University of California • Berkeley
The Theodore P, Hill Collection
of
Early American Mathematics Books
KEY
TO THE
PROGRESSIVE
PEACTICAL ARITHMETIC.
INCLUDINa
ANALYSES OF THE MISCELLANEOUS EXAMPLES
PROGRESSIVE INTELLECTUAL ARITHMETIC.
FOR TEACHERS ONLY.
IVISON, BLAKEMAN, TAYLOR & CO..
NEW YORK AND CHICAGO.
1877.
ROBINSON^S
Mathematical Series.
Graded to tlie wants of Primary, Intermediate, Grammar,
Normal, and High Schools, Academies, and Colleges.
Progressire Table Boob.
Progressive Primary Aritliraetic.
ProgressiYO Intellectual Arithmetic,
Rudiments of Written Arithmetic.
JUNIOR-CLiSS ARITHMETIC, Oral and Written. NEW.
Progressive Practical Aritlimetic.
Key to Practical Arithmetic.
Progressive Higher Arithmetic.
Key to Higher Arithmetic.
Nevr Elementary Algebra.
Key to New Elementary Algebra.
New University Algebra.
Key to New University Algebra.
New Geometry and Trigonometry, In one vol.
Geometry, Plane and Solid. In separate vol.
Trigonometry, Plane and Spherical. In separate vol.
New Analytical Geometry and Conic Sections.
New Surveying and Navigation.
New Differential and Integral Calculus.
University Astronomy— Descriptive and Physical.
Key to Geometry and Trigonometry, Analytical Geometry and Conie
Sections, Surveying and Navigation.
Entered, according to Act of Congress, in the year 1860, by
HORATIO N. ROBINSON, LL.D.,
in the Clerk's Office of the District Court of the United States for the Northern
District of New York.
P R E I^ 1 1 .
A Key to any Matliematical work is not intended to sa-
persede labor and study, but to give direction to the latter and
make it more effective and useful.
In many examples and problems tbe same results may be
obtained by different processes, but the shortest and most
simple method is to be desired ; hence the object of a Key
should be to give not results only, but the explanation of
methods, and a full analysis of such questions as contain a
peculiar application of principles involved.
It is supposed, of course, that every teacher is fully com-
petent to solve all the questions, but with the multiplicity
of duties ordinarily put upon the teacher, time cannot always
be had to answer or solve all the questions presented by the
pupil. Therefore the Key is intended to lessen the labor
and save the time of the teacher by presenting the shortest
solution, and the best form of analysis as a standard to which
the pupil should be required to conform.
IV PREFACE.
In compliance with the wishes of many teachers, brief
analyses of the Miscellaneous Examples in the Intellectual
Arithmetic have been added to the latter part of this work.
Much labor has been bestowed upon the present work
to give a full, complete, and logical analysis of all difficult
rxamples^ and of g'jch questions as contain the application
of a new principle, The arrangemeut i& such as to be
easily understood.
KEY.
NOTATION.
ROMAN NOTATION .
(17, page 9.)
Ex.
Ex.
Ex.
Ex.
1. Ans XL
S, Ans, XXV.
5. Ans. XLVm.
Y. Ans. CLIX.
9. Ans. MDXXXYIII.
Ex. 2. Ans. XV.
Ex. 4. Ans. XXXIX.
Ex. 6. Ans. LXXVII.
Ex. 8. Ans. DXCIV.
Ex. 10. Ans. MDCCCCX,
I
Ex.
Ex.
Ex.
1. Ans. 125.
4. Ans, 900.
1. Ans. 505.
ARABIC NOTATION.
(26, page 12.)
Ex. 2. Ans. 483.
Ex.5.
Ex. 8.
Ans. 290.
w4?i5, 557.
Ex.3. Ans. 716.
Ex. 6. ^/i5. 809.
(28, page 13.)
Ex. 2. ^W5. 5160.
Ex. 5. ^715. 2090.
Ex. 8. Ans. 9427.
Ex. 3. Ans. 3741.
Ex. 6. ^715. 7009.
Ex. 9. Ans. 4035.
Ex. 1. Ans. 1200.
Ex. 4. ^715. 8056.
Ex. 7. Ans. 1001.
Ex. 10. ^»5. 1904.
Ex. 11. Ans. Seventy-six; one hundred twenty-eight ; four
hundred five ; nine hundred ten ; one hundred sixteen ; three
thousand four hundred sixteen ; one thousand twenty-five.
Ex. 12. Ans. Two thousand one hundred; five thousand
forty-seven ; seven thousand nine ; four thousand six hundred
6 SIMPLE NUMBERS.
seventy; three thousand nine hundred ninety seven; one
thousand one.
(29, page 14.)
Ex. 1. Ans. 20000. Ex. 2. Ans. 47000. Ex. 3. Ajis. 18100.
Ex. 4. Ans. 12350. Ex. 5. Ans. 39522. Ex. 6. Ans. 15206.
Ex. 7» Ans. 11024. Ex. 8. Ans. 40010. Ex. 9. Ans. 60600.
Ex. 10. Ans. 220000. Ex. 11. Ans. 156000.
Ex. 12. Ans. 840300. Ex. 13. Ans. 501964.
Ex. 14. Ans. 100100. Ex. 15. Ans. 313313.
Ex. 16. Ans. 718004. Ex. 17. Ans. 100010.
Ex. 18. Ans. Five thousand six; twelve thousand three
hundred four ; ninety-six thousand seventy-one ; five thousand
four hundred seventy ; two hundred three thousand four hun-
dred ten.
Ex. 19. Ans. Thirty-six thousand seven hundred forty-one ;
four hundred thousand five hundred sixty ; thirteen thousand
sixty-one ; forty -nine thousand ; one hundred thousand ten.
Ex. 20. Ans. Two hundred thousand two hundred ; seventy
five thousand six hundred twenty ; ninety thousand four hun-
dred two ; two hundred eighteen thousand ninety-four ; one
hundred thousand one hundred one.
(31 5 page 16.)
Ex 1. Ans. 140. Ex. 2. Ans. 30201. Ex. 3. Ans. 8050.
Ex 4. Ans. 2900417. Ex. 5. Ans. 300040.
Ex. 6. Ans. 96037009. Ex. 7. Ans. 4064200150,
Ex. 8. Ans. 846009350208.
(34, p. 19.)
Ex. 1. Ans. 436. Ex. 2. Ans. 7164.
Ex. 3. Ans. 26026. Ex. 4. Ans. 14280.
Ex„ 5. Ans. 176000. Ex. 6. Ans. 450039.
Ex. 7. Ans, 95000000. Ex. 8. Ans. 433816149.
NOTATION AND NUMERATION. 7
Ex.9. Ans, 900090. Ex. 10. Ans. 10011010.
Ex. 11. Ans. 61005000000. Ex. 12. Ans, 5080009000001.
Ex. 13. Ans, Eight thousand two hundred forty.
Ex. 14. Ans, Four hundred thousand nine hundred.
Ex. 15. Ans. Three hundred eight.
Ex. 16. Ans, Sixty thousand seven hundred twenty,
Ex. 17. Ans, One thousand ton.
Ex. 18. Ans, Fifty-seven million four hundred sixty-eight
thousand one hundred thirty-nine.
Ex. 19. Ans, Five thousand six hundred twenty- eight
Ex. 20. Ans, Eight hundred fifty million twenty-six thou-
eand eight hundred.
Ex. 21. Ans, Three hundred seventy thousand five.
Ex. 22. A71S, Nine billion four hundred million seven hun-
dred six thousand three hundred forty-two.
Ex. 23. Ans, Thirty-eight million four hundred twenty-nine
thousand five hundred twenty-six.
Ex. 24. Ans, Seventy-four billion two hundred sixty-eight
million one hundred thirteen thousand seven hundred fifty-
nine.
Ex. 25. Ans, 7000036.
Ex. 26. Ans, 563004.
Ex.27. Ans, 1096000.
Ex. 28. Ans, Nine billion four million eighty-two thousand
five hundred one.
Ex. 29. Ans, Two trillion five hundred eighty-four billion
five hundred three million nine hundred sixty-two thousand
forty-seven,
Ex. 30. Ans. 3064159,
Ex, 31. A71S. Two of the sixth order, 9 of the fifth, 6 of the
third, 4 of the second, and 8 of the first.
Ex. 32. Ans. One of the seventh order, 3 of the fifth order,
7 of the fourth order, and 5 of the second order.
Ex. 3
Ex.6.
Ans. 6i>8.
Ans. 898.
SIMPLE NUMBElia
ADDITION.
(40, page 21.)
Ex. 4. Ans. 967.
(43, page 24.)
Ex. 7. ^7i5. 1807. Ex. 8. Ans. 27246.
Ex. 9. Ans. 4945. Ex. 10. ^7Z5. 78313.
Ex. 19. Ans. 145. Ex. 22. Ans. 69585.
Ex. 23. Ans. 566. Ex. 24. Ans. 3746.
Ex. 27. ^^^5. 4619. Ex. 28. Ans. 4915.
Ex. 29. Ans. 4320. Ex. 30. Ans. 4623.
Ex. 31. Ans. 3871. Ex. 35. ^?Z5. 101500.
Ex.37, ^ns. 50000000. Ex.40, ^w*. 1194.
Ex. 44. Ans. 2773820. Ex. 45. Ans. 4403241.
Ex. e.
Ex. 8.
Ex. 19.
Ex. 24.
Ex. 26.
SUBTKACTION.
(49, page 30.)
Ans. 353.
Ans. 205.
Ans. 123.
Ans. 4202.
Ans. 16348755.
Ex. 7.
Ex. 9.
Ex. 22.
Ex. 25.
Ex. 27.
Ans. 210.
Ans. 320.
Ans. 2113.
Ans. 11425,
Ans. 4014580.
(51, page 32.)
Ex. 3. Ans. 721.
Ex. 4. Ans. 561.
Ex. 5. ^ws. 3769.
Ex. 6. Ans. 269,
SUBTRACTION.
Ex.
7.
Arts. 4509.
Ex. 8.
Ans. 3449.
Ex.
9
Arts. 1288.
Ex. 10.
Ans. 30616.
Ex.
11.
Ans. 21078.
Ex. 12.
Ans. 142.
Ex.
13.
Ans. 762301.
Ex. 19.
Ans. 224130.
Ex
20.
Ans. 220874.
Ex. 25.
Ans. 181972.
Ex.
31.
Ans. 529509693.
Ex. 34.
Ans. 1902001
EXAMPLES COMBINING ADDITION AND SUBTRACTION,
(52, page 33.)
Ex. 1. 2500+ 175 = 2675
6200—2675 = 2525 dollars, Ans.
Ex. 2. 235 + 275 + 325 + 280 = 1115;
1300 — 1115 = 185 miles, Ans.
Ex. 3. 4234 + 1700 + 962 + 49 = 6945,
87,^^-6945 = 1807 dollars, Ans.
Ex. 4. 47(^5 + 750=5515;
5515 — 384 = 5131 dollars, Ans.
Ex. 6. 1224 + 1500 + 1805 = 4529;
7520 — 4529 = 2991 barrels, Ans.
Ex. 6. 450 + 175= 625, B's;
450 + 625 = 1075, A'sandB's;
1075 — 114= 961, C's sheep, Ans.
Ex. 7. 1575 — 807 = 768, bushels of wheat,
900 — 391 = 509, " " corn,
Ex. 8. 2324 + 1570 + 450 + 175=4519;
6784—4519 = 2265 miles, Ans.
at, )
Ex. 9. 7375, first paid;
7375+ 7375 = 14750, second paid;
7375 + 14750 = 22125;
36680 — 22125 = 13555, dollars, Ans.
Ann,
10
SIMPLE NUMBERS.
Ex. 10. '750-{-3'79 + 450 = 1579;
1579 — 1000=579, dollars, Ans.
Ex. 11. 6709 + 3000=9709;
9709—4379 = 5330 dollars, Ans,
Ex.12. 10026402+ 9526666 = 19553068, total ;
8786968+ 8525565 = 17312533, native;
19553068 — 17312533= 2240535, foreign, Ans.
MULTIPLICATION.
(61, page 38.)
Ex.
5
Ans. 247368.
Ex. 6.
Ans.
648998.
Ex.
7.
Ans. 224744.
(64,
Ex. 8.
page 41.)
Ans.
416223.
Ex.
5.
Ans. 2508544.
Ex. 6.
Ans.
15731848,
Ex.
7.
Ans. 16173942.
Ex. 9.
Ans.
78798.
Ex.
13.
Ans. 937456.
CONTRACTIONS.
Ex.
2.
Ex.
3.
Ex.
4.
Ex. 5.
(675 page 43.)
3472x6 = 20832; 20832x8 = 166656, Ans.
14761 X 8 = 118088 ; 118088 x 8=944704, Ans.
87034 X 3 = 261102 ; 261102 x 3 = 783306 ;
783306x9 = 7049754, Ans.
47326 X 6 = 283956 ; 283956 x 5 = 1419780 •
1419780x4 = 5679120, Aiis.
MULTIPLICATION, II
Ex. 6. 60315x8X3x4 = 5790240, Ans,
Ex. 7. 291042x5x5x5 = 36380250, Ans.
Ex. 8. 430 X 7 X 8 = 24416 miles, Ans.
Ex. 9. 124 X 6 X 3 X 4 = 8928 dollai-s, Ans.
Ex.13. 5280x7x3x4 = 443520 feet, Ans.
Ex. 11. 120x5x5x5 = 15000 dollars, Ans.
{61>5 page 44.)
Ex. 3. Ans. 13071000. Ex. 4. Ans. 890170000.
(70, page 45.)
Ex. 12. 296
3000
888000 dollars, Ans.
EXAMPLES COMBINING ADDITION, SUBTRACTION, AHD
MULTIPLICATION.
Ex. 1. 4x45 = 180; 13x9 = 117;
180 + 117=297 dollars, Ans.
Ex. 2. 31x6 = 186; 39x6 = 234;
234 — 186 = 48 dollars, Ans,
Ex. 3. 288x9 = 2592;
2592 — 1875 = 717 dollars, Ans.
Ex. 4. 240 + 125 + 75 + 50 = 490;
500 — 490 = 10 dollars, Ans,
Ex. 5. 184x2=368; 67x4 = 268;
368 — 268 = 100 dollars, Aiis
Ex. 6. 36 X 320 = 11520, A received ;
48 X 244 = 11712, B received ;
11712 — 11520 = 192 dollars, Ans.
Ex, 7 35 + 29 = 64 miles, in one day ;
64 X 16 = 1024 miles, ^^.
12 SIMPLE NUMBERS.
E%. 8. 14 X 26 X 43 = 15652 yards, Am*
Ex. 9. 4 X 365=: 1400, yearly expenses ;
3700—1460 = 2240 dollars, Ans,
Ex.10. 2475, first;
2475— 840 = 1635, second;
2475 + 1635 = 4110, third;
8220 dollars, Ang.
Ex. II. 336 — (28 X 10) = 56 miles. Am.
Ex.12. 23 X 14= ^22, cost of cows;
96 X 7= 672, " " horses;
57 X 34 = 1938, " " oxen;
2x300= 600, " " sheep;
3532, " " whole.
3842 — 3532 = 310 dollars, Ans.
Ex. 13. 36X164 = 5904
Sx 850 = 2550
8454 dollars, Ans.
Ex. 14. 14760— (1575 x 5)=6885 dollars, An9»
Ex. 15. 936 X 9=8424, cost ; 480 x 10 = 4800
456 X 8=3648
Flour sells for, 8448.
8448 — 8424 = 24 dollars, Ans.
DIVISION.
(77, page 50.)
Ex. 2. Ans, 16358. Ex. 3. Ans, 17827.
Ex. 4. Ans. 29822. Ex. 5. Ans. 672705.
Ex. 6. Ans. 182797. Ex. 7. Ans. 829838.
DIVISION. 13
Ex. 13. Ans, 1048795f. Ex. 14. Ans, 635926f .
Ex. 15. Ans. 2379839I-. Ex. 16. Ans. 9355Y51f
Ex. 17. Ans, 245640}-!. Ex. 18. Ans. 70141321.
Ex. 19. 47645-^5 = 9529 dollars, Ans. •
Ex. 20. 17675-^7:=:2525 weeks, Ans.
Ex. 21. 6756^6 = 1126 barrels, Ans.
Ex. 22. 46216464^12:^3851372 dozen, Ans.
Ex.23. 347560-^5 = 69512 barrels, Ans.
Ex.24. 3240622-7-11 = 294602 acres, Ans.
Ex. 25. 38470-T-5=7694 dollars, Ans.
(80, page 54.)
Ex. 5. Ans. 212//4-. Ex. 14. Ans. 1489f|.
Ex. 15. Ans. 121522%. Ex. 16. Ans. 508301yV2-
Ex. 17. Ans. 12109001$ Jf Ex. 18. Ans. 9974^9-.
Ex. 19. Ans. 1343if Jf. Ex. 20. Ans. 5473|fff
Ex. 21. Ans. 7500yVTTV
Ex.27. 1850400-^18504 = 100 dollars, Ans.
Ex. 28. 72320060-^10735 = 6736yVT3J <iollars, Ans.
Ex.29. 942321-^213 = 44243^? 3- volumes, Ans.
Ex. 30. 5937120^22320 = 266 dollars, Ans.
CONTRACTIONS.
(81 5 page 56.)
Ex. 2. (3690-T-3)-4-5 = 246, Ans.
Ex.3. (3528^4)-^6 = 147, Aiis.
Ex.4. (7280-^5)^7 = 208, Ans.
Ex. 5 (6228-^6)-^6 = l73, Ans.
Ex.6. (33642^3)-^9 = 124d, Ans.
Ex, 7. (153160-^7)^8 = 2735, Ans.
Ex. 8. [(15625-^5)-^5]^5 = 125, An9.
14 SIMPLE NUMBERS
(82, page 57.)
Ex. 2. 6)34712
7)5785 2
826 ---3x6 = 18^
20, Ans.
Ex. 3. 8)401376
8)50172
6271 ---4x8=32, Ans.
Ex. 4. 3)139074
4)46358
6)11589 2x3= 6
1931 ---3x4x3 = 36^
42, Ans^
Ex. 5. 3)9078126
5)3026042
6)605208 - - - 2 X 3=6, Ans.
100868
Ex. 6. 4)18730627
5)4682656 8
6)936531 1x4= 4
156088 ---3x5x4 = 60
67, Am.
Ex. *i 2)7360479
6)368C239 1
8)613373 1 x2= 2
76671 ---5x6x2 = 60^
63, Ana.
SIMPLE NUMBERS. 15
Ex. 8. 2)24726300
6)12363150
7)2472630
353232 -"GxbX 2 = 60, Am
Ex. 9. 7)5610207
2)801458 1
6)400729
66788--- 1x2x7 = 14
15, Ans,
(83j page 58.)
Ex. 2. Ans. 476. Ex. 3. Ans. 2Q20j\\.
Ex. 4. Ans. 1306y3-2-i7. Ex. 5. Ans. 976yf JJ^.
Ex. 6. Ans. 2037lyVoVoV-
(8 5, page 59.)
Ex. 6. Ans. 14556yVoVoV- Ex. 7. ^ns. 106099VoVo-
Ex. 8. Ans. 114304^1^15. Ex. 10. Ans. Q84J^^\%\\%\%
Ex. 11. 24898^50 = 497ff hours, Ans.
Ex. 12. 350000-r-14000=25 dollars, Ans.
EXAMPLES IN THE PRECEDING RULES.
(Page 60.)
Ex. 1. 1732 + 67 = 1799, Ans.
Ex. 2. 1095-^365=3 dollars, Ans.
Ex. 3. 141+47 = 188 dollars, ^ns.
Ex. 4. 500 + 17 + 98 + 121 = 736 acres owned;
736 — 325=411 acres, Ans.
Ex. 6. 2300 — 625 = 1675 dollars, ^719.
16 SIMPLE NUMBERS.
Ex. 6. 60 — 45 = 15 dollars, saved in one month;
900-^15r^60 months, Ans.
Ex. 7. 87 X 9 = 783 days, Ans.
Ex. 8. 4 first number ;
4x8= 32 second;
32x9 = 288 thh'd;
324, Ans.
Ex. 9. 2x2x7 = 28;
364^28 = 13, Ans.
Ex. 10. 78 + 104 = 182;
182 X 2 = 364 acres, Ans.
Ex.11. 90-1-30 + 12-1-5 + 7 = 144;
144 X 27 = 3888 dollars, Ans.
Ex. 12. (2250 X 4)^3 = 3000 dollars, Ans.
Ex. 13. 35 + 40=75 miles in one day;
75 X 6 = 450 miles, Ans.
Ex. 14. 40—35 = 5 miles in one day;
5 X 6 = 30 miles, Ans.
45 — 19=26 years, Ans.
1000000000-7-25000=40000 acres, Ans.
384 + 1562 + 25 + 946 = 2917 ; 2917— rr23~l94
194-f-97=2; and 2 x 142 = 284, JLns.
5280^3 = 1760 steps, Ans.
2375 + 340 =2715 dollars, cost;
867 + (235x8) = 2747 " sold foi
2747 — 2715=32 dollars gain, Ans.
Ex.20. 4500 — 1350 = 3150 to gain;
800— 450 = 350 yearly savings;
3150^350 = 9 years, Ans.
Ex. 21. 1600 X 75 = 120000 ;
120000-^40 = 3000 bushels, Ans.
Ex. 22. 325 X 50 x 2 = 32500 dollars, Ans.
Ex.
15.
Ex.
16.
Ex.
17.
Ex.
18.
Ex.
19.
SIMPLE NUMBERS. 17
Ex. 23. 225 — 75 = 150;
150 X 52 = 7800 cents, Ans.
Ex. 24. 31383450^4050 = 7749, Ans.
Ex. 25. 31647000-^700 = 45210 dollars, Ans.
Ex. 26.- Reversing fourth operation, 100—40= 00;
Reversing third operation, 60 x 5 = 300 ;
Reversing second operation, 300-r-3 = 100;
Reversing first operation, 100 x 7 = 700, Ans,
Ex. 27. (54 X 17)4-27 = 34 cows, Ans.
Ex. 28. 56 — (2 X 2 6) = 4 dollars, Ans.
Ex. 29. 98 X 7 = 686 days, Ans.
Ex. 30. 5301212-M1137=476 dollars, Ans.
Ex. 31. 60 — 39 = 21 gallons, gained hourly ;
840^21 = 40 hours, Ans.
Ex. 32. 4500 X 24 = 108000, Ans.
Ex. 33. 1900—1492=408 years, Ans.
Ex.34. Maine, 31766;
New Hampshire, 9280 ;
Vermont, 10212;
Massachusetts, 7800 ;
Rhode Island, 1306 ;
Connecticut, 4674 ;
65038—47000 = 18038, Ans,
Ex. 36. 25000^8 = 3125 pounds in the thread ;
3125 + 235 = 3360 pounds, Ans.
Ex. 37. 8546 + 342 = 8888 ;
8888-^4 = 2222 dollars, Ans.
Ex. 38. 245x37 = 9065 ;
9065 + 230 = 9295, Ans.
Ex. 39. 6190048-^72084 = 72, Ans.
18 PRIME NUMBERS.
Ex. 40. 109 X 73 = 7957, greater number;
28 X 17^:476, difference;
7957-476 = 7481 less, Jns.
Ex 41. 360 — 114 = 246, greater;
246x114 = 28044, Ans.
Ex: 42. 2568754 — 2473248 = 95506, A7i8.
Ex. 43. Wheat, 35 x 2 = 70 dollars ;
Wood, 18x3 = 54 "
124 "
Cloth, 9x4 = 36 "
88 dollars, Ans.
Ex. 44. 684 — 375 = 309 yearly savings ;
309 X 5 = 1545 dollars, Ans.
Ex. 45. 58 + 10-1-5 + 28 + 3 = 104, cost of one barrel;
125 — 104 = 21 cents, Ans,
Ex. 46. 286000-6000 = 280000 ;
280000-^14 = 20000 dollars, Ans.
Ex.47. 256x25 = 6400; 6400—625=6775;
5775-v-35 = 165, Ans.
Ex. 48. 189-=-(4 + 5) = 21 hours, Ans.
PRIME NinVIBERS.
(91 5 page 68.)
Ex 2. Ans, 2, 2, 3, 5, 19. Ex. 3. Ans. 3, 3, 5, 5, 7, II'
Ex. 4. Ans, 11, 13, 17. Ex. 5. Ans, 19, 23, 29.
Ex. 6. Ans, 2, 3, 5, 7, 11 Ex. 7. Ans, 3, 3, 5, 7, 7.
Ex. 8. Ans, 11, 31, 41.
PRIME NUMBERS.
19
Ex. 2. 24= i
Ex. 4. 40 r= ^
(92, page 69.)
<
^^'^^ Ex.3. 12o=r
3x8
(5x25
( 5x6x6
4x6
2x3x4
2x2x6
2x2x2,>. 3
2x20
4x10
5x8
2x2x10
2x4x5
2x2x2x5
^2x36
3x24
4x18
6x12
8x9
2x2x18 -
2x3x12
12= <
2x4x9
3x4x6
2x6x6
3x3x8
2x2x2x9
2x2x3x6
2x3x3x4
*>x2x2x3xa
20
PROPERTIES OF NUMBERS.
CANCELLATION.
Ex.
3.
u
(95, pa
n
u
ige 72.)
Ex. 4.
1$
n 8
11
^0
14, Ans.
33, Ans.
Ex.
5.
1 $4
X$
H
7
£$ 16
.%$ 4
$
64
Ex. 6.
30
$
n
n
W0
13
t
3
13
9|, Jns.
41, -4ms.
Ex.
7.
8
8
00 3
9
X$ 3
81
Ex, 8.
tn
1$
u
200
?0 2
$0
2, Ana.
lOi, Ans.
Ex.
9.
n
00 2
Ex. 10.
4
$
t
i?2
*<
S,Ans.
^^2
4, u4»».
Ex.
11.
it
1 t.nns_ AnR
ka. 12
00
4
xn
6
. >, ww--^,
(
i firkms, A7if
El. 13.
GREATEST COMMON DIVISOR
^ Ex. 14. 3 U
t
21
20 suits, Ans.
3
115
00 2
230
76|days,-4w«.
GREATEST COMMON DIYISOR.
(98, page 74.)
Ex. 1, 3 X4=:12, ^?25.
Ex. 3. 4x6 = 24, Ans,
Ex. 5. 2x7=1 14, Ans.
Ex. 7. 7x10 = 70, ^?Z5.
Ex.9. 2x7x9 = 126,^715.
Ex. 12. 5x5 = 25, Ans,
Ex.2.
Ex. 4.
Ex. 6.
Ex.8.
2x3 = 6, Am.
3 X 6 = 18, Ans.
4 X 4 = 16, Am.
3 x5 x5 = 75. Am,
Ex. 10.4x8 = 32, ^/i5.
(OOj page 78.)
Ex. 8. To arrange a nnmber of things in equal parcels, the
parcel must be a divisor of the number ; and to arrange two
numbers in equal parcels, the parcel must be a common divisor
of the two numbers. Ans. 5 in a parcel.
Ex. 9. The lots, to be equal, must be a common divisor of
the three fields, and to be the greatest possible, must be the
greatest common divisor. Ans. 2 acres.
Ex. 10. To avoid mixing, the capacity of a bin must be a
common divisor of the two numbers of bushels ; and to have
the least number of bins, will require the yreatest common di-
visor of the two numbers of bushels. Ans. 21 bushels.
Ex. 11. Tlie greatest common divisor of 124, 116, and 92
rods, the three fronts. Ans. 4 rods.
22 PROPERTIES OF NUMBERS.
Ex 12. The greatest common divisor of the three lengths,
3013, 2231, and 2047 feet. Ans. 23 feet.
Ex. 13. The greatest common divisor of the three num-
bers of bushels is 2, which must be the capacity of the bag.
Now there are to be forwarded 2722-1-1822 + 1226 = 5770
bushels; and 5770-f-2zii:2885. Ans, 2885 bags.
Ex. 14. The greatest common divisor of $120, $240, and
$384, is $24, the price of the cows; and $120-t-$24=:5, A's
number; $240-^$24 = 10 B's number; and $384-r-$24 = 16,
C's number.
MULTIPLES.
(104, page 81.)
Ex. 2. 2 X 5 X 7 X 7=490, Ans.
Ex. 3. 2x2x2x3x7x 17 = 2856, Ans.
Ex. 4. 2x2x2x3 x 3 = 72, Ans.
Ex. 5. 2x2x2x3x5x7x11 = 9240, Am.
Ex.6. 2x3x3x5X5 = 450, ^7Z5.
Ex. 7. 2x2x3x3x5x 7 = 1260, Ans.
Ex. 4. 2, 3
2,5,7
(10^5 page 83.)
42 .. 60
7 .. 10
2x2x3x5x 7=420, Ans,
Ex. 5. 3, 7
2,5
21 .. 35 .. 42
5 .,
8x7x2 X 5 = 210, Ans,
Ex. 6.
Ex. 7.
Ex. S.
Ex. 9
Ex. )'>.
LEAST COMMON MULTIPLE.
28
2,5
3, 2, 5, 5
60 .. 100 .. 125
6 .. 10
25
2x5x3x2x5x 5==1500, Ans.
2,2,5
40 .
. 96 .. 105
2, 2, 2, 3
2 .
.24.. 21
1
1
2x2x5x2x2x2x3x '7 = 3360, An^
2,2,3
48 .
.60.
.12
2, 2, 3, 5
4 :
. 5 .
. 6
2x2x3x2x2x3x5 = 720, Ans.
2,2,3
84 .
. 224 .
. 300
7,5
7 .
. 56 .
. 25
2, 2, 2, 5
8 .
. 5
2x2x3x7x5x2x2x2x 5 = 16800, Ans.
3,3
270 .. 189
. 297 .
. 243
3, 3,7
30.. 21.
. 33 .
. 27
2,5,11,3
10 ..
11 .
. 3
3x3x3x3x7x2x5x11 x 3 = 187110, Ans.
Ex. 11.
2,3,5
5 .
.6 .
.7 .
. 8 .
.9
2, 2, 1, 3
,
. 7.
.4 .
.3
2x3x5x2x2x7x 3 = 2520, Ans.
Ex. 12. To purchase books at 5 dollars, or 3 dollars, or 4
dollars, or 6 dollars, the sum of money must be some common
mvliiple of 5, 3, 4 and 6 ; and the least sum will be the teasi
iommor, multiple^ which is 60 dollars, Ans,
E.c. 13. The least common multiple of 12, 15, and ly
barrels, whicb is 180 barrels, Ans.
Ex. 14. The least common multiple of the prices, $30,
$o5, and $105, which is $2310, Ans.
24 FRACTIONS.
Ex. 15. The least common multiple of 41, 63, and 64
sheep, which is 15498 sheep, Ans,
Ex. 16. He must spend in the purchase of each kind of
fowls a sum equal to the least common multiple of the prices
paid. Suppose he takes the cheaper turkeys ; the least com-
mon multiple of 12, 30, and 75 is 300; and 300^12=25,
number of chickens; 300-^301=10, number of ducks; 300
-i-75 = 4, number of turkeys; and 25 -flO + 4 = 39, the whole
number of fowls purchased. Next suppose he takes the tur-
keys at the higher price; the least common multiple of
12, 30, and 90 is 180; and 1 80-^-12 ==15, number of chickens ;
180-f-30 = 6, number of ducks; 180-^90 = 2, number of tur-
keys; and 154-6 + 2 = 23, whole number of fowls. But
39 — 23 = 16, number of fowls purchased more than was n^
cessary; and 16x5 = 80 cents, Ans.
FRACTIONS.
(ISO, page 88.)
Ex. 2. Ans, /j,
Ex. 4. Ans, ij.
Ex. 6. Ans, JJ.
Ex. 8. Ans. p^fy.
Ey 10 An<f — OJL^
Ex.11. Nine tenths; seven twelfths; five twentieths,
jwelve twenty-eights; fifteen seventy-fifths ; nine one hun
tf.red twelfths; forty-five two hundred twentieths; one hup-
dred twenty -five /oz^r hundred twenty-eighths.
Ex. 12. Ninety one hundredths ; three hundred twenty-
five one thousandths ; four hundred fifty one thousand tufo
Ex.
1.
Ans.
h
Ex.
Ex.
3.
5.
Ans,
A71S,
Ex.
Ex.
7.
9.
Ans,
Ans,
Ts 6
REDUCTION. 26
hundred fortieths • twenty-five one thousand Jiie hundredths ;
twelve two thousandths ; seven hundred twenty-six three
thousand four hundred seventy-fifths.
Ex. 13. Seventeen one hundred fourths ; one ten thousand
one hundred tenths ; nine hundred fifteen eighty-four thousand
six hundred twenty-firsts; thirty-eight thousand sixty-five
four million five hundred thirty-one thousand four hundred
twenty -ninths.
REDUCTION.
(126, page 90.)
Ex. 5. Ifl^i, Ans, Ex. 6. ^VtVt^/f^ ^ris.
Ex. Y. tVYt^^ ^^^- Ex. 8. tV¥8=/3, ^^*-
Ex. 12. fM=e. -^^^- Ex. 13. i||=Ai ^^^*
Ex. 14. \m=h ^^^^
(127, page 91.)
Ex. 4. A jx:^ 1532^ ^^5. Ex.6, ^p=5^i, Ans.
Ex.1. ^8^=41, ^715. Ex.9. 3^2L— 430i|, :247i«,
(128, page 92.)
Ex.4. 140=:2.2.AA ^^5. Ex. 6. 94=8.fi ^715.
Ex. 7. 180:=-i^^|^ Ans, Ex. 9. 247—^^31, ^w^.
(129.)
Ex. 3. 7lf rrri-99, ^715. Ex. 5. l^j^^^h^^ ■^^^'
Ex.9. 96yVo==HM% ^^«- Ex. 11. 400||=-L4_8pL,^««.
^,^:
FRACTIONS.
(130, page 93.)
Ex.2.
15^5
= 3;
Ex. 3.
l=A» ^^^.
Ex.4.
51~1Y=3;
if =jf, Ans.
Ex. 5.
Ex. 6.
3488—436=8;
Ex.1.
(ISSj
page 95.)
Ex. 2.
5,5
2,3
50 . . 75
2 .. 3
35-4-7 = 5;
f =11, Ans.
150-v-30=5;
3 ISO* -"'*'•»•
1000—125=8;
T2s~ToVir« Ans,
Ex.3.
Ex 5.
Ex. 7.
Ex. 8.
Ex. 9.
Ex. 10.
Ex. 11.
Ex. 12.
5 X 5 X 2 X 3 = 150, least common denominator.
^_ Ji 47 4 — 12 45 141 _1_ A^i*
2,2,2
2,3,7
16 .. 21
21
2x2x2x2x3x 7=336, least com. denom.
± 4 _3_ JL— 15.1 1.3.3. _6_3_ _3_2_ Ans
2»T) 16) 21 — 336) 336» 336) 336> -^'*^«
Ex. 4. 3, 3, 2
7,2
9 ..21 ..4
3x3x2x7x 2 = 252, least com. denom.
a. _1_ 3 3. — _5JB_ Jl_2_ J.19. _1_5 1.5. J770
¥) 21) 4) 1 — 252) 252) 252) 252 » -^'*''»»
?JL SL JJ.— AA j_a. j_i. Ans
T)4) 8 — 8) 8) 8) ■^'t'^'
3 1 2 i_7 _5_ — 2.3. A _aJ_ JL8_ ±l.ft JLO yl-w*
4) 8') T) 6 ) 14 — T6 8) 16 8) 16 T) 16 8) 168) ^^^»
4 7 115. 1 — 3i. 31 1_6_5 4JL5 35. y|^o
5) TJ) 3) 1) 9—45) 4 5) 45 ) 45 ) iJ) ■^'*^«
2 1 J 3. 1 11 6. — 4.2. i_o_4 45. 4 A Vie Ans
2" 8") T) 8) 14)1 56) 56)56»56) 56) -^'*'^«
2 1 31 A 5. 11 5 32.1 111 A 8.0 2Cr 4 4_ 7J ^yij
10)40) 1)3)30)8 120) 120) 120)1^^*120) I t^l ^"''
A 2 1 _7_-
9) 3) ¥) 12-
7 5 1_1 1-
fi) 7) 4)2"
-1^ 2 A _ft_
' 36) 36) 36)36
"56) 56) 16 ) 5
, Ans.
ADDITION. 27
iliX. Id. ^f, fo, 3, -'j— fQ, l^o, 3^, f 0", ^7i6.
Fv Ifi 7 2 5 _$ 13 3 7 J2_5.18 140l2iI0 AnS^
UiX. ID. 2 0> 4 > 10 » 1> S> 2 — 2 0> 20 » 2 0") "2 0-? 2 0» 2 0» ^'^*
ADDITION.
(133, page 96.)
7 + 3+1+54-9 25 ^5 ^1 ^
Ex.2. --±-^ =-=2-=2-,^..
4 + 5 + 7 + 1 + 3 + 11 31 7
^''•^' 12 -12^^2' "^'''•
^ , 7 + 9 + 2 + 13 + 16 + 21 68 ^18 ^
Ex.4. ^^ =.-=2-,Ans.
41+63 + 71+89 + 109 373 13 ,
Ex. 5. = =:3 , Ans,
120 120 120'
^ ^ 13 + 76 + 140 + 181+223 683 61 ^
Ex.6. ■ ■ ■ ■ = — =2 — , Ans.
225 225 75'
(134, page 97.)
„ „ 3 2 27 + 8 35 .
Ex. 2. T + - = ■ — = — » -4rwr.
4 9 36 36'
^ „ 4 11 56+55 111 41 .
^,3125
^^•'^ 4 + 8 + 7 + 12=
126+214-48+70 _265_, 97
168 -168-^168' "***'*
18 FRACTIONS.
14 . 9 2
1274 + 2835 + 390 4499 404 ,
= ~1- , Ans,
4095 4095 4095*
42 9 Y 1
^'- li0+70 + 28+n=
42 + 18 + 35 + 10 105 3 .
--, Ans
140 140 4'
^ ^ bl 131 24 1 2
^^•'' Y5 + 160 +2^ + 2 + 3 =
102 + 131 + 144 + 75 + 100 _ 17
150 "" 25'
„« 3124.56 78 9
'^•«- 4 + 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10 =
1^90 + 12 60 + 1680 + 2016 + 2100 + 2160 + 2205 + 2240 + 2268
2520
17819 ^179 ,
2520 2520'
T. ,^ 4 9 2 19 ^19
^^•^^- 5-^10 + 3 + 20=^60
14 + 3+ 1 = 18
21U, Ans.
Ex.11. f + T\ + f= 2/j
1 + 10 + 5 = 16
18^, -4/i«.
Ex. 12. f + A + ,-V = lA
17 + 18 + 26 = 61
62/^, Ans.
Ex. 13. /8+H + i + H = HJ
1 + 3 = 4
5^J, -4««.
SUBTRACTION. 2^
Ex. 14. ;}+ t\+ 1 = l^T
125 + 327 + 25— 477
4782V, -^^*-
Ex.15. in + U+H + ii-^\n=^Hih^'^'
Ex. 16. /o+H+T + To= 4
3 + 2 + 40 + 10^:55
55f , ^ws.
Ex. 17. f + f + 1 = 2/:,
125 + 96+48=:269_
27I2V yards, Arts.
Ex. 18. | + ^ + J=zlJ^
5 + 3 — 8
9jV yards, Ans,
Ex. 19. ^V + i^ + 1 + 3 + 2 ^ 311
26+40 + 51+59 + 62 = 238
241^f acres, Ans,
Ex. 20. | + 4+±i + _7_ := 23«,
175 + 325 + 270 + 437 = 1207
35
1209/5 bushels.
205 + 296 + 200 + 156 = 857
$859||, dollare.
SUBTRACTION.
(135, page 99.)
Ex. 2. —-—=-=-, Ans,
9 93'
14-11 3 1 .
Ex.3. -^-=^^=-,Ans.
80 FRACTIONS.
20 — 6 14 ,
Ex.5. ^±Z?lJ^^,Ans.
^ ^ YS-ll 64 1 .
Ex. 6. -— ^^~ ^7i^.
128 128 2'
»
^ ^ 182-110 72 6 ^
Ex. 7. = — =r — , Ans.
348 348 29'
(I365 page 99.)
Ex. 2. -— -=-^^= — , Ans,
2 9 18 18'
15 2 75-48 27 9
^'''^' 24-5=~12Cr=T20 = 40'"^'**-
^,34 51 — 32 19 .
Ex. 4. -:=z = — , Ans,
8 17 136 136'
^ ^ 84 4 49-8 41 .
Ex. 5. ■ = -i: — =ir-, Ans,
120 35 70 70'
1500 50 125 — 100 25 .
^^•'- 1728-7^==— 4-4-=U4'^"^-
^ ^ 60 332 720 — 83 637 ,
Ex. 7. — -= = , Ans,
89 4272 1068 1068'
Ex. 9. 8i = 8y\ Ex. 10. 25| = 25^^
4|f, ^?Z5. 163^ = 16^3,^/1*.
Ex. 11. 4f =4|| Ex. 12. 6
3 jj, ^725. 4-f, ^715.
Ex. 13. 450J =450^1 Ex. 14. ^^J = Z^^
i20Jv=i2oii tVs= m
330jf, Ans, 33Y5, ^n».
MULTIPLICATION. 81
Ex. 15. 751 Ex. 16. 227f
49 1961=1961
26i; Ans. 30f , Ans.
Ex. 19. $Y|-~$6i=$lf^2, Arts.
Ex. 20. 4 + 31 =4V\ Ex. 21. 6i + 2i+f= 9^^
5i-4i =- H $255-$92«o =$16Ht
4_3_ - |i = 3|f , ^n^. -^^^*"
Ex. 22. n-2f =4|f, ^7^5. Ex. 23. ^f-H=^» ^^^«-
Ex. 24. 9121 + 5451 =145^^
|2000-$145YyV=$542^^, Ans.
Ex. 25. $136yV + $350|=$487|i cost.
$184i + $416J = $6011 receipts.
$601i-$487ii=$llV8» ^^^•
MULTIPLICATION.
(I375 page 101.)
Ex. 4. tV X '^=iT=lTf» ^^^•
Ex. 5. 1^4 X 12 = VV='7f» ^^^•
Ex. 6. /t X 63=5 X 3 = 15, ^W5.
Ex.8. '7|xl2 = *|«=91i, ^^5.
Ex. 9. tVt X 8=f It^SyVr, ^^^•
Ex.10, ^l^x51=f = 2,^r^5.
Ex.11. 15fx 16 = 125 X 2=250, ^w«.
Ex.12. mx22 = 'i'=lH,Ans.
Ex.13. $8Axl2=$Hf'=^106i, ^n#.
Ex. 14. |iix9 = $H=^S-I' ^''^'
Ex. 15. $J X 2'7=$H^=$23f, ^w«.
32
Ex. 2.
FRACTIONS.
(138, page 103.)
Ex. 3.
14
100
9
450
Ex. 4.
21
li, Ans.
105
17
Ex. 5.
47
85, Ans,
47
64f, Ans,
19
13
247
Ex. 7.
42
39
Ex. 8.
16
5if, Ans.
80
233
819
'1165, Ans.
409^, Ans.
Ex. 9.
39
156
27
Ex. 10. $8xf = 6| dollars, ^/MT.
Ex. 11, 36 X 10|=384 miles. Ana.
108, Ans.
Ex. 12. $450 X ^2 =$262|, Ans.
Ex. 13. $16 X 2J=$44^, Ans.
(139, page 104.)
I Ex. 3. 8
Ex. 2.
10
Ex. 4.
24
65
10
^, Ans.
11
36
Ex.
5. 6
7
y\, Ans.
21
6
18
3|, Ans.
MULTIPLICATION.
33
Ex.
6.
10
9
1
2
9
5
4
1
28
1
aV, ^ns.
E^. 9
Ex. 11.
Ex. 13.
15
8
4
9
5
1
3
22
25
44
lif, Arts.
3
2
1
n
4
5
4
4
13
35
78
2A, -4^5.
8
7
2
1
9
11
2
3
1
8
12
77
6y5_ Arts.
Ex. 15 $ixf=$i u4w5.
Ex.17. ix^=^J Ans,
:. 7. 6
11
6
3
2
3
16
15
176
1111, Ans.
Ex. 10.
7
2
1
16
10
7
3
80
3
256
851, Am
Ex. 12.
2
6
4
3
5
4
3
4
2, Ans.
2
25
2
11
4
27
16
7425
Ex. 14.
464i-V» -^^**
Ex. 16. 4x|=:J^, ^W5.
Ex.18. V x$|=$ff, ^'/wj
84
Ex.20.
4
2
51
17
8
867
FRACTIONS.
Ex. 21.
Ans., 108f.
8
6
51
14
20
357
Ans,^ 11^1 dollai*.
Ex. 22.
Ex. 23.
Ex. 24.
Ex. 25.
Ex.
Ex.
Ex.
26.
27.
28.
f xf x$VV=$21y\-, Ans.
V x$V-=^22iJ, Ans.
f X V- X I X P^^-=$25^j, Ans.
Xx|=i4, Ans.
$A.^xi=$25-^\,Ans.
■^^ acres x | x 1=49-,-^ acres, Ans^
f-f X f barrels =6| barrels, An>s.
DIVISION.
(I4O5 page 107.).
Ex. 6. j\^j---25z=^.l-^,Ans.
Ex. 9. $|-r-6=$i, Ans. Ex. 10. J-v-7=|-, ^7i5.
Ex. 11. -f -v-5 = 3«j, -47^5. Ex. 12. $^-^-9=$^, Ans.
Ex. 14. V-v-3 = 5yV ^^5- Ex. 15. ;2 x Y-^^ -|,^w*,
Ex. 16. $2.±|iL_^4 = $\2_4._$24||, ^7^s.
(141 5 page 109.)
Ex, 7. I X 9-^1=15, ^715.
Ex. 8. 121-x-^— $49, u4w5.
Ex. 9. 16xf = 10; 10-^|-=:22i, .^WA.
Ex. 11. 75-i-Y-=^H» •^^^•
Ex. 12. 149-^J4i=6yVj, ^w«-
Ex. 18. 15^-f =9, Ans.
Ill VISION.
B6
Ev 14. f X 320=200 ; f x Y=¥ ;
200— V =254, Ans.
Ex.15. $32x1=8; Yxi=f;
Ex. 16. 183~J^i=4, Ans.
Ex. 2.
Ex. 4.
Ex. 6.
Ex. 8.
Ex. 10.
(143,
page 110.]
)
8
7
Ex.3.
9
6
3
4
1
6
6
r"
3
10
1 1^, Ans.
31, Ans.
1
4
Ex. 5.
2
1
9
10
7
14
13
63
40
13
ih ^^«-
\h -4^*'
3
2
Ex.7.
6
5
27
28
4
5
81
56
•
24
25
ih ^ns.
l^V, ^^*'
3
5
Ex. 9.
19
17
8
7
7
19
9
35
7
17
3f , Ans.
2f , ^»«.
20
13
Ex. 11.
7
2
5
16
2
6
25
52
1
3
2
4
2^, ^»5.
21
40
l^f, Ans.
FRACTIOOTL
Ex. 12.
10
9
4
5
6
13
4
325
432
6f 56 3 28 .
Ex. 15. -^=— - X TTTT^rrr, Arts.
81 9 26 39'
Ex. 16.
Ex. 17.
Ex. 18.
Ex. 19.
"'a
80 1
— ^=— -x-=20, Ans.
4 Y 4 '
^5 5 5 25 .
ii= — X — = — , Ans*
4J 11 22 242'
_-x-x--l,^n.,
2 5 9 2 1.
— - X - X - X -=-, Ans.
|xf5629 3'
2 X A
Ex. 20. V x|=14, Ans.
Ex. 21. 3_3 X f = Y=6|, ^M5.
Ex. 22. 8 35 • Ex. 23.
8
35
1
2
6
6
2
21
3
14
98
6
35
'uT^Ans.
Ex.24. $J43.xixf=$14f ;
$17— $141=$ 2^, Ans.
Ex. 25
10
2
20
$/^, Ans.
Ex. 26.
3
10
PROMISCUOUS EXAMPLES.
Ex. 27. 1
87
10
2
3
2 bu., Arts.
10
1905
8
127
12j-V, Ans
PROMISCUOUS EXAMPLES,
(Page 112.)
Ex.2. 91^7 = 13; 4^; 1 3=f^, ^w*.
Ex. 3. 3, 40 I 3 :: 40
3x40 = 120, Ans,
Ex. 4. 4 + 3 =7
i + J + f of f = 2|f|_
Ex.6. |xn=fi=4J,
li|}, Ans.
Ex. 6. 47561- + 1281=4885 3V, ^w«.
Ex.7, f xix|xY-=H=VTr
Yxfxfx |=H=3li_
31J, ^ris.
Ex. 8. f X f =20, ^W5. ; f x f = V =1t» ^^*-
Ex. 9. 18251=1^1-^; iAjiLixf=^4|i^=3043i, .4rw.
Ex. 10. i+i=/o ; 1-2V=H; ^V-i^ = 140, ^715.
Ex. 11. ^ X $ V X V =^24^1, ^ws.
Ex. 12. $V X |-=$23i, ^715.
88
Ex 13.
Ex. 15.
Ex. 16.
Ex. 17.
Ex. 18.
Ex. 19.
Ex. 20.
Ex. 21.
Ex. 22.
Ex. 23.
Ex. 24.
Ex. 25.
Ex. 26.
Ex. 27.
FRACTIONS.
8
14701
14701 Ex. 14. 8
2 471
37803
2
4
1
628
$12601, Ans,
$1, Ans.
P-Y^ X
^ Xf =$40551, ^ws.
1 42 10 2,22,,
^*1 (
y x|xf=27, Ans.
V X V X l=¥/=24-i-V, ^ns.
V xf x|=34i, ^715.
lx\^=^=1j\,Ans.
3_3 X _3_ xifi=iy-a=589|, ^W5,
H^ X yaVo X ^V^=2500, Ans.
$204- f =$50, ^715.
4 5 — 20J-'- 20 — 20? ^^'20 — ^^) -n/«^,
$i7^j.Xf=$4608, Ans.
320 X $21=$ 720 755x|X$lf= 528^
435x$lJ=:$ 815f
755x|x$2i-:
755 $1535|
$15351— $1491^
962f
$14911
l:^^, Ans.
^ ^^ 14-5 12 12 7 5 ,
£x. 28. - -= — : = — , Ans.
8 + 5 13' 13 8 104'
_ ^^ 8 + 5 13 8 13 6 ^
Ex. 29. J- ,=— -; 71— — =ri, Ans.
7 + 5 12' 7 12 84'
Ex. 30. f X f X f X |=7-J, Ans.
Ex. 31. J X $V X f X ^= V =$3J, Ans.
DIVISION. 39
Ex 32. 16f-3J = 12l=i?/
H^x'V X 2%=^ W^ =95311, Ans.
Ex.33. -Vxy = 'LV = 12ff, -4w5.
Ex. 34. 6-^-1846=^1^, Ans.
Ex 35. $V X f X 3 J =$3, ^W5.
Ex 36. $1-6 X f X i X J X f-=$5, ^W5.
Ex SY 5 4-J'- — 17. 1 11 — _3_- _3 1—1
20-^^V = 800, ^715.
Ex. 38. Jf cents xy xf = 100 cents, Ans.
Ex. 39. i+f =H ; 1- H = 2T, remainder.
^f i X 2^ X $2.1^=^1 i-^=$45YYW, ^^^•
Ex. 40. If the horse cost 1\ times as much as the wagon,
ihe horse and wagon must cost 2 J times the wagon. Hence,
$270-T-2i=$120, Ans.
Ex.41. Y^xf=32; 32-20f = lli, ^W5.
Ex. 42. P^-i^ X aV X $1=126, Ans,
Ex. 43. If A can do f as much as B, he can do the work
in 4. of the time that B will require, and in 1+|=|- of the
time be Ah will require. Hence
14 days x |=32| days, A's time ; )
32| days x J=24i days, B's time ; ) ^*
Ex. > 4. -V^ X } X f =11 J, Ans.
Ex. 45. A, B, and C can do J of the work in a day ;
B and C can do \ of the work in a day ; hence
A alone can do \--\z=:^^ of it in a day ; and
be wiP therefore require ^^=:1Z\ days, Ans.
fix. 46. 1 + 1+1=/^; 1-^^=J^, remainder;
tV-tV=3V; $24~3V=$720, Ans.
Ex 11 -V^ X aV X V =Hh ^^5-
10
DECIMALS.
Ex. 48.
Ex. 49.
Ex. 50.
Ex. 51.
Ex.52.
i-i=j\ ; 30 feet-^y3__ioo feet, Ans,
i+ i =T2> fraction of the post below water,
1-t'2=tV " " " (^^ove "
21-h/^=36 feet, Ans.
^= eldest son's fraction;
^x^=l^= youngest son's fraction;
1 — (-? + If) == Jf = daughter's fraction ;
n-if = 4V; ll'723f--/^=$21}U;f ^Twr.
DECIMAL FRACTIONS.
(145, page 118.)
Ex. 1.
Ans, .38.
Ex. 2. Ans (.
Ex. 3.
Ans. .325.
Ex. 4. Ans 04.
Ex. 5.
Ans. .016.
Ex. 6. Ans. .00074.
Ex. 7.
Ans. .000745.
Ex. 8. Ans. .4232.
Ex.9.
Ans. .500000.
Ex. 10
. Five hundredths;
twenty-four hundredths; six
hundred seventy-two thousandths; six hundred eighty-one
thousandths ; twenty-four thousandths; eight thousand four and
seventy-one ten-thousandths; nine thousand thirty-four ten-
thousandths; five ten-thousands ; one hundred thousand two
hundred forty-eight millionths ; nineteen thousand two hun-
dred forty-eight hundred- thousandths ; opc thou&dnd three
hundred eighty-five millionths ; one million eighty-pere^ t^D-
millionths.
NOTATION AND NUMERATION. 41
(I465 page 118.)
Ex. 1. Ans,
18.027.
Ex. 2. Ans. 400.0000019.
Ex. 3. Ans.
54.000054.
Ex. 4. Ans. 81.0001.
Ex. 5. Ans»
100.0067.
Ex. 6. Eighteen, and twenty-seven thousandths; eighty*
one, and one ten-thousandth ; seventy-five, and seventy-fiv<,
thousandths ; one hundred, and sixty-seven ten-thousandths ;
fifty-four, and fifty-four millionths ; nine, and two thousand
eight hundred six ten-thousandths ; four hundred, and nine-
teen ten-millionths ; three, and three hundredths ; forty, and
forty thousand four hundred four hundred-thousandths.
(148, page 120.)
Ex. 1. Ans. ,000^25, Ex. 2. Ans. .6000.
Ex. 3. Ans. .01859. Ex. 4. Ans. .000260008.
Ex. 5. Six thousand three hundred twenty-one ten thous-
anths ; five million four hundred thousand twenty-seven ten-
millionths ; seven hundred forty-eight thousand two hundred
forty-three millionths ; sixty million hundred-millionths ; two
million nine hundred sixty-two thousand nine hundred ninety-
nine ten-millionths ; six hundred-millionths.
Ex. 6. Ans. 502.001006. Ex. 7. Ans. 31.0000002
Ex. 8. Ans. 11000.00011.
Ex. 9. Ans. 9000000.000000009.
Ex. 10. Ans. 10.2. Ex. 11. Ans. 124.315.
Ex. 12. Ans. .700. Ex. 13. Ans. .00007.
Ex. 14. Twelve, and thirty-six hundredths ; one Lundred
forty-two, and eight hundred forty-seven thousanths ; one, and
two hundredths ; nine, and fifty-two thousandths ; thirty-two,
and four thousandths ; four, and five ten-thousandths ; six-
ty-two and nine thousand nine hundred ninety-nine ten-
thousandths ; one thousand eight hundred fifty-eight, and four
^2 DECIMALS.
thousand five huadred eighty-three ten-thousandths ; twenty
seven, and forty-five hundred-thousandths.
REDUCTION.
(149,
page 121
Ex. 2. .I'ZOOOOO
Ex.3.
.'TOOOOO
24.6000000
.024000
.0003000
.000187
84.0000000
.000500
721.80002'71
108.450000
Ex. 4. 1000.001000
841.'780000
2.600400
90.000009
6000.000000
(150, page 122.)
Ex. 2. TV¥o=i» ^^«- Ex. 3. tVo = 2V» ^^«-
Ex.4. jWo=m^^^^' Ex.5. y»JV¥o =H» -4^-
Ex.6. j^i^^=^^\^^ Arts.
(151, page 123.)
Ex. 4. Arts. .4. Ex. 6. Ans. .875.
Ex. 9. Ans. .375. Ex. 10. Ans. .0375.
ADDITION.
(152, page 124.)
Ex, 6. 26.26 Ex. 7. 36.015
.7 300.0605
6.083 5.000003
4.004 60.0000087
87 047, Ans. 401.0755117, Ans,
ADDITION.
43
Ex. 8.
64.34
1.0009
3.000207
.023
8.9
4.135
71.399107; Ans.
Ex.
9. 18.375
41.625
35.5
95.500, Ans.
Ex. 10
61.843
143.75
218.4375
21.9
Ex. 11.
445.9305, Ans.
Ex. 12.
^=
2.5
5f =
5.75
H=
3.625
3.0642
8.925
23.8642 barrels, Ans.
12J =12.75
18f =18.4
9=9
241 =24.125
4|f= 4.8125
8yV= 8.9
15iJ=15.65
93.6375, Ans,
Broadcloth.
Ex. 13. First suit, 2.125
Second " 2.25
Third " 5.0625
Sums
Total
Cassimere.
Satin.
3.0625
.875
2.875
1.000
1.125
9.4375 5.9375 3.000
9.4375 + 5.9375+3 = 18.375, Ans,
44
DECIMALS.
SUBTRACTION.
(I535 Pag^ 126.)
Ex.
4.
6.
8.
10.
12.
714.000
.916
Ex. 5.
Ex. 7.
Ex. 9.
Ex. 11.
2.000
.298
Ex.
713.084, Ans.
21.004
.75
1.702, Ans.
10.0302
.0002
Ex.
20.254, Ans,
900.
.009
10.03, Ans
2000.
.002
Ex.
899.991, Ans.
1.
.000001
1999.998, Ans.
.427
.000427
Ex.
.999999, Ans.
.34
.034
.426573, Ans.
.306, Ans.
MULTIPLICATION.
\ (1*3^4:5 page 127.)
Ex. 4. 274.855, Ans. Ex. 8. 243.6, Ans.
Ex. 12. .000030624, Ans.
NOTATION AND NUMBKATION. 46
\
DIVISION.
(15>5, page 129.)
Ex. 5. .111, Ans. Ex. 6. 11.1, Ans.
Ex. 8. 15,21 -\-, Ans. Ex. 9. 1; 10; 100; 1000, ^ws,
Ex.10. 5,68Ui-,Ans. Ex.12. 3020,^^5.
Ex. 17. 3.65, Ans.
PROMISCUOUS EXAMPLES.
(Page 130.)
Ex. 2. 6188.311478, Ans. Ex. 3. 86.913, Ans.
Ex. 6. .00012, Ans. Ex. 9. 4, Ans.
Ex. 11. 70.6755-T-6.35r^ll.l3, Ans.
Ex. 12. tWo -f, ^ns. Ex. 13. 26yVVo=26i, Am,
^■'''^^=-''^^-
^ ,^ .25x17.5 ^ .
Ex. 16. —-—=5, Ans.
.5x1.75
Ex. 17. 3.625 X 36.75 x $.85=$113.2359375, Ans.
Ex.18. 56.925-^4.6 = 12.375 = 12f, ^n«.
DECIMAL CURRENCY.
NOTATION AND NUMERATION.
(I6O5 page 132.)
Ex. 2. Ans. $2.09. Ex. 3. Ans. $10.10.
Ex. 6. Ans. $.032. Ex. 7. Ans. $100,011
46 DECIMAL CURRENCY.
Ex. 8. Seven dollars ninety-three cents/; eight dollars two
cents ; six dollars fifty-four cents two mills.
Ex. 9. Five dollars twenty-seven cents two mills ; ono
hundred dollars two cents five mills; seventeen dollars five
mills.
Ex. 10. Sixteen dollars twenty cents five mills ; two hun-
dred fifteen dollars eight cents one mill ; one thousand dol-
hrs one cent one mill ; four dollars two mills.
REDUCTION.
(161, page 133.)
Ex. 2. Ans. 3600 cents. Ex. 3. Ans. 524800 cents.
Ex. 6. Ans. 160 mills. Ex. 1. Ans. 3008 mills.
Ex. 8. Ans. 890 mills.
(162, page 134.)
Ex. 2. Ans. $15.04. Ex. 3. Ans. $138.75.
Ex. 4. Ans. $16,525. Ex. 5. Ans. 52.4 cents.
Ex. 6. Ans. $6,524.
ADDITION.
(163,
page 134
.)
Ex. 2. $ 50.07
Ex.
3.
$ 364.541
1000.75
486.06
60.003
93.009
.184
1742.80
1.01
Ans.
3.276
25.458
$2689.686, -
$1137.475,
ADDITION.
Ex. 1 $ .92 Ex. 5. $89.74
.104 13.03
.357 6.375
.186 19.625
.125
.99
$ai2e Ans.
Ex. 6. $ 9.17 Ex. 7. $2175.75
.875 240.375
.0625 605.40
.04 140.125
.08
.11
$3161.65, Ans.
$10.3375, Ans.
Ex. 8. $ 6.08 Ex. 9. $7425.50
26.625 253.96
16.000 170.09
7.40
156.105, Ans.
$7849.55, Ans.
Ex. 10. $3,625
1.75
1.375
.625
.875
$8.25, Ans.
4^ DECIMAL CURRENCY.
SUBTRACTION.
(164
5 page 136.)
Ex. 2. $365,005
Ex. 3. $50.
267.018
.60
$97,987, Ans.
$49.50, Ans.
Ex. 4. $100.
Ex. 5. $1000.
.001
.037
$ 99.999, Ans.
$ 999.963, Ans.
Ex. 6. $1834.16
Ex. 7. $145.27
1575.24
37.69
$ 258.92, Ans.
$107.58, Ans.
Ex. 8. $6.84
Ex. 9. $14725
5.625 $3560-|-$'3^015.875=:10575.876
$1,215, Ans.
$4149.125, Ans
Ex. 10. $13.75
5.25
1.375
.875
$25-$21.25=:
$3.75, Ans.
Ex. 11. $480
80.50
$560.50-
-$200=$360.50, Ans.
MULTIPLICATIOK
(16^5 page 137.)
Ex. 2. $4,275 X 300=:$1282.50, Ans.
Ex. 3. $2.45 X 176=:$428.75, Ans.
DIVISION. 49
Ex. 4. 11.28 X 800:=$1024, Ans.
Ex.5. $.15 x3Y2=:$55.80
.125x434= 54.25
.33 X 16= 5.28
$115.33, Ans.
Ex. 6. $.56 X 3=:$1.68
.07x15= 1.05
.08x27= 2.16
$5--$4.89 = $.ll, Ans.
Ex. 7. $.375 X 125 =$46,875
.09 X 75 = $6.75
.60 X 12= 7.20 = 13.95
$32,925, Ans.
Ex.8. $32.50 X 80 r
34.25x70= 2397.50
$4997.50
3975
$1022.50, Ans.
DIVISION.
(166, page 138.)
Ex. 2. $41.25-^33=$1.25, Ans.
Ex. 3. $94.50-^27=$3.50, ^715.
Ex. 4. $136-h64 = $2.125, Ans.
Ex. 5. $1.32-^$.12 = 11, Ans.
Ex. 6. $405-f-$.54 = 750, Ans.
Ex. 7. $180-^12=$15, Ans.
Ex. 8. $2847.50 -M00=$28.475, -4715.
Ex 9. $80.46-j-894=$.09. Arts.
Ex. 10. $1,125 x 120=$135 ; $135-4-27=$5, Ans.
K.P, 3
so DECIMAL CUiaiENCY.
Ex. 11. $3.20 X 4=:$12.80
.08x37= 2.96
$15.76
6.80
$8.96-^$.16=56, Ans.
Ex. 12. $4.50 4- $2.75 =$7.25 ;
$166.75-^$7.25 = 23, Ans,
Ex. 13. $18.48-M54=:$.12, Ans.
Ex. 14. $560
106.75
$453.25-M4=$32.37i, Ans.
ADDITIONAL APPLICATIONa
(I685 page 139.)
Ex.2. 693x$i=$321, ^7i6\ Ex.3. 478x^=$2S9, Ans.
Ex. 4. 4266 X $yV=$355.50, Ans,
Ex. 5. 1250 X $i=$156.25, Ans,
Ex. 6. 3126 X $yV=$195.375, Ans.
Ex. 7. 1935 X $i=$322.50, Ans.
Ex. 8. 56480 x $^=$7060, Ans.
Ex. 9. 1275 X $i=$255, Ans,
(169, page 140.)
Ex. 2. $.09 X 864=177.76, Ans.
Ex. 3. $1.25 X 87=$108.75, Ans.
Ex. 4. $1.45 X 400 = 1580, Ans,
Ex. 5. $.44 X 52 X 16=$366.08, Ans.
(170, page 141.)
Ex. 2. $l75-4-25=$7, Ans,
Ex. 3. $200-^48=$4.16|, ^/^^.
ADDITIONAL APPLICATIONS. 51
Ex. 4. $1200-f-96 = $12.50, Ans.
Ex. 5. $56.25-f-10 = $5.62J, Arts.
Ex. 6. m.^O-MS^r^-GS, Ans.
Ex. V. llO.O'Z-f-SS^S.lQ, Ans.
Ex. 8. $1016-T-800 = $1.2Y, Ans.
Ex. 9. $8'74.65-^343=$2.55, Ans.
Ex. 10. $684.3'75-^36o=:$1.8'75, Ans.
(in, page 142.)
Ex. 2. $5.55-f-$.15=37, ^lw5.
Ex. 3. $2 16 ^$12 — 18, Ans.
Ex. 4. $21Y8.'75-^$1.25 = 1'743, ^W5.
Ex. 5. $643.50^$19.5=ir33, ^?^5.
Ex. 6. $52.65-f-$.45 = ll'7, ^?i5.
Ex. Y. $6336-f-$132=:48, ^n5. ^
Ex. 8. $117715-^165 = 1811, ^w«.
(ITS, page 143.)
Ex. 2. $4.50 X 42.65 = $191.925, Ans.
Ex. 3. $.85 X 24.89 = $21.156.+, Ans.
Ex. 4. $17.25 X 7.842=$135.274+, ^w«.
Ex. 5. $12.50 X 23.48 =$293.50, Ans.
Ex. 6. $3 X 1.728=$5.184, Aiis.
Ex. 7. $7 X 2.40 =$16.80
6.40 X .865= 4.671
.80X12.56 = 10.048
$31,519, Alls.
Ex. 8. $4,375 X 14.76 =$64,575, Ans.
(173, page 144.)
Ex.2. $7-4-2 = $3.50;
$3.50 X 1.495=$5.2826, Ans.
52 DECIMAL CURRENCY.
Ex. 3. $S.75~-2=z$4.Sl5;
$4,375 X .325 = $1.421 -f , Ans.
Ex. i. $3.84-^2=$1.92;
$1.92 X 3.142=$6.032 +, An8.
Ex. 5. $5.60-^2=$2.80;
$2.80 X 1.848 = $5.1744, Ans.
Ex. e $18-^2=$9;
$9 X 125 X .148=r:$33.30, Ans.
Ex. 7 $3.054-2r:r$1.525;
$1,525 X 31.640=$48.251, Ana,
(174.5 page 145.)
Ex. 1. $3.60 X '7=r$25.20
1.125 X 9= 10.125
•.90 xl2= 10.80
1.375x24= 33.00
.65 x32=: 20.80
Ex. 2.
Ex. 3.
$99,925,
Ans.
$3.75
X 67 =
:$251.25
2.62
xl08 =
: 282.96
1.12
X 75 =
: 84.00
.86
X 27 =
: 23.22
.70
X 35 =
: 24.50
1.04
X 50 =
: 52.00
$717.93,
Am.
$.07
X325:
=$22.75
.0625x148:
= 9.25
.05
X286:
= 14.30
.125
X 95 =
= 11.875
2.75
X 50:
= 1-37.60
3.625
X 75:
= 271.875
2.85
X 12r
= 34.20
$501.75, Ans.
PROMISCUOUS EXxiMPLES. ti
Ex. 4. $15 X 20 =8300.
9.50 X 7.5 = 71.25
C.25 X 10.75 = C7.1875
2.625 X 3.96 = 10.395
3.00 X 5.287 = 15.801
$464.6935, Atu
Ex.6. $.11 X 25=12.75
.625 X 5= 3.125
.0625x26=: 1.625
.42
X 4= 1.68
.09
x46= 4.14
.14
x30= 4.20
.04
X 6= .24
.12
X 4= .48
$18.24, Ans.
PEOMISCUOUS EXAMPLES.
(Page 146.)
Ex. 1 $124.35 X 62.75=$7802.9625, Am.
Ex. 2. $.17x15=82.55, ^?i5.
Ex. 3. $1406.25-v-2250=$f, Arts.
Ex. 4. $48.96-M2=$4.08, Ans. ^
Ex. 5. 325 miles x .45 = 146.25 miles, Ans,
Ex. 6. 657-^36.5 = 18,^725.
Ex. 7 $105+$125 + ($35x4)=$370
$400 — $370 =$30, Ans.
Ex. 8. $19— $15 = $4; $4x28=$112, ^?i«.
Ex. 9. ■V-X2\ = ¥-='^A •^^^•
Ex. 10. $9-^$.3125 = 28.8, Ans.
Ex. 11. $3.50 X 365=$1277.50
$2000-1277.50 = $722.50, ^/w?.
M DECIMALS.
Ex. 12. $687.25 + $943.64=:$1G30.89
$1630.89 — $875.29 = $755.60, Ans.
Ex. 13 $l728-7-2=:$864 1st half sold for;
144x8 = 1152 2d " " "
$2016,^725.
Ex. 14. $3.75 X .875 = 13.281 +, Ans.
Ex. 15. $65.42 — $46.56 = 118.86, gain per head;
$3526.82-^-$18.86 = 187, Ans,
Ex. 16. $54.72-^36.48 = $1.50 ;
$1.50 X 14.25 = 821.375, Ans.
Ex. 17. $3548-^4 = $887, Ans.
Ex.18. 112.34-^$.82 = 137, ^n5.
Ex. 19. $3461.50-^46 = $75.25 ;
$75.25 X 5 = $386.25, A718.
Ex. 20. $24000 X | x j = $3200, Ans.
Ex.21. $1.25x160= $200
$5 X 80= 400
$600
$2.50x240= 600
Loss 000, Ans,
Ex. 22. $1.70x48 = 881.60
72.90
$ 8.70, Ans.
Ex.23. 122|4-75i = 197|; 197f — 60 = 137f ;
$.9375 — $.8125 = $.125, loss per bushel;
$.125 xl37f = $17.218 -f loss;
12.50 gain;
$4,718 + , loss, Ans.
Ex 24. $1.40 X e =$8.40 wages ;
$ .75x7= 5.25 expenses,
$3.15 savings, Ans.
PROMISCUOUS EXAMPLES. 55
$.08 X 39
Ex.25, -jj^ =19^, Ans.
Ex. 26. $4.50 X 23.487=:$105.6915, Ans.
Ex. 27 $1200-^365:=z$3.2874f, Ans.
Ek 28. $.17 X 56 X 28 = 1266.56, ^W5.
Ex 29. $.07 X 26 X 13 X 16=$378.56, Ans,
Ex. 30. $4.75 X 4.868=$23.123, Ans.
Ex. 31. $.33ix27 = $9.00
.25 x28= 7.00
.50 xl9= 9.50
$25.50, Ans.
Ex.32. 44—32 = 12;
— - — -=21^ minutes, Ans,
12 ^ '
$32.3 4 15 ^
Ex. 33. — -- X — X —=$51, Ans.
J It/ z
Ex 34. $5.d35-f-.875=$6.44; $6.44 x 9^ =$59.57, Ans.
Ex. 35. $5000
$1200.25 x3 = $3600.75
1800.62x8= 5401.86
$14002.61
$950.87x2= 1901.74
$12100.87, Ans.
Ex 36. $4.50xl86.40=$838.80, J7i5.
Ex. SS. $96.40-v-2=$48.20 ;
$48.20 X 1.375=$66.275, Ans.
Ex.39. ,Vo¥t=JL, ^W5
5Q COMPOUND NUMBERS.
Ex.40. 3\=.09375; .62i=.625; .37y^ ::^,370625 ;
|=::.375 ;
.09375 + .625 + .370625 + .375 = 1.464375, Ans.
Dr. Cr.
Ex. 41. $4,745 $2,765
2.625 1.245
1.27 .625
.45 3.45
5.285 1.875
Ex. 42.
$14,375 - $9.9C
$.125 X 120=115.00
.625 X 18= 11.25
.07 X 47= 3.29
.18 X 6= 1.08
J=$4.415, Ans.
$1.50
1.27
1.87
2.30
$30.62 -
$6.94 =$23.68,
• REDUCTION.
(183, page 152.)
Ex. 1. 14194 far.-^4 = 3548 d. 2 far. ; 3548 d.-^12
= 295 s. 8 d. ; 295 s.-f-20=14 £ 15 s.
Ans, 14 £ 15 s. 8 d. 2 far.
Ex. 2. 14£x 20 + 15 s.=295s.; 295s. xl2 + 8d.=3548d,:
3548d.x4 + 2 far. = 14194 far., Ans.
Ex.3. 15 £x 20 + 19 s. = 319 s.; 319 s.xl2 + lld.
= 3839 d. ; 3839 d.x 44-3 far.= 15359 'far., Ans
Ex. 4 15395 far.-~4 = 3839 d. 3 far.; 3839 d.-7-12
= 319 s. 11 d. ; 319 s.-v-20 = 15£ 19 s.
Ans. 15 £19 s. 11 d. 3 far.
Ex.5. 46 sov.x 20 + 12 s. = 932s.; 932s.xl2+2d.
= 11186 d., Ans
REDUCTION. 57
Ex. 6. 11186 d.-M2 = 932 s. 2d.; 932 s.-f-20==:46 sov
12 s. Ans. 46 sov. 12 s. 2 d
(IStJ, page 153.)
Ex.3. 5lb. X12 + 7 oz.=:67 oz. ; 67 oz.x 20 + 12 pwt.^
1352 pwt.; 1352 pwt. x 24 + 9 gT.=: 32457 gr., Aiis
Ex.4 43457 gr.-^24=rl810 pwt. 17 gr. ; 1810 pwt.^
20 = 90 oz. 10 pwt.; 90 oz. +-12 = 7 lb. 6 oz.
Ans, 7 lb. 6 oz. 10 pwt. 17 gr.
Ex. 5. 41760 gr. -^24 = 1740 pwt. ;l740pwt.-^20=::87oz.;
87 oz.-r-12i=7 lb. + 3 oz., Ans,
Ex. 6. 14 lb. 10 oz. 18 pwt. = 3578 pwt. ;
3578 X ^$.75=$2683.50, Ans,
Ex. 7. 5 lb. 6 oz.=1320 pwt. ; 2 oz. 15 pwt.=:55 pwt. ;
1320-^55 = 24, Ans,
Ex. 8. 1 lb. 1 pwt. 16 gr.=5800 gr. ; 4 pwt. 20 gr.= 116 gr. ;
5800-M16=:50 ;
$1.25 X 50 ==$62.50, Ans,
(I865 page 155.)
Ex.3. 16 lb. X 12 + 11 oz. = 203 oz. ; 203 oz. x 8 + 7 dr.
= 1631 dr.; 1631 dr.x3 + 2 sc. =4895 sc;
4895 sex 20+19 gr. = 97919 gr., Ans,
Ex.4. 471b. X 12 + 6 | =570 3 ; 570 | x8 + 4 3
= 4564 3 ; 4564 3 x 3 = 13692 3, Ans.
Ex. 5. 20 gr.x 5x365 = 36500 gr. ;
36500 gr.-^20 = 1825^; 1825 3-^-3 = 608 31 3;
608 3 +-8 = 76 5 ; 76 | -M2 = 6 lb. 4 ^ .
Ans, 6 lb. 4 3 13.
(187 5 page 156.)
Ex.3. 3 T.X20 + 14 cwt.=74 cwt. ; 74 cwt.xlOO + 74
lb.=7474 lb.; 7474 lb. x 16 + 12 oz.= 11959e
oz.; 119596 oz.x 16 + 15 dr.= 1913551 dr., Ans.
68 COMPOUND NUMBERS.
Ex. 4. 1913551 dr.~16 = 119596 oz. 15 dr.; 11959G 02.
-M6 = 7474 lb. 12 oz. ; 7474 lb. -^100=74 cwt.
74 lb.; 74 cwt.-^20=:3 T. 14 cwt.
Ans, 3 T. 14 cwt. 74 lb. 12 oz. 15 dr.
Ex 5. 3 T. 15 cwt. 20 ]b.=:7520 lb.
Ans. $.22 X 7520=:$1654.40.
Ex. 6, 115 lb.-=-2000=.0575 T. ; $10 x .0575=i$.575,
Am
Ex. 7
217 lb. X 10 = 2170 lb. @ $.06 =$130.20
306 1b. X 5=:1530 lb. @ $.07A = 114.75
3700 lb. @ $.08 =$296.00, which — $244.95
=$51.05, Ans.
Ex. 8. 2 T. X 2000 = 4000 lb. ; 4000 x $.12i-=$500 ;
$500 — $360 = $140, Ans.
Ex. 9. 10 T. X20 + 6 cwt. = 206 cwt.; 206 cwt. x 4 + 3 qr
= 827 qr. ; 827 qr.x 28 + 14 lb.=23l70 lb.
$.06 buying price.
$130-r-2000 = .065 selling price.
$.005 gain per pound.
$.005x23l70 = $115.85, Ans.
Ex. 10. 2352 lb.-v-56=42 bu.;
$.90 X 42 X 2 = $75.60, Ans.
Ex. 11. 300 bbl. X 196 = 58800 lb., Ans.
Ex. 12. $1.25 X 3 = $3.75 cost^
$.0075x280x3= 6.30
$2.55, Ans.
REDUCTION. 59
(191, page 157.)
Ex. 1. 6 lb. 10 oz.=90 oz. ; $.50 x 90 = $45.0C .ost.
$J2_x_8_><_437^ x^^^^8.Y5 sold lor.
480
$33. 75, Ans.
Ex. 2. 424 dr. -^8 =53 oz. ; 53 oz.-t-12 = 4 lb. 5 oz., Jps
Ex. 3. 20 lb. 8 oz. 12 pwt. = 119328 gr.
119328 gr.^7000=:l7gV3 lb., Ans.
Ex.4. $.40x16x20 =$128 cost
$.50x320x437.5
480
.=1 145.83^
$ 17.831, Ana,
(193, page 159.)
Ex. 3. 7912 mi. x 63360 = 501304320 ia., Ans.
Ex.4. 168i74 ft.-^3 = 56158 yd.; 56158 yd. -^5^ = 10210
id. 3 yd.; 10210 rd. -4-40 = 255 fur. 10 rd. ; 255 fur.-^-8 = 31
mi. 7 fur. Ans. 31 mi. 7 fur. 10 rd. 3 yd.
Ex.5. 31 mi.x84-7 fur.=255 fur.; 255 fur.x40-fl0
rd.=10210 rd.; 10210 rd. x 5i + 3 yd.=56158 yd.; 56158
yd. X 3 = 168474 ft., Ans.
Ex.6. 2500 fathoms X 6 = 15000 ft.; 15000 ft. -^161 =
909 rd. l|ft. ; 909 rd.-4-40 = 22 fur. 29 rd. ; 22 fur.-f-8 =
2 mi. 6 fur. Ans. 2 mi. 6 fur. 29 rd. 1| ft.
Ex. 7. 2200 mi. X 5280 = 11616000 ft.;
$.10 X 11616000 = $1161600, Ans.
Ex.8. 4 fathoms X 6 + 3 fl.=27 ft.; 27 ft. xl2+8in.=
832 in Ans.
Ex. 9 200 mi. = 12672000 in. ; 18 ft 4 in = 220 in. ;
12672000-4-220 = 57600 times, Ans.
Ex. 10. 120 lea. x 3 = 360 geo. mi ; 360 geo. mi x 1.15 -
4 14 Eng. mi., Ans,
Ex. 11. 141 hands x 4=58 in., Ans.
60 COMPOUND NUMBERS.
(1045 page 160.)
Ex.1. 3 mi. X 80 + 51 d\.^291 cb. ; 201 ckxlOO-i-
1S l. = 291'73 1., Ans.
Ex, 2. 291'73 l.-^100--291 ch. 73 1. ; 291 ch,-v-80 = 3 ml
51 ch. Ans. 3 mi. 51 ch. 13 L
Ex 3. 17 cL 31 1. = 17.31 ch.
12 ch. 87 l. = 12.87 ch.
30.18 ch. half romid the field.
30.18 ch. X 2 X 66 = 3983.76 ft., Ans,
(lOGj page 163.)
Ex.3. 87 A.X4 + 2 R. = 350 R. ; 350 R. x 40 -f-38 sq. rd
= 14038 sq. id. ; 14038 sq. rd. x 30^ + 7 sq. yd.==424656i sq
yd.; 4246561 sq. yd. x 9 + 1 sq. ft. = 3821909i sq. ft.;
38219091 sq. ft. X 144 + 100 sq. in=:550355068 sq. in., Ans,
Ex. 4. 550355068 sq. in. +-144 = 3821910 sq.ft. 28sq.in.;
3821910 sq. ft. -+9 = 424656 sq. yd. 6 sq. ft. ; 424656 sq. yd.-+
301 = 14038 sq. rd. 61 sq. yd.; 14038 sq. rd.+-40 = 350 R.
38 sq. rd.; 350 R.+-4 = 87 A. 2 R.
Ans, 87 A. 2 R. 38 sq. rd. 6i sq. yd. 6 sq. ft. 28 sq. in.
But (-J- sq. rd.)=:4 sq. ft. 72 sq. in.
Hence, Ans, 87 A. 2 R. 38 sq. rd. 7 sq. yd. 1 sq. ft. 100 sq in.
Ex. 5. 100 X 30 = 3000 sq. rd.=:18 A. 3 R., Ans,
Ex. 6. 4 mi. x 320 = 1280 rd., Ans,
Ex. 7. 2 mi. x 320 = 640 rd., Ans,
Ex. 8. 100000 sq. ft. +- 9 =11111 sq. yd. 1 sq. ft.;
IIIU sq. yd.+-30i = 367 sq. rd. 91 sq. yd
367 sq. rd.+-40 = 9 R. 7 sq. rd.
9 R.+-4 = 2 A. 1 R.
Ans. 2 A. 1 R. 7 sq. rd. 9i sq. yd. 1 sq. ft. ; or
2 A 1 R. 7 sq. rd. 9 sq. yd. 3i sq. ft.
REDUCTION. 61
Ex. 9. 181x16 = 296 sq.ft.;
296 sq. ft.-f-9=:32f sq. yd., Ans.
Ex. 10. (18 4- 161) X 2 = 69 ft., distance round the room ;
69x9
-69 sq. yd., in tlie walls;
9
18x161
= 83 sq. yd., in ceiling;
69 sq. yd. + 33 sq. yd. = 102 sq. yd.
$.22xl02=:$22.44, Ans,
Ex. 11. 40x20x2 = 1600 sq. ft. = 16 squares;
$10xl6=$160, Ans.
(lOr, page 164.)
Ex. 2. 3686400 P.-M02400 = 36 sq. mi., Ans.
Ex. 3. 94 A. X 10 + 7 sq. ch. = 94'7 sq. ch. ;
947 sq. ch.xl6 + 12 P.= 15164 P.;
15164 P. X 625 4-118 sq. 1 = 9477618 sq. l, A^^s.
Ex. 4. 4550000 sq. l.-M0000 = 455 A.
$50x455=$22750, Ans,
(lOO, page 166.)
Ex. 1. 125 cu. ft. X 1728 + 840 cu. in. = 216840 cu. in., ^ai5.
Ex.2. 5224 cu. ft.-M28=40if Cd., ^715.
Ex. 3. 3ft.2in.=38in.; 2ft. 2 in.=26 in.; 1 ft. 8 in. =20 in.;
88 X 26 X 20 = 19760 cu. in., Ans.
Ex. 4. 6x6x6 = 216 cu. ft.;
216 cu. ft. X 1728 = 373248 cu. in., Ans.
Ex. 5. 60 X 20 X 15 = 18000 cu. ft. ;
18000 cu. ft.-^128 = 140| Cd., Ans.
Ex. 6. 10 X 31 X 3i = 113f cu. ft., Ans,
Ex. 7. 128^(3 X 12) = 3f ft. high, Ans
62 COMPOUND NUMBERS.
Ex. 8. 21x115 lb.=4725 lb.=2 T. 1 cwt. 25 lb., Ann
Ex. 9. 32 ft. + 24 ft.=i56 ft.; 56 ft. x 2=^112 ft. girt;
112 X 1| X 6 = 1008 cu. ft.; 1008 cu. ft.-^24^—A0j\ Pch.
$1.25 X 40y\=:$50.909 -f , Arts.
32 X 24 X 6
Ex.10. $.15 X — =$25.60,^715.
Ex. 11. 10x9x8 = 720 cu. ft.;
'720+-10 = Y2 minutes, Ana.
Ei. 12. 30 X 20 X 10=6000 cu. ft. ;
6000 ,, . ,
-=12 minutes, Arts,
50x10 '
(SOO, page 168.)
Ex. 3. 3 hhd. X 2016 = 6048 gi., A7is,
Ex. 4. 6048 gi.+2016 = 3 hhd., Ans.
Ex.5. 13 hhd. X 63 + 15 ga]. = 834 gal.; 834 gal. x 4 x 1
qt. = 3337 qt. ; 3337 qt. x 2 = 6674 pt., Ans.
Ex. 6. 6674 pt.-^2 = 3337 qt. ; 3337 qt.-v-4 = 834 gal. 1
qt.; 834 gal.-v-63 = 13 hhd. 15 gal.
13 hhd. 15 gal. 1 qt. Ans.
Ex. 7. 1 hhd.=2016 gi. ; $.06 x 2016=$120.96, Ans,
Ex. 8. $2 X 10 = $20 cost ; $.05x 4 x 31^ x 10=$63 reed.
$63 — $20 = $43 gain, Ans.
Ex. 9. $3.84-T-$.06 = 64 pt.=8 gal., Ans.
Ex. 10. 2 gal. 2 qt. 1 pt.=21 pt.; 1 hhd.=504 pt. ;
604-4-21 = 24, Ans.
(SOI, page 169.)
Ex.1. 49 bu. x4 + 3 pk.=199 pk. ; 199 pk. x8 + 7 qt,=
1599 qt. ; 1599 qt. X 2 + 1 pt.=3199 pt., Ans.
REDUCTION. 68
Kx. 2. 3199 pt.-^23:=1599 qt. 1 pt. ; 1599 qt.-^8 = l99 pk.
7 qt; 199 pk.^4r=:49 bu. 3 pk.
Ans. 49 bu. 3 pk. 7 qt. 1 pt.
Ex. 3. 1 bu. x4 + l pk.=5 peck; 5pk. xS + l qt.— 41 qt, ;
41 qt.x2 + l pt.=83 pt., Ans.
Ex. 4. 83 pt.-^2rr41 qt. 1 pt.; 41 qt.-f-8=:5 pk. 1 qt.
5 pk.-^4 = l bu. 1 pk. Ans, 1 bu. 1 pk. 1 qt. 1 pt.
Ex. 5. $.Q5 X 60=$32.50 cost ;
$.25 X 4 X 50 = $50.00 sold for ;
$17.60 Ans.
(205, page 170.)
Ex. 1. 1 bu. (Dry Measure) =21501 cu. in.;
2150| cu. in.-^57j = 37;^J wine quarts;
37±| qts.— 32:=5i| qts., Ans,
Ex. 2. 40 qt.-f-4=rl0 gal. ; 10 gal. x 282 = 2820 cu. in. ,
2820 cu. in.-^57f = 48|| qts. Wine Measure :
48^ qts.~40 qts. = 8f| qts., Ans,
Ex. 3. 1 bu. Dry Measure =2150f cu. in.
32 qt. Wine Measures 1848 cu. in.
302| cu. in., Ans.
(206, page 171.)
Ex. 1. 365 da. X 24 4-5 h.=8765 b. ; 8765 h. x60f48
min.=525948 mi»;t 625948 min.x 60+46 se us=s
31556926 sec, Ans,
Ex. 2 31556926 sec. -^60 = 525948 min. 46 sec;
525948 min-v-60 = 8765 h. 48 min.; 8765 h.-^24
=365 da. 5 b. Ans. 365 da. 5 b. 48 min. 46 sea
64 COMPOUND NUMBERS.
Ex.3. 5 wk. x7 + l da. = 36 da.; 36 da. x 24+1 h,=
865 h. ; 865 li. x60 + l niin.:=51901 mm.; 51901
min. x60 + l scc. = 31 14061 sec, Ans,
Ex. 4. 3114061 sec. -^60 = 51901 mm. 1 sec; 51901 mm.
-^60:=865 h. 1 min.; 865 h.-^24 = 36 da. 1 h. ;
36 da.-^-7 = 5 \vk. 1 da.
Ans, 5 wk. 1 da. 1 h. 1 min. 1 sec
Ex. 6. 10 mi.=:l'7600 yd.;
17600 sec-^60==293 min. 20 sec ; 293 min.-^60
=4 h. 53 min. Ans, 4 h. 53 min. 20 sec
Ex.7. 29 da.x24+12 li.=708 h. ; 708 h. x 60 + 44 min,
=42524 min. ; 42524 min. x 60 + 3 sec =
2551443 sec, Ans,
Ex.8. 40 yr.x365^z=:14610 da.; 14610 da. x45 =
657450 min. gained.
657450 min.-f-60=rl0957 h. 30 min.; 10957 h.-^
24=456 da. 13 li. Ans, 456 da. 13 h. 30 min.
(207, page 173.)
Ex.1. 10 S. x30 + 10°=310°; 310° x 60 + 10'=18610' ;
18610'x60 + 10''=1116610^ Ans,
Ex. 2. 1116610'' -^60 = 18610' 10''; 18610'-^60 = 310° 10';
310°-^30=:10 S. 10°. Ans. 10 S. 10° 10' 10".
Ex. 3. 11400'-^60 = 190°, Ans,
Ex. 4. 190° X 691 = 13148 miles, Ans.
Ex. 5. 360° X 60 = 21600', Ans,
Ex. 6. 397'-^60=6° 37', Ans,
(210, page 174.)
Ex.1. 150000000-^12 = 12500000 doz.;
12500000 doz. -^12 = 1041666 gross 8 doz.;
1041666 gross-^12 = 86805 great gross+6 gross,
Ans, 86805 great gross 6 gross 8 doz.
REDUCTION. 65
Ex. 2. 100000 sheets-^24r=4166 quires 16 sheets;
4166 quires-^20 = 208 reams 6 quires;
208 reams-^2r=104 bundles ;
104 bundles-4-5 — 20 bales 4 bundles.
Ans. 20 bales 4 bundles 6 quires 16 sheets.
Ex, 3. 20 years x 4 + 10 years =90 years, Ans.
Et 4. 8 sheets x 8=64 leaves ; 64 leaves x 2 = 128 pages,
Ans,
Ex. 5. 32 pages xl0x2 = 640 pages, Ans,
PROMISCUOUS EXAMPLES IN REDUCTION.
Ex. 1. 6 yd. 3f qr. = 27J qr.; 333 yd. = 1332 qr. ;
1332-^273 =48 suits, Ans,
Ex. 2. 1 oz. 15 pwt.i=:35 pwt. ;
I.YOx 35 =$24.50, Arts.
Ex. 3. 2 lb. 3 3 5 3 13 10 gr.=13290 gr.;
13290-^15 = 886, Ans,
Ex. 4. 1 T. 11 cwt. 12 lb. = 3112 lb.;
3112x$.01i=$38.90, Ans,
Ex. 5. 1456 lb.-^32 = 45.5 bu. ;
$.375 X 45.5=$17.0625, Aiis.
Ex. 6. 45 lb. X 1000 = 45000 lb.
45000 lb.-M96 = 229 bbl. 116 lb., Ans.
Ex. 1, 2430 lb.-T-60c±40.5 bu. ;
$1.20 X 40.5=$48.60, Ans,
Ex. 8. $12.50-^200=$.06i, Ans,
Ex. 9 360° X 69.15=24897.6 stat. mi. ;
24897.6 X 63360=1577511936, Ans.
Ex. 10. 10 mi. X 80 + 7 ch. + l eh. (4 rd.)=808 ch. ; 808
nh. X 100 + 20 1.= 80820 L, Ans.
1j6 compound numbers.
Ex. 11. 25 X 100 X 144 = 360000 sq. in. ;
$.01 X 360000 = 13600, Ans.
Ex. 12. 50 X 25 X 10 = 12500 cu. ft. ;
12500 cu. ft. -^1 6 = 781 cd. ft. 4 cu. ft.;
'781 cd. ft.^8=97 Cd. 5 cd ft. ;
Ans, 97 Cd. 5 cd. ft. 4 cu. ft
Ex 13, 10 X 10 X 10 X 1728 = 1728000 cu. in. ;
1728000 cu. in.-v-231 = 74804^ gal.;
1i80^ gal.-^63 = 118 hhd. 464-f gal., ^W5.
Ex. 14. 8 X 5 X 41=180 cu. ft.=311040 cu. in. ;
311040 cu. in.-T-2150.4 = 144y\ bu., Ans.
Ex. 15. Mar. 31 da. June 30 da. Sept. 30 da,
Apr. 30 da. July 31 da. Oct. 31 da.
May 31 da. Aug. 31 da. Nov. 30 da.
Spring, 92 da. Summer, 92 da. Autumn 91 da.
92 da.— 91 da. = l da. = 86400 sec, Aiis.
Ex. 16. 1296000 sec.-=- 86400 = 15 da., Ans.
^ ,^ 20x13 ,^ , .
Ex. 17. =40 yd., Am.
Ex 18. 4 reams X 20 + 10 quires= 90 quires; 90 quires x
24 + 10 sheets=2l70 sheets, Ans.
Ex.19. 16 ft. 6 in.=l rd. ; 1 mi. = 320 rd. ; 320^1 =
Ifto times in 1 mi. 320 x 42 = 13440 times, Ans.
)|Ex. 20. 1000000 sec.-^60 = 16666 min. 40 sec. ;
16666 min.-f-60 = 277 h. 46 min. ;
277 h.-f-10 = 27 da. 7 h. ;
Ans. 27 da. 7 h. 46 min. 40 sec.
Ex. 21. 6x4^ =27 sq. mi ; — — — =216 farms, Ans.
Ex. 22. 10 mi. 176 rd.=3376 rd.
$21.75 x 3376 ^$73428, Ans.
REDUCTION. til
(21 1 5 pagelTG.)
Ex. 2. j-oV f iJ X V X -V = 2«5 d., A71S,
Ex. 3. y^ ^ Wk. X I X V- X -¥- = f ^^^-j ^'^*-
Ex. 4. y-,V2 ^^^^' X r- X f X f- X f - 1 gi., ^ws.
Ex 5. ^J-o oz. X -/ X V — i gr., J.71S.
Rv fi 1 mi X ^ X ^-" X -3-3- X J^= J-9-®-, in.. Ans
Ex. 7. f X i X f lb. X Y =1 oz-j ^^^5«
Ex. 8. elo hhd. X V- X f X 2. = f J pt., ^?^5.
Ex. 9. yyVo A. X f X V-=i- rd., ^n5.
(SIS, page 177.)
Ex. 2. i ft. X 3=3 = ri2 r^- ■^^^^•
Ex. 3. f dr. X j\ X tV^tAo lb., ^7^5.
Ex. 4. J ct. X y^Vo = 2 oVo E., ^?^5.
Ex. 5. i ft. X 3 J-^=y^i^^ mi., Ans.
Ex. 6. -f X t pwt. X ^V X T2 = 2 « lt>., Ans.
Ex.7, f pt.x-Lxix^V=8io l^bd.
^ hhd.—- gJ liiid.=:f J-f hhd., Ans.
Ex. 8. I in. X c 3 k =To jVo i^i-, ^'^s-
Ex. 9. f oz. X tV=:2\ lb— 2T <^^' 2 lb. ; and ^V of 2 lb. is
I of f of 2V of 2 lb., or I of ^ of 2 lb., Ans.
Ex. 10. f oz. X tV=2"t lb. = 2T of 2 lb. ; and ^\ of 2 lb. ia
J. of f of J, of 2 lb., or I of f of 2 lb., Ans.
(2185 page 178.)
Ex.2. 4mo.x30 = l7| da.; -Ida. x24 = 3f h. ; ^ h. x
60 = 254 rain. ; 4 min. x 00 = 424 sec.
A71S. 17 da. 3 b. 25 min. 42^ sec.
Ex. 3 |£ X 20 = 84 s. ; 4 s. X 12 = 64 d. ; 4 d. x 4 = 34 ^
A71S. 8 s. 6 d. 34 far.
68 COMPOUND NUMBERS.
Ex. 4. § bu. x4 = lj pk. ; J pk.x8 = 4f qt.; f qt x2-=
1| pt. Ans, 1 pk. 4 qt. If pt.
Ex.5. 4 of 15 cwt. = 124 cwt. ; f cwt. xl00=:85f lb.;
f lb. X 16 = llf oz. ; f oz. x 16 = 6^ dr.
Ans. 12 cwt. 85 lb. 11 oz. Of dr.
Ex.6. |x-}x V^3=4|f oz. ; |^ oz. x IGnzllf^ di.
Ans. 4 oz. 11|^ dr.
Ex. 7. I A. X 4 = 3i R. ; ^ R. x 40 = 131 P.
^m\ 3 R. 131 p.
Ex. 8. If da. X 24r=16/3 h. ; j\ b. x 60 = 36|| min. ; \i
inin. X 60 = 55 j^^ sec.
Ans, 16 b. 36 min. 55-^^^ sec.
Ex. 9. I lb. X 12 = '7i oz. ; i oz. x 20 = 4 pwt.
Ans, 7 oz. 4 pwt,
Ex. 10. i ofVT.=4/_ T. ; /^ T.x20=5f cwt; f cwt
X 100 = 55f lb. Ans. 4 T. 5 cwt 55| lb.
Ex. 11. I ofY A. = lf A.; f A. X 4 = 11 R. ; i Rx40 =
20 P. Ans. 1 A. 3 R. 20 P.
(214, page 179.)
Ex. 2. 6 fur. 26 rd. 3 yd. 2 ft=4400 ft ; 1 mi.=5280 ft;
UU mi.=f mi., Ans.
Ex. 3. 13 s. 7 d. 3 far.=655 far. ; l£ = 960 far. ;
960 ^ — 192 "^i -^rx/Cis,
Ex. 4. 10 oz. 10 pwt 10 gr. = 5050 gr.; 1 lb. = 5760 gr.;
HU Ib.-Hf lb., Ans.
Ex. 5. 2 cd. ft 8 cu. ft =40 cu. ft ; 1 Cd. = 128 cu. ft ;
tVV Cd.=tV Cd., Ans.
Ex.6. lbbl.lgal.lqtlptlgi. = 1053gi.; lLbd.=2016gi»^
i^fl bbd.=iif hbd., Ans.
Ex. 7. 4 yd. li ft = 27 half-feet; 2 rd.=66 half-feet;
REDUCTION. 89
Ex. 8. -p bu- =—-=:- bu., Ans.
4 20 5
Ex. 9. -^j = ij Ans.
Ex. 10. 2 yd. 2 qr.=10 qr ; 8 yd. 3 qr.=:35 qr. ;
H yd.==f yd., Ans.
(215>5 page 180.)
Ex.2. .217° X 60=13.02'; .02'x 60 = 1.2^
Ans, 13' 1.2^
Ex.3. .659wk. X 7=4.613 da. ; .613 da. x 24 = 14.712 h.;
•712 h. X 60 = 42.72 min. ; .72 min. x 60 = 43.2 sec.
Ans, 4 da. 14 h. 42 min. 43.2 sec.
Kx. 4. .578125 bu. X 4 = 2.3125 pk.; .3125 pk. x 8=2.5 qt.;
.5 qt. X 2 = 1 pt. Ans, 2 pk. 2 qt. 1 pt.
Ex. 5. .125 bbl. X 31.5 = 3.9375 gal. ; .9375 gal. x 4 =
3.75 qt. ; .75 qt. x 2 = 1.5 pt. ; .5 pt. x 4 = 2 gi.
Ans. 3 gal. 3 qt. 1 pt. 2 gi.
Ex. 6. .628125 £ x 20 = 12.5625 s. ; .5625 s. x 12 = 6.75 d.;
.75 d. X 4 = 3 far. A7is, 12 s. 6 d. 3 far.
Ex. 7. .22 bhd. x 63 = 13.86 gal. ; .86 gal. x 4 = 3.44 qt. ,
.44 qt. X 2 = .88 pt. ; .88 pt. x 4 = 3.52 gi.
Ans, 13 gal. 3 qt. 3.52 gi.
Ex. 8. .67x3.45 = 2.3115; .3115x320 = 99.68;
.68x16^=11.22; .22x12 = 2.64.
Ans. 2 rai. 99 rd. 11 ft. 2.64 in.
Ex. 9. .42857 mo. x 30 = 12.8571 da.; .8571 da. x 24 =
20.5704 h. ; .5704 h. x 60 = 34.224 min. ;
.224 min. x 60^13.44 sec.
Ans. 12 da. 20 h. 34 min. 13^} sec.
Te
COMPOUND NUMBERS.
Ex. 10. .78875 T. x 20 = 15.775 cwt.; ,775 cwt. x 4=3.1 qr.;
.1 qr. X 28 = 2.8 lb. ; .8 lb. x 16 = 12.8 oz.
Ans. 15 cwt. 3 qr. 2 lb. 12.8 oz.
Ex. 11. .88125 A. X 4 = 3.525 R. ; .525 E.x 40 = 21 P.;
Ans, 5 A. 3 R, 21 P.
Ex 12. .0055 T.x 2000 = 11 lb., Ans.
Ex. 13. .034375 bundles x 40 = 1.375 quires; .-^>75 quires x
24=9 sheets; Ans, 1 quiie 9 sheets.
(916, page 181.)
Ex.
1.00 gi.
1.250 pt.
3.625 qt.
Ans. .90625 cral.
Ex. 3.
24
20
12
9.000 gr.
13.375 pwt.
10.66875 oz.
Ans. .8890625 ib
Ex.4.
2
4
1.2 pt.
.6 qt.
63
.150 gal.
Ans. .00238 + hhd.
Ex. 5.
1.12 qt.
3.14 pk.
Ans. .785 bu.
Ex. I
40
4
12.56 P.
3.314 R.
Ans. .8285 A.
Ex. 7. 12
3
5,5
40
8
6 in.
1.5 ft.
iT^yd.
3.1818181+ rd.
.07954545 + fur
Ans. .00994318 + mi.
Ex. 8.
Ex. 9.
Ex. 10.
Ex. 11.
Ex. 12.
.32 pt.-h64=.005 bu., Ans.
4.875 ft.+6 = .8125 fathoms, Ans.
150 sheets-^480 = .3125 Rm., Ans.
47.04 lb.-f-196=.24 bbl flour., An^
.33 ft.-^5280 = .0000625 mi., Ans,
ADDITION.
n
Ex. 13.
60
51.6 sec.
60
36.96 min.
24
5.616 h.
Ans. .234 da.
ADDITION.
(Sir, page 183.)
Ea 3. 43 10. 1 3 2 ^ 16 gr., Ans.
Ex. 5. 68 bu. 3 pk. 1 qt. 1 pt., Ans,
Ex. 6. 21 mi. 5 fur. 23 rd. 1 yd. 4 in., Ans.
Ex. 10. 627 hbd. 1 gal. 1 qt. 1 pt., Ans.
Ex. 11. 187 bu. 3 pk. 1 pt., Ans.
Ex. 16. 152 cu. yd. 9 cu. ft.=:152i- cu. yd. )
$.16xl52J- ===$24,371. )
Ex. 17. 2564 lbs.
2713 "
3000 "
3109 "
~" =203.3214+ bu.
2.657 + , At^.
Ex.
An3»
11386 lb«i,-
-56
=:
$.80 X
203.3214:
bbls. gal.
qt.
pt.
gi.
18. 1 4
1
30
2
1
2 15
3 49 2 1 1=:4613 gi.
$.09x4613=:$415.l7, Ans.
(21 85 page 185.)
Ex. 2 i rd. = 12 ft. 4i in.
I ft. = 9_ "
13 ft. 11 in., Ans.
T2 COMPOUND NUxMBERS.
Ex, 3. 1 mi. = 7 fur.
I fur.= 26 rd. 11 ft.
I rd. = 13 " 9 in.
7 fur. 27 rd. 1^ ft. 9 in. ; or
1 fur. 27 rd. 8 ft. 3 in., Am.
Ex.4. |£=13s. 4 d.
^ s. = 6 " 2| far.
13 s. 10 d. 2| far., A7is,
Ex. 6.
f T. =12 cwt.
^ cwt.= 42 lb. 134 oz.
12 cwt. 42 lb. 13f oz., An4.
Ex. 6.
f da. = 9 h.
i h. = 30 min.
9 h. 30 min., Ans,
Ex. 7. 1 wk. = l da. 4 h.
f da. = 18 "
^ h. = 15 min.
1 da. 22 h. 15 min., Arts,
Ex. 8. -f lihd. = 54 gal.
f gal. = ^qt.
64 gal. 3 qt., Ans.
Ex. 9. 4 cwt. =64 lb.
8f lb. = 8 " 13 oz. 5i dr.
3yV oz. = 3 " 14| "
73 lb. 1 oz. 3|i dr., Ana.
Ex. 10. I mi. = 3 fur.
|yd.= 2 ft.
J ft. = 9 in.
3 fur. 2 ft. 9 in., An8.
SUBTRACTION. T8
Ex. 11. 1 of } A.=:i A. - 26 P. 181 ^ sq. ft.
60f rd. = l K. 20 " 204^^ «
f A.=l " 5 " 194 If "
Y^ A. == 13 " 90 f "
3 K. 26 P. 126yV2 sq. ft., Am,
Ex. 12. 11 T. =1 T. 3 cwt. 33 lb. 5^ oz.
If^ T. =1 " 3 " 75 "
18f cwt.= 18 " 83 " 6i '••
3 T. 5 cwt. 91 lb. 10| oz., Ans.
SUBTRACTION.
(2195 page 187.)
Ex 4. 3 T. 18 cwt. 70f lb., Ans.
Ex. 6. 2953 £ 2 s. 7f d., Ans.
Ex. 11. 365 da. X 5 4-2 da.=1827 da.
1 bhd.=63 gal.
1827 gi. =57 " qt. pt. 3 gi.
6 gal. 3 qt. 1 pt. 1 gi., Ans.
Ex. 12. 196 A. 2 R. 16.25 P.
200 " 1 " 20 "
177 " " 36 "
1st, 2d, and 3d own 574 A. R. 32.25 P.
640 "
4th owns 65 A. 3 R. 7.75 P., A71S.
Ex. 13. 16 Cd. 5 cd. ft. 75 Cd. 6 cd. ft.
24 " 6 " 12cii.ft. 69 " 2 " 12 cu. ft.
^^ * 6 Cd. 3 cd. ft. 4 cu. ft.
69 Cd. 2 cd.ft. 12cu.ft. Ans.
K. P.
^ COMPOUND NUMBERS.
Ex. 14. 10 gal. 1 qt. 1 pt. 63 gal
15 " 1 pt. 40 " 1 qt.
14. « Q «
22 gal. 3 qt., Ans,
40 gal. 1 qt., Ans.
(32O5 page 189.)
yr. mo, da. yr. mo. da.
Ex. 2. 1799 12 14 Ex. 5. 1815 G 18
1732 2 22 1775 6 17
67 9 22, Ans. 40 1, Ana
jT. mo. da. h. min.
Ex. 6. 1861 1 3 8 50
1856 6 24 14 20
4 6 8 18 30, Ans,
Ex. 7. 122 da. ; 244 da. ; 306 da., Ans.
Ex. 8. Erom Nov. 6 to April 6, 151 da.
From Apr. 6 to Apr. 15, 9 "
160 da., Am,
Ex. 9. From Aug. 20 to June 20, 304 da.
Subtract 5 "
299 da., Aru
(221 5 page 190.)
Ex. 2. i rd.=:8 ft. 3 in.
|ft. = 9 «
7 ft. 6 in., Ans.
Ex. 3. f £=11 s. 1 d. li far.
4s.= 6"
10 8. 7 d. 11 far., Aiu.
MULTIPLICATION^. 75
Ex. 4. I of 3.45 mi.=2.3 mi ;
2.3 mi.— .7 mi.=:1.6 mi.
1.6 mi. = l mi. 4 fur. 32 rd., Ans.
Ex. 5. 8yV cwt. =8 cwt. 3 qr. 16 lb. 12 oz. 12f dr
1 qr. 2^ lb. = 1 qr. 2 " 6 " 13 4 "
Ans. 8 cwt. 2 qr. 14 lb. 5 oz. 15/j dr
Ex. 6. 1 wk.=l da. 9 h. 36 min.
1 da. = 4 " 48 "
1 da. 4 li. 48 min., Ans.
f I of 120 mi.=41 mi. 7 fur. 9 rd. 8 ft. 7^ in.. Am
Ex.8. l-i=J; ioff=/^;
y^ of 96 gal.=25 gal. 2 qt. 3i gi., Ans
MULTIPLICATION.
(222, page 192.)
Ex. 4. Ans. 23 £ 13 s. 4 d.
Ex. 5. Ans. 23 lb. 4 oz. 6 pwt. 10 gr.
Ex. 6. ^7i5. 163 T. 1 cwt. 36 lb. 4 oz.
Ex. 7. ^W5. 128° 35' 15"^.
Ex. 9. Ans. 20fi) 1 | 3 3 1 3 16 gr.
Ex. 10. Ans. 235 mi. 6 far. 7 rd. 4^ ft.
Ex. 13
bu. pk. qt pt.
45 3 6 1
8
Ex. 14.
1
a.
9
d.
12
367 2 4
8
17
16
6
10
2941, Ans 178 5, ^w«.
76 COMPOUND NUMBERS.
Ex. 15. U800-T-$80 = 60 — 6 x 10.
A. R- P. 8q. yd. 8<i. ft
4 3 26 20 3
6
29 2 1
10
295 10, Alls,
Ex. 17. Ans, 359° 45'40.45^ Ex. 18. Arts, 6 libd.
DIVISION.
(223, page 194.)
Ex. 7. Ans. 1 oz. 17 pwt. 4 gr.
Ex.11. 5£ 10 s. 10d.=1330d.;
537£lOs. 10d.=129010 d.
129010-^1330=^97, Ans,
^ ,^ 50x30x6 ^^ . ^ .^ .
Ex. 12. — =11 cu. yd. 3 cu. ft., Ans.
2i X o X b
^ ^^ 6x5x640 ^^^^ .
Ex. 13. __ = 106|A.
106 A. 2 R. 26 P. 20 sq. yd. 1 sq. ft. 72 sq. in., Ans.
Ex. 14. 4 bu. 3 pk. 2 qt.=154 qt. ;
336 bu. 3 pk. 4 qt. = 10780 qt.
10780-^-154 = 70, Ans,
Ex 15. 60 yd. 2.25 qr. = 242.25 qr.; 242.25 qr.x4 =
969 qr. ; | of 969 qr.=646 qr. ; 9 yd. 2 qr.=:38 qr. \
6464-38 = 17, Ans,
LONGITUDE AND TIME.
LONGITUDE AND TIME.
(325, page 196.)
Ex. 2. 84°
24'
Ex. 3. 155°
74
1
18
28'
10°
23'
173°
28'
4
, Ans,
4
41 min.
32 sec,
11 li. 33 min. 52 sec., Ans.
Ex.4. ir
1'
Ex. 5. 118°
+ 122°=240°;
SO
19
360°
-240° = 120°;
107°
20'
4
120°H
rl5=:8 h., Ans.
7 h. 9 min. 20 sec., Ans,
Ex. 6. 12 h.
77° 1'= 5 " 8 min. 4 sec.
6 h. 51 min. 56 sec, A.M., Ans,
Ex. 7. 90° 15'
63 36 4 li.
26° 39'= 1 li. 46 min. 36 sec.
2 h. 13 min. 24 sec, P.M., Ans,
Ex. 8. 124°— 67°:=57° = 3 li. 48 min., Ans,
Ex. 9. 99° 5
68 47
30° 18'=2 h, 1 min. 12 sec. difference of time.
Time at Bangor, 1859 yr. 1 mo. 1 da. 1 h. min. sec. a. m
Subtract 2 h. 1 min. 12 sec.
Time at Mexico, 1858 yr. 12 mo. 31 da. 10 h. 58 min. 48 sec. p.m.
Ans,
Note. In the above subtraction, borrow 31 days, the month being
December.
78 COMPOUND NUMBERS.
(236, page 197.)
Ex. 2. 11 h. 33 min. 52 sec. = 693 rain. 52 sec;
(693 min. 52 sec.) -^4 = 173° 28', Ans.
Ex. 3. 7 h. 9 rain. 20 sec.=429 rain. 20 sec. ;
(429 min. 20 sec.)-r-4 = 107° 20', Ans.
Ex. 4. 16 h. 30 rain. at St. Petersburgh ;
8 li. 32 rain, 36 sec. at New Orleans ;
7 h. 57 rain. 24 sec.=477 rain. 24 sec.
(477 rain. 24 sec.)-T-4 = 119° 21', Ans.
Ex. 5. 74° 1' West; 8 h. 40 rain.=130°
4h.=60° 74° r West,
1st Ans, 14° 1' West. 2d Aris, 55° 59' East
13 h. 25 rain. = 201° 15'
74° I'West,
3d Ans. 127° 14' East
DUODECIMALS.
Ex. 2
13 ft 9'
11'
MULTIP
(229,
LIGATION.
page 200
Ex. 3.
11 ft 9'
1 3'
12 ft. 7' S
r, Ans.
2 11'
11 9
3^^
14 ft 8'
4
68 ft 9',
Ans.
DUODECIMALS. T9
Ex. 4. 12 ft. 11' 6 ft. 2'
4 2 ft. 4'
51 ft. 8' length of walls; 2 0' 8'
9 ft. 3' 12 4'
12 11 14 ft. 4' 8"
465 3
477 ft. 11' area of walls ; 43 ft. 2', windows and door;
43 ft. 2'
434 ft. 9'=48 sq. yd. 2 sq. ft. 9', Arts.
Ex. 5 30 ft. 4' Ex. 6. 18 ft. 6'
12 ft.
25 ft.
6'
15
2
758
4
773 ft.
6'
12 ft.
5'
322
3 6"
9282
9604 ft. 3' 6% Ans,
Ex. 7. 36 ft. 10' Ex. 8.
22 ft. 3'
9
2
6
810
4
819 ft.
6'
e''
5 ft.
2'
136
7
1
4097
8
6
222 ft.
5 ft. 6'
111
1110
1221 ft.
1221 cu. ft.
-^-
128 =
9 Cd. 69 cu.
ft.
, Ans,
32 ft. 8
9 ft.
294 sq. ft.
294 sq.ft.-v-9 = 32|sq.yd
$.l7x32|=$5.55i Ans
4234 ft. 3' V ;
156 cu. yd. 22 cu ft. 3' 1\ Ans.
80 COMPOUND NUMBERS.
Ex. 9. 33 ft. 9' 27' 3
48 ft. 12x3 4
aJ^'
36
1584 180-^f = 240 yd,, Ans.
1620ft. = 180 8q. yd.
DIVISION.
(330, page 201.)
2. 16 ft. 8^)44 ft. 5' 4''(2 ft. 8', Ans.
33 4
11 1 4
11 1 4
Ex. 3. 40 ft. 11' 4'')184 ft. 3' 0''(4 ft. 6', Ana.
163 9 4
20 5 8
20 5 8
Ex.4. 2 ft. r)14ft. 6' (5 ft. r 4'' + , ^w».
12 11
1
7 0^
1
6 1
11
0'"
10
4
8'", rem.
PROMISCUOUS EXAMPLES. 81
Ex. 5. 3 ft. f 8 ft. 11' 6^)64 ft. 2' b\l ft. 2^ Arts
2 it. 6' 62 8 6
7 2
1 9 6
1
1
5 11
5 11
8 ft. 11' Q"
PROMISCUOUS EXAMPLES.
(Page 202.)
Ex 1. 115200 g^.-^5760 = 20 lb., Ans,
Ex. 3. 1560 bu. X 4=6240 pk. ; 3 bu. 1 pk.=13 pk. ,
6240-^-13=480, Ans.
Ex.4. 295218 in.-^12 = 24601i ft.; 24601i ft.-r-16i=-
1491 rd.; 1491 rd.-^40=37 fur. 11 rd. ;
37 fur.~8=4 mi. 5 fur.
Ans, 4 mi. 5 fur. 11 rd.
Ex. 6. 3 X 20 X 24 = 1440, Ans.
^ ^ $3.25x4x20x6x10 ^ ^ ,
Ex. 7. ^ =$121.87^, Ans.
Ex. 8. 1 bbl.=1008 gi. ; 1 qt. 1 gi.=9 gi. ;
1008^9 = 112, Ans.
Ex. 9. $.6.40 X Y X f =$980.10, Ans.
Ex. 11. 336 bu. 3 pk. 4 qt.= 10780 qt. ; 4 bu. 3 pk. 2 qt.=
154 qt.; 10780-^-154=70, Ans.
Ex. 12. 3 qt. 1 pt., Ans.
Ex. 13. 1 mi.=l760 yd.; 2 fur. 36 rd. 2 yd.=640 yd. ;
j%f^^^ mi.=y\ mi., Ans.
82 COMPOUJSTD NUMBERS.
Ex (4. 2 da.= 172800 sec; 13 h. 26 min. 24 sec.=
48384 sec. ; rVVVVo = 21, ^^'
Ex 1.5. 2G A. 2 R.=106 R.; 5 A. 3 R.=:23 R.,
106 R.— 23 R.=83 R.; j%\, Ans.
Ex. 16. f£ = 12 s.
51 d.
Alls, 11 s. 6J d.
Ex. 17. I yd. =5| in.
^ ft. =lf "
1 in . = 4 ^ ^
7 in., ^715.
Ex. 19. 1732 yr. 2 mo. 22 da.
1706 " 1 " 18 "
26 V- 1 rno. 4 da.
Ex. 20. 87o 30
71° 4' 12 h.
16'' 26'= 1 h. 5 min. 44 sec.
10 h. 54 min. 16 sec, A. M., Aru^
Ex 22. I mi. =5 fur. 13 rd. 5 ft. 6 in.
^fur.=: 20 "
rd. = 2 " 9 "
TJ
5 fur. 33 rd. 8 ft. 3 in., Ans.
Ex. 23 20 bu. 3 pk. 6 qt.=20.9375 bu. ;
$.80 v: 20.9375=116.75, A7is,
Ex. 24. .875 gross x 12 = 10.5 = 10| doz., Ans,
Ex, 25. 56.5x24.6 = 1389.9 P. = 8 A. 2 R. 29.9 P., Ans.
Ex. 26. 20f (lix2) = 23 ft., length of one side;
23 X 8 X 11 X 4 = 1104 cu. ft., by the mason's rule ;
(see note 5, page 166).
1104-^24.75 = 44.6+ Pch., Ans.
PROMISCUOUS EXAMPLES. 88
El. 27, 640 bu. 3 pk. produce of the farm.
160 " ** 6 qt., i for the rent.
480 bu. 2 pk. 2 qt. to be shared among A^B, and G.
170 " 2 " 6 " A's share.
309 bu. 3 pk. 4 qt. left for B and C.
147 " 3 ** 6 " B's share.
161 bu. 3 pk. 6 qt., C's share, Ans,
Ex, 28. 13 lb. 8 oz. 11.4 dr. = 13.54453125 lb. Troy.
13.54453125 lb. X |Jf = 16.46036783 +lb. Av.=
16 lb. 5 oz. 10 pwt. 11.7 4-gr., Arts.
Ex. 29. 154 bu. 1 pk. 6 qt. = 154.4375 bu.
$j73.7£>a^ $m.74x^^ ^^3^^ ^^^^
154.4375 30.8875
Ex. 30. .0125 T.x 2000 = 25 lb., Ans.
Ex. 31. j\ of 2 bu. 3 pk.= Jl pk. ; 3 bu.=12 pk. ;
^1^12 =^^\, ^715.
Ex. 3^ 4- X I X I X -i-V-^ X 2 K=H?^^=1'747V, Ans.
Ex. 33. V- X J/ x^3Q-xi-V-^x^yio,T = Vr32'= 301.339 f ,
Ans
„ ^^ $.26x36x20 ^^^^^ .
Ex. 34. =$20.80, Ans.
y
^ ^ 46x20x2x144
Ex 35. = 13248,^715.
4x5
<ix. 86. 1864 yr. 4 mo. 20 da. 18 h. 30 min
1836 " 9 " 4 " 3 " 45 "
27 vr. 7 mo. 16 da. 14 h. 46 min.
84 COMPOUND NUMBERS.
Ex. 37. 28 ft. 9' = 28f ft. ; 22 ft. 8' = 22| ft. ; 1 ft. 6=1^ ft
i^i X -V X y X 2V=-H-=1S1 5V cu. yd., Ans.
Ex. 38. 30 bu. 54 lb. = 30.9 bu.
$1,375 X 30.9c^$42.4875, Ans,
Ex 39. 24 ft. 8'==:24| ft. ; 18 ft. 6'=18i ft. ;
Ex. 40. 54 bu. 8 lb.=:54^ bu.
$.84x541 =$45.50, ^715.
Ex. 41. 18720-r-120=:156, Ans.
^ ,^ 21x5280x12 ^^^^^ .
Ex. 42. =44352, Ans.
30 '
Ex. 43. 90^=324000''; 3' 12^ = 192';
324000-^192 = 1687| mm.=
1 da. 4 h. 7 min. 30 sec, Ans.
Ex. 44. 65 mi.=65 x 63360 in. ; 9 ft. 2 in.=110 m.;
65x63360
= 65x576 = 37440, Ans.
110
Ex. 45. 10 bu. X 2150.4 = 21504 cu. in.
21504x4 4096 ^^^4 ^
21504^571 =--3-^=.— =372- qte.
$.22 X 372yV=$81.92 sold for ;
$ 5 X 10 =$50.00 cost.
$31.92 gain, Ans,
„ _ 240x6x3x1728 ,,,,,^, .,
Ex. 46 ■ — =116640 bricks;
8x4x2
$3.25 X 116.64=$379.08, Ans.
PERCENTAGJfl.
85
Ex. I
PERCENTAGE.
(S34,
, page 206.)
.03
Ex. 2.
.0625
.06
.0875
.09
.3333 +
.14
.075
.24
.104
.40
.09625
1.125
1.035
1.50
2.25
Ex. 3.
Ex 4 -3- • J- • -3- • -O- • 14.1 • 113. •
^^» '*' SOI Js> 26» 200) 800) tof)
.0025
.0075
.00666 +
.008
.00625
.0125
.028
.04333 +
.0575
.07125
.122
.25375
11 3 .*
4 0)
^8 .
26 )
(S355 page 207.)
Ex. 9. ^W5. 63 sheep. Ex. 10. Ans, 620 men.
Ex. 12. Ans, $22692.25.
Ex. 20 1.00 — .25=:.75; 760 sheep x .75=570 sheep, ^w«
Ex.21. .18+.30=:.48; 1.00-.48=.52;
$24500 X .52=$12740, Atis.
1576 barrels x.l 2 5 = 197 barrels, Ans.
Ex, 22
Ex.2?
.75=1; .33i=i;
^ — 1=
1^)
$2760 X y\=$1150, A71S.
Ex.24. jXyVo=/5 sold; |-^\=i| left, ^n«.
86 PERCBNTAGB.
Ex. 25. I, owed after the 1st payment,
fxf, " " *' 2d
I X f X I, " " " 3d "
$^f ^ X f X f X ■i=$226.5G|, Ans.
(S36, page 208.)
Ex. 2. 90-^450 = .20=:20 per cent., Ans,
Ex 3 175^1400=.125 = 12i per cent., Atv*.
Ex. 4. 165-v-'750r=.22 = 22 per cent., Ans.
Ex. 5. 13.20-^240=. 055 = 51 per cent, Ans,
Ex. 6. .15-f-2 = .075 = 7^ per cent., Ans,
Ex. 1. 6 bu. 1 pk.=200 qt; 4 bu. 2 pk. 6 qt.= 150 qt.
150^200 = .'75 = 75 per cent., Ans,
Ex. 8. 15 lb. = 240 oz. ; 5 lb. 10 oz.=90 oz. ;
90-r-240 = . 375 = 37^ per cent., Ans,
Ex. 9. 40-T-250=.16 = 16 per cent., ^?i5.
Ex. 10. 100 + 90 = 190;
190-f-760=.25 = 25 per cent., Ans,
Ex. 11. I of f =|=.50=50 per cent., Ans
(237, page 209 )
Ex. 2. 16^.08 = 200, Ans,
Ex. 3. 42-r- .07=600, Ans,
Ex. 4. 75-^.125 = 600, ^?i5.
Ex. 5. 33^.0275 = 1200, Ans.
Ex. 6. $281.25-^.375 = $750, Ans.
Ek, 7. 50-^.20 = 250, Ans,
Ex.8. $59.75-T-.125 = $478, ^W5.
Ex. 9. $975^.15=16500, Ans.
Ex. IC .40x.25=.l
$1246.50-^.l =$12465, Ans,
Ex.11. 2000-r- .40 = 5000; 5000-2000 = 3000, Jn.<f.
Ex.
2.
Ex.
3.
Ex.
4.
Ex.
5.
Ex.
6.
Ex.
7.
co3iMissioi;r akd brokerage. 87
(238, page 211.)
1.00-f .18 = 1.18; 14754-1.18 = 1250, Ans,
1.00-h.25=3l.25 ; $4.00-^1.25 = $3.20, Ans.
1.00 + . 15 = 1. 15; $6900-T-1.15 = $6000, Ans.
1.00 + .08,V-1.08A^;
$432250^1.080625 = 1400000, Ans.
1.00 + .041 = 1.0425 ;
$8757^1.0425=18400, Ans.
Since he increased his capital the first year by
20 % of itself he mast have had 100 ^ + 20 ^, or
120 % of original capital for new capital the sec-
ond year ; and since he increased his new capital
by 20 % he must have had 120 % of 120 %, that
is, 144 ^, of original capital; therefore
$9360-f-1.44 = $6500, Ans.
(239^ page 212.)
Ex.2. 1.00-.15 = .85; 340 -f-. 85 = 400, Ans.
Ex.3. 1.00 — .20 = . 80; $1000-^.80 = $1250, ^7i5.
Ex.4. 1.00-.24 = .76; $4028 -f-. 76 = $5300, ^^5.
Ex. 5. 1.00-.00l=.995 ; 298^--.995 = 300, Ans.
Ex. 6. $198, his selling price, is 90 ^ of his asking price ;
therefore $198-^.90 = $220, his asking price; and
$220, his asking price, is 110 ^ of the cost ; there-
fore $220-f-1.10=$200 cost, Ans.
COMMISSION AND BROKERAGE.
(243, page 213.)
Ex. 2. $6756 X .02=$135.12, Ans.
Ex. 3. $17380 X .035 = $608.30, Ans.
87a PEECEKTAGE.
Ex. 4. $.75x4700=$3525;$3525x.015=$52.875,^n5.
Ex. 5. $25875 x .0075 = 164.6875, Ans,
Ex.6. $32844-12176. 50 = $5460.50;
$5460.50 x.0225=:$122. 86 +, ^W5.
Ex. 7. $2890 X 54^=:$23.12, Ans.
Ex. 8. $.32 X 26750=$8560 ;
$8560 X .02| = $235.40, Ans,
Ex. 9. 400 X 570 x $.09 x .0225 = $461.70, Ans.
Ex. 10. $7.60x450 =$3420
.25x56x38 = 532
.09x48x105= 453.60
$4405.60 X .055 = $242,308,^/15.
Ex.11. $950x.06i = $61.75, fee; $950— $61.75
= $888.25, remitted.
Ex.12. $30456.50 x.06 = $1827.39
19814.15
$10642.35 X .04= 425.694
« $2253.084, Ans.
(244, page 215.)
Ex. 2. $3246.20^]. 02 = $3182.55 (nearly) invested;
$3246.20 — $3182.55 = $63.65, Ans.
Ex. 3. $9682-M.03 = $9400, Ans.
Ex. 4. $10246,50-^1.035 = $9900 invested ;
$9900-^$5.50 = 1800, Ans.
Ex. 5. $4908^1.045 = $4695.69 + , Ans.
Ex. 6. $603.75-7- 1.05 = $575 invested;
$575-^$5 = 115, Ans.
Ex. 7. .03 + .015 = . 045
$9376.158-+1.045 = $8972.40, to pay out;
$9376.158 — $8972.40 = $403.758, fees, Ans.
STOCKS. 876
Ex. 8. $13842.0'7-^1.01Y5=:$13604, invested ;
$13842.07 — $13604=$238.07, commission, Ans.
Ex. 9. $10650^1.0025==$10623.44 + , Ans,
STOCKS.
(261, page 218.)
Ex.2. $1200x.95 = $114'0, ^715.
Ex. 3. $3500 X .85=^$2975, Ans.
Ex. 4. $1.00 + $.05l+$.00l=:$1.06,costofdoll.ofstock;
$150x48 = $7200, nominal amount;
$1.06 X 7200=:$7632, ^?Z5.
Ex. 5. $1.09| X 5364=:$5853.465, Ans.
Ex. 6. $6275 X .12=:$753, Ans.
Ex. 7. 1.00 + .04| + .00|r= 1.05;
$25000 X 1.05=r$131250, Ans.
Ex. 8. .14 + .125 = .265, rate of gain ;
$4200 X .265 = $11 13, Ans.
Ex. 9. $17500 X 1.0075 = $l7631.25, Ans.
Ex. 10. .03 +.0225 = .0525, rate of gain;
$50 X 75 =$3750, nominal amount of stock ;
$3750x.0525=$196.875, Ans.
Ex.11. $50x28 = $1400; $1400 X 1.07 = $1498 ;
$100x25 = $2500; $2500X.875 = $2187.50;
$2187.50 — $1498=:$689.50, Ans.
Ex. 12. $3600x.95 = $3420;
$2700xl.03=:$2781;$3420-$2781=$639,^n^
Ex. 13. $12000 X.145 = $l 740, Ans.
(262, page 219.)
Ex. 2. $6300 = 1. 05 = $6000 = 60 shares, Ans.
Ex. 3. $6187.50-+.90=$6875, Ans.
88 PERCEISTAGE.
Ex. 4. $53500-^1. 07 = $50000, Ans,
Ex. 5. $1150-^-.92 — $1250, nominal amount;
$1250-^50:=$25, Ans,
(265, page 222.)
Ex. 2. #867^1.02=z$850, stock purchased ;
$850 X .06 = $51, income, Aiis.
Ex. 3. $8428-^.98 = $860Q, stock pm'chased;
$8600 X. 05 = 1430, income, Ans.
Ex. 4. 1.04|-f .00| = 1.05;
$10500-^1.05c=:$10000, Ans,
Ex.5. .87 + .00i = .875 ;
$4795-f-.875 = $5480, stock purchased ;
$5480 X. 05 = $274, income, Ans,
Ex. 6. 1.07i + .00i- = 1.08 ; .96i + .00-i=.97 ;
$10476-M.08=:$9700, purchase of G's ;
$10476-^.97 = $10800, " " 5-20's;
$9700 X .06 =$582, income from 6's ;
$10800 X. 05 = $540, " " 5-20's;
$42, Ans.
Ex. 7. 1.08| + .00| = 1.09;
$125xl09 = $13625;
$13625-M.09 = $12500, stock purchased;
$12500 X .06 =$750, income from stock ;
$750 — $681.25=$68.75, income increased, Ans
(266, page 223.)
Ex. 2. $840-^.06 =$14000, stock required ;
$14000 X .95 = $13300, investment, Ans.
Ex. 3. $1860-^.05 =$37200, stock required.
$37200 X .98^ = $36642, investment, Ans.
Ex. 4. $1 080 -f-. 05 = $21600, stock required;
$21600 X 1.08l = $23436, investment, Ans.
GOLD IKYESTMEKTS. 88a
Ex, 5. $25000 x.93| 3=123437.50, proceeds of 5-20's ;
$960^.06 = 116000, 6's required ;
$16000 xl.OQi =$17480, investment in U. S. 6's I
$23437.50-$l7480 = $5957.50,costofhouse,^w5.
(267^ page .223.)
Ex. 2. .06-^.87=:.06|g=:6ff %, -^ns.
Ex. 3. .06-M.05 = .05f =:5| %, Ans.
Ex. 4. .06^.75 = .08=:8 ^, Ans.
Ex.5. .06-^1.08=:.05| = 5| ^, ^715.
Ex. 6. .05-.985 = .05J/^ = 5J^, %;
,0Q-~1.0d = .05j%% = 5j%% %, better, Ans.
(268^ page 224.)
Ex 2. .06-^.09 = .66| = 66| ^, -^715.
Ex, 3. .06-f-. 05 = 1.20;
1.20 — 1.00 = .20 = 20 fo premium, Ans.
Ex. 4. .05^.06 = .83l = 83l %, ^ns.
Ex. 5. .05^.07 = . 7l|;
1.00 — .711 = . 28| = 28| ^, Ans.
GOLD Il^rYESTMEKTS.
(270^ page 225.)
Ex. 2. $1.47 X 150 = $367.50, Ans.
Ex. 3. $1.37| X 320.50 = $440.68|, Ans.
Ex. 4. $1.33 X 2500 = $3325, Ans.
Ex. 5. $8000 X .05 = $400, income in gold ;
$1.38x400 = $552, " " ciivvency, Ans.
Ex. 6. $9500 X .06 = $570, income in gold ;
$1.40x570 = $798, " " currency, ^^^5.
886 PEECEKTAGB.
Ex. 1. $1,475 X 3000=$4425, cost of house in currency
^ by latter offer ;
$4500 — $4425 = $75, gain, Ans.
271, page 226.)
Ex. 2. $1.00^$1.38i=r:$7225^6_^ Ans.
Ex. 3. $].00-~$1.45i=.68|| ;
1.00 — .68||=.31^iy=:31^^y %, Ans. to first.
$1.00-^$1.47 = .68y|^;
1.00-.683|^=r.31j||=:311||^,^w5, to second.
$1.00-^$1.955 = .5l3W ;
1.00-.51-3\\=3.48||f = 48|§f %, Ans, to third.
$1.00^$2.85-— .35|;
1.00 — .35| = .64f = 64| %t ^^s. to fourth.
Ex. 4. $4181-^$1.48=:$2825, Ans.
Ex. 5. $.24-^$1.60 = $.15, Ans.
Ex. 6. $5900 X .90 = $53 1 0, proceeds of sale of 10-40's ;
$5310-^$1.47^ = $3600, gold purchased, Ans.
Ex. 7. $1.00^$1.45=$.68|f, Ans.
Ex. 8. $792-f-$1.65 = $480, Ans.
Ex.9. $1.00-^$.30:=-3.33^ ==333^ ^, /4w5. to first.
$1.00-^$.45 = 2.22| =222f ^, *• " second.
$1.00^$.54=:1.85f^ = 1853/^^, " " thu'd.
$1.00-r-$.60 = 1.66| =:166f %, " " fourth.
$1.00~$.74=rl.353\ = 135/^ ^, *• "fifth.
Ex. 10. $126-^$1.40=$90, value of currency in gold;
$90^$3.50 = 25f yards, Ans.
Ex.11, $11.75-^$1.50=|7.83l,pu^ch. price of flour,gold;
$10.35-^$1.35=:S7.66|, selling " " " "
$7.831 — $7.66| = $.161, loss in gold on one barrel ;
$.16§ x300 = $50, entire loss, Ans.
PROFIT Ais^D LOSS. . 89
Ex 12. $.07-^$1.40 = .05 = 5 %, rate of income in gold
from mortgage ;
.06 — .05=:1 ^, 5-20's better, Ans.
Ex. 13. $51100 X. 073=: $3730.80, inc. from Y.SO's, in cur.;
$3'730.30^$1.46=$2555, '" " '/.30's in gold;
$51100 X 1.04 =$53144, proceeds of 7.30's;
$53144-^$1.46 = $36400, gold purchased;
$36400-f-$.70 = $52000, 10-40's purchased;
$52000 X .05 = $2600, inc. from 10-40's in gold;
$2600 — $2555 = $45, income increased, Ans.
PROFIT AKD LOSS.
(27 3^ page 228.)
Ex. 2. $84.80 X .125 = $10.60, Ans.
Ex. 3. $1.15 x500=:$575, cost of wheat;
$575x.l6~$95.83i, Ans,
Ex.4. $3,625 X 76 =$275.50, cost of wood; *
$275.50 X .26=$71.63, Ans,
Ex. 5. $1.75 X 40 = $70, cost ; $70 x .14f =$10, Ans,
Ex. 6 $.0825x230 x3 = $56. 925, cost; .18y\ = -,\;
$56,925 X j\ =$10.35, gain ;
$56,925 +$10.35 =$67,275; 230 lb. X 3 = 690 lbs.
$67.275 -f-690 = $.0975, selling price, Ans.
Ex. 7. $.625 X 3840 x .375 = $900, Ans.
Or $.625=$^; .375 =|; ssjj). x | x | = $900, ^/25.
90 PERCENTAGE.
Ex. 8. $4720 X .125=:$590, loss in the bargain ;
14720— $590=$4130 ; $4130 x .15=$619.50 loss in bad debts
$590-l-$619.50=$1209.50, Ans.
Ex. 9. 1 + .225 = 1.225, bis per cent, after 1 year ;
1.225 X 1.30 = 1.5925, bis per cent, after 2 years ;
1.5925 X I =1.3270f, bis per cent, after 3 yeara
$3000 x.3270f =$981.25, Ans.
(27 4, page 229.)
Ex. 2. $330— 1275 = $55, gain ;
$55-^$275=.20, Ans.
Ex. 3. $.75— $.60=$.15, gain ;
$.15-^$.60=.25, Ans.
^ ^ $114,885 ^^ ,
Ex.4. - ^ ^, ^^ —.23, Ans.
108 x $4,625 '
Ex. 5. $.095— $.08 =$.01 5, gain on 1 lb. ,
$.015-^$.08=. 1875 = 183 per cent., Ans.
Ex. 6. $42 X 150=$6300 ; $6300— $5400=$900-
$900-r-$6300=.14f, Ans.
Jfix. 7. $25— $15=$10 ; $10-t-$25=.40, Ans.
Ex. 8. $.25 X 20=$5.00, received per ream ;
$5.00— $2.00 = $3.00, gain per ream ;
$3.00-^-$2.00 = 1.50=150 per cent., Ans.
Ex. 9. If I sells for J its cost, 1 sells for | = | its cost
f — 1=1^, gain on 1, =.50 = 50 per cent., Ans.
Ex. 10. i-T-f =f ; or tbe whole would be sold for f of itfl
cost ; hence f of the cost Avas lost. And,
1 = 371 per cent., Ans.
Ex. 11, One peck is gained on 3 pecks ; hence
l-f-3=.33i per cent., Ans,
PKOFIT AND LOSS. 91
Ex. 12.
8 7 J- per cent., Ans.
Ex. 13.
342 lb. @$.08 ==$27.36
3781b. @ .081= 32.13
$59.49 sold for ;
720 lb. X .07 = 60.40 cost ;
$9.09 gain ;
$9.09-^$50.40=.182V per cent., Ans,
Ex. 14. $1.60 — $1.25=1.35, gain per gal. ;
$.354-$1.25=.28, gain per cent. ;
$1.25 X 63 X 2 X .28 =$44.10, whole gain, Ans,
Ex. 15. $.66— $.55 = $.ll, gain per bushel on the corn ;
$.ll-f-$.55=.20, gain per cent, on the corn.
$1.375 -$1.10=$.275, gain per bushel on the wheat ,
$.275-r-$1.10 = .25, gain per cent, on the wheat;
.25— .20=. 05 per cent, on the wheat, Ans.
(275, page 231.)
^ ^ $140x1.25 ^^^ ,
Ex. 2. =$.14, Ans,
1250 '
Ex. 3. $.30 X 1.16| X 1.33i=$.46|, Ans.
Ex. 4. $.105 X l.l7i=$.1232, Ans.
Ex. 5. 1.00-.15 = .85
$.62ix.85=$ .531 y
1.20 x.85= 1.02 I Ans.
3.875 X. 85= 3.29f )
Ex. 6. $3240 X .82 = $2656.80, Ans.
Ex.7. $28xl20=$3360; $3360 + $480 = $3840, whole
cost; $3840xl.l2i=$4320; $4320 — $3840=$480, gain ;
$4320-^120 = $36, an acre, Ans,
Ex. 8. $2.60 X 52 = $135.20 ; $135.20 x If =$185,90 ;
52-^7 = 45; $185.90-4-45=$4.13i, ^?i5.
92 PERCENTAQB.
Ex.9. 1.18f = if; 23i=-V-;
$3.±s. X If X j\ X yV=82.85, An9.
(276, page 232.)
Ex. 2. $.08-^.80 = 1.10, Ans,
Ex. 3. $6.125-^.875 = $7.00, ^/i5.
Ex. 4. $.96-^1.28 = $.75, Ans.
Ex. 5. 1.18Jr=if ; $1881-^if = $1584, Ana.
Ex.6. $69.75 ^^^, .
Ex. 7. $ V- X -y- X VV=$86.25, ^W5.
Ex. 8. $96-^80=$120, cost;
$120xl.l5 = $138, Ans.
Ex.9. 1.125 = 1; 1.18f = if ;
$570 X If X f =$426,661, Ans.
Ex. 10. $24x4=$96 whole, proceeds.
$24 X 2=$48 ; $48-M.20 = $40, cost of 1st 2 bW.
$48-^ .80 = $60, cost of 2d 2 bbi.
$40 + $60 = $100; $100— $96=$4, lost.
Ex. 11. $4900-f-1.40=$3500 = 3timeswhatLe began with;
and $3500-r-3=$1166.66|, Ans.
mSURANCK
(282, page 234.)
Ex. 2. $750 X .04=$30, Ans.
Ex. 3. $4572.80 x .025=$114.32, Ans.
Ex. 4. $5700 X .01 75 = $99.75, Ans.
TAXES. 98
Ex. 5. $28400 X .035 = $994, Ans,
Ex. 6. $55800 X .028zr:$1562.40, premium ;
$55800 -$1562.40z=$54237.60, Ans.
Ex. 7. $47500 X ^-f ^==$356.25, Ans,
Ex.8. $8000 + 14000=:$12000;
$12000 X .021 = 8285, Ans,
Ex. 9. $1.20 X 4000=$4800, worth of wheat;
$4800 X I =$3200, amount insured for;
$3200 X f X 2? — *3^> premium ;
$3200— $36==$3164, saved by insuring;
$4800 — $3164=$1636, owner's loss, Ans,
Ex. 10. $21000 X 5^0=^^168;
$15400x^^0=^ 9^-25;
$264.25, Ans,
TAXES.
(289, page 23G.)
Ex. 3. Property tax ==$26.95
1 poll .75
$27.70,^^15.
Ex. 4. $.50 X 2981 =$1490.50, poll tax ;
$9190.50— $1490. 50=$7700, properly tax ;
$7700-^-$l,400,000 = .0055, rate of taxation.
$12450 x.00553=$68.475
2 polls =$ 1.00
C's tax $69,475, Ans,
Ex. 5 $5375 X .0055 =$29.5625, Ans.
K. p. 5
M PERCENTAGE.
Ex. 6. $.625 X 30 = ^18.75, poll tax ;
$4342.75— $18.75 = $4324, property tax;
$4324-^$188000=:.023, rate of taxation.
$2500 X. 023 + $.625 =$58,125, Ans.
Ex. 7. $.30 X 25482 = $7644.60, poll Jpx ;
$103294.60 — $7644.60=$95650, property tax ;
$95650-J-$38260000=.0025, rate of taxation.
$9470 X. 0025 -f-$.90 = $24.575, Ans.
Ex. 8. $10000 X 1.025 = $10250, whole tax;
$1.25x225=:$281.25, poll tax ;
$10250-$281.25=$9968.75, property tax;
$9968.75-^$1246093.75=:.008, rate of taxatioL .
$11500 X .008+ $1.25 = $93.25, E's tax, Am.
Ex. 9 $275.57 — $98=$177.57, tax ;
$l77.57-^3946=$.045, rate per day;
$.045 X 118 X 2=$10.62, Ans.
CUSTOM HOUSE BUSINESS. 95
CUSTOM HOUSE BUSINESS.
(303, page 239.)
Ex. 2. $-95 X 224=$212.80, value of the silk ,
$212.80 X.19=$40.432, Ans.
Ex. 3. $.54 X 31.6 X 50=$850.50, gross value ;
deduct $850.50 x .02=5 17.01 for leakage ;
$833.49, net value;
$833.49 X .24 =$200.0376, Ans.
Ex. 4. $.15 X 115 X 175=$3018.75, value of the coflfU?
$3018.75 X .15=$452.81J, Ans.
Ex. 5. $.36 X 63 X 25 =$567, gross value ;
deduct $567 x .005= 2.835, for leakage ;
$564,165, net value;
-^ $564,165 X .24=$135.3906, Ans.
PERCENTAGB.
SIMPLE INTEREST.
(311, page 241.)
Ex. 3. $45.92, Ans, Ex. 8. $093.83+, Ara
Ex. 11. $607.50, Ans, Ex. 15. $440,625, Ans.
Ex. 17. $605.70 + liit. for 3 yr.=$751.068, Ans.
(313, page 245.)
Ex. 7. $106,855, Ans. Ex. 8. $1.72+, Ans.
Ex. 11. $91.85 + , Ans. Ex. 15. $2,138+, Ans. '
Ex. 18. $24.87+, Ans. Ex. 19. $282.75+, Ans.
Ex. 22. $82.36+, Ans.
Ex. 23. Time, 7 yr. 7 mo. 2 da.; $51.98 + , Ans.
Ex. 24. Time, 2 yr. 1 mo. 4 da. ; $4,474, Ans.
Ex. 25. Time, 11 yr.3mo.27 da.; $19.818f + , Ans.
Ex. 26. Time, 9 mo. 19 da.; $408,957 + , Ans.
Ex. 27. First payment, $2000
Second payment, $3157.50
Third payment, $1105.
$6262.50, Ans.
Ex. 28. $350. + int. for 11 mo. 21 da.= $373,887 +
$150. +int. for 8 mo. 16 da.= 157.466 +
$550.50 + int. for 3 mo. 11 da.= 561.310 +
Total = $1092.663 + , Ansi
PARTIAL PAYMENTS.
97
PARTIAL PAYMENTS.
(314, page 249.)
Kx 3. Amt.ofnotetoNov.l2, 1858, (4mo. 22da.)$535,27-f"
Payment, 105.50
New Principal, $429.77-+-
Amt., Mar. 20, 1860, (16 mo. 8 da.) 488.03 +
Payment, 200
New Principal, $288.03 +
Amt., July 10, 1860, (3 mo. 20 da.) 296. 83 -j
Payment, 75.60
New Principal, ..$221.23 +
Amt., June 20, 1861, (11 mo. 10 da.) $242.12+,
Ann
Ex. 4. Amt. of note, Jan. 1, 1860, (7 mo. 24 da.) $3136.50
Sum of payments to tliis date, 525.00
New Principal, $2611.50
Amt., April 4, 1861, (15 mo. 3 da.) 2841.53 +
Sum of payments, 1575.00
' New Principal, $1266.53 +
Amt., Feb. 20, 1862, (10 mo. 16 da.) $1344.35+,
Ex. 6. Amt. of note, Jar.. 1, 1852, (16 mo. 28 da.)$977.15 +
Payment, 250.00
New Principal, $727.15 -
Amt. May 4, 1853, (16 mo. 3 da.) 775.93 +
Payment, 316.75
New Principal $459,18 4
Amt. Sept. 15, 1853, (4 mo. 11 da.) $467.53+,
5 Ans
98 PERCENTAGE.
Ex. 6. Interest commence 1 Aug. 2, 1860.
Amt. of note, May 6, 1861, (9 mo. 4 da.),. $192.988 -f
Payment, 50
New Principal, $142,988 +
Amt. Aug. 26, 1862, (15 mo. 20 da.) $154,188+.
Ex. 1. Amt. of mortgage, Jan. 1, 1852, (3 mo.). .$6120.00
Payment, 500
New Principal, $5620.00
Amt. Sept. 10, 1852, (8 mo. 9 da.) 5930.98 +
Payment, 1126.00
New Principal, $4804.98 +
Amt. March 31, 1854, (18 mo. 21 da.) . . . 5404.00 -f-
Payment, 2000.00
New Principal, $3404.00 +
Amt Aug. 10, 1854, (4 mo. 9 da.) 3501.57 +
Payment, 876.50
New Principal, .$2625.07 +
Amt. Oct. 1, 1857, (37 mo. 21 da.) $3284.84 +
(315, page 251.)
Ex. 1, Amt. from Jan. 1, 1858, to Jan. 1, 1859,
(1 yr.) $487.60
Amt. of 1st pay't from Apr. 16, 1858, to
Jan. 1, 1859, (8 mo. 15 da.) 154.29
New Principal, $333.31
Amt. from Jan. 1, 1859, to Mar. 11, 1860,
(14 mo. 10 da.) 357.19 +
Payment, 75.00
New Principal, $282.19 +
PARTIAL PAYMENTS. H
Ami. from Mar. 1 1, 1860, to Dec. 11, 1860,
(9 mo.) $294. 89 +
Amt. of 3d pay't from Sept. 21, 1860, to
Dec. 11,1860, (2 mo. 20 da.) 56.74 +
Ans, $238.15 -h
(316, page 251.)
Ex. 1. Amt. of Principal, Jan. 1, 1859,
(2 yr. 8 mo. 20 da.) $698.00
A.mt. of 1st endorsement, (for 2 yr.
4 mo. 21 da.) $178,386
Amt. of 2d endorsement, (for 1 yr.
10 mo. 19 da.) 222.633
Amt,of3dendorsement,(for7mo.) 191.475 592.494
Ans. $105.50 + ,
(SIT, page 252.)
Ex 1. Amt. of note, Aug. 4, 1859, (1 yr.) $609.50
Amt. of 1st pay't, Aug. 4, 1859, (9 mo.) . . 66.88
New Principal, $542.62
Amt. of new Principal, Aug. 4, 1860, (1 yr.) 575.17
Amt. of 2d pay't, Aug. 4, 18G0,
(7 mo. 21 da.) $49.85
Amt. of 3d pay't, Aug. 4, 1860,
(4 mo. 18 da.) 253.70 303.55
New Principal, $271.62
Amt. of new Prin., Nov. 4, 1860, (3 mo.). 275.69
Amt. of 4tli pay't, Nov. 4, 1860, (1 mo.
6 da.) 60.30
Ans, $215.33.
100 PERCENTAGE.
(31 85 page 253)
Ex. 1. 1st installment of interest, due Feb. 2, 1856, ^30
2d " " " '' " " 1857, 30
3d " " " " " « 1858, 30
4th « " " " " " 1859, 30
5th " " . " " Aug. 2, 1859 15
4135
1st installment draws int. 3 jr. 6 mo.
2d " " " 2 jr. 6 mo.
3d " " " 1 jY, 6 mo.
4th " " " 6 mo.
Int. of $30 for 8 yr. mo $14.40
Principal, $500.00
Ans, $649.40
PROBLEMS IN INTEREST.
(320., page 253.)
Ex. 2. Int. of $1 for 6 yr. 3 mo. at 6 per cent., is $.395 ;
$28.125^.375 = $75, Ans.
Ex. 3. Int. of $1 for 4 mo. 18 da. at 4 per cent., $.015^ ;
$9.20-^.015i=z$600, Ans.
Ex. 4. $1260^.07 = $18000, Ans.
Ex. 5. $33'70-^.10=z:$33'700, Ans.
(321, page 254.)
Ex. 2. $1 for 8 mo. at 6 per cent., amounts to $1.04}
$655.20-^1.04=:$630, Ans.
Ex. 3. Amt. of $1 for 5 yr. 5 mo. 9 da. at 5 per cent.,
1.27201 ; $106.855 4-1.2720f =$84, Ans.
.^tyh
PROBLEMS IN INTEREST. 101
Ex. 4. Amt. of $1 for 8 yr. 5 mo. at 5 J per cent.,
$1.462916+ ;
$1897.545^1.462916 + =$1297.09+, Ans.
Ex. 6 Amt. of $1 for 3 yr. 4 mo. at 7 per cent., $1.23^ ;
$221.075-+1.23i = $l79.25, Ans.
Ex. 6. Amt. of $1 for 11 yr. 8 da. at IQi per cent., $2.1 57 -J ;
$857.54 + 2.157iirr$397.50, principal ;
Int. of $397.50 for 11 yr. 8 da., at 10^ per cent., =
$460.04, Ans.
(322, page 255.)
Ex. 2. Int. of $500 for 3 yr. at 1 per cent., $15 ;
$45 -+$15 = 3 per cent., Ans.
Ex. 3. Int. of $180 for 1 yr. 2 mo. 6 da. at 1 per cent.,
$2.13 ; $12.78-+$2.13=:6 per cent., Ans.
Ex. 4. Int. of $2000 for 6 mo. at 1 per cent., $10 ;
$75-^$10 = 7^ per cent, per annum, Ans.
Ex. 5. Int. of $1000 for 3 yr. 3 mo. 29 da. at 1 per cent,
$33,305+ ;
$183.18-+$33,305=:5.5 per cent., Ans.
Ex. 6. Int. of $21640 for 1 year at 1 per cent, $216.40 ;
$2596.80-^$216.40=rl2 per cent, Ans.
(323, page 256.)
Ex. 2. $325 X .06 =$19.50, int for 1 yr. ;
$58.50-+$19.50 = 3 yr., Ans.
Ex.3. $1600x.06 = $96; $2000-$1600=$400;
$400+-$96 = 4J yr-=4 yr. 2 mo., Ans.
Ex.4. $204x.07--$14.28; $217.09 — $204=$13.09;
$13.09 -; $14.28 = f J- yr. = ll mo.. Am,
102 PERCENTAGE.
Ex. 5. $750 X .Oe = $45 ; $942 — S750=$192 ;
$192-v-$45 = 4y4j yr. = 4 yr. 3 mo. 6 da., Am.
Ex.6. $200 X. 06 =$12;
$200^$12=:16 I yr.=16 yr. 8 mo., An8.
Ex.7. $675x.05=r$33.75;
$675-^$33.75 = 20 years, Ans.
COMPOUND INTEREST.
(324, page 257.)
Ex. 2. $500.00 Prin. for 1st year.
35.00 Int. " " "
$535.00 Prin. " 2d **
37.45 Int. " " **
$572.45 Amt. " 2 years.
600. Given Prin.
Ajis, $ 72.45 Compound interest.
fix. 3. $312.00 Prin. for 1st year.
18.72 Int. " " "
$330.72 Prin. " 2d "
19.84 Int. " " "
$350.56 Prin. " 3d "
21.03 Int.
Ans. $371.59 + , Arat. " 3 year^
COMPOUND INTEREST. 103
Ex. 4. $250.00 Prin. for 1st. half year.
7.50 Int. " " "
$257.50
7.72
$265.22
7.96
Prin.
Int.
Prin.
Int.
Prin.
Int.
Amt
u
u
u
u
2d
3d
$273.18
8.19
4tli
u
$281.37
250.00
2 years.
Ans, $31.37 + Compound interest
Ex. 5. $450.00 Prin. for 1st quarter.
7.87 Int. " " "
$457.87
Prin. "
2d
8.01
Int. "
Prin. "
u
$465.88
3d
8.15
Int. "
Prin. "
((
$474.03
4th
8.30
Int. "
((
^?6S. $482.33+ Amt. " 1 year.
Ex. 6. $236.00 Prin. for 1st year.
14.16 Int. " " "
$250.16 Prin. " 2d
15.01 Int. " "
$265.17 Prin. " 3d
15.91 Int. " "
$281.08 Prin. " 4th •*
16.86 Int. « « -
IGil PERCENTAGE.
1297.94 Prin. " 1 mo. G d*.
10.72 Int. " 7 " 6 "
$308.66 Arat. " 4 yr. 7 mo. 6 dfi,
230.00 Given principal.
Ans, $72.66+ Int. 4 yr. 7 mo. 6 da.
Ex. 7 $700.00 Prin. for 1st year.
49.00 Int. " " "
$749.00 Prin. " 2d "
52.43 Int. " " "
$801.43 Prin. " 3d "
56.10 Int. " " *'
$857.53 Prin. " 9 mo. 24 da.
49.02 Int. " 9 " 24 "
Ans, $906.55 -f,Amt." 4 yr. 9 mo. 24 da.
Ex. 9. $120 X 2.078928==$129.47 + , ^/2S.
Ex.10. $.10x3.86968=$.386968, ^^5.
DISCOUNT.
(326, page 259.)
Ex. 2. $180-M.20r=:$150, Ans,
Ex. 3. $1315.389^1.175 = $1119.48, Ans,
Ex. 4. $866.0384-1.281i = $675.888-f,pref. worth. [
$866.038— $675.888 + =$190.15+. viuv-ft ] '**
^x. 5. $1005 — $475=$530
$475-^1.05 =$452.38 +
$530-^1.075 = 493.02 +
$945.40 f , Ans.
PROMISCUOUS EXAMPLES, 10t5
Ex. 6. Term of discount, 6 mo. 24 da.
$529.925-^1.034 = $512.50 present worth.
$529.925 — $512.50 = $17.425 discount, Ans.
Ex. 7. $3675 cash offer.
$4235-^1.21=$3500 cash value of note.
Am, $ 175, loss.
Ex. 8 $550-^1.10 = $500, present value of note ;
$480, cash offer ;
Ans, $ 20, gain.
Ex. 9. $517.50-^1.035:=$ 500
$793.75-M.05f=:$ 750
$1326.47-f-1.105 = $1200.426-f-
$2450.426 + , entire pres. worth
$2637.72-$2450.426 + =:$187,29-f,^/^s.
Ex. 10. $.10 X If r=$y^^,int. of $1 for 10 mo. at 10 per cent
$130^1y'2 =::$120 ; $130 — $120 = $10.00, discount.
$130 X yV =$10,831, interest.
Ans, $.83^.
PROMISCUOUS EXAMPLES IN PERCENTAGE.
(Pag^ 260.)
Ex. 1. .02 4- .25 = . 27 gain per cent, on cost.
V' ^ fo =^tV <^ents, selling price of what remains
of every pound, after transportation ;
Ex. 2. $200 X .40 = $80 gain on one ;
$200 X .20 $40 loss on the other;
Ans, $40.
106 PEKCENTAGB.
Ex. 3, $425-^1.03=$412.62+ cash value of sale;
$425 — $25 =$400.00 cost ;
Ans, $1^.624, profit.
Ex.4 $.13-f-1.04 = $.125; $.13— $.125=$.005 ;
$.005-r-$.125=.04, ^?i5.
Ex. 5. $150-^-1.25=$120 costof one;
$150^ 15= 200 cost of the other;
$320 cost of both;
$320 — $300 = $20, Ans.
Ex. 6. Amt. of $1 for 2 yr. 8 mo. at 9 per cent., $1.24 ;
jiyji X |« J X f =$3750, Ans.
Ex. 1. 1.00 — .07 = 14f years, Ans.
Ex. 8. 3 yr. 4 mo.=r3i yr. ; .121 x 3i=:.4is., whole nito
of gain; $5000-r-.41| = $12000, capital ;
$12000 X f = $7500, ^'s, )
$12000 X f — $4500, B's, J
Ex. 9. $800 X .15 = $120, gain on groceries ;
500 X .20:= 100, loss on dry goods ;
Whole gain $20, Ans.
Ex.10. 1.00-.08i=.91|;
$1100h-.91| = $1200, Ans.
Ex. II. $667-f-1.04=$C41.346 +, cash value of goods ;
600x1.06= 636.^ pres. valuation of goods ,
True gain, $5,346 + , Ans.
Ex. 12 $18xi=$6, profits; $18— $6=$12, cost;
$6-7-12 = 50 per cent., profit, Ans,
Ex. 13. If f sell for f of cost, the whole would sell for f x |
>f the cost, which is 1]| times cost. Hence JJ = .40f, is the
S|;ain per cent.
PROMISCUOUS EXAMPLE? 107
Ex. 14. $1.30, received for lumber originally wortli $1.00 ;
$1.06|, valuation of ditto, after 16 mo. int. accrues ;
■ $.231 gain on $1.06|;
$.23iH-$1.0G|=.21|, Ans.
Ex.15. 1-1 = 1; ix^^i,Ans.
Ex. 16. $1,121 x728 = $819, expended in wheat;
.60 X .30 X .50=1.09 ; $819^.09 =:$9100,in bank;
1.00 — .60^.40; $9100x.40=:$3640, Ans.
Ex. 17. 6x5=i30sq. yd., area to be covered ; 4 per cent.
IS 2^, and 5 per cent, is ^V ; hence every yard purchased will
be, after shrinking, || of 1 yd. long, and if off yd.==|-J yd.
wide, and will contain || x f J sq. yd. Therefore, as many
yards must be purchased as |i x f i is contained times in 30.
Ex. 18. 1.00 per cent.=B's money;
1.28 " =A's "
.28-4-1.28-. 211, Ans,
Kx. 19. $1200-T-.12:=$10000, f of his capital;
$10000-4-|=$25000, whole capital;
$25000 X f X yi^=$750, loss on | of the capital ;
$1200— $750 = $450, gain.
Ex. 20. Amt. of $1 for 3 yr. 9 mo. at 10 per cent., $1,375 J
$1933.25^-1. 375=:$1406=i of C's money.
$1406x2=±$2812, C's,
$2812
X2=±$2812, C's, )
x#=$4218,D's, ) •
108 PERCENTAGE.
BANKING.
(336^ page 264).
Ex. 1. Int. of $450 at 6 per cent., for 63 da.=$4.725 lisc't :
$450— $4,725=1445.275, proceeds.
Ex. 2. Int. of$368at7percent.,for93da. = $6.654+,disc't;
$368 — $6,654+ =$361,345 + , proceeds, Ans,
Ex.3. Int. of $475.50 at 5 per cent., for 63 da.=$4.16 + ,
discount; $475.50— $4.16 + =$471.33 + , proceeds.
Ex. 4. Int. of $10000 at 6 per cent., for 93 da.=$155, disc't;
$10000— $155 = $9845, proceeds, Ans.
Ex. 5. Proceds of the note, disc'ted at 6 per cent., $247,375 ;
$247.375— $240=$7.375, Ans,
Ex. 6. Int. of $360.76 at 6 per cent., for 93 da. =$5,591 +,
disc't; $360.76— $5.591 + =$355,168+, proceeds.
Ex. 7. Proceeds of the note, $529.2355 ;
$530 — $529.2355 = $.7645, Ans.
Ex. 8. From Mar. 2 to Apr. 7 is 36 da., term of discount;
Int. of $500, at 6 per cent., for 36 di., is $3.00, disc't ;
$500-$3.00 = $497, proceeds.
Ex. 9. From Nov. 15 to Dec. 15 is 30 da term of discount ;
Amt. of $750 on interest for 6 no. 3 da. at pel
cent, is $772,875 ;
Bank discount of $772,875 for 30 aa., x\ iO km^ g^^av
is $6,440+ ; $772.875— $r.44U^ ^ -^
$766,434+, proceeds
EXCHANGE. 109
(337, page 2G6.)
Ex. 2. $680-^.9895 — $687,215+, Ans.
Ex. 3. $1000-^.9870| = $1013.085+, Ans,
Ex. 4. $o00-+.9935f =:$503.22+, Ans,
Ex. 5 $1256+- .96441=11302.341 +, Ans.
EXCHANGE.
(350, page 269.)
Ex. 3. $1000 X 1.03 = $1030, Ans.
Ex. 4. $400 X 1.0075 :=:r$403, Ans.
Ex. 5. 8530 X 1.0275rrr$544.575, cost of draft ;
20. transportation ;
Ans. $564,575.
Ex. 6. $1— $.01225 =$.98775, proceeds of $1 at b'k disc'tj.
Add .02 premium ;
$1.00775, cost of exchange for $1 ;
$800 X 1.00775 =$806.20, Ans.
Fa. 7. $1— $.0055 = $.9945
Subtract .015 discount;
$.9795, cost of exchange for $1 ;
$420 X. 9795 =$411.39, Ans.
Ex. 8. $1 — $.0225=$.9775; 320 x $10=$3200;
$3200x.9775 = $3128, draft;
312, transportation ;
400, gain ;
To be sold for $3840
$3840^320=$12, Ans.
110 PERCENTAGE.
(351, page 271.)
Ex. 2. $243.60^1.015=1240, Ans.
Ex. 3. $79.20-^.99=:$80, Ans.
Ex. 4. $1— $.0105 = $ .9895
Add .02
$1.0095, draft for $1;
$282.66^1.0095 =$280, An€.
Ex. 5. $1— .0055=$.9945
Subtract .0125
$.982, draft for $1 ;
$240-^.982 = $244.399 + , draft hot. for $240;
1240 — ($240 X. 005)= 238.80 current money for $240 ;
Ans. $ 5.599 + , saved.
Ex. 6. $1.00000
.01225, bank disc't at 7 per cent. (63 da^
$ .98775
.0075 premium.
$3C00-h .99525 = $3617.181 +, draft required.
$3600^ 1.0075 =$3573.200 + , draft sent.
Add int. for 60 da. $ 35.732+ (at 6 per cent.)
Amt. at time req'd, $3608.932 +
$3617.181 -$3608.932 =$8.24+, loss, Ans.
EQUATION OF PAYMENTS. Ill
EQUATION OF PAYMENTS.
(356, page 273.)
Ex. 2. $700x20 = 114000
400x30= 12000
700x40= 28000
$1800 $54000
54000^1800 = 30 da. Average credit
Sept. 25-f30 da. = Oct. 25, equated time.
Ex. 3. $250x4 = $1000
750x2= 1500
500x7= 3500
$1500 $6000
6000 -T- 15 00 =4 mo. average credit,
July 1+4 mo. = Nov. 1, Ans,
Ex. 4. $1x0=$
2x1= 2
3x2= 6
4x3= 12
5x4= 20
6x5= 30
7x6= 42
$28 $112
112-r-28=4 da.
Monday -I- 4 da. = Friday, Ans,
Ex. 5 $650 X 4=$2600
725 X 8= 5800
500x12= 6000
$1875 $14400
14400^1875 = 7.68 mo.=7 mo. 20 da.
Jan. 1 + 7 mo. 20 da. = Aug. 21, Ans,
112
AVERAGINa ACCOUNTS.
Ex. 2.
£x. 3.
(362, page 276.
Due.
da.
Items.
Prod.
Jan. 1
16
Feb. 4
March 3
15
34
61
150
200
100
160
3000
3400
9760
610
16160
16160^610 = 26 da.
Jan. l-}-26 da. = Jan. 27, Ans.
Due.
da.
Items.
Prod.
March 1
April 4
Aug. 18
Aug. 8
34
170
160
300
240
100
400
8160
17000
64000
1040
89160
89160 — 1040 = 86 da.
March 1 + 86 da.=May 26, Ans.
Ex. 4.
Due.
da.
Items.
Prod.
June 1
600
" 12
11
300
3300
" 15
14
832
11648
" 25
24
760
18240
30
29
750
21750
8242 ,
54938
54938-^3242 = 17 da.
June 1 + 17 da.=June 18, Ans.
AVERAGING ACCOUNTS.
118
Ex. 5.
Due.
da.
Items.
Prod.
Jan. 16
Feb. 20
March 4
April 24
35
48
99
536.78
425.36
259.25
786.36
14887.60
12444.00
77849.64
2007.75
105181.24
j 105181.24-^-2007.75 = 52 da.
^^' \ Jan. 16, 1856 + 52 da.=March 8, 1856.
Ex.6.
Due.
da.
Items.
Prod.
April 1
" 28
June 15
27
75
420
135
1800
3645
135000
2355
138645
A71S.
( 139650-^2355 = 59 da.
( Apr. 1 + 59 da.=May 30.
Ex. 2.
(36 3, page 278.)
Br.
Or,
Due.
da.
Items.
Prod.
Due.
da.
Items.
Prod.
Jan. 1
Feb. 4
*' 20
34
50
448
364
232
12376
11600
Jan. 20
Feb. 16
" 25
19
46
55
560
264
900
10640
12144
49500
104423976|
172472284
1044'23976|
Balances
68048308
Ans,
48308-^-680 = 71 da.
Jan. 1+71 da.=March 13
114
RATIO.
Ex.3.
Dr.
Cr.
DnSf
da.
Items.
Prod.
Due.
da.
Items.
Prod.
Apr. 1
June 12
Sept. 3
Oct. 4
72
155
186
145.86
37.48
12.25
66.48
2698.56
1898.75
12365.28
May 11
July 12
Oct. 12
40
102
194
11.00
15.00
82.00
440.001
1530.00
15908.00
262.07
108.00
154.07
—
16962.59
108.00
17878.00
16962.55
Bal acct.
Bal. Prod.
915.41
( 915.41-f-154.07 = 6 da.
' I Apr. 1 — 6 da.=:Marcli 26, 1858.
RATIO.
Ex. 3.
3V = 1, Ans.
Ex. 5.
^ = 5l,Ans.
Ex. 7.
\*~Xj% = l2, Ans,
Ex.9.
ixi=i,Ans.
Ex. 11.
3-0x-^-z=5, Ans.
(379^ page 281.)
Ex. 2. 2\={, Ans,
Ex.4. ^f^=1,Ans.
Ex. 6. 2J.x}={i,Ans,
Ex.8, ^f = 41, Ans.
Ex. 10. -j4_x|=:|, ,Ans,
Ex. 12. 3 gal. = 24 pt. ; 2 qt. 1 pt. = 5 pt. ;
5-^24 = 2^7 ■^^^'
Ex. 13. 8 s. 6 d.=:8.5 s. ; |;f =|f = 1||, Ans.
Ex. 14. -/I^/Zo =tV, ^ris.
Ex. 15. 19 lb. 5 oz. 8 pwt.=4668 pwt. ; 25 lbs. 11 oz. 4
pwt. = 6224 pwt. |||i = li, Ans.
Ex. 16. ||=f, Ans. Ex. 17. f x l = j\y Ans.
Ex. 18. 2^3 x^^rzz^, Ans. Ex. 19. 16-r-2f ==7, Ans.
SIMPLE PROPORTION. 115
Ex. 20. 14.5 X 3 = 43.5, Ans. Ex. 21. J X f =| = li, Ans.
Ex. 22. ^xiz=j\,Ans.
PROPORTION.
(388, page 283.)
„ , 48x50 , ^ . ^ ^ 70x3 ^ .
Ex. 1. =120, Ans, Ex. 2. ■— — = 5, Ans,
20 ' 42 '
Ex. 3. =6, Ans. Ex. 4. -^=12, Ans.
_ ^ 201.75x48 yd. ^^^ , .
Ex. 6. 3 lb. 12 oz. = 60 oz.
10.50x60 oz.
3.50
= 180 oz. = ll lb. 4 oz., ^n5.
Ex. Y. 8 bu. 2 pk.=34 pL; 1Q bu. 2 pk.: ?06 pk.
$38.25x34 ^, , ,
=$4.25, ^n5.
306
Ex. 8. V X ^^ X ^1 ^= Y-=8i, ^Tis.
Ex. 9. V X I X 5 = 7, ^?25. Ex. 10.^ X I X f =|, ^ns.
SIMPLE PROPORTION.
(3995 page 287.)
Ex 1. 48 Cd. : 20 Cd. : : $120 : ( )
, , $120x20 ,^ .
( )=^-jg =$50, ^/w
Or. $120xf? = $50, Ans,
116 PKOPORTION.
Ex. 2. 6 bu. ; 75 bu. : : $4.75 : ( )
, , $4.75x75 ^ ^ ^, ,
( ) = =$59,371, ^W5.
Or, $4.75 X V =$59.37^, Ans.
Ex. 3. $3i : $50 : : 8 yd. : ( )
, , 50x8 yd. ^^^, , .
Ex. 4. 12 : 20 : : 42 bu. : ( )
, , 42 bu. X 20 . ^ , .
( )=z -^ :^10hu., Ans.
Or, 42 bu. X ^:=^10 t>u., Ans.
Ex. 5. $.75 : $9.00 : : 7 lb. : ( )
, , 900 X 7 lb. ^^ ,^ ,
( )=-Y^ ^84lb., ^7Z5.
Ex. 6. 3 lb. 12 07. : 11 lb. 4 oz. : : $3.50 : ( )
60 oz. : 180 oz. :: $3.50 : ( )
, , $3.50x180 ^ -^ ,
( ) = - =$10.50, Ans.
Ex. 7. 1 ft. 6 in. : 75 ft. : : 3 ft. 8 in. : ( )
lift. : 75 ft. :: 3| : ( )
( ) = Y x-y ft. x|==183J ft. = 183 ft. 4 in., ^n*.
Ex. 8. $2.75x VV =$19-^^7 ) ^^^^•
Ex. 9. $13.32 : $51.06 :: 12 bu. : ( )
, , 51.06 x 12 bu. ^^,
( ) — — 46 bu., Ans.
^ ' 13.32 '
Ex. 10. 15hhd.i=945 gal.
945 gal : 28.5 gal. : : $236.25 : ( )
, , $236.25x28.5 . ^^, .
( ^" ^45" -'^.ISi, ^/^s.
SIMPLE PROPORTION. ' 117
Ex. 11. 6 mo. : 11 mo. : : 1 bbl. : ( )
, . 11 xY bbl. ,^, ,, , .
( )=~ =l2^hhl,Ans.
Ex. 12. 5 £ 12 s. : 44 £ 16 s. : : 9 yd. : ( )
, . 9 yd. X 896 .^ , .
( )=-^-u^ = 72 yd., ^715.
Ex. 13. $3100 X J7//^=$310, Ans.
Ex. 14. 100 lbs. cofFee=100 X 1 = 160 lbs. sugar;
2 : 160 : : $.25 : ( )
, , $.25x160 ^^^ ,
( )= =$20,^715.
Ex. 15. 13° 10' SO' : 360° : : 1 da. : { )
47435'' : 1296000" : : 1 da. : ( )
( )r=J-f i| jiiL da. = 27 da. 7 h. 43 min. 6.06 + sec, Ans.
Ex. 16. 8J : 13^ : : $4.20 : ( )
( ) =$4-f-o X V- X 3V=$6.48, Ans.
Ex. 17. 6J d. : 10 £ 6 s. 8 d. : : If yd. : ( )
( )=2.4_8jL X J yd. X 2T = 694| yds., Ans.
Ex. 18. 121 cwt. : 48f cwt. : : $421 : ( )
( ) = $i|5.xi|ix 22j=$163.50 + , Ans.
Ex.19. $lf : $317.23 :: 8| lb. : ( )
( )=:317.23 X 8.4 lb. X 4 = 1522.7+ lb.=
15 cwt. 22.7+ lb., Ans.
Ex. 20. $1561 : $95.75 : : 15f bu. : ( )
( ) = ^J^■LS-x^^ bu. 615=9.575 bu.=
9 bu. 2 pk. 2 1 qt., Ans.
HJx. 21. I bar. : 1 bar. : : $^| : ( )
( )=$^ xixi = $j^j, Ans.
K. P 6
118
Ex. 22.
Ex. 23.
Ex. 24.
Ex. 25.
PROPORTION
4 rd. : llf rd. :: f A. : ( )
( )=:} A. X V- X i = 2^\ A.=2 A. 28 rd., ins
13cwt. : 12 cwt. :: $421 : ( )
16 oz. : 12 oz. :
3x12
( y-
16
: $28 : ( )
=$21, ^/i5.
16 oz.— 1411 Q2.=ly5_ oz., cheat in 16 oz,
16 oz. : l/g oz. : : $30 : ( )
( )=:$3jP X 2.1 XyV=$HI ==$2.46 + , Ant
Ex. 26. 1 yr. 6 mo. : 3 yr. 9 mo. : : $750 : ( )
, , $750x45 ^ ^
18
Ex. 27. 10 mo. x V/r-^^^ ^o-> ^^^•
Ex. 28. $25 : $30^ : : $28 : ( )
( )=$2J- X V- X 2V = $34.16, Ans.
Ex. 29. 1 yr. 4 mo. = li = | yr. ;
i yr. X|J| = 2J yr. = 2 yr. 9 mo. 10 da. Am
COMPOUND PROPORTION.
\
(401 5 page 292.)
. 1. 16
50 J
/o[-^^« = ( ) -
128
5
( )
90
( ) = 72 bii., Am.
COMPOUND PROPORTION.
119
Ex. 2. 3
12
h<4
120
; 360 120
10
( )
300
12
54
( ) = 10f days, Ans.
Ex.3.
6
10
( )
34
20
15
( ) = 170 yards, Ans.
Ex. 4
450 ( )
12 V : 9
12 ) 8
1 : 1
8
9
( )
12
12
450
( )=900, Jtis.
Ex. 5. 1200 ;
Ex. 6.
Ex.7.
wi
500 : 960
m--
4
^
8J
■■v\--
48
6|:( )
500
960
1
8
4
5
( )
1200
1
23040
( )=3291^, An9
36
12
12
( )
48
9
9
8
( )=:6 men, Ans,
V-xVx*xYxixi>< 7V=¥i^ = 40tt» ^^*'
120 PROPORTIDN.
Ex.8. 41 ) 4| }
6 ^ : 9 V : : 540 : GOO
20 ) ( ) )
Ex. 9. 2^
If
[ :^;| j ::$3.27i:( )
.6.75 73 3 2 5 J478.25 ^_ ^^
Ex. 10. 5
6
[ • ^ 12 [ • • ^^-^ • ^^'^•^
417.6x5x6 ^^
— ^ttt: — 7^ — = 20 men, Ans,
52.2x12 '
Ex. 11. 6 ) 9 ) 22.5 ) 45
2.5 y : ( ) [ :: 17.3 [ : 34.6
12.3 ) 8.2 ) 10.25 ) 12.3
6x2^x12^3x^5 X 34.6 X 12.3 _
9x8.2x22.5x17.3x10.25 *"^^ ^^^^' ^''''
Ex. 12. 54 J 75
24J f : ( )
121 ) lOJ
V- X V X V X T J X /t=21 days, Ans.
Ex. 13. 24 ^ 217 J 33J ) 23i
189 [ : 51 V : : 5|
U) { ) ) 31) 21
Vx^l^XVxVxy XlX2lTXi^TXHjX^YXf = 16,
^7lS.
PARTNERSHIP.
121
Ex.2,
$ 8000
12000
20000
$40000
PARTNERSHIP.
(407, page 295.)
TVoW=i» A's fraction.
±2 0.0.0. _3_ TJ'q U
40000 10> -^^
2.00.00. — i n\ a
-$336, A's; $1680 x tV=$504, B's
$1680xi==:$840, C's.
Ex. 3. $1200+$1000 + $600=::$2800;
$2800
: $1200 :
' $224
: (
)=:$96, A's share;
$2800
: $1000 : :
$224
'{
) = $80, B's "
$2800
: $ 600 : ,
$224
(
)=$48, C's "
Kx. 4. $20000 : $13654 :: $3060 : ( )=$2089.062
$20000 : $13654 :: $1530 : ( )=$1044.531
Ex. 5. 16 + 24 + 28 + 36 = 104
$13XyV4=^2, A pays;
'x/oV=^3, B pays;
$13 X tVt ==^3.50, C pays ; $13 x j\\=$^,50, D pays.
Ex.6. 14 + 6 + 12 = 32 shares.
$2240 X i|=:$980. Captain's share ;
$2240 X 3\=:$420, Mate's share ;
$2240 X i|=$840, divided among the sailors ^
$840-r-12=$'70, each sailor's share.
fix.)
$3475.60— $2512=$963.60, lost to the owner* ;
$963.60 X i=$120.45, A's \
$963.60 X 1 = $240.90, B's i Jns.
$963.60 X f = $602.25, C's )
Proof 1 =$963.60
122 PARTNERSHIP.
• Ex 8. 6, C's proportional share.
6 f 4=10; 10x1= 2, E's ** **
6 + 4 + 2 = 12
$2571.24 X ^2 =$1285.62, C's ; \
$2571.24 X yV=$ 857.08, D's ; V Ans.
$2571.24 Xy2-=$ 428.54, E's;)
Ex. 9 $7500-($2000 + $2800.75 + $1685.25) = $1014,
D's gain ;
gain. cap. gain. cap.
$1014 : $3042 :: $2000 : ( )=$6000 Am
$1014 : $3042 :: $2800.75 , ( )=$8402.25, B; i Am,
$1014 : $3042 :: $1685.25 : ( ) = $5055.75, C ; )
(408, page 297.)
Ex. 2. $250 X 6 =$1500, B's product ;
275x8= 2200, C's "
450x4= 1800, D's "
$5500
$825 X If =$225, B's share of gain ;
825 X 11= 330, C's " " "
825X;?-|= 270, D's "
((
$1000 X 8=$ 8000
1600x10= 16000
Ex.3. $1000 X 8=$ 8000 $1500 x 4 = $ 600C
1200x14= 16800
A's product, $24000 B's product, $22800
$2l000 + $22800 = $46800, sum of products.
$46800 : $24000 :: $1394.64 : ( ) =$715.20, A's gain •
$46800 : $22800 :: $1394.64 : ( )=$679.44, B's "
PARTNERSHIP.
123
Ex. 4. 4x5 days =20 days' work A furnished ;
3x6 " =18 " " B "
6x4 " =24 " . " C "
62 " " all "
372 buslie]s-f-4 = 93 bushels to be divided.
62 : 20 :: 93 : ( )=30 bu., A's ; j
62 : 18 :: 93 : ( ) = 27 " B's ; [ Ans.
62 : 24 :: 93 : ( )=36 " C's; J
Ex. 5.
From Jan. 1, 1856, to Apr. 1, 1858, is 27 mo., Gallup's time ;
" Mar. 1, 1856, " Apr. 1, 1858, " 25 " Decker's "
" July 1, 1856, " Apr. 1, 1858, " 21 " Newman's"
$3000 X 27 = $81000, Gallup's product ;
2000x25= 50000, Decker's "
1800x21= 37800, Newman's "
$168800, sum of products.
$4388.80 : ( ) = $2106, Gallup's gain;
$4388.80 : ( ) =$1300, Decker's ''
$4388.80 : ( ) =$982.80, Newman's ".
$168800
$168800
$168800
$81000
$50000
$37800
Ex. 6. $560— 8=$70, A's monthly profit;
$800^10=$80, B's " "
$150
Since the gains of the partners are proportional to their
amounts of capital when the times are equal, we have
$150 : $70 : : $5600 : ( ) =$2613.331-, A's gain;
$150 : $80 : : $5600 : ( ) = $2986.6G|, B's gain.
Ex. 7. If we allow 2 parts of the gain to A, 3 parts to
B, and 4 parts to C, | of A's gain will be equal to I of B's,
and to J of C's, and the proportion of the shares will corres-
Dond to the conditions.
124
ANALYSIS.
2+3+4=9
$117 X |=:$26, A's gain .
$ll7xf=$39, B's "
|ll7x^riz$52, C's "
If we now divide the proportional shares of the gain, 2, 3, 4,
by the respective times, 3, 5, 7, we shall obtain the piopor-
tional monthly shares of the gain, which must be in the sa.i:*e
proportion as the respective shares of the capital ;
2-f-3 = |, A's proportional share of capital •
3-+5 = f, B's " " "
4-h7=4, ^'s " " " "
l"^! + 7=Tol» ^^™ of proportional shares.
: ( ) = $700, A's capital ;
:( )=$630, B's "
:( )=$600, C's
tf f : I : : $1930
19 3 . 3
105*5
HI: 4
$1930
$1930
ANALTSia
(412, page 300.)
Ex. 5. We multiply the number of
casks by the number of pounds per
cask, and this product by the number of
pence per pound, and obtain the cost in
pence; which, divided by 56, the number
of pence in a dollar, gives $27, answer.
56
3
126
4
$27, Ans,
Ex. 6. We multiply 19 (pence) by 28 7
for the cost of the butter (in pence), and j_^_
livide by 1 times 12 (pence) the price of _£_
the tea.
28
19
"l9
61, Ans.
ANALYSIS.
125
Ex 7. 10 s. 6d=126d. The
product of 2, 72, and 4 is the num-
ber of quarts. Multiply this oy
126, the selling price per quart,
and divide by 96, to reduce the
result to Decimal currency, and we $756 —
obtain $756. Subtracting the cost
we obtain the profit.
96
2
72
4
126
$756
= $108, An9.
Ex.8. 2 s. 6d.=:30d. Then 2 x 3 x
7 X 30= cost in pence. Divide by 60 to
reduce to Decimal currency.
60
2
3
7
30
$21, Ans.
Ex. 9. 20 X 3 X 12r=value of the ap-
ples in pence. Divide by 6 s. 3 d.
(=75 d.) to find the number of days'
work to be given in exchange.
20
3
75
12
5
48
9|, Ans.
Ex. 10.
96
160
18
Cost, $30
$42.66f-
90
160
24
sold for $42.66|.
)=$12.66|, ^/i5.
Ex. 11. 431= V; 10 s. 6 d.=
8 s. 3 d. = 99 d.
:126d.
Ex. 12. 9 s. 4 d. = 112 d. ;
$1=96 d., Mich, currency
2
99
87
126
11
609
65y*i- A718,
96
300
112
$350, Am
126
ANALYSIS.
128
5
90
Ex.16. Dividing 128 by 16, we ob-
tain what 1 horse Avill consume in 50 iq
days ; dividing this result by 50, we ob- 50
tain what 1 horse will consume in 1 day. 72, Ans,
Multiplying by 5, we find what 5 horses
will consume in 1 day ; and multiplying this result by 90, we
find what 5 horses will consume in 90 days.
Ex. 11. Divide 4| (rr:Y) ^7 l^i
(=%') to find what amount of wood $1
will buy ; then multiply by 24:^[=:\^) to
find how much 824J will buy.
Ex. 18. 52 X 3 X ^^= the money
given for the cloth. Divide this result by
65, the number of yards, to find the price
per yard.
Ex. 19. A shadow of 1 foot will require an object J of 3
feet in length ; and a shadow of 46| feet will require an object
46 J times ^ of 3 feet in length ; hence
^xlx^^=28ieet,Ans.
s
14
21
2
4
99
U.Ans.
52
3
3
100
65
Ans
. 80 cents.
Ex. 20.
8 sheep x Y|= 60 sheep for 1 mo., A's use of the pasture ;
12
15 " vfi2_inn « " " P.'a " " «
X4i=: 50
X6|=100
u
u
" ,B's "
" ,C's "
" , total "
210
(C
Each man should pay such part of the whole cost as his use
of the pasture is part of the total use ; hence
$63 X 2^=118, A must pay ;
$63x^y=:$15, B " "
$63xif==|30, C " "
ANALYSIS.
127
Ex. 21. 1 bu. oats =1 dollars ;
1 bu. rye =-y bu. oats = Y ^ i <iollars ;
1 bu. wheats Y bu. rye=Y X Y xf <iollais.
If we divide $30 by the price of 1 bushel of wheat, we shall
have the number of bushels which $30 will buy ; hence
V- X f X y\ X tV=15 bu. Ans.
Ex. 22. If $480 gain $84 in any time,
Ic gain $21 in the same time will require
II of $480 ; and if |i of $480 gain $21
in 30 mo., to gain the same amount in 15
mo. will require f | of |i of $480.
84
15
480
21
30
$240, Ans,
Ex. 23. 28 X f = 21 sq. yd., contents of the 28 yd. ;
21-h| = 31i yd. of that which is | yd. wide.
Ex. 24. If 130 miles require 3 days, 390
miles will require f fj of 3 days ; and if 14 I^^
hours a day require 3 days, 7 hours a day will
require y- of 3 days.
3
390
14
18,
Ans.
Ex. 25. If 6 men cut 45 cords in any time, 8
men can cut | of 45 cords in the same time ; ^
and if in 3 days any number of cords be cut, in _
9 days there will be cut | times as many cords.
45
8
9
180,
A71S,
Ex. 26.
A's age + 13's age= 1 -f- 1| =2^ times A's ;
C*s age = 2 yV times this sum —.5} " "
And the sum of all their ages, or 93 yr. = 7^ " *'
Hence, 93^7f = 12 yr., A's age
12x1^ = 18 " B's " [Ans.
12x5 J- = 63 " C's "
128 ANALYSIS.
Ex. 27.
1 clay ofC =j% da. ofD.;
1 day of B= V da. of C = ^ x i% ^^' o^ ^^ i
1 day of A^l daofB =f x V x A ^a. of D;
lience 5 days of A=^ X | X Y x A ^^- ^f D=8 da.
of D, Ans.
Ex. 33. If the cost of 12 oranges and O- ^- ^
10 lemons is 54 cents, the cost of one half ,o tq 54
the lot, or 6 oranges and 5 lemons, will be ~^ r — -^
27 cents. 2 6
But the cost of 6 oranges and 7 lemons 1 lemon =3 cts.
is 33 cents. And, by subtracting, we 1 orange=2 cts.
find the cost of 2 lemons to be 6 cents,
which gives the cost of 1 lemon 3 cents. From the first ex-
pression, 6 oranges and 21 cents (equal to 7 lemons) is equal
to 33 cents ; hence 6 oranges are worth 12 cents, and 1 orange
is worth 2 cents.
Ex. 34. 18x20x1000 = the whole
number of ounces of provisions ; and since
this quantity is to supply 1600 men 30
days, we divide by 30 to find the daily al-
lowance for the army, and this result by ^^* 2» ^^*
1600 to find the daily allowance to each
man.
Ex. 35. If we add 6 bushels to the smaller bin, there will be
60 bushels in both ; but as the larger will then contain 2 time«
as many bushels as the smaller, the two together wi'l contaiB
three times the number in the smaller ; hence
3 times the smaller =60
The smaller =20
The larger =20 x 2 = 40, Ans,
18
30
20
1600
1000
15
ANALYSIS.
129
Ex. 36. We take the difference of two numl)<)r« l>rom tlve
greater to find the less. The greater diminished by i of the
greater equals the less, which must be | of the greater. And
if the less be | of the greater, their sum, 20, is 1| times tb<?
greater. Hence we have
20-
^3
:12, the greater, Ans.
Ex. 37. 1 day of C
1 day of A:
6 days of A
6 weeks of A
f da,
= f da. of B ;
ofC=: fxf da. of B; hence
= 6 X f X f da. of B ; and,
=6x1x1 wk. ofB.
6 X f X f wk. = lli wk., Ans,
Ex. 38. 36 X 1J=45 sq. yd. to be lined.
45 yd.-f-J = 60 yds.. Arts.
Ex. 39. 80 X 31 X 96 = value of the
broadcloth, in pence ; 104 x 10= value of
one sack of coffee, in pence ; and to obtain
the number of sacks we divide the former
product by the latter.
80
4
13
104
96
10
24, Ans.
Ex. 40. If the time past since noon is equal to J of the timo
to midnight, both intervals, or 12 hours, must be 1;^ times the
time to midnight ; hence
12 h.-^H =10 h. to midnight.
12 h.— 10 h.= 2 h. p. M., Ans.
Ex. 41 She bought one half for i cent apiece ;
And the other half for i cent apiece.
(2 + 3)"^^ — t\» average buying price ;
3-T-5 =1, selling price.
A —ih g^^^ o^ ^°® peach.
55-r 11 = 300," ^71.9.
I~r^
6*
130 ANALYSIS.
Ex. 42. A can build the boat in 18 x 10 = 180 hours ;
B " " " " " 9x 8= 72 "
A " " yI of the boat in an hour ;
BU U 1 (( u u u
A and B can build TsT + TV^se o ^^ ^^^ ^^^^ '^
an hour ;
A and B can build 3 J^ x 6=/^- of the boat in a day
of 6 hours.
It will, therefore, require as many days as 7 is con-
tained times in 60 ; hence
60-^7=^84 days, Ans,
Ex. 43. He spent at first ^, and he had ^ left. He then
spent A of this ^, and he had | of this ^ left ; hence | of ir=
^ of his money, which is $10, and the whole is $30, Ans.
Ex. 44. 4 times the work will require 4 times as many men,
and 1 of the time will require 5 times as many men ; hence
30x4x5 = 600, Ans,
Ex. 45. If $3.25. buy 16.25 lb., $1.25 will buy Jf | of 16.26
lb. ; hence
16.25 lb.xi|} = 6.25 = 6i lb., Ans,
Ex. 46. On every idle day he lost the forfeit, $1, and his
wages, $2.50, which together amount to $3.50. Had he la-
bored every day, he would have received $2.50 x 20 = $50.
$50 — $43 = $7, what he lost by being idle ;
and $7-f-$3.50 =2, the number of idle days. Hence
20 — 2 = 18, the number of days he labored, Ans,
Et. 47. A, B, and C perform ^j in an hour
AandB " yV " "
Hence, C performs ^j — ^=zj\ " "
Again, A, B, and C perform ,'^ " "
AandC '• yV " "
Hence, B performs tj—tV^sV " "
Therefore, B and C perform ^V + 3V ==tJ4 " **
And in 9^ hours they will perform yj^ x Y = J, Atvs.
ALLIGATION. V61
ALLiaATION.
(414, page 305.)
Ex. 2. 61.00 Xl2 = $12.00
1.50 X 5= 1,50
3
20 )$19.50
$ .975, Ans.
Ex. 3. $1.25x52 = 165
13
65 65
Mixture, $1 per gal.
65 X 32 X $.06J=$130, receipts ; $130— $65=$65, gain,
Ex. 4. 8x10= 80 Ex.6. 12 x 1^= 90
9x12 = 108 101 X 8 = 84
11x16 = 176 11 X 9 = 99
- — 10 xl0_i = 105
38x10 = 380 35 378
A "TT 4. 378x11=567
Ans, 16 cents. , ^
Ans. 567-^35=16icts.
Ex.6.
50 X 4=200 lbs.; $.13 x200=$26.00
4Cx 10 = 400 lbs.; .10x400= 40.00
25x24 = 600 lbs.; .07x600 42.00
1200) $1Q8.0
$.09 average cost per lb
$.095 — $.09 = $.005 ; 1200 x $.005 =$6, Ans.
132
ALLIGATION.
(416, page 310.)
10
i
1
1
11
1
2
2
14
i
4
1
1
2
Ex. 3. 12
Ans, 1 lb. at 10, and 2 lbs. at 11 and 14 cents.
Ex. 4. 90
( 120
9
_!_
30
Ans. 1 gal. of water to 3 gal. of wine.
Ex.5. 21 5 i
'200
tV
3
3
250
ii
1
1
300
ii
1
1
400
rh
5
5
Ans, 5 of the 1st kind, 1 of each of the 2d and 3d, and 3 of
the 4th.
Ex. 6. 90 <
fso
84
87
94
6
6
i
6
6
i
4
4
i
3
3
\
10
6
16
-4w5. 6 of the first 2 kinds, 4 of the third, 3 of the fourth,
and 16 of the fifth.
(ilT, page 311.)
Ex. 2. 80 <
'40
tV
2
2 20
60
1
US
2
2 20
75
i
2
2 20
90
1
TIT
tV
tV
8
4
1
13 130
Ans, 20 lbs. of each of the first three kinds, aL s 130 Ib«. lA
tlje fourth.
Ex. 3. 4 ]
2
-i
1
1
3
1
1
1
6
1
1
2
1
3
24
24
72
Ans. 24 at $3, and 72 at i
ALLIGATION.
183
7V
4
4
60
sV
4
4
60
tV
1
40
9
3
12
180
Ex. 4. 90 hJ 60
130
Ans. 60 gallons each of alcohol and water.
Ex. 5. 40
35 1 }
50
TIT
150
15
Ans, 150.
Ex. 6.
f 10 I
f
4
4
8
80
6
6
60
f
2
2
20
Ans. 60 lbs. at Si cts. and 20 lbs. at 10 cts.
(418. page 312.)
Ex. 2. 14 <
(9
i
6
6
60
12
1
2
4
4
40
18
1
4
2
2
20
20
i
5
5
IT
50
170
Ans, 60 at 9 s., 40 at 12 s. ; 20 at 18 s., and 50 at 20 a
Ex. 3. 22 <
Ans, 6 ounces each of the first three, and 33 ounces of the la* t
Ex. 4. $l78.50-4-210=$.85, average price
ri6
i
2
2
6
18
i
2
2
6
21
1
2
2
6
24
i
i
i
6
4
1
11
17
33
51
' 50
1
3 J
13
1
13
78
85 H
: ^5
tV
13
13
78
(
[ 150
eV
aV
7
2
9
54
' 1
35
210
Ans, 78 bu each of oats and corn, and 54 bu. of wheat
184
EVOLUTION.
( 45
4
\
1
2
3
3600
Ex. 5.
48 .
51
\
1
1
1200
J54
I
6
1
1
1200
1
5
6000
Arts. A 3600 bu. ; B and C each 1200 bu.
Ex. 6. $84 -T- 5 6 =$1.50, average daily wages.
150^
f 60
rh
6
6
24
15
tV
1
1
4
115
Vt
3
3
12
300
Th
4
4
16
14
5G
Ans, The boys 24, 4, and 12 days, respectively, and the
man 16.
EYOLUTION.
SQUARE ROOT.
(434, page 319.)
Ex. 9. Ans. 234135.
(441, page 321.)
Ex. 3. 200 X 1|=225 sq. yd. ; V225=15 yd.=45 ft, Ans.
Ex.4. 10 A.=1600 rd. ; i/l600=40 rd., length of ono
side ; 40 x 4 = 160 rd., Ans.
Ex. 5. 45^=2025, square of the base ;
60'=3600, square of the perpendicular;
5625, square of the hypotenuse.
V'5625 = 75, Ans.
SQUARE ROOT. 135
Ex. 6. 39^—1521, square of the hypotenuse
15'= 225, square of the base ;
1296, square of the perpendicular.
V'l296=:36, height of the stump.
36 ft. + 39ft.=:'75 ft., Ans.
Er. 1 |/40'— 33'=22.60 + ,fronafootofla(idertocjncsidei
4/40' — 21'=34.04-f " " " " " other"
56.64 + , ^^5.
Kx. 8 5 2' =2 704, square of hypotenuse ;
48^=2304, square of perpendicular ;
400, square of the base ;
|/40b=20, Ans.
Ex. 9 1 mi. =320 rd., length of 1 side of the park.
820''=102400
102400
204800; |/204800=452.5-f , diagonal.
320 X 2 = 640, distance around the park to the opposite corner.
640— 452.5 = 18*7.5, distance between A and B, when A ar-
rives at the opposite corner. 187.5 4-^2 = 93 7+, Ans.
Ex.10. 20Hl6'=square of the diagonal of the floor. The
diagonal of the floor and the height of the room will form the
base and perpendicular of a right-angled triangle, of which the
diagonal from the lower corner to the opposite upper corner
IS the hypotenuse. Ilence
ini c^^^ r square of the diasjonal of the floor ;
lb =iioo j .
1 2*= 144 square of the perpendicular ;
800 square of the required diagonal ;
i''80b = 28.28427l+ feet, Ans,
186 EVOLUTION.
Ex. 11. 2:3: : (63.39)' : ( )=:6027.43815
1/6027.43815 = 77.63 -|-rods, Ans.
Ex.12 1:2::5':( ) = 50 ;
4/50 = 7.07106 +feet, Am.
CUBE ROOT.
(444, page 327.)
APPLICATIONS IN CUBE ROOT.
Ex. 1. Vl331=:ll ^t,,Ans.
Ex. 2. V373248=:72 iii.=6 ft., Ans.
Ex, 3. V474552 = 78 in.=6i ft., length of 1 side ;
61x61 = 421 sq. ft., Ans.
Ex. 4. V||=f ft., length of 1 side ;
J X I sq. ft. = area of 1 side ;
fxjx6xi=f sq. yd., area of 6 sides, Ans.
Ex. 5. If the bin be divided by a vertical section equi-
distant from the ends, the two parts will be cubes, each of a
capacity of one half of the bin.
125 X 2150.4 = 268800 cu. in. in the bin ;
268800-^2 = 134400 cu. in., contents of one halfl
Vl34400=51.223+in., width and depth.
61.223 X 2 = 102.446 in., length.
Ex.7. 1 :2 = (14.9)»: ( ) = 6615.898
^''¥6157898= 18.7 + inches, Ans,
Ex. 8. 16 : 25 • : 8 : ( )=12.5 cube of the diameter.
Vl2^ = 2.32+ ft.
ARITHMETICAL PROGRESSION. 137
ARITHMETICAL PEOGRESSION.
(451, page 329.)
Ex.1. (19~l)x3=54; 54 + 4=58, ^W5.
Ex. 2. (13 — 1) X 5 = 60 ; 75 — 60 = 15, Ans.
Ex 3. 2=firstterm; 3 = com. diff. ; 18=No. terms.
(18 — l)x3 = 51; 514-2 = 53 cents, ^/i5.
Ex.4. (40-l)xi=9f; 9} -hi = 101, Ans.
Ex. 5. 20= first terra ; 5= com. diflf. ; 9= No, terms.
(9 — l)x5 = 40; 40 + 20 = 60, ^W5.
Ex. 6. 100=first term ; 7=com. difF. ; 46=No. terms.
(46 — l)x 7 = 315; 315 + 100=$415, ^W5.
(452, page 330.)
Ex. 1. 17-2 = 15 ; 15-^5 = 3, Ans.
Ex. 2. 14—2 = 12 ; 12-~6 = 2 years, Ans.
Ex. 3. 501-1 = 49^ ; 49i-v-33 = li, Ans.
Ex. 4. 3=first term ; 9J=last term ; 14 =No. terms.
91-3=61; 61-^-13 = 1 com. dif.
(453.)
Ex.1. 43—7—36; 36-^4=9; 9i-l = 10, Ans.
Ex. 2. 40-21 = 371 ; 37iv-7i=5 ; 5 + 1=6, Ans.
Ex. 3. 6=first term ; 226 = last term ; 4= com. diff.
226 — 6 = 220; 220-^4 = 55 ; 55 + 1=56, ^»#.
(454, page 331.)
Ex. 1. (5+32)xV=222, Ans.
Ex.2. (1+12) x\^=1 8, Ans.
18b GEOMETRICAL FfioQRESSION.
Ex. 3. 24=:first term; 1224=:last term; 52=:No. terms.
($1224 + 124) X ^/=$32448, Ans.
Ex. 4, 4= first term, or twice the distance to the first apple,
400 = last " " " " " " last "
100= No. terms.
(400 yd. + 4 yd.) x 4^^^ = 20200 yd., Ans.
GEOMETRICAL PROGRESSION.
(458, page 333.)
Ex. 1. 4x3'=26244, Ans.
Ex. 2. 1024 X (iy=jV3%\=T\, ^ns.
Ex.3. 1= first term; 2= ratio; 9= No. terms.
1 mill + 2''=256 mills=$.256, Ans.
Ex. 4. 1 X U) =77J = ;n=r-7T7n;j ^^s.
Rx. 5. 1= first term; 1.07 = ratio ; 5= No. terms.
1 X (1.07)*=|1.40255+, Ans.
Ex. Q. 3= first term; 3= ratio; 7= No. terms.
$3x3"=$2187, Ans.
(45 9, page 334.)
Ex. 1. (512x3)--2 = 1534; 1534—2 = 767, Am.
Ex 2 (262144 x4)-4 = 1048572;
1048572-^3 = 349524, Ans.
Ex. 3. (162 X 3)--2=484 ; 484-^2=242, Ans.
Ex. 4. J^-^^=l, ratio;
(1x5)— = 1; l^4=i, Ans.
PROMISCUOUS EXAMPLES. 139
Ex. 5. 2— first term; 6-^2 = 3, ratio ; 12=^0. terms.
$2 X 3" = $354294, last term ;
($354294 X 3) — 2r=$1062880 ;
$1062880-v-2=$531440, Ans.
Ex. 6. 7:= first term, or yield of the first year;
7= ratio ;
12 ^No. terms, or the number of years to yield.
7x'7" = '7^'=:13841287201, last term, or 12tb
year's produce.
(13841287201 x7) — 7 , „
^^ ^ =16148168400, sum of all terms.
16148168400-4-1000 = 16148168.4 pt.
16148168.4 pt. = 252315 bu. 41 qt., Ans,
Ex. 7. 200-^20 = 10, the number of times the family
doubled its number.
10+1 = 11, No. terms; 2 ratio.
6x2^'»=6144, ^/i5.
PROMISCUOUS EXAMPLES
(Page 334.)
Ex. 1. 800 x 2 = 1600, the sum ; and
200 x 2= 400, the diflference.
The greater of any two numbers is equal to the less -f- ths
difference ; and the greater and the less, or the sum of the
numbers, must be composed of twice the less and the differ-
ence. Hence
1600 -400 = 1200, twice the less;
1200-^ 2= 600, the less ;
6004-400=1000, the greater.
140 PROMISCUOUS EXAMPLES.
Ex. 2. I of j\=fy. If ^<L. of a number be added to itsell,
the result must be l/j times the number. Hence,
61-M/y = 55, Ans.
Ex.3. 3 h. 21 min. 15 sec.=:12075 sec; 1 da. =86400 sec;
ifnMa.=TTV2 dx,Ans,
Ex. 4. 3 bu. 3 pk. 3 qt.
10
38 bu. 1 pk. 6 qt.
1
269 bu. pk. 2 qt., Ans.
Ex. 5. A and B together have 3 times A*s ;
C and D together have $300 + $500=$800 ;
And they all have 3 times A's+$800.
Therefore, $1100— $800=1300=3 times A's.
$300-^3 = $100, Ans.
Ex. 6. B has A's votes H- 20O
C has A's votes + 1000
- A B and C have 3 times A's votes + 1200
Therefore, 3000 — 1200 = 1800, 3 times A's voteii.
1800-^3 = 600, Ans.
Ex. 7. —-7=:—-, Ans.
I7i 70'
Ex. 8. J— ^ = j'g. Hence 10 is j\ of the number; and
the number must be 560, Ans.
Ex. 9. $28.35-r-$.35 = 81 gal. mixture.
81 — 63 =18 gal. water added.
Ex. 10. When A had gained }, he had f of the original
stock. B, after his loss, had ^ as much, or f of the original
etock ; hence he had lost | ; the $200 which he lost was |
of his stock; and his whole stock must have been 200-r-} =
8500.
PROMISCUOUS EXAMPLES. 141
Ex. 11. 1.35 X 13 ==$4.55; $31.55 — $4.55=:$27, cost of the
whole, if the wheat had been at the same price as the barley.
17 + 13 = 30, whole number of bushels.
$27-^30 = 1 .90, price of barley, ) ,
':\-
$.90 + $.35 =$1.25, price of wheat,
Ex. 12. 4 mo. 11 da. 7 h. 5 min.
3 20 15 21
21 da. 15 h. 44 min., Ans.
Note. — Borrow 31 days for March.
Ex. 13. The point of time divides the whole 12 hours into
two intervals, which are in the ratio of 9 to 11. Hence, by
Partnership,
9
11
20 : 9 : : 12 h. : ( ) = 5 h. 24 min., Ans.
Ex. 14. The least common multiple of 63, 42 and 31| ; or,
since 63 is 2 times 31^, the least common multiple of 63 and
42, which is 126, Ans,
Ex. 15. The least common multiple of 8, 9, 15 and 16,
which is 720, Ans.
Ex. 16. Since B gets in debt $10 yearly, his income would
enable bin: to spend $30 — $10 = $20 a year more than A
8})ends. Hence $20 is } of the income ; and
$20 X 8=:$160, income, Ans.
Ex. 17. $2.19 xf-f If = $2.40, Ans,
Ex. 18. I \: ]\ \', $3.37| : ( )=$52.779, Ans,
I5 ) I2 )
Ex. 19. $1000 : $200 : : 6 mo. : ( ) = li mo.. Am,
K. P. 7
142
PROMISCUOUS EXAMPLES.
Ex.20. |2350.80-^.40 = $5892, left;
$5892 + $2356.80=18248.80, Ans.
Ex. 21. — ^^=tVo X f X f =1 = 121 per cent., Ans.
Ex. 22. 1 private has 1 share ; 60 privates have 60 shares .
1 subaltern" 2 " 6 subalterns " 12 '*
1 lieut. " 6 " 3 lieut's "18 "
1 commander has 12 "
All have 102 shares,
$1 0200 -f- 102 = $ 100, share of a private.
$100 X 12 =$1200, share of the commander.
Ex.23. 19 — 16 = 3; 51-^3 = 17 hours, ^?Z6\
Ex. 24. 40 <
20
2V
14
14
7
133^
30
tV
10
10
5
95
50
tV
10
10
5
95
54
tV
20
20
10
190
Ans,
Ex. 25. $33.75-v-22.5=$1.50, selling price per bu.
$22.50-^18 =$1.25, buying " "
$ .25, profit on 1 "
$.25x240=$60, Ans,
Ex. 26. Tlie wagon is worth 4 times the harness ; the hursu
is worth 8, times the harness; hence the horse, wagon and
harness together are worth 84-4 + 1 = 13 times the harness
Tnerefore, $169-^13 = $13, harness, ^?Z5.
Ex. 27. 18 in : 42 ft. : : 40 in. : ( ) = 93i ft., Ans.
Ex. 28. 25 rd. : 40 rd. : : 4 rd. : ( ) = 6| rd., Ans.
PROMISCUOUS j:xamples. 143
Ex. 29. J, /j, and I2=|f, if, and if
And since fractions having a common denominator are pro-
portional to their numerators, we have
1 5 shares for A and B ;
18 " " A " C;
13 " " B " C;
46, twice the number of shares for A, B and C.
4b -,'2 =23 shares for A, B, and C.
23-13 = 10 " " A;
23-18= 5 " " B;
23-15= 8 " " C.
$26.45 X if = $11.50, A.'s portion ;
$26.45 x^^-^zS 5.Y5, B's " j
$26.45 x/^=$ 9.20, C's "
Ans,
Ex. 30.
6^ [ • ^^4 [ • • 12 : ( )=480 oz. = 30 lb. Ans.
Ex. 31. $6300 x I = $ 900, A's,
$6300xi=$1260, B's;
$6300x| = $1400, Cs;
$900 + $1400 =$2300, D's ;
$6300— $5860=$440
$440 xf=$l 65, E's,
$440xf= 275, Fs.
Ex. 32. $200xl.593848=$318.769 + ,^w*.
Ei, 33. At the time of the dismissal, the provisions on
hand would supply 360 men 1 month ; they would supply ^
us many men 5 months.
Hence 360-4-5 = 72, the number that remained;
360-72 = 288, dismissed, Ans.
144
PROMISCUOUS EXAMPLES.
Ex. 3i. $1.338220 ^amt. of $1 at compound int. for 6 yra.
at 6 per cent.
$669,113-^1.338226 =$500, principal.
$669.113— $500 = $169.1 13, interest.
$500 X. 06 = $30, simple int. of $500, for 1 year
at 6 per cent.
169.113-^30=5.6371 yr. = 5 yr. 7 mo. 19.356 + da., Am,
Ex. 35. $148.352-f-9.728=$15.25, Ans,
V
Ex. 36. It is evident that the product of two numbers must
»'ontain each common factor to the two numbers twice, and
'^ach factor not common once.
483-f-23 = 21, product of the factors not common.
23x23x21 = 11109, Ans.
Ex. 37.
^}
12)
2
n
: 9
:: 1 :2
12
8
15
( )
9
( )
1
15
9
140
Ex. 38.
Ex. 39.
15f, Ans.
36 ) . 60 ) 36 ) ( ) )
9r •27[-- 1^- 11 f
Ans, 144 yards.
1 X 4 = 4, A's product ;
3 x2 = 6, B's "
7ixl = 7|, C's "
A ' 2
171 : 4 : : $52.50 : ( )=$12, A's share ;
17^ : 6 : : $52.50 : ( ) = $18, B's share;
I7i : 7i : : $52.50 : ( )=$22.50, C's sham
Ex.40. |andf = fandf.
4 + 5 = 9.
9 : 4 :: $45 : ( )=$20, A's;
9:5:: $45 : ( )=$25, B's.
PROMISCUOUS EXAMPLES. H5
Ex. 41. $.35, interest of |1 for 5 years at 7 per cent.;
$33.25-^. 35=$95, Ans.
Ex. 42. 6 X VS = 6 X 'VY^O X 1 = 3, Ans.
Ex.43. 2 + 3+4 = 9.
9:2:: $360
9:3:: $360
9:4:: $360
( ) = $ 80
( )i=$120 [• Ans.
{ )=r$160,
Ex.44. 8i:6f ::12i :( )=9^^, Ans.
Ex.45. 9 X 9=::81 sq. in. in 1 stone ;
144 X 9 = 1296 sq. in. in 1 sq. yd. ;
1296-^81 = 16 stones for 1 sq. yd. ;
16x40 = 640, Ans.
Ex.46. 1.00— .08=3.92; $2 3 -+.92== $25, cost of the calves
and sheep sold. Since the price multiplied by the quantity
gives the cost, we have these two conditions, viz. ; 3 times the
number of calves, plus 2 times the number of sheep, equals
76 ; and 3 x J = J of the number of calves, plus 2 x f = f of
the number of sheep, equals 25. Expressing these conditional
as in Analysis, page 294, we have
C. 8.
1st condition 3 2 76
2d condition f I 25
2dx4= 3 Y 1^^
Subtract the 1st from the 3d= | 24
That is, I of the sheep are equal to 24 ; hence
24-7-| = 20, number of sheep.
76 — (2 X 20)^:36 ; 36-7-3 = 12, number of calve*.
Ex. 47. Hi I 16
lOi J- : 7 ^ : : 546 : ( ) = 384, Am.
13 ) 15
146 moMiscuous examples.
Ex. 48. 12 ) ( ) )
15f V : 15 V :: 2 : 1.
9 1 1)
Reducing the statement, ( ) = 8, the number of men re*
quired to finish the job.
12 — 8 = 4 men withdrawn.
Ex. 49. It is evident that to increase the number in both
rank and file by 1 man, would require twice the number in
rank or file at first, plus 1 (for the man at the corner). And
since to eftect this, required 59 + 84 = 143 men, ^^^^=^=^1 is
the number of men in rank or file at first. Hence
IV -\- 59 = 5100 men under command, Ans»
Ex. 50. Cost of the corn was 2 times the cost of the barley
" " wheat " 4 " " " " "
Cost of corn and wheat " 6 " " " " "
Hence, cost of barley was } of the cost of com and wheat
Again,
Wheat cost $243 -f J of the whole.
Corn " 153 + yV " "
Wheat and corn " $396+1 « «
Barley " QQ-^t\ " " ( J of w. iSz; c.)
Cost of the whole was $462 + j\ " "
Hence, $462 was |f— y\r=|i of the whole cost •
And $462-+||=$'756, the whole cost.
And since the barley, corn, and wheat cost in the proper
lion of 1, 2, and 4, we have
1+2+4=7
$756x4 =$108, cost of barley; $108-+$ .60 = 180 bu. barley;
$756 X ^ = $216, cost of corn; $216^$ .75 = 288 bu. corn ;
1*756 X 4 =$432, costof wheat; $432-+$1.50 = 288 bu. wheat;
Ans, 756 bu. grain.
6
a
u
and 7
(i
u
21
$630 X 2V
= $240,
1st;
$630 X/y
=$180,
2d;
$630 X 2V
=$210, 3d.
PROMISCUOUS EXAMPLES. 147
Ex 51 As often as the first has 1,
" second " f,
and " third " iofl5=|;
1, f, and f =1, f, and f.
We therefore assume 8 as the proportional number for the first ;
" • " second;
" " third
Ans,
Ex.52. $28xl.20=$33.60 what the 56 remaining gal-
lons must be sold for. $33.60-^-56=$.60, Ans.
r. .0 rVoXf 20 4 1 4 2 16 ^^. .
^^'''' ^^^irT^0><5^2><3><3-225=-^^
Ex. 54. $3500— $2100=$1400, what B owns now ;
$1400^ 1.40=$1000, B put in ;
$2100-T- 1.40 = $1500, C put in.
Ex. 65. i/S'-f 16^ = 17.88 +ft., Ans.
Ex. 56. 12 ) 10 ) ^. ^ . V J
Ex. 57. $12-v-1.09=$11.0091-f, worth of sugar;
$12— $11.0091=$.9909 + , grocer's profit;
$6^1.10=$5.4545+, worth of beef;
$6— $5.4545=$.5455+, farmer's profit;
$.9909— $.5455 =$.445+, grocer gains more,
Ex. 58. $336.42 — $311.50 = $24.92, interest;
$4.15i int. of $311.50 for 1 yr. 4 mo. at 1 per cent
$24.92-i-$4.15|^=6 per cent, Ans.
148 PROMISCUOUS EXAMPLES.
Ex 50 A 2 «,-,/] 2 — 12 10 anri -8-
r^A. OJ. 513? '*"'^ 5 — 15) 1 5> ^l^'-i 1 5»
These fractions are to each other as their nuiaerators, 12, 10,
and 6 ; and these numerators are to each other as 6, 5, and 3,
Hence, we have 6 shares for A and B ;
5 shares for A and G ;
3 shares for B and C.
14, twice the number of shares for A, B, an J C.
14 -7- 2 ==7 shares for A, B, and C.
7 — 3 = 4 shares for A ;
7 — 5 = 2 shares for B ;
7 — 6 = 1 share for C.
Hence $20 x -f =$llf , A's ;
$20xf = $54,B's;
$20x|=$24, C's.
Ex. .60. $375-^.025 = 815000, Ans.
Ex. 61. Interest commenced Apr. 1, 1857.
Amt. of note July 1, 1857, (3 mo.) $1015
Payment, 560
New Principal, $ 455
Amt., Dec. 1, 1857, (5 mo.) 466.37 -f
Payment, 406
New Principal, $ 60.37 4-
Amt., Aug. 23, 1859, (1 yr. 8 mo. 22 da.). .$ 66.63+,
Ans.
Ex. 62. B has | x ^ of A's =f of A's ;
C has 4 X f of B's=-^ x f x f of A's =|f of A's ;
D has I X f of C's= J x f x |? of A's=-; f of A's
And since D has $45 more than C,
If of A's— 1^ of A's=$45; or ^% of xVs -$45.
Hencc; $45-^/^ =$378, A's;]
$378 X I =$336, B's;
$378x|^=$360, C's; ^^^^•
$378 X If =$405, D's.
PROMISCUOUS EXAMPLES.
149
Ex. 63. B had the use of $300 for 2Y months before it waa
dne, which was equivalent to the use of $1 for 27 x 800=8100
months. But the use of $1 for 8100 months is equal to the
use of $600 for VoV-=13i months, the time he should wait.
Ex. 64. His savings and expenses together, or his sal&ry
is If + A = Tf ^^ what he saves ; hence $800 = if rf That ^ie
aves; and $800-^|f =$550, ^/i5.
Ex. 65. $.87i=$J ; $1.00 H- 1.1 0=:$|^, cost per yd.;
$-;^— $1 — $JL., loss at $.871 per yd. ;
Tj^TT^/o^.OSf = 3^ per cent., ^;25.
Ex. 66.
9 1 JL 2 4
^^ 1 I 11 .
240
'320
3 /3\* 27 ,
^ ^^ 63 189 27 /27 3 ,
^"'^- 1491=448=64' / 64=? ^"^•
Ex. 68. 50 -
20
tV
20
2
2
20
35
tV
10
2
2
20
60
To^
15
3
3
30
70
^V
30
3
3
"lo"
30
100
Ans, 20 of oats and corn, and 30 of rye and wheat.
Ex. 69.
40 + 500
= 270, half the sum of the extremes.
And since the sum of all the terms is equal to half the sum of the
extremes multiplied by the number of terms, 6480-7-270=24,
the number of creditors.
500 — 40=460, difference of extremes.
$4604-23 =$20, common difference.
Ex. 70. Vl28''-f 72' = 146.86-h, Ans.
150
PROMISCUOUS EXAMPLES.
10
11
14
4
10
550, Ans.
Ex. 71. If 7 pounds of butter are equal
to 10 pouuds of cheese, 11 pounds of but-
ter are equal to y of 10 pounds of cheese ;
and if 2 bushels of corn are equal to 11
pounds of butter, 14 bushels of corn are
eqial to 'g*- of 11 pounds of butter, or y-
of y of 10 pounds of cheese ; and if 8 bushels of rye are equal
to 14 bushels of corn, 4 bushels of rye arc equal to f of 14
bushels of corn, or | of Jg^ of Jy of 10 pounds of cheese ; and,
finally, if 1 cord of wood is equal to 4 bushels of rye, 10 cords
of wood are equal to 10 times 4 bushels of rye, or \^ of f of
V ^^ V ^^ ^^ pounds of cheese=550 pounds of cheese, Ans,
Note. — ^Instead of the fractional form the vertical lino may bo iSQd^
as above.
Ex. 72. $18-^f = $45 A's gain.
f : ^ :: $45 : ( ) =$37.50, B's gain.
$45-^.06 =$750, A's stock ;
$37.50-^.06 = $625, B's stock
'. 73. 20 ) ( ) ) 30 '
45^
21 V : 25 V :: 15
16
10) 8) 12
^- 18
>■
3^
5
Reducing the statement, we have
( )-=84
, Ans,
Ex. 74. 2| : 27^ :: 10 ft. : ( ) = 103i it., Ans.
Ex. 75. A can do ^ of the work in 1 day ;
J^ u ^ a u a -j^ u
C " ^^ " " " 1 "
They all " i + 1 + 7*2 = f of the piece in 1 day.
Hence it will require 1-^f =f of a day.
PROMISCUOUS EXAMPLES. 151
Ex. Y6. $1890-^1.25=:$1512, true value of 1st.
$1890— .75=$2520, " " "2d.
$4032, « " " both
$1890 X 2 =$3780, received for both.
Ans, $ 252, lost.
Ex. 77. If C paid one half the cost, A and B together paid
$50. Since C cows eat as much as 4 horses, 12 cows eat as
much as 8 horses. Therefore, A put in 9 horses for 1 unit of
time, and B put in the same as 8 horses for 2 units of time.
Hence,
A's use of the pasture was 9x1= 9 horses for 1 unit of time ;
B's " " " " 8x2 = 16 " •« 1 " « "
A andB's use of the pasture was =25 " " 1 " " "
$50x^5 =$18, A paid ;
$50 X if =$32, B paid.
Again, C's time was 2i times B's time, or 2 x 2^ = 5 units of
time. And since C paid half the cost, his use of the pasture
must have been as much as A's and B's together, which is 25
horses for 1 unit of time ; and this is equal to 5 horses for C's
5 units of time, Now the pasturage of 1 horse is f times the
pasturage of 1 cow, and the pasturage of 1 cow is J/- times the
pasturage of one sheep ; hence the pasturage of 5 horses (for
which C paid) is '/xf x5=25 times the pasturage of 1
sheep. Therefore C put in 25 sheep.
Ex. 78
$350C-4-1.0175=$3439.803-f , pres. worth of 1st installment;
13500 -M.02i =$3420.195+, " " 2d **
|3500-^l.04| =$3343.949+, " " 3d **
$10203.947 -f, Ans.
Ex. 79. Had the farmer sold both geese and turkeys at $.75
%piece,thc50 fowls would have brought $.75 x 50 =$37.50,
152 PROMISCUOUS EXAMPLES.
which is $52.50 — $87.50 = $15 less than they really brought ;
consequently the difference between the two estimates of the
turkeys, reckoned at $.75 and $1.25 apiece, is $15. Hence.
$].25-$.75 = S.50; $15-^$.50=:30, the number of turkeys;
and 50 — 30=::20, the number of geese.
Ex. 80. B gains of A 3 miles an hour, and C 5 mih:a
an hour. Hence B will pass A every ^ hours=24 h. ?0
inin.=:1460 min. ; and C will pass A every y hours=14 h,
36 min. rr: 876 min. Now the least common multiple of 1460
and 876 will express the number of minutes in which B and
C will first pass A together.
2,2
1460 .
. 876
73
865 .
. 219
5,3
5 .
. 3
2x2x73x5x 3 = 4380 min.=:6 da. 1 h., Ans.
Ex. 81. i, -} and i^f f, ii and i|.
And since fractions having a common denominator are to eacV
other as their numerators. A, B and C were to share in the
proportion of 20, 15 and 12. But C dying, his 12 parts must
be shared by A and B in the proportion of 20 and IS, or 4
and 3. 4 + 3 = 7.
7 : 4 : : 12 : ( ) = 6f , A's share of C's 12 parts ;
7 : 3 :: 12 : ( ) = 5i, B's " " " " "
20 + 6-|==26|, A's number of parts of the money ;
15 + 51 = 20 j, B's " " " " "
47
: 26
f ::
$100000 :
(
)=
:$57l42.85f A's, )
.$42857.14f, B's, ) •
47
: 20
\'
: $100000 :
(
>
Ex.
82.
A'
C
c
s + B's=:5
5-fB's=:7
s-B's=l
Since A's + B's are t<
B's + C's as 5 to 7, A
and B together have 5
i.±J- =.4, C's
proportion ; as often as B and C to-
7-
-4r=3, B's
u
gether have 7. And
5-
-3 = 2, A's
u
since C's -B's are to C's
PROMISCUOUS EXAMPLES.
158
f B's as 1 to 7, 7 is the sum and 1 the difference of B and C's
proportionate shares. Hence we find the proportionate share
of each. Then
2 + 3 + 4 = 9, the sum of their proportions.
And A has | of 135 = 30 sheep ;
B " f " 135=45 «
C " ^ " 135 = 60 "
Ex. 83. 250x4 =1000
300x4^ = 1350
369x5 =1845
919
4195
4195-4-919 = 4f II, Ans.
Ex. 84. The relative values of the work performed by one
of each class, in the same number of days, are as follows :
1 boy 3x 8 = 24 ) (2
1 woman 4 x 9 = 36 >■ or •] 3
1 man 6 x 12 = 72 ) ( 6
The relative amounts of wages received by the whole num-
ber of boys, women and men, in the same number of days, are
as follows :
Boys, $5 ; women, $10 ; men, $24.
Hence the proportions of boys, women and caen ar*^ ex-
pressed by the following quotients :
5-^2 = 21 boys;
10-^3=3i women ;
24-
i-6=4 men.
n
:2i ::
59 : ( ) = 15 boys;
H
: 31 :
59 : ( ) = 20 women;
n
; 4 :
59 : ( ) = 24 men.
7*
164 PROMISCUOUS EXAMPLEa
Ex. 85. A, B and C fill ^ of it in 1 hour ;
B, C " D " 1 " " "
C, D " A " tV " " "
D, A " B " j\ " " «
A, B, C and D fill i of J/^=:-JJ^ of it in 1 hour.
W, X and Y empty | of it in 1 hour ;
X, Y " Z « } " " **
Y, Z '' W " \ " " "
Z, W " X " 1 " " "
W, X, Y and Z empty J of 111= Jt?^ of it in 1 hour.
yVo"~TVo =tVo> ^^ emptying pipes gain of the filling pipes
m 1 hour. Therefore, to exhaust the fountain will require
120-M9 = 63V hours, Ans,
Ex. 86. 1 A. 1 R. 6 P. 181 gq. yd.=56250 sq. ft.;
56250 ^ ,
Ex. 87.
75x125 '
U^lbO,
$ Ix
= $
2x
1 =
2
4x
2 =
8
8x
3 =
24
16 X
4=
64
32 X
5 =
160
64 X
6 =
384
128 X
7 =
896
256 X
8 =
2048
512X
9 =
4608
1024 X
10 =
10240
2048 X
11 =
22528
Cost, $4095. $40962, sum of products.
40962 -f-4095 = 10 mo., average term of credit.
Jan. 1 4-10 mo. = Nov. 1, average time.
PROMISCUOUS EXAMPLES. 155
Ex. 88. 44.32 x 36 = 1595.52 sq. ch. = 159 A. 2 R. 8.32 P..
Ans,
Ex.89 l-|-i4-3-=lf. That is, if a number be increased
by ^ and ^ of itself, the result will be If times the number.
Hence, by the conditions. If times the number, plus 18, is
equal to 2 times the number; consequently, 18 is 2 — l| = i
of the number, and 18 x 6 = 108 is the number.
Ex. 90. If that which is worth $.621 be rated at $.56, what
ought that which is worth $.25 be rated at ?
$.621 : $.25 :: $.56 : ( )=$.224
Therefore, m the barter for a pound of coffee at $.22, the mer-
chant obtains that which is worth, ratably, $.224; hence he
gains $.004 on $.22 ; and $.004-f-$.22 = .01/y, ^^s.
ANOTHER SOLUTION.
A pound of tea, in the barter, will buy 56-4-22 = 2/y pounds
of coffee; and this is worth $.25 x 2y«y=:$.63yV But the
pound of tea given is worth $.621; and $63/y~$.62i =
$.01^2, the gain on $.621. Hence, lJL^62i=.01yV, the
gain per cent.
Ex. 91. First find how much ready money will cancel a debt
of $1, due in 4 equal installments, for the times and at the
rate mentioned,
$ .25-^-1.011 =$.245901 -f, pres. worth of 1st installment;
$ .25-^1.0375=$.240965+, *' " 2d "
$ .25^1.05 =$.238095+, " « 3d "
$ .25^1.081 =$.230769+, " « 4th "
$1.00 $.955730 + , present worth of $1.
Now ii' $1, payable as by the conditions named, be worth
$.95573 in ready money, what sum will be worth, in readj
money, $750, on the same conditions ?
$1 : ( ) : : $750 : $.95573 ; $784.74 + , Ans.
156
PROMISCUOUS EXAMPLES.
Ex. 92 |— 11=1 of the capital of either must be equal to
$500. Therefore $500 x 8 = $4000, Ans.
Ex.93.
Due.
Dec. 4
Feb. 9
Feb. 29
March 4
May 12
da.
00
67
87
91
160
Items.
240.75
137.25
65.64
230.36
36.00
710.00
Prod.
9195.75
5710.68
20962.76
5760.00
41629.19
41629.19-^710 = 59 da., average term of credit.
Dec. 4, 1859 + 59 da.=Feb. 1, 1860, Arts.
Ex. 94. When he had spent i of his fortune, he had f lelt
I ofi = /^. He had spent, in all, i + 2^=4 of his fortune;
consequently, the $2524 which he had left was ^ of his for-
tune. $2524-^^ = $5889.33-f , whole fortune, Ans.
Ex. 95 > By the conditions, the payments consist of five
several parts or installments of the $3000 and interest on the
first part or installment for 1 yr., on the second installment
for 2 yr., on the third for 3 yr., on the fourth for 4 yr., and
on the fifth for 5 yr. ; and the payments, each of which con-
sists of the sum of one installment and its interest, are equal
to each other. That is, the amount of first installment for
1 yr.= amount of second installment for 2 yr.= amount of
third payment for 3 yr., and so on. But, as the payments are
made annually, the interest must be added to the principal at
the end of each year ; consequently the second year's interest
is less than the first by the interest on the first installment,
and the second installment must exceed the first by this in-
terest, or by .07 times the first ; therefore, the second install*
ment =1.07 times the first. For similar reasons the third
installment =1.07 times the second, =1.07 x 1.07 times the
first, and sa on to the last. Hence, the installments form a
Geometrical Series, of which
MENSURATION.
157
Ist installm.=: 1st term ; 5, (numb. ofpaym'ts)=No. of terms;
1.07=ratio; and $3000 = sum of the series.
That is, we have the ratio, the number of terms, and the sura
of the series, to find the first term.
$3000 X (1 . 07-1'=).07 _^
(i.ovyi^i
1521.674 (1st installment) + $210 (int. on $3000 for 1 yr.)
= 1^731.674, annual payment, Ans,
:1st term=:$521.674-|- ;
Ex.96.
Due.
mo.
Itoms.
Prod.
Jan. 1, 1859
Sept. 1, "
Apr. 1,1860
8
15
200
350
500
2800
7500
1050
10300
10300-7-1050 = 9 mo. 24 da.
Jan. 1, 1859 4-9 mo. 24 da.=Oct. 24, 1859, Ans,
Ex, 97. Make the notes given the Dr. side of an account,
and the note received the Cr. side ; the balance will be the
other note received, and the average maturity, its date.
Br.
Cr.
Dae.
da.
Items.
Prod.
Due,
da.
Items.
Prod.
July 7, 1859
Oct. 4, "
Feb. 20,1860
00
89
228
600
530
400
47170
91200
Novl5,1859
131
730
95630
1530
730
138370
95630
730956301
Balances
800
42740
42740-^800 = 53 da.
July 7-^53 da.=Aug. 29, 1859, note due.
158
MENSURATION.
MENSURATION OF LINES AJN'D SURFACES.
(460, page 342.)
Ex. 1. 3 X 12=36 inches long; 36 x 20=720, Ans.
$32x198x150
Ex. 2.
Ex. 3.
160
1000x100
=$5940, Ans.
= 10 A., Ans,
10000
(469, page 343.)
Ex.1. 20x12=240 sq. ch. ; 240x16 = 3840 sq. rd. m
iie meadow, requiring 3840 rain, for mowing.
3840 min. = 6 da. 4 b., Ans.
Ex. 2.
15
\/15^— 9^=12, the perpendicular;
15x12 = 180 sq.ft., Ans.
13
Ex. 1.
Ex. 2.
Ex.3.
16 + 9_
2
16 + 8_
2
40 + 22
(471.)
= 12i in.=lJj ft, average width ;
12 xl2V = 12i sq.ft., Ans.
= 12 in.=l ft., mean width ;'
1 x8 = 8 sq. ft., Ans,
X 25=115 sq. ch.=77 A. 5 sq. ch., Ans.
(47 5.)
Ex. 1. ^^ X 45=3330 sq. fl.=370 sq. yd., Ans.
Ex. 2. The two ends together are equal to a rectangle, 2H
feet by 7 feet; hence 28 x 7 = 196 sq. ft.. Ans.
MENSURATION. 159
(47 9^ page 344.)
Ex. 1. 5x3.1416 = 15.708 ft. = 15 ft. 8.4 + in., Ans.
Ex. 2. 721 X 3.1416 = 2265.0936 rd. = 7 mi. 25 rd. 1.54 + ft
Ans,
Ex. 3. 33 X .3183 = 10.5 + yards, Ans.
(480, page 344.)
Ex.1. 11^211^=10028.15, Ans.
4
Ex. 2. Reversing the rule (II,)
1 sq. mi.-^.7854= 1.2732 +, square ot aiamet«r,
^.2732 = 1.1284 +mi. = l mi. 41 rd. 1.4 +ft.
Ex. 3. 84' X. 07958 = 561.5 +P. = 3 A. 81.5 + P., Ans.
k: E y
MISCELLANEOUS EXAMPLES
PROGRESSIVE INTELLECTUAL ARITHMETIC.
97.
2. Since he sold -^ of his share,he had | left; and | of J is ^,
Therefore, if a man owning f of a share in the Central Rail-
road sold -f of it, he had | of a share left.
3. Since he gave J of it away, he had | of it left; and § of
\ is i. Therefore, etc.
4. Since he gave | of it for a knife, he has | of it left, >»nd
I of I is y^^. Therefore, etc.
5. Since $18 was | of what the watch cost, he lost J of the
cost, which is i of $18 or 6. Therefore, etc.
6. Since $45 was f of the cost, he gained i, which is J ot
$45, or $5. Therefore, etc.
'7. Since -f of the cost was sacrificed, $120 is | of the cost;
f of $120, which is $30, is |, and 3 times $30 is $90, the whole
loss. Therefore, etc.
8. Since he lent i of the remainder, $22 is |, and i of ^22,
which is $11, is i ; 3 times $11 is $33, or the whole of the
remainder; and $33 is J^ of 4 times $33 which is $132.
Therefore, etc.
9. Since $80 v^as ^ of | of 2 times, or | the cost, the losa
was i which is J of $80, or $40 Therefore, etc.
(US)
162 MISCELLANEOUS EXAMPLES IN THE
10. Since S^54 was ^ of 2 times, or -f the selling price, lie
gained |, which is J- of $54, or $9. Therefore, etc.
11. Since 15 is f, ]• of 15, which is 3, is } ; 8 times 3 's 24,
and i of 24 is 8. Therefore, etc. Or, i of that number of
which \ of 15 is i.
12. Since 4 is |, i of 4, which is 2, is ^; 3 times 2 is 6,
which is I of 2 times 6, or 12. 4 from 12 leaves 8, aL^d
limes 8 is 16. Therefore, etc.
13. Since he sold J of his flock, 20 must be f ; | of 20,
which is 10, is ^, and 5 times 10 is 50. Therefore, etc.
14. Since | of the remainder, or | of J which is |, was in
the water, the 3 feet in the mud must be the remaining a ; and
3 feet is i of 5 times 3 or 15 feet. Therefore, etc.
15. J + 1 = 1- j, and the 4 years equals the remaining y^, and
4 years are /_ of 30 years. Therefore, etc.
16. Since $20 = 1 the cost of the coat plus $12, $20— $12 =
$8, or I the cost; I of $8, or $4 = i, and 3 times $4 is $12.
Therefore, etc.
17. If i the number + 80 was 5 more than 3 times the num-
ber, 80 — 5 = 75, when added to I the number must have been
3 times the number. 3 — |, and J — i = f ; and 75 must have
been f times the number. 75 is f of 2 times } of 75, or 30.
Therefore, etc.
18. Since 16 is |, | of 16 or 8 is i, and 3 times 8 is 24,
which is twice as many as James has ; then i of 24 or 12 is what
James has after losing 16, and 12 + 16 = 28, or the number
James had at first ; since J of John's equaled ^ of James's,
^ of ^ or I of James's=^ of John's ; | of 28 is 4, and 5 timea
4 is 20, or what John had at first. Therefore, etc.
19. Since J + 10 ycars=:l^ or J of his age, f — 3 — J or J
must =10 years; and 2 times 10 years is 20 years. There-
fore, etc.
20. Since | of $40 is /_, i of |, or i of $40 is j\ ; i of $40
is $8, and 11 times $8 is $88, which is 2| or | times i ; } of
(145, 146)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 163
$88 or $11 is i of 3 times §11 or $33, and $33 is i of 4 times
$33, or $132. Therefore, etc.
21. 4 times f is f ; and 25 is f of 4 times i of 25 which is
20. Therefore, etc.
22. Since he made away with f of | of it, the $10 left was
the remaining | of | or I of the whole ; and 5 times $10 is $50.
Therefore, etc.
23. Since | of $1500 is f, J of |, or j% of $1500, which is
$200, is 1 ; 8 times $200 is $1600, which is 4 times the cost
of the barn, and J- of $1600 is $400. Therefore, etc.
24. Since | of 500 or 300 men was | of yf ^, or -^^ of the
force, 50 times 300 men or 15000 men, was the whole force.
Therefore, etc.
25. Since f of 100 or 60 was 7i or J/» tj of 60 which is 4
is J ; 2 times 4 is 8, which is y^^ of 100 times 8 or 800.
Therefore, etc.
26. Since f of 60 or 50 was 2^ or f times yi^, i of 50 or 10
men is |, and 2 times 10 is 20, which is j^q of 150 times 20
or 3000 men. Therefore, etc.
27. Since f of 100 or 80 was 1^ or y-, yV of 80 or 8 is
|. 1 times 8 is 5Q, less 6 = 50 or ^\. And 20 times 50 is
1000. Therefore, etc.
28. Since 2i times 30 or 70 is 3i or V", rV of 70 or 1 is i,
3 times 7 is 21, which is j\ of 10 times 21 or 210. There-
fore, etc.
29. Since i of 1200 or 1000 is 8^ or V', sV of 1000 or 40
is 1 ; 3 times 40 is 120, which is yf^ and i of 120 or 20 ia
^^0 ; 100 times 20 or 2000 is j of 10000, which lacking
1000 of being the whole army, 10000 + 1000 = 11000. There-
fore, etc.
(146,147)
104 MISCELLANEOUS EXAMPLES IN THE
98.
2. Since ^6 + $4 or $10, bouglit 40 bushels, $1 would bujf
1*0 of 40 bushels or 4 bushels, aud $6 would buy 6 times
4 bushels or 24 bushels, and |4, 4 times 4 bushels or 16
bushels. Therefore, etc.
3. Since they approach each other 4 miles + 3 miles or 7
miles an hour, they will meet in Y- or 7 hours, and the one who
traveled 4 miles per hour would travel 7 times 4 or 28 miles,
and he who traveled 3, Y times 3 or 21 miles. Therefore, etc
4 Since 3 weeks +2 weeks or 5 weeks' hire is $25, 1 week
IS i of $25 or $5, 2 weeks 2 times $5 or $10, and 3 weeks 3
times $5 or $15. Therefore, etc.
5. Since 5 cows -f 3 cows or 8 cows' pasture cost $24, 1
cow's pasture cost | of $24, or $3, 5 cows' 5 times $3 or $15,
and 3 cows' 3 times $3 or $9. Therefore, etc.
6. 2 horses for 2 weeks=l horse 4 weeks, and 2 horses 4
weeks = l horse 8 weeks; and since 12 weeks' pasture cost
$12, 1 week's pasture costs y*2 of $12, or $1, 4 weeks' 4 times
$1 or $4, and 8 weeks' 8 times $1 or $8. Therefore, etc.
7. Since 9 cents + 7 cents or 16 cents bought 32 figs, 1 cent
would buy y*g of 32 or 2 figs, 9 cents would buy 9 times 2 or
18 figs, and 7 cents would buy 7 times 2 or 14 figs. There-
fore, etc.
8. A's $10 for 5 months=:$5 for 1 month, B's $5 for 8
months=:$40 for 1 month; and since $50 + $40 or $90 gain
$45, $1 will gain J^ of $45, or $i, $50 50 times $^, or $25,
and $40, 40 times $i, or $20. Therefore, etc.
9. Since they paid in the proportion of $5, $4, and $3j.
they own in the same proportion ; consequently the gain ii
divided into 5 plus 4 plus 3, or 12 parts, and j\ of $24 is $2.
A's portion is 5 times $2 or $10; B's, 4 times $2 or $8;
and C's, 3 times $2 or §6. Therefore, etc.
10. Since A does 2 times 3 days, or 6 days' work, B 3
times 3 days, or 9 days' work, and C 3 times 2^ or 5 days'
(147, 148)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 165
work, it takes 20 days to mow the field ; 1 day^s work c/jsts
5?o of |40, or $2 ; and A should receive 6 times $2, or $12;
B 9 times $2, or 18 ; and C 5 times $2, or §10. Thei'efore, etc.
11. Since C took $10, or :^2 — -^V of the gain, he must have
put in 2^^ of the stock, and A's $30 plus B's $50, or $80 = |f ;
^V of $80, or $5=^\, and 5 times $5 or $25=:iC's siock
$42— $10=$32,A's + B'sgain; A's=ff or f of $32, or $12;
end B'sr=f J or ^ of $32, or $20. Therefore, etc.
12. Since he put in f^J or f of the capital, he should also
lake f of the gain; f of $240rr$150, and $150-$145=$5
loss. Therefore, etc.
13. Since 2 colts consume as much as 3 calves, 4 colts, or 2
times 2 colts=:2 times 3 calves, or 6 calves, and 5 calves plus
6 calves = ll calves. If 11 calves cost $11, 1 calf cost j\ of
$11, or $1 ; 5 calves 5 times $1, or $5; and 6 calves 6 times
$1, or $6. Therefore, etc.
14. Since C pays |- of the rent, he puts in i of the cows.
Then A's 5 cows + B's 3 cows =8 cows--=| of the cows, and
|- of 8, or 4 cowsr^C's number. And since C's 4 cows cost i
of $42 or $14, 1 cow cost { of $14, or $3i ; 5 cows cost 6
times $31, or $17^; and 3 cows 3 times $31, or $10^.
Therefore, etc.
15. Since 4 cows=3 oxen, 8 cows, being twice 4, = 2x3,
or 6 oxen ; and since 5 calves = 4 cows, 10 calves, being twice
6, =2 X 4 or 8 cows. But 8 cows=:6 oxen; and 9 oxen + 6
oxen -f- 6 oxen==21 oxen, which cost $5B. 1 ox cost ^^ of
$5G or $2| ; 9 oxen 9 times $2|, or $24 ; and 6 oxen 6 times
$2 1, or 16 ; etc.
16. Since Mary wrote | as many lines as Melissa, Melissa's
work IS divided into 8 parts, 7 of which =Mary's; then 8 + 7
= 15 ; and j\ of 60 is 4 ; j\ 8 times 4, or 32 ; and y''^ 7 times
4, or 28. Therefore, etc.
17. Since the boys received as many pears as the girls,
tbey received i of 24, or 12. There were as many boys as
8.R.P. ' (148, 149)
166 MISCELLANEOUS EXAMPLES IN THE
} is contained tioies in 12, whicli is 4 times; as many girls
is 4 is contained times in 12, which is 3 times ; and 4 + 3 = 7,
rherefore, etc.
18. Since each son received ^ as much as each daughter, the
2 sons received as much as 1 daughter ; then we have $96
divided into 3 + 1=4 parts; } of $96=$24=each daughter's
portion ; and J of $24=$12=each son's portion. Therefore, etc.
99.
1 The 1st has 1 part, the 2d 1 part +2, and the 3d 1
part-T'2 + 6; then 3 parts + 2 + 2 + 6 = 76, or 76 = 3 parts
+ 10; and 76 — 10, or 66 = 3 parts; J of 66 or 22=what
1st boy had ; 22 + 2, or 24 = what 2d boy had ; and 22 + 2 + 6,
or 30 = what 3d boy had. Therefore, etc.
2. Henry has 2 more than James, and Joseph having 2
more than Henry, has 4 more than James ; hence 72, the
sum of all, is 2 + 4, or 6 more than if each had no more
than James. l2 — Q = G6yAns,
3. If Henry had 2 more he would have as many as Joseph ;
and James + 2=Henry, and +2 more= Joseph ; and 72 + 2
+ 2 + 2 = 78, Ans.
4. If Joseph give James 2, Joseph's number will be dimin-
ished and James's increased 2, when each will=Henry's.
James will now have ^ of 72 — 6, or 22 ; Henry 22 + 2, or 24 ;
and Joseph 24 + 2, or 26. Therefore, etc.
5. Since C paid as much as A and B, he paid ^ of $600,
or $300; and B and A paid $300. And as B paid $100
more than A, $300 — $100==$200, or what each would have
paid if they had paid no 'nore than A. J of $200 is $100,
or what A paid ; and $100 + $100 = $200, what B paid.
6. The drum cost T part, the rifle twice as much, or 2
parts., and the watch twice as much as the rifle, or 4 parts ;
hence $42 is divided into 1 part + 2 parts + 4 parts, or 7
parts. I of $42, or $6 = costof drum; 2 times $6, or $12 =
cost of rifle; and 2 trmes $12, or $24 —cost of watch
Therefore, etc. (149)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 167
Y. The harness cost 2 parts, the horses 4 times as much
or 8 parts, and the wagon IJ times the harness, or 3 parts;
and the harness 2 parts 4- the horses 8 parts, plus the wagon
5 parts=15 parts=$225. j\ of $225 is $15, and 2 times
$]5, or $30==harness ; 8 times $15, or $120=:cost of horses ;
and 5 times $15, or $75z=cost of wagon, etc.
8. Since he traveled ^ as far the 1st as the last 2 daya,
the last 2 days' travel is divided into 2 parts ot which 1 =
first day ; hence ^ of 114 miles, or 38 milesi^lst day; the
same reasoning applied to the last day gives 38 miles, and
leaves 38 miles for the 2d day.
10. The note of $20 was less than ^ of what remained
due after the 1st payment, by the $20 that exceeded \ ; hence
$20 + $20, or $40=r|. 2 times $40, or $80 = what remained
after 1st payment, and $80 was less than ^ the debt, by the
$10 the payment exceeded i ; $80 + $10, or $90= J ; and 2
times $90, or $180= the whole debt.
11. The 4 pennies left is less than \ of the remainder by
the 1 penny more than ^ paid for the whip; then 4+1=5
pennies, or ^, and 2 times 5, or 10 pennies = remainder after
purchasing top ; and since he paid 2 pennies more than \
of all for the top, 10 pennies + 2 pennies, or 12 peunies=| ;
^ of 12 or 6 pennies=i ; and 3 times 6 = 18 pennies. There-
fore, etc.
12. Since he sold the whole, the \ gallon more than | the
remainder sold was ^ of the remainder, and 2 times ^ or 1
gallon = remainder. The gallon left after 1st sale was less
than 1 the keg by the | gallon more than the \ gallon sold ;
then 1 gallon + i gallon or 1| gallons = ^ the keg, and 2
times 1 1^ or 3 gallons=the contents of the keg.
14. Since | of John's = J of Mary's, \ of John's=-J- of J or
f of Mary's, and | or all of John's =3 times f or | of Mary's ;
hence xMary's are divided into 8ths and John's =9 of them,
and the whole = J/ of Mary's, y^y of 34 or 2 is | ; 9 times 2
or 18= John's, and 8 times 2 = 16 = Mary's.
(1195 150)
168 MISCELLANEOUS EXAMPLES IN THK
15. Since | of A's plus 8=B's, B's— 8 = | of A's; and if 8
be taken from B's, the sum of both flocks will be 83 — 8 or 75.
A has 3 parts, B 2, and both 5. ^ of 75 is 15. 3 times 15
or 45= A's; and 2 times 15 or 30, + 8 = 38, B's.
16. Since f of Mary's less 10 cents = Susan's, Susan's f 10
cents=:f of Mary's, and then both would have 39 + 10 or 49
cents. Mary having 4 parts, and Susan 3, they both have 7
parts. I of 49 or 7 = 1 part; 4 times 7 or 28=Mary's; and
3 times 7 or 21 — 10 = 11 =:Susan's.
17. Since J of Homer's=-f of Silas's, J of Homer's will=J-
of f or f of Silas's, and f or the whole of Homer's, 5 times f
or Jy" of Silas's; and since Homer's exceeds Silas's by ^ of
Silas's, the 3 marbles must=^ of Silas's ; hence Silas has 1
marbles and Homer 10.
100.
1. Since the first drink a gallon in 8 days, he will drink i
of a gallon in 1 day, and since the second drink a gallon in 4
days, he will drink J of a gallon in 1 day ; both will drink
i+i or -{fj of a gallon in 1 day, and 1 gallon will last as many
days as y'g, what they drink in 1 day, is contained times in j|
or 1 gallon ; y\ is in y| 1-f times. Therefore, etc.
2. Since Julia can do it in 7 hours, in 4 hours she can do 4
of it, and Jane must do the remaining ^ ; and since Jane does
if in 4 hours, she will do | in i- of 4 or li hours, and ^, or tht
whole, in 7 times li, or 9^ hours. Therefore, etc.
3. Since the first can do it in 9 hours, he can do f in 5
hours, and the second must do the remaining |- ; and since the
second pitches |^ in 5 hours, he can pitch J in | of 5, or 1 1
hours, and f, or the whole, in 9 times 1^, or 11; hours.
4. 3f=Y- and 7{=Y-. Since the second pipe can empty
t in \^ hours, it can empty j\ of it in |, and |f in Y" l^ours,
and the first must empty the remaining || ; and since the first
can empty || in ^■^- hours, it can empty -^\ in gV of W or j
hour, and |-|, or the whole, in 58 times |, or 7| hours.
(151)
PROGRESSIVE INTELLECTUAL ARITHMEIKJ. 169
6. Since A can make a vest in | of a day, he can make
as many vests in a day as f is contained times in |, or 1^
vests ; B as many as f is contained times in J, or 1-^ vests ;
and l| + li, or 3 vests=:what A and B can both do. C can
make as many as f is contained times in |, or 1} vests, and
3— li = l|. Therefore, etc.
6. Susan can knit as many pairs as | is contained timei
m I, or 1| pairs; Sarah can knit as many as -3 is contained
times in ^, or 2i pairs; and l| + 2^ = 4 pairs.
7. Since Sarah can knit 2i or J pairs in a day, she can
knit i of a pair in | of a day, which is the part she must
knit for Susan.
8. Since Susan can knit 1| or f pairs in a day, she can
knit ^ of a pair in i of a day, which is the part she must knit
for Sarah.
9. Since Jason can hoe 10 rows in J of an hour, he can
hoe 1 row in y^ of J, or j% of an hour, and as many rows
in an hour as 3 is contained times in 40 or 131 rows.
Since Jesse can hoe 10 rows in j of an hour, he can hoe 1
row in y'^ of f or j\ of an hour, and as many rows in an
hour as 3 is contained times in 50, or 16| rows; and
both can hoe 13i + 16|, or 30 rows, in an hour; 1 row in
^^ of an hour ; and 10 rows in i§ or ^ of an hour.
10. Smce Jesse can hoe 16| or \o rows in an hour, in
I of an hour he can hoe | of */ or V=^3 rows; leaving
1| rows for Jason, who can hoe 13i or ^« rows in an hour,
1^ of a row in y^^ of an hour, and 1| or f rows in 5 times yV
or I of an hour.
11. Since Jason can hoe 13^ or ^^^ rows in an hour, in J of
an hour he can hoe ^ of */, or V=4f rows ; leaving 5f rowa
for Jesse, who can hoe 16|, or Y=-^p rows in an hour, J of
a row in yJ^ of an hour, and \% or 5f rows in 50 timea , jj
of an hour.
12. See analysis of Example 9.
13. Since A and B can clear the field in 15 days, thev can
(151. 152)
170 MISCELLANEOUS EXAMPLES IN THE
clear f^ of it in 1 day, and -fj or J of it in 9 days ; and sinco
A and B clear | of it in 9 days, C must clear the remaining
} ; and if he clear f in 9 days, he will clear i in ^ of 9 or 4j
days, and f or the whole field in 5 times 4^ or 22| days.
14. Since A and B can dig it in 6 days, they can dig } of
it in 1 day ; since A and C can dig it in 8 days, they can dig
J of it in 1 day ; and |— J or j\ of it, is what B does more
in a day than C. As B and C dig it in 9 days, they can
di^ i ^^ it in 1 day, and since B's day's work exceeds C's by
^■f of the well, J — aV ^^ 1^=^ ^^ ^'® ^^7^ ^.nd J of yV ^^ yf j
=what C can do in 1 day ; hence C can do it in as many days
as 5 is contained times in 144 or 28|^ days. Since B and G
dig I of it in 1 day, and C digs yf y of it in 1 day, i — yf 4- or
yU_=what B digs; hence B can dig it in as many days as 11
is contained times in 144 or 13tV davs. Since A and B dis
^ of it in 1 day, and B digs jW of it in 1 day, J — yVv or tV\
=:what A digs in 1 day ; hence A can dig it in as many days
as 13 is contained times in 144 or llyj days.
15. Since A digs y^^^, B y^^, and C yf y of it in 1 day
they will all dig yW + yW -f if?? or y^y of it in 1 day ; and
it will take as many days as 29 is contained times in 144 01
m <^ays.
16. Since Patrick and Peter can dig it in 15 days, they
can dig y'y of it in 1 day, and |{ or | in 10 days, and Philc
must dig the remaining third ; and since Philo digs ^ in 10
days, he can dig f or the whole in 3 times 10 or 30 days.
Since Philo can dig it in 30 or */■ days, he can dig ^V of it
»n 1 of a day, and in 13^ or ^/ days he can dig 40 times ■^\
or |- of it, and Peter must dig the remaining f ; and since he
digs f in 4Ji days, he will dig | in ]^ of V" or f days, and f or
the whole in 9 times f or 24 days. Since Peter can dig it
in 24 days, in 15 days he can dig ^f or f of it, and Patrick
must dig the remaining | ; and since he digs | in 15 daysi
be will dig | in i^ of 15 or 5 days, and f in 8 times 5 oz
40 days. As Patrick can dig 40 rods in 24 days, he can dig
(152)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 171
iV of 40 or 1| rods in 1 day, and since Peter ,ian dig 40
rods in 40 days, lie can dig 1 rod a day, and it will take liirn as
many days asl|-fl = 2|is contained times in 28, or 10|^ days.
17. Since 30 rods is JJ or J of 40 rods, it will take each
man J as long to dig it. Since Patrick could dig it in 40 days,
he can dig 30 rods in J of 40 or 30 days ; since Peter can
dig it in 24 days, he can dig 30 rods in f of 24 or 18 days
and since Philo can dig it in 30 days, he can dig 30 reds in
I of 30 or 22i days.
18. Henry's work is divided into 4 equal parts, and since
Harlan's exceeds Henry's by 1 of these parts, he must do 5
parts, and both of them 4 + 5 or 9 parts. Since Henry cuts
f of it in 6| or ^^ days, he can cut ^ in ^ of -^- or f days,
and f in 9 times | or 15 days. Since Harlan cuts f of it in
6| or 2-0 days, he can cut |^ in ]^ of ^/- or ^ days, and f in 9
times 1^ or 12 days.
19. Since the 3d does | as much as the 1st and 2d, the
work of the 1st and 2d is divided into 5 parts; and since
the 3d does | as much,. the whole is divided into 5-f 2 or
7 parts. Since the 3d does f of the whole in 10 days, he
can do I in |- of 10 or 5 days, and -J in 7 times 5, or 35
da;'s. Since the 1st and 2d do ^ in 10 days, they can do
4 in } of 10 or 2 days, and ^ in 7 times 2 or 14 days. And
since the 1st does J as much as the 2d, the whole is divided
hto 7 parts, of which the 1st does 3, and the 2d 4 parts.
Smce the 1st does ^ in 14 days, he can do | in |^ of 14 or 4 1
days, and ^ in 7 times 4| or 32| days. Since the 2d does \
in 14 days, he can do | in {^ of 14 or 3^ days, and ^ in 7
times 3i or 24^ days.
20. See Analysis of example 19.
21. Since the 1st can do it in 32| or *3^ days, he can do
j'j of it in |- of a day or -^j in a day ; and since the 3d cau
do it in 35 days, he can do j'j of it in 1 day ; and both can do
^8 + 3 J ^^ -Ns ^^ 1 ^^y» *^^ ^^^ whole in as many days ag
29 is contained times in 490 or l^-^j days.
(152)
172 MISCELLANEOUS EXAMPLES IN THE
22. Since the 2d can do it in 24^ or Y" ^^ys, he can dc j^
of it in i of a day, or /^ in a day ; and since the 3d can io ^^-g
of it in 1 day, the 2d and 3d can do /g +-3J, or g'^^ of it in 1
day, and they can do all of it in as many days as 17 is con-
tained times in 245, or 14^'^^ days.
23. Since B and C can do it in 12 days, they can do y"^ cr
I of it in 8 days, and A must do the other i ; and since A can
do "I" in 8 days, he can do ^ in 3 times 8 or 24 days. Since A
and B can do it in 10 days, they can do j% or J of it in 8
days, and C mast do the other \ ; and since C can do | in 8
days, he can do | in 5 times 8, or 40 days. Since A can do
it in 24 days, he can do i| or /^ of it in 10 days, and B must
do the remaining y^g ; and since B can do y\ i^ ^^ ^^J^i ^g can
do j\ in I of 10 or 1^ days, and -ff in 12 times If or 17 j dayt.
24. Since the 1st and 2d will discharge it in 8 hours, they
'discharge f or i of it in 4 hours, and the 3d must discharge
the other ^ ; and since it discharges -^ in 4 hours, it will dis-
charge I in 2 times 4, or 8 hours. Since the 3d will discharge
it in 8 hours, it discharges f or | of it in 6 hours, and tho
1st must discharge the other \ ; and since the 1st discharges
} of it in 6 hours, it will discharge | in 4 times 6 or 24 hours.
Since the 1st and 3d discharge it in 6 hours, they will dis-
charge f or I of it in 4 hours, and the 2d must discharge the
other i ; and since the 2d discharges J in 4 hours, it will dis^-
charge f in 3 times 4 or 12 hours.
25. Since A and B can do it in 20 days, they do ij or f
of it in 10 days, and C does the other i ; and since C does |
in 10 days, he can do | in 2 times 10 or 20 days. Since B
and C can do it in 15 days, they do j| or | of it in 10 days,
and A does the other third ; and since A does ^ in 10 days,
he can do f in 3 times 10 or 30 days. Since A can do it in
80 days and C in 20 days, they can both do 3V + 2V or y'^ of
it in 1 day, and if in 12 times 1 or 12 days.
26. Since it would last them aH 30 days, they would eat 3'^
of it in 1 day, and 20 times gV or | of it in 20 days, leaving \
(152, 153)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 173
of it to be eaten by the sister. Since the brother and servant
would eat it in 45 days, they would eat |f or | of it in 30
days, and the sister must eat the other J in 30 days.
101.
2. Since 2 plums was tbe increase given to 1 playmate, and
9—1 or 8 plums the increase given to all, there were as many
!d\ ay mates as 2 is contained times in 8, which is 4 times.
Therefore, etc.
3. Since the difference between 6 times and 3 times a num-
ber is 3 times the number, 15 must be 3 times the number, and
J- of 15, or 5 must be the number. Therefore, etc.
4. Since the difference per yard was 12 cents — 8 cents, oi
4 cents, she wanted as many yards as 4 is contained times in
the whole difference, 11 cents + 17 cents, or 28 cents, which is
7 times.
5. Since the difference between 6^ times and 4 times a
number is 21 times or f times the number, 15 must be |
times the number; J of 15 or 3, i ; and 6, the number.
6. Since | — i = |, i of 4 or 2 must be | and 9 times 2 or
,18 = t. • •
7. Since the difference between 5^ times and 3| times a
number is 2.1 times the number, -^j of 21, or 1, must be y^,
and 10 times 1, or 10, |J, or the number.
9. If we let 1 or | represent the whole number of chickens,
I times 5 + J- times 3=-'/, will represent the whole number of
grains, that is, "3^=26. And since 26 is -y- times the number
of chickens, | or the whole number of chickens, was 3 times |
of 26, or 6.
10. Since 26 is 5 times | + 3 times ^, or ^-f of the number;
}^ of 26 or 2 is ^, and 3 times 2 or 6 is f or the number.
Therefore, etc.
11. Since the 1st condition gives 5 times ^ of a number,
plus the 2d condition, which gives 3 times J of the same num-
ber, plus the 3d, which gives 2 times | of the same n amber,
(153, 151)
174 MISCELLANEOUS EXAMPLES IN THK
plus the 4th, which gives once J the same number ; we have
^ of the class equal to 29, and ^^^ of 29 or 1 is ^ of 9 times
1 or 9. Therefore, etc.
13. Since 4 times ^ of a number, plus S^ times the num-
ber, or -y- times the number, is equal to 28 plus 5, or 33, f\
of 33, or 3 is ^ of the number, and 2 times 3 or 6 is the num-
ber. Therefore, etc.
14. Since the 1st condition gives ^ of his age plus 4, the
2d gives f, and the 3d gives ^ less 4, we have the sum -'J
equal to 50 ; y^ ^^ ^^ ^^ 5 is ^ of his age, and 3 times 5 or
15 is J. Therefore, etc.
15. Since he paid $5 a head for ^ of the flock, the cost=:5
times ^ or 1^ of his flock; $4 a head for as many more = 4
times I or J of his flock; $3 a head for | of the remainder,
or ^ = 3 times J or i of his flock ; and 82 a head for the rest,
or J- = 2 times | or i- of his flock; and f + J + i + i = V'
That is, the number of dollars the flock cost=:\3 of f}^Q num-
ber of sheep, and 115 is ^ o^ ^ times gV of 115 which is 30
Therefore, etc.
16. Since he received 6 dimes
each for I =1x6=1
for ^ of the remaining | and 3 more,
4 dimes each =(f + 3) x4=f + 12dime»
for 1 of the rest (which is f less 1
on each eight), or 1 + 1 = (} + l)x3=:f+ 3 **
for the rest (which is f less 1 on
each eight, also the 2 of last sale),
or f less 4 =(|— 4) x 2=| — J **
the whole number of dimes is equal
to -5/ of the baskets and 7 dimes besides, . . =-\^+ 7 **
hence $10 or 100 dimes less 7 dimes=93 dimes=3_i^ _i_ qj
93 or 3 = |-, and 8 eighths, 8 times 3 or 24. Therefore, etc.
I 18. 6 times a number equals \^y Y times J of it plus 5
dmes 1 of it equals ^/, and Y less Y- = J or ^ of it, which,
(154, 155)
PROQKESSIVB INTELLECTUAL ARITHMETIC. 175
according to the condftion of the question, is 4 ; and 4 is |
of 2 times 4 or 8. Therefore, etc.
19. 6 times the number, or Y» left 4 cents, but 5 times J oi
it, or y , plus 7 times ^ oi it, or Y- was it all of it ; and by the
condition of the question ^^ less y or \ equals 4 ; and f is 2
times 4 or 8. Therefore, etc.
20. 4 times a number equals Y, 5 times 4 of it equals y,
and Y less Sy^- equals f of it, which-by the question is 6 ; an.l
6 is I of 7 times ^ of 6, which is 14. Therefore, etc.
21. 2 times a number equals | of it, 5 times i of it equals
I, and this plus 2 times \ of it — which is | — equals | of it,
and J less f , equals J of it, which by the conditions of the
question is 60 ; and 60 is | of 2 times -J- of 60 which is 40.
Therefore, etc.
22. 2 times } of a number equals f of it, which is 8 more
than J' hence 8 is f or i of it, and 2 times 8 or 16 is the
whole of it Therefore etc,
103.
2. Since 19 is the sum of two numbers whose difference is
8, 19 less 3, or 16, is twice the less number; ^ of 16 is 8, the
less number, which, increased by 3, equals 11, the greater
number. Therefore, etc.
3. Since 31 is the sum of two numbers whose difference is
9, 31 less 9 or 22, is twice the less number ; | of 22or 11 is*
the less number, which, increased by 9, equals 20, the greater
number. Therefore, etc.
4. Since 37|^ is the sum of two numbers whose difference it
5^, 3Y| less 5^ or 32, is twice the less number ; | of 32 is 16
the less number, which, increased by 5^, equals 21J-, the
greater number. Therefore, etc.
5. Since 21 is the sum of two numbers whose difference is
5, 21 less 5 or 16, is twice the less number; |^ of 16 is 8;
the number Homer had at first, plus 3, equals 11, or what
(155, 156)
176 MISCELLANEOUS EXAMPLES IN THE
he lias now ; and 21 less 11, or 10, equals what Horace h^
now. Therefore, etc.
6. Since Mary has twice as many as Martha, she has 2
parts, and Martha 1, they both have 3 parts; ^ of 12 quarts or
4 quarts, equals what Martha has, and twice 4 or 8 quarto
equals what Mary has. Therefore, etc.
7. Since 47 is the sum of two numbers, one of which is S
nore than twice the other, 47 less 5, or 42, equals 3 times
he less number ; i of 42 or 14, equals the less, and twice 14
or 28 plus 5, which is 33, equals the greater. Therefore, etc.
8. If the small bin held 6 bushels more, it would contain J
as much as the other, and both would hold 60 bushels, or 3
times as much as the small one ; ^ of 60 or 20, less 6, which
is 14, equals the number of bushels in the smaller bin, and 2
times 20 or 40, equals the number in the larger bin. There-
fore, etc.
9. Had the watch cost $4 more, both would have cost $100,
or 4 times the cost of the chajn ; J- of $100, or $25, equals
the cost of the chain, and $96 less $25, or $71, equals the
cost of the watch. Therefore, etc.
10. Since Hiram received 11 times 2, or 22 dimes more
than Harvey, 253 dimes, what both received, less 22 dimes, or
231 dimes, equals twice what Harvey received; | of 231, or
1151 dimes equals what Harvey received, and 115^ dimes,
■plus 22 dimes, or 137|^ dimes equals what Hiram received;
y'y of 115 J dimes, which is $1.05, equals what Harvey received
per day; and $1.05, increased by 2 dimes, equals $1.25, what
at Hiram received. Therefore, etc.
11. Since B's age was 2 times A's 6 years since, 48 years^
the sum of their ages then, must have been 4 times A's age ;
I of 48, which is 12, plus 6, or 18 years, equals A's age ; and
60 less 18; or 42 years, equals B's age. Therefore, etc.
12. Since the horse cost $4 more than 3 times the cost of
the cow, $124 less $4, or $121, is 4 times the cost of the
cow; \ of $121, or $30.25, equals the cost of the cow
(156, 157)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 177
and $124 less $30.25, or $93.75, equals the cost of the lorso
Therefore, etc.
13. Since the product is the same whichever factor be taken
for the multiplicand, we will use } of the cost of the cow,
which taken 4 times, equals J or the whole cost ; hence, i of
the cost of the colt must be $4 ; twice $4, or $8, equals what
he paid for the colt; and $24 less $8, or $16, equals what
he paid for the cow. Therefore, etc.
14. Since the cost of the cover (which, by a condition of
the question is ^ as much as the dish plus the difference),
increased by the difference, equals the cost of the dish, the
dish costs twice the difference plus I of itself, or the differ-
ence equals J of the cost of the dish ; and f less |, or § of
the cost of the dish equals the cost of the cover ; and 24
dimes equals | of the cost of the dish. | of 24 dimes=3
dimes, 5 times 3 = 15 dimes, the cost of the dish; and 24
dimes— 15 dimes =9 dimes, the cost of the cover.
15. Since the less number, — which by the question equals |
of the greater plus the difference, — increased by the difference
equals the greater, we have the greater equaling -i of itself
plus twice the difference, or the difference equaling j\ of the
gi-eater, and t6~"tV — tV ^^ ^^^ greater equals the less; |J
4-y«g=f-f ; 2 J of 25 pounds, or 1 p )und, is j\ of 16 pounds,
the greater number, and 9 times 1 pound or 9 pounds is the
less. Therefore, etc.
16. Since the sum of the difference and the less number
equals the greater, the less must equal | of the greater, and
both numbers f of the greater ; | of 10 = 2 is i of the greater
number, 3 times 2 is 6, the greater ; and 10 less 6, or 4 is the less,
17. Since the cost of ironing, plus ^ of the difference, equals
I*-, of the cost of the wood-work, the remaining ^j must equal
i of the difference, and the difference equals j\ of the cost
of the wood-work ; || less y\ equals j\ of the cost of irou-
mg ; II plus Y®, or |^ times the cost of the wood-work equali
(157)
178 MISCELLANEOUS EXAMPLES IN THK
|38. i-'g of $38 or $2, is yV» ^ ^^^n^^s $2 or $22, is the cost o(
the wood-work, and $38 less $22 or $16 is the cost of ironing,
18. Since the cost of the ribbon, — which by the question
equals ^ of the cost of the lace, plus ^ the difference, increased
by the difference between the cost of the lace and ribbon, —
equals the cost of the lace ; we have -^ of the cost of the lace
equal to f of the difference, or the lace costing a sum equal
Ic } of the difference, and the ribbon ^ of the difference, and
both 30 cents, or 5 times the difference. | of 30 cents, or
cents, is the difference between the cost of the two ; 30
cents less 6 cents or 24 cents, is twice the cost of the ribbon,
and ^ of 24 cents or 12 cents is the cost of the ribbon ; and
30 cents less 12 cents, or 18 cents is the cost of the lace.
19. Since the whole of the cost of the knife and once the
difference equals the cost of the skates, and by the question ^
the cost of the knife plus twice the difference equals the same,
once the difference must equal ^ the cost of the knife, twice
the difference the whole cost, 3 times the difference the cost of
the skates, and 5 times the difference equals 20 shillings, or
the cost of both ; | of 20 shillings is 4 shillings, 2 times 4
shillings is 8 shillings, the cost of the knife ; and 3 time?* 4
shillings is 12 shillings, the cost of the skates.
20. Had the harness cost $1 more, both would have cost i J5,
and the horse would cost ^ of $35 or $20, and the harnesi* ^
of $35 or $15, less $1 or $14. Therefore, etc.
103.
2. Had all been old sheep, he would have paid $84, or $8
more than he did ; each yearling made a difference of $1,
hence there were as many yearlings as $1 (the difference on
1) is contained times in $8 (the difference on all), which i» 8
times ; and 28 less 8 equals 20, the number of old sheep.
3. Had all been first quality, he would have paid $9( ^ ox
$8 more than he did ; and since the difference per barrel Ht»
(157, 458)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 179
$•1, he bought as many barrels of poor quality as $1 is contained
times in $8, or 8 barrels; and 20 less 8 equals 12 first quality.
4. Since he lost | of the cost, ^ of $18 or $9, must have
been J of the cost, and 5 times $9 or $45, was the whole cost.
5. There were as many of each as 12 dimes (the number
it took to pay one of each) is contained times in 72 dimes
(the number paid to all). 12 is in 72 6 times, and 2 times 6
12, the whole number.
6. Since »he received 8 dimes for 1 of each, she sold as
many of each as 8 dimes is contained times in 40 dimes, which
is 5 times ; twice 5 is 10, the number of fowls she sold.
7. He bought as many bushels as $.50, the difference on 1
bushel of each is contained times in $7, the difference on all ;
$.50 is in $7 14 times. Therefore, etc.
9. He was idle as many days as $3.50 (the difference made
by 1 idle day) is contained times in $7 (the difference made by
all the idle days), which is 2 times ; 20 days less 2 days is 18
days. Therefore, etc.
11. Since she gave | of the remainder to her teacher, the
2 left must be the other ^ ; 4 times 2 is 8, which was the J
left after division among the playmates, and 4 times 8 or 32
equals the number she had at first. Therefore, etc.
12. J of 1 1 is y\ ; hence 12 is y\ of his flock, and 10 times
12, or 120 equals the number of sheep he had at first
13. Since he paid | of the remainder, $3 must be | of it.
J of $3 is $1, and 5 times $1 is $5, the remainder, which by
thefirst payment wants $5 of being i of the whole ; $5 plus
$5 is $10, 1 of the whole, and 4 times $10 is $40, the whole
Therefore, etc.
li. Since he lent |, $3 plus $5 or $8, must have been i.
8 times $8 or $24 is what he had after paying for the watch ;
$24 plus $12 or $36 equals what he had after paying for hia
clothes, which lacks $10 of being i of his wages; $36 plua
llO, or $46 is ^ ; and 2 times $46, or $92 equals his wages.
(158,159)
180 MISCELLANEOUS EXAMPLE? IN THE
15. iSince in $1 there are 10 climes, he could be idle tm
many days, for each day he worked, as 2 dimes, what he paid
a day for board, is contained times in the amount his daily
wages exceeded $1, which is once; hence he worked ^ of the
time, 9vd was idle 10 days.
104.
2. The part standing was divided into 4 equal parts, 3 of
ir!iich equaled the part broken off; the sum of both piec^
uras 7 equal parts, 1 of which was | of 56 feet or 8 feet, 3
parts were 3 times 8 or 24 feet, which was the part broken off;
and 4 times 8 or 32 feet was the part standing. Therefore, etc.
3. Since Henry has 5 parts and Horace 4 parts, both have
9 parts ; ^ of 45 is 5 ; 4 times 5 or 20 equals the number
Hor? ',e had, and 5 times 5 or 25 equals the number Henry
had. Therefore, etc.
4. Since he left 5 parts and took out 3, he left f of 160, or
100 pounds. Therefore, etc.
5. Since he paid 5 parts for his lodging and 4 for his sup-
per, his supper cost J of 63, or 28 cents. Therefore, etc.
6. Since 9 times | = | times the cost of wagon, equals the
cost of the horse, both cost 8 plus 9, or y- times the wagon ;
iV of $170 is $10 ; 8 times $10, or ^80 was the cost of the
wagon ; and 9 times $10, or $90 the cost of the horse.
Therefore, etc.
7. Since the second day's travel was 1^^=^ times the first,
both equaled f times the second ; | of 140 miles is 20 miles;
3 times 20 equals 60 miles, the second day's travel; and 4
times 20 equals 80 miles, the first day's travel.
8. Since Bergen is 50 miles from Buffalo, 280 miles less 60
or 230 miles equals the distance from Bergen to Schenectady;
and as the distance from Utica to Schenectady is l^zzz^j^ times
the distance from Bergen to Utica, the whole distance from
Bergen to Schenectady equals y plus |, or Y '^ _i_ of 230
miles is 10 miles, and 15 times 10, or 150 miles equals the
(159, 160)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 181
distance from Schenectady to Bergen ; and 150 miles plus
60 miles, the distance from Bergeu to Buffalo, gives 200 miles
from Buffalo to Utica.
9. Since the head was 3 inches long, 17 inches less 3 or 14
inches equals the length of the body and the tail; and as the
body was divided into fifth's, 2 of which equaled the tail, we
have body and tail divided into 7 parts ; | of 14 inches is 2
inches, and 2 times 2 or 4 inches equals the tail.
10. Since the less has 7 parts and the greater 11, both have
18 ; jV o^ '^6 is 2 ; 7 times 2 = 14, the less part; and 11 times
2 = 22, the greater.
12. If the distance from Victor to Rochester were 4 miles
less, it would equal the ^j mentioned, and the whole distance
would be 52 miles less 4 miles, or 48 miles ; from Geneva to
Victor is 11 parts, from Victor to Rochester 5 parts, in all 16
parts; y^- of 48 is 4 ; 11 times 4 miles=44 miles, the distance
from Geneva to Victor ; and 52 miles less 44 miles =18 miles,
the distance from Rochester to Victor.
13. If the church were 6 feet lower, the whole distance
would be 140 feet, of which the steeple would be 4 part's, the
church 3 parts, and both 7 parts; | of 140 feet is 20 feet;
and 4 times 20 is 80 feet, the height of the steeple. There-
fore, etc.
14. Since the jar (which, by a condition of the question,
weighs as much as i the cover plus 12 pounds) and the cover
weighs 18 pounds, we have the cover, J the cover and 12
pounds equal to 18 pounds, or f of the cover weighing 6
pounds; ^ of the cover, ^ of 6 pounds, or 2 pounds; and |,
times 2 pounds, or 4 pounds ; and 18 pounds less 4 pounds, i
14 pounds, the weight of the jar. Therefore, etc.
15 Had the vest cost ^3 less, both had cost but $16, of
which the coat cost 3 parts, the vest 1, both 4 parts ; } of ^16
18 ^4 ; 3 times $4 is $12, the cost of the coat ; and $4 plus $3,
or $7 is the cost of the vest. Therefore, etc.
17 Since | of George^s equaled J of Abel's, 2 halves would
(160, 161)
182 MISCELLANEOUS EXAMPLES IN THE
equal twice f , or J ; then Abel had 4 parts, George 6 parta^
and both 10 parts ; j\ of 50 cents is 5 cents ; 6 times 5 cents
= 30 cents, George's money ; and 4 times 5 cents =20 cents,
Abel's money. Therefore, etc,
18. Since f equaled 4, i would equal J of 4, orf, and f 5
times f or y ; then the black ones were 7 parts, the gray ones
10, and both 17 ; j\ of 34 is 2 ; 10 times 2 is 20, the num-
ber of gray ones ; and 7 times 2 is 14, the number of black
ones. Therefore, etc.
19. Since | equaled f, | would equal | of |, or i^g, and J,
3 times /^ or |f ; one number is divided into sixteenths, 15 of
which equals the other, and ^ equal both ; Jy of 62 is 2 ; 16
times 2 is 32, the larger number; and 15 times 2 is 30, the
smaller number. Therefore, etc.
20. Since J equals |, J would equal J of |, or -j^, and J, 4
times ^y or y"j ; the value of the contents is 15 parts, of thfl*
purse 8 parts, and of both 23 paits ; ^3 of 46 shillings is 2
shillings ; 15 times 2 shillings is 30 shillings, the value of
the contents ; and 8 times 2 shillings is 16 shillings, the value
of the purse.
22. Since from midnight to 10 o'clock is 10 hours, and the
past time is divided into 3 parts, the future into 2, and the
whole iuto 5, we have 1 part equal to ^ of 10 hours, or 2 hours ;
and 3 times 2 hours is 6 hours, the past time ; hence it was 6
o'clock.
23. Since | equals J, ^ must equal ^ of J, or f, and f, 3
times f , or f ; from midnight to 5 o'clock, p.m., is 17 hours,
and as past time is 8 parts, future 9 parts, and the whole 17
parts, 1 part equals 1 hour, and 8 parts 8 hours ; hence it is 8
o'clock, A. M.
24. Since J equaled J, { would equal \ of J, or J, and }, 4
times I, or |. John's age was divided into fifths, 4 of which
equaled Peter's, and both equaled f of John's ; J of 36 years
= 4 years, J of John's; 5 times 4 years =20 years, John'i
age; and 4 timcf 4 years =16 years, Peter's age,
(161, 162)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 183
25. Since J equaled J^*, f would equal f ; and we have what
:ras wanting divided into 5 parts, what was in the bin into 6
parts of the same size, and the whole capacity of the bin into
11 parts; ^j of 44 bushels=4 bushels, 1 part; and 5 times
4 bushels = 20 bushels, what was wanting to fill the bin.
26. Since f of what it exceeded equaled V of what it lacked.
I would equal y- ; and we have what it lacked divided into 7
parts, what it exceeded into 15, or the whole, 83 miles— 39
miles=44 miles, divided into 22 parts ; ^V ^^ ^^ ™^^^^ ^^ ^
oiles; 7 times 2 miles is 14 miles, the distance it lacked of
being S? miles ; and 83 miles less 14 miles is 69 miles, the dis-
tance to Cincinnati.
27. Since f of what it lacks of being 150 miles equals
what it exceeds 100 miles, we have, the excess, 3 parts plus
the deficiency, 2 parts, or 5 parts in all, equal to 150 miles less
100 miles, or 50 miles ; } of 50 miles is 10 miles ; 3 times 10
miles is 30 miles ; and 100 miles plus 30 miles =130 miles, the
distance from Charleston to Columbia.
105.
2. Since f equal 4 + 9, J will equal ^ of ^ 4- 9, which is f
4-8, I, and 4 times f + 3, which is ^ -h 12 ; hence, the mother's
age is divided into 7 parts, and 8 of the same size + 12 years
equals the father's age, or 15 parts + 12 years equals 72 years ;
72 years less 12 years is 60 years, j\ of 60 years is 4 years,
and 7 times 4 years equals 28 years, the mother's age.
3. Since | equal f less 4 rods, -J- will equal ^ of J less 4
rods, which is | less 2 rods, and §, 3 times | less 2 rods, which
is I less 6 rods ; hence what one built equals 6 rods less than )
of what the other built, and both built -y of the amount the
eeccud did, less 6 rods ; 38 rods plus 6 rods, or 44 rods equals
V ? fV ^^ ^4 r^^s ^^ ^ Todsy is A of 5 times 4 rods or 20 rode,
what the second built; and 38 rods less 20 rods, or 18 rods
fquftis what the first built.
4. Since \ was 4 more than J^ | would be 1 more than J,
(162, 163)
184 MISCELLANEOUS EXAMPLES IN THE
and -J, 7 more than J ; hence what Richard sheared are divided
into 5 parts,- Hiram's into Y parts plus 7 sheep, and both into
12 parts plus 7 ; 67 less 7 is 60 ; /_ of 60 = 25, the number
Richard sheared ; 67 less 25=42, the number Iliram sheared.
5. Since | of future time equaled | of the past 4- f f
hours, j- would equal J of | + 1| hours, which is ^ -f j\, and |,
5 times ^4-tj ^ours, which is -3 +f hours; hence the future
time equals f hours more than f of the past, and both past
and future time equa. f of the past-hf hours, or 24 hours;
24 hours less |- hours is 21^ hours, and | of 21^ hours is 8
hours, or the past time; hence it was 8 o'clock a. m.
6. Since | of what his age lacked of being 100 years equaled
f of what it exceeded 64 years, + 9 years, | of his age would
equal i of J + 9 years, whicb is J--|- 1 year, and f , 8 times } + l
year, which is | -f 8 years ; hence, what his age lacked of
being 100 years equaled 8 years more than | of what it ex-
ceeded 64 years, and \^ of what it exceeded 64, is 8 years
less than the difference between 100 years and 64 years, or 36
years; 36 less 8 is 28 years, j\ of 28 years is 2 years, and
6 times 2 or 12 years, is what his age exceeded 64 years.
8. Since the body is as long as the head and tail, it must
be I of the length of the fish ; the tail being as long as the
head and ^ the body, must be ^ of the length of the fish plus
7 inches, and the 7 inches it exceeds the { with the 7 inches of
the head, must equal the other ^ ; 14 is J of 4 times 14 or
66. Therefore, etc.
9. The first price plus the second, equal to J 4- 3 pounds,
equals the third price; 2 times j+3, equal to | of it + 6, equals
the whole of it, and 6 pounds must be f of it ; | of 6 pounds 01
2 pounds is } of it, and 5 times 2 = 10 pounds is the whole of it.
Ji^^ Or it may be solved like the following,
10. Since the third dug as many as the other two, he dug ^ ,
and as the first two dug \ less 2 bushels + 5 bushels, or 3
bushels more than |^, those 3 bushels must equal the difference
(163, 164)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 185
Detween |- and ^ of them, or } of tliem ; and 3 bushels is J oi
6 times 3 bushels which is 18 bushels.
11. Since the distance from Avon to Bath is 12 miles more
than the sum of the other two distances mentioned, we have
the whole distance equal to f of itself+ 60 miles ; hence |
»f 60 miles or 20 miles is \ of the distance ; and 5 times 20,
OT 100 miles is the whole distance from Batavia to Corning.
12. Since he took $24 more than i of the whole for shee]»
and swine, and $7 less than f as much for cattle, he took for
the cattle |2 more than \ of the whole ; and we have $18 4-3
of the whole, +$6, +1 of the whole + $2, or ^^ of the
whole + $26, equal to the whole amount; hence -^^ of $26,
3r $2 is 2V, and 24 times $2, or $48 is what he took for all.
13. Of ^ that number t)f which |, of J and ^ of ^ of 12 is
1. 1 and i of ^ of 12 is 6, ^ of 6 is 2, and 2 is 1 of 8 times 2,
or 16. Therefore, etc.
14. Since he earned | as much as he had spent, he only
lacks I of |=J of the whole, of having as mucn as he had at
first ; $16.50 is ^ of 6 times $16.50, or $99. Therefore, etc.
15. Since | equal f, | will cost i of | of an eagle, or $2.
16. Since C is | as old as A, he is 4 years more than ^ as
old as B ; and as B's age equals the sum of A's and C's, we
have i of it plus 6 years, plus ^ of it plus 4 years, or | of it
+ 10 years, equal to itself; hence 10 years must be } of B^s
age, and 6 times 10 years is 60 years B's age ; ^ of 60 is 30,
30 + 6 is 36, A's age ; and | of 36, or 24 is C's age.
Il, Since C owns ^ as much as A, he owns 6 acres more
than 3 as much as B; and we have what A owns, 12 more
than J as many acres as B, + what C owns, 6 more than f at
many acres as B, equal to 18 more than f as many acres as B
owns, or 24 acres more than his farm ; hence 6 acres equals
J- of B's farm, 8 times 6 is 48 acres, B's ; f of 48 is 36 acres,
and 36 + 12 equals 48 acres or A's ; and ^ of 48, or 24 acres
equals C's.
(164J
186 MISCELLANEOUS EXAMPLES IN THE
106.
2. Since the son's age is | of the father's, the 22 years tho
father's age exceeds the son's must be ^ of the father's age ;
3 times 22 years is 66 years, the father's age, and 66 years
less 22 years equals 44 years, the son's age ; at the son's birth
the father was 22 years old, in 22 years from that time each
would be 22 years older, and the son being 22, and the
father 44 years of age, would answer the condition of the ques
tion, and as the son is 44 now, 44 years less 22 years, or 22
years since, he was | as old as his father.
3. At Helen's birth her sister was 22 less 9, or 13 years
of age, and in 13 years from that time Helen would be 13
and her sister 2 times 13, or 26 years of age; and as Helen
has advanced through 9 of 13 years, she has 13 less 9 or 4
years more to advance.
Or, for brevity, 2 times 9 is 18 ; 22 less 18 is 4. Therefore, etc,
5. Since he took as many from one field and put in the other
as were there, and now both have twice as many as were there
at first, the 60 sheep must have been three times ths number
before removing ; ^ of 60 is 20, the number in the smaller ;
and 20 plus 60, the number in the larger flock, equals 80, the
whole number.
6. Since both bins now contain the same quantity, and each
2 bushels more than twice what was in the less at first, the
larger must have had 4 bushels more than 3 times the less ;
62 less 4 is 48 bushels, ^ of 48 is 16 bushels, what was in the
less; and 16 bushels + 52 bushels, what was in the larger,
equals 68 bushels. Therefore, etc.
7. J less I is 2*;j-, which by the condition of the question, is
6 more than | of his age; ^*y less | is j'^, 6 is y'j of 72.
Therefore, etc.
8. Since he received { of his wages for his summer's la-
bor, \ as much, or } of them in fall -f S20, and $20 in springs
$20 -f $20=^40 must be f or i of his wages, and 3 time*
|40=$120, must be the whole amount.
(165)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 187
10. Since A sold B J as much as B had, B now has J o)
what he had i*t first, which is J of what A has left ; ^ of |
=y''^ is I ; and 4 times j\, or | what A has now — f f, what A
has left ; f | plus the J sold B, gives A f } of B's before the
Bale; -^\ of 74 is 2, and 12 times 2 is 24, the number of acres
B had before the sale ; and 24 acres plus f of 24 acres equals
42 acres, what he now has; and 74 less 18 acres leaves 5Q
acres, what A has left.
11. Since | equal i, i will equal ^ pf J or J, and f, 3 times
J or I ; hence as the turkeys equal J of the chickens, 10 must
be the remaining \ ; 4 times 10 is 40. Therefore, etc.
12. Since | of the price of the coat equaled J of the price
of the suit, ^ would equal ^ of j- or J-, and f , 3 times | or J of
the suit ; and since the coat cost f of the price of the suit,
$15 must be f of it ; ^ of $15, or $3 is |, and f are 8 times
$3 or $24. Therefore, etc.
13. Since f times his brother's equaled J of his, ^ would
equal | of J or y^^ of his ; f , which is 3 times jV, or j\ and
the 14 more must be the remaining ^^^ | of 14 is 2 ; 10
times 2 or20 equals what Daniel caught, and 20 less 14 equals
6, what his brother caught.
14. Since by the conditions of the question we have ^ of
3^ times a number +15, equal to once the number +15, or,
to avoid fractions, 2 times a number +30, equal to 3^ times
he same number + 15, 2 times the number is equal to 2 times
the same number, leaving 30 equal IJ times the number + 15,
or 15 equal to J of the number; ^ of 15 or 5 is i of it; 2
times 5, or 10 is the less number; and 3^ times 10, or 35 is
the larger.
15. Since f equal /j, | will equal i of j%, or /^ ; and }, 5
♦.irnes /^ or | ; and as the buggy cost | as much as the horse,
the difference, $40, must be i of the cost of the horse ; 3 times
$40 or $120 is the value of the horse ; and | of it, or $80, is
the value of the buggy.
16. 5 years since the mother's ago was 5 times Alice's, ana
(166)
188 MISCELLANEOUS EXAMPLES IN THE
by the first condition we have 5 times Alice's age +5 (the
mother's age) equal to 3 times Alice's age +15 ; and since
3 times Alice's age equals 3 times her age, we have 2 times
her age + 5 equal to 15, or 2 times her age equal to 10 ; ^
of 10 is 5, her age; 5 years since + 5 equals 10, her age now ;
and 3 times 10 or 30 is her mother's age ; 2 times 10 is 20,
and 30 less 20 is 10, the number of years in which she will be
I as old as her mother. See Ex, 3, in this lesson,
17. Since Hobart has but | of his left, he lost \ of them
to Dwight, which, by the condition of the question, was equal
to ^ of D wight's ; % must have equaled all of D wight's, and
the 20 Hobart's exceeded Dwight's must have been \ of Ho-
bart's; 3 times 20 = 60 marbles Hobart had ; and 2 times 20
= 40 marbles Dwight had.
18. Since the difference between the numbers is 16, if 4 be
taken from the larger difference will be 12, then added to the
less it will be but 8 ; 2 J times this difference, or 19, is equal to
3J- times less 2f times = if times the less number; y^ of 19
or 1 is -^^ ; 24 times 1=24, the less number; and 24-fl6 =
40, the larger number.
19. Since he paid twice as much for the rifle as for the
watch, and the watch cost $20, the rifle cost- 2 times $20, or $40.
20. Since C's age at A's birth was 5^ times B's, and is now
equal to the sum of A's and B's ages ; and as the increase of
C's age w^ould just equal A's age, and B's increase being the
same, the increase must have been what B's age lacked of being
equal to C's at first, or 4^ times B's age then ; hence we have
A's age now equal to 41 times B's age at first ;
J^'g « *4 a u gx ** " " " "
Q)g u u u <' 10 ^^ ^^ ^^ ^^ ^^
^ow if 4 years be added to B's age, | of the sum, or 4} times
/i's age as first + 3 years, is equal to A's age, and 4^ times B's
it first, which gives the 3 years, is equal to f of B's age at first;
hence B was 8 years old then, and is now 5^ times 8, or 44
vears old ; A is 4| times 8, or 36 years old ; and C is 10 times
8, or 80 years old.
(166, 167)
PROaRESSIVB INTELLECTUAL ARITHMETIC. 189
107.
2. In as many hours as 2 miles, the number he gained in I
hour, is contained times 2 times 5 miles, the distance to be
gained ; 2 times 5 is 10 miles, and 2 is in 10 5 times. There-
fore, etc.
3. As many times 9 rods as 2, the number of rods he gaina
in running 9, is contained times in 28, the number to be
gained; 2 is in 28 14 times, and 14 times 9 is 126 rods.
Therefore, etc.
4. John will have as many times $7 as $2, what he gains
on $7, is contained times in $30, the whole gain ; $2 is in $30
15 times, and 15 times $7 is $105, what John has saved ; and
$105 less $30 is $75, what Henry has saved.
5. Since the distance B ran is divid'^d into eighths, 1 of
which equaled the distance he was ahead of A, A must have
run J as far as B ; | of 84 = 12 rods is \ of the distance B ran,
and 8 times 12 = 96 rods, is B's distance
6. Since $25 is \ of what B and C paid, they paid 4 times
$25 or $100, which, with the $25 A paid, makes $125, th©
cost of the horse ; and since B paid | as much as A and C,
they paid 3 parts and he 2 parts ; that is, 5 parts equal the
whole cost ; f of $125 is $50, what B paid ; and $50 plus
the $25 A paid equals $75 which, taken from $125, leaves $50,
what C paid.
8. Since the minute hand passes over 12 spaces while the
hour hand passes over 1, tbe minute hand gains 11 spaces on
the hour hand for every 12 spaces it passes over, and it would
pass as many times 12 spaces as 11, the number it gains in
passing 12, is contained times in 45 spaces, the number to be
gp^ned after 3 o'clock before they are opposite ; 11 is in 45 4 p"|
timus, and 4Jy times 12 is 49 jV spaces. Therefore it would
be 49y*y minutes past 3 o'clock.
9. Since 3 of the hound's leaps equal 6 of the fox's, 1 will
equal I of 6, or 2 of the fox's, and 4, 4 times 2, or 8 of the
fox's ; hence the fox will take as many times 7 leaps as 1, the
9.R.P. (167, 168)
190 MISCELLANEOUS EXAMPLES, ETC.
number the hound gains on the fox in making 7, is contained
times in 40, the number of leaps to be gained ; 1 is in 40, 40
times, and 40 times 1 is 280. Therefore, etc.
10. Since the distance the sheep ran was divided into 5 parts,
8 of which equaled the distance between them, the whole dis-
tance equaled 8 parts ; | of 80 rods or 10 rods is 1 part, and
3 times 10 or 30 rods eq^ials 3 parts, or the distance betwee
thorn.
11. Since the interest at 5 per cent., for 2 years 7 months
and 6 days, is y'^^ of the principal, the amount will be m ;
j{^ of $2260 is $20, and 100 times $20r=$2000, the sum at
interest ; and since B's money equaled f of A's, the whole
equaled | o^ A's ; } of $2000 is $400 or i of what A had in,
which is I of all, and 5 times $400 or $2000 equals A's ;
$400 equaled \ of what B had in, which is | of all, and
8 times $400 or $3200 equals B's.
12. Sinc(> B's fortune is 1^ times A's, \ of A's is equal to ^
of Ws, and Ih*^ mterest on it for 5 years at 6 per cent, would
equal j\ of it ^ of $600, or $200, is yV of 10 times $200,
or $2000; i ot $2000, or $1000*, equals what each had in ; 2
times $1000, or $2000, equals A's fortune ; and 3 times $1000,
or $3000, equals B'c.
13. Since he lost 8 per cent, or -fj of the cost on the sale,
he sold for || of the ca^ ; hence \ of his calves and | of his
shf^ep cost $25, and 4 times $25, or $100 is the cost of all
the calves and | of the sheep ; this exceeds the whole cost by
$24, which must equal the cost of the | of the sheep ovei
the whole number ; -J- of $24 is $C, and 5 times $8 is $40
which would buy 20 sheep at $2 ; ^76 lesj. $40 givep $36
for calves, which would buy 12 calves at $3. Therefore, etf
(168)
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