ans. 37173 1. Ex. 9. Ans. 111111 sq. yd. Ex. 10. 3 gal. 1 qt. 1 pt.=27 pt.; 3 hhd.:=1512 pt.j 1512 pt.-f-27 pt.=56, Ans. (222) 96 COMPOUND NUMBERS. Ex. 11. 2 T.=5000 Ib.; $.095x5000 =$475, sold for; $475 $375.75=$99.25, Am. Ex. 12. 3 Ib. 9 oz.=900 pwt.; 1 oz. 5 pwt.=25 pwt.; 900-r-25=36=3 &>z., Ans. Ex. 13. 4 gal. 3 qt.=152 gi.; 2 qt. 1 pt. 2 gi.=22 gi.; Ex. 14. |X T 4 i rd.xy=f yd., Ex. 15. 26^X20=530 sq. ft.=58| sq. yd.=58| yd., Ex. 16. 15 T. 3 cwt. 3 qr. 24 Ib. long ton weight=34044 lb.= 17.022 T. snort ton weight; hence $140xl7.022=$2383.08, sold for; $ .06 X 34044= $2042.64, cost; $340.44 gained, Ans. Ex. 17. (40 ft.-f 36 ft.)X2=153 ft., round the room; 153X22|=3404| sq. ft. in the walls; 36^X40=1460 sq. ft. in the ceiling; 3404| sq. ft.+1460 sq. ft. 1375 sq. ft.=3489^ sq ft. to be paid for, at ^ 8 =2 cents per sq. ft.; hence 3489|X2=$69.785, Ans. Ex. 18. 78 Ib. 9 oz.x23=18 cwt. 6 Ib. 15 oz., Ans. Ex. 19. 33 2'+30 41'=63 43', Ans. Ex. 20. 1 gross 4 doz.=16 doz. ; 16 doz.X31=496 doz., Ans. Ex. 21. 4 sq. rd. 120 sq. ft. 84 sq. in.XlS =79 sq. rd. 264| sq. ft. 72 sq. in. ; 160 sq. rd. 79 sq. rd. 262^ sq. ft. 72 sq. in. =80 sq. rd. 7 sq. ft. 72 sq. in., Ans. Ex. 22. $12.025-f-$13 = .925 T.=1850 Ib., Ans. Ex. 23. 1 pk. 4 qt.=.375 bu.; $.72-f-.375=$1.92, Ans. Ex. 24. 36244 lb.-:-60=604 T L bu., Ans. (222, 223) PROMISCUOUS EXAMPLES. 97 rix. 25. 32X24X6=4608 cu. ft. = 170f cu. yd.; $.20xl70f =$34.13+, Ans. Ex. 26. (32 ft.+24 ft.)X2=112 ft.,' mason's girt; 112X6X1^=1008 cu. ft.=40 T 8 T Pch.; ?, Ans. Ex. 27. 1 mi.=5280 ft.; 10 ft. 4 in. = 10| ft.; i26720 times, hind wheels; r _. 84480 forward 42240 times, Ex. 28. $3.75X15.22=$57.075 4.25X 7.36= 31.28 $88.355 cost; 1045X2258= 101.61 sold for; $13.255, Ans. Ex. 29. f of 3 T. 10 cwt. =5000 Ib. of 7 T. 3 cwt. 26 lb.=4408 Ib. 592 Ib., Ans. Ex. 30. 16 Ib. 5 oz. 10 pwt. 13 gr. Troyr=94813 gr ; 94813 gr.^-7000=13.5447+lb. Avoirdupois; 13.5447+lb.=13 Ib. 8 oz. 11.4+dr., Ans. Ex. 31. 1 ft. 7.8 in. =1.65 ft. = .l rd., Ans. Ex. 32. 9 in.=| ft.; 6 in.=| ft.; fX|Xf=8ft., Ans. Ex. 33. 16 in.=| ft.; 7X|=5| ft., Ans. Ex. 34. 17280 cu. in.H-268|=:64? gallons, dry measure, 17280 cu. in.^-282 =61|| beer Difference ^s|-g gallons, (223, 224) 98 COMPOUND NUMBERS. Ex. 35. 6 T Xf XT! ft.X5 1 o= 2 - 2 75 tons, Ans. Ex. 36. 7^ bu. = 3 pk. 44 qt. ; X%X 3 q^*- If q^*j 3pk.4f qt.+lfqt.=3pk.6lfqt; 5 bu. 3|i qt. 3 pk. 6|f qt. =4 bu. 5| qt.^=16| pk., ^.TIS. sq. mi. A. R. P. sq. yd. mi. A. R. P. sq. yd, Ex.37. 5 250 3 456 3 14 25 7 o^ 8)37 475 1 2 90 2 4 14 114 33 21 4 459 1 25 2 204 2 38 5| 2 204 2 38 2 254 2 26 24|, Ans. Ex. 38. 14.2878X5.6=80.01168 Ib. = 80 Ib. 2 pwt. 19.2768 gr., Ans. Ex. 39. | of 24=15 carats fine, Ans. Ex. 40. 2416=8 carats=^=| alloy, Ans. Ex. 41. 384| A.=384 A. 3 R. 8 P. 22 " 1 " 20 " 2) 362 A. 1 R. 28 P. (See Prob. 33, p. 64.) 181 A. R. 34 P., younger ; 203 2 u 14 " , elder. Ex. 42. 4000 bu.Xf =3586^ bu., Ans. Ex. 43. 110 bu.x64=7040 Ib. clover seed exchanged; 7040 lb.-4-60=117| bu. clover seed, N. Y. measure; 1171 Xf X| 704 bu. corn received, N. Y. measure; 704 X 5 T 8 X 5^=7294 bu. corn, N. J. measure; (224) PROMISCUOUS EXAMPLES. 8f X7294=$486 3 2 T , corn brought in N. J.; $4X^10 =$440, clover seed worth in N. J. ; $ 46^, the N. J. farmer gained ; HOxfXf 660 bu. corn, he would have received had the standards of measure been the same ; hence 729$ bu. 660 bu.=694 bu. corn, gained by the N. J. farmer in the reckoning. Ex. 44. 763.4X763.4=582779.56 sq. ft. =13 A. 1 R. 20 P. 164.56 sq. ft., Arts. Ex. 45. 20^ ft.X2=41 ft., width of both sides; 42X41=1722 sq. ft.=17.22 squares; $4.62|X17.22=$79.64|, An*. Ex. 46. 17 T. 15 cwt. 62' lb.=17.78125 T.; $1333.593--17.78125=$75, nearly, Ans. Ex. 47. 4 3 231 4 15 5 1728 Ex. 48. 96 300 112 $350, Ans. 77 I 14400 > Ans - Ex. 49. 72 300 112 Ex. 50. I 300 90 I 112 3 1400 3 1120 .66|, Ans. $373.33^, Ans. Ex. 51. 56 300 112 Ex. 52. 60 300 112 , Ans. 8560, Ans (224, 225) 100 COMPOUND NUMBERS. Ex. RO i g \/ r 20 ft. 5'db 24 ft. 6' 6" 8' 2ft. 4" 8' 3ft. 40 10' 73 6' 13 7' 16 4' 10' 8' 55 ou. ft. 3'zb ; Am. 90 ft. 6', An*. (229, 230) DIVISION. 106 DIVISION. (392, page 230.) Ex. 1. 17 ft.) 287 ft. T (16 ft. 11', An*. 17 117 102 15 ft. 7' 15 1' Ex. 2. 6 ft. 8') 29 ft. 5' 4" (4 ft. 5', Ans. 26 8' 2 ft. 9' 4" 2 " 9' 4" Ex. 3. 48 ft. 6') 1176 ft. 1' 6" (24 ft. 3', An*. 1164 12 ft. V 6" 12 " 1' 6" Ex.4. 38 ft. 10' 362 ft. 5' 4")275 cu. yd. 5 cu. ft. 1' 4"= 9ft. 4' 362 ft. 5' 4")7430 C u. ft. 1' 4"(20 ft. 6', Ans. 7248 cu. ft. 10' 8" 12 11" 4" 349 6' 181cu.ft. 2' 8' 181cu.ft. 2' 8" 362ft. 5' 4" (230, 231) 106 DUODECIMALS. CONTRACTED METHOD. (393, page 231.) Ex. 1. 2 ft. 10' 7") 7 ft. 7' 3" (2 ft. 7' 8", Ans. 5 9' 2" 1 ft. 10' V 1ft. 8' 2" i' ii" i' ii" Ex.2. 7 ft. 2' 4" 33 ft. 5' 6") 64 ft. 9' 8" (1 ft. 11' 3", 8'" 9" 7' 4 ft. 33 5' 6" 28 9' 4" 31 ft. 4' 2" 4 2' 4" 30 8' 1" 5' &" 5" 8'1" 8' 4" 33 ft. 5' 6" Ex, 3. 7 ft. 2' 11") 36 ft. 4' 8" (5 ft. 3" T",An$. 86 ft. 2' 7" 4" 4" (231) SUBTRACTION. 107 Ex. 1. 1000 756 SHOET METHODS. FOR SUBTRACTION. (395, page 232.) Ex. 2. 4000000 8576 244, Ans. Ex. 3. 10.0000 .5768 9.4232, Ans. Ex. 5. 64000.00000 90.59876 63909.40124, Ans. Ex. 7. 1000 100000 271 18365 3991424, Ans. Ex. 4. 1700000 13057 1686943, Ans. Ex. 6. 1000000 599948 400052, Ans, 10000000 3401250 729; 81635; 6598750, Ans. Ex. 1. 78400 784 FOR MULTIPLICATION. (396, page 233.) Ex. 2. 5873.000 5873 77616, Ans. Ex. 3. 478300000 4783 5867.127, Ans. Ex. 4. 75000.000 75 478295217, Ans. (232, 233) 74999.925, Ans. 108 SHORT METHODS (397, page 233.) Ex. 1. 78600 Ex. 2. 432700 1572 17308 77028, An*. 415392, Ans Ex.3. 7328000 Ex.4. 7873586000 21984 39367930 7306016, Ans. 78342.18070, Ans. Ex.5. 437890000 Ex.6. 7077364000000 262734 49541548 437627266, Ans. 7077314.458452, Ans. (398, page 234.) Ex.1. 567X13 Ex.2. 439603x10.5 7371, Ans. 4615831.5, Ans. Ex.3. 7859X107 Ex.4. 18075x1008 840913, Ans. 18219600, Ans. Ex. 5. 3907X10.002 39077.814, Ans. (399, page 235.) Ex. 1. 56783x71 Ex. 2. 47.89x60.1 4031593, Ans. 2878.189, Ans. Ex. 3. 3724.5X.901 Ex.4. 103078x40001 3355.7745, Ans. 4123223078, Ans. (233-235) FOB MULTIPLICATION. 109 Ex.1. Ex.3. Ex. 1 Ex. 2 Ex.3 28 2 = 39 2 = 37 2 = (4OO, page 236.) 432711000 Ex. 2. 432711 9)432278289 48030921 2 96061842, Am. 673200000 6732 Ex.4. 9)673193268 74799252 8 59.8394016, Ans. Ex.5. 4444400000 444.4.4. J. J- JL J. X 9)4444355556 493817284 8 3950538272, Ans. (401, page 237.) 24xBO+3 2 =729, Ans. 48X50+1 2 =2401, Ans. 5780000 578 9)5779422 642158, Ans. 8675000 8675 9)8666325 674047.5, A** = 784 26*=22x30+4 2 = 676 38X40+F=1521 38 2 =36x40+2* =1444 34X40+3 2 =1369 36 2 =32x40+4 2 =1296 35 2 =30X40+5 2 =1225. (236, 237) 10 110 SHORT METHODS Ex.4. 77*:= 74X80+3* =5929; 88 2 =86x90+2 2 =7744 ; 8.6 2 = 8.2x9.0+.4 2 =73.96 ; 99* =98x100+1* =9801; 98 2 =96xlOO+2 2 =9604 ; 69 2 =68X 70+1 2 =4761; 68 2 =66X 70+2 2 =4624^ 6.7 2 =6.4x7.0+.3 2 =44.89; 62*=60x 64+2 2 =3844. (4O3, page 237.) Ex. 1. 43700--4=10925, Ans. Ex. 2. 68720-f-4=17180, Ans. Ex. 3. 5734154000--3=:1911384666f ; Ans, Ex. 4. 75864200-r-8=9483025 ? Ans. Ex. 5. 78563000--8 =9820375, Ans. Ex. 6. 57687000-f-7=8241000, Ans. (4O4, page 238.) Ex. 1. 43789X100X81=36125925, Ans. Ex. 2. 58730X1000X7^=418451250, Ans. Ex. 3. 7854X34|=268999.5, Ans. Ex. 4. 30724X10000X7^=22530933331, Ans. Ex. 5. 47836X100X7^=34083150, Ans. Ex. 6. 53727X100X24^=129840250, Ans. (4O5, page 239.) Ex. 1. $568-M=$142, Ans. Ex. 2. $51-^-6=$8.50; 33x8=264 yd.; $264^-16 $16.50; $18-f-3=86; $8.50+$16.50=$25; $25-46=$19, f Am Ex. 3. $28-f-8~ $3.50, Ans. Ex.4. $576-4-9=$64, Ans. Ex. 5. $7.875-f-63=8.125=$, gain on 1 gal.j $576-r-8=$72, Ans. (237-239) FOR MULTIPLICATION. Ill (4O6, page 241.) Ex. 2. $15.46 46 $711.16, cost of 3 Ib. 10 oz.=46 oz.j .773 lpwt.=^of loz.; 5.411 " 7 " .129 " 4gr.= i oflpwt.; .032 " " 1 " .016. " " i " $717.521, Ex. 3. 90 lb.=.9 cwt.; $.56x5.9=$3.304 ; Art*. Ex.4. $4.48 3 $13.44, cost of 3 bu.; 1.12 " " lpk.=jbu.; .14 lqt.=4pk.; 28 " " 2 " $14.98, An*. Ex. 5. 8 Ib. 5 oz. 6.74 dr. 5 41 Ib. 11 oz. 1.7 dr., weight of 5 gal. ; 2 " 1 " 5.685 " 1 qt.r^| ^1, . 4 " 2 11.37 " " 2 1 " 10.842+" " " lpt.=^qt.; 4 " 2.71 " " " lgi-=i PM 8 5.42 2 " 49 Ib. 12 oz. 5.73 dr., (241) 112 SHOKT METHODS Ex.6. $17.50 3 - $52.50 , cost of 3 A.; 4.375, " " IE.; .547, " " 5 P= K.; 3.282, " " 30 P.; 043, " " .4P.= T ^ $60.747, ^TIS. Ex. 7. 3 17 s. 10.5 d. ;.. 7 27 5s. 1.5 d., val. of 7 oz.; 1 18 " 11.25 " " 10 pwt=,J oz.; 19" 5.625" " 5 " 3 " 10.725 " " " 1 " 1" 11.3625" " 12gr.=^pwt.; 11.68125" " " 6 "=|of!2gr.; 30 10 s. 4.14375 d., Ans. Ex. 8 4 36' 40" 5 23 3' 20", in 5 da.; 2 18' 20", " 12h.= i da.; 34' 35", " 3 " 5' 45|", " 30 min.=J of 3 h.; 23 T ^", " 2 " 5f|", " 30sec.=* of 2min.; " 15 " 10 26 2' 34|||", Ans. (24X) FOE DIVISION. 1 1 3 Ex. 9. 7 gal. 1 qt. 1 pt. 3 gi.=7.46875 gal.; hence, $7.46875, cost at 8 s.; $3.734375, "4s.; .622395+, " 8 i; $4.35+, Ans. Ex. 10. $12.50 , at 6 s. per day; 2.083+, 1 s. " $10.416 +, " 5 s. " " .521, 3 d. " $10.93+, FOR DIVISION. (4O7, page 242.) Ex. 1. 634.75X4=2539, Ans. Ex. 2. 785.6X8=6284.8, J.TIS. Ex. 3. 5.16X3=15.48, Ans. Ex. 4. .167324X8=1.338592, ^TW. Ex. 5. 1748X7=12236, Ans. Ex. 6. 57.634X6=345.804, Ans. (4O8, page 248.) Ex. 1. 2575 ) 64375 4 4 103/00 ). 2575/00(25, 206 515 515 (241-243) 114 SHORT METHODS. Ex. 2. 3625 ) 76394 29/000) 611/152 (2l$$s, Am. 58 31 29 2152-5-8=269 Ex. 3. 4331 ) 7325 3 3 13/00 ) 219/75 (16f |, 13 78 * _ 1175x4=47/00 1300x4=52/00 Ex.4. 431.25 ) 5736 3450 45888 2 2 69/00 ) 917/76 (IStff, Ant. 69 227 207 207612=173 (243) RATIO. 115 Ex. 5. 566f ) 42.75 3 3 17/00) 128/.25 (j07|f, * Ans - Ex. 11. $if:8 3 V=*bn.:(?); (?) = bu.X 5 7 Xfi=T 7 3z bn., Ans. Ex. 12. 46 A. 3 K. 14 P. =7494 P. ; 35 A. 2 E. 10 P. = 5690 P.; $374.70xf!M=$284.50, Ans. Ex. 13. 1 yr. 3 mo.=15 mo.; 2 yr. 8 mo.=32 mo.; $1870.65Xff =$3990.72, Ans. H Ex. 14. 164.5X=246.75 ; Ans. 5 Ex. 15. 12 A. 3 R. 36 P.=2076 P.; 2076 P.x V^=17127 P.=107 A 7 P., Ans. (252, 253) COMPOUND PKOPORTION. 119 Ex. 16. $325 : $2275=$26.32 : (?); $26.32X2275 (?)= =$184.24, Ans. 325 Ex. 17 60.5X44=2662 sq. ft.=295| sq. yd. ; 14| sq. yd. : 295| sq. yd.=$34i : (?); (?)=$JLp-X 2 -V- XA=$709.86|, Ans. Ex. 18. 1 doz. = 12; lOf gross=1548; .0625x i f|- a =$8.06|, Ans. Ex. 19. 7 s. 6 d. = 7^s.= 1 3 6 s.; and since the weight of the loaf should vary inversely as the price of wheat, we have 7 s. : 6s. :=(?) :9 oz.j (?)=9 oz.XV 5 X^=Hi oz., Ans. Ex.1. Ex.2. COMPOUND PKOPORTION. (439 5 page 256.) (12 ((?) : J = 11 : I 5 (18 33 Ex.3. 12 16 16 = 1260 : 4728 (?) (?) 1260 (253-256) 18 11 12 5 33 (f)=10, Ans, 12 4728 10 | 197 (?)=19.7 da., Ans. 120 PROPORTION. Ex.4. 144 6 12 30 ( 200 (?)=] 3 7 (2 350 6 3 144X6X12X350X6X3 (?)= =259.2 da., An*. 30XTX200X3X2 Ex. 5. 5 bu.=20 pk.; 3 bu. 3 pk.=15 pk.; 22 da.XlXff^ 44 da -> Am. Ex. 6. 10| . 4 3000 =1:1 (0 32 3 3 4 3000 3 38 2 11 Ex. 7. T300 ("300 ("(?) (?) \ : ] = 1 : 8 .90 (1.25 (.90 1 300 1.25 3 Ex. 8. 468 (?)=1250 bu., An*. [120 6 468X120X6 (f) = =3240, An*. 26X4 Ex. 9. io l3 i (16 * : 17 =546 : (15 Ex. 10. 7 = 384 bbl, Ant. {^=22 da., Ant. (256) PROMISCUOUS EXAMPLES. 121 PROMISCUOUS EXAMPLES IN PROPORTION. (Page 257.) Ex. 1. 7 ft. : 198 ft. =4 ft. : (?) 4 ft. X 198 (?) = =1134 ft., Ans. 7 Ex. 2. $972 : $11|=$607^ : (?) Ex. 3. 3 cwt. X Vo X =99 cwt., Ans. Ex. 4. 18 da.XT%=14 da., Ans. Ex. 5. ( (?) ( 140 (16.50 * (24.75 140 A.X24.75 (?) = _ -210 A., Ans. 16.50 Ex. 6. (1728 (750 4 : 1 =$155.52 : (?) (18 (54 $155.52X750X54 (?)= =$202.50, Ans. 1728X18 Ex. 7. 28 oz ( 2 2 ( . : (?)= J : 4 (6 ( Ex. 8. 15 menX 2 5=60 men, Ans. Ex. 9. 12s. 7d.=151d.; 1 oz.=240 gr.; 15 Ib. 11 oz. 13 pwt. 17 gr.=92009 gr.; 240 gr.: 92009 gr.=151d.: (?) 151d.x92009 (?)= - =28944.497+d. ; 240 28944.497d.-7-Q6=$301.50+, Ans. (257) 11 122 PROPORTION. Ex. 10. (16 (17 1 (?) $36.72 1 7 : \ 10i=$36.72 : ( 16 17.5 (15 (16* 7 10.5 15 16 (?)=$64.26, Ana. Ex. 11. 2 yr. 5 mo. 5 da.=29| mo.; 1 yr. 8 mo. =20 mo.; and since the money lent should vary inversely as the time, we have 291 mo. : 20 mo.=(?) : $1200; $1200X291 v .'J 20 :tjpjLiju, jtins. Ex. 12. < 59 : f ?5=$10.50 : $5.83| ; Ans. Ex. 13. (12 1 9 (CO : \ 7=2 : I (?) 7 12 9 15 f (15 15 2 9 140 12 8=4 men, Ans. (?) 8 Ex. 14. 1 yr. 8 mo.=20 mo.; 3 yr. 4 mo. 24 da.=40.8 mo.; '300 210.25 =$30 : (?) 40.8 $30X210.25X40.8 (?)= --- =$42.891, Ans. Ex. 15. ( (?) 300X20 2 =1 : 1 Or, 4 19yd. (?)=y> yd.X|X|-15| yd, Ans. ( )=15| rd., Ans, Ex. 16. (24 (38 =$95.60 : (?) 18 22 ; Ans. (257, 258 PROMISCUOUS EXAMPLES. 123 Ex. 17. 16^ Cd. : 0)=11A T. : 15 T. 0) = W Cd.xWX 3 %=22| Cd., Am. Ex.18. 8 18 s=75 : 450 0) ?V = 7 I& ft '> AUS - Ex. 19. ( 4 ( 15 0)=V A. Ex. 20. 600 X T % XV s =450 men, Ans. Ex. 21. 120 gal. 80 gal.=40 gal. gained in 1 hour; 20 bar. =630 gal. to be be filled; 40 gal. : 630 gal.=l h. : 15| h., Ans. Ex. 22. 16 oz. 14 T 9 g oz.=l T 7 s oz.=f | oz. Now, if every 16 ounces of groceries have been lessened f | ounces, how much should the price, 838.40, be lessened ? 16 oz.: 1/5 oz.=$38.40 : $3.45, Ans. Another Solution. If every 16 oz. be reduced to 14 T 9 g , to what sum should the cost, $38.40, be reduced ? 16 oz. : 14 T 9 =$38.40 : $34.95 $38.40 $34.95=$3.45, Ans. Ex. 23. 75 Ib. coffeeXf=120 Ib. sugar; 5 Ib. : 120 lb.=$.625 : $15, ^7^. Ex. 24. $28.75X3|u=$35.38+, Ans. Ex. 25. ( 12000 ( 3000 J8 : -|l2 =859|:(1) ( 550 ( 320 =83 Am (258) PERCENTAGE. '24 f 7 PERCENTAGE. NOTATION. (442, page 260.) Ex 1. .03 ; .09 ; .12 ; .16 ; .23 ; .37 ; .75 ; 1.25 ; 1.84; 2.05. Ex. 2. .15 ; .11 ; .045 ; .0525 ; .0875 ; .205 ; .25625 ; .356; .24875; 1.305. Ex. 3. .0025 ; .0075 ; .005 ; .004 ; .00375 ; .0028 ; .00288; .013125; .102. ^.4. 2*; f; iVs; i; f; TT; if>^- Ex. 6. .065 =.06^=6^ per cent., Ans. Ex. 7. .14375=.14|=14| per cent., Ans. Ex. 8. .0975 = .09|=9| per cent., Ans. Ex. 9. .014=.01f =1| per cent., Ans. Ex. 10. .1025=.10|=10^ per cent., Ans. Ex. 11. .004=.00|=f per cent., Ans. Ex. 12. .028=.02|=2| per cent., Ans. Ex. 13. .1324=.13^=13 5 6 5 %, Ans. Ex/ 14. .084f=.083-83 %, Ans. Ex. 15. ,004 T 6 T =.00 T \= f \ %, Ans. Ex. 16. .003 T ^=.00 T 4 3= T 4 ^ %, Ans. (258260) GENERAL PROBLEMS. 125 GENERAL PROBLEMS IN PERCENTAGE. (449, page 261.) Ex. 5. Ans. 70.65 Ib. Ex. 6. An*> 240 mi. Ex. 8. Ans. 919 men. Ex. 9. 756x1-25=945, Ans. Ex. 12. iX,fo= T fo> Ans - Ex. 13. 14f %=4; VX4 Ex. 14. $375x.05=$18.75, Ex. 15. $536+$450+$784=$1770, debts; $1770X.54=$955.80, ^s. Ex. 16. 15 %+5 %+6 %+8 %=34 %=.34; $1500 X- 34 $510, 4*w. Ex. 17. $3500X3=$10500, received in 3 years; 10 %+l2 %+lS %=40%; $3500X.40=$1400, spent in 3 years; $10500 $1400 =$9 100, Ans. Ex.18. 1.00 .25=.75; .75X.30 = .225; .25+.225=.475 drawn; .475X-10=.0475 deposited again; 1.00 .4275 =.5725 in bank; $6000 X- 5725-4-13435, Ans. Ex. 19. 2 %+3i %+2 %+2 %+l^ %+2| %+4 % +3 %=21 % entire gain; - 4 % loss; 17 % net profits; $5400 X- 17 =$918, Ans. (45O, page 263.) Ex. 1. $21.60-5-$720=.03=r:3 %, Ex. 2. 234-^-1560=.15=15 Ex. 3. 49-f-980=.05=5 %, ^TW. (261-263) 126 PERCENTAGE. Ex. 4. 320 10 s.=320.5; 25 12.8 s.=25.64j 25.64--320.5=.08=:8 %, Ans. Ex. 5. 5 gal. 3 qt. = 5.75 gal.; 5.75-r-46=.125=12 3 | %, Ans. Ex. 6. 5.495-5-7.85=.7 = 70 %, Ans. Ex. 7. f-r- T 1860 ' e(luated time - Ex 2. 1+|= T 7 ,; !-&=&; i y 3=1 i X 8=2 T 5 2 X 12 = 5 8-5-1=8 mo., Ans. I 8 Ex. 3. $480X90=843200 320X60= 19200 $800 $62400 62400-^-800^-78 da., average term of credit; hence the note for the whole amount will run 365 da. 78 da.=287 da. Again, (347-350) 176 PERCENTAGE. $480-4-1.015 =$472.906, present worth of $480 ; $320-1- 1.01= 316.831, " 320; $789.737, present worth both debts ; Amount of $800 for 287 da. =$838.266 " $789.737 for lyr.= 837.121 $ 1.14+, AM. Ex.4. $280x3=$ 840 300x4= 1200 6400-^-1340 =4 mo. 23 da., average Or.; 200X5= 1000 Apr. l-f4 mo. 23 da.=Aug. 24, Ans. 560x6= 3360 $1340 Ex. 1. $6400 (618, page 352.) Due. Da. Items. Prod. Mar. 4 25 Apr. 16 " 30 May 17 21 43 57 74 250 960 96 200 850 7560 4128 11400 25900 48988 1256 489881256=39 da.; Mar. 4+39 da.= Apr. 12, 1860, Ans. Ex.2. Due. Da. Items. Prod. Oct. 12,1859 Nov. 6, " Jan. 15,1860 Mar. 19, " Feb. 24, " Mar. 25, 25 95 159 135 165 ISO 300 150 350 130 MO 7500 14250 55650 17f50 mop K50 118050 Ex.3. Due. Da. Items. Prod. Nov. 30, I860 240 Jan. 11, 1861 42 500 21000 Feb. 23, Mar. 17, " Apr. 25, " 85 10T 146 436 325 436 37060 347T5 63656 May 16, " 16T 537 89679 2474 246170 118050-^-1250=94 da.; Oct. 12, 1859 +94 da.= Jan. 14, 1860, Ans. 246170-^2474 =100da.; Nov. 29, 1860+ 100 da.= Mar. 10, '61, Am. (350-352) EQUATION OF PAYMENTS. 177 Ex.4. Due. Da. Items. Prod. Aug. 16, 1859 Oct. 15, " Dec. 14, " Feb. 12, 1860 60 120 180 o50 250 300 248 15000 36000 44640 1148 95640 9564Q-*-1148==88 da.; Aug. 16, 1859 +83 da.=Nov. 7, 1859, Ans. (63O, page 356.) 3x. 2. Due. Da. Items. Prod. Due. Da. Items. Prod. Mar. 1 Apr. 12 July 16 Sept. 14 42 137 197 436 548 312 536 23016 42744 105592 May 27 " 9 June 20 Aug. 3 87 69 111 155 400 650 200 84 34800 44850 22200 13020 Balances, 1832 1334 171352 114870 56482^-49 1334 114870 8 = 113 da.; 498 56482 Mar. 1, 1860+113 da. = June 22, I860, due. Ex.3. Focal date. Due. Da. Items. Int. Due. Da. Items. Int. Apr. 1 " 12 Mar. 16 June 25 85 74 101 54.36 28.45 95.75 26.32 .77 .35 1.61 Apr. 1 June 18 ' 12 " 20 85 7 13 5 50.00 3000 125.00 150.00 .71 03 .27 .13 Ba 204.88 2.73 1.14 355.00 204.88 1.14 ances, 1.59 150.12 Int. on $150.12 for 1 da. =$.02502 ; $1.59^-1.02502 =64 da. ; June 25, 1858+64 da. = Aug. 28, 1859. Ex.4, Due. Da. Item?. Int. *.58 .51 .24 .08 Due. Da. Items. Int. Jan. 1 Feb. 1 Mar. 17 Apr. 1 95 64 20 5 36.72 48,25 72.36 98.48 ' Jan. 10 " 21 Mar. 23 Apr. 6 86 75 14 98.72 25.84 15.17 8.96 1.42 -32 .04 1.78 1.41 talance of Items, 255.81 148.69 107.12 1.41 148.69 Balance of Int., .37 Int. of $107.12 for 1 da. = .018 ; $.37-^,018=20 da., Ans. (352-358^ 178 PERCENTAGE. Ex.5, Due. Da. Items. I Prod. Due. Da. Items. Prod. Apr. 25 24 1000 24003 1 Apr. 1 " 21 20 SCO 324 6480 Balances, 1000 884 24(00 6480 L " 116 17520 Ex.6. 17520-5-116=151 da.; Api l.-f 151da.=Aug. 30. Due. Da. Items. Prod. Due. Da, Items. Prod. Aug. 12 Oct. 15 11 75 684 468 7524 35100 Aug. 1 839 Balances, 1152 839 42624 839 42624-^-313 = 136 da. j 313 42624 Aug. l.-|-136 da. Dec. 15, Am. Ex. 7. Due. Da. Items. Prod. . Due. Da. Items. Prod. Mar. 1 June 1 Aug. 1 92 153 500 800 600 73600 91800 Apr. 1 31 1000 31000 Balances, 1900 1000 1^5400 31000 134400- 1000 31000 -900=149 900 134400 Mar. 1.+149 da.= July 28, Ans. Ex 3. Due. Da. Items. Prod. Apr. 12 Sept. 20 161 500 1000 161000 1500 161000 161000-s- 1500 = 107 da.; Apr. 12+ _ 107 da. = July 28, the average maturity of the two debts. Hence to discharge the whole obligation by two equal payments at an interval of 60 da., these payments must be made, one 30 da. before July 28, and the other 30 da. after July 28; and we have July 2830 da. = June 28, date of 1st payment; July 28+30 da. = Aug. 27, 2d (356, 357) EQUATION OF PAYMENTS. 179 Ex.9. Due. Da. Items. Prod. Duo. Da. Items. Prod. Sept. 12 " 20 " 30 Oct. 5 " .6 " 29 8 18 23 34 47 530.84 236.48 739.C6 273.44 194.78 536.42 1892 13312 6289 6623 25212 Sept. 14 " 25 Oct. 3 ,t 17 Nov. 16 " 24 13 21 35 65 73 436 320 ' 560 370 840 660 872 41CO 11760 l.9fSO 5: POO 40880 2511.52 53328 308 > 2511.fti? U5-22 Balances, 574.48 1 71894^-574.48 = 125 da.; Sept. 12, 1859+125 da.=Jan. 15, 1860, Ex. 10. Due. Da. Items. Prod. Due. Da. Item-. Prod. June 1 July 16 Aug. 12 Sept. 12 Nov. 18 45 72 103 170 500.78 21 6.94 843.75 * 94.37 856.48 97063 60750 61220 145602 June 3 July 1 Nov. 1 2 30 153 500 1000 1500 1000 30000 229500 Balances, 4988.32 3000.00 364635 260500 3000 1 260500 1988.32 104135 104135-v-1988.32=:52 da.; June l-j-52 da. = July 23, Ans. ACCOUNT SALES. (625, page 358.) Ex. 1. We first average the sales, using Apr. 1, the earliest maturity of the en- tire account, for a focal date, thus : 69511513362.80 = 52 da.; Apr. 1.-J-52 da. May 23, date for commission and guaranty. We now average the entire account, using the same focal date as before, and putting (357, 358) Due. I>a. from Apr. 1. Items Pro, 1 . May 15 " 5 June 28 44 34 88 43R8.iV) 5344.80 650.0 192192 181723 o21200 = 52 da.: 13362.80 695115 180 PERCENTAGE. Freight, Primage, Wharfage and Cartage, as one item; and also, commission and guaranty. Due. Da. 1 Items. Prod. Due. Da. Items. Prod. Apr. 1 June 3 June 28 May 23 63 88 52 185.20 3207.07 37.68 443.27 202055 ofel5 2S050 May 15 " 5 June 28 44 34 88 48680.00 5344.80 3t50.0 1921 0-2 181723 o-l-CO | 3873.32 228410 13362.80 3873.32 695115 228 11 4667059489.48=49 da.; Balances, ipr. 1+49 da.inMay 20, proceeds due. 9489.48 466705 Ex. 2. Equation for the average of sales. 424900^-6275= da. Mayl +68 da. == July 8, date for com- mission; $175.48+$ 56.25 + $8.37= $240.10, charges made May 1, for freight, etc. Storage on 200 bbl. 5 wk.= storage on 1 bbl. 1000 wk.; Due. Da. from May 1. Items. Prod. June 3 " 30 July 29 Aug. 6 33 60 89 97 1250 2275 2450 00 41250 136500 218050 29100 424900 6275 " " 350 " 9 " = " " " 400 " 13 " = " " " 50 " 14 " = " " " 3150 " " " 5200 " " " 7CO " 10050 wk. 10050X.02=$201 storage, due Aug. 6; $6275 X- 035 = $219.63 commission, due July 8. Due. Da. Items. Prod. Due. Da. Items. Prod. - May 1 Aug. 6 JulyS 97 68 240.10 201.00 219.62 19497 14935 June 3 ! 30 July 29 Aug. 6 33 60 89 97 1250.00 2275.00 2450.ro 300.00 41250 136500 2180fO 29100 390468--5 660.72 34422 6275.00 660.72 5614.28 42100 34432 390 168 614.28= Balances, 70 da.; May 1+70 da.=July 10, proceeds due. (358) PARTNERSHIP. SETTLEMENT OF ACCOUNTS CURRENT. (626, page 360.) 181 Ex.1. Due. Da. | Items. Int. Cashval. Due. Da. Items. Int. Cash val. Jan. 12 " 26 Feb. 13 Mar. 16 Apr. 25 140 126 108 77 37 500.36+11.67 260.48+ 5.26 400.00+ 7.20 750.00+ 9.63 200.00+ 1.23 512.03 Jan. 1 255.74 Feb. 3 407.20 Mar. 26 759.63 Apr. 20 201.23 May 12 151 118 67 42 20 636.72+13.51 486.57+ 9.57 1260.78+14.08 756.36+ 6.29 248 79+ .83 550/23 496.14 1274.86 761.65 249.6? 2135.83 8332.50 $3332.50 $2135.83=$1196.67, Am. Ex.2. Due. Da. Items. Int. Cash val. Due. Da. Items Int. Cash value. ! Sept. 3 Jan. 2 < 21 Feb 12 Dec. 15 119 2 21 43 16 478.36+11.07 256.37- .10 375.26-- 1.53 8000- .67 148.76+ .46 489.43 256.27 373.73 79.33 149.22 Sept. 17 " 20 Oct. 3 Nov. 17 Bee. 27 105 102 89 44 4 96.54+1.P7 200.00+3.97 325.00+5.62 50.00+ .43 84.00+ .07 98.61 0o.97 .330.62 50.43 84.07 1347.98 767.G>J $1347.98 $767.60=1580.38, Am. PARTNERSHIP. (629, page 361.) Ex. I. $6470+$3780-f$9860=$20110 $20110: $6470=17890: (?)=$2538.453, A'sshaio $20110: 83780=$7890: (?) = $1483.053,B's " $20110 :$9860=$7890: (?) = $3868.493, C's " Ex. 2. $1847.50 $739=$1108.50, C pays; 73900 2 "R^a frflpfinn 110850 3 PV TS4750 5? -E traction ; iH?fw f* ^ 8 $375X|=$225, C's ' (360, 361) 16 182 PARTNERSHIP. Ex. 3. $10000 888 = IHb A ' s fraction ; 12800 ii{HJ* = lt,B's " 3200 $26000 $9400 ($1500+$3400)=$4500, net profits; $4500xff =$1730.77, A ' s s]bare of net profits; $4500 X if = 2215.38, B's " " $4500x<^= 553.85, C's " $553'.85+$1500=$2053.85, C ; s whole income. Ex. 4. 115A.32P. = 115.2 A.; $3.75x115.2 =$432, rent; 144+160+192+324=820 slieep; 820 : 144=$432 : (?) = 75.86, A pays; 820 : 160 = $432 : (?)=84.29, B 820 : 192 = $432 : (?) = $101.15, C " 820 : 324=$432 : (?)=$170.69, D Ex.. 5. $6+$4+$2 = $12 $2640 X T %=$1320, B's share ; $2640 X T 4 3 = 880, C's $2640 X A = 440, D ; s " Ex. 6. $6300X4=^900, A ? s; $6300 x = 1260, B s; $6300 X =$1400, C's; $900+$1400= $2300, D's $6300 ($900+$1260+$1400+$2300) =$440 for E and F; $440X|=$165, EV; $440x| Ex.7, f ": 8 = J: |=4: 5; 4+5=9; $90X|=$40, share of first ; 50, " second. Ex. 8 . If the younger has 8 shares, the elder will have 9 " 17, sum of shares ; (362) PARTNERSHIP. 183 $5463.80 X I 8 ? = $2571.20, the younger receives; $5463.80X T 9 7= 2892.60, the eider Ex.9. $1680 $840=8840, A and B's gain; and since C's gain is equal to A and B's gains together, his stock must be $12000+$8000=$20000 ; 8840 X~ 8 o =$336, A's gain; Ex. 10. Since the portions of the stock which they severally put in, are proportioned to their gains respectively, we have $2000+$2800.75+$1685.25+$1014 =$7500, whole gain; $7500 : $2000 = $22500 : (?) = $0000, A's stock ; $7500 : $2800.75 = $22500* (?)X $8402.25, B's $7500 : $1685.25=$22500 : (?)=$5055.75, C's " $7500 : $1014 =$22500 : (?) = $3042, D's Ex. 11. J : f : | :i=i* : fMg : |{| = 15 : 36 : 40 : 30 15+36+40+30=121, sum of proportional terms; $30000X T T=* 3719 T27^ share of lst ; $30000 X$h= 8925 T \ 5 T , " " 2d; $30000x T 4 3r= 9917 TV^ " " 8d ^ 3 3 ,= 7438 T | T , " 4th. Ex. 12. $71.27+$142 t 54=$213.81, gain of A and B; $1200 $500 = $700, stock of A and B; $700 : .$500=1213.81 : (?)=$152.724, C's gain; $213.81 : $71.27=$700 : (?)=$233.33J, A's stock $213.81 : $142.54=1700 : (?)=$466.66f, B's Ex. 13. 3+5+7 = 15; $1 8840 X T 7 s= $8792, C's stock; $8792X T ^=$ 175 S.40, A's gain; $8792X T 5 5= 2930.66f, B's $8792X T 7 5= 4102.931, C's " (33?, 363) 184 PARTNERSHIP. Ex. 14. I , J% , |=|f , {j , ^ ; and these fractions are proportioned to 15, 18 and 13. Hence, A and B receive 15 shares, A C " 18 B C " 13 " ~~ ^ A, B and C receive 4 of 46=23 shares; and A will receive 23 13 = 10 shares, B 2318= 5 C 2315= 8 . Hence A has ^{ of $26.45=811.50; B ^ of $26.45= 5.75; C 2 % of $26.45= 9.20. (63O 5 page 364!) Ex. 1. $357X 5=$1785, A's product; 371X 7= 2597, B^s 154X11= 1694, C's $6076 : $1785=$347.20 : (?) = $102, A's gain; $6076 : $2597 = $3 47.20 : (?) = $148.40, B's " $6076 : $1694=$347.20 : (?)=$ 96.80, C's " Ex. 2. 5x12=60 cows for 1 week=A^s use of pasture; 4X10=40 " =B's " 6X 8=48 =C's " 148 " " " $55.50 Xi^^ 22 - 50 * A pays 55.50X T 4 A= 15 > B " $55.50 Xi 4 4 8 5 = 18 ,C " (363, 364) PARTNERSHIP. 185 Ex. 3. $4500 $1800=$2700, B's gain ; $15000X12=1180000, B's capital for 1 mo.; $2700-*-180000=$.015 gain per month on $1 ; $.015X9=1135, gain on $1 for 9 mo., or C's time; $1800-7-.135:=$13333.33|, value of C's land; $13333|-f-125=$106f , value of land per acre, An*. Ex.4. $4200x9=137800 $1500x 6=$ 9000 $4400x7= 30800 $1000x10= 10000 A's product, $68600; B's product, $19000 $68600+$19000=$87600, sum of products; 876 : $686=1772.20 : (?) =$604.71, A's gain; : $190=$772.20 : (?)=$167.49, B's Ex. 5. 30X12X 9=3240 hours' work by 1st company; 32X1^X10=4800 " " 2d 28X18X11=5544 ' 3d 20X15X12=3600 4th 17184 " *^ all. $1500X T 3 7 2 T 4 84 -$282.82, wages of 1st company ; $1500XT 4 7 8 T84= 418.99, . J< 2d $1500Xffr 4 s\= 483.94, ^ 3d $1500 x T 3 T 6 T g4= 314.25, " 4th Ex. 6. To avoid fractions, suppose A had $5X^=$30, and Bhad$8x6=$48. Then $30X4=1120 $48 x 4= $192 $15x8-f- 120 $16X8= 128 $240, A's prod. ; $320, B's prod. ; $240+$320=$560, sum of products; = 2285f, B's. (364, 365^ 186 PARTNERSHIP. Ex. 7. $15800X4=$ 63200 14600X2= 29200 13100X4= 52400 14100x8= 112800 $257600, B's prod.; $25000X6=$150000 23000X4= 92000 21500x1= 21500 22300X7= 156100 419600, C's prod. ; $30000XT=$210000 31800X3= 95400 26800X8= 214400 519800, D's prod. ; $1197000, sum of products; $15000x T \Wo=$3228.07, B receives; k $15000 X T 4 TWo= 5258.15, C $15000 x T 6 T W%= 6513.78, D Ex. 8 C put in 1 (i+|j=t\j f r 1 J ear ; hence X|= J? A's product; But | ? i ^nd T V=|i 4% and 4% 5 and these frac - tions are proportional to 15, 8 and 4 ; hence 15+8+4=27, sum of proportional terms; $5400X;Mf =$3000, A's share of the gain; $5400X2 8 T = 1600, B's " "" $5400X 3 4 7 = 800, C's " $5400 X i= 2700, stock A put in; $5400 X |= 2160, B $5400 X 7 1 iy= ^40, " G $3000+$2700=$5700, A's share of entire stock; $1600+$2160= 3760, B's " " $800+$540 = 1340, C's ' (365) PARTNERSHIP. 187 Ex. 9. $456-=-10 =$45.60, A's monthly profit; $343.20-r-8 = 42.90, B's " " $750-f-12 = 62.50, C s " Now, their respective amounts of capital must be pro- portional to their monthly profits ; hence $45.60+$42.90+$62.50=$151 ; $151 : $45.60 =$14345 : (?)=$4332, A's capital; $151 : $42.90=814345 : (?) = $4075.50, B's " $151 : $62.50=$14345 : (?)=$5937.50, C's Ex. 10. C will have 1 ( T ^+|) ^ of the profits; y 1 ^ -5-4=^5, B may claim for 1 mo.; i-f-8=A,0 |-j-6 = T J B , D " " " " ; hence j\ : ^=$600.0 : (?) = $2250, B put in; (?) = $5625, C Ex. 11. $2400 $1920 =$480, A's gain; $2080 $1280= 800, C's Now, since the gain varies as the product of the cap- ital and time, we have the compound proportion 1920 ( 1280 : ) =$480 : $800 6 / (?) (?)=iffg^f 00^15 mo., O's time. Again, since A gained $480 in 6 mo., he would have gained $480x2=$960 in 12 mo., B ; s time; and his stock and gain together would have been $1920 +$960=$2880; hence $2880 : $1920=$4800 : (?)=$3200 ; B's stock. (365) 188 ALLIGATION. Ex.1. ALLIGATION. (633, page 366.) $ .60X4=$2.40 .70X3= 2.10 1.10X1= 1.10 1.20X2= 2.40 Ex.2. SO X14=$ 0.00 .75X12= 9.00 .90X24= 21.60 1.10X16= 17.60 10 ) $8.00 66 ) $48.20 Ans. $.80 Ex. 3. 3 Ib. 6 oz.=42 oz. ; 4 " 8 " =56 " 3 " 9 " =45 " 2 " 2 " =26 " Ex.4. $1.20xl5=$18.00 1.10X 5= 5.50 .90 X 5= 4.50 .70X10= 7.00 Ans. $.73 Jg. 23X42= 966 carats; 21X56=1176 20X45= 900 0X26= -0 " 169 ) 3042 18 carats ; Ans. 35 ) $35.00 worth of Ibu., $1.00 \ $1.25 $1=$.25, Ans. Ex. 5. 8.05x17=$ .85 .08X51= 4.08 .10X68= 6.80 .12X17= 2.04 153 $13.77 (366, 367) $13.77_,_153~$.09 ; average selling price ; $.09-f-1.33|=$.0675, Ans. ALLIGATION. 189 Ex.6. Ex.7. Ex.8. Ex.2. Ex.3. $2.70X42=1113.40 2.85X48= 136.80 8.24X65= 210.60 155 $460.80, cost; $460.80X1^=1552.96, to be sold for; $552.9C-:-155=$3.567if, Ans. As the degrees and minutes of all the observations are alike, we need to average only the seconds, thus : 25.4"+24.5"+27.8"+26.9"+25.4"+24.7" +24.2"+26.3"+25.8"+26.7"=257.7"; 257.7"-*-10=25.77". Ans. 36 17' 25.77". Since the third trial is entitled to twice the degree of reliance to be placed upon either of the others, it is equivalent to two trials giving the same result ; hence 37 min. 54.16 sec., 1st trial; 37 55.56 2d 37 " 54.82 " 3d " 37 " 54.82 3d 44 ) 151 min. 39.36 sec. 37 min. 54.84 sec.=9 28' 42.6", Ans. (634, page 370.^ 40 8 A 15 3 A 15 3 75 15 Ans. 3 Ib. of the first kind, 2 Ib. of the second, 3 Ib. of the third, and 5 Ib. of the fourth. Ans. 8 gal. of water to 3 gal. at $.60, 3 gal. at $.90, and 15 gal. at $1.15. (367370) 190 ALLIGATION. Ex.4. ' , 1 40 17^ 17^ 35 7 22. * 1 62^ 60 17 i V* 35 7 1 f 80 22^ 2 2 25 50 10 fc 17* 17 i Sx. 5. Ex.6. Ans. 7 A. from each of the first two tracts, and 10 A. from the other. {60 _i_ 15 15 3 50 ^ I 7 3 10 2 42 i 5 5 1 38 4 5 5 1 30 _i 15 15 3 Ans. 3 pt. of the first kind, 2 pt. of the second, 1 pfc. of the third, 1 pt. of the fourth and 3 pt. of fifth. Another Solution 1 1 i i 7 1 8 5 5 i 57 5 5 T5 3 3 Ans. 1 pt. of first kind, 8 pt. of second, 5 pt. of third, 5 pt. of fourth and 3 pt. of fifth. 40 24- 24 8 Ans. 3 Ib atf 24 40 120 24 160 3 20 pure, 3 Ib. at pure, 20 Ib. iV NOTE. In the above solution, we multiply the first ratio, 8 : 40, by 3, making 24 : 120. Then by adding-, we get 24 : 24 : 160, which reduced gives 3 : 3 : 20. (370) ALLIGATION. 191 (635, page 371.) Ex.1 58 - 25 j& 12 4 4 NOTE. Divide the 50 1 4 4 terms in the 3d en! - 62 8 8 umn by 3, and to 70 A 4 33 11 11 the result add the terms in the 4th. Ans. 4 -gal. of the first two kinds, 8 gal. of the third, and 11 gal. of the fourth. Ex.2. Ans. 5, 5, 5, and 2. NOTE. --Multiply the terms in the third col- umn by 5, and add the result to the terms in the fourth column. Ex.3. For the first mixture. C 30 70-^ 50 (.100 3TJ Ans. 1, 1, and 2. For the second mixture. 3 3 1 3 12 12 4 2 8 12 4 Ans. 1, 4 and 4. 70 \ 50 (100 NOTE. Multiply the terms in the fourth column by 4, and to the result add the terms in the third column, and then divide by 3. (636, page 372.) Ans. 10 at $16, 10 at $18 and 60 at $24. Ex. 2. $15-r-12r=$l|, price per yard of the ingredient whose quantity is limited. Ex. 1. ( 16 I 1 1 f 10 22 J 18 22 J20 1 , 1 1 1 10 10 [24 I \ | 3 2 1 6 60 1 11 12 4 16 Ans. 16yd. 9 9 9 9 3 3 12 12 at$|, and 12 yd.$: ff- (371, 372 N 192 ALLIGATION. i:x. 3. 9 gal. 2 qt. 1 pt.=77 pt. "{$ SI* 11 1 77 1 Ans. 11 pt. = l gal. 1 qt. 1 pi Ex.4. ("30 45-} 55 A A T5 2 3 5 10 15 15 15 60 60 (70 sV 3 3 12 Ans. 60 Ib. (637, page 373.) Ex. 1. $2.20X7=115.40 $2.00X7-*- 14.00 14 ) 29.40 $2.10, average price of the 14 yd. (160 180 i 175 (210 A 14 28 14 Ans. 14 yd, at $1.60, and 28 yd. $1.75. Ex. 2. $1.14X60= $68.40 $1.26X30= 37.80 90 ) $106.20 $1.18, average price of the 90 gaL 157 { 118 175 18 39 90 I 195 I Ans. 195 gal. Ex. 3. $2.00X40= $80 $ .50X70= 35 110 ) $115 u. 22 $ IT^, average price of the 110 bu. 4 I 2 | 6| 66 33 110 Ans. Q'i bu. NOTH. Multiply the terms in the fourth column by * = * (372, 373) INVOLUTION. (638, page 374 ) 193 Ex. 1. f 8 T2 1 1 40 Ans. 40 Ib. at 8 16 20^ 24 I j ^ 1 3 1 1 4 ^ cts., 40 Ib. at 16 4 cts., and 160 Ib. I 6 240 at 24 cts. Ex. 2. $154--154=$l, average price. ' 1 31 1 !-< 1* 3 1 14 i 2 2 5 3 7 42 98 Ans. 14 calves, 42 sheep, and 98 lambs. 11 154 Ex. 3. $165-i-55= $3, average weekly wages. f5 ^ 2 5 1 6 30 1 ^ 1 1 5 4 1 3< j 4 4 20 2 2 Ans. 30 men, 5 11 55 women, and 20 boys. INVOLUTION. (644, page 376.) Ex. 3. Ans. 2102500 Ex. 13. Ans. 2023.37890625 Ex. 14. Ans. 3838.28125 Ex. 16. Ans. 1871688 T 7 T 17 (374376) 194 EVOLUTION. EVOLUTION. SQUARE ROOT. (657, page 381.) Ex. 3. AM. 7502; 24.4315+. Ex. 9. 9922563504=35721 ; 1/35721=189, AM. Ex.10. 1/.126736=.356; 1/.045369 = .213; .356 .213 = .143, Ans. 1/169 _ 1/7056 Ex. 11. 1/jf |= -=11 ; I/Iff |= -=11 ; 1/196 1/9216 1 3\/ 84 _ 1 3 - Ex. 12. 1/81 3 X625 2 X2 4 =81X625X2 2 =202500, AM.. CONTRACTED METHOD. (658, page 382.) Ex. 1. |5.6568542+, Ans. Ex. 2. |3.4641016+, Jns. 32.000000 12.000000 25 9 106 7.00 64 300 636 256 1125 6400 686 4440 5625 4116 11306 77500 6924 28400 67836 27696 (381-383) 11312 SQUARE 9664 9050 ROOT. 6928 195 704 693 1131 614 566 693 11 7 113 48 45 7 Ex.4. .5= 144 4 4 |.745355+, Ans. 11 Ex.3. 107 3 2 J57.3322+, Am. 3286.9835 25 .555555+ 49 786 749 655 576 1143 3798 3429 1485 7955 7425 1146 369 344 1490 530 447 115 25 23 149 83 75 12 Ex.5. 64= 45 2 2 |2.563479+, Ans. 15 Ex.6. 1.06 = 21 8 8 |1.156817+, An* 6.571428+ 4 1.338226:+: 1 257 225 33 21 (383) 196 506 EVOLUTION. 3214 225 3036 1282 1125 5123 17828 15369 2306 15726 13836 5126 2459 2050 2312 1890 1850 513 409 359 231 40 23 51 Ex.7. 201 50 46 1.01258=1.0380+ |1.0188+, Am. 23 Ex.8. 201 17 16 |1.011620 ,An*. 1.023375 1 1.0380+ 1 0233 201 380 201 2021 3275 2021 202 179 162 2022 1254 1213 20 Ex.9. 17 16 202 41 40 20 1 CUBE BOOT. (661 , page 388.) -y 134217728=512; 1^512=8, Ans. Ex.10. 39304 2 =1544804416; ifl 544804416= 1156, Am. (383-388) CUBE ROOT. 197 EX. 11. Ex. 12. 1^50=3.6840+ ^31=3.1413+ 6.8253+, sum of cube roots; 1^50- r 31=^ / 8T=4.3267+, cube root of sum; 2.4986+, AIM. Ex.1. CONTRACTED METHOD. (622, page 390.) [2.8844992:+, Am. 24.000000 68 544 1200 1744 16000 13952 848 6784 235200 241984 2048000 1935872 864 346 248832 249178 112128 99671 4 24952 24956 12457 9982 2496 2475 2246 250 229 225 25 4 5 (388390) 198 Ex.2. Ex.3. EVOLUTION. Ans, |22.894801334, 62 124 1200 1324 12000.812161 8 4000 2648 668 5344 145200 150544 1352812 1204352 6849 61641 15595200 15656841 148460161 140911569 6867 2747 15718563 15721310 7548592 6288524 69 55 1572406 1572461 1260068 1257969 157252 2099 1573 1573 526 472 157 54 47 16 7 6 |.555554730r, Am. .171467000 125 155 775 7500 8275 46467 41375 1655 8275 907500 915775 5092000 4578875 (390) CUBE ROOT. Ex.4. Ex.5. 1665 833 924075 924908 513125 462454 17 9 33 99 92574 92583 50671 46292 9259 4379 3704 926 675 648 93 27 27 9 300 399 |1.34442zb, Am. 2.429990 1 1429 1197 394 1576 50700 52276 232990 209104 402 161 53868 54029 23886 21612 4 2 66 396 5419 5421 2274 2168 542 1200 1596 106 108 |2.6888, An*. 19.440 8 11440 9576 (390) 200 EVOLUTION. Ex.6. Ex.7. 78 60 2028 2088 1864 1670 1 1 274 1096 215 216 194 173 22 21 18 2 1= 24300 25396 3 |.941035, Ans. .833333 729 104333 101584 282 28 26508 26536 2749 2654 242 41 2656 95 80 27 19200 34 19684 15 14 |.829826686, An*. .571428888 512 59428 39368 2017200 2469 22221 2039421 20060888 18354789 2061723 2487 1990 2063713 1706099 1650970 206570 25 5 206575 55129 41315 (390; CUBE ROOT. 201 Ex.8. Ex.9. 20658 13814 12395 2066 1419 1240 207 179 166 21 13 13 |1.Q57023:, Am. 1.08674325*= 1.181011+: 1 805 1525 30000 31525 181011 157625 315 221 308 2464 33075 33296 23386 23307 3352 79 67 34 1.05 5 = 30000 32464 12 10 |1.084715:, An* 1.276282 1 276282 259712 824 130 34992 35122 16570 14049 3 2 3525 3527 2521 2469 353 52 35 35 17 18 (391) 202 EVOLUTION. ROOTS OF ANY DEGREE. (664, page 391.) Ex. 1. 6=3X2; ^6321363049 = 1849; 1/1849=43, Ex.2. 4=2X2; 1/5636405776=75076; 1/75076=274, An*. Ex. 3. 8=2X2X2; 1/1099511627776=1048576 ; 1/1048576=1024; 1/1024=32, ' Ans. Ex.4. 6=3X2; 1/25632972850442049=160103007; ^160103007=543, Ans. Ex. 5. 9=3X3; ^1.577635=1.164132+ ; if 1.164132=1.051963+, Ans. Ex.6. 12=2X2X3; 1^16.3939=2.5404+; 1/275404=1.5938+; 1/L5938=1.2624+, Ans. Ex.7. 18=2X3X3; 1/104.9617=10.24508+; ^10.24508=2.171893+ ; ^2171893=1.2950+, Am. (665, page 393.) Ex 2. Vl20=3.31+; ^120=2.22+; 3.31+2.22 = 5.53; 5.53-4-2=2.76, assumed root; 2.76 4 =58.06+ ; 120-4-58.06=2.0669+; 2.76x4+2.0669=13.1069; 13.1069-4-5=2.6214 ; 1st approximation. 2.6214 4 =47.2203+ ; 120-^47.2203 = 2.54128+ ; 2.6214 X 4 + 2.54128 = 13.02688 ; 13.02688-^-5=2.60537, 2d approximation. (391-393) ROOTS OF ANY DEGKKE. 203 2.604378+ ; 2.60537X4+2.604378=13.025858; 13.025858-5-5=2.605171, 3d approximation, which is correct to the last decimal place. Ex. 3. Vl.95u7b=1.11838+ ; ^1.95678=1.08292+ ; 1.11838+1.08292=2.20130 ; 2.20130-5-2=1.10065, assumed root; L10065 6 =1.77785129+ ; 1.95678-5-1.77785129 = 1.10064324+ ; 1.10065x6+1.10064324= 7.70454324; 7.70454324-*-7=1.10064903, 1st ap- proximation, correct to the last decimal place. Ex.4. 10=2X5; 1/743044=862. Take 4 =the assumed 5th root of 862; then 44=256; 862-5-256=3.36; 4x4+3.36=19.36; 19.36-j-5=3.872, 1st approximation. 3.8724=224.771579^; 862-5-224.77 1579 =~~"~ 3.83500443+ ; 3.872x4+3.83500443= y 19.32300443; 19.32300443-5-5=3.8646008, 2d approximation, Ans. Ex.5. 15=3X5; ^15 =2.466212+. Take 1.2=the assumed 5th root of 2.466212 ; then 1.2 4 =2.0736; 2.466212-5-2.0736=1.189339+; 1.2x4+1.189339 = 5.989339 ; 5.989339-5-5 = 1.197868, 1st approximation. 1.197868 4 =2.058898+ ; 2.466212-5-2.058898 = 1.197831+ ; 1.197868 x4+1.197831=5.989303 ; 5.989303^-5=1.197861, 2d approximation, correct to the last place ; hence Ans., 1. 197861 +. Ex. 6. Since 25 = 5x5, we might extract the 5th root of fye 5th root, for the 25th root. A more convenient method is as follows : (393,) 204 EVOLUTION. - 1/100=10; VlOO =1/10=3.1622+; VlOO =1/3.1622=1.7782+; VTo6=f / 1.7782=1.2115+. Now, as the 25th root must be less than the 24th root, take 1.2= the assumed root. 1.2 24 =79.49684+ ; 100-5-79.49684=1.25792+ ; 1.2x24+1.25792=30.05792 ; 30.05792-^-25=1.2023168, 1st approximation. 1.2023168 24 =83.2677184+ ; 100--83.2677184= 1.2009492+ ; 1.2023168x24+1.2009492= 30.0565524 ; 30.0565524-5-25=1.202262+, Ans.j correct to 5 decimal places. Ex. 7. VT-1.5+; V~5 =1.3+; 1.5+1.3=2.8; 2.8-r-2=1.4, assumed root 1.44=3.8416; 5-5-3.8416=1.3016+; 1.4x4+ 1.3016=6.9016; 6.9016-^-5= 1.38032, 1st approximation. 1.38032 4 =3.63011+; 5-^-3.63011=1.37721+ ; 1.38032x4+1.37721 = 6.89849; 6.89849-^-5 = 1.37970+, Ans. APPLICATIONS OF SQUARE AND CUBE ROOT. (689, page 396.) Ex. 1. 256 2 =65536, square of hypotenuse ; 75 2 = 5625, " altitude; 59911, square of base; (393-396) APltlCATIONS OF ROOTS. 205 1/599H= 244.76+, base; 244.76 ft. 22 ft.=222.76 ft., Ans. Ex. 2. l/84*+50 2 =97.75+ lea.=337.23+ mi., Ans. Ex. 3. 1/50* 30* =40 ft. from foot ladder to one side; 1/50 2 40^=30 " " " a the other " 40 ft.+30 ft.=70 ft., Ans. Ex.4. V (3 ~+6 2 = diagonal of one side; and since this di- agonal and the adjacent edge form a right-angled triangle whose hypotenuse is the required diagonal, we have i/6*+6 2 +6 3 =1/108=10.39+ ft., Ans. (69O, page 397.) Ex. 1. 208X13=2704; 1/2704=52 rods, Ans. Ex. 2. (216+24) x 2=480 rd. of fence to inclose the farm in rectangular form. $312-i-480 = $.65, price per rd. ; 216X24=5184; 1/5184=72 rods, in one side of square farm of same area ; hence $.65 X 72 X4= $187.20, cost offence for square farm; $312 $187.20 = $124.80, Ans. Ex. 3. 1/588> - 4 ^- Ex. 3. ll=N"o. terms ; 220=last term ; 17=com. diff. ; (11 1) XI 7= 170; $220 $170=$50, first term, ($220+$50)XV =11485. (404-406) 210 SERIES. GENERAL PROBLEMS IN GEOMETRICAL PROGRESSION. (711 , page 407.) Ex.1. 6X4 6 =6144, Ans. Ex.2. 192-*-2 <*= 3, Ans. 122 Ex. 3. 6X- = = - , Ans. 3 7 3 6 729 1 25 5* 1 Ex.4. 25X ===, An*., 5 4 5 4 5 4 25 (712, page 408.) Ex.1. 512n-2=256; V256==4, Ex. 3. 7--.0112=:625 ; V625=5, Ans. Ex 4. 50008=625; Vg25 = 5s ra tio; 8, 40, 200, 1000, 5000. (713, page 408.) Ex 1. 1458-^2=729 Ex. 2. 100-4-1=1000 3 729 3 243 3 81 3 27- 3 9 3 3 1 Ans 7. 10 10 10 1000 100 10 (407 408) GEOMETRICAL PROGRESSION. 211 Ex. 3 Ex. 4. 196608-4-6=32768 128 64 32 16 8 32768 8 4096 8 512 8 64 8 8 1 Ans. 6. i.8. (714, page 409.) Ex. 1. 384X2=768; 7683=765; 765-f-l=765, Ans. Ex.2. 1080X6=6480; 6480 5=6475; 6475^-5=1295, Ans. Ex.3. 4f X 3=14|; 141-^=14^1; Uj{j|-f-2=7tffr, Ans. Ex. 4. The least extreme is 0, and the ratio is 8-j-4=2 ; hence, 8x2=16, Ans. (7 15, page 410.) Ex.1. 34=81; 81 1=80; 80X7=560; 560-j-2=280, Ans. Ex. 2. 375-^-5 3 =3, least term; 5 4 =625; 6251=624; 624X3=1872; 1872-f-4=468, Ans. Ex. 3. 1.06 5 =1.338226+; 175X-338226= 59.189550 ; 59.18955-f-.06=986.49+, Ans. (409, 410) 212 (716, page 410.) Ex. 1. 8002=798; 800686=114; 798--114=7 ; ^n*. Ex.2. 127f 1=-127; 127J 64=63| ; 127^-r-63j=2, Ans. Ex.3. 4 0=4^; 4^3=11; COMPOUND INTEREST BY GEOMETRICAL PROGRESSION. (718, page 411.) Ex. 1. $3 50= first term, 5= number of terms, and 1.06= ratio ; the last term is required. $350 XL06 4 =$441.86+, Ans. Ex. 2. To find the compound interest of $1, we have $1= first term, 1.07=ratio, 3=No. terms; $lXl.07 2 =$l'.1449, amount; $1.1449 $1 = 1.1449, int. of $1; $150-r-.1449=$1035.196+, Ans. Ex. 3. We have given $1000=last term, 1.06=ratio, 4= No. terms, to find the first term. Reversing the rule, $1000-*-1.06=$839.62 , Ans. Ex, 4. $40=first term; $53. 24= last term; 1.10=ratio. We find the number of terms by Prob. Ill, thus : 53.24--40=1.331; 1.331-r-1.10 = 1.21 ; 1.21-j- 1.10=1.10; 1.10-5-1.10=1. Hence 3-f-l=4=No. terms ; and Ans., 3 years. Ex. 5. Let the principal be $1 ; then 1= first term ; 2= last term ; 9=No. terms. We find the ratio by Prob. II, thus : 2--l=2 ; V~2" == 1.0905+ ; 1.0905 1=.0905=9.05+ %, Ans. (410-412) ANNUITIES. 213 Ex. 6. $322.51= last term, 1.05= ratio, 25= No. terms. We find the first term by Prob. I, thus : 1.05 2 <= 8.225100+ ; $322.51-f-3.2251=$100, Ans. ANNUITIES AT SIMPLE INTEREST (726, page 414.) Ex.3. $150=the last term; $150X.015=$2.25=common difference; 5^X4=22=No. terms. We find the sum of the series by TOG and 7O9, thus : $150+($2.25X21)=$197.25, first term; $197.25+1150 X 22 =$3819.75, Ans. 2 Ex. 4. $500= last term, $3450= sum of series, $500 X -06= $30 = common difference; we are required to find the number of terms. Now 3450-^500=6.9; that is, if the pension did not draw interest, the time re- quired for it to amount to the given sum would be less than 7 yr. ; by trial we find the time to be 6 yr. Ex. 5. $6000=last term; $59760=the sum of the series; 8= No of terms. We find the common difference, thus : according to 7O9. 2 times the sum of the series = the sum of the extremes multiplied by the number of terms. Therefore $59760X2 =$14940, the sum of the extremes; 8 $14940 $6000=88940, first term. Hence by 7O7, $8940 $6000 =$2940; $2940-^-7 =$420, common difference. Then $420-^46000 =.07 =7 %, Am. (412-414) 214 SERIES. PROMISCUOUS EXAMPLES IN SERIES. (729, page 415.) Ex. 1. Its present amount will be the sum of the geometric- al series in which $200= first term; 1.06= ratio, and 20= No terms. Hence, by 715, 200X(1.06 2 1) 200X2.207135+ = =$7357.11+. 1.061 .06 Ex. 2. We have given $16459.35=sum of the series, 25= No. of terms, 1.06= ratio, to find the first term. According to 715, the sum of the series is equal to 1.0635 1- the first term multiplied by the fraction 1.061 ; consequently the first term will be found by dividing the sum of the series by the same fraction ; and we 1.061 $987.5613 have $16459.35 X = =$300. 1.06^ 51 3.291871 Ex. 3. We first find the amount of the annuity in arrears for the 7 years. We have given $500 = first term, 1.06= ratio, and 7= No. terms. Hence by 715. $500 X (1.06 7 1) $251 .815 = =$4196.911, sum of 1.061 .06 series. We now find, by 553, what sum will amount to $4196.91| in 7 years, at 6% compound interest; thus: $4196.91f-=-1.503630=$2791.18+.' Kx. 4. We first find the value of the annuity in arrears for the 20 years, or its worth when it expires. We have given $100= first term, 20= No. terms, and 1.05= ratio, to find the sum of the series. By 715. (415) EXAMINES. 215 $100x(1.05 20 1 -=$3306.596, sum of series. 1.051 This is what the lease is worth 20-{-14=34 years hence ; therefore its present value, by Ot>3, is $3306.596-^-5.253348 = $629.426+, An*. 21= No. terms, 5=firstterm, | = common diff. By 7O6, 5 (X20)=0, last term; by TOO, + X 21=52|, Am. Ex. 6. 80 last term, 5=common diff., 13=No. terms; to find the first term, and the sum of the series. Reversing the rule under 7OG, we have (13 1)X 5=60 ; 8060=20, first day's journey. Then by 7OO, ^^^X 13 =650, whole distance traveled. 1 Ex. 7. 15-r-30=|,theratio;by711,30X =j Ex. 8. By 714, <^*f-a = 2 -3 4 - = 682 > Ans - Ex. 9. We have given 360=sum of an arithmetical series, 27= first term, and 45= last term, to find the num- ber of terms. By TOO, twice the sum of the series is equal to the sum of the extremes multiplied by the number of terms; conversely, ||^||=10 ; Ans. Ex. 10. By 712, Vl5625=5, Ans. Ex. 11. $500=first term, 10= No. terms, 1.06= ratio; to find the sum of the series. By $500 X (1.06 1 i) $395.424 ---- = -- =$6590.40, Ans. 1.061 .06 ^ Ex. 12. Reversing the rule under 7OO, we have ^oi^ia^lO^ the sum of the extremes. Now, since 6 is the com- mon difference, and 8 the number of terms, (8 1) (415, 416) 216 SERIES. X6=42 is the difference of the extremes; hence by Prob. 33, 127, ui^-hi^ 72, last term; and UlAf** =30, first term. Ex. 13. We have $1196= the sum of an arithmetical series, 13=No. of terms, and $12=. the common difference, to find the first and last terms. Proceeding as in the last example, mffZ3=tIS4, the sum of the extremes ; (131) X 12=$144, the difference of the extremes. *l5*il.4=$20, first payment ; .LS4+ill.4=$164, last payment. Ex. 14. $2= first term of a geometrical series, $512= last term, and 4= ratio ; to find the number of terms, and the sum of the series. Firstly 713, 512--2=256; 256--4=64; 64-5-4 =16; 16--4=4; and 4 --4=1. Hence 4+1=5, number of payments. Second, by 7 14$ Hi*>=*==$682, indebtedness. Ex. 15. $4800= first term, 5= No. terms, 1= ratio; to find the last term, and the sum of the series. By 711 , $4800x(li) 4 =$24300, share of eldest; ($24300x1-1) $4800 By 714, =$63300, property, 2 . Ex. 16. $2818.546= sum of series, 5= No. terms, 1.06= ra- tio ; to find the first term. Reversing the rule under 71t>, as in Ex. 2, we have 1.061 $169.11276 $2818.546 X = -=$500, An*. 1.06 6 1 .338226 Ex. 17. $10= first term, $7290= last term, 3= ratio, to find the number of terms, and the sum of the series. By 713, 7290-f-10=729; 729--3=243; (416) PROMISCUOUS EXAMPLES. 217 243-j-3=81; 81-5-3=27; 27--3=9 ; 9-^3=3; 3-j-3=l ; 6+1=7, payments. % By 714, (3^gA|JO=lLO = 4l0930, debt. Ex. 18. If we take the distance traveled in going to the several stations and returning, we shall have an arithmetical series, of which 10 mi.=the first term, 50 mi.=the last term, and 180 mi.=the sum. We are to find the number of terms and the common difference. Reversing rule under *7OO, we have lgax2__(^ N O> stations. And reversing rule under 7OG. ^-Zj^-=8, common difference; hence 8-v-2 =4 mi., distance between the stations. B*. 19. We first find the amount of the annuity in arrears for 12 years, by 7 15. $200 X (1-06 * 31) - =$3373.99+, amount. 1.061 Since this will be the value of the annuity when it expires, we must find its present worth, at 6% com- pound interest for 6-f- 12=18 years. By *>*>*$, $3373.99--2.854339-f-$1182.05+, Am. Ex. 20. We have a geometrical series, in which $6:=first term, 1.06=ratio, and 60 16=44=No. terms. Hence, -- =$1198.548, saved by dis- 1.061 pensing with tobacco ; and $1198 544+$500= $1698.548, Ans. Ex. 21. The value of a perpetuity when entered upon, is a sum whose annual interest=the perpetuity ; hence, S100-J-.05 %=$2000, value when entered upon. (416, 417) 19 218 SERIES. Since this is the value 30 years hence, we find its present value hy OO# 5 thus : $2000-r-4-321942=$462.75+, Am. Ex. 22. First; $2000 is the amount of a certain sum at sim- ple interest 2112=9 years at 7%. That is, $2000 is the last term of an arithmetical series, .07 times the first term is the common difference, and 9-}-l is the number of terms ; and from these data we are to find the first term. Assuming a series of which $1 is the first term, $.07 the common difference, and 10 the number of terms, by 78O ? we find the last term to be $1+($.07X10 1)=*1.63. Now, since $1.63 is the last term of an arithmetical series oi which $1 is the first term, $2000 must be the last term of a similar series, the first term of which is $2000-5-1.63=$1226.993+, the sum left at 7% simple interest. Second; $2000 is the sum of a geometrical series, 1.03 is the ratio, and (21 12x2) +1=19 is the number of terms; and we are required to find the first term. By 7 1 1 , we have 2000-f-1.03 1 - 1 = $2000-*-1.702433=$1174.789+, the sum left at 6% compound interest payable semi-annually. Ex. 23. The prices of the several pieces form an arithmetical series, of which $136=sum, $4=the com. diff., $31 =the last term, and 8=the No. terms. By 7O6. 31 (8^1X4)=3, first term, or the value of the the first piece; and as the price of this piece was $1. per yard, $3-r-$l=3 yards in the first piece. Now, the number of yards in the several pieces form a series, the first term of which is 3, the common dif- ference is 2, and the number of terms is 8. Hence, (417) MISCELLANEOUS EXAMPLES. 219 g_f_2 (8 1)=17 yd. in longest piece; Ut*x8=80 " whole quantity. We now have 3 yd., 5 yd., 7 yd., 9 yd., 11 yd., 13 yd., 15 yd., 17 yd., lengths; $3, $7, $11, $15, $19, $23, $27, $31, prices; $1, $1|, $14, $1|, $1 T 8 T , $1{|, $lf, $1}4, prices per yard. Ex. 24. 600=sum of series, 2 ratio, and 320=greatest ex- treme. First, to find the No. of bushels of the first kind. According to the rule under T14. the sum of the* series multiplied by the ratio less 1, is equal to the difference between the last term multiplied by the ratio and the first term; hence, 600 X (2 1)=600, difference between the last term multiplied by the ratio, and the first term. Then, 320x2=640 ; and 640600=40 bu. of the first kind. Next, to find the number of terms. 320-^40=8 ; 8-5-2=4 ; 4-j-2=2 ; 2-j-2=l ; hence, 3+1=4 kinds. Ans. MISCELLANEOUS EXAMPLES. (Page i!8.) Ex. 1. 18X236; 36x36=1296 sq. ft. in both sides of the roof. But 1296 sq. ft.=12.96 squares of 100 ft. each; and since 1000 shingles make 1 square, (282, Note 3), there will be 12.96 M. shingles. Ex. 2. 70 mi.Xf X 4X1=3.80952+ mi. 1.375 mi. X-73 =1.00375 mi. 2.80577 mi., Ans. (417, 418) 220 MISCELLANEOUS EXAMPLES. Ex. 3. Ex.4. Ex.5. | = 137|, the whole remainder; Ans. 4612 =-X-X-=-, 4 9143 s'o i. 3 gal. of water, 2 of cider, 4 of wine, and 5 of brandy. Ex. 6. A number increased by ^, ^ and | of itself, will be ^ 1+i+i+i ^-z times the number; hence 125-=- 2 T ^r=60, Ans. Ex. 7. From noon to midnight is 12 hours; and since the time past noon is f of the time to midnight, the whole 12 h. must be If times the time to midnight Hence 12 h.-f-lf=7 h. 12 min., time to midnight; 12 h. 7 h. 12 min.=4 h. 48 min., P. M., Ans. Ex. 8. $10X12=1120 8|X 3= 26 8= 60 23 ) $206 Ex. 9. Int. on $5.84-s-$.97j=6 $8|f , Ans. for 4 mo. 26 da., at 1 Ans. 24 189 217 (331 5*= I 2f 234 2| 31 Ex.10. |XV^ 16 ^., Ans. Ex. 11 $450X.05X6|=$153.75, interest; $450-~1.34|=$335.40, present worth; (418) MISCELLANEOUS EXAMPLES. 221 $450 $335.40=$114.60, discount; $153.75 $114.60=$39.15, Am. Ex. 12. $6300xf=$7200, eldjer brother received : $6300+17200=413500, An*. Ex. 13. .7=3; .88=fi=l; -727=313; Ex.14. $438 X - -= $547.50, 90 Ex. 15. Tnt. on $1 for 4 mo. 3 da.=$.0239 ; $1 $.0239=$.9760f, proceeds of $1; $875.50--.9760f=$896.95-f-, Ans. Ex. 16 $228.00^- 5=445.60, A's monthly gain ] 266.40-7- 8= 33.30, B ? s 830.00-^-12= 27.50, C ; s $106.40, entire $2128 X T W4^* 912 ^ A's- stock; 2128x T 3 o 3 A= 6.66, B'B 2128Xy 2 o%= 550, C ? s tflx. I 1 ; $.35X300=:$105, due Oct. 27 ; .40X150= 60, 31; .38x500= 190, Nov. 7; .42X200= 84, " " 12; .40X250= 100, 25; Hence, taking Oct. 27 for a focal date, 105X 0= 000 60 X 4= 240 190=11=2090 6574^-539=12 da. ; 84X16=1344 Oct. 27+12 da. = Nov. 100x29=2900 539 6574 (418) 222 . MISCELLANEOUS EXAMPLES. Ex. 18. A, B, and C do ^ of it in 1 day ; G does 75*4 of it, and A ^ ; hence, B can do J 2 (^+^\)=^ in l da 7 > and 408-r-5=81f days, Ans. Ex. 19. We here have an arithmetical series, of which 7= the first term, 51= last term, and 4= com. diff. ; by 7O8, 5 -V- 7 +l = 12 da., time; and by 7O9, M^X 12 =348 mi., distance. Ex. 20. $2500 X-OlSOf =$45.21 , bank discount; $2500-4-1.0175=82457.00+, true present worth; $2500 $2457=$43, true discount; $45.21 $43 =$2.21, Ans. Ex. 21. $5-5-1.035=$4.830+, present worth of $5; hence this is the better condition by $4.875 $4.83 =$.045. Ex. 22. Since, if sold at cost, ^ of the lot should have sold for only ^ of the cost, the gain on ^ was | i == g of the cost; and ^-f-^=25 %, Ans. Ex.23. C$500 C$960 ] : $50= 1 : $60 (IF. ( (0 (?) = g2f=i yr.=7 mo. 15 da., Ans. Ex. 24. The remainder, which is 88 % of the whole, must be sold for 125 % of the cost of the whole ; hence 125 -5-88 = 1.42^; 1.42^1=42^ % above cost. Ex, 25. 3_6oiKa6j7J> 182.53375, greater; 3 5-^0 6 * s -,182.46625, less * Ex. 26. 7 yr.=84 mo. ; 3 yr. lOmo. 15 da. =46 ma.; (445.625 (650 _ 12 ooo. m i 84 '{ 46^=128.99 . (0 x lx ||.^ == $io4.15+, Am. (418, 41?) MISCELLANEOUS EXAMPLES. 223 Ex.27. JJi* 1*4*1=: $10125, invoice; $10125Xl.26=$12757.50, sold for; $12757.50X-015=$191.36, commission ; $12757.50 $191.36=$12566.14, net proceeds; $12566.14-*-.995=$12629.28+, Ans. Ex. 28. $.194X5000 =$970, draft on Paris; 5000-:-5.20=$961.53+, remittance from Paris. Ex. 29. $1000-r-.065=$15384.61+, stock required; $15384.61 Xl.05=$16153.84, Am. Ex. 30. $25000x T 7 2%= :: $ 14583 - 33 ib stock at 8 % ', $25000 X -06= 1500.00 , income of 6 % stock; 14583.333 X -08= 1166.66|, " " 8 % $333.33^, Ans. Ex. 31. $15190--.28 =$54250, money of all. Suppose A has $1, } B will have | X| = $f > \ proportionate terms. o " " !x|x|-$|?,) Multiplying these proportioned terms by 81, we have 81, 72 and 80, the proportional terms of A, B, and C respectively. 81-|_72+80=233, sum of proportional terms; 854250 X^Vs^ 18859 - 4 ^ A's money; $54250X2%= 16763.95+, B's " 54250 X 3%= 18626.61+, C's Ex. 32. 21 14=7 payments, or deposits. Now the last de- posit will not draw interest; the last but one will! draw compound interest for 2 half years, and th< amount will be $250xL03 2 ; the last but two wil draw compound interest for 4 half years, and th amount will be $250xL03 4 ; and so on. Hence we have $250=first term, 7=No. terms, and 1.03 (419> 224 MISCELLANEOUS EXAMPLES. = the ratio, to find the sum of the series. Now (1.03 2 )^=1.03 1 4; hence by 715~ (1.03 14 1)X250 =$2104.227+, Ans. 1.031 Ex. 33. | f=3nj cents > profit on 1 peach ; 420-r-,&=1200, Ans. Ex. 34. Since he is to cover 7% of the cost for expenses, and still clear 12^% on the cost, the sales, without allowing for credit or bad debts, must be 100% + 7%+12|%=119|% of the cost. But, since the sales are on credit of 6 mo., the 119.5% is the present worth of the sales; and 119.5x1-03 = 123.085%, the % of the cost, which the sales would be, without allowing for bad debts. Now, since 5 % of the sales is lost by bad debts, the 123.085% of the cost is 95% of the sales. Hence 123.085-i-.95=129.56+%, what the sales must be to cover all conditions ; and 129.56100=29.56+%, Ans. Ex. 35. 28X20X10=5600 days' work by 1st; 25X15X12=4500 " 2d; 18X25X11=4950 " 3d; . 15X24X 8=2880 4th; 17930 " all. : ^ 2686 - 00 +j lst receives, = 2158.39+, 2d = 2374,24+, 3d " = 1381. 37+, 4th Ex.36. 75XMX T V^2284,^3. (419, 420) MISCELLANEOUS EXAMPLES. 225 Ex, 37. $35.26+int. for 101 da.=$36.15 $48.65+ 70 = 49.50 $ 6.48+ " " 56 = 6.57 $50.00+ " 30 " = 50.38 $142.60, J.ws. Ex. 38. Had he received $.35 per bushel less for the barley, the whole cost would have been $.35x56=$19.60 less, or $63.10 $19.60=$43.50. But in this case, the prices of barley and corn would be equal. Hence 56+34=90 bu., whole quantity; $43.50-r-90=$.48|, price of corn; $.48|+$.35=$.83^, barley. Ex. 39. Since $12950 is the proceeds of the note discounted for 6 mo., (allowing 3 days grace), $12950-*- .96441666=813427.81,- face of the note re- ceived for the flour, Dec. 1. $13427.81-- .85=$15797.42, value, Nov. 1; 815797.42-5-1.30= 12151.86, " Oct. 1; 812151.86-^1.25= 9721.49, paid for flour; $12950 89721.49=83228.51, Ans. Ex.40. $660-^120 =$5.50, mean price; 5.50- J5.75UV 15.0&I.V 2 1 80 40 3 120 Ans. 80 bbl. From the conditions old par value; $2000 $1777|=$222f, premium; $222f-5-$17773=12 %, An*. Ex. 57. 21 mi. 7 mi. =14 mi., B gains of A daily; 84-s-14=6 ; hence A and B are together every 6 days. 7 mi.-{-14 mi. =21 mi., A and C approach each other daily; 14 mi.-f-14 mL--28mi., B and approach each other daily; 84-j-21=4; that is, A and C meet every 4 days ; 84-r-28=3 ; that is, B and C meet every 8 days ; Now, since A and B meet every 6th day, A and (421) MISCELLANEOUS EXAMPLES. 222 every 4th day, and B and C every 3d day, the in- terval of time required fcr all to meet, is the least common multiple of 6 da., 4 da. and 8 da. = 12 da. Ex. 58. 10463 9436 1027 963 64 41852 1348 1027 1 4 0463 _ 1_ 321 The approximating fractions 320 are , &, , f, and J$f. Hence, there must be a leap year once in four years, 7 times in 28 years, 8 times in 33 years, 31 times in 128 years, or 163 times in 673 years. Ex. 59. $14071-5-10000=81.4071, the amount of $1 for the time he owned the farm. By reference to the table of Compound Interest, we find that the time in which $1 at 5% will amount to $1.4071, is 7 years. Now, 11 yr. 7 yr.=4 yr., the perpetuity in rever- sion when he purchased it. $14071 X- 06=1844.26, the perpetuity which his money would purchase, if it were to be entered upon immediately; and the equivalent perpetuity in rever- sion 4 years, is the amount of $844.26 at 6% com- pound interest, for 4 years. Hence, $844.06X1. 26247=$1065.85, An*. Ex. 60. $l-!-1.035=$.966=, cash value of $1; $1 1966=1.034, gain on $.966; $.034-=-1966=.035+, Ans. Ex. 61. The several investments will form a geometrical se ries, of which the last will be the first term ; the last but one multiplied by 1.05, the second term-, and so on. Hence 1.05= ratio, 50 21=29= No. terms, (421, 422) 20 230 MISCELLANEOUS EXAMPLES. and $30000 the sum of the series; we are to find the first term. Reversing the rule under T15 5 we have $30000 X (1.05 1) $1500 - -= =$481.37, Ans. 1.05 29 1 3.116136 Ex. 62. The shares of the other two must be as 4 to 5 ; hence 4+5=9; $10000X| =$4444|, stare O f 2d; $10000xf=$5555f, " 3d. Ex. 63. (?)hhd.=1500bbl.; 50bbl. =125 yd.; 80yd. =6 bales; 13 bales=3| hhd.; (?)=75 T ^ hhd., Ans. Ex. 64. 11| 5 =6| mi., the 7th gains of the 1st, daily ; =5 " " " 2d " 3i.| 3d a =3 " 4th " 1^ 9| =1| 5th 1| 10|=1 " 6th 120 Hence, The 7th will pass the 1st once in - da. f 61 ft (( 2d " 120 3(J a a _ it 5th " 6th " a a 12^ et 120 (422) MISCELLANEOUS EXAMPLES. 281 And the time required for the 7th to pass all the others at the same place, will be the least com- mon multiple of the ahove intervals, which is found by dividing the least common multiple of the nume- rators by the greatest common divisor of the denom- inators. Least common multiple of 120 120 ; Greatest common divisor of 6|, 5, 3{^, 3, 1|, and 1 =^2 ; hence 120-^ = 1440 da., Ans. Ex. 65. 1 da. =86400 sec. 8640020=86380, No. beate 1st makes in 1 da.; 86400+15=88415, " 2d " " " ||4og_4j2_Q sec ^ time } n which l s t makes 1 beat; 6400 4 1 5 Now, the least common multiple of ||f sec. and l^go sec. is -L12^2. sec., which is the interval of time that must elapse before the two pendulums will beat in unison. But in this time the 1st pendulum will make Ii28JL-=-||fj = 2468 beats; and the 2d pen- dulum will make !lf &-*- | gf = 24 69 beats. There- fore, 2468 sec.=41 min. 8 sec. P. M., time by 1st; 2469 sec. =41 min. 9 sec. P. M., " 2d. Ex.66. (3 30 l)x^ 3 30 1 ---- = -- =343,518868244U in. ; 31 6 343,5188682441i-:-63360==541590730f!imi. Ex. 67. 105 % : 85%=(?) : 6 % (?) == u^ == 7 1 ^ %. Ex.68. A. P. 6 + 7 =$ .33, 1st condition ; 10 + 8 = .44, 2nd Istx5=30 +35 =$1.65 = 1.32 11 =$ .33 (422) 282 MISCELLANEOUS EXAMPLES. $.33-^-11= 3 cents, price of one peach; $.03X7 = 1.21; $.33 $.21 = $.12, price of 6 apples; $.12-:-6=2 cents, price of 1 apple. Ex. 69. If A has $9 B will have 5 and C will have | of $5= 24 $164 - =T I y 5 3> O's part of the whole estate: 164 $3862.50-T- T ' T 5 1 j=$29097.50, Ans. Ex. 70. C. R. 16 +20 =$30, 1st condition ; 24 +10 = 27, 2d 1stX|=24+30=i45 20= 18 $18-;-20 =$.90, price per bu. for rye; hence, from 1st or 2d, $.75, " " corn. Ex. 71. For 1 ox worth $28, there were 2 cows " 34 and 6 sheep " 45 $107 =$196, sum paid for oxen 749Xi%V= 238, cows; 749X T %\= 315, sheep; hence, $196-^$28 = 7 oxen , 238-^- 17 =14 cows; 315-f- 7.50=42 sheep. Ex. 72. $25000-7-.76=$32894.73+,"^w. Ex. 73. The difference between the coat and $20, and the coat and $9, is $20 $9=$11, which must be the (422, 423) MISCELLANEOUS EXAMPLES. 2S8 ratable wages for 20 12 8 weeks. Hence $11-H8 = $1|, one week's wages ; $lf X20=$27.50, wages for 20 weeks ; and $27.50 $20= $7.50 > value of coat. Ex. 11. 540 A. 36 P. = 86436 sq. rd.; 1/86486 294: rd., one side of square piece; 86436-j-42=2058 sq. rd., in each of equal square; 1/2058 = 45. 3 -f- rd., side of one of equal squares. Ex. 75. 25 A.=4000 sq. rd.; 1/4000 = 63.245+ rd., one side of the square field; 63.245 rd.X4=252.980 rd., perimeter of the square field. 4000-^2=2000; l/2jUU=44.721 rd., width of the rectangle; 44.721 rd.X2:=89.442 rd., length of the rectangle; (44.721 rd., +89.442 rd.) X2=268.326 rd., perimeter of the rectangle; 268.326 rd. 252.980 rd. = 15.346 rd., difference of the perimeters ; 1625X15.346 = 19.59+, Ans. Ex. 76. The hour and minute hands start together at 12 o'clock ; at 1 o'clock the hour hand has passed over 1 twelfth of the dial, while the minute hand has passed over the whole dial. Therefore, the minute hand gains of the hour hand at the rate of {^ of the dial in 60 min. But, at 5 o'clock the minute hand has T \ of the dial to gain of the hour hand, before passing it. Hence ji : T 5 2 =60 min. : (?)=27 min. 16 T 4 T sec. Ans 27 min. 16 T 4 T sec. past 5 o'clock. Ex. 77. It is evident that, to increase the numher in both rank and file by 1 man, would require twice the number in rank and file at first, plus 1 (for the man (423) 234 MISCELLANEOUS EXAMPLES. at the corner). And, since to effect this requires 284+25309 men, ==154 is the number of men in rank or file at first. Hence 154* +284=24000,^?. Ex. 78. 7 =first proportional term, 9 = second " " 7 X| =9= third 25^ = sum of " terms. 25^ : 7 =$3648 : (?) =$1008 25^ : 9 =$3648 : (?) =$1296 25| : 9|=$3648 : (?) =$1344 Ex. 79. 6 A. 3 R. 12 P. =1092 P. Then by 691, 13X21=273; 1092^-273=4; 1/4=2; 13X2=26 rd., width 21x2=42 rd, length ; (26 rd.+42 rd.)X2=136 rd. of fence. Ex. 80. A, B, 0, and D, do J^-^jf^ O f it in 1 day; A, B, C, and E, ^=7*1*1* " A,B,D,andE, T \, = T l4fiF " AP T) anrl "R I -- 5460 (t , C/, u, ana Jii, yg TosT4TJ BP T) pnrl TT, " 1 __ 744 , U, JJ, ana Jii __ 14 Taking the sum of the fractions, we shall have 4 days' work of each man. Consequently A, B, C, D, and E do y^ViV of Jt in 4 da J s ; and A, B, C, D, and E ^4^ j day> Hence, 414960^-3641 1=1 1^ 8 ^ da. required for all. If, now, from 4 3 T ^VffU> *^ e P art ^ t ^ ie wor ^ which all could do in 1 day, we take, successively. the parts which B, C, D, and E ; A, C, D, and E ; A, B, D, and E ; etc., can do in one day, we shall have the parts which A, B, C, D, and E can do, separately, in 1 day, thus : (423^ MISCELLANEOUS EXAMPLES. 235 A Cim do i 36411 6916 __ 8747 T* U (I JttGTj - TZT3T40 -- 4r4"" Therefore,414960-v-4491 = 92|f|f da. req'rd for A ; 414960--8747= 47|||f " B; 414960^-1931=214i?|f 5 414960-^-4571= 90|f^ " D; 414960-T-6771 61^ 9 T " E. Hence B will do the work in the shortest time. .Ex.81 The first has 1 share; the second has 1 Xli=H " the third has All have 4| 4| : 1 =$500 : (?)=-4105 T \, 1st has ; 4| : U=$500 : (?)=$157j|, 2d 4| : 2i=$ Ex. 82. $2500X5 =$12500 for 1 mo. ; $5500X7 = 38500 " B's investments=$51000 a S1333^---51000=$ T |3,B's (and also A's) gain per month on each $1 of capital invested. $5000 Xyfs X4=$522if|, A>s g ain during the first 4 months ; $1066| 8522i||=$543i|i, A's gain during the last 8 months ; $ T | 3 x8 = $ T %, gain on $1 in 8 mo.; $543if!-f- T % 2 3=$2600, A's capital for last 8 mo.; S5000 $2600=$2400, Ans. (423) f ' 236 MISCELLANEOUS EXAMPLES. Ex. 83, There will be 4 intervals of time each ^ year, and the rate 5 % ; hence $12750-=-1.215506=$10489.459-, Ans. Ex. 84. The proceeds of $1 for 93 da. are $.9819 ; and since he sells at 120 % of the purchase price, he realizes 120 %X.9819i=ll7.83 % in ready money. De- ducting 5 %+2 %=7 % for expenses, we have 117.83 % 1 $,=110.83 %, his rate of net receipts on each investment. Hence $1500 X 1.1083=81662.45, net proceeds 1st sale; $1662.45x1-1083= 1842.53+, " 2d " $1842.53X1.1083=2042.07+," " 3d " $2042.07X1.1083= 2263.23+, 4th $2263.23 $1500 = 763.23, whole gain, Ans. Ex. 85. $.80X1.20=1.96, what it must sell for; and since the selling price is 100 % 10 %=90 % of the asking price, we have |.96-*-..90==fl.06|, Ans. Ex. 86. 25X20=500 yd. $4JX500 $2187.50, purchase price; $4|X500=: 2312.50, selling " $2312.50-f-1.02 =$2267.15, cash value of sale ; $2187.50-i-1.045= 2093.30, purchase. $173.85, Ans. Ex. 87. $1200^-200=6, number of payments ; (1.08 6 1)X200 =$1437. 185, amount of the annuity .08 at the maturity of the last installment. And sin ce this sum will be at compound interest for 10 6=4 years, $1467.185xl.360489=$1996.07+, Ans. (423, 424) MISCELLANEOUS EXAMPLES. 2f,7 Ex. 88. The amount of an annuity of $3000 in arrears for (1.06 10 1)X$3000 10 years, is =$39542.40. .06 The present worth of this sum, due 10 years hence, is $39542,40-1-1.790848=822080.28, the value of the bequest to the eldest son. The present value of the same sum due 20 years hence is ^39542.40--3.207136=;$12329.51, the val- ue of the bequest to the second son. The worth of the perpetuity when entered upon by the institution, was $3000-f- .06 =$50000. But, as the perpetuity is not to be entered upon till 20 years after the man's decease, its present value is $50000-i-3.207136=$15590.23. Ex. 89. It requires 5-|-l=6 days for the horse team to per- form the trip and rest a day, 7+1=8 days for the mule team, and 11-|-1=12 days for the ox team to do the same. Now, the least common multiple of 6 da., 8 da. and t2 da is 24 da. Therefore the three teams will rest together on the 24th day, and conse- quently 24 1=23 da. must elapse before this day comes. Ex. 90. The four payments must be treated according to the U. S. rule for Partial Payments ; that is, the pay- ment each year will cancel the interest which has accrued on the principal for that year, together with a certain portion, or installment, of the principal it- self. Now, the principal for the second year will be less than the principal for the first year, by the value of the first installment; and consequently the inter- est to be canceled by the second payment will be (424) 238 MISCELLANEOUS EXAMPLES. less than the interest canceled by the first payment, by one year's, interest, or 6 %, of the first install- ment. Hence the second installment will exceed the first, by the same sum ; that is, the second install- ment will be 1.06 times the first. For a similar reason, the third will be 1.06 times the second ; and so on. Therefore, these installments of the princi- pal form a geometrical series, of which the 1st in- stallment == first term, 4= No. terms, and $4500= sum of the series. We may find the first term by reversing the rule under 715 5 thus: (1.06 1)X $4500 -- =$1028.67+= 1st term, or that 1.06 4 1 part of the principal canceled by the first payment. But the payment must also cancel the interest of the principal for that year, which is $4500X-06=$270; hence $1028.67+$270=$1298.67, Am. Ex. 91. Of the payments coming due after Jan. 1, 1864, one is to be discounted for 2 half years at 6 % per annum, or for 2 intervals at 3 % ; and the other for 4 intervals at 3$. Hence $1050, 4th payment; $1050-7-1.0609=$ 989.725+,present val. 5thpa/t. $1050-4-1.125509= 932.911 +, " 6th " $2972.636+, Ex. 92. 2X3=6 ; 864-4-6=144 ; T/144=12 ; 3x12=36, No. of rows; 2x12=24 " trees in each row. And since the spaces between the trees are 1 less, each way, than the number of trees, (424) MISCELLANEOUS EXAMPLES. 289 (36 1)X7=245 yd., length of the orchard; (24- -1 ) X 7 = 161 yd., breadth 245X161=39445 sq. yd., Ans. Ex. 93. $2500X1.06=12650, first cost; $2650x1-225043= $3246.36, amount for 3 yr.; $3246.36X1.0175= 3303.17, " 3 yr. 3 mo, which is to be considered as the whole cost of the investment. Now, there were received 6 dividends of $100 each, received 2 yr. 9 mo., 2 yr. 3 mo., 1 yr. 9 mo., 1 yr. 3 mo., 9 mo., and 3 mo., respectively, before the sale of the stock. Hence, reckoning compound interest, Amt. of $100 for 2 yr. 9 mo. =$120.50 " . 2 " 3 = 116.49 1 9 " = 112.62 " 1 " 3 " = 108.87 9 = 105.25 3 " = 101.75 Value of dividends, $ 665.48 Sale of stock, $2500x1.11= 2775.00 Whole sum realized, $3440.48 $3440.48 $3303.17=4137.31, Ans. Ex. 94. Firstj the number in the smallest company is 4, and in the largest 64, and the number in each company is double the number in the preceding company. Hence the companies are, respectively, 4, 8, 16, 32, 64. There are therefore 5 companies, and the total number of men is - =124. 21 Second^ The smallest company received $.50 each C424) 24:0 MENSURATION. per day, tin largest $1.50 each, and the common difference of the rates of daily wages is $.25 Hence the daily rates in the respective companies are $.50, $.75, $1.00, $1.25, $1.50. Now, if we multiply these rates of daily wages, each by the number of men in the respective companies, and add the products, we shall obtain the whole sum paid per day, thus : $ .50, $ .75, $ 1.00, 8 1.25, $ 1.50 4 8 16 32 64 $2.00 +$6.00 +$16.00 +$40.00 +$96.00 =$160 paid per day \ $160X6=?960, weekly payment. MENSUKATION. (731 5 page 425.) Ex (. (24+15) X2=78 ft., perimeter of the room; 78X83 =663 sq. ft. in the walls; 24X15X2=720 " in floor and ceiling ; 1383 sq. ft. = 153| sq. yd., Arts. Ex. U. Let A B D E F represent the farm. B u If we extend the line D E to Q-, the G land will be divided into two rectangles, G B C D, and A O E F. Since B 0= A _ 25.14 ch., and C D =12.08 ch., we have 25.14X 12.08=303.6912 sq. ch., area of G B C D. Again, since E F =26.12 ch., and F A =16.84 ch., 26.12 X 16.84=439.8608 sq. ch., area of A a E F. Then (424, 425) LINES AND SUPERFICIES. 2-n 439.8608+303.6912=743.552 sq. ch.=74.3552 A. =74 A 56.8+ P., Ans. Ex. 3. 240-f-20 = 12 rd., Ans. Ex. 4. 70 A.=11200 sq. rd. ; 11200-5-120=93^ rd., Am. Ex. 5. 40 A. =400 sq. ch. ; 4008=50 ch., Ans. 72 (732, page 426.) .fix. 1. 28 ft. 9 in.=28| ft.; 36X28| = 1035 sq. ft. =3 sq. rd. 218| sq. ft., Ans. Ex.2. 1/20 2 122=16; 72X16=1152 P. =7 A. 32 P., Ans. Ex. 3. 14 A?72 P.=2312 P. Now, by 691, 1X2=2, product of the proportional terms; 2312-1-2=1156; 1/1156=34; 34X1=34, shorter side; 34x2=68, longer side; and 68x30=2040 P.=12 A. 120 P. (733, page 426.) * Ex. 1. 12^+8jbJL(2=24 ft., Ans. Ex. 5. -*-^=135 sq. rd., Ans. Ex.6. 1/18 2 9 2 =15.588 + ft.=BD; 15.588 X V 8 =140.29+sq. in. Ex.1. Ex.2. Ex. 3. Ex.4. * (735, page 428.) 8 ft.X3.14159=25.13272 ft.=25 fy 1.59+ in. 49.52X.31831=15.762+ rods, Ans. 4 ft. 8 in.=56.5 in.; 56.5x2=113 in., diameter; 113x3.14159=355 in., circumference; 18 =3^ = 2*0 of a circumference; 355 in.X 2 V= 17 - 75 in 1 ft - 5 - 75 in ^ Ans - 66 ch.=264 rd. ; 264 X. 31831=84.03384 rd.,.diameter of garden; 66X.31831=21.00846 " " "pond; 2) 68.02558 rd., twice the width of ring ; 31.51+ rd., Ant. 16.5X-31831=5.25+ ft.=5 ft. 3+ in.,. Ex.5. (736, page 428.) Ex. 1. ^21x^10^40115, Ans. Ex.2. 25* X- 7854=490.875 sq. in., Ans. Ex. 3. 6 ft. 10 in.=82 in. ; 82*X-07958=535.1+sq. in.=3 sq. ft. 103.1 sq. in. (427, 428) SOLIDS. 2-13 Ex. 4. 6.44598-r-.07958=81 ch., square of circumference ; l/8l=9 ch.=9 ch.=36 rd., ^ns. Ex. 5. 8 2 X-7854=50,2656 sq. rd., area of circle ; 50.2656-=-4=12.5664 sq. rd., Ans. (744, page 429.) Ex 1. 16 3 =4096 cu. ft., Ans. Ex. 2. 15X3X Ji=41i cu. ft., Ans. Ex. 3. l/i a .5 a =V776"= .866 ft.+, altitude of triangle ; .866X.5=-433 sq. ft., surface of the end; 24x.433=10.39+cu. ft., Ans. Ex. 4. .?854X9 2 X9^X1728=1044343.2384 cu. in. ; 1044343.2384-f-231=4520.96+gal.= 71 hhd. 47.96+gal., Ans. Ex. 5. Let A B C D be tlie end of tlie log when hewn square. B~C 2 +BA 2 = 2 BO^AC^^ Hence, BG 2 =rr50 sq. ft., the area the square end. .7854X10* X20X12=18849.6 cu.in. in whole log; 50 sq. ft. X 20x12=12000 " sq.timber; 6819,6 cu. in.=3 cu. ft. 1665.6 cu. in., Ans. Ex. 6. 1 h. 20 min.=4800 sec. Hence the quantity of water discharged will be equal to the contents of a tube 2 in. in diameter, and 12 in. X 4800 =57600 in, in length.; .7854 X2 2 X 57600=180956.16 cu. in. ; 180956.16 cu. in.-f-231=783.36+gal., Ans. Ex. 7 29 23=6;f of 6=4; 23 +4=27 in., mean diameter. ,7854x27^X36 =.0034X27*X36=89.2296 gal. 231 (428-430) 244 MENSURATION. Ex. 8. 2825=3; T 6 of 3=1.8; 25+1.8=36.8 in., mean diameter. And since .7854--231=.0034, . .0034X26.8* X35=85.47056 gal., Ans. (745, page 430.) Ex. 1. 17 3 X12=3468 cu. ft., solid contents. From the foot of the perpendicular to the middle of one side of Che base is 17-4-2=8.5 ft. Hence, 1/36* +8.5 2 =36.99+ft., the slant height; 36.99 X 8.5x4=1257.66+sq. ft., lateral surface. Ex. 2. 5i| ft.=5.916666+ ft. ; 5.916666 2 X- 07958=2.78584+ sq. ft., area of base; 2.78584X|=4.64306+ cu. ft., Ans. Ex. 3. From the solution of Ex. 5, 74:4 5 we learn that the square of the diameter of a circle is equal to twice the area of the square that may be drawn with- in it. Hence, 72 =24.5 sq. ft., area of the base of the pyramid ; 2 24;5X 1 /=24.5X 6=147 cu. ft., Ans. Ex. 4. &!^-X4==1500 sq. ft. in the convex surface ; 302 = 900 " " " 2400 sq. ft. in entire surface. Now, since the slant height is 25 ft., and a line from the middle of one side of the base to the foot of the perpendicular is 15 ft., we have 1/25 2 15 2 =20 feet, the altitude; and 30 2 X 2 3 : = : 6000 cu - ft -> solid contents. Ex. 5 30X V 8 =270 s q- in 1 sq. ft - 126 s q- in -> convex surface; 30X-31831=9.5493 in., diameter of the base; 9.5493 in. -=-2 =4. 7 7465 in., radius. Now, (430, 431) SOLIDS. 245 since the slant height of the cone and the radius of the base form the hypotenuse and base of a right- angled triangle, of which the other side is the altitude of the cone, .77465 2 =17.3580+ in., altitude ; 30 2 X- 07958=71.622 sq. in., area of base; 71- fi?3g i?-35 8; [243.21+ cu. in., solidity. (746, page 431.) Ex. 1. 4 2 X3.1416=50.2656 sq. ft., Ans. Ex. 2. 8.5 3 X.5236=321.55+cu. in., Ans. 12 3 x.5236 Ex. 3. -- =452.39-|-cu. in., Ans. Ex. 4. Since the corners of the inclosed cube will touch the surface of the sphere, the diagonal of the cube, E C, will be the diameter of the sphere, or 12 ft. Now, remembering that A E, A B, and B C, are all equal, being each a side of the cube, we have (A E) 2 +(A C) 2 =(E C) 2 . But (A C) 2 =(AB) 2 +(BC) 2 , or 2(AB) 2 , or 2(A E) 2 . Hence, 3 (A E) 2 =(E_C) 2 =12 2 =144, and 144-- 3=48=(A E) 2 ; 1/48=6.928+ ft.= A E., Ans. Ex. 5. 65.45-4-;5236=125, cube of diameter; ^125=5 in., Ans. Kx, 6. Keversing the rule for finding the surface, 78.54-r- 3.1416=25, the square of the diameter; and 1/25 =5 in., the diameter. Then, by the rule -for finding the solid contents, 5 3 X.5236=65.45 cu. in., the capacity. (431) 24:6 MENSURATION. Ex. 7. 14 3 X- 5236 =i 1436.7584 cu. in. in larger globs, 12 3 x. 5236= 904.7808 " smaller 531.9776 cu. in., (747, page 432.) ffix. 1. 1 gal.=231 cu. in. Hence 231-^-2=115.5 cu. in., the contents of the lead; $.15x115.5 = 117.325. Ex. 2. 6 gal. 3 qt. 1 pt.=6| gal. "; 8| gal. 6| gal.=l| gal., the solidity of the anvil. Hence, 231 cu. in.Xlf =317f cu. in., Ans. Ex. 3. 8 3 =512 cu. in., contents of box; 8j qt.=j| gal.=216 T 9 g cu. in., deficiency; 295 T 7 g cu. in., Ans. (748, page 432.) Ex. 1. The diameters will be to each other as the cube roots of 64 and 512, as 4 to 8, or as 1 to 2. Ex. 2. Since the blocks are to each other as 105_to 2835, or as 1 to 27, the lengths will be as ty \ to "^27, or as 1 to 3. Hence 7 in. X 3=21 in., Ans. Or, 105 : 2835=7 3 : (?)=9261 1^9261=21 in., Ans. Ex. 3. iTT : lTir=:2 ft. : ('0=2 ft. X 1.4422=2.8844 ft. =2 ft. 10.6+ in., Ans. Ex. 4. Since the contents of the required cellar will be 18 -4-6=3 times the contents of the given cellar, its seve- ral dimensions will be ty 3 =1.4422 times the given dimensions. Hence 14 ft. X 1.4422=20.1908 ft., length; 12 ft. X 1.4422=17.3064 ft., width; 6 ft.Xl.4422= 8.6532 ft., depth. (431, 432) SOLIDS. 247 Ex. 5. 9000217=41.474+, ratio of the contents ; =3 .46-}-, ratio of respective dimensions ; 20 in.XB.46=69.2+ in., length; 15 in.x3.46=51.9+ in., breadth; 8 in.X3.46=27.6+ in., thickness. Ex. 6. A will take off a pyramid containing 4 tons ; A and B, a pyramid containing 8 tons ; and A, B and C, a pyramid containing 12 tons. And, since these pyra- mids are similar figures, they will be to each other as the cubes of their altitudes. Hence, comparing the whole pyramid with each pyramid taken off, Tons. ft. .16 : 4=16 3 : (?)=1024, cube of height taken by^ ; 16 : 8=163 : (?)=2048, " " A and B; 16 : 12=16 3 : (?)=3072, " " ^1024=10.079 ft., taken off by A; ^2048=12.699" " " AandB; . ^3072=14.537 " " A, B and 12.69910.079=2.620 ft., taken off by B ; 14.537__f2.699=1.838 " C; 16.00014.537=1.463 " P (432) A UNIVERSITY OF CALIFORNIA LIBRARY