LIBRARY
OF Tin:
UNIVERSITY OF CALIFORNIA.
GIKT
Mrs. SARAH P. WALS WORTH.
Received October, 1894.
^Accessions No . 3Lr Class No.
KEY
TO THE
PROGRESSIVE
PRACTICAL ARITHMETIC,
INCLUDING
ANALYSES OF THE MISCELLANEOUS EXAMPLES
IN THE
PROGRESSIVE INTELLECTUAL ARITHMETIC,
FOR TEACHERS ONLY.
IVISOST, BLAftl^^rraYLOR & CO.,
NEW YORK AND CHICAGO.
1875.
ROBINSON'S
Mathematical Series.
Graded to the wants of Primary, Intermediate, Grammar,
Normal, and High Schools, Academies, and Colleges.
sv$t!4
Progressive Table Book.
Progressive Primary Arithmetic.
Progressive Intellectual Arithmetic.
Rudiments of Written Arithmetic.
JUiYlOR-CLASS ARITHMETIC, Oral and Written. NEW.
Progressive Practical Arithmetic.
Key to Practical Arithmetic.
Progressive Higher Arithmetic.
Key to Higher Arithmetic.
New 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 Conic
Sections, Surveying and Navigation.
Entered, according to Act of Congress, in the year 1860, by
HORATIO N. ROBINSON, LL.D.,
Jn the Clerk's Office of the District Court of the United States for the Northern
District of New York.
PREFACE
A KEY to any Mathematical work is not intended to su-
persede labor and study, but to give direction to the latter and
make it more effective and useful.
In many examples and problems the 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
examples, and of s^ch questions as contain the application
of a new principle. The arrangement * such as to be
easily understood.
NOTATION.
ROMAN NOTATION.
(17, page 9.)
Ex. 1. Ans XL
Ex. 3. Ans. XXV.
Ex. 5. Ans. XLVHI.
Ex. 7. Ans. CLIX.
Ex. 9. Ans. MDXXXVIII.
Ex. 2. Ans. XV.
Ex. 4. Ans. XXXIX.
Ex. 6. Ans. LXXVII.
Ex. 8. Ans. DXCIV,
Ex. 10. Ans. MDCCCCX.
Ex. 1. Ans. 125*
Ex. 4. Ans. 900.
Ex. 7. Ans. 505.
ARABIC NOTATION.
(26, page 12.)
Ex. 2. Ans. 483.
Ex. 5. Ans. 290.
Ex. 8. Ans. 557.
(28, page 13.)
Ex. 2. ^4ns. 5160.
Ex. 5. Ans. 2090.
Ex. 8. Ans. 9427.
Ex.3. Ans. 716,
Ex. 6. Ans. 809.
Ex. 3. Ans. 3741.
Ex. 6. Ans. 7009.
Ex. 9. ^W5. 4035.
Ex. 1. Ans. 1200.
Ex. 4. ^tm\ 8056.
Ex. 7. ^ws. 1001.
Ex. 10. Ans. 1904.
Ex. 11. ^4ws. 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 tbousand 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. Ans. 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, 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. 406420015vJ,
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 ten.
Ex. 18. Ans. Fifty-seven million four hundred sixty-eight
Jiousand one hundred thirty-nine.
Ex. 19. Ans. Five thousand six hundred twenty-eight.
Ex. 20. Ans. Eight hundred fifty million twenty-six thou-
sand eight hundred.
Ex. 21. Ans. Three hundred seventy thousand five.
Ex. 22. Ans. 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. Ans. 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.5.
Ans. 698.
Ans. 898.
SIMPLE NUMBERS.
ADDITION.
(4O, page 21.)
Ex. 4. Ans. 967.
(42, page 24.)
Ex.
7.
Ans.
1807.
Ex.
8.
Ans.
27246.
Ex.
9.
Ans.
4945.
Ex.
10.
Ans.
78313.
Ex.
19.
Ans.
145.
Ex.
22.
Ans.
69585.
Ex.
23.
Ans.
566.
Ex.
24.
Ans.
3746.
Ex.
27.
Ans.
4619.
Ex.
28.
Ans.
4915.
Ex.
29.
Ans.
4320.
Ex.
30.
Ans.
4623.
Ex.
31.
Ans.
3871.
Ex.
35.
Ans.
101500.
Ex.
37.
Ans.
50000000.
Ex.
40.
Ans.
1194.
Ex.
44.
Ans.
2773820.
Ex.
45.
Ans.
4403241
SUBTRACTION.
(49, page 30.)
Ex. 6. Ans. 353.
Ex. 8. Ans. 205.
Ex. 19. Ans. 123.
Ex. 24. Ans. 4202.
Ex. 26. Ans. 16348755.
Ex.
Ex.
Ans. 210.
Ans. 320.
Ex. 22. Ans. 2113.
Ex. 25. Ans. 11425.
Ex. 27. Ans. 4014580,
Ex.
Ex.
(51, page 32.)
Ans. 721.
Ans. 3769.
Ex.
Ex.
Ans. 561.
Ans. 269.
SUBTRACTION.
Ex. 7.
Ex. 9
Ex. 11.
Ex. 13.
Ans. 4509.
Ans, 1288.
Ans. 21078.
Ans. 762301.
Ex 20. Ans. 220874.
Ex. 8. Ans. 3449.
Ex. 10. Ans. 30616.
Ex. 12. Ans. 142.
Ex. 19. Ans. 224130.
Ex. 25. Ans. 181972.
Ex. 31. Ans. 529509693. Ex. 34. Ans. 1902001
EXAMPLES COMBINING ADDITION AND SUBTRACT ION.
(52, page 33.)
Ex. 1. 2500+ 175 = 2675
5200 2675 = 2525 dollars, Ans.
Ex. 2. 235 + 275 + 325 + 280 =
1300 1115 = 185 miles, Ans.
Ex. 3. 4234 + 1700 + 962 + 49=6945;
87^6945 = 1807 dollars, Ans.
Ex. 4. 4765 + 750 = 5515;
f>515-34^5131 dollars, Ans.
Ex. 5. 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,
Ans.
Ex. 8. 2324 + 1570 + 450 + 175=4519;
67844519=2265 miles, Ans.
Ex. 9. 7375, first paid ;
7375+ 7375 = 14750, second paid;
7375 + 14750=22125;
35680-22125 = 13555, dollars, Ans.
10 SIMPLE NUMBERS.
Ex.10. 750 + 379 + 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. Ex. 8. Ans. 416223.
(64, page 41.)
Ex. 5. Ans. 2508544. Ex. 6. Ans. 15731848.
Ex. 7. Ans. 16173942. Ex. 9. Ans. 78798.
Ex. 13. Ans. 937456.
CONTRACTIONS.
(67, page 43.)
Ex, 2. 3472x6=20832; 20832x8 = 166656, Ans.
Ex. 3. 14761x8=118088; 118088x8=944704,^5.
Ex. 4. 87034x3=261102; 261102x3 = 783306;
783306x9 = 7049754, Ans.
Ex. 5. 47326x6 = 283956; 283956 x 5 = 1419780*
1419780x4 = 5679120, Ans.
MULTIPLICATION. 1 1
Ex. 6. 60315x8x3x4 = 5790240,
Ex. 7. 291042x5x5x5 = 36380250, Ans.
Ex. 8. 430x7x8 = 24416 miles, Ans.
Ex. 9. 124 x 6 x 3 x 4 = 8928 dollars, Ans.
Ex. 10. 5280 x 7 x 3 x 4 = 443520 feet, Ans.
Ex. 11. 120 x 5 x 5 x 5 = 15000 dollars, Ans.
(69, page 44.)
Ex. 3. Ans. 13071000. Ex. 4. Ans. 890170000.
(7O, page 45.)
Ex. 12.
888000 dollars, Ans.
EXAMPLES COMBINING ADDITION, SUBTRACTION, AND
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, Ans
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 ;
tf4 x 16 = 1024 miles, Ans.
b SIMPLE NUMBERS.
Ex. 8. 14 x 26 x 43 = 15652 yards, Ans.
Ex. 9. 4 x 365 = 1460, yearly expenses ;
3700 1460 = 2240 dollars, Ans.
Ex.10. 2475, first;
2475 840=1635, second;
2475 + 1635=4110, third ;
8220 dollars, Ans.
Ex.11. 336 (28x10) = 56 miles, Ans.
Ex. 12. 23 x 14= 322, 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
3x850 = 2550
8454 dollars, Ans.
Ex. 14. 14760 (1575 x 5)=6885 dollars, Ans.
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.
Ex. 13. Ans. 10487951. Ex. 14. Ans. 635926f.
Ex. 15. Ans. 2379839f. Ex. 16. Ans. 9355751*.
Ex. 17. Ans. 245640}!. Ex. 18. Ans. 7014132f
Ex. 19. 47645-^-5 = 9529 dollars, Ans.
Ex. 20. 17675-4-7 = 2525 weeks, Ans.
Ex. 21. 6756-=-6 = 1126 barrels, (Ans.
Ex. 22. 46216464-^-12 = 3851372 dozen, Ans.
Ex. 23. 347560-4-5 = 69512 barrels, Ans.
Ex.24. 3240622-=-!! =294602 acres, Ans.
Ex. 25. 38470-4-5 = 7694 dollars, Ans.
(8O, page 54.)
Ex. 5. Ans. 212 T 9 /4. Ex. 14. Ans. 1489|f.
Ex. 15. Ans. 12152^. Ex. 16. Ans. 608301^.
Ex. 17. Ans. 1210900-LfH. Ex. 18. Ans. 997f J$.
Ex. 19. Ans. 1343iff|. Ex. 20. Ans. 5473|fff
Ex. 21. Ans. 7500 T y T T V-
Ex.27. 1850400-4-18504 = 100 dollars, Ans.
Ex. 28. 723 20060-4- 10735 = 6736 T VWs dollars, Ans.
Ex.29. 942321-4-213 = 4424^^ volumes, ^Lns.
Ex. 30. 5937120^-22320 = 266 dollars, Ans.
CONTRACTIONS.
(81, page 56.)
Ex. 2. (3690-7-3)-^5 = 246, Ans.
Ex.3. (3528H-4)-J-6=147, Ans.
Ex.4. (7280-f-5)-4-7=208, Ans.
Ex. 5 (6228-^-6)^-6 = 173, Ans.
Ex. 6. (33642-=-3)-J-9 = 1246, Ans.
Ex. 7. (l53160-^-7)-^8 = 2735, ^4s.
Ex.8. 15625^-5^-5-4-5 = 125, Ans.
11 SIMPLE NUMBERS.
(8, page 57.)
Ex. 2. 6)34712
7)5785 2
826---3x6 = 18_
20, Arts.
Ex. 3. 8)401376
8)50172
6271 ---4x8=32, Ans.
Ex. 4. 3)139074
4)46358
6)11589 ...... 2x3= 6
1931- --3x4x3 = 36^
42, Am.
Ex, 5. 3)9078126
5)3026042
6)605208 ---2x3 = 6, .4ns.
100868
Ex. 6. 4)18730627
5)4682656
6)936531 ....... 1x4= 4
156088 ---3x5x4=60^
67, Ant.
Ex. 7 2)7360479
6)368C239 ............ 1
8)613373 ---- --1 x2= 2
76671 ---5x6x2 = 60
63, Ans.
SIMPLE NUMBERS. 16
Ex. 8. 2)24726300
5)12363150
7)2472630
353232 - - - 6 x 5 X 2=6Q, An*
Ex. 9. 7)5610207
2)801458 1
6)400729
66788--- 1x2x7 = 14
15, Ans.
(83, page 58.)
Ex. 2. Ans. 476. Ex. 3. Ans. 3620 7 W
Ex. 4. Ans. 1306 T : VVo- Ex* 5 - Ans - 9>76 iiUo-
Ex. 6. Ans. 2037l T 8 oWoV
(85, page 59.)
R*. 6. Ans. 14556 T \VoVV Ex. 7. ^ins. 10609 VoVo-
Ex. 8. Ans. 114304^ffJ. Ex. 10. Ans.
Ex. 11. 24898-^50=497|f hours, Ans.
Ex. 12. 350000^-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, Ans.
Ex. 4. 500 + 17 + 98 + 121 = 736 acres owned?
736 325=411 acres, Ans.
Ex. 5. 2300 625 = 1675 dollars, Ans.
SIMPLE NUMBERS.
Ex. 6. 60 45 = 15 dollars, saved in one month;
900 -M 5 60 months, Ans.
Ex. 7. 87 x 9 = 783 days, Ans.
Ex. 8. 4 first number ;
4x8= 32 second;
32x9 = 288 third;
324, Ans.
Ex. 9. 2x2x7 = 28;
364-^-28 = 13, Ans.
Ex. 10. 78 + 104 = 182;
182x2 = 364 acres, Ans.
Ex. 11. 90 + 304-12 + 5 + 7 = 144;
144x27 = 3888 dollars, Am.
Ex. 12. (2250 x 4)-f-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.
Ex. 15. 45 19=26 years, Ans.
Ex. 16. 1000000000-^25000=40000 acres, Ans.
Ex. 17. 384 + 1562 + 25 + 946=2917 ;2917 2723 104
194^97 = 2 ; and 2 x 142=284, Ans.
Ex.18. 5280-f-3 = 1760 steps, Ans.
Ex. 19. 2375 + 340 =2715 dollars, cost;
867 + (235 x 8)=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.
SIMPLE NUMBERS. 17
Ex. 23. 225 75 = 150;
150 x 52 = 7800 cents, Ans.
Ex. 24. 31383450-7-4050=7749, Ans.
Ex. 25. 31647000-r-700=45210 dollars, Ans.
Ex. 26 Reversing fourth operation, 10040= GO;
Reversing third operation, 60x5 = 300;
Reversing second operation, 300-^3 = 100 ;
Reversing first operation, 100 x 7 = 700, Ans.
Ex. 27. (54 x l7)-r-27=34 cows, Ans.
Ex. 28. 56-(2 x 26) = 4 dollars, Ans.
Ex. 29. 98 x 7 = 686 days, Ans.
Ex. 30. 5301212-r-11137 = 476 dollars,
Ex. 31. 60 39 = 21 gallons, gained hoi
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 ;
651)38-47000 = 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. 5190048^-72084 = 72, Ans.
18 PRIME NUMBERS.
Ex. 4 . 109x73 = 7957, greater n umber ;
28x17=476, difference;
7957476 = 7481 less, Am.
Ex.41. 360 114 = 246, greater;
246x114 = 28044, Ans.
Ex. 42. 2568754 2473248 = 95506, Ans.
Ex. 43. Wheat, 35x2 = 70 dollars ;
Wood, 18x3 = 54 "
124 "
Cloth, 9x4 = 36 "
88 dollars, Ans.
Ex. 44. 684375 = 309 yearly savings ;
309 x 5 = 1545 dollars, Ans.
Ex. 45. 58 + 10 + 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; 6400625=5775;
5775^-35 = 165, Ans.
Ex. 48. 189 + (4 + 5) = 21 hours, Ans.
PRIME NUMBERS.
(91, page 68.)
Ex 2. Ans. 2, 2, 3, 5, 19. Ex. 3. Ans. 3, 3, 5, 5, 7, 19
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=
(92, page 69.)
Ex. 3.
Ex. 4. 40 r= i
2x12
3x8
4x6
2x3x4
2x2x6
2 x 2 x 2 >. 3
2x20
4x10
5x8
2x2x10
2x4x5
2x2x2x5
Ex. 5 72 =
125
5x25
5x5x6
2x36
3x24
4x18
6x 12
8x9
2x2x18
2x3x12
2x4x9
3x4x6
2x6x6
3x3x8
2x2x2x9
2x2x3x8
2x3x3x4
O x2x2x3x3
20
PROPERTIES OF NUMBERS.
CANCELLA1
(95, p--e 72.)
Ex. 3. n
$
3 Ex. 4. 4
n 3
It
$
nt y$
y> dp
11
u
t
14
20
14, Ans.
33, Ans.
Ex. 5. 7 $4
4$ 16 Ex. 6. 30
,*rt "# '
*4
K
Kf' ft
n 4 20
Pi?
13
t
7
64 3
13
9|, ^4^5.
4i, Ans.
EX. 7. n
00 3 Ex. 8. ^0 200
9 40 $0 2
8 40
It 3 X$ $0
a:4 ^a:
8
81 2, As.
10}, A*.
Ex. 10. 4
102
Ex. 9. 00
240 4
n
at * 2
~
8, Ans. %
42 2
4, ^4n*.
Ex. 11. X2
30 10 ta. 12. 00
4
4 *
10 tons, ^4n.<?
6 firkins,
GREATEST COMMON DIVISOR.
EA. 13. 1$
EX.
20 suits,
115
230
76f days, Ana.
GREATEST COMMON DIYISOR.
(98, page 74.)
Ex.1. 3x4 = 12, As.
Ex. 3. 4 x 6 = 24, Ans.
Ex. 5. 2x7 = 14, Ans.
Ex. 7. 7x10 = 70, As.
Ex.9. 2x7 x9 = 126, As.
Ex. 12. 5x5 = 25, Ans.
Ex.2. 2x3 = 6, Am.
Ex. 4. 3 x 6 = 18, As.
Ex. 6. 4x4 = 16, Ans.
Ex. 8. 3 x 5 x 5 = 75, As.
Ex. 10.4x8 = 32,
(99, 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 greatest common di-
visor of the two numbers of bushels. Ans. 21 bushels.
Ex. 11. The 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 204*7 feet. , Ans. 23 feet.
Ex. 13. The greatest common divisor of the three num.
burs of bushels is 2, which must be the capacity of the bag.
Now there are to be forwarded 2722 + 1822 + 1226=5770
bushels; and 5770-=-2 = 2885. Ans. 2885 bags.
Ex. 14. The greatest common divisor of $120, $240, and
$384. is $24, the price of the cows; and $120+424 = 5, A's
number; $240-=-$24 = 10 B's number; and $384+-$24 = 16,
O's number.
MULTIPLES.
(1O4, page 81.)
Ex. 2. 2 x 5 x 7 x 7=490, Ans.
Ex. 3. 2x2x2x3x7x 17=2856, Ans.
Ex. 4. 2 x 2 x 2 x 3 x 3 = 72, Ans.
Ex. 5. 2x2x2x3x5x7x 11 = 9240, ATM.
Ex.6. 2x3x3x5X5=450,^4^5.
Ex. 7. 2x2x3x3x5x 7 = 1260, Ans.
(1O5, page 83.)
Ex.4.
2,3
2,5,7
42 .. 60
7 .. 10
2x2x3x5x 7=420, Ans,
Ex. 5. 3, 7
2,5
21 .. 35 .. 42
3x7x2x5=210, Am.
LEAST COMMON MULTIPLE.
Ex. 6.
2,5
3, 2, 5, 5
60 .. 100 .. 125
6.. 10.. 25
2x5x3x2x5x5 = 1500, Ans.
Ex. 7. 2, 2, 5
2,2,2,3
40 .. 96 .. 105
2 . . 24 .. 21
2x2x5x2x2x2x3x 7 = 3360, An*.
Ex. 8. 2, 2, 3
2, 2, 3, 5
48 .. 60 .. 72
Ex. 9
2x2x3x2x2x3x5 = 720, Ans.
2, 2, 3 I 84 .. 224 . . 300
7, 5 I 7.. 56.. 25
2, 2, 2, 5 I 8_ 1: 5
2x2x3x7x5x2x2x2 x 5 = 16800, Ans.
Ex. ]'. 3, 3 | 270 .. 189 . . 297 . . 243
3, 3,7
2, 5, 11, 3
30 .. 21 ., 33 . 27
10 ..
11 . 3
3x3x3x3x7x2x5xllx 3 = 187110, Ans.
Ex. 11. 2, 3, 5
2, 2, 7, 3
5 .. 6 .. 7 .. 8 .. 9
. 7. .4 .. 3
2x3x5x2x2x7x3=2520, Ans.
Ex. 12. To purchase books at 5 dollars, or 3 dollars, or 4
dollars, or 6 dollars, the sum of mouey must be some common
multiple of 5, 3, 4 and 6 ; and the least sum will be the least
sommon multiple, which is 60 dollars, Ans.
Ex. 13. The least common multiple of 12, 15, and 1
barrels, which is 180 barrels, Ans.
Ex. 14. The least common multiple of the prices, $30,
$55, and $105, which is $2310, Ans.
24 FRACTIONS.
Ex. 15. The least common multiple of 41, 63, and 54
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-f-12=25,
number of chickens; 300-r-30 = 10, number of ducks; 300
-i-75=4, number of turkeys ; and 25 + 10 + 439, 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 180-f-12 = 15, number of chiokens ;
180-j-30 = 6, number of ducks ; 180-f-90=2, number of tur-
keys; and 15 + 6 + 2 = 23, whole number of fowls. But
3923 = 16, number of fowls purchased more than was ne
cessary; and 16x5 = 80 cents, Ans.
FRACTIONS.
(ISO, page 88.)
Ex. 1. Ans. i. Ex. 2. Ans. ^.
Ex. 3. Ans. jfo. Ex. 4. Ans. if.
Ex. 5. Ans. T V . Ex. 6. Ans. .
Ex. 7. Ans. ff. Ex. 8. Ans. | T .
Ex. 9. Ans. T VWo- Ex. 10. Aw. WWW-
Ex. 11. Nine tenths; seven twelfths; five twentieths.
awelve twenty-eights; fifteen seventy-fifths; nine one hun
dred twelfths; forty-five two hundred twentieths; one hui>-
dred twenty-five four hundred twenty-eighths.
Ex. 12. Ninety one hundredths ; three hundred twenty-
five one thousandths; four hundred fifty one thousand tieo
REDUCTION. 25
hundred fortieths twenty -five one thousand fiiv hundredth^ ;
twelve two thousandths j 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 Jive hundred thirty-one thousand four hundred
twenty-ninths.
REDUCTION.
(126, page 90.)
Ex. 5. Iff =4, An*. Ex. 6.
Ex. 7. TWV=f> Ans - Ex - 8 -
Ex. 12. tH=fi ^*- Ex - 13 -
Ex. 14. fHf =}, ^s.
(127, page 91.)
Ex.4. i.|J.=i53|, -4*. Ex.6. i-|p=:54if,
Ex.7. J T V= 41 > Ans - ' Ex - 9- 7 H J -= 43 Tf
(138, page 92.)
Ex.4. 140=*^ Ans. Ex.6. 94 = t|
Ex. 7. 180:=-4f-fi-S ^. ws . Ex. 9. 247=a.p, Ans.
(129.)
Ex. 8. 7lf =A$S ^ws. Ex. 5.
Ex.9. 96 4 = ^Iws. Ex. 11.
K P
FRACTIONS.
(13O, page 93.)
Ex. 2. 15-^5 = 3;
= T <L, Ana.
Ex. 4. 51-^-17=3;
|f 1 4, Ans.
Ex.6. 3488-7-436=8; Ex.7. 1000-7-125 = 8;
Ex. 3. 35-h7=5;
f =|f, ^n.
Ex. 5. 150-^30=5;
5 page 95.)
Ex. 2. 5, 5
2,3
50 . . 75
2 .. 3
5 x 5 x 2 x 3 = 150, least common denominator,
ft, ft, H, ft=iWir, TO, 1H, rlo,
Ex.3.
2,2,2
2,3,7
16 .. 21
2 21
2x2x2x2x3x 7 = 336, least com. denom.
Ex.4.
3,3,
2
9
.. 21
. . 4
7 2
7
2
3x3x2x7x 2 = 252, least com. denora.
I, 2V, *, f =ftV, 1%, ill
Ex 5.
Ex. 7.
Ex. 8.
Ex. 9.
EX. 10.
Ex. 11.
Ex. 12.
f, v-=
f, }, f , V 7 , ft=H*, ftV, AV, H*,
f, ft, V, f, *= H, W, VV, ft,
||, V, J, H, f = H, VeS If, H,
, 4, y , i=
, ft, t*,
, VVS H,
ADDITION. 27
Ex. 13. |f, ft, }, Y-=ft, fl f, |f, ,4n.
Ex. ID. T v, Y, If, f =H, W, , W, ^ 5 -
Ex. 16. ft, v-, A. i f 1= A. W, H, W. . .
ADDITION.
(133, page 96.)
7 + 9 + 2 + 13 + 16 + 21 68 18
86- =26 = 2 26'
41+63 + 71+89 + 109 373 13
13 + 76 + 140 + 181 + 223_633_ 61
225 ~~225~~ 75 1
(134, page 97.)
126 + 21+48 + 70 265 97
, _ rjj 1 >4l?4
168 ~" 168 168'
3 FRACTIONS.
Ex. 5. + 1 =
45 13 21
1274 + 2835 + 390 4499 404
T j
~~
4095 4095 4095 ' '
42 9 7 1
42 + 18 + 35 + 10 105 3 A
=777;=7 Ans
140 140 4'
61 131 24 1 2
fcx ' 7 - 7^ + 150 +25 + 2 + 3 =
102 + 131 + 144 + 75 + 100 17
"" l3 ' '
31245678 9
EX ' 8 - 4 + 2 + 3 + 5 + 6 + 7 + + 9 + IO =
1C90 + 1260 + 1680 + 2016 + 2100 + 2160 + 2205 + 2240 + 2268
2520
4 j^ 2 19_ 19
K "I f\ O OA AA
O JL \J o *i\J DU
14 + 3+ 1 = 18
Ex. 11.
17819 _ 179
2520 ~" 2520'
1+10 + 5 = 16
Ex. 12.
17 + 18 + 26=61
, Ans.
Ex. 13. _ V
1+3 = 4
SUBTRACTION.
Ex. 14.
125 + 327 + 25=477
478^y , Ans.
Ex. 15. iAo +| 7 +u + H+ i n=3 i f i, An8 .
Ex. 16. T V + H-K + T'o = 4
3 + 2 + 40 + 10 = 55
554, ^4w5.
Ex. 17. i + f + f = 2^
125 + 96 + 48 = 269
27l 7 T yards, Ans.
5 + 3 = 8
9, 1 7 yards, ^4/i5.
Ex. 19. A + tf + f + f + | = s
26 + 40 + 51+59 + 62 = 238
241 if acres, Ans.
Ex. 20. ! + 4+i2 + 7 7. = 2 ^
175 + 325 + 270 + 437 = 1207
1209/j bushels.
205 + 296 + 200 + 156= 857
$859||, dollara.
SUBTRACTION.
(135, page 99.)
85 3 1
. 8. -=-=-, Ans.
14 11 3 1
80 FRACTIONS.
20 6 14
49 36 13
-To-=75'
75-11 64 1
182 110 72 _ 6
"848" ~348~29' *
(136, page 99.)
1 2 94 5
Ex. 2. --- = - = , Ans.
2 9 18 18'
15 2_75-48_ 27 _ 9
~-~~- == A
3 4 51-32 19
EX ' 6 '
84 4 49-8 41
T2b-3F = -7b- = 7-0'
1500 50 125 100 25
1728-72= 4T- = 144'
60 332 _720 83_ 637
~~ 1068 ~1068' A
Ex. 9. 8^ = 8-^ Ex. 10. 25f =
4}f , ^n. 163*0 =
Ex.11. 4f=4|f Ex. 12. 6
J li
3}|, ^4ns. 4f, Ans.
Ex. 13. 4501 =450$ Ex. 14. 3-^ =
330|f,
I
MULTIPLICATION. 31
Ex. 15. 75i Ex. 16. 227|
49 196| = 196|
261, Ans.
Ex. 19. $7f $6 =$!-&, Ans.
Ex. 20. 4 + 31 =4j\ Ex.21.
5? 4^ 12"
Ex. 22. 7|-2f =4|f, ^5. Ex. 23. H~U= Ai Ans '
Ex. 24. 9121 + 5451 =
$2000 $1457^=1542-^, Ans.
Ex. 25. $136 T ^ + $350!=$487H cost.
= $6011 receipts.
Ans.
MULTIPLICATION.
(137, page 101.)
Ex.4. J> T x>I=&=l
Ex. 5. T 9 x 12 = W =H Ans '
Ex.6. 2 5 T x63 5 x3 = 15, Ans.
Ex. 8. 7| x 12 = *J 6 =
Ex. 9. T Vr x 8 = m^
Ex.10. T ^x51=f=2
Ex.11. 15fx 16 = 125x2 = 250, Ans.
Ex.12. ifix22 = 1 | 1 =
Ex.13. $8/oXl2=$ i n
Ex. 14. $H x9 = $tt=
Ex. 15. $f x27^$ L | 9 =$23f, Ans.
32
FRACTIONS.
(138, page 103.)
Ex. 2.
Ex. 4.
Ex. 7.
3
Ex. 3.
100
9 4
14
9
3 4
7
450
1J, Ans.
64f, ^*.
105
Ex. 5.
19
21 17
47
13
85, Ans.
47
247
-
5H, Ans.
42
Ex. 8.
80
4 39
16
233
2 819
1165, Ans.
t
Ex. 9.
39
40 9 , Ans.
156 Ex. 10. $8xf=6f dollars, Ans.
27 Ex. 11. 36 x lOf =384 miles, Ann,
108, Ans.
Ex.12. $450 x T 7 2
Ex.13. $16x2*=
Ex. 2. 4
(139, page 104.)
4
9
3
4
3
1
Ex. 3. 8
10
Ex. 4. 24
55
10
11
36
Ex. o. 6
7
F 7 o> <*n*.
21
6
18
3|, Ant.
MULTIPLICATION.
Ex. 6. 10
9
7
2
9
5
4
1
28
1
5 ! T , As.
E^. 9 15
8
4
9
5
1
3
22
25
44
l^f, As.
Ex. 11.
3
2
1
7
4
5
4
4
13
35
78
Ex. 13. 8
7
2
1
9
11
2
3
1
8
12
77
Ex. 7. 6
, Ans.
Ex. 10.
7
1
10
3
2
16
7
80
3 256
851, Ant
Ex. 12. 2
4
5
3
Ex. 14. 2
2
4
6
3
4
4
2, As.
25
11
27
16 7425
^7ei
6 T 5 2 Ans.
Ex. 15 $ x f =8|, As. Ex. 16.
Ex. 17. f x li=$f, As. Ex. 18.
x =
x $f =$, A*
84
FRACTIONS.
Ex. 20. 4
8
51
17
867
Ans., 108f.
Ex. 21. 8
5
20
51
14
357
Ans.,
dollars.
x f x
y x$-
i x V 3 x f x $Y-=
Ex. 22. ? x 4 x I W =121 A, Ans.
Ex. 23.
Ex. 24.
Ex. 25.
Ex. 26. $*fa x =$25 A,
Ex. 27. ^f- 3 - acres x f x |=49 T ^ acres, Ans.
Ex. 28. H x f barrels =6 barrels,
DIVISION.
(14O, page 107.)
EX. 6. Tj
Ex. 9. $f-
Ex. 11. -f ^-
Ex. 14. *J
Ex. 16.
, Ans. Ex.10.
Ex. 12.
Ans. Ex. 15.
(141, page 109.)
Ex, 7. | x 9^f=
Ex, 8. $21^-^ $49, Ans.
Ex. 9. 16 x | = 10; 10-f-f =22^,
Ex. 11.
Ex. 12.
Ex. 13. 15^-f =9, Ans.
=1, Ana.
x ^9-=
iU VISION.
Ev 14. f x 320 = 200;
200-S-V=S
Ex. 15. $32x1 = 8; ^
Ex. 16. 183-^J-p=4,
(149, page 110.)
Ex. 2. 8 | 7
Ex. 4.
63
, Ans.
10
40
Ex. 3. 9
10
Ex. 5.
14
13
13
f?, Ans.
Ex. 6.
3
2 Ex. 7. 6
5
27
28 4
5
81
56 24
25
f f , ^.
1ft. ^
Ex. 8.
3
5 Ex. 9. 19
17
3
7 7
19
9
35 7
17
01 /i
Ex. 10.
20
13 Ex. 11.
7
7? *
2
5
16
2
5
25
52
A
4 2
g
A
02 Ans
2fi7
21
40
FRACTIONS.
Ex. 12. 10
13
325
432
HH,
Fx 15 ?|_!! X 3_?!
8|~9 X 26~39'
J?v 1A J y Of)
!* V7. . A. a ^(V.
T 5 T 5 5 25
=
Ex. 18.
|x|_2 3 2_.
25921
Ex.20. V x f= 14
Ex. 21. \ 3 xf=Y=6|,
Ex. 22. 8 35
21
Ex.24.
Ex. 23. 3
14
$17 $14f=$ 2$, An*.
Ex. 25 10
20
98
35
11 1, Ant.
PROMISCUOUS EXAMPLES.
37
Ex. 26.
10
10
2 bu., Ans.
Ex. 27. 16
75
10
1905
127
Ans
PROMISCUOUS EXAMPLES.
(Page 112.)
Ex.2. 91-^7 = 13; 4* j jj-ff, Ans.
Ex. 3. 3, 40 | 3 :: 40
3x40 = 120, Ans.
Ex. 4. 4+3
Ex.5. fxH==4A
m>== 2 H
l}ff, Ans.
Ex. 6. 4756f + 128f =4885^, An*.
Ex.7, f xfx|xV-=H=^A
V xfxfx |^H= 4 . 2 ?
Ex. 8. f x f =20, ^W5. ; f x J = V =1 8
Ex. 9. 1825f=-^f^; lAfJiixf^ 2 - 4 1^=3043}
Ex.10. i+J=A? l 9 o-H; 77-^=140,
Ex. 11. i x $V x V =*24|f
Ex, 12. $V-xt=l23i,
38
FRACTIONS.
Ex 13.
8
14701
14701
Ex. 14. 8
471
37803
628
$1, Ans.
$12601, Ans.
Ex. 15.
Ex. 16.
'_eji x i x =
, As.
1 42 10 2 22
O_ _ _ v _ _ ^ __ _ _ . "I A _ _ - I A
~2 X 5 X 147~7' 7 7~
X f xf = 27, Ans.
xyxf = W=24 1 V, Ans.
xfx f=34i, Ans.
Ex. 17. V
Ex. 18. V
Ex. 19. x y=-
EX. 20. !-2 9 <T = ;
Ex. 21. fi+J- =
Ex. 22. Y x T 3 T
Ex. 23. 221. x T -
Ex. 24. i -f- v
$20-f-f=$50,
Ex.25, f f = 2 3 oi ! *V=i
Ex. 26. $J-\
Ex. 27. 320 x $2i=$ 720 755x|x$lf =
435x$lJ=$ 815f 755x|x$2i= 962f
755 $1535f $14911
$1535f $1491}=$44, As.
, 7 + 5 12 12 7 5
Ex.28.- -=;__-=__, As.
Ex 29 8 + 5 _ 13 . 8 13 - 5
KL.V9. 7 + 5 -^, 7-l2~84 J
Ex. 30. f x f x f x | = 7J, As.
Ex. 31. | x $ V x 2 x T 2 T = V =$3}, As.
DIVISION. 89
El 32. 16f 3l = 12l=
A X ' ' x =
Ex. 33. V x y -<u>_3 =
Ex. 34.*6-hl846= lr f 3,
Ex 35. $ 2 T x x &=$
Ex 36. $ y x i x i x | x f-=$5,
Ex.37. J + T V=H; 1-=A; A-i=A
Ex. 38. if cents x y x = 100 cents,
Ex. 39. i+f=i3. ; i_ 11 = ,, remainder.
H* x 2T x $2-1^=^1 ^i=$4577
Ex. 40. If the horse cost li times as much as the wagon,
the horse and wagon must cost 2i times the wagon. Hence,
$270 -T- 21 =$120, Ans.
Ex.41. JL. X =32; 32 20f = lli, ^1*.
Ex. 42. $4-^1 x aV x $f=126, Ans.
Ex. 43. If A can do as much as B, he can do the work
in A of the time that B will require, and in l+|=i of the
time be \h will require. Hence
14 days x i=32| days, A's time ; ) .
32| days x f = 241 days, B's time ; ) ^
Ex. '4. J^-xf xf=lli, Ans.
Ex. 45. A, B, and C can do 1 of the work in a day ;
B and C can do 1 of the work in a day; hence
A alone can do i 1=^ of it in a day ; and
be wW therefore require -*/-=13i days, Ans.
Ex, 46. f + i+i=JL; 1--^=^, remainder;
rV-rV^sV; $24-r-3-V=^20, Ana.
Ex 4-1 V x A x V =4f
DECIMALS.
Ex. 48. i x i
Ex. 49. A-i= T 3 o ; 30 feet-5- 1 ^ = 100 feet,
Ex. 50. $i| x f x | =$6 J J, ^W5.
Ex. 51. + =7 S 2 fraction of the post below water,
1& =7*3- " " " above "
21 -=--^=36 feet, Ans.
Ex. 52. ^n= eldest son's fraction;
f x 4 H= youngest son's fraction;
1 (? + i)= H= daughter's fraction;
DECIMAL FRACTIONS.
(145, page 11 8.)
Ex. 1. Ans. .38. Ex. 2. ^te f.
Ex. 3. .4ns. .325. Ex. 4. Ans 04.
Ex. 5. Ans. .016. Ex. 6. Ans. .00074.
Ex. 7. Jbts. .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; ore thouhdnd three
hundred eighty-five millionths ; one million eighty-peret) ten-
millionths.
NOTATION AND NUMERATION. 41
(146, 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-fivt
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. .000425. 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 hundred
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
DECIMALS.
thousand five hundred eighty-three ten-thousandths ; twenty
seven, and forty-five hundred-thousandths.
REDUCTION.
(149, page 121.)
Ex. 2, .1700000 Ex. 3. .700000
24.6000000 .024000
.0003000 .000187
84.0000000 .000500
721.8000271 108.450000
Ex. 4. 1000.001000
841.780000
2.600400
90.000009
6000.000000
(15O, page 122.)
Ex.2. MJ v =i,Ans. Ex.3. T ^=^ Ans
Ex. 4. TW T =H* Ans. Ex. 5.
Ex. 6. =
(151, page 123.)
Ex. 4. .4/1*. .4. Ex. 6. Ans. .875.
Ex. 9. Ans. .375. Ex. 10. Ans. .0375.
ADDITION.
(158, page 124.)
Rx, 6. 26.26 Ex. 7. 36.015
300.0605
5.000003
60.0000087
87 04^7, Ans. 401.0755117, Ans.
ADDITION,
43
Ex. 8.
Ex. 10
Ex. 9.
71.399107, Ans.
61.843
143.75
218.4375
21.9
445.9305, Ans.
Ex. 12. 21= 2.5
5 = 5.75
3 = 3.625
3.0642
8.925
Ex. 11.
23.8642 barrels, Ans.
18.375
41.625
35.5
95.500, Ans.
12f =12.75
18| =18.4
9=9
241 =24.125
4}f = 4.8125
8 T 9 o= 8.9
151^=15.65
93.6375, Ans.
Broadcloth.
Ex. 13. First suit, 2.125
Second " 2.25
Third " 5.0625
Sums
Total
Gassimere.
3.0625
2.875
Satin.
.875
1.000
1.125
9.4375 5.9375 3.000
9.4375 + 5.9375+3 = 18.375,
44
DECIMALS.
SUBTRACTION.
(153, page 126.)
Ex. 4. 714.000
.916
713.084, Ans.
Ex. 6. 21.004
.75
20.254, Ans.
Ex. 8. 900.
.009
899.991, Ans.
Ex. 10. 1.
.000001
.999999, Ans.
Ex. 12. .34
.034
.306, Ans.
Ex. 5. 2.000
.298
1.702, Ans.
Ex. 7. 10.0302
.0002
10.03, Ans
Ex. 9. 2000.
.002
1999.998, Ans.
Ex. 11. .427
.000427
.426573, Ans.
MULTIPLICATION.
(154, page 127.)
Ex. 4. 274.855, Ans. Ex. 8. 243.5, Ans.
Ex. 12. .000030624, Ans.
NOTATION AND NUMERATION. 45
DIVISION.
page 129.)
Ex. 5. .111. Ans. Ex. 6. 11.1, Ans.
Ex. 8. 15.27 +, Ans. Ex. 9. 1; 10; 100; 1000, Ans.
Ex. 10. 5.6814 + , Ans. Ex. 12. 3020, Ans.
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^-6.35-11.13, Ans.
Ex. 12. T 6 oVo=i, Ans. Ex. 13. 26 T Vo s o=26i, Am.
Ex. 17. 3.625 x 36.75 x $.85-$113.2359375,
Ex. 18. 56.925-r-4.6 = 12.375 = 12f, Ans.
DECIMAL CURRENCY.
NOTATION AND NUMERATION.
(16O, 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 ; one
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-
lars 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. 7. Ans. 3008 mills.
Ex. 8. Ans. 890 mills.
(163, page 134.)
Ex. 2. Ans. $15.04. Ex. 3. Ans. $138.75.
Ex. 4. Ans. $16.525. Ex. 5. Ans. 52 .4 centa.
Ex. 6. Ans. $6.524.
Ex.2.
ADDITION.
(163, page 134.)
$ 50.07
Ex. 3. $ 364.541
1000.75
486.06
60.003
93.009
.184
1742.80
1.01
3.276
25.458
$2689.686, Ans.
$1137.475,
Ans.
ADDITION.
Ex. 4. $ .92
.104
.357
.186
.444
.125
.99
126 Ans.
Ex. 5. $89.74
13.03
6.375
19.625
$128.77, ]An*.
Ex. 7. $2175.75
240.375
605.40
140.125
$3161.65, Ant.
$10.3375, Ans.
Ex. 8.
$ 6.08
26.625
16.000
7.40
156.105, Ans.
Ex. 9. $7425.50
253.96
170.09
$7849.55, Ans.
Ex. 10. $3.625
1.75
1.375
.625
.875
$8.25, Ant.
48 DECIMAL CURRENCY.
SUBTRACTION.
(164, page 136.)
Ex. 2. $365.005 Ex. 3. $50.
267.018 .50
$97.987, Ans. $49.50, Ans.
Ex. 4. $100. Ex. 5. $1000.
.001 .037
$ 99.999, Ans. $ 999.963, An*.
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 + $7015.875 = 10575.875
$1.215, Ans. $4149.325, Am,
Ex. 10. $13.75
5.25
1.375
.875
$25 -$21.25 = $3.75, Ans.
fcx. 11. $480
80.50
$560.50 $200 = $360.50, Ans.
MULTIPLICATION.
(165, page 137.)
Ex. 2. $4.275 x 300 = $1282.50, Ans.
Ex. 3. $2.45 x 175=$428.75, Ans.
DIVISION. 49
Ex. 4. $1.28 x 800 =$1024, Ans.
Ex.5. $.15 x 372 = $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, An*.
Ex. 8. $32.50 x80 = $2600
34.25 x70= 2397.50
$4997.50
3975
$1022.50, Ans.
DIVISION.
(166, page 138.)
Ex. 2. $41.25^-33=$!. 25, Ans.
Ex. 3. $94.50 -=-27 =$3. 5 0, Ans.
Ex., 4. $136-=-64 = $2.125, Ans.
Ex. 5. $1.32^-$.12 = 11, Ans.
Ex. 6. $405-r-$.54 = 760, Ans.
Ex. 7. $180-^12=$15, Ans.
Ex. 8. $2847.504-] 00=$28.475, Ans.
Ex. 9. $80.46-=-894=$.09, Ans.
Ex. 10. $1.125 x 120=$135 ; $185-^-27 $5, Ans.
K. P. 3
50 DECIMAL CURRENCY.
Ex.11.- $3.20 x 4=$12.80
.08x37= 2.96
$15.76
6.80
J-h$.16=56, Ans.
Ex.12. $4.50 + $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 APPLICATION
(168, page 139.)
Ex. 2. 693 x $=$321, Ans. Ex. 3. 478 x =$239, Ans.
Ex. 4. 4266 x $ T V=$355.50, Ans.
Ex. 5. 1250 x $i=$156.25, Ans.
Ex. 6. 3126 x $ T V=$195.375, Ans.
Ex. 7. 1935 x $i=$322.50, Ans.
Ex. 8. 56480 x$i=$7060, Ans.
Ex. 9. 1275 x $}=$255, Ans.
(169, page 140.)
Ex. 2. $.09 x 864=$77.76, Ans.
Ex. 3. $1.25 x 87=$108.75, Ans.
Ex. 4. $1.45 x 400 = $580, Ans.
Ex. 5. $.44 x 52 x 16 =$366.08, Ans.
(17O, page 141.)
Ex. 2. $l75^-25=$7, Ans.
Ex. 3. $200^-48=$4.16|, Ans.
ADDITIONAL APPLICATIONS. 51
Ex. 4. $1200^96 = $12.50, Ans.
Ex. 5. $56.25-^-10 = $5.62, Ans.
Ex. 6. $11.70-M8=$.65, Ans.
Ex. 7. $10.07 -=-53 =$.19, Ans.
Ex. 8. $1016-r-800=$1.27, Ans.
Ex. 9. $874.65-^-343 =$2.55, Ans.
Ex. 10. $684.375-:-365=$1.875, Ans.
(171, page 142.)
Ex. 2. $5.55-i-$.15=37, Ans.
Ex. 3. $216-^$12 = 18, Ans.
Ex. 4. $2178.75-J-$1.25 = 1743, Ans.
Ex. 5. $643.50-^-$19.5 = 33, ^ln*.
Ex. 6. $52.65-^$.45 = ll7, Ans.
Ex. 7. $6336-=-$132=48, ^rcs.
Ex. 8. $H77l5-^-$65 = 1811, Ans.
page 143.)
Ex. 2. $4.50 x 42.65 = $191.925, A.
Ex. 3. $.85 x 24.89 = $21.156+, Ans.
Ex. 4. $17.25 x 7.842=$135.274+, Anf.
Ex. 5. $12.50 x 23.48 = 1293.50, Ans.
Ex. 6. $3 x 1.728=$5.184, Ans.
Ex. 7. $7 x 2.40 =$16.80
6.40 x .865= 4.671
.80X12.56 = 10.048
$31.519, Ans.
Ex. 8. $4.375 x 14.76=$64.575, Ans.
(173, page 144.)
Ex.2. $7-j-2 = $3.50;
$3.50 x 1.495 =$5.2825, Ant.
52 DECIMAL CFRRENCY.
Ex, 3. $8.75-^-2=$4.375;
$4.375 X .325=$1.421 -f, An*.
Ex. 4. $3.84-:- 2 = $1.92;
$1.92 x 3.142 =$6.032 + , Ans.
Ex. 5. $5.60^-2=$2.80;
$2.80 x 1.848=$5.1744, Ans.
Ex. 6 $18-h2=$9;
$9 x 125 x .148=$33.30, Ans.
Ex. 7 $3.05-5-2=11.525;
$1.525 x 31.640=$48.251, Ans<
(174, page 145.)
Ex. 1. $3.60 x 7=$25.20
1.125 x 9= 10.125
.90 Xl2= 10.80
1.375x24= 33.00
.65 x32= 20.80
$99.925, Ans.
Ex. 2. $3.75 x 67=$251.25
2.62x108= 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, Ans.
Ex. 8. $.07 x325=$22.75
.0625x148= 9.25
.05 x286= 14.30
.125 x 95= 11.875
2.75 x 50=137.50
3.625 x 75 = 271.875
2.85 x 12= 34.20
$501.76, Ant.
PROMISCUOUS EXAMPLES.
Ex. 4.
Ex. 5.
$15 x20
9.50 x 7.5
G.25 x 10.75
2.625 x 3.90
3.00 x 5.287
$.11 x25=$2.75
.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,
= 8300.
= 71.25
= 67.1875
= 10.395
= 15.861
$464.6935, Am
Ans.
PROMISCUOUS EXAMPLES.
(Page 146.)
Ex. 1. $124.35 x 62.75 =$7802.9625, Ans.
Ex. 2. $.17 x 15=$2.55, Ans.
Ex. 3. $1406.25-=-2250=$f, Ans.
Ex. 4. $48.96-^12=$4.08, Ans.
Ex. 5. 325 miles x. 45 = 146.25 miles, Ans.
Ex. 6. 657-^36.5 = 18,^4^.
Ex. 7 $105 + $125 + ($35x4)=$370
$400 $370 =$30, Ans.
Ex. 8. $19 $15 = $4; $4x28=$112, Ans.
Ex. 9. e 2 9 -x^3=V-=^ 5 Ans -
Ex. 10. $9-f-$.3125 = 28.8, Ans.
Ex.11. $3.50x365=$1277.50
$2000-1277.50=$722.50, Ans.
64 DECIMALS.
Ex. 12. $687.
$1630.89 $875.29 = $755.60, Ans.
Ex.13 $!728-+2 = $864 1st half sold for ;
144x8 = 1152 2d " " "
Ex. 14. $3.75 x .875 = $3.281 +, Ans.
Ex. 15. $65.42-$46.56 = $18.86, gain per head;
$3526.82^$18.86 = 187, Ans.
Ex. 16. $54.72-^36.48=$1.50 ;
$1.50 x 14.25 = $21.375, Ans.
Ex. 17. $3548-r-4 = $887, Ans.
Ex.18. 112.34-^$.82 = 137, Ans.
Ex. 19. $3461.50^-46 = $75.25 ;
$75.25 x 5 = $386.25, Ans.
Ex. 20. $24000 x | x i =$3200, Ans.
Ex.21. $1.25x160= $200
$5 x 80= 400
$600
$2.50x240= 600
Loss 000, Ans.
Ex. 22. $1.70x48 = $81.60
72.90
$ 8.70, Ans.
Ex.23. 1221 + 751 = 197^; 197|-60 = 137f ;
$.9375 $.8125 =$.125, loss per bushel;
$.125 x 137f = $17.218+ loss ;
12.50 gain ;
$4.7 18 + , loss, Ans.
Ex 24, $1.40x6 =$8.40 wages;
$ .75x7= 5.25 expenses,
$3.15 savings, Ans.
PROMISCUOUS EXAMPLES.
55
$.08 x 39
Ex. 25. r-- =
Ex. 26. $4.50 x 23.487 =$105^6915, Ans.
Ex. 27 $1200-=-365=$3.287^f, Ans.
Ex 28. $.f 7 X 56 X 28 = $266.56, Ans.
Ex 29. $.07x26xl3xl6=$378.56,
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;
32 x 8
^ = 21^ minutes, Ans.
$32.3 4 15
" X T9 X T= $51 '^'
$5.635 -r-.875 =$6.44; $6.44 x 9 =$59.57, Ans.
$5000
$1200.25 x3 = $3600.75
1800.62x3= 5401.86
Ex. 33.
Ex 34.
Ex. 35.
$14002.61
.87x2= 1901.74
$12100.87, Ans.
Ex 36. $4.50 x 186.40 =$838.80, Ans.
Ex. 58. $96.40-^2=$48.20 ;
$48.20 x 1.375=$66.275, Ans.
Ex. 39. r s o 6 oV7r=T 9 6>
5*5 COMPOUND NUMBERS.
Ex.40. ^=.09375; .62i=.625; .37^ =-.370625;
f =.375 ;
.09375 + .625 + .370625 + .375 1.464375, Am.
Dr. Cr.
Ex. 41. $4.745 $2.765
2.625 1.245
1.27 .625
.45 3.45
5.285 1.875
$14.375 $9.96 = $4.415, Ans.
Ex. 42. $.125 x 120 = 115.00 $1.50
.625 x 18= 11.25 1.27
.07 x 47= 3.29 1.87
.18 x 6= 1.08 2.30
$30.62 $6.94=$23.68, Ans.
REDUCTION.
(183, page 152.)
Ex. 1. 14194 far. -7-4 = 3548 d. 2 far. ; 3548 d.-^-
= 295 s. 8 d.; 295 s.-f-20 = 14 15 s.
Ans. 14 15 s. 8 d. 2 far.
Ex. 2. 14x20 + 15s.=295s.;295s.xl2 + 8d.
3548d.x4 + 2 far. = 14194 far., Ans.
Ex.3. 15 x 20 + 19 s. = 319 s.; 319 s. x 12-flld.
= 3839 d. ; 3839 d. x 4 + 3 far.=15359 far., Am
Ex. 4. 15395 far. -1-4 = 3839 d. 3 far.; 3839 d.-i-12
= 319 s. 11 d. ; 319 s.-=-20 = 15 19 s.
Ans. 15 19 s. 11 d. 3 far.
Ex.5. 46 sov.x 20 + 12 s. = 932s.; 932s. x!2f2d.
= 11186 d M Ans
REDUCTION. 57
Ex. 6. 11186 d.-M2 = 932 s. 2 d. ; 932 s.-f-20=46 sov
12 s. Ans. 46 sov. 12 s. 2 d
(185, page 153.)
Ex.3. 5lb. x 12 + 7 oz.=67 oz. ; 67 oz. X20 + 12 pwt.=
1352 pwt.; 1352 pwt. x 24 + 9 gr. = 32457 gr., Ans
Ex. 4 43457 gr. -7-24 = 1810 pwt. 17 gr. ; 1810 pwt.-f-
20=90 oz. 10 pwt.; 90 oz. -r-12 = 7 Ib. 6 oz.
Ans. 7 Ib. 6 oz. 10 pwt. 17 gr.
E* 5. 41760 gr. -f-24 = l740 pwt. ; 1740 pwt. -f- 20 = 87 oz.;
87 oz.-7-12 = 7 lb. + 3 oz., Ans.
Ex. 6. 14 Ib. 10 oz. 18 pwt.=3578 pwt. ;
3578 x $.75=12683.50, Ans.
Ex. 7. 5 Ib. 6 oz.=1320 pwt.; 2 oz. 15 pwt.=55 pwt. ;
Ex. 8. 1 Ib. 1 pwt. 16 gr.=5800 gr. ; 4 pwt. 20 gr.= 1 16 gr. ;
5800-7-116 = 50;
$1.25x50=$62.50, Ans.
(1 86 5 page 155.)
Ex.3. 16 Ib. X12 + 11 oz.=203 oz. ; 203 oz. x 8 + 7 dr.
-* =1631 dr.; 1631 dr.x3 + 2 sc. =4895 sc.;
4895 sc. x 20+ 19 gr.=97919 gr., Ans'.
Ex. 4. 47 Ib. x!2 + 6 =570 ; 570 xS + 43
= 45643 5 45643 x 3 = 13692 3, Ans.
Ex.5. 20 gr. x5 x 365 =36500 gr. ;
36500 gr.-j-20=18253; 18253-^3=60831 3;
608 3 -^-8=76 ; 76 -1-12=6 Ib. 4 | .
Ans. 6 lb.4| 13.
(187, page 156.)
Ex.8. 3 T.X20 + 14 cwt.=74cwt.; 74 cwt.xlOO + 74
lb.=7474 Ib.; 7474 Ib.xl6 + 12 oz.=119596
oz. ; 119596 oz. x!6 + 15 dr.= 1913551 dr., Ans.
58 COMPOUND NUMBERS.
Ex. 4. 1913551 dr. -4-16 = 1 19596 oz. 15 dr.; 119596 oz.
-f-16 = 7474 Ib. 12 oz. ; 7474 Ib. -=-100 = 74 cwt.
74 Ib.; 74 cwt.-r-20=3 T. 14 cwt.
Ans. 3 T. 14 cwt. 74 Ib. 12 oz. 15 dr.
Ex 5. 3 T. 15 cwt. 20 lb. = 7520 Ib.
Ans. $.22 x 7520=$1654.40.
Ex. 6. 115 lb.-j-2000=.0575 T. ; $10 x .0575 =$.575,
Ans ,
Ex. 7.
217 Ib. x 10=2170 Ib. @ $.06 -$130.20
306lb.x 5 = 1530 Ib. @ $.07 =114.75
3700 Ib. @ $.08 =$296.00, which -$244.95
=$51.05, Ans.
Ex. 8. 2 T. x 2000=4000 Ib. ; 4000 x $.12 = $500 ;
$500 $360 = $140, Ans.
Ex. 9. 10 T. X20 + 6 cwt. = 206 cwt. ; 206 cwt. x4 + 3 qr,
= 827 qr. ; 827 qr. x 28 + 14 Ib.=23l70 Ib.
$.06 buying price.
$130-f-2000 = .065 selling price.
$.005 gain per pound.
$.005 x 23170 = $115.85, Ans.
Ex. 10. 2352 Ib.-f- 56=42 bu.;
$.90 x 42 x 2 =$75.60, Ans.
Ex. 11. 300 bbl. x 196 = 58800 Ib., Ans.
Ex. 12. $1.25 x 3 = $3.75 cost.
$.0075 x 280x3= 6.30
$2.55, Ans.
REDUCTION. 59
(191, page 157.)
Ex. 1. 5 Ib. 10 oz. = 90 oz. ; $.50 x 90=$45.0C cost.
480
$33.75, Ans.
Ex. 2. 424 dr.-=-8 = 53 oz. ; 53 oz.-f-12 = 4 Ib. 5 oz., Ans
Ex.3. 20 Ib. 8 oz. 12 pwt. = 119328 gr.
119328 gr.-f- 7000 = 1 7^ lb ->
Ex.4. $.40x16x20 =$128 cost
$.50x320x437.5 ia .. fi o,
---------- --- r= 145.834-
480 __ L
$ 17.831,
(193, page 159.)
Ex. 3. 7912 mi. x 63360 = 501304320 in., Ans.
Ex.4. 168474 ft. +-3 = 56158 yd.; 56158 yd.+-5i = 10210
d. 3 yd.; 10210 rd. +-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. x 8 + 7 fur. = 255 fur.; 255 fur. x 40 + 10
rd.=10210 rd.; 10210 rd. x 5^ + 3 yd.-56158 yd.; 56158
yd. x 3 = 168474 ft,, Ans.
Ex.6. 2500 fathoms x 6 = 15000 ft.; 15000 ft. -+16i =
909 rd. lift.; 909 rd.-=-40 = 22 fur. 29 rd. ; 22 fur. -+8 =
2 mi. 6 fur. ' , Ans. 2 mi. 6 fur. 29 rd. 11 ft.
Ex. 7. 2200 mi. x 5280 = 11616000 ft. ;
$.10 x 11616000 = $1161600, Ans.
Ex. 8. 4 fathoms x 6 + 3 ft.=27 ft. ; 27 ft. x 12 +8 in.=
832 in Ans.
Ex. 9 200 mi.=12672000 in. ; 18 ft. 4 in = 220 in. ;
12672000-^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. HI bands x 4 = 58 in., Ans.
60 COMPOUND NUMBERS.
(194, page 160.)
Ex.1. 3 ini.x 80 + 51 ch. = 291 ch. ; 291 ch. x 100 -I-
73 1. = 29173 1., Ans.
Ex. 2. 29173 1. + 100- 291 ch. 73 1. ; 291 ch. + 80 = 3 mi
51 ch. Ans. 3 mi. 51 ch. 73 L
Ex 3. 17 ch. 31 1. = 17;31 ch.
12 ch. 87 1 = 12.87 ch.
30.18 ch. half round the field.
30.18 ch. x 2 x 66 = 3983.76 ft., Ans.
(196, page 163.)
Ex.3. 87 A. x 4 + 2 R.= 350 R. ; 350 R. x 40 -f 38 sq. rd
= 14038 sq. rd. ; 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. 28 sq.in.;
3S21910 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. 61 sq. yd. 6 sq. ft. 28 sq. in.
But (| 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.;
111.1 sq. yd.-+30i = 367 sq. rd. 9| 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. 9} sq. yd. 1 sq. ft. ; or
2 A 1 R. 7 scj. rd. 9 sq. yd. 3J sq. ft.
REDUCTION. 61
Ex. 9. 181x16 = 296 sq.ft.;
296 sq. ft.-:-9 = 32 sq. yd., Ans.
Ex. 10. (18 + 16i) x 2 = 69 ft., distance round the room ;
69 x 9
= 69 sq. yd., in the walls;
y
18x161
-=83 sq. yd., in ceiling;
y
69 sq. yd. + 33 sq. yd. = 102 sq. yd.
$.22 x 102 =$22.44, u&w.
Ex. 11. 40x20x2 = 1600 sq. ft.=16 squares;
$10xl6=$160, Ans.
(197, page 164.)
Ex. 2. 3686400 P. +102400 = 36 sq. mi., Am.
Ex.3. 94 A. x 10 + 7 sq. ch.=947 sq. ch. ;
947 sq. ch. X16 + 12 P. = 15164 P.;
15164 P. x 625 + 118 sq. l. = 9477618 sq. 1., A *s.
Ex. 4. 4550000 sq. 1. -=-10000=.455 A.
$50 x 455=122750, Ans.
(199, page 166.)
Ex.1. 125 cu. ft. x 1728 + 840 cu. in. = 216840 ou. in., Ans.
Ex. 2. 5224 cu. ft.-M28=40i Cd., Ans.
Ex. 3. 3ft.2in.=38in.; 2ft. 2 in.=26 in.; 1 ft. 8 in. =2 Om.;
88 x 26 x 20 = 19760 cu. in., Ans.
Ex.4. 6x6x6 = 216 cu. ft.;
216 en. ft. x 1728 = 373248 cu. in., Ans.
.Ex.5. 60x20x15 = 18000 cu. ft. ;
18000 cu. ft.-f-128 = 140f- Cd., Ans.
Ex. &. 10 x 3i x 3i=113f cu. ft., Ans.
Ex. 7. 128-r-(3xl2) = 3f ft. high, Ans
62 COMPOUND NUMBERS.
Ex. 8. 27 x 175 lb.=4725 lb.=2 T. 7 cwt. 25 lb., Ann.
Ex. 9. 32 ft. + 24 ft.=56 ft. ; 56 ft. x 2 = 112 ft, girt;
112 x 1^x6 = 1008 cu. ft; 1008 cu. ft.^-24f--40 T 8 1 Feb.;
$1.25 x 40 T 8 T =$50.909 +, Ans.
Ex.10. $.15x 32X 2 2 7 4X g=$25.60,
Ex 11. 10x9x8 = 720 cu. ft. ;
720-^-10=72 minutes,
EJC. 12. 30 x 20 x 10 = 6000 cu. ft. ;
6000
-=12 minutes, Ans.
60x10
(2OO, page 168.)
Ex. 3. 3 bbd. x 2016 = 6048 gi., Ans.
Ex. 4. 6048 gi.-r 2016 = 3 hhd., Ans.
Ex. 5. 13 hhd. x 63 + 15 gal. = 834 gal. ; 834 gal. x 4 x 1
qt. = 3337 qt. ; 3337 qt. X 2 = 6674 pt., Ans.
Ex. 6. 6674 pt.-h2 = 3337 qt. ; 3337 qt.-h4 = 834 gal. 1
qt. ; 834 gal. -^-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. $3.84-h$.06 = 64 pt. = 8 gal, Ans.
Ex. 10. 2 gal. 2 qt. 1 pt. = 21 pt.; 1 hhd. = 504 pt. ;
504-^21 = 24, Ans.
page 169.)
Ex.1. 49 bu. x4 + 3 pk.=199 pk. ; 199 pk. x 8 + 7 qt.=
1599 qt. ; 1599 qt. X 2 + 1 pt. = 3199 pt, Ans.
REDUCTION. 63
Ex. 2. 3199 pt. -=-2 = 1599 qt. 1 pt. ; 1599 qt.-^-8 = 199 pk.
7 qt.; 199 pk.^-4 = 49 bu. 3 pk.
Ans. 49 bu. 3 pk. 7 qt. 1 pt.
Ex. 3. 1 bu. x4-f-l pk.=5 peck; 5 pk. x8 + l qt. 41 qt. ;
41 qt. X 2 + 1 pt. = 83 pt., Ans.
Ex. 4. 83 pt.-^-2 = 41 qt, 1 pt.; 41 qt.-j-8 = 5 pk. 1 qt.
f pk.-j-4 = l bu. 1 pk. Ans. 1 bu. 1 pk. 1 qt. 1 pt.
Ex. 5. $.65 x 50=$32.50 cost ;
$.25 x 4 x 50=450.00 sold for ;
$17.50 Ans.
page 170.)
Ex. 1. 1 bu. (Dry Measure) =21 50f cu. in.;
2150| cu. in.-:-57 = 37if wine quarts;
3711 qts. 32 = 5i| qts., Ans.
Ex. 2. 40 qt.^4 = 10 gal. ; 10 gal. x 282 2820 cu. in. ,
2820 cu. in.^57f = 48f4. qts. Wine Measure:
48ff qts. 40 qts. = 8^4 qts., Ans.
Ex. 3. 1 bn. Dry Measure =2150| cu. in.
32 qt. Wine Measure = 1848 cu. in.
302| cu. in., Ans.
(2O6 5 page 171.)
Ex. 1. 365 da.x24+-5 h.=8765 h. ; 8765 h.x604-48
min.=525948 min. ; 525948 min. x 60+46 se u=^
31556926 sec., Ans.
Ex. 2 31556926 sec. ^-60 = 525948 min. 46 sec.;
525948 min-j-60 = 8765 h. 48 min.; 8765 h.-f-24
= 365 da. 5 h. Ans. 365 da. 5 h. 48 rain. 46 sec,
64 COMPOUND NUMBERS.
Ex.3. 5 wk.x7;fl da. = 36 da.; 36 da. x 24+1 h.-
865 h. ; 865 h. x60 + l rain. = 51901 niin. ; 51901
inin. x 60 + 1 sec. = 31 14061 sec., Ans.
Ex. 4. 3114061 sec.-:- 60 = 5 1901 rain. 1 sec.; 51901 min.
-7-60 = 865 h. 1 min.; 865 h.-7-24 = 36 da. 1 h. ;
36 da. -=-7 = 5 \vk. 1 da.
Ans. 5 wk. 1 da. 1 h. 1 rain. 1 sec.
Ex. 6. 10 mi. = 17600 yd.;
17600 sec.-f-60 = 293 min. 20 sec. ; 293 rain.-7-60
=4 h. 53 min. Ans. 4 h. 53 min. 20 sec.
Ex.7. 29 da. X24+12 h.=708 h. ; 708 h. x 60 + 44 min.
=42524 min. ; 42524 min. x 60 + 3 sec.=
2551443 sec., Ans.
Ex.8. 40 yr.x 365^ = 14610 da.; 14610 da.x45 =
657450 min. gained.
657450 min.-^-60 = 10957 h. 30 min. ; 10957 h.-r-
24 = 456 da. 13 h. Ans. 456 da. 13 h. 30 min.
(2O7, 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';
310-7-30 = 10 S. 10. Ans. 10 S. 10 10' 10".
Ex. 3. 11400'-7-60 = 190, Ans.
Ex. 4. 190 x 69} = 13148 miles, Ans. '
Ex. 5. 360 x 60 = 21600', Ans.
Ex. 6. 397'-f-60 = 6 37', Ans.
(31O, page 174.)
Ex. 1. 150000000-^12 = 12500000 doz.;
12500000 doz. -M2 = 1041666 gross 8 doz.;
1041666 gross-f- 12 = 86805 great gross+-6 gross.
Ans. 86805 great gross 6 gross 8 doz.
REDUCTION. 65
Ex. 2. 100000 sheets -=-24 = 4166 quires 16 sheets;
4166 quires -=-20 = 208 reams 6 quires;
208 reams -7- 2 = 104 bundles;
104 bundles^ 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 page*,
Ans.
Kx. 5. 32 pages x!0x2=640 pages, Ans.
PROMISCUOUS EXAMPLES IN REDUCTION.
Ex. 1. 6 yd. 3f qr. = 27 qr.; 333 yd. = 1332 qr. ;
1332-=-27f =48 suits, Ans.
Ex. 2. 1 oz. 15 pwt.=35 pwt. ;
$.70x35 =$24.50, Ans.
Ex. 3. 2 Ib. 3 5 5 3 13 10 gr.=13290 gr.;
13290-^15 = 886, Ans.
Ex. 4. 1 T. 11 cwt. 12 lb.=3112 lb.;
3112x$.01i=$38.90, Ans.
Ex. 5. 1456 lb. -=-32 = 45.5 bu. ;
$.375 x 45.5=$17.0G25, Ans.
Ex. 6. 45 lb. x 1000=45000 lb.
45000 lb. -7-196 = 229 bbl. 116 lb., Ans.
Ex. 7. 2430 lb.H-60=40.5 bu. ;
$1.20x40.5=$48.60, Ans.
Ex. 8. $12.50-f-200 = $.06}, 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 ch.(4 rd.)=808 ck ; 808
nh. x 100 + 20 L= 80820 L, Ans.
i)6 COMPOUND NUMBERS.
Ex. 11. 25x100x144 = 360000 sq. in.;
$.01 X 360000 = ^3600, Ans.
Ex.12. 50x25 x 10 = 12500 cu. ft. ;
12500 cu. ft. -=-16 = 781 ccl. ft. 4 cu. ft.;
781 cd. ft. -=-8=97 Cd. 5 cd ft. ;
Ans. 97 Cd. 5 cd. ft. 4 cu. ft.
Ex 13. 10x10x10x1728 = 1728000 cu. in. ;
1728000 cu. in.-=-231 = 74804f gal.;
7480ff gal.-+63 = 118 bhd. 46^ gal., Ans.
Ex. 14. 8 x 5 x 4^ = 180 OIL ft. = 3 11040 cu. in. ;
311040 cu. ^.+2150.4 = 144/4 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., Ans.
Ex. 16. 1296000 sec.+-86400 = 15 da., Ans.
9.0 v 1 9
Ex.17. ~=40 yd., Ans.
Ex 18. 4 reams x 20 + 10 quires =00 quires; 90 quires x
24 + 10 sheets=2l70 sheets, Ans.
Ex.19. 16 ft. 6 in.=l rd. ; 1 mi. = 320 rd. ; 320-=- 1 =
320 times in 1 mi. 320 x 42 = 13440 times, Ans.
Ex. 20. 1000000 sec. + 60 = 16666 min. 40 sec. ;
16666 min.-r-60 = 277 h. 46 min. ;
277 h. -7-10 = 27 da. 7 h. ;
Ans. 27 da. 7 h. 46 min. 40 sec.
Ex. 21. 6 x 4i =27 sq. mi ; -= 216 farms, Ans.
80
Ex. 22. 10 mi. 176 rd. = 3376 rd.
$21.75 x 3376 ^$73428, Ans.
REDUCTION. 67
11, page 176.)
Ex. 2. n * 5f x V x -V 2 = & d., Ans
Ex. 3. TT { yy wk. x T x V- x V-=T
Ex. 4. T o^ h lld - x V- x t x \ x f =1
Ex 5. ff i 7 oz. x V x V =\ g r -> Ans -
Ex 6. TO o o o c mi. x f x V x V x Jf
Ex. 7. | x | x f- Ib. x Y=l oz - ^ W5 *
Ex. 8.' F ^- hlid. x 6 T 3 x T x f- = f | pt.,
Ex. 9. TT 7 7 A. x | x V=-2- rd - Ans -
page 177.)
Ex. 2. J- ft. x ^3 = T h rc l- Ans -
Ex. 3. f dr. x T V x T V~ T/O"
Ex. 4. | ct. x T J--=r^^^ E.,
Ex. 5. % ft. x 3 T V o = TJ ^ T o mi.,
Ex. 6. ^ x f- pwt. x T V x j2 = 2K
Ex.7. Jpt.x'ix|x^= T
i hlid. T J bhd.=fU hhd., Ans.
Ex. 8. f in. x ^_i ? _=: TJ Vo o mi -> ^*-
Ex. 9. | oz. x T V=jft lb. = 2-V <> f 2 lb - ; and 2T of 2 lb - is
| of | of J T of 2 lb., or | of of 2 lb., ^ra.
Ex. 10. f oz. x T V= 2 T lb.- aV o f 2 lb - ; and T of 2 lb - ia
i of f of J T of 2 lb., or J- of f of 2 lb., ^w*.
(SI 3, page 178.)
Ex. 2. 4 mo.x30 = l7j da.; } da. x 24 = 3^ h. ; ^ h. x
60 = 254 rain. ; f min. x 60 = 42-f sec.
Ans. 17 da. 3 h. 25 min. 42-fi sec.
Ex. 3 f x 20=84 s - ? 4 s - x 12 = 6^ d. ; 4 d. x 4=34
8 s. 6 d. 34 far.
68 COMPOUND NUMBERS.
Ex. 4. bu.x4 = lf pk. ; f pk. x8 = 4f qt.; qtx2~
12 pt. Ans. I pk. 4 qt. If pt.
Ex.5. ^ of 15 cwt. = 12$ cwt. ; 4 cwt. x 100 = 854 lb.;
| Ib. x 16 = 11$ oz. ; 5 oz. x 16 = 6$ dr.
.d/w. 12 cwt. 85 Ib. 11 oz. 6f dr.
Ex. 6. | x | x 1 a=4|$ oz. ; f$ oz. x 16 = llf$ di.
Ans. 4 oz. 11 |f dr,
Ex. 7. | A. x 4=3i R. ; $ R. x 40 = 131 p.
^tws. 3 R. 13i ?.
Ex. 8. if da. x 24 = 16 T 8 3 h - 5 A b - x 60 = 36if min. ; }a
min. x 60 = 55^ sec.
Ans. 16 h. 36 min. 55 T 5 3 sec.
Ex. 9. | Ib. x 12 = 7i oz. ; 1 oz. x 20 = 4 pwt.
Ans. 7 oz. 4 pwt.
Ex. 10. % f-y-T.-4 T 5 T. ; & T.x20 = 5f cwt.; cwt.
x 100 = 55 Ib. Ans. 4 T. 5 cwt. 55f Ib.
Ex. 11. f ofY A.= lf A.; f A.x4 = li R. ; 1 Rx40=
20 P. Ans. 1 A. 1 R. 20 P.
(214, page 179.)
Ex. 2. 6 fur. 26 rd. 3 yd. 2 ft. =4400 ft. ; 1 mi. =5280 ft.;
|f mi.=f mi., Ans.
Ex. 3. 13 s. 7 d. 3 far. = 655 far. ; l = 960 far. ;
Ex. 4. 10 oz. 10 pwt. 10 gr.=5050 gr.; 1 lb.=5760 gr.;
.= lb - Ans -
Ex. 5. 2 cd. ft. 8 cu. ft.=40 cu. ft. ; 1 Cd. = 128 en. ft. ;
T W Cd. = t V Cd., Ans.
Ex.6. lbbl.lgal.lqt.lpt.lgi.=1053gi.; lhhd.=2016gi.;
i^ff hhd.=H} hhd., Ans.
Ex. 7. 4 yd. li ft.=27 half-feet; 2 rd. = 66 half-feet;
REDUCTION. 69
Hi 82,
Lx. 8. - bu.= =- bu., Ans.
Ex. 9. ^V = , Ans.
Ex. 10. 2 yd. 2 qr.=10 qr ; 8 yd. 3 qr. = 35 qr. ;
i| yd.=f yd., Ans.
(215, page 180.)
Ex.2. .217 x 60 = 13.02'; .02' x 60 = 1.2*.
Ans. 13' 1.2*.
Ex.3. .659 wk. x7 = 4.613 da. ; .613 da. x 24 = 14.712 h.;
'712 h. x 60=42.72 min.; .72 min. x 60 = 43.2 sec.
. 4 da. 14 h. 42 min. 43.2 sec.
Ex. 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. x4 = 2gi.
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." Ans. 12 s. 6 d. 3 far.
Ex. 7. .22 hhd. 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.
Es 8. .67 lea. x 3 = 2.01 mi. ; .01 mi. x 8 =.08 fur. ;
.08 fur. x 40 = 3.2 rd.; .2 rd. x 5.5 = 1.1 yd.;
.1 yd. x 3 = .3 ft. ; .3 ft, x 12 = 3.6 in.=3f in.
Ans. 2 mi. 3 rd. 1 yd. 3 j 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.
70
COMPOUND NUMBERS.
Ex. 10. .78875 T.x 20=15.775 ewt.;. 775 cwt.x 4=8.1 qr.;
.1 qr. x 28 = 2.8 Ib. ; .8 Ib. x 16 = 12.8 oz.
Ans. 15 cwt. 3 qr. 2 Ib. 12.8 oz.
Ex. 11. .88125 A. x 4=3.525 R. ; .525 R. x 40=21 P.;
Ans. 5 A. 3 R, 21 P.
Ex 12. .0055 T.x 2000 = 11 Ib., Ans.
Ex. 13. .034375 bundles x 40 =1.3 75 quires; .-<>75 quires x
24 = 9 sheets ; Ans. I quue 9 sheets.
, page 181.)
Ex. 2.
Ex.4.
Ex.6.
. 4
1.00 gi.
2
1.250 pt.
4
3.625 qt.
Ans. .90625 gal.
. 2
1.2 pt.
4
.6 qt.
63
.150 gal.
Ans. .00238 + hhd
. 40
12.56 P.
4
3.314 R.
Ex.
Ex.
3. 24
20
12
9.000 gr.
13.375 pwt.
10.66875 oz.
An
5. 8
4
*. .8890625 Ib
1.12 qt.
3.14 pk.
Ans.
.785 bu.
40 12.56 P.
Ex. 7. 12
3
5.5
40
8
6 in.
4 3.314 R.
1.5 ft.
Ans. .8285 A.
17.5 yd.
3.1818 81+ rd.
.0795 545+ far.
Ans. .0095 &]
Ex. 8. .32 pt.-=-64=.005 bu., Ans.
Ex. 9. 4.875 ft.-5-6 = .8125 fathoms, Ans.
Ex. 10. 150 sheets-f-480=.3125 Rm., Ans.
Ex. 11. 47.04 lb.-M96=.24 bbl flour., An*
Ex. 12. .33 ft.-=-r>280 = .0000625 mi., AM.
ADDITION.
Ex. 13. 60
60
24
57.6 sec.
36.96 min.
5.616 h.
Ans. .234 da.
ADDITION.
(217, page 183.)
Ex 3. 43 10. 1 3 2 3 16 gr., Ans.
Ex. 5. 68 bu. 3 pk. 1 qt. 1 pt., Ans.
Ex. 6. 21 mi. 5 for. 23 rd. 1 yd. 4 in., Ans.
Ex. 10. 627 hhd. 7 gal. 1 qt. 1 pt., Ans.
Ex. 11. 187 bu. 3 pk. 1 pt., Ans.
Ex. 16. 152 en. yd. 9 cu. ft. = 152i cu. yd.
$.16x1521 =$24.37f
Ex. 17. 2564 Ibs.
2713 "
3000 "
3109 "
11386 Ibs, -=-56 = 203.3214+ bu.
$.80x203.3214=:$162.657 + , ATM.
bbls. gal. qt. pt. gi.
Ex. 18. 1 4 1
30 2 1
2 15
Ana.
3 49 2 1 1=4613 gi.
$.09 x 4613=1415.17, Ans.
(318, page 185.)
Ex. 2
$ rd. = 12 ft. 41 in.
| ft. = _9_ "
13 ft. li in., Ans.
72 COMPOUND NUMBERS.
Ex. 3. i mi. = 7 fur.
I fur.= 26 rd. U ft.
rd. = 13 " U in.
7 fur. 27 rd. 7 ft. in. ; or
7 fur. 27 rd. 8 ft. 3 in. Ant.
Ex.4, | =13 s. 4d.
s. = 6 2| far.
13s. 10 d. 2| far., Ans.
Ex. 5. f T. =12 cwt.
| cwt.= 42 Ib. 134 oz.
12 cwt. 42 Ib. 13f oz.,
Ex. 6. da. =9 h.
i h. = 30 min.
9 li. 30 min.,
Ex. 7. wk. = l da. 4 h.
da. = 18 "
| h. = 15 min.
1 da. 22 h. 15 min., Ans.
Ex. 8. $ hhd.=54 gal.
gal. = 3 qt.
54 gal. 3 qt., ^ws.
Ex. 9. 4 cwt.=64 Ib.
8f Ib. = 8 " 13 oz. 5 dr.
. = 3 " 14 "
73 Ib. 1 oz. 3} dr., Ant.
Ex. 10. ^ mi. =3 fur.
|yd.= 2 ft.
ft. = 9 in.
3 fur. 2 ft. 9 in., Ans.
SUBTRACTION. 78
Ex. 11. i of A. = i A.- 26 P. 181 | sq. ft.
60f rd. = l R. 20 " 204^ "
f A. =1 " 5 " 194 if "
A. = 13 " 90
3 R. 26 P. 126 T VV sq. ft.,
Ex. 12. li T. 1 T. 3 cwt. 33 Ib. 5 oz.
ly^ T. =1 " 3 " 75 "
18f cwt.= .18 " 83 " 51 '-
3 T. 6 cwt. 91 Ib. lOf oz., Am.
SUBTRACTION".
(919, page 187.)
Ex 4. 3 T. 18 cwt. 70| Ib., Am.
Ex. 6. 2953 2 s. 7f d., Ans.
Ex. 11. 365 da. X5 + 2 da. = 1827 da.
1 hhd. = 63 gal.
1827 gi. =57 " qt. pt. 3 gi.
5 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., Ans.
Ex. 13. 16 Cd. 5 cd. ft. 75 Cd. 6 cd. ft.
24 " 6 " 12cu.ft. 69 " 2 " 12 cu. ft.
27 " 7 " 6Cd.3cd.fT4cn.ft.
69 Cd. 2 cd.ft. 12cu.ft. Ans.
K. p.
74 COMPOUND NUMBERS.
Ex. 14. 10 gal. 1 qt. 1 pt. 63 gal.
15 " 1 pt. 40 " 1 qt.
14 " 3 "
40 gal. 1 qt, Ans.
I, page 189.)
22 gal. 3 qt., Ans.
yr. mo. da. yr. mo. da.
Ex. 2. 1799 12 14 Ex. 5. 1815 18
1732 2 22 1775 6 17
67 9 22, Ans. 40 1, Am
yr. mo. da. h. inin.
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. From Nov. 6 to April 6, 151 da.
From Apr. 6 to Apr. 15, 9 "
160 da., Ans.
Ex. 9. From Aug. 20 to June 20, 304 da.
Subtract 5 "
299 da., AIM
(891, page 190.)
Ex. 2. \ rd. = 8 ft. 3 in.
ft. = 9 "
7 ft. 6 in., Ans.
Ex. 3. f = 11 s. 1 d. li far.
s. = 6 "
10 s. 7 d. li far. An*.
MULTIPLICATION.
Ex. 4. lea.=2 ini.
75
TO mi.=
5 fur. 24'rd.
1 mi. 2 fur. 16 rd., Ans.
Ex. 5. 8 T 9 7 cwt. =8 cwt. 3 qr. 16 Ib. 12 oz. 12f dr
1 qr. 2f Ib. = 1 qr. 2 " 6 " 13 4 "
Ans. 8 cwt. 2 qr. 14 Ib. 5 oz. 15/j d>-
Ex. 6. i wk.^rl da. 9 h. 36 min.
da. = 4 " 48 "
1 da. 4 h. 48 min., Ans.
f| of 120 mi. = 41 mi. 7 fur. 9 rd. 8 ft. 74 in., An*
Ex.8. 1-i-f; Joff = A;
T 4 3 of 96 gal.=25 gal. 2 qt. 3} gi., Ans
MULTIPLICATION.
(222, page 192.)
Ex. 4. Ans. 23 13 s. 4 d.
Ex. 5. Ans. 23 Ib. 4 oz. 6 pwt. 10 gr.
Ex. 6. Ans. 163 T. 1 cwt. 36 Ib. 4 oz.
Ex. 7. Ans. 128 35 r 15".
Ex. 9. Ans. 20ft> 1 3 3 1 3 16 gr.
Ex. 10. Ans. 235 mi. 6 fur. 7 rd. 4 ft.
bu. pk. qt pt.
Ex. 13 45 3 6 1 Ex. 14.
367 2 4
8
2941, Ans
178
12
17 16 6
10
5, Ans.
76 COMPOUND NUMBERS.
Ex. 15. $4800-f-$80 = 60 = 6 x 10.
A.
R.
P.
Bq. yd.
sq.fl
4
3
26
20
3
6
29
2
1
10
295 10, Ans.
Ex. 17. Ans. 359 45' 40.45*. Ex. 18. Ans. 6 hhd.
DIVISION.
, page 194.)
Ex. 7. Ans. 1 oz. 17 pwt. 4 gr.
Ex.11. 5 10s. 10d.=1330d.;.
537 10s. 10d.=129010 d.
129010-^1330 = 97, Ans.
Ex. 12. - -- =11 cu. yd. 3 cii. ft.. Ans.
27 x5 x 6
K, 13 . ^-
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. ; f of 969 qr.=646 qr. ; 9 yd. 2 qr. = 38 qr. ;
646-^-38 = 17, Ana.
LONGITUDE AND TIME. 77
LONGITUDE AND TIME.
(229, page 196.)
Ex. 2. 84 24' Ex. 3. 155
74 1 18 28'
10 23' 173 28'
4 4
41 min. 32 sec., Ans. 11 h. 33 min. 52 sec., Ans.
Ex.4. 77 1' Ex.5. 118 + 122=240 ;
30 19 360 240 = 120;
" 7 120-M5 = 8 h
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 h.
26 39'= 1 h. 46 min. 36 sec.
2 h. 13 min. 24 sec., P. M., Ans.
Ex. 8. 124 67^57 = 3 h. 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.
(26, page 197.)
Ex. 2. 11 h. 33 min. 52 sec. = 693 inin. 52 sec. ;
(693 min. 52 sec.)-;- 4 = 173 28', Ans.
Ex. 3. 7 h. 9 rain. 20 sec.=i429 min. 20 sec. ;
(429 min. 20 sec.)-h4 = 107 20', Ans.
Ex.4. 16 h. 30 min. at St. Petersburgh ;
8 h. 32 min. 36 sec. at New Orleans ;
7 h. 57 min. 24 sec.=477 min. 24 sec.
(477 min. 24 sec.) -4-4 = 119 21', Ans.
Ex.5. 74 1' West; 8 h. 40 min.=130
4 h.=60 74 l f West,
1st Ans. 14 1' West. 2d Ans. 55 59' East
13 h. 25 min.=r201 15'
74 1'
3d Ans. 127 14' East.
DUODECIMALS.
MULTIPLICATION.
(939, page 200.)
Ex. 2 13 ft 9' Ex:. 3. 11 ft. 9'
11' 1 .3'
12ft. 7' 3*, Ans. 2 11'
11 9
14ft. 8' 3'
4
58 ft. 9', Ans.
DUODECIMALS. 79
Ex. 4. 12 ft. 11' 6 ft. 2'
4 2 ft. 4'
51 ft. 8' length of walls; 2 0'
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
Ex. 5 30 ft. 4'
25 ft. 6'
sq. yd. 2 sq. ft. 9', Ans.
Ex. 6. 18 ft. 6'
12 ft.
-7-128 =
32| sq. yd
.551, Ans
15 2
758 4
222 ft:
5 ft. 6'
773 ft. 6'
12 ft. 5'
111
1110
322 3 6"
9282
1221 ft.
1221 cu. ft.-
Ans. 9 Cd ' 69 cu '
Ex. 8. 32 ft, 8
9 ft.
9604 ft. 3' 6",
Ex. 7. 36 ft. 10'
22ft. 3'
926
810 " 4
294 sq. ft.
294 sq. ft. ^9 =
&.l7x32f = $5
819 ft. 6' 6"
5 ft. 2'
136 7 1
4097 8 6
4234 ft. 3' 7" ;
156 cu. yd. 22 cu. ft. 3' 7', Ans.
80 COMPOUND NUMBERS.
Ex.9. 33ft. 9' 27' __3
48ft. 12x3~4
36
1584 180-^-f = 240 yd., An*.
1620 ft. = 180sq. yd.
DIVISION.
(23O, page 201.)
Ex. 2. 16 ft. 8')44 ft. 5' 4"(2 ft. 8', Am.
33 4
11 1 4
11 1 4
Ex. 3. 40 ft. 11' 4*)184 ft. 3' 0*(4 ft. 6', Ans.
163 9 4
20 5 8
20 5 8
Ex.4. 2 ft. 7')14 ft. 6' (5 ft. 7' 4* + , Ans.
12 11
170'
1 6 1
11 0'
10 4
8'", rem.
PROMISCUOUS EXAMPLES. 81
Ex. 5. 3 ft. 7' 8 ft. 11' 6")64 ft. 2' 5"(7 ft. 2',
2ft. 6' 62 8 6
72 1 5 11
196 1 5 11
8 ft. 11' 6"
PROMISCUOUS EXAMPLES.
(Page 202.)
Ex 1. 115200 gr. 4-5760=20 lb., Ans.
Ex. 3. 1560 bu. x 4-6240 pk. ; 3 bu. 1 pk.=13 pk. ,
62404-13=480, Ans.
Ex.4. 295218 in.4-12 = 24601i ft.; 24601| ft.4-16=-
1491 rd. ; 1491 rd. 4-40=37 fur. 11 rd. ;
37 fur. 4-8=4 mi. 5 fur.
Ans. 4 mi. 5 fur. 11 rd.
Ex. 6. 3 x 20 x 24=1440, Ans.
$3.25 x4x20x6 x 10
Ex. 7. =$121.871, Ans.
Ex. 8. 1 bbl.=1008 gi. ; 1 qt. 1 gi.=9 gi. ;
10084-9 = 112, Ans.
Ex. 9. $26.40 x \ 3 x f =$980.10, Ans.
Ex. 11. 336 bu. 3 pk. 4 qt.= 10780 qt. ; 4 bu. 3 pk. 2 qt.:
154 qt.; 107804-154=70, Ans.
Ex. 12. 3 qt. 1 pt, Ans.
Ex. 13. 1 mi.= l760 yd.; 2 fur. 36 rd. 2 yd.=640 yd. ;
T y 6 <V mi.= T 4 T mi., Ans.
82 COMPOUND NUMBERS.
Ex. 14. 2 da.= 172800 sec.; 13 h. 26 min. 24 sec.
48384 sec.; iWftV =
Ex. 15. 26 A. 2 R.-106 R.; 5 A. 3 R.=23 R.,
106 R.-23 R.=83 R. ; J&, Ans.
Ex. 16. = 12 s.
5 d.
11 s. 6 d.
Ex. 17. ^ yd.=5j in.
| ft. =14
4 in. = 4 "
7 in.,
Ex. 19. 1732 yr. 2 mo. 22 da.
1706 "1 " 18 "
26 v- l mo . 4 da.
Ex. 20. 87 30
71 4' 12 h.
16 26'= 1 h. 5 min. 44 sec.
10 h. 54 min. 16 sec., A. M., Am.
Ex 22. I mi. =5 fur. 13 rd. 5 ft. 6 in.
far.= 20 "
T 3 rd. = 2 " 9 "
5 fur. 33 rd. 8 ft. 3 in., Ans.
Ex. 23 20 bu. 3 pk. 6 qt.= 20.9375 bu. ;
$.80 x 20.9375=116.75, Ans.
Ex. 24. .875 gross x 12 = 10.5 = 1 0| doz., Ans.
Ex. 25. 56.5 x 24.6 = 1389.9 P.=8 A. 2 R. 29.9 P., Ans.
Ex. 26. 20 4- (Hx 2) = 23 ft., length of one side;
23 x 8 x 1| x 4=1104 cu. ft., by the mason's rule ;
(see note 5, page 166).
1104-r-24.75 = 44.6 -f Pch., Ans.
PROMISCUOUS EXAMPLES. 88
Ex. 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 0.
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 Ib. 8 oz. 11.4 dr. = 13.54453 125 Ib. Troy.
13.54453125 Ib. x }f = 16.460367.83 + lb. Av.=
16 Ib. 5 oz. 10 pwt. 11.7-f-gr., Ans.
Ex. 29. 154 bu. 1 pk. 6 qt. = 154.4375 bu.
$173.74x^.5 $_m.74x .3
154.4375 30.8875
Ex. 30. .0125 T. x 2000 = 25 IB., Ans.
Ex. 81. T 7 2 of 2 bu. 3 pk.=;$ pk. ; 3 bu.=12 pk. ;
tt+12=ffr, Ans.
EX. 30,. 4- X I X | X -L3JLI X i^T=- L H =
Ex. 33. V- x x x ^V^ x irrlo.T^ HV =301.339 f ,
Ex. 34. ^ X36X2 =$20.80. An,.
46x20x2x144
Ex 35. - -=13248, Ans.
4x5
<ix. 86. 1864 yr. 4 mo. 20 da. 18 h. 30 min
1836 " 9 " 4 " 3 " 45 "
27 vr. 7 nio. 16 da. 14 h. 45 min.
84 COMPOUND NUMBERS
Ex. 37. 28 ft. 9' = 28f ft. ; 22 ft. 8' = 22f ft. ; 1 ft. 6' = 7J ft
ii x x x = i=181l cu -
Ex. 38. 30 bu. 54 lb. = 30.9 bu.
$1.375 x 30.9 = 142.4875, Ans.
Ex. 39. 24 ft. 8'=24f ft. ; 18 ft. 6' = 18 ft. ;
-V x x iif 50 ^ Ans -
Ex. 40. 54 bu. 8 lb.=54i bu.
$.84 x 54| =$45.50, Ans.
Ex. 41. 18720-5-120 = 156, Ans.
Ex. 42. 21X528 X12 = 44352, An,.
30
324000-^192 = 1687^ rain.=
1 da. 4 h. 7 rain. 30 sec., Ans.
Ex. 44. 65 mi.=65 x 63360 in. ; 9 ft. 2 in. = 110 in.;
!2^ =65X 676-37440,^
Ex. 45. 10 bu. x 2150.4 = 21504 cu. in.
11
$.22 x 372 T V=$81.92 sold for ;
$5 x 10 =$50.00 cost.
$31.92 gain, Ans.
t 240x6x3x1728
Ex.46 -=116640 bricks;
8x4x2
$3.25 x 11 6.64 =$379.08, Ans.
PERCENTAGE.
PERCENTAGE.
(234, page 206.)
Ex. 1 .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. A; ft; A? tffr; w. in;
m; if; f-
.0025
.0075
.00666 +
.008
.00625
.0125
.028
.04333 +
.0575
.07125
.122
.25375
; I;
, page 207.)
Ex. 9. Ans. 63 sheep. Ex. 10. Ans. 620 men.
Ex. 12. Ans. $22692.25.
Ex. 20 1.00 .25=. 75; 760 sheep x .75 570 sheep, Ans
Ex. 21. .18 + .30=.48; 1.00 .48^.52;
$24500 x .52=$12740, Ans.
Ex, 22 1576 barrels x.l 25 197 barrels, Ans.
Ex.23, .75=|; .33J=t; f-J=A?
$2760 x T 5 2=r$1150, ^4?w.
Ex.24, XTY^/jtold; |-A=H loft
86 PERCENTAGE.
Ex. 25. |, owed after the 1st payment.
f x f, " " " 2d "
fxfxf, " " 3d "
> x f x | x |=$226.56f , .471*.
page 208.)
Ex. 2. 904-450 = .20 = 20 per cent., Ans.
Ex 3 175-7-1400=.125 = 12| per cent,, Ans.
Ex. 4. 165-7-750 = .22 = 22 per cent., Ans.
Ex. 5. 13.20^240 = . 055 = 51 per cent, Ans.
Ex. 6. .154-2":=.075 = 7i per cent., Ans.
Ex. 7. 6 bu. 1 pk.=200 qt.; 4 bu. 2 pk. 6 qt.= 130 qt.
150-f-200 = .75 = 75 per cent., Ans.
Ex. 8. 15 lb. = 240 oz. ; 5 Ib. 10 oz. = 90 oz. ;
90-^-240 = . 375 = 37i per cent., Ans.
Ex. 9. 40^-250^.16 = 16 per cent., Ans.
Ex. 10. 100 + 90 = 190;
190-^-760n:.25 = 25 per cent., Ans,
Ex. 11. | of =i = .50 = 50 per cent., Ans.
(337, page 209 )
Ex. 2. 164- .08 = 200, Ans.
Ex. 3. 42-f- .07=600, Ans.
Ex. 4. 75 -=-.125 = 600, Ans.
Ex. 5. 33-=- .0275 = 1200, ^Iras.
Ex. 6. $281.25-^.375 = 8750, Ans.
Ex. 7. 50-=-.20 = 250, Jws.
Ex.8. |59.75-=-.125=$478, Ans.
Ex. 9. $975 -Kl 5 =$6500, ^4ws.
Ex. 1C .40x.25 = .l
$1246.50-=-.! =$12465, Ans.
Ex. 11. 2000^- .40 = 5000; 5000-2000 = 3000, Am.
COMMISSION AND BllOKERAGE. 87
(238, page 211.)
Ex.2. 1.00 + .18 = 1.18; 1475-M. 18 = 1250, Ans.
Ex.3. 1.00 + .25 = 1. 25 $4.00-=- 1.25 = $3. 20, Ans.
Ex. 4. 1.00 + .15 = 1.15; $6900-^-1. 15=$6000, Ans.
Ex. 5. 1.00 + .08 T 1 g = 1.08 T 1 ff ;
$432250-M.080625 = $400000, Ans.
Ex. 6. 1.00 + .04J = 1.0425;
$8757-=- 1.0425 =$8400, Ans.
Ex. 7. Since he increased his capital the first year by
20 % of itself he must have had 100 % + 20 %, or
120 <f 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-M.44 = $6500, Ans.
(239, page 212.)
Ex.2. 1.00 .15 = .85; 340^.85 = 400, Ans.
Ex.3. 1.00 .20=.80; $1000H-.80 = $1250, Ans.
Ex.4. 1.00-.24 = .76; $4028-^.76 = 15300,^5.
Ex. 5. 1.00-.00l = .995 ; 298^-i- .995 = 300, Ans.
Ex. 6. $198, his selling price, is 90 f of his asking price ;
therefore $198-^.90=$220, his asking price; and
$220, his asking price, is 110 % of the cost ; there-
fore $220-^1. 10=$200 cost, Ans.
COMMISSION AND BROKER AG
(243, page 213.)
Ex. 2. $6756 x .02=^135.12, Ans.
Ex. 3. $17380 x.035 '
a PERCENTAGE.
Ex. 4. $.75 x 4700=$3525; $3525 x .015=852.875, Ans.
Ex. 5. $25875 x .0075=$64.6S75, Ans.
Ex.6. $3284+$2l76.50=$5460.50 ;
$5460.50 x .0225 = 1122.86+, Ans.
Ex. 7. $2890 x 5^-^23.12, Ans.
Ex. 8. $.32 x 26750 = $8560 ;
$8560 X .02 1 = $235.40, Ans.
Ex. 9. 400 x 570 x $.09 x. 0225 = $461.70, Ans.
Ex. 10. $7.60 x 450 =$3420
.25x56x38 = 532
.09x48x105= 453.60
$4405.60 X .055 = $242.308,
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^- 1.02 = $3182. 55 (nearly) invested;
$3246.20 $3182.55 = $63.65, Ans.
Ex. 3. $9682^-1.03 = $9400, Ans.
Ex. 4. $10246.50-^1.035 = $9900 invested ;
$9900-f-$5.50 = 1800, Ans.
Ex. 5. $4908-^1. 045 = $4695.69+, Ans.
Ex. 6. $603.75-r-1.05 = $57o invested;
$575-r-$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. $13842.07 -7-1.0l75 = $13604, invested ;
$13842.07 $13604 = $238.07, commission, Ans.
Ex. 9. $10650-^-1. 0025 = $10623.44 + , Ans.
STOCKS.
(261, page 218.)
Ex. 2. $1200 x .95 = $1140, Ans.
Ex. 3. $3500 x. 85 = $2975, Ans.
Ex. 4. $1.00 + $.051 +$.001 = $1.06, cost of doll, of stock;
$150x48 = $7200, nominal amount;
$1.06 x 7200 = $7632, Ans.
Ex. 5. $1.09| x 5364 = 85853.465, Ans.
Ex. 6. $6275 x .12 = $753, Ans.
Ex. 7. 1.00 + .04f + . 00-J = 1.05;
$25000 x 1.05 = 8131250, Ans.
Ex. 8. .14 + .125 = . 265, rate of gain ;
$4200 x .265 = $11 13, Ans.
Ex. 9. $17500 x 1.0075=817631.25, Ans.
Ex. 10. .03 +.0225 = .0525, rate of gain;
$50 x 75 =$3750, nominal amount of stock ;
$3750 x .0525 =$196.875, Ans.
Ex.11. $50x28 = $1400 ; $1400 x 1.07 = $1498 ;
$100 x 25=$2500 ; $2500 x .875=82187.50 ;
$2187.50 $1498 = 8689.50, Ans.
Ex. 12. $3600x.95=$3420;
$2700x1.03=82781 ; $3420 $2781 =$639,-4/w
Ex. 13. $12000 x.H5 = $l 740, Ans.
(262, page 219.)
Ex. 2. $6300 = 1.05 = $6000 = 60 shares, Ans.
Ex. 3. $6187.50-r-.90 = $6875, Ans.
88 PERCENTAGE.
Ex. 4. $53500-^1.07 = $50000, Ans.
Ex. 5. $1150-^.92 = $1250, nominal amount;
$1250-=-50 = $25, Ans.
(265, page 222.)
Ex. 2. $867-7-1.02 =$850, stock purchased ;
$850 x .06 = $51, income, Ans.
Ex. 3. $8428^.9S = $8600, stock purchased;
$8600 x .05 = $430, income, Ans.
Ex.4. 1.04f +.00| = 1.05;
$10500-f-1.05 = $]QOOO, Ans.
Ex. 5. .87 + .OOi = .875 ;
$47 95 -4-. 8 75 = $54 80, stock purchased ;
$5480 x. 05 = $274, income, Ans.
Ex.6. 1.07-1 + . 00^ = 1.08; .96 + .00=.97 ;
$10476-7- 1.08 = $9700, purchase of 6 ? s ;
$10476-r-.97 = $10800, " " 5-20's :
$9700 x .06 =$582, income from 6's ;
$10800 x.05 = $540, " " 5-20's ;
$42, Ans.
Ex. 7. l.OSf + .00-] = 1.09;
$125 x 109 = $13625;
$13625-j-l. 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-K05 =$37200, stock required.
$37200 x .98^ = $36642, investment, Ans.
Ex. 4. $1 080 -f-. 05 =$21600, stock required ;
$21600 x 1.08^ = $23436, investment, Ans.
GOLD INVESTMENTS. 88<*
Ex. 5. $25000 x.93f =$23437.50, proceeds of 5-20's ;
$960-h. 06 =$16000, 6's required ;
$16000 x 1.09| =$17480, investment in U. S. 6's I
$23437.50 $l7480 = $5957.50,
(267, page 223.)
E*. 2. .06~.87 = .06| = 6ff $, -4s.
Ex/3. .06-M.05 = .05f = 5f %, Ans.
Ex. 4. .06-v-.75 = .08 = 8 %, Ans.
Ex. 5. .06-j-1.08 = .055 = 5f ^, ^4w5.
Ex. 6. .OS-^.QSo^.Ooy^rirSj 1 ^ ^;
.06^-1.09 = .05f u 5 y = 5^\ J ^, better, Ans.
(268, page 224.)
Ex.2. .06-f-.09 = .66 = 66 %, Ans.
Ex. 3. .06-^.05 = 1.20;
1.20 1.00 = .2 = 20% premium, Aug.
Ex. 4. . 05 -f-. 06 = .83 1=83] %, Ans.
Ex. 5. .05-f-.07 = .7lf ;
1.00 .7l| = . 28^ = 28.^ ^, Ans.
GOLD INVESTMENTS.
(27 O, page 225.)
Ex. 2. $1.47 x 150 = 1367.50, Ans.
Ex. 3. $1.37i x 320.50 = $440.68|, Ans.
Ex.4. $1.33x2500 = $3325, Ans.
Ex. 5. $8000 x .05 = $400, income in gold ;
$1.38x400 = $552, " "currency,^*.
Ex. 6. $9500 x .06 = $570, income in gold ;
$1.40x570 = $798, " " currency. Ans.
886
PERCENTAGE.
Ex. 7. $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^-$1.38^ = $72 2 5 ^, Ans.
Ex.3. $1.00-f-8l.45 = .68|| ;
1.00 .68|f =.$1^ = 31^ %, Ans. to first.
$1.00-7-$1.47=.68 T | 7 ;
1.00 .68 T | 7 = .31 r|| 31{|| %, Ans. to second.
1.00-.51fy 9 T = .48||f = 48f f %, Ans. to third.
$1.00-r-$2.Q5=.35f ;
1.00 .35 f = .64f = 64| %. Ans. to fourth.
Ex. 4. $4181-f-$1.48 = $2825, Ans.
Ex. 5. $.24-j-$1.60=$.15, Ans.
Ex. 6. $5900 x .90 = $5310, 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-T-$1.65=$460, Ans.
Ex.9. $1.008.30=3.33] =333| %, Ans. to first.
$1.00-8.45 = 2.22| =222! ^ *' " second.
$1.00 $.54 = 1. 85W\ = 185 3 5 T %, " "third.
$1.00 $.60 = 1.66| =166f %, " " fourth.
$1.00-$.74 = 1.35A=135 3 5 7 ^, fifth.
Ex. 10. $126-r-$1.40=$90, value of currency in gold;
$90-=-83.50 = 25| yards, Ans.
Ex. 11. $11.75-h$1. 50=87.83], purch. price of flour,gold;
8l0.35-7-$1.35 = 87.66|, selling " " " "
$7.83] $7.66| = 8.16f, loss in gold on one barrel ;
$.16f x 300 = $50, entire loss, Ans.
PROFIT AND 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 = $3730.30, inc. from 7.30's incur. ;
$3730.30+41.46 = $2555, " " 7.30's in gold;
$51100 x 1.04 =$53144, proceeds of 7.30's ;
$53144^$1.46 = $36400, gold purchased;
$36400-h$.70 = $52000, 10-40's purchased;
$52000 X .05 = $2600, inc. from 10-40's in gold;
$2600 $2555 $45, income increased, Ans.
PROFIT A^D LOSS.
(27 3, page 228.)
Ex. 2. $84.80 x .125=$10.60, Ans.
Ex. 3. $1.15 x 500 =$5 75, cost of wheat;
$575 X.I 6f =$95.83^, 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 $.0825 x 230 x3 = $56. 925, cost; .18 T 2 T = T 2 T ;
$56.925 X T 2 T =$10.35, gain ;
$56.925 +$10.35 = $67. 275; 230 lb. x 3 = 690 Ibs.
$67.275+690 = 1.0975, selling price, Ans.
Ex. 7. $.625 x 3840 X .375 =$900, Ans.
Or$.625=$|; .375=|; 3 - 8 T 4 - ^x f x | = $900, Ans.
90 PERCENTAGE.
Ex. 8. $4720 x .125 $590, loss in the bargain ;
$4720 $590=14130 ; $4130 x .15=$619.50 loss in bad debts
$590 + $619.50=$1209.50, Ans.
Ex. 9. 1 +.225 = 1.225, his per cent, after 1 year ;
1.225 x 1.30 = 1.5925, his per cent, after 2 years ;
1.5925 x i =1.3270$, his per cent, after 3 years
$3000 x.3270f =$981.25, Ans.
(274, page 229.)
Ex. 2. $330 $275 = $55, gain ;
$554-$275=.20, Ans.
Ex. 3. $.75 $.60=$.15, gain ;
$.15-^-$.60=.25, Ans.
$114.885
Ex. 5. $.095 $.08=$.015, gain on 1 lb. ,
$.015^-$.08=.1875 = 18f per cent., Ans.
Ex. 6. $42 x 150=$6300 ; $6300 $5400 = $900
$900-7-$6300=.14f, Ans.
Ex. 7. $25 $15=$10; $10+-$25=.40, Ans.
Ex. 8. $.25 x 20 =$5.00, received per ream ;
$5.00 $2.00 = $3.00, gain per ream ;
$3.00^$2.00 = 1.50 = 150 per cent., Ans.
Ex. 9. If i sells for its cost, 1 sells for = | its cost
| 1 = ^, gain on 1, =.50 = 50 per cent., Ans.
Ex. 10. -i-f =7 ; or the whole would be sold for of ita
cost ; hence f of the cost was lost. And,
f = 37^ per cent., Ans.
Ex. 11. One peck is gained on 3 pecks ; hence
l-f-3=.33i per cent., Ans.
PROFIT AND LOS3. 91
Ex. 12. 87 per cent., Ans.
Ex. 13. 342 Ib. @$.08 $27.36
378 Ib. @ .081= 32.13
$59.49 sold for ;
720 Ib. x .07 = 50.40 cost ;
$9.09 gair. ;
$9.09-;-$50.40=.182V per cent, Ans.
ET. 14. $1.60 $1.25 = 8.35, gain per gal. ;
8.35-h8l.25=.28, gain per cent. ;
$1.25 x 63 x 2 x .28 =$44.10, whole gain, Ans.
Ex. 15. $.66 8.55 = 8.11, gain per bushel on the corn ;
8.11-J-8.55 =.20, gain per cent, on the corn.
81.37581.10=8.275, gain per bushel on the wheat j
8.275-r-8l.lO=.25, gain per cent, on the wheat;
.25 .20 = . 05 per cent, on the wheat, Ans.
(275, page 231.)
$140x1.25
1250
Ex. 3. $.30 x 1.16| X 1.33i=$.46f, Ans.
Ex. 4. $.105 x 1.171=8.1232, Ans.
Ex. 5. 1.00 .15 = .85
$.621 x. 85=$ .531 ^
1.20 x.85= 1.02 [Ans.
3.875 X. 85= 3.29f )
Ex. 6. 83240 x. 82 = $2656.80, Ans.
Ex.7. $28xl20=$3360; $3360 + $480 = $3840, whole
cost ; $3840 x 1.121=84320 ; $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-^45=$4.13i, Ans.
PERCENTAGE.
Ex.9. 1.18 = f; 23i=y;
$*P x | x ^ x T 3 o =82.85, At*.
(276, page 232.)
Ex. 2. $.08-K80 = $.10, ^ws.
Ex. 3. $6.1 25 -^875 =$7.00, Ans.
Ex. 4. &96-T- 1.28=&75, ^w*.
Ex. 5. 1.182=|f ; 81881 4-|f = $1584, ^in*.
Ex. 6. $69.75
Ex. 7. I 3 /- x -V*- x V/ ^=$86.25,
Ex. 8. $96-^80=$120, cost;
$120xl.l5 = $138, Ans.
Ex.9. 1.125=f; 1.18f = |f;
$570 x }f x f =$426.66|, Ans.
Ex. 10. $24x4=$96 whole, proceeds.
$24x2 = $48; $48^-1.20=$40, cost of 1st 2 bbl.
$48^ .80=$60, cost of 2d 2 bbi.
$40 + $60=$100; $100 -$96 =$4, lost.
Ex. 11. $4900^- 1.40 =$3500 = 3 times what lie began with;
and $3500-7-3 = $1166.66|, Ans.
INSURANCE.
(282, page 234.)
Ex. 2. $750 x .04 = $30, Ans.
Ex. 3. $4572.80 x .025=$114.32, Ans.
Ex. 4. $5700 x .0175 =$99.75, Ans.
TAXES. 93
Ex. 5. $28400 x .035 =$994, Ans.
Ex. 6. $55800 x .028 =$1562.40, premium ;
$55800 -$1562.40=$54237.60, Ans.
Ex. 7. $47500 x T o =$356.25,
Ex. 8. $8000 + $4000=$12000 ;
$12000 x .02=$285, Ans.
Ex. 9. $1.20 x 4000=$4800, worth of wheat ;
$4800 xf =$3200, amount insured for;
$3200 x | x 2l^ = $36, premium ;
$3200 $36=13164, saved by insuring;
$4800 $3164 = 11636, owner's loss, Ans.
Ex.10. $21000x^0 =$168;
$15400 X ? o=$ 96.25; -
$264.25, Ans.
TAXES.
page 236.)
Ex. 3. Property tax =$2 6. 9 5
1 poll .75
$27.70, Ans.
Ex. 4. $.50 x 2981 =$1490.50, poll tax ;
$9190.50 $1490.50=$7700, property tax ;
$7700-r-$l,400,000=.0055, rate of taxation,
$12450 x .0055=$68.475
2 polls =$ 1.00
C's tax $69.475, Ans.
Ex. 5 $5375 x .0055 =$29.5625, Ans.
K. p. 5
94 PERCENTAGE.
Ex. 6. $.625 x 30 = $18.75, poll tax ;
$4342.75 $18.75 =$4324, property tax;
$4324-^$18SOOO = .023, rate of taxation.
$2500 x. 023 + $.625 =$58.1 25, Ans.
Ex. 7. $.30 x 25482 = $7644.60, poll tax ;
$103294.60 $7644.60=$95650, property tax;
$956504-$38260000=.0025, rate of taxation.
$9470 x. 0025 + $.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-f-$1246093.75 = .008, rate of taxation;
$11500 x .008+$1.25 = $93.25, E's tax, Ans.
Ex. 9 $275.57-$98=$l77.57, tax ;
$l77.57-4-3946=$.045, rate per day;
$.045 x 118 x 2 =$10.62, Ans.
CUSTOM HOUSE BUSINESS. 0$
CUSTOM HOUSE BUSINESS.
(3O3, page 239.)
Ex. 2. $-95 x 224=$212.80, value of the silk ,
$212.80 x. 19 =$40.432, Ans.
Ex. 3. $.54 x 31.5 x 50 =$850.50, gross value ;
deduct $850.50 x .02= 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 coffee ;
$3018.75 x .15=$452.81{, 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.
WJ PERCENTAGE.
SIMPLE INTEREST.
(311, page 241.).
Ex. 3. $45.92, Ans. Ex. 8. $693.83+, Ant.
Ex. 11. $607.50, Ans. Ex. 15. $440.625, Ans.
Ex. 17. $605.70 + int. for 3 yr.=$751.068, Art*.
(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. 3 mo. 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.
\ i. 28. $350. r 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 + , Ans.
PARTIAL PAYMENTS. I f
PARTIAL PAYMENTS.
(314, page 249.)
Rx. 3. Amt.of note to Nov. 12, 1858, (4 mo. 22 da.)$535.27 +
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 + ,
Ans
Ex 4. Amt. of note, Jan. 1, 1860, (7 mo. 24 da.) $3136.50
Sum of payments to this 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+,
Ans
Ex. 5. 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 $459.1 8 4
Amt Sept. 15, 1853, (4 mo. 11 da.) $467.53 + .
5 An&
98 PERCENTAGE.
Ex. 6. Interest commenced Aug. 2, 1860.
Amt. of note, May 6, 1861, (9 mo. 4 da.),. $192.988 +
Payment, 50
New Principal, $142.988 +
Amt. Aug. 26, 1862, (15 mo. 20 da.) ....$154.188 +
Ex. 7. 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 +
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, v 75.00
New Principal, $282. 19 -f
PARTIAL PAYMENTS. .99
Ami. from Mar. 11, 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 +
(316, page 251.)
Ex. 1. Amt. of Principal, Jan. 1, 1859,
(2 yr. 8 mo. 20 da.) $698.00
Arnt. 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. of 3d endorsement, (for 7 mo.) 191.475 592.494
Ans. $105.50 +
(317, 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, 1860,
(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 4th pay't, Nov. 4, 1860, (1 mo.
6 da.) 60.36
Ans. $215.33.
100 PERCENTAGE.
(318, 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
$135
1st installment draws int. 3 yr. 6 mo.
2d " " 2 yr. 6 mo.
3d " " " 1 yr. 6 mo.
4th " " " 6 mo.
Int. of $30 for 8 yr. mo $14.40
Principal, $500.00
Ans. $649.40
PROBLEMS IN INTEREST.
(32O, page 253.)
Ex. 2. Int. of $1 for 6 yr. 3 mo. at 6 per cent., is $.395 ;
$28.125-i-.375 = $75, Ans.
Ex. 3. Int. of $1 for 4 mo. 18 da. at 4 per cent., $.01 5 ;
$9.204- .015i = $600, Ans.
Ex. 4. $1260-5- .07 = $18000, Ans.
Ex. 5. $33704-.10:=$33700, Ans.
(321, page 254.)
Ex. 2. $1 for 8 mo. at 6 per cent., amounts to $1.04 ;
04-1.04 = $630, Ans.
Ex. 3. Amt. of $1 for 5 yr. 5 mo. 9 da. at 5 per cent.,
1.2720| ; $106.855 -M.2720 -$84, Ans.
PROBLEMS IN INTEREST. 101
Ex. 4. Amt. of $1 for 8 yr. 5 mo. at 5 per cent.,
$1.462916+ ;
$1897.545-r-1.462916 + =$1297.09+, Ans.
Ex. 5 Amt. of $1 for 3 yr. 4 mo. at 7 per cent., $1.2 3\ ;
$221.075-i-1.23i = $l79.25, Ans.
Ex. 6. Amt. of $1 for 11 yr. 8 da. at IQi per cent., $2.1 57 ;
$857.54-=-2.157i = $397.50, principal ;
Int. of $397.50 for 11 yr. 8 da., at 101 per cent., =
$460.04, Ans.
(322, page 255.)
Ex. 2. Int. of $500 for 3 yr. at 1 per cent., $15 ;
$45-j-$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-f-$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-i-$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 = 12 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. $1600 x .06=$96 ; $2000-$1600=$400 ;
$400-^$96 = 4 yr.=4 yr. 2 mo., Ans.
Ex.4. $204 x. 07 =$14.28; $217.09-$204=$13.09;
$13.09-j-$14.28 = H yr. = H mo., An*.
102 PERCENTAGE.
Ex. 5. $750 x .06 = 145 ; $942 $750=$192 ;
$45=4 T ^ yr. = 4 yr. 3 mo. 6 da., Ans.
Ex. 6. $200 x. 06=$12;
$200-^112 = 16 | yr.=16 yr. 8 mo., Ans
Ex. 7. $675 x. 05=$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.
500. Given Prin.
Ans. $ 72.45 Compound interest.
Ex. 3. $312.00 Prin. for 1st year.
18.72 Int. " " "
$330.72 Prin. 2d "
19.84 Ink " " u
$350.56 Prin. " 3d "
21.03 Int.
Ans. $371.59 + , Amt. " 3 yeara.
COMPOUND INTEREST. 108
Ex. 4. $250.00 Prin. for 1st. half year.
7.50 Int. " " tt
$257.50 Prin. " 2d
7.72 Int. " " "
$265.22 Prin. " 3d
7.96 Int. " " "
$273.18 Prin. " 4th
8.19 Int. " " "
$281.37 Amt " 2 years.
250.00
Arts. $3 1 .3 7 + Compound interest
Ex. 5. $450.00 Prin. for 1st quarter.
7.87 Int. " " "
$457
.87
Prin.
u
2d
u
8.01
Int.
u
M
u
$465
.88
Prin.
<t
3d'
u
8
.15
Int.
u
u
u
$474
.03
Prin.
a
4th
u
8
.30
Int.
"
u
u
Ans. $482.33 + Arnt. " 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. " "
104 PERCENTAGE.
$297.94 Prin. " 7 mo. 6 da.
10.72 Int. " 7 " 6 "
.66 Arnt. " 4 yr. 7 mo. 6 da.
236.00 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+, Amt." 4 yr. 9 mo. 24 da.
Ex. 9. $120x2.078928=$129.47
Ex. 10. $.10 x 3.86968=$.386968, Ans.
DISCOUNT.
(326, page 259.)
Ex.2. $180 -7- 1.20 =$150, Ans.
Ex. 3. $1315.389-^-l.l75 = $1119.48, Ans.
Ex. 4. $866.038^-1.281i = $675.888 + ,pres. worth. (
$866.038 $675.888 + =$190.15 +, disc't. 1
^.x. 5. $1005 $475=:$530
$475-M.05 =$452.38 +
$530-=-1.075 = 493.02 +
.40 f, Ans.
PROMISCUOUS EXAMPLES. 105
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.
Ans. $ 175, loss.
Ex. 8 $550-j-1.10=:$500, present value of note ;
$480, cash offer ;
Ans. $ 20, gain.
Ex. 9. $51 7.50 -=-1.035 = $ 500
$793.75-=-1.05=$ 750
$2450.426 + , ent.irepres.wonh
$2637.72 $2450.426 + -$187,29 + , Ans.
Ex. 10. $.10 x ||=$ T ^,int. of $1 for 10 mo. at 10 per qent
$130-M T V = $120 ; $130 $120 $10.00, discount,
$130 x T V =$10.33^, interest
Ans. $.83i.
PROMISCUOUS EXAMPLES IN PERCENTAGE.
(Page 260.)
Ex. 1. .02-1- .25 =.2 7 gain per cent, on cost.
sj- x ff = J T V cents, 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.
5*
706 PERCENTAGE.
Ex. 3, $425-+1.03=$412.62 + cash value of sale ;
$425 $25 =$400.00 cost ;
Ans. $12.624, profit.
Ex.4 $.13-^1.04 = $.125 ; $.13 $.125=$.005 ;
$.005 -=-$.125 = .04, Ans.
Ex L. $150-=- 1.25 =$120 cost of one;
$150 + 75= 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 ;
$JL_I_ OJL x } x f =$3750, Ans.
Ex. 7. 1.00-^.07 = 14f years, Ans.
Ex. 8. 3 yr. 4 mo.=3i yr. ; .12 x 3i=.41f, whole rate
of gain; $5000-f-.41f =$12000, capital ;
$12000 x f =$7500, A\ \
$12000 xf = $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=.91f ;
$1100-r-.91f =$1200, Ans.
Ex. 11. $667+-1.04=$641.346 + , cash value of goods;
600 x 1.06= 636. pres. valuation of goods ,
True gain, $5.346+, Ans.
Ex. 12. $18xi=$6, profits; $18-$6=$12, cost;
$6-^12=50 per cent., profit, Ans.
Ex. 13. If f sell for f of cost, the whole would sell for f x jf
^f the cost, which is IJf times cost. Hence if = .40f, is the
gain per cent.
PROMISCUOUS EXAMPLES 107
Ex. 14. $1.30, received for lumber originally worth $1.00 ;
$1.06f, valuation of ditto, after 16 mo. int. accrues ;
$.231 gain on $1.06|;
$,23i-^-$1.06f =.21f,- Ans.
Ex.15. 1-1 = 4; | x |=i, Ans.
Ex. 16. $1.121 x 728=$819, expended in wheat;
.60 x .30 x .50=.09 ; $819-^.09=$9100,in bank ;
LOO Uo=.40 ; $9100 x .40=$3640, Ans.
Ex. 17. 6 x 5=30 sq. yd., area to be covered ; 4 per cent.
is ^, and 5 per cent, is ^V ; hence every yard purchased will
be, after shrinking, || of 1 yd. long, and if off yd. |i yd.
wide, and will contain f x f sq. yd. Therefore, as man/
yards must be purchased as f 1 x f | is contained times in 30.
V-xff x=43if yd., An*.
Ex. 18. 1.00 per cent.=B's money;
1.28 " =A's "
.28-j-l.28-.21i, Ans.
Ex. 19. $1200-=-.12 = $10000, f of his capital;
$10000 -=-f =$25000, whole capital ;
$25000 x | x T f^=$V50, loss on f of the capital ;
$1200-$750 = $450, gain.
Ex. 20. Amt. of $1 for 3 yr. 9 mo. at 10 per cent, $1.375 ;
$1933.25-r-1.375=r$1406= of C's money.
$1406x2=$2812, C's, )
$2812 x i=$421f; D's, J
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 =$445.275, proceeds.
Ex. 2. Int. of $368 at 7 per cent, for 93 da. =$6.654 +,disc't;
! $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, discV,
$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., terra of discount ;
Int. of $500, at 6 per cent., for 36 da., 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 mo. 3 da. at 6 pel
cent, is $772.875 ;
Bank discount of $772.875 for 30 da., at 10 per cent
is $6.440+ ; $772.875 $6.440+ =
$766.434?-, proceeds.
EXCHANGE. 109
(337, page 2G6.)
Ex. 2. $680-^.9895 = $687.2154-, Ans.
Ex. 3. $1000-~.9870 = $1013.085 + , Ans.
Ex. 4. $500-K9935 =$503.22 +, Ans.
Ex. 5 $1256-h.9644j=$1302.341+, Ans.
EXCHANGE. *
(35O, page 269.)
Ex. 3. $1000 x 1.03 =$1030, Ans.
Ex. 4. $400 x 1.0075=1403, Ans.
Ex. 5. 8530 x 1.0275=$544.575, cost of draft ;
20. transportation ;
Ans. $564.575.
Ex. 6. $1 $.01225=$.98775, proceeds of $1 at b'k disc't ,
Add .02 premium ;
$1.00775, cost of exchange for $1 ;
$800 x 1.00775 =$806.20, Ans.
Ex. 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-hl.015=$240, Ans.
Ex. 3. $79.20 -v-.99 =$80, Ans.
Ex, 4. $1 1.0105=$ .9895
Add .02
$1.0095, draft for $1 ;
$282.66-M.0095=$280, Ans.
Ex. 5. $1 .0055 = 1.9945
Subtract .0125
$.982, draft for $1 ;
$240-+.982 =$244.399 + , draft bot. for $240;
$240 ($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 v
$ .98775
.0075 premium.
$3000-7- .99525 = $361 7.1 81 +, 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 +
83617.181 $3608.932 =$8.24 + , loss, Ans.
EQUATION OF PAYMENTS. ill
EQUATION OF PAYMENTS.
(356, page 273.)
Ex. 2. $700x20=$14000
400x30= 12000
700x40= 28000
$1800 $54000
54000-^-1800=30 da. average credit
Sept. 25+30 da.=0ct. 25, equated time.
Ex. 3. $250x4 = $1000
750x2= 1500
500x7= 3500
$1500 $6000
6000-^-1500=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-^-28=4 da.
Monday + 4 da. = Friday, Ans.
Ex. 5 $650 x 4=$2600
725 x 8= 5800
500x12= 6000
$1875 $14400
14400-7-1875 = 7.68 mo.=7 mo. 20 da.
Jan. 1+7 mo. 20 da. = Aug. 21, Ans.
112
AVERAGING ACCOUNTS.
Ex. 2.
(362, page 276.
Due.
da.
Items.
Prod
Jan. 1
150
" 16
15
200
3000
Feb. 4
34
100
3400
March 3
61
160
9760
610
16160
16160-j-610 =
Jan. 1 + 26 da. = Jan. 27, Ans.
Ex. 3.
Due.
da.
Items.
Prod,
March 1
300
April 4
Aug. 18
Aug. 8
34
170
160
240
100
400
8160
17000
64000
1040
89160
89160-M040 = 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
3242
54938
54938-^3242 = 17 da.
June 1 + 17 da. = June -18, Ans.
AVERAGING ACCOUNTS.
133
Ex. 5.
Due.
da.
35
48
99
Items.
Prod.
Jan. 16
Feb. 20
March 4
April 24
536.78
425.36
259.25
786.36
14887.60
12444.00
77849.64
2007.75
105181.24
Ans.
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
j 139650-^2355 = 59 da.
' ( Apr. 1 + 59 da, = May 30.
Dr.
Ex. 2.
(363, page 278.)
dr.
Due.
da.
Items.
Prod.
Due.
da.
Items.
I'rod.
Jan. 1
Feb. 4
" 20
34
50
448
364
232
12376
11600
Jan. 20
Feb. 16
" 25
19
40
55
560
264
900
10640
12144
49500
1044
23976
Balances
1724
1044
72284
23976
680
48308
Ans. \
48308-^680 = 71 da.
Jan. 1 + 71 da. = March 13
114
RATIO.
Ex. 3.
Dr.
O.
Due.
da.
Items.
Prod.
Due.
da.
Items.
Prod.
Apr. 1
145.86
May 11
40
11.00
440.00
June 12
72
37.48
2698.56
July 12
102
15.00
1530.00
Sept. 3
155
12.25
1898.75
Oct. 12
194
82.00
15908.00
Oct. 4
186
66.48
12365.28
262.07
16962.59
108.00
17878.00
108.00
16962.55
Bal acct.
154.07
Bal. Prod.
915.41
Ans.
915.41-5-154.07=6 da.
\ Apr. 16 da. = March 26, 1858.
Ex. 2.
Ex. 4.
Ex. 6.
Ex. 8.
Ex. 10.
RATIO.
(379, page 281.)
= 7, Ans.
=4|, Ans.
Ex. 12. 3 gal. = 24 pt. ; 2 qt. 1 pt.=5 pt. ;
Ans.
Ex.3. & = i,Ans.
Ex. 5. ff = 51, Ans.
Ex. 7. J T -x^ T = 12, Ans.
Ex.9. fxf=f, Ans.
Ex. 11. 0-x T 3 o=5, Ana.
Ex. 13. 8 s. 6 d. = 8.5 s. ; |:f =|f =
Ex. 14. /|=AV = 1 V, Ans.
Ex. 15. 19 Ib. 5 oz. 8 pwt.=4668 pwt. ; 25 Ibs. 11 o& 4
pwt. = 6224 pwt. .a||i =
Ex. 16. U= ^^ Ex. 17. f x f = T 9 4,
Ex. 18. ji x /ff=i ^ ws - ^x- 19. 16-r- 2f = 7,
SIMPLE PROPORTION.
Ex. 20. 14.5 x 3 = 43.5, Ans. Ex. 21. | x ^=
Ex. 22. f x T V, ^s.
11, Ann.
Ex.1.
20
PROPORTION.
(388, page 383.)
= 120,^. Ex.2.
42
= 5, Ans.
Ex. 3. - =6, Ans. Ex. 4. y=12, Ans.
201.75x48 yd.
Ex.5. > == 144 yd., Ans.
Ex. 6. 3 Ib. 12 oz. = 60 oz.
10.50 x 60 oz.
3.50
= 180 oz. = ll Ib. 4 oz.,
Ex. 7. 8 bu. 2 pk. = 34 pk. ; 76 bu. 2 pk. = 306 pk.
$38.25x34
Ex. 8. V 1 x H a x rf 3 =
Ex. 9. x x =
Ex. 10. T V x I x f f
SIMPLE PROPORTION.
(399, page 287.)
Ex. 1. 48 Cd. : 20 Cd. :: $120 : ( )
$120x20
( ) = =$50, Ans.
Or. $120 x = $50, Ans.
r ^ ^>J
&> <*
*
*4
%ri
116 PROPORTION.
Ex. 2. 6 bu. : 75 bn. : : $4.75 : ( )
( )= !i!l>iL 5 = $ 5 9.37i, An,.
\ Or, $4.75 x Y L =$59.37, Ans.
Ex. 3. $3i : $50 : :. 8 yd. : ( )
( ) = 0^ =sll4f yd, Ans.
Ex. 4. 12 : 20 : : 42 bu. : ( )
42 bu. x 20
( ) ---- =70 bu., Ans,
Or, 42 bu. x H ' 70 bu " Ans.
Ex. 5. $.75 : $9.00 : : 7 Ib. : ( )
900 x 7 Ib.
( )=- - =84lb.,^w.
Ex. 6. 3 Ib. 12 oz. : 11 Ib. 4 oz. : : $3.50 : ( )
60 oz. : 180 oz. :: $3.50 : ( )
Ex. 7. 1 ft. 6 in. : 75 ft. : : 3 ft. 8 in. : ( )
1| ft. : 75 ft. : : 3| : ( )
ft. x f-1831 ft.-183 ft. 4 in., Ant.
Ex. 8. $2.75 x VV=$ 19 - 64 7> Ans.
Ex. 9. $13.32 : $51.06 : : 12 bu. : ( )
51.06 x 12 bu.
Ex. 10. 15hhd. = 945 gal.
945 gal : 28.5 gal. : : $236.25 : ( )
SIMPLE PROPORTION. 117
Ex. 11. 6 mo. : 11 mo. :: 7 bbl. : ( )
11x7 bbl.
( )= --- = 12 bbl., Ans.
Ex. 12. 5 12 s. : 44 16 s. : : 9 yd. : ( )
x 9 yd. x 896
- -- =
Ex. 13. $3100x^W=$ 310 Ans -
Ex. 14. 100 Ibs. coffee=:100x } = 160 Ibs. sugar;
2 : 160 : : $.25 : ( )
$.25 x 160
( )= - - - =$20,
Ex. 15. 13 10' 35" : 360 : : 1 da. : ( )
47435" : 1296000" : : 1 da. : ( )
( ) J-|^IAIJL da. = 27 da. 7 h. 43 mm. 6.06 + sec., Ans.
Ex. 16. 8f : 131 : : 84.20 : ( )
x y- X ,& = $6.48, Ans.
Ex. 17. 6| d. : 10 6 s. 8 d. : : If yd. : ( )
( ) =2.0.9. x yd. x 2 4 j = 6 94| yds., Ans.
Ex. 18. 121 cw t. : 48f cwt. : : $42i : ( )
= $-L x -3-i x 2 2 $163.50 + , Ans.
Ex.19. $lf : $317.23 ::8|lb.: ( )
( ) = 3l7.23x8.4 Ib.x4-1522.7+ lb.
15 cwt. 22.7+ lb., Ans.
Ex. 20. $1561 : $95.75 : : 15f bu. : ( )
( ) fljyii x ifi bu. | y ^9.575 bu.=
9 bu. 2 pk. 2 1 qt., Ans.
Ex.21. | bar. : $ bar. :: fy$ : ( )
K. P.
118
PROPORTION
Ex. 22. "4 rd. : llf rd. : : A. : ( )
( ) = a A. x V- x i=2^ A.=2 A. 28 rd.,
Ex. 23. 13 cwt. : 12 cwt. : : $421 : ( )
( )=*4*. x -V 2 - x T V =$39, Ans.
Ex. 24. 16 oz. : 12 oz. : : $28 : ( )
Ex. 25. 16 oz. 14}i oz.^lfV oz., cheat in 16 oa.
16 oz. : 1 T 5 F oz. : : $30 : ( )
Ex. 26. 1 yr. 6 mo. : 3 yr. 9 mo. : : $750 : ( )
, An,
Ex. 27. 10 mo. x Yjo 30 mo ->
Ex. 28. $25 : $30 : : $28 : ( )
( )=$5 T 8- x V- X V $34.10, Ans.
Ex. 29. 1 yr. 4 mo. = l = f yr. ;
I yr. x f| = 2 yr.r=2 yr. 9 mo. 10 da. Am
COMPOUND PROPORTION.
(4O1, page 292.)
Ex 1. 16
50
16
50
128
90
) = 72 bu., Am.
COMPOUND PROPORTION.
119
Ex. 2.
3 i <
12 J '
: i<H :: 1^0: 360
120
10
u,
5
360
'6
12
54
= 10f days, Ans
Ex.3.
Ex. 4
450) ()
12 V : 9
12 8
1 : 1
12
12
450
= 900, ATM
Ex. 5. 1200 | ( ) )
H : H :<
Ex. 6. 6 ) ( )
9V : 12
9 ) 12
500 : 960 500
36 : 48
960
1200
i 7 23040
36
12
12
48
( )=6 men, Ans.
Ex.7. 4
8} ]
15
'SI
120 PROPORTION.
Ex. 8. 41 ) 4| 1
6 V : 9 V : : 540 : 600
20) ( )]
H x f x f x V x fV x x T J T =-4A= 14^, AM.
6.75 73 3 2 5 .1478.25
X Y X 2 X 5 X 7 =$ ^8~
6
417.6x5x6
Ex. 11. 6 ) 9 ) 22.5 ) 45
2.5 I :( ) [ ::17.3 : 34.5
12.3 ) 8.2 ) 10.25 ) 12.3
6 X 2.5 x 12.3 x 45 x 34.6 x 12.3 _
9 x 8.2 x 22.5 x 17.3 x 1CX25~ ays> W
Ex. 12. 54 } 75
^ X ? / X -^5 X -j = *.
Ex. 13. 24 ^ 217 j 33
189 [: 5i V : : 5f } : 3|
14) ( ) ) 31
PARTNERSHIP. 121
PARTNERSHIP.
(4O7 5 page 295.)
Ex. 2, $ 8000 T 8__<LP- i, A's fraction.
12000 if=: T 3-, B's "
20000 =i C's "
$40000
$1680 x =$336, A's; $1680 xtV=$ 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 "
Ex. 4. $20000 : $13654 :: $3060 : ( )=$2089.062
$20000 : $13654 :: $1530 : ( )=$1044.531
Ex. 5. 16 + 24 + 28 + 36 = 104
$13x T Y T =$2, A pays; $l3xftt=$3, B pays;
$13 x y 2JL-$3.50, C pays ; $13 x ^=$4.50, D pays.
Ex. 6. 14 + 6 + 12 = 32 shares.
$2240 x if = $980. Captain's share ;
$2240 x ^=$420, Mate's share ;
$2240 x if =$840, divided among the sailors,
$840-7- 12 =$70, each sailor's share.
Ex. 7 $3475.60 $2512=$963.60, lost to the owners ;
$963.60 x|=$120.45, A's
$963.60 x i = $240.90, B's [ Ans.
$963.60 xf = $602.25, C's
Proof 1 =$963.60
1 22 PARTNERSHIP.
Ex 8. 6, C's proportional share.
4, D's
6 f 4=10; 10x1= 2, E's " "
$2571.24 x W=$1285.62, C's ;
$2571.24 x yV =$ 857.08, D's ; Ans.
$2571.24 XfV=$ 428
.62, s ; \
.08, D's ; (
.54, E's;)
Ex. 9 $7500-($2000 + $2800.75 + $1685.25) = $1014,
D's gain ;
gain. cap. gain. cap.
$1014 : $3042 :: $2000 : ( )=$6000 A;)
$1014 : $3042 :: $2800.75 , ( )=$8402.25, B; ( Ans t
$1014 : $3042 :: $1685.25 : ( )=$5055.75, C ; )
(4O8, page 297.)
Ex. 2. $250 x 6 =$1500, B's product ;
275x8= 2200, C's "
450x4= 1800, D's "
$5500
$825 x || =$225, B's share of gain ;
825 x f|= 330, C's " " "
825x-]-|= 270, D's u " "
Ex.3. $1000 x 8=$ 8000 $1500 x 4=$ 600C
1600x10= 16000 1200x14= 16800
A's product, $24000 B's product, $22800
$2 4000 + $22800 = $46800, sura 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 bushe1s-M=93 busliels to be divided.
62 : 20 : : 93 : ( )=30 bu., A's; ^
62 : 18 :: 93 : ( )=27 " B's ; ) Am.
62 : 24 :: 93 : ( )=36 " C's; )
Ex. 5.
From Jan. 1, 1856, to Apr. 1, 1858, is 27 mo., Gal lap's time ;
" Mar. 1, 1856, " Apr. 1, 1858, " 25 " Decker's "
" July 1, 1856, u Apr. 1, 1858, " 21 " Newman's"
$3000 x 27=$81000, Gallup's product ;
2000 x 25= 50000, Decker's
1800x21= 37800, Newman's "
$168800, sum of products.
$168800 : $81000 : : $4388.80 : ( )=$2106, Gallup's gain;
$168800 : $50000 : : $4388.80 : ( )=$1300, Decker's "
$168800 : $37800 : : $4388.80 : ( ) =$982.80, Newman's u
Ex. 6.
-T- 8 =$70, A's monthly profit ;
-M0=$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 : ( )=$2986.66|, 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 \- of B's,
and to i of C's, and the proportion of the shares will corres-
pond to the conditions.
124
ANALYSIS.
2+3+4=9
$117 xf=$26, A's gain
$117 x =$39, B's "
$mx$ = |52, C's "
If we now divide the proportional shares of tht 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 same
proportion as the respective shares of the. capital ;
2-+3=|, A's proportional share of capital ;
3-f-5=, B's " " " "
4-5-7=4, C ' s " " " "
Iff* sum of proportional shares.
: a' :: $1930 :( )= $700, A's capital;
: f : : $1930 : ( )=$630, B's "
: A : : $1930 :( )= $6 00, C's "
ANALYSTS.
(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.
Ex. 6. We multiply 19 (pence) by 28
for the cost of the butter (in pence), and
iivide by 7 times 12 (pence) the price of
the tea.
56
7
3
126
4
$27, Ans>
28
19
T9
ANALYSIS.
125
Ex 7. 10s. 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
$648=$108, Ana.
Ex.8. 2s. 6d.-30d. Then2x3x
7 x 3 Oncost in pence. Divide by 60 to
reduce to Decimal currency.
60
2
3
7
30
$21, Ans.
Ex. 9. 20 x 3 x 12= 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.
75
20
3
12
5 | 48
9|, Ans.
Ex. 10.
96
160
18
90
160
24
Cost, $30 sold for $42.66f.
$42.66f-$30=*$12.66f, Ans.
Ex. 11. 431 = V ; 10 s. 6 d.=126 d. ;
8 s. 3 d. = 99 d.
Ex. 12. 9s. 4 d. = 112 d. ;
$1=96 d., Mich, currency
11
96
87
126
609
Ans.
300
112
$350, Am
."56 ANALYSIS.
Ex. 16. Dividing 128 by 16, we ob-
tain what 1 horse will consume in 50 IQ
days ; dividing this result by 50, we ob- 50
128
5
90
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. 17. Divide 4f (=/) b 7 10 * 3 | u
i'=V) to find what amount of wood $1 21 j 2
4 i QQ
will buy ; then multiply by 24ff=y) to '
find how much $24 will buy. U *
Ex. 18. 52 x 3 x J-f *= the money
given for the cloth. Divide this result by
65, the number of yards, to find the price
3
65
52
3
100
Ans. 80 cents.
Ex. 19. A shadow of 1 foot will require an object of 3
feet in length ; and a shadow of 46| feet will require an object
46 1 times of 3 feet in length ; hence
x } x J-f A 28 feet > -A ns -
Ex. 20.
8 sheep x 7|= 60 sheep for 1 mo., A's use of the pasture ;
12 " x4| = 50 " " " ,B's " " "
15 " x6f=:100 " " " , C's " " * .
210 " " " , total " " "
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 8 T $18> A. must pay ;
$63 x |f = 130, C " u
ANALYSIS.
127
Ex. 21. 1 bu. oats = dollars ;
1 bu. rye =-'/ bu. oats = J / x | dollars ;
1 bu. wheat= y bu. rye=Y x J / x e dollars.
If we divide $30 by the price of 1 bushel of wheat, we shall
have the number of bushels which $30 will buy ; hence
3 T n - x f x fV x T 7 o =15 bu. Ans.
Ex. 22. If $480 gain $84 in any time,
tc gain $21 in the same time will require
fJ of $480 ; and if fi of $480 gain $21
in 30 mo., to gain the same amount in 15
mo. will require ff of f 1 of $480.
| 480
$ 240 ' AnS '
Ex. 23. 28 x = 21 sq. yd., contents of the 28 yd. ;
21-f-f =31 yd. of that which is f yd. wide.
Ex. 24. If 130 miles require 3 days, 390
miles will require ff of 3 days ; and if 14
hours a day require 3 days, 7 hours a day will
require J T 4 - of 3 days.
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.
3
390
14
Ex. 26.
A's age + B's age=
C's age=2 T 1 times this sum
And the sum of all their ages, or 93
Hence, 93-j-7f = 12 yr., A's age ;
12x11=18 " B's "
12x51 = 63 " C's "
=2| times A's;
=5| " "
Ans.
J -8 ANALYSIS.
Ex. 27.
1 day of = T k - da. of D. ;
1 day of B= '/ da. of C = V 2 x T ^ da. of D. ;
1 day of A= da of B =f x V 2 x A da. of D ;
hence 5 days of A=f x x y x T 8 T da. of D=8 da.
of D, Ans.
Ex. 33. If the cost of 12 oranges and - L - cta -
iO lemons is 54 cents, the cost of one half ^ jo 54
the lot, or 6 oranges and 5 lemons, will be -g r ^7
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.
18
20
1000
Ex. 34. 18x20x1000 = the whole
number of ounces of provisions ; and since
this quantity is to supply 1600 men 30 16QQ
days, we divide by 30 to find the daily al- '
lowance for the army, and this result by ^ ns ' H oz<
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 times
as many bushels as the smaller, the two together will contain
three times the number in the smaller ; hence
3 times the smaller =60
The smaller =20
The larger =20 x 2=40, Ans.
ANALYSIS.
12f
Ex. 36, We take the difference of two numbers from the
greater to find the less. The greater diminished by 1 of the
greater equals the less, which must be f of the greater. And
if the less be | of the greater, their sum, 20, is 1| times ths
greater. Hence we have
20 -^- If = 12, the greater, Ans.
tix. 37. 1 day of C = da. of B ;
1 day of A=f da. of C= f x f da. of B ; hence
6 days of A =6 x f x f da. of B ; and,
6 weeks of A =6 xf x wk. of B.
6 x f x f wk. = 1 1 1 wk., Ans.
Ex. 38. 36 x 1 j=45 sq. yd. to be lined.
45 yd.-j- = 60 yds., Ans.
Ex. 39. 80 x 3i 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.
4
104
10
80
13
96
24, Ans.
Ex. 40. If the time past since noon is equal to 1
to midnight, both intervals, or 12 hours, must be
time to midnight ; hence
12 h.-^-li =10 h. to midnight.
12 h. 10 h.= 2 h. P. M., Ans.
Ex. 41 She bought one half for \ cent apiece ;
And the other half for \ cent apiece.
(^+) -7-2 = ^, average buying price ;
3-^-5 =|, selling price.
f ^ =J-i, gain on one peach.
55 -f 11 = 300, Ans.
of the time
times the
1 30 ANALYSIS.
Ex. 42. A can build the boat in 18x10 = 1 80 hours ;
B " " " " " 9 x 8= 72 "
A " " T { of the boat in an hour ;
Bu i a a a a
72
A and B can build yio' + TV s~lo f the boat m
an hour ;
A and B can build -g^- Q x 6 = g- 7 - of the boat in a zlay
of 6 hours.
It will, therefore, require as many days as 7 is con-
tained times in 60 ; hence
60 -f- 7 =8| days, Ans.
Ex. 43. He spent at first , and he had left. He then
spent i of this , and he had f of this \ left ; hence | of \ =
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 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 if} of 16.25
Ib. ; hence
16.25 Ib.x iff = 6.25 = 61 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-
bred every day, he would have received $2.50 x 20 =$50.
$50 $43 =$7, what he lost by being idle ;
and $7-^-$3.50 =2, the number of idle days. Hence
20 2 = 18, the number of days he labored, Ans.
Ex. 47. A, B, and C perform yL- in an hour
A and B " T V " "
Hence, C performs r V rV^T? " "
Again, A, B, and C perform ^ "
A and C < T V " "
Hunce, B performs ^j TV=sV " *'
Therefore, B and C perform T ^ + -U y i T
And in 9^ hours they will perform T 7 x y =,
ALLIGATION.
ALLIGATION.
(414, page 305.)
Ex. 2. $1.00xl2
1.50 x 5= 7.50
20 )$19.50
.975, Ans.
Ex. 3. $1.25x52 =
13
65 65
Mixture, $1 per gal.
65 x 32 x $.061=^130, receipts ; $130 $65=$65, gain.
Ex. 4.
8x10= 80
Ex. 5. 12 x 7i= 90
9x12 = 108
. lOix 8 = 84
11x16 = 176
11 X 9 = 99
38 364
10 x 101 = 105
88x10 = 380
35 378
Ans. 16 cents.
378x11=567
Ans. 567-^-35 = 161
cts.
Ex.6.
50 x 4=200 Ibs. ; $.13 x 200 = $26.00
4CX 10 = 400 Ibs.; .10x400= 40.00
25x24 = 600 Ibs.; .07x600 42.00
1200)$108.00
$.09 average cost per Ih
f.095-$.09=$.005 ; 1200 x $.005 =$6, Ans.
ALLIGATION.
(416, page 310.)
Ex. 3. 12
10
11
14
1
1
2
Ans. 1 Ib. at 10, and 2 Ibs. at 11 and 14 cents.
1 I
Ex. 4. 90
120
TfV
Ans. I gal. of water to 3 gal. of wine.
Ex.5. 275^
f200
250
300
[400
1
Ans. 5 of the 1st kind, 1 of each of the 2d and 3d, and 3 of
the 4th.
(80
To"
6
6
84
)
6
6
87
*
4
4
94
*
3
3
96
*
}
10
6
16
Ans. 6 of the first 2 kinds, 4 of the third, 3 of the fourth,
and 16 of the fifth.
(417, page 311.)
Ex. 2. 80
Ans. 20 Ibs. of each of the first three kinds, at / > DO Ibs.
the fourth.
40
T'O
2
2
20
60
.
A
2
2
20
75
i
2
2
20
90
rV
T'O
TV
8
4
1
13
130
Ex. 3. 4
24
24
72
Ans. 24 at $3, and 72 at $5.
ALLIGATION.
Ex. 4. 90 1 60
130
40
4
4
12
60
60
180
Ans. 60 gallons each of alcohol and water.
Ex. 5. 40
35
50
150
75
Ans. 150.
Ex. 6. 7i >
81 !
10 '
II
6
80
60
20
Ans. 60 Ibs. at 81 cts. and 20 Ibs. at 10 cts.
(418, page 312.)
Ex. 2. 14
Ans. 60 at 9 s., 40 at 12 s. ; 20 at 18 s., and 50 at 20
Ex. 3. 22
Ans. 6 ounces each of the first three, and 33 ounces of the la,t
9
i
6
6
60
12
i
2
4
4
40
18
*
2
2
20
20
i
5
5
50
17
170
16
1
2
2
6
18
2
2
6
21
1
2
2
6
24
i
i
6
4
1
11
33
IT
51
Ex. 4. $178.50-^210 $.85, average price ;
( 50
i
3J
1113
13
78
85-^ 75
TV
13
13
78
( 150
61
eVll
2
9
54
35
210
Ans. 78 bu each of oats and corn, and 54 bu. of wheat
184
EVOLUTION.
Ex. 5. 48
(45
51
( 54
21
3600
1200
1200
6000
Ans. A 3600 bu. ; B and C each 1200 bu.
Ex. 6. $84-:- 56 =$1.50, average daily wages.
(50
lio
6
6
24
i8
7
1
3
1
3
4
12
300
TJo
4
4
16
14
56
Ans. The boys 24, 4, and 12 days, respectively, and the
man 16.
EVOLUTION.
!
SQUARE ROOT.
(434, page 319.)
Ex. 9. Ans. 234135.
(441, page 321.)
Ex. 3. 200 x lj = 225 sq. yd. ; 1/225 = 15 yd. = 45 ft., Ans.
Ex.4. 10 A.=1600 rd.; 1/1600=40 rd., length of one
side ; 40 x 4 = 160 rd., Ans.
Ex. 5. 45 3 =2025, square of the base ;
60 a =3600, square of the perpendicu.ar ;
5625, square of the hypotenuse.
= 75, Ans.
SQUARE ROOT. 135
Ex. 6. 39 2 =1521, square of the hypotenuse
15 a = 225, square of the base ;
1296, square of the perpendicular.
4/1296=36, height of the stump.
36 ft. + 39 ft. =75 ft., Ans.
Ex. 7. 4/40 2 33 2 = 22.60 + , from foot of ladder to one side ;
4/40' 21 2 =34.04+ " " " " " other"
56.64 + , Ans.
Kx. 8 52 a =2704, square of hypotenuse ;
48 a =2304, square of perpendicular ;
400, square of the base ;
4/400 = 20, Ans.
Ex. 9 1 mi. =3 20 rd., length of 1 side of the park.
S20 2 =102400
102400
204800; 4/204800=452.5+, diagonal.
320 x 2 = 640, distance abound the park to the opposite corner.
640452.5 = 187.5, distance between A and B, when A ar-
rives at the opposite corner. 187.5 + -^-2=937+, Ans.
Ex. 10. 20 a + 16 a = 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
!s the hypotenuse. Hence
Ifi2 _ I square of the diagonal of the floor ;
12* =144 square of the perpendicular;
800 square of the required diagonal ;
4''80b = 28.28427l + feet, Ans.
136 EVOLUTION.
Ex. 11. 2:3:: (63.39) 3 : ( )=6027.43815
V / 6027.43815=77.63 + rods, Ans.
Ex. 12 1 : 2 :: 5 3 : ( ) = 50 ;
4/50 = 7.07106+ feet, Ans.
CUBE ROOT.
(444, page 327.)
APPLICATIONS IN CUBE BOOT.
Ex. 1. 1/1331 = 11 ft., Ans.
Ex. 2. 1/373248 = 72 in.=6 ft., Ans.
Ex. 3. V474552 = 78 in.=6i ft., length of 1 side ;
6 x 6=42i sq. ft., Ans.
Ex. 4. Vf|=f ft-, length of 1 side ;
x f sq. ft. = area of 1 side ;
xx6x= 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 half.
Vl34400 = 51.223+in., width and depth.
51.223 x 2 = 102.446 in., length.
Ex. 7. 1 : 2=(14.9) 8 : ( )=6615.898
V6615.898 = 1 8.7 + inches, Ans.
Ex. 8. 16 : 25 : 8 : ( ) = 12.5 cube of the diameter.
= 2.32+ ft.
ARITHMETICAL PROGRESSION. 137
ARITHMETICAL PROGRESSION.
(451, page 329.)
Ex.1. (19 l)x 3 = 54; 54 + 4=58, Ans.
Ex.2. (13 I)x5 = 60; 75 60=15, Ans.
Ex 3. 2=firstterm; 3=com. diff. ; 18=No. terms.
(18 I)x3 = 51; 51+2=53 cents, Ans.
Ex. 4. (40 1) x i=9f ; 9f +i = 10i, Ans.
Ex. 5. 20= first term ; 5= com. diff. ; 9= No. terms.
(9 I)x5=40; 40 + 20=60, Ans.
Ex. 6. 100=first term ; 7=com. diff. ; 46=No. terras.
(46 1) x 7 = 315; 315 + 100=$415, Ans.
(452, page 330.)
Ex.1. 172 = 15; 15+5 =3, Ans.
Ex. 2. 14 2 = 12 ; 12-^6=2 years, Ans.
Ex. 3. 501 1=491 ; 49i-:-33 = li, Ans.
Ex. 4. 3=first term ; 91= last term ; 14=No. terms.
9i3 = 6i; 61-7-13 = 1, com. dif.
(453.)
Ex.1. 43 7 36; 36-i-4=9; 9 + 1 = 10, Ans.
Ex.2. 40-21 = 371; 37i~7i=5; 5 + 1 = 6, Ans.
Ex. 3. 6=first term ; 226 = last term ; 4=com. diff.
2266 = 220; 220-^-4=55; 55 + 1=56, Ant.
(454, page 331.)
Ex. 1. (5-h32)x-^= 222 Ans.
Ex. 2. (l+12)xY=^ 8 . Ans >
1&5 GEOMETRICAL rivuGRESSION.
Ex. 3. 24=first term; 1224=last term; 52==No. terms.
($1224 + $24) x */=$32448, Ans.
Ex. 4. 4 =first term, or twice the distance to the first apple ,
400= last " " " u " " last "
100 = No. terms.
(400 yd. + 4 yd.) x ^=20200 yd., Ans.
GEOMETRICAL PROGRESSION.
(458, page 333.)
Ex. 1. 4 x 3 8 = 26244, Ans.
Ex. 2. 1024 x U) T =TVWr =A. Ans -
Ex.3. 1 first term; 2=ratio; 9 = No. terras.
1 mill + 2 8 = 256 mills=$.256, Ans.
Ex.4. 7x'==
Ex. 5. 1= first term ; 1.07= ratio ; 5 = No. terras.
1 x (1.07) 4 =$1.40255 + , Ans.
Ex.6. 3= first term; 3= ratio; 7= No. terms.
$3 x 3 9 =$2187, Ans.
(459, page 334.)
Ex. 1. (512x3)-2 = 1534; 1534^-2=767, Ant.
Ex 2 (262144 x4)-4 = 1048572;
1048572^-3 = 349524, Ans.
Ex. 3. (162 x 3)-2=484 ; 484-^2 = 242, Ans.
Ex. 4. _u^-^i, ratio;
(iL X 5) = 1; 1-^-4 = 1, Ans.
PROMISCUOUS EXAMPLES. 139
Ex. 5. -2= first term; 6-r-2 = 3, ratio ; 12= No. terms.
$2 x 3 n = $354294, last term ;
($354294 x 3) 2 = $1062880 ;
$1062880-^2=1531440, Ans.
Ex, 6. 7= first term, or yield of the first year;
7 = ratio ;
12 = No. terms, or the number of years to yield.
7x7 n =7 13 =13841287201, last term, or 12th
year's produce.
(13841287201x7) 7
^ -'- =16148168400, sum of all terms.
6
16148168400-i-1000 = 16148168.4 pt.
16148168.4 pt.=252315 bu. 4-t qt., Ans.
Ex. 7. 200^20 = 10, the number of times the family
doubled its number.
10+1 = 11, No. terms; 2 ratio.
6x2 10 =6144, Ans.
PROMISCUOUS EXAMPLES.
(Page 334.)
Ex. 1. 800 x 2 = 1600, the sum ; and
200 x 2= 400, the difference.
The greater of any two numbers is equal to the less 4- the
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;
1200-7- 2= 600, the less ;
600 + 400=1000, the greater.
140 PROMISCUOUS EXAMPLES.
Ex. 2. f f T 3 T sV If A f a number be added to itseh,
the result must be 1/j times the number. Hence,
61-hl/ T 55, Ans.
Ex. 3. 3 h. 21 min. 15 sec. = 12075 sec. ; 1 da.=86400 sec. ;
., Ans.
Ex. 4. 3 bu. 3 pk. 3 qt.
10
38 bu. 1 pk. 6 qt.
7
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=$300=3 times A's.
$300-^-3=$100, Ans.
Ex. 6. B has A's votes H- 200
C has A's votes + 1000
A B and C have 3 times A's votes +1200
Therefore, 30001200=1800, 3 times A's votes.
1 800 -r- 3 =600, Ans.
Ex. 8. i -1=-^-. Hence 10 is ^ of the number; and
the number must be 560, Ans.
Ex. 9. $28.35^$.35 = 81 gal. mixture.
8163 =18 gal. water added.
Ex. 10. When A had gained J, he had | of the original
stock. B, after his loss, had as much, or f of the original
stock ; hence he had lost f ; the $200 which he lost was |
of his stock; and his whole stock must have been 200-7-f =
$500.
PROMISCUOUS EXAMPLES. 141
Ex. 11. $.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.
$27-^-30=$ .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
v Arts.
21 da. 15 h. 44 rain., 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 Imr to spend $30 $10 = $20 a year more than A
spends. Hence $20 is } of the income ; and
$20 x 8 = $160, income, Ans.
Ex. 17. $2.19xf^o_$ 2 .40, Ans.
Ex. 18. * : 3 * - : :
. j : - : : $3.37f : ( )=$52.779, Ans.
Ex. 19. $1000 : $200 : : 6 mo. : ( ) = ! mo., Ans.
K. r. 7
142
PROMISCUOUS EXAMPLES.
Ex.20. $2356.80-=- .40 = 15892, left;
$5892 + $2356.80=|8248.80, Am.
Ex. 21. = ^. x a x f =!=12i 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.
$10200-M02=$ 100, share of a private.
$100 x 12 =$1200, share of the commander.
Ex. 23. 19 16 = 3 ; 51-^3 = 17 hours, Ans.
Ex. 24. 40
30
50
54
14
20
10
133
95
95
190J
[Ans.
Ex. 25. $33.75-^22.5 =$1.50, selling price per bu.
$22.50-M8 =$1.25, buying "
$ .25, profit on 1 "
$.25x240=$60, Ans.
Ex. 26. The wagon is worth 4 times the harness ; the horse
is worth 8 times the harness ; hence the horse, wagon and
harness together are worth 8 + 4 + 1 = 13 times the harness
Therefore, $1 69 -^ 1 3 = $1 3, harness, Ans.
Ex. 27. 18 in : 42 ft. : : 40 in. : ( )=93i ft., Ans.
Ex. 28. 25 rd. : 40 rd. : : 4 rd. : ( )=6| rd., Ans.
PROMISCUOUS EXAMPLES. 148
Ex. 29. }, ft, and if =if, if, and if.
And since fractions having a common denominator are pro-
portional to their numerators, we have
15 shares for A and B ;
18 " " A " C;
13 " " B " C;
46, twice the number of shares for A, t> and C.
462 =23 shares for A, B, and C.
no TO 1 A U U A
Xi. j
23 18= 5 " " B;
23-15= 8 " " C.
$26.45 x if =$11.50, A's portion ;
$26.45 x /3=$ 5.75, B's " } Ans.
$26.45 x V=$ 9.20, C's "
Ex. 30. 2^
6'
j : 15 j- : : 12 : ( )=480 oz.=30 Ib. Ans.
Ex. 31. $6300x|=$ 900, A's,
$6300 x =$1260, B's;
$6300 x =$1400, C's;
$900+41400 =$2300, D's;
$5860
$6300 $5860=$440
$440 xf =$165, E's,
$440 xf= 275, F's.
Ex. 32. S200xl.593848=$318.769 + ,
Ex. 33. At the time of the dismissal, the provisions on
hand would supply 360 men 1 month ; they would aupplv \
us many men 5 months.
Hence 360-^-5 = 72, the number that remained;
360-72 = 288, dismissed, Ans.
144
PROMISCUOUS EXAMPLES.
Ex. 34 $1.338226 =amt. of $1 at compound int. for 5 yrs,
at 6 per cent.
$669.113-^1.338226 = $500, principal.
$669.113 $500 = $169.113, interest.
$500x.06 = $30,-simple int. of $500, for 1 year
at 6 per cent.
169.113-r-30=5.637l yr. = 5 yr. 7 mo. 19.356 + da., Ans.
Ex. 35. $148.352^-9.728 -$15.25, Ans.
Ex. 36. It is evident that the product of two numbers must
ontain each common factor to the two numbers twice, and
sach factor not common once.
483-7-23 21, product of the factors not common.
23 x23 x 21 =11 109, Ans.
Ex. 37.
8
7
15
12
9
:: 1 : 2
12
9
2
8
7
15
9 140
154,
Ex. 38. 36
) 60 ) 36
9 j ' 27 j ' 1
Ans. 144 yards.
Ex. 39.
Ex. 40.
1 x 4 = 4, A's product ;
3 x 2 = 6, B's
in
I7i : 4 : . $52.50 : ( )=$12, A's share ;
I7i : 6 : : $52.50 : ( )=$18, B's share;
17 : 71 : : $52.50 : ( )=$22.50, C's share,
| and |=| and f.
4 + 5 = 9.
9 : 4 :: $45 : ( ) = $20, A's;
9 : 5 :: $45 : ( )=$25, B's.
PROMISCUOUS EXAMPLES. 146
Ex. 41. $.35, interest of $1 for 5 years at 7 per cent,;
$33.25-h. 35 =|95, Ans.
Ex. 42. 6
Ex.43. 2 + 3 + 4 9.
9:2:: $360 : ( )=$ 80 }
9:3:: $360 : ( )=$120 ( Ans.
9:4:: $360 : ( )=$160;
Ex. 44. 8i : 6| : : 12| : ( )=9 T <V, 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=.92; $2 3 -f-.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 |= of the number of calves, plus 2 x f = of
the number of sheep, equals 25. Expressing these conditions,
as in Analysis, page 294, we have
c. s.
1st condition 3 2 76
2d condition 25
2dx4= 3 y 10
Subtract the 1st from the 3d= f 24
That is, | of the sheep are equal to 24 ; hence
24-r-| = 20, number of sheep.
76 (2 x 20) = 36 ; 36-^3 = 12, number of calve*.
Ex. 47. 17A J 16
10H : 7 : : 546 : ( ) = 384, Ans.
I'd 15
146 PROMISCUOUS EXAMPLES.
Ex.48. 12 ) ( )|
15* } : 15 } :: 2 : 1.
9 j 1}
Reducing the statement, ( ) = 8, the number of men ro
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 effect this, required 59 + 84 = 143 men, -*^Mp^=7l is
the number of men in rank or file at first. Hence
7l 3 -f 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 + j of the whole.
Corn " 153 + T^ " "
Wheat and corn " $396 + i " "
Barley " 66+ T V " M (ofw.&c.)
Cost of the whole was $462 + T \ " "
Hence, $462 was }f -^ = 11 of the whole cost
And $462-i- T }=$756, the whole cost.
And since the barley, corn, and wheat cost in the propor
lion of 1, 2, and 4, we have
1+2+4=7
$756x-| =$108, cost of barley; $108^-$ .60=180 bu. barley;
$756 x f =$216, cost of corn ; $216-f-$ .75=288 bu. corn ;
$756 x 4 =$432, cost of wheat ; $432-j-$1.50=288 bu. wheat ;
Ans. 756 bu. grain.
PROMISCUOUS EXAMPLES. 147
Ex 51 As often as the first has 1,
" second " $,
and " third " ofl|=;
1, , and =, f, and $.
We therefore assume 8 as the proportional number for the first ;
6 " " " " second;'
and 7 " " " " third
21
$630x T =$240, 1st; )
$630x^=^180, 2d; [ Ans.
$630x/ T =$210, 3d. )
Ex.52. $28xl.20=$33.60 what the 56 remaining gal-
lons must be sold for. $33.60-^-56 =$.60, Ans.
Ex. 54. $3500 $2100=$1400, what B owns now ;
$1400-^ 1.40 =$1000, B put in ;
$2100-:- 1.40 = $1500, C put in.
Ex. 55. |/
Ex.56. 12 10
Ans.
12) 8
Ex. 57. $12-7-1.09=$11.0091+, worth of sugar;
$12 $11. 0091 =$.9909 + , grocer's profit;
$6 -j-1.10 =$5.45 45 + , 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;
$4.15 int. of $311.50 for 1 yr. 4 mo. at 1 per cent
$24.92-h$4.15i=6 percent., Ans.
148 PROMISCUOUS EXAMPLES.
Ex. 59. f, |, and f = }f, }f, and ft.
These fractions are to each other as their nutaerators, 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 C ;
3 shares for B and C.
14, twice the number of shares for A, B, and 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 $20 x 4=$lLf , A's ;
$20xf=*5$,B'8;
$20x j=$2|, C's.
Ex. 60. $375-j-.025 = $15000, Ans.
Ex. 61. Interest commenced Apr. 1, 1857.
Amt. of note July 1, 1857, (3 mo.) 81015
Payment, 560
New Principal, $ 455
Ami, 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 f x f of A's =f of A's ;
C has -f x f of B's=f x f x of A's =f f of A's ;
D has f x f of C's=f x f x f of A's=-J f of A'R
And since D has $45 more than C,
H of A's-H of A's=$45; or ^ of A's ^$45.
Hence. $45-f- T 5 2 =$378, A's; 1
$378 x | =$336, B's; I
$378x|=$360, C's;
$378 x If =$405, D's.
PROMISCUOUS EXAMPLES.
149
Ex. 63. B had the use of $300 for 27 months before it was
dne, which was equivalent to the use of $1 for 27 x 300 = 8100
months. But the use of $1 for 8100 months is equal to the
use of $600 for yjL9.=i3 months, the time he should wait.
Ex. 64. His savings and expenses together, or his salary
is 1 1 -t-JL.=: js. of what he saves ; hence $800 =}& rf what Ue
aves; and $800-^}f =$550, Ans.
Ex. 65. $.87i=$f. ; $1.00-M.10=$}f, cost per yd.';
$-!-?$ $a\i l ss at $-87^ per yd. ;
/j-4-lf =8^ = .03f = 3f per cent., Ans.
Fx 66 21 - W 240 3. /3x 27
2^=^ = 320-4' W ^64' A
63 189 27 /27 3
149i 448 64' r 64 4'
Ex. 68. 50
Ans. 20 of oats and corn, and 30 of rye and wheat.
20
A
20
2
2
20
35
TV
10
2
2
20
60
rV
15
3
3
30
70
A
30
3
3
30
10
Too
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.
50040=460, difference of extremes.
$460-r-23=$20, common difference.
Ex. 70.
, Ans.
150
PROMISCUOUS EXAMPLES.
10
11
14
4
10
Ex. 71. If 7 pounds of butter are equal
to 10 pounds 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
equal to -y 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 are equal to f of 14
bushels of corn, or f- of -y of -y 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
y. of y of 10 pounds of cheese =550 pounds of cheese, Ans.
NOTE. Instead of the fractional form the vertical line may be ased,
as above.
Ex.72. $18^f=$45 A'sgain.
f : :: $45 : ( )=$37.50,B's gain.
$45 -=-.06 =$750, A's stock ;
837.50-i-.06=:$625, B's stock
Ex. 73. 20 ) ( ) ) 30 ] <
21 [ : 25 V :: 15 I 16
10) 8) 12 f : 18
3J 5
Reducing the statement, we have
Ex. 74. 2| : 27i :: 10 ft. : ( )=103J- ft., Aits.
Ex. 75. A can do of the work in 1 day ;
J u j| U u << i u
They all " + f + W = f of the piece in 1 day.
Hence it will require 1 -*-=$. of a day.
PROMISCUOUS EXAMPLES. 151
Ex. 76. $1890-M.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 6 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 and B's use of the pasture was = 2 5 " " 1 " " "
$50 x ^=$18, A paid ;
$50 x if = $32, B paid.
Again, C's time was 21 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 -| times the
pasturage of 1 cow, and the pasturage of 1 cow is -y- times the
pasturage of one sheep ; hence the pasturage of 5 horses (for
which C paid) is ^ xf x ^=25 times the pasturage of 1
sheep. Therefore C put in 25 sheep.
Ex. 78
$350C -7-1.0175= $3439.803+, pres. worth of 1st installment;
$35004-1.Q2 =$3420.195+, " " 2d "
I3500-M.04! =$3343.949+, " " 3d "
$10203.947+, Ans.
Ex. 79. Had the farmer sold both geese and turkeys at $.75
ipiece, the 50 fowls would have brought $.75 x 50 =$37.50,
152 PROMISCUOUS EXAMPLES.
which is $52.50 $37.50 = $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.
$1.25 $.75 $.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 milts
an hour. Hence B will pass A every 7 ^ hours=24 h. ?0
min. = 1460 min.*; and C will pass A every 7 j 3 hours=14 h,
36 min. = 876 min. Now the least common multiple of 1460
and 876 will express the number of minutes in which B and
will first pass A together.
2,2
73
5,3
1460 . .
876
365 ..
219
5 ..
3
2x2x73x5x 3=4380 min. = 6 da. 1 h., Ans.
Ex. 81. i, } and |=f $, $fr and if.
And since fractions having a common denominator are toeact
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 15, or 4
and 3. 4 + 3 = 7.
7 : 4 : : 12 : ( )=6f, A's share of C's 12 parts ;
7 : 3 : : 12 : ( )=5|, B's " " " " "
20 + 6|=26f, A's number of parts of the money ;
20 B's " " " "
47 : 26f : : $100000 : ( )=$57l42.85f A's, )
47 : 204 : : $100000 : ( )=$42857.14f , B's, )
Ex. 82. A's + B's=5 Since A's-f B's are t<
C's + B's=7 B's + C's as 5 to 7, A
C's B's=l and B together have 5
ii =4, C's proportion ; as often as B and C to-
7-4=3, B's gether have 7. And
5-3=2, A's " since C's -B'sare toC's
PROMISCUOUS EXAMPLES, 158
f B's as 1 to 7, 7 is the sum and 1 the difference of B and C's
proportion ate- shares. Hence we find the proportionate share
of each. Then
= 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
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 3 x 8 = 24 ) C 2
1 woman 4 x 9 = 36 V or -! 3
1 man 6x12 = 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 men aro ex-
pressed by the following quotients :
5-=-2 = 21 boys;
10-f-3=3i women;
24-f-6 = 4 men.
9
: 2i
:: 59 : (
) 15 boys;
: 3i
: : 59 : (
)=20 women;
: 4
:: 59 : (
)=24 men.
7*
1-54
PR3MISCUOUS EXAMPLES.
Ex. 85. A, B and G fill J- of it in 1 hour ;
B, C " D " | " " "
C, D " A " T V
D, A " B " T V
of it in 1 hour.
" "
" "
A, B, C and D fill 1 of &\=
W, X and Y empty | of it in 1 hour ;
X, Y " Z " |
Y, Z u W " |
Z, W " X " i
" "
" "
" "
W, X, Y and Z empty of iif^/JL. of it in 1 hour.
fW~~~TW rW the emptying pipes gain of the filling pipes
m 1 hour. Therefore, to exhaust the fountain will require
120-7-19 = 6^ hours, Ans.
Ex. 86. 1 A. 1 R. 6 P. 18| sq . yd.=56250 sq. ft. ;
56250
Ex. 87.
$
1 x
2x
4x
8x
16 x
82 x
64 x
128 x
256 x
512 x
1024x10=
2048x11=
0=$
1=
2=
3=
4=
5=
6=
7=
8=
9=
2
8
24
64
160
384
896
2048
4608
10240
22528
Cost, $4095. $40962, sum of products.
40962-r-4095=10 mo., average term of credit.
Jan. 1 -|-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 1+i + =! That is, if a number be increased
by i 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 If =
of the number, and 18 x 6=108 is the number.
Ex. 90. If that which is worth $.62 be rated at $.56, what
ought that which is worth $.25 be rated at ?
$.62 : $.25 :: $.56 : ( )=$.224
Therefore, in 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 $.004-4-$.22=.01 1 r , Ans.
ANOTHER SOLUTION.
A pound of tea, in the barter, will buy 56-^22 = 2^ pounds
of coffee; and this is worth $.25 x2 1 r =$.63 T V But the
pound of tea given is worth $.62 ; and $.63 T 7 T -$.62i=
$.01 3 2, the gain on $.62. Hence, l^-r-62 =.01^, 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-M.Olf =$.245901 +,pres. worth of 1st installment;
$ .25-^-1.0375=$.240965+, * " 2d "
$ .25^-1.05 =$.238095+, " 3d "
$ .25-T-1.081 =$.230769 + , " " 4th "
$1.00 $.955730 + , present worth of $1.
Now i f $1, payable as by the conditions named, be worth
$.95573 in ready money, what sum will be worth, in ready
money, $750, on the same conditions ?
$1 : ( ) : : $750 : $.95573 ; $784.74 + , Ans.
156
PROMISCUOUS EXAMPLES.
Ex. 92 f H I of the capital of cither must be equal lo
$500. Therefore $500 x 8-_$4000, Ans.
Ex. 93.
Due.
da.
Items.
Prod.
Dec. 4
Feb. 9
Feb. 29
March 4
May 12
00
67
87
91
160
240.75
137.25
65.64
230.36
36.00
9195.75
5710.68
20962.76
5760.00
710.00
41629.19
41629.19-^-71059 da., average term of credit.
Dec. 4, 1859 + 59 da.=Feb. 1, 1860, Ans.
Ex. 94. When he had spent j of his fortune, he had left
^ ofz=-3j. He had spent, in all, 1 + ^=4 of his fortune;
consequently, the $2524 which he had left was % of his for-
tune. $2524 -=-f = $5889.33 + , 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 so on to the last. Hence, the installments form a
Geometrical Series, of which
MENSURATION.
157
1st installm.= lst 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 sum
of the series, to find the first term.-
$30 Voffrr = 1St term=te2I.6V4+ ;-
6521.674 (1st installment) + $210 (int. on $3000 for 1 yr.)
sn |731. 674, annual payment, Ans.
Ex. 96.
Due.
mo. j Items.
Prod.
Jan. 1, 1859
Sept. 1, "
Apr. 1, 1860
8
15
200
350
500
2800
7500
1050
10300
10300-4-1050 = 9 mo. 24 da.
Jan. 1, 1859 + 9 mo. 24 da. = 0ct. 24, 1859, Ans.
Ei. 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.
Dr.
Cr.
Dae.
da.
Items.
Prod.
Due.
da.
Items.
Prod.
July 7, 1859
Oct. 4,
Feb. 20,1 860
00
89
228
600
530
400
1530
730
47170
91200
Nov.15,1859
131
730
95630
Balances
138370
95630
42740-j-S
730
95630
00 = 53 da.
800
42740
July 7 + 53 da.=Aug. 29, 1859, note due.
158 MENSURATION.
MENSURATION OP LINES AND SURFACES.
(466, page 342.)
Ex. 1. 3 x 12 = 36 inches long; 36 x 20=720, Ans.
$32x198x150
Ex. 2. --- =$5940, Ans.
160
1000 x 100
(469, page 343.)
Ex.1. 20x12=240 sq. ch. ; 240x16 = 3840 sq. rd. in
ho meadow, requiring 3840 rain, for mowing.
3840 min. = 6 da. 4 h., Ans.
Ex. 2.
N/15 2 9 2 =12, the perpendicular; A 11
15 X 12 = 180 sq. ft., Ans.
12
9
(471.)
Ex. 1. - =121 in.=lJ T ft, average width ;
12 x l^ T = 12i sq. ft., Ans.
i.= l ft., mean width ;
1 x 8=8 sq. ft, Ans.
Ex. 2. - =12 in.= l ft., mean width ;
40 -I- 22
Ex. 3. -y x 25 = 775 sq. ch.=77 A. 5 sq. ch., Ans.
(475.)
Ex. 1. -4* x 45 = 3330 sq. ft.=370 sq. yd, Ans.
Ex. 2. The two ends together are equal to a rectangle, 28
feet by 7 feet; hence 28 x 7 = 196 sq. ft.. Ans.
MENSURATION. 159
(479, page 344.)
Ex. 1. 5 x 3.1416=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.
(48O, page 344.)
Ex.1. 113 * 355 = 10028.75,.te.
4
Ex. 2. Reversing the rule (II,)
I sq. mi. -i-.7854= 1.2732 +, square ot <uameter.
f 1.2732=1.1284 +mi.=l mi. 41 rd. 1.4+ ft.
Ex. 3. 84 s x. 07958=561.5 +P.=3 A. 81.5 + P., Ans.
METRIC SYSTEM.
REDUCTION DESCENDING.
(Page 444.)
Ex. 1. 3 M32.58X1000000= 3 M 32580000, Ans.
Ex. 2. 4 M5X100000=!M 500000, Ans.
Ex. 3. G402X1000= 3 G 402000, Ans.
Ex. 4. 2 A42.3X10000= 2 A 423000, Ans.
Ex. 5. 3 L 93.2X10000=^ 932000, Ans.
Ex. 6. 4 S 895 X 1000 = X S 895000, Ans.
Ex. 7. ' 2 A903.2X100000= 3 A 90320000, Ans.
Ex. 8. 1 G539X1000= 2 G 539000, Ans.
REDUCTION ASCENDING.
Ex. 1. 2 A 5-j-1000000= 4 A .000005, Ans.
Ex. 2. 3 M 403-=-1000000= 3 M .000403, Ans.
Ex. 3. 1 S42.3-f-1000= 2 S .0423, Ans.
Ex. 4. 3 A 7.2-5-10000=^ .00072, Ans.
Ex. 5. 3 O 3-v-1000000= 3 G- .000003, Ans.
Ex. 6. 2 L 5.7-^10000= 2 L .00057, ^s.
Ex. 7. 3 M 9-f-10000000= 4 M .0000009,
Ex.8. 2 S47.3^-1000r=, 1 S .0473,^5.
MEASURE OF SURFACES.
(Page 447.)
Ex. 1. 1.25X8.7 = 10.875 sq. M or 2 A 10.875 *
2 A10.875-v-100 = A.10875, Ans.
Ex. 2. A.10875XlOOr= 2 A 10.875, Ans. to first
A.10875-i-1000= 3 A. 00010875, Ans. to second
A.10875-J-100= 2 A .0010875, Ans. to third
* See Note under Rule, page 447, and preceding illustrations.
(160)
MEASURE OF SURFACES. 161
Ex. 3. 2 M47-r-100=M.47
5.32x.47=2.5004sq. M or 2 A 2.5004*
2 A2.5004-i-100= A .025004, Ans.
Ex. 4. A .025004X1000= 3 A 25.004, Ans. to first
A .025004-=-10000= 4 A .0000025004, Ans. to second
A .025004-f-100= 2 A .00025004, Ans. to third
Ex. 5. 5.4X6= 2 A32.4*
32.4-r-.9=M36 length of carpet, Ans. to first
$2.50X36=$90, Ans. to second
Ex. 6. 342X273= 2 A 93363,* area of first field
634X350= a A 221900, " " second field
450X329= a A 148050, " " third field
632.7X730= 2 A 461871, " " fourth field
2 A 925187, total area
2 A925187-=-10000= 2 A 92.5187, Ans.
Ex.7. !M4-5-10=M.4
4X-4= 2 A1.6*
2 A 1.6X 150= 2 A 240, area of boards of 1st dimensions
2 M52-f-100=M.52
6.2X.52= 2 A 3.224 [mensions
2 A3.224X225= 2 A 725.4, area of boards of 2d di-
2 M43--100=M.43
5.2X-43= 2 A 2,236 [dimensions
2 A2.236X642= 2 A 1435.512, area of boards of 3d
2 A 240+ 2 A 725.4-f- 2 A 1435.512= 2 A 2400.912
2 A 2400.912-^ 100= A 24.00912
$42x24.00912 = $1008.38-i-, Ans.
Ex. 8. !M 2.2-r-10=M.22 ; ^ l.l-5-10=M .11
.22X.H= 2 A .0242,* area of one brick
842. 6X2.2= 2 A 1853.72, area of walk
2 A1853.72-v- 2 A .0242=76600, bricks, Ans. to first
$.60X1853.72=$1112.23+, Ans.
* See Note under Rule, page 447, and preceding illustrations.
162 METRIC SYSTEM.
MEASURE OF VOLUMES.
(Page 450.)
Ex. 1. 2 M5.2-j-100=M.52
24X8.5X-52=S 106.08*
$4.25X106.08=$450.84, Ans.
Ex. 2. 15.7X3X7.52=8 354.192*
$1.50X354 192=$531.288, Ans.
Ex. 3. 18.3.X10.73X3.4=S 667.6206*
$.15X667.6206=$100.14+, AM.
Ex. 4. 7.36-f-3.2=M 2.3=^ 23, Ans.
Ex. 5. !M 5.2-r-10=M .52 ; jM 7.3-^10=M .73
.52X.73X13=S4.9348,* volume of 1st stick
2 M43-r-100=M.43; 2 M 65-^-100 =M .65
.43X.65X17.5=S 4.89125, volume of 2d stick
.53X. 37X15.42=8 3.023862, volume of 3d stick
2 M39-r-100=M.39; 2 M 56-r-100=M .56
.56X-39X14=S a.0576, volume of 4th stick
1 M4.52-7-10=M.452; ^3.78^-lO^M .378
.452X- 378X15=82.56284, volume of 5th stick
8 4.9348+8 4.89125+8 3.023862+8 3.0576+
82.56284=818.470352, Ans.
Ex. 6. 3.2X1.1=3.52
4.0128-r-3.52=M1.14, Ans.
MEASUREMENT OF ANGLES.
(Case I., page 452.)
Ex. 1. 25* 34 X 42 U =25* .3442
25 .3442-^-l=22.80978=22 48' 35.208", Ans.
(See 370.)
* See Note under Rule, page 450, and preceding illustrations.
MEASUREMENT OF ANGLES. 163
Ex. 2. 57*93 u = g .5793
.5793-s-l J=.52137=31' 16.932", Am.
Ex.3. 83 g 13 x 87 u = 83 g .1387
83 g .1387-i-l J=74.82483=74 49' 29.388", Ans.
Ex.4. 36 g 98 x 15 u =36 g .9815
36 g .9815-=-l=33.28335 =33 17' .06", Ans.
Ex. 5. 14 g 15 v 60 u = 14 g .1560
14 g .156-J-H=12.7404:=12 44' 25.44", Ans.
Ex. 6. 90* 90 X 90 u =90 g .9090
90* .9090-r-l J=81.8181 = 81 49' 5.16", Ans.
Ex. 7. 18 g 50 x 25 u =18 g .5025
18*.5025-v-H=16.65225=:16 39' 8.1", Ans.
(Case II., page 452.)
Ex. 1. 36 18' 27"=36.3075 (See 374)
36.3075XH=40 g .3416| =4034 N 16 u , Ans.
Ex. 2. 56'54"=.94833
Ex. 3. 27 36' 45"=27.6125
27.6125Xli=30 g .6805|=30 g .68 x 5| u , Ans.
Ex. 4. 189 15' 20"=189.25555f [Ans.
189.25555|Xli=210 g .2939|}=210 g 29 x 39|V v ,
Ex. 5. 63 14' 58"=63.24944|
Ex. 6. 14724'48"=:147.41333i [Ans.
147.41333^XU = 163^
Ex.7. 117 36' 54"= 117.615
104 METRIC SYSTEM.
TO CHANGE METRIC TO COMMON SYSTEM.
(Page 453.)
Ex. 2. 2 M 472-f-100=M 4.72
39.37 in X 4.72=185.8264 in. = 15.4855^ ft, Ans.
Ex. 3. S S 2X1000 = 82000
1.308 cu. yd.x2000=2616 cu. yds.
27 cu. ft.X2616=70632 cu. ft., Ans.
Ex. 4. jL325-f-10=L 32.5 [qts., Ans.
1.0567 qts. X 32.5=34.34275 qts. = 8 gal. 2.34275
Ex. 5. 1.0567 qts.Xl08.24=114.377208qts.=28.594+
gals., Ans,
Ex.6. 3 L3262X1000=L 3262000 {Ans.
.908 qts.X3262000=2961896 qts. =92559. 25 bush ,
Ex. 7. 119.6 sq. yds.X436 = 52145.6 sq. yds., Ans.
Ex. 8. 2 L 942325-f-100=L 9423.25 [bush., Ans..
.908qt.X9423.25 = 8556.311qts.=267.3847+
Ex. 9. 4 G 436X 10000=G 4360000
15.432 gr. X4360000=67283520 gr.
67283520 gr.-*-7000=9611.9314 Ibs., Ans.
TO CHANGE COMMON TO METRIC SYSTEM.
(Page 454.)
Ex. 2. 7000gr.X6172.8=43209600gr. [Ans.
43209600 gr.-s-15.432gr. = G 2800000=^280000,
Ex. 3. 2392 sq. yds. -r-119.6sq. yds. = A20= 2 A .2, Ans.
Ex. 4. 1 sq. m. = 3097600 sq. yds. \_Ans.
3097600 sq.yds.-r-119.6sq.yds.=A25899.665552
Ex. 5. 18924 cu. yds.-f-l.308 cu. yds.=S 14467.8899082
S 14467.8899082X1000= 8 S 14467889.9082, Ans.
MISCELLANEOUS EXAMPLES. 165
Ex. 6. 892 gr.-^15.432gr.=a57.8019= 2 a. 578019, Ans.
Ex. 7. 2 mi. 6 fur. 39 rds. 5 yds. = 182142 in. \Ans.
182142 in. -7-39.37m.=M4626.416= 3 M4.626416+
Ex. 8. $3X454=$1362, cost of wheat
454 bush. = 14528 qts.
14528 qts.-r-.908 qt.=L16000= 2 L160, Ans. to 1st
$8.75X160=$1400, selling price of wheat
$1400 $1362=$38, Gain, Ans.
MISCELLANEOUS EXAMPLES.
Ex. 1. $6.00X122 =$732.000
$.35X320 = 112.000
$.30X230 = 69.000
$.31X206.5 = 64.015
$2.40X107.9= 258.960
$1235.975, Ans.
Ex. 2. $2.50X30.5X40=$3050.00
$40X12 = 480.00
$.32X110X25 = 880.00
$.38X95X10 = 361.00
$.50X12X30 = 180.00
$4951.00 Ans.
Ex. 3. 2 M40X100=M4000; 2 M20XlOO=M2000
4000X2000= 2 A 8000000 *
2 A 8000000-^100 = A 80000
A 80000X3= A 53333.33^, Ans to first
119.6 sq. yds. X 80000 =9568000 sq. yds.=
' 1976.8595 acres
$100X1976.8595=$197685.95, Ans.
* See Note under Rule, page 447, and preceding illustrations,
K P. 8
166 METRIC SYSTEM.
Ex. 4. $6.00X540=$3240.00, received for wheat
$3240-f- $8 =405, tons coal purchased
405X2000X7000 = 5670000000 gr.
5670000000 gr.-s-15.432gr.=G 367488351.47
G 367418351.47-i-10000= 4 G 36741.835147, Ans.
Ex. 5. 42.5X2X1.9=8 161.5*
$2X161.5 = $323, Ans.
Ex. 6. $1.20X300 = $360, received for oats
$360-r-$.25=M1440, Ans.
Ex. 7. $.28-=-7000 = $.00004, cost of 1 grain butter [ 3 G
$.00004X15432 (No grain in 3 G) = $.61728, cost of
$.61728-$.60=$.01728, loss on 8 G
$.01728-*-$.61728=.02JJ=2JJt % lost, Ans.
Ex. 8. 5X5X2.5=862.5 [Ans. to first
Since 8 = 3 L, 8 62.5= 3 L 62.5=L 62500= 2 L 625
.908 qt.X62500=56750 qts. = 1773.4375 bush.
$2X1773.4375 = $3546.875, Ans.
Ex. 9. 15.432 gr.XlOO=1543.2 gr., No grains in 2 G
$2. 50-7-1543. 2= $ T *VW, cost of one grain
$iA 5 ^X7000 = $11.34, Ans.
Ex. 10. $2.00X240 = $480, cost of silk
39.37 in.X240 = 9448.8 in.=262.4| yds.
$1.95X262.4| =$511.81, received for silk
$511.81 $480 = $31. 81 gain, Ans.
Ex. 11. V 5 u = g .0105 .0105-f-l4=. 00945, Ans.
Ex. 12. 2501bs.X50 = 125001bs. = 125cwt.
12500X7000 = 87500000 grs.
87500000 grs.-r-15432 gi-.= 3 G 5670.036
$.34X5670.036 = $1927.812, received for sugar
$2Xl25=$250, cost of transportation
$1927.812-$250=$1677.812, to be invested in cloth
$1677.812-j-$4=M 41 9.453 [yds., Ans.
39.37 in.X419.453=16513.86461in.=458.71+
* See Note under Kule, page 450, and preceding illustrations.
MISCELLANEOUS EXAMPLES. 167
Ex. 13. 10M-H=9 ; (45+9)-j-2=27, greater angle
(45-9)-J-2=:18 , lesser angle
Ex. 14. 66f g -f-l^ = 60 ; and, since 20 units of angular measure
are equal to 60, one unit is equal to ^ of 60 =
3, AnS.
Ex. 15. When the unit of length is a Metre, the unit of surface
is a Centiare (see Note to Rule, page 447, and pre-
ceding illustrations) ; therefore,
(M6.2-f-M4.7)X2=M 21.8, around first room
21.8X3.8= 2 A 82.84, in walls of first room
6.2X4.7= 2 A 29.14, " ceiling "
(M4.52+M 4)X2=M 17.4, around second room
17.4X3.8:= 2 A 64.752, in walls
4.52X4= 2 A 18.08 "ceiling "
(M6+M 5.2)X2=M 22.4, around third room
22.4X3.8= 2 A 85.12, in walls
5.2X6= 2 A31.2, "ceiling
(M3.82-f-M 3.82)X2=M 15.28, around fourth room
15.28X3.8= 2 A 58.064, in walls
3.8X3.8 2 A 14.5924 "ceiling
(M 6.2-f-M 7) X 2 M 26.4, around fifth room
26.4X3.8= 2 A 100.32, in walls " "
6.2X7= 2 A43.4 "ceiling
(M4.5+M4.25)X2=M 17.5, around sixth room
17.5X3.8= 2 A 66.5, in walls " "
4.25X4.5 2 A 19.125, in ceiling
2 A 82.84+ 2 A 29.14+ 2 A 64.752+ 2 A 18.18+ 2 A 85
.12+ 2 A 31.2+jA 58.64+ 2 A 14.5924+ 2 A 100
.32 + 2 A 43.4+ 2 A 66.5+ 2 A 19.125= 2 A 613
.1334, area of walls and ceilings
2 A 613.1334-j-12= 2 A 51.09445 [first
2 A613.1334- 2 A51.09445= 2 A 562.03895, Ans. to
$.38X562.03895=213.57+, Ans.
MISCELLANEOUS EXAMPLES
PROGRESSIVE INTELLECTUAL ARITHMETIC.
97.
2. Since lie sold % of his share,lie had % left; and ^ of $ is .
Therefore, if a man owning of a share in the Central Rail-
road sold -f of it, he had of a share left.
3. Since he gave 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
| of | is T 4 y. Therefore, etc.
5. Since $18 was of what the watch cost, he lost \ of the
cost, which is \ of $18 or 6. Therefore, etc.
6. Since $45 was f of the cost, he gained j, which is \ ot
$45, or $5. Therefore, etc.
7. Since ^ of the cost was sacrificed, $120 is 4 of the cost;
\ of $120, which is $30, is |, and 3 times $30 is $90, the whole
loss. Therefore, etc.
8. Since he lent \ of the remainder, $22 is f , and \ of $22,
which is $11, is \ ; 3 times $11 is $33, or the whole of the
remainder; and $33 is \ of 4 times $33 which is $132.
Therefore, etc.
9. Since $80 was \ oi' | of 2 times, or | the cost, the los&
was i, which is \ of $80, or $40 Therefore, etc.
(H5)
170 MISCELLANEOUS EXAMPLES IN THE
10. Since $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 , of 15, which is 3, is 1 ; 8 times 3 's 24,
and i of 24 is 8. Therefore, etc. Or, 1 of that number of
which 1 of 15 is 1.
12. Since 4 is f, 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, ard 2
limes 8 is 16. Therefore, etc.
13. Since he sold f of his flock, 20 must be f ; of 20,
which is 10, is i, and 5 times 10 is 50. Therefore, etc.
14. Since f of the remainder, or | of f which is f , was in
the water, the 3 feet in the mud must be the remaining 1 ; and
3 feet is i of 5 times 3 or 15 feet. Therefore, etc.
15. i -|_| 1| ? and the 4 years equals the remaining -^j, and
4 years are ^ of 30 years. Therefore, etc.
16. Since $20 = f the cost of the coat plus $12, $20 $12 =
$8, or | the cost; \ of $8, or $4 = -|, and 3 times $4 is $12.
Therefore, etc.
17. If the number + 80 was 5 more than 3 times the num-
ber, 80 5 = 75, when added to the number must have been
3 times the number. 3 f, and f =i ] and 75 must have
been times the nnmber. 75 is f of 2 times | of 75, or 30.
Therefore, etc.
18. Since 16 is f, ^ of 16 or 8 is *, and 3 times 8 is 24,
which is twice as many as James has ; then 1 of 24 or 12 is what
James has after losing 16, and 12 + 16 = 28, or the number
James had at first ; since | of John's equaled -f of James's,
i of -^ or | of James's=i of John's; | of 28 is 4, and 5 time*
4 is 20, or what John had at first. Therefore, etc.
19. Since + 10 years=li or f of his age, f f =f or
must =10 years; and 2 times 10 years is 20 years. There-
fore, etc.
20. Since f of $40 is ft, | of f , or 1 of $40 is T y ; 1 of $40
is $8, and 11 times $8 is $88, which is 2| or | times | ; } of
(145, 146)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 171
$88 or $11 is i of 3 times $11 or $33, and $33 is $ of 4 times
$33, or $132. Therefore, etc.
21. 4 times is f- ; and 25 is of 4 times | of 25 which is
20. Therefore, etc.
22. Since he made away with of of it, the $10 left was
the remaining of | or j of the whole ; and 5 times $10 is $50.
Therefore, etc.
23. Since f of $1500 is f, of f, or & of $1500, which is
$200, is i ; 8 times $200 is $1600, which is 4 times the cost
of the barn, and of $1600 is $400. Therefore, etc.
24. Since f of 500 or 300 men was f of T $ , 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 7 or J/ rV f 60 which is 4
is ; 2 times 4 is 8, which is T i of 100 times 8 or 800.
Therefore, etc.
26. Since f of 60 or 50 was 2 or f times T i , of 50 or 10
men is , and 2 times 10 is 20, which is T ^ of 150 times 20
or 3000 men. Therefore, etc.
27. Since f of 100 or 80 was If or L-, T V of 80 or 8 is
|. 7 times 8 is 56, less 6 = 50 or ^. And 20 times 50 is
1000. Therefore, etc.
28. Since 21 times 30 or 70 is 31 or y, r ' 7 of 70 or 7 is \ t
3 times 7 is 21, which is -J^- of 10 times 21 or 210. There-
fore, etc.
29. Since f of 1200 or 1000 is 8 or a *-, J- 5 - of 1000 or 40
is i- ; 3 times 40 is 120, which is T and of 120 or 20 is
T j ; 100 times 20 or 2000 is of 10000, which lacking
1000 of being the whole army, 10000 + 1000 = 11000. There-
fore, etc.
(146,147)
172 MISCELLANEOUS EXAMPLES IN THE
98.
2. Since $6 + $4 or $10, bought 40 bushels, $1 would buy
r L of 40 bushels or 4 bushels, and $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 *f- 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, 7 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 + 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 1 horse 4 weeks, and 2 horses 4
weeks^l horse 8 weeks; and since 12 weeks' pasture cost
$12, 1 week's pasture costs T '^ 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 -^ of $45, or $1, $50 50 times $1, or $25,
and $40, 40 times $|, or $20. Therefore, etc.
9. Since they paid in the proportion of $5, $4, and $3,
they own in the same proportion ; consequently the gain is
diyided 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 1
(147, 148)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 173
work, it takes 20 days to mow the field ; 1 day's work costs
5-V 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. Therefore, etc.
11. Since C took $10, or !j /y of the gain, he must have
put in -/ T of the stock, and A's $30 plus B's $50, or $80= if ;
-J-g- of $80, or $5=-2 1 T > and 5 times $5 or $25=C's stock.
$42-$10=$32,A's + B'sgain; A's=f or f of $32, or $12;
and B's= $ or of $32, or $20. Therefore, etc.
12. Since he put in or of the capital, he should also
take f of the gain; f of $240=$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 =11 calves. If 11 calves cost $11, 1 calf cost T V 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 i of the rent, he puts in 1 of the cows.
Then A's 5 cows + B's 3 cows=8 cows | of the cows, and
% of 8, or 4 cows=C's number. And since C's 4 cows cost -i
of $42 or $14, 1 cow cost of $14, or $3 ; 5 cows cost 5
times $3, or $l7i; and 3 cows 3 times $3, 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
5, =2 x4 or 8 cows. But 8 cows =6 oxen; and 9 oxen + 6
oxen + 6 oxen = 21 oxen, which cost $56. 1 ox cost ^ T of
$56 or $2f ; 9 oxen 9 times $2f, or $24 ; and 6 oxen 6 times
$2|, or 16 ; etc.
16. Since Mary wrote J as many lines as Melissa, Melissa's
work is divided into 8 parts, 7 of which = Mary's ; then 8 + 7
= 15 ; and y 1 ^ of 60 is 4 ; T 8 T 8 times 4, or ? ' md T 7 ^ 7 times
4, or 28. Therefore, etc.
17. Since the boys received as many pears as the girla,
they received of 24, or 12. There were as many boys aa
(148, 149)
174 MISCELLANEOUS EXAMPLES IN THE
) is contained times in 12, which is 4 times ; as many girla
is 4 is contained times in 12, which is 3 times; and 4 + 3 = 7.
Fherefore, 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 $9 6 =$2 4 Breach daughter's
portion ; and of $24=$12 Breach son's portion. Therefore, etc.
99.
1 The 1st has 1 part, the 2d 1 part +2, and the 3d 1
part-r2 + 6; then 3 parts + 2+2 + 6 = 76, or 763 parts
+ 10; and 76 10, or 66=3 parts; 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. 72 6 = 66, Ans.
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 1 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 more than A. of $200 is $100,
or what A paid ; and $100 + $100 =$200, whatB paid.
6. The drum cost 1 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. 4 of $42, or $6 = costof drum; 2 times $6, or $12 =
cost of rifle; and 2 tr'mes $12, or $24 -cost of watch
Therefore, etc. (149)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 175
7. The harness cost 2 parts, the horses 4 times as much
or 8 parts, and the wagon 1 times the harness, or 3 parts;
and the harness 2 parts + the horses 8 parts, plus the wagon
5 parts=15 parts=$225. -Jj of $225 is $15, and 2 timea
$15, or $30=harness ; 8 times $15, or $120=cost of horses;
and 5 times $15, or $75 = cost of wagon, etc.
8. Since he traveled ^ as far the 1st as the last 2 days,
the last 2 days' travel is divided into 2 parts of which 1 =
first day; hence of 114 miles, or 38 miles = 1st 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 1 of what remained
due after the 1st j^yment, by the $20 that exceeded \ ; hence
$20 + $20, or $40=. 2 times $40, or $80=what remained
after 1st payment, and $80 was less than the debt, by the
$10 the payment exceeded \ ; $80 + $10, or $90=| ; 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 1 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 pennies=|;
\ 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 \ the keg by the \ gallon more than the \ gallon sold ;
then 1 gallon + 1 gallon or \\ gallons = the keg, and 2
times 11- or 3 gallons=the contents of the keg.
14. Since | of John's = f of Mary's, of John's=i of f or
f of Mary's, and or all of John's=3 times f or f of Mary's ;
hence Mary's are divided into 8ths and John's =9 of them,
and the whole = J J L ^ Mary's. T ' T of 34 or 2 is | ; 9 times 2
or 18 John's, and 8 times 2 = 16=Mary's.
(149, 150)
176 MISCELLANEOUS EXAMPLES IN THK
15. Since f 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 of Mary's less 10 cents= Susan's, Susan's + 10
cents = 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. | 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 of Homer's =-f of Silas's, of Homer's will=i
of -f or f of Silas's, and f or the whole of Homer's, 5 times $
or J T of Silas's ; and since Homer's exceeds Silas's by ^ of
Silas's, the 3 marbles must=^ of Silas's ; Jience Silas has 7
marbles and Homer 10.
too.
1. Since the first drink a gallon in 3 days, he will drink i
of a gallon in 1 day, and since the second drink a gallon in 4
days, he will drink of a gallon in 1 day ; both will drink
-L+i or T 7 j of a gallon in 1 day, and 1 gallon will last as many
days as T \, what they drink in 1 day, is contained times in j|
or 1 gallon ; T 7 is in }| If times. Therefore, etc.
2. Since Julia can do it in 7 hours, in 4 hours she can do -|
of it, and Jane must do the remaining | ; and since Jane does
^ in 4 hours, she will do | in ^ of 4 or 1^ hours, and ^, or thi
whole, in 7 times 1, or 9 hours. Therefore, etc.
3. Since the first can do it in 9 hours, he can do in 5
hours, and the second must do the remaining ; and since the
second pitches f in 5 hours, he can pitch | in | of 5, or 1 1
hours, and , or the whole, in 9 times 1|, or 11{ hours.
4. 3f =\ 9 - and 7| = - 5 8 8 -. Since the second pipe can empty
it in ^8- hours, it can empty jV of it in |, and f f in *Ji hours,
and the first must empty the remaining |f ; and since the first
can empty |f in \ 9 - hours, it can empty j\ in -j of 2 T *> or J
hour, and f f , or the whole, in 58 times |, or 7| hours.
(151)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 177
5. Since A can make a vest in of a day, he can make
as many vests in a day as | is contained times in f, or 1
vests ; B as many as is contained times in , or 1^ vests ;
and l^ + lj, or 3 vests what A and B can both do. C can
make as many as is contained times in $, or Ij- vests, and
3 1=1. Therefore, etc.
6. Susan can knit as many pairs as f is contained times
m |, or 1| pairs; Sarah can knit as many as f is contained
times in ^, or 2 pairs ; and If + 2^=4 pairs.
7. Since Sarah can knit 2i or 1 pairs in a day, she can
knit ^ of a pair in \ of a day, which is the part she must
knit for Susan.
8. Since Susan c|n knit If or f pairs in a day, she can
knit ~ of a pair in 1 of a day, which is the part she must knit
for Sarah.
9. Since Jason can hoe 10 rows in of an hour, he can
hoe 1 row in T ! 7 of , or -fa 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 | of an hour, he can hoe 1
row in T ' 7 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 + 16f, or 30 rows, in an hour; 1 row in
j\ of an hour ; and 10 rows in if or i of an hour.
10. Since Jesse can hoe 16| or ^ rows in an hour, in
i of an hour he can hoe | of 5 j or ^5 gi rows; leaving
If rows for Jason, who can hoe 131 or 4 ^ rows in an hour,
i of a row in ^ of an hour, and 1| or f rows in 5 times ?\
or | of an hour.
11. Since Jason can hoe 13^ or ^ rows in an hour, in of
an hour he can hoe 1 of 4 /, or 4 / 4 t rows 5 leaving 5f rows
for Jesse, who can hoe 16|, or *^-=% rows in an hour, of
a row in T ]^ of an hour, and ^-, or 5f rows in 50 times , J 3
of an hour.
12. See analysis of Example 9.
13. Since A and B can clear the field in 15 days, they can
(151. 152)
178 MISCELLANEOUS EXAMPLES IN THE
clear T ' f of it in 1 day, and y 9 ^ or of it in 9 days ; ana since
A and B clear f of it in 9 days, C must clear the remaining
| ; and if he clear f in 9 days, he will clear in of 9 or 4|
days, and 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
f of it in 1 day ; and | or ^ of it, is what B does more
in a day than C. As B and C dig it in 9 days, they can
dig of it in 1 day, and since B's day's work exceeds C's by
J I T of the well, 5*4 or j 5 2=^ of C's days, and of -f^ or T f T
=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 C
dig -i of it in 1 day, and C digs T T of it in 1 day, T f f or
T y4=whatB digs; hence B can dig it in as many days as 11
is contained times in 144 or ISyL days. Since A and B dig
of it in 1 day, and B digs T y T of it in 1 day, T y 4 - or y 1 ^
=what A digs in 1 day ; hence A can dig it in as many days
as 13 is contained times in 144 or lly 1 ^ days.
15. Since A digs yW, B T W, and C T T of it in 1 day
they will all dig T V 3 4 +TT4 ^T4T? or rVf f ^ * n * day ; and
it will take as many days as 29 is contained times in 144 01
4|| days.
16. Since Patrick and Peter can dig it in 15 days, they
can dig y'j of it in 1 day, and }| or f 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
in of a day, and in 13^ or \ days he can dig 40 times ^
or f of it, and Peter must dig the remaining ; and since ha
digs in 4J>- days, he will dig in % of *g- 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 f ; and since he digs | in 15 dayS|
he will dig | in A- of 15 or 5 days, and | in 8 times 5 or
40 days. As Patrick can dig 40 rods in 24 days, he can dig
(152)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 179
gij- of 40 or If rods in 1 day, and since Peter jan dig 40
rods in 40 days, he can dig 1 rod a day, and it will take him as
many days as 1| + 1 = 2f is contained times in 28, or 10 days.
17. Since 30 rods is f or of 40 rods, it will take each
man ^ as long to dig it. Since Patrick could 'dig it in 40 days,
he can dig 30 rods in 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 rods in
f of 30 or 2 2i 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
of it in 6| or % days, he can cut in j of 2 / or f days,
and f in 9 times f or 15 days. Since Harlan cuts of it in
6 1 or Y days, he can cut in } of \- or f days, and in 9
times | or 12 days.
19. Since the 3d does f as much as the 1st and 2d, the
work of the 1st and 2d is divided into 5 parts; and since
the 3d does f as much, the whole is divided into 5 + 2 or
7 parts. Since the 3d does & of the whole in 10 days, he
can do ^ in % of 10 or 5 days, and ^ 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 as much as the 2d, the whole is divided
nto 7 parts, of which the 1st does 3, and the 2d 4 parts.
Since the 1st does -f in 14 days, he can do | in ^ of 14 or 4|
days, and % in 7 times 4| or 32| days. Since the 2d does |
in 14 days, he can do | ii: | of 14 or 3^ days, and % in 7
times 3| or 241 days.
20. See Analysis of example 19.
21. Since the 1st can do it in 32 1 or \ 8 - days, he can do
jL of it in i of a day or -/j in a day ; and since the 3d can
do it in 35 days, he can do ^ T of it in 1 day ; and both can do
tfV + ST or TVo i n 1 d av an d ^ ne whole in as many days as
29 is contained times in 490 or 16/ days.
(152)
180 MISCELLANEOUS EXAMPLES IN THE
22. Since the 2d can do it in 24 or ^ days, he can dc T *j
of it in i of a day, or fa in a day ; and since the 3d can io Jj
of it in 1 day, the 2d and 3d can do fa + Jj, or J^. 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 T 7 T days.
23. Since B and C can do it in 12 days, they can do j% or
| of it in 8 days, and A must do the other 1 ; and, since A can
do i in 8 days, he can do f in 3 times 8 or 24 days. Since A
and B can do it in 10 days, they can do ^ or | of it in 8
days, and C must do the other ; and since C can do in 8
days, he can do f in 5 times 8, or 40 days. Since A can do
it in 24 days, he can do if or f^ of it in 10 days, and B must
do the remaining T 7 ; and since B can do ^ m 10 days, ^ e can
do j\ in I of 10 or If days, and jf in 12 times If or 17 j days.
24. Since the 1st and 2d will discharge it in 8 hours, they
discharge f- or \ of it in 4 hours, and the 3d must discharge
the other 1 ; arid since it discharges ^ in 4 hours, it will dis-
charge | 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 the
1st must discharge the other \ ; and since the 1st discharges
\ of it in 6 hours, it will discharge f in 4 times 6 or 24 hours.
Since the 1st and 3d discharge it in 6 hours, they will dis-
charge f or | of it in 4 hours, and the 2d must discharge the
other i ; and since the 2d discharges \ 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 \l or \
of it in 10 days, and C does the other \ ; and since C does \
in 10 days, he can do f in 2 times 10 or 20 days. Since B
and C can do it in 15 days, they do j| or f 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
30 days and C in 20 days, they can both do ^V + aV or TI ^
it in 1 day, and |f in 12 times 1 or 12 days.
26. Since it would last them all 30 days, they would eat ^
of it in 1 day, and 20 times ^ or f of it in 20 days, leaving \
(152, 153)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 181
of it to be eaten by the sister. Since the brother and servant
would eat it in 45 days, they would eat f f or f of it in 30
iays, and the sister must eat the other in 30 days.
101,
2. Since 2 plums was the increase given to 1 playmate, and
9 1 or 8 plums the increase given to all, there were as many
playmates as 2 is contained times in 8, whicL 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
of 15, or 5 must be the number. Therefore, etc.
4. Since the difference per yard was 12 cents 8 cents, 01
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 2^ times or f times the number, 15 must be |
times the number; of 15 or 3, ; and 6, the number.
6. Since f f, ^ of 4 or 2 must be % and 9 times 2 or
18=|.
7. Since the difference between 5| times and 3| times a
number is 2.1 times the number, -^\ of 21, or 1, must be T V,
and 10 times 1, or 10, }, or the number.
9. If we let 1 or | represent the whole number of chickens,
| times 5 + i- times 3=-^, will represent the whole number of
grains, that is, 0^=26. And since 26 is -L- 3 - times the number
of chickens, | or the whole number of chickens, was 3 times
of 26, or 6.
10. Since 26 is 5 times f + 3 times J, or ~y of the number,
V L - 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 -i- of the same num-
ber, plus the 3d, which gives 2 times f of the same number,
(153, 154)
182 MISCELLANEOUS EXAMPLES IN THE
plus tho 4th, which gives once the same number ; we have
V of the class equal to 29, and ^ f 29 or 1 is of 9 times
1 or 9. Therefore, etc.
13. Since 4 times 1 of a number, plus 3 times the num-
ber, or -y- times the number, is equal to 28 plus 5, or 33, T \
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 i of his age plus 4, the
2d gives , and the 3d gives less 4, we have the sum -L
equal to 50 ; r \ of 50 or 5 is i of his age, and 3 times 5 or
15 is f. Therefore, etc.
15. Since he paid $5 a head for ^ of the flock, the cost =5
times i or of his flock; $4 a head for as many more =4
times i or | of his flock; 83 a head for | of the remainder,
or J-r=3 times or \ of his flock ; and $2 a head for the rest,
or | = 2 times or J of his flock; and f + f + | + i = V-
That is, the number of dollars the flock cost = \ 3 of the num-
ber of sheep, and 115 is ^- 3 of 6 times ^ of 115 which is 30
Therefore, etc.
16. Since he received 6 dimes
each for i ={ x 6=f
for i of the remaining | and 3 more,
4 dimes each r=(| + 3) x4=f + 12 dimes
for i of the rest (which is f less 1
on each eight), or i + 1 = (i + l)x3 = f+ * "
for the rest (which is f less 1 on
each eight, also the 2 of last sale),
or f less 4 =(%*) X 2 ^1~1_? "
the whole number of dimes is equal
to 2J- of the baskets and 7 dimes besides, . . =- 3 g L + 7 "
hence $10 or 100 dimes less 7 dimes = 93 dimes^Y* ^ T ol
93 or 3rr|, and 8 eighths, 8 times 3 or 24. Therefore, etc.
18. 6 times a number equals y, 7 times of it plus 5
times | of it equals ^ and ^ less V i or i f ^ which,
(154, 155)
PROGRESS 13 INTELLECTUAL ARITHMETIC. 183
according to the condition of the question, is 4 ; and 4 is \
of 2 times 4 or 8. Therefore, etc.
19. 5 times the number, or \, left 4 cents, but 5 times 01
it, or y , plus 7 times oi it, or *- was it all of it ; and by the
condition of the question *- less \ or equals 4 ; and f is 2
times 4 or 8. Therefore, etc.
20. 4 times a number equals 2 T 8 > 5 times 4 of it equals Y-,
and zf less A equals ^ of it, which by the question is 6 ; and
6 is ^ of 7 times 1 of 6, which is 14. Therefore, etc.
21. 2 times a number equals f of it, 5 times of it equals
|-, and this plus 2 times i of it which is f equals of it,
and f less f, equals f of it, which by the conditions of the
question is 60 ; and 60 is f of 2 times of 60 which is 40.
Therefore, etc.
22. 2 times f of a number equals f of it, which is 8 more
than f ; hence 8 is f or ^ of it, and 2 times 8 or 16 is the
whole of it. Therefore etc.
1O2.
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 greatei
number. Therefore, etc.
4, Since 37^ is the sum of two numbers whose difference is
5, 37 less 5 or 32, is twice the less number ; of 32 is 16
the less number, which, increased by 5, equals 21 , 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)
184 MISCELLANEOUS EXAMPLES IN THE
he Las now ; and 21 less 11, or 10, equals what Horace has
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 quarts
equals what Mary has. Therefore, etc.
7. Since 47 is the sum of two numbers, one of which is 6
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 |
as much as the other, and both would hold 60 bushels, or 3
times as much as the small one ; i 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 84 more, both would have cost $100,
or 4 times the cost of the chain ; 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
115^ dimes equals what Harvey received, and 115^ dimes,
plus 22 dimes, or 137| dimes equals what Hiram received;
T ' T of 115i 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 ;
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. 185
and $124 less $30.25, or $93. 75, equals the cost of the horse
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 or the whole cost ; hence, 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.
J.4. 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 } of itself, or the differ-
ence equals f of the cost of the dish ; and f less f , or f of
the cost of the dish equals the cost of the cover ; and 24
dimes equals f 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 | of itself
plus twice the difference, or the difference equaling T 7 g- of the
greater, and |f T 7 g-= T 9 F of the greater equals the less; |
+yV ff i IT of 25 pounds, or 1 pound, is T 'g- 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 f of the greater, and
both numbers f of the greater ; | of 10 = 2 is 1 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 1 of the difference, equals
ft of the cost of the wood-work, the remaining T 2 T must equal
- of the difference, and the difference equals T 3 T of the cost
of the wood-work ; || less T 3 T equals T 8 T of the cost of iron-
ing ; || plus T 8 i or |f times the cost of the wood-work equali
(157)
186 MISCELLANEOUS EXAMPLES IN THB
$38. r V of $38 or $2, is T ' T , 11 times $2 or $22, is the cost of
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
to | of the difference, and the ribbon f 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 times 4
shillings is 12 shillings, the cost of the skates.
20. Had the harness cost $1 more, both would have cost & 15,
and the horse would cost 4 of $35 or $20, and the harness $
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 wero as many yearlings as $1 (the difference OH
1) is contained times in $8 (the difference on all), which is 8
times ; and 28 less 8 equals 20, the number of old sheep.
3. Had all been first quality, he would have paid $9( -, or
$8 more than he did ; and since the difference per barrel *aa
(157, 158)
PROGRESSIVE ^NTELLECTUAL ARITHMETIC. 187
$0., 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 f of the cost, ^ of $18 or $9, must have
been 1 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
is 12, the whole number.
6. Since she 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. | of 1 1 is T \ ; hence 12 is ^ of his flock, and 10 times
12, or 120 equals the number of sheep he had at first.
1 3. Since he paid f of the remainder, $3 must be f of it.
i of $3 is $1, and 5 times $1 is $5, the remainder, which by
thefirst payment wants $5 of being | of the whole ; $5 plus
$5 is $10, i of the whole, and 4 times $10 is $40, the whole
Therefore, etc.
14. Since he lent f , $3 plus $5 or $8, must have been A.
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 his
clothes, which lacks $10 of being 1 of his wages ; $36 plus
110, or $46 is 1 ; and 2 times $46, or $92 equals his wages.
(158, 159)
188 MISCELLANEOUS EXAMPLES IN THE
15. Since in $1 there are 10 dimes, he could be idle as
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, *d was idle 10 days.
1O4.
2. The part standing was divided into 4 equal parts, 3 of
which equaled the part broken off ; the sum of both pieces
was 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 ; i of 45 is 5 ; 4 times 5 or 20 equals the number
Horp ',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 of 63, or 28 cents. Therefore, etc.
6. Since 9 times |=f times the cost of wagon, equals the
cost of the horse, both cost 8 plus 9, or y- times the wagon ;
fo 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 l=f times the first,
both equaled |- 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 50
or 230 miles equals the distance from Bergen to Schenectady;
yzd as the distance from Utica to Schenectady is 1| J T S times
the distance from Bergen to Utica, the whole distance from
Bergen to Schenectady equals Y plus f, or 2 3 ; gV f ^30
tniles is 10 miles, and 15 times 10, or 150 miles equals the
(159, 160)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 189
distance from Schenectady to Bergen ; and 150 miles plua
60 miles, the distance from Bergen 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 arid the tail ; and as the
body was divided into fifths, 2 of which equaled the tail, we
have body and tail divided into 7 parts ; | of 14 inches is 2
mches v and 2 times 2 or 4 inches equals the tail.
10. Since the less has 7 parts and the greater 11, both have
18 ; T V of 36 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 milea
less, it would equal the / T 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 ; T L of 48 is 4 ; 11 times 4 iniles=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 parts, 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 the cover plus 12 pounds) and the cover
weighs 18 pounds, we have the cover, i 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 f ,
times 2 pounds, or 4 pounds ; and 18 pounds less 4 pounds, i
\\ 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
is $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 \ of Abel's, 2 halves would
(160, 101)
K. P. Q
190 MISCELLANEOUS EXAMPLES IN THE
equal twice , or ; then Abel had 4 parts, George 6 partis,,
and both 10 parts ; T ' 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 j, } would equal of 4, orf, and f 5
times f or y ; then the black ones were 7 parts, the gray ones
10, and both 17 ; T ' T 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 f equaled f, | would equal of , or -^-, and f,
3 times -j^ or {f ; one number is divided into sixteenths, 15 of
which equals the other, and f equal both ; ^ T 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 equals f , would equal i- of f , or /j, and -, 4
times ^_, or T \ ; the value of the contents is 15 parts, of the
purse 8 parts, and of both 23 parts ; -fa 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 into 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 f equals f, must equal of f, 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 equaled , 1 would equal -f of |, or , and J-, 4
times i, or |. John's age was divided into fifths, 4 of which
equaled Peter's, and both equaled f of John's ; of 36 years
4 years, } of John's ; 5 times 4 years =20 years, John's
age ; and 4 timeF 4 years' =16 years, Peter's age.
(161, 162)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 191
^5. Since f equaled J T 4 ) f would equal f ; and we have what
Tf'ds 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; T \ of 44 bushels=4 bushels, 1 part; and 5 times
4 bushels 20 bushels, what was wanting to fill the bin.
26. Since | of what it exceeded equaled f of what it lacked,
| would equal J T 5 - ; 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 ; ^ of 4-4 miles is 2
miles; 7 times 2 miles is 14 miles, the distance it lacked of
being 8? miles ; and 83 miles less 14 miles is 69 miles, the dis-
tance to Cincinnati.
27. Since f of Avhat 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.
2. Since equal -f + 9, { will equal of -f 4-9, which is $
-f8, , and 4 times f + 3, which is -| + 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 f equal less 4 rods, J- will equal ^ of f less 4
rods, which is f less 2 rods, and f, 3 times f less 2 rods, which
is f less 6 rods ; hence what one built equals 6 rods less than $
of what the other built, arid both built -y of the amount the
second did, less 6 rods ; 38 rods plus 6 rods, or 44 rods equals
u ; J T of 44 rods or 4 rods, is } of 5 times 4 rods or -20 rods,
what the second built; and 38 rods less 20 rods, or 18 rods
what the first built.
4. Since $ was 4 more than A, | would be 1 more than ,
(162, 163)
192 MISCELLANEOUS EXAMPLES IN THE
and |, 7 more than f ; hence what Richard sheared are divided
into 5 parts, Hiram's into 7 parts plus 7 sheep, and both into
12 parts plus Y ; 67 less 7 is 60 ; ^ of 60 = 25, the number
Richard sheared ; 67 less 25=42, the number Hiram sheared.
5. Since f of future time equaled f of the past + {
hours, | would equal ^ of 1 4- }| hours, which is ^ 4 T 8 5 , and |,
5 times ^+ T 8 j hours, which is | -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-f-f hours, or 24 hours;
24 hours less f hours is 211 hours, and f of 21 i hours is 8
hours, or the past time ; hence it was 8 o'clock A. M.
6. Since f of what his age lacked of being 100 years equaled
| of what it exceeded 64 years, + 9 years, } of his age would
equal of f + 9 years, which is + 1 year, and f , 8 times J- + 1
year, which is f + 8 years ; hence, what his age lacked of
being 100 years equaled 8 years more than f of what it ex-
ceeded 64 years, and ty 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, 7 ! T 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 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 | of 4 times 14 or
56. Therefore, etc.
9. The first price plus the second, equal to -f- 3 pounds,
equals the third price; 2 times j+3, equal to f of it + 6, equals
the whole of it, and 6 pounds must be f of it ; of 6 pounds or
2 pounds is } of it, and 5 times 2 = 10 pounds is the whole of it.
Jp~ 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 bush els + 5 bushels, or 3
bushels more than , those 3 bushels must equal the difference
(163, 164)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 193
between and | of them, or | of them ; and 3 bushels is of
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 i
of 60 miles or 20 miles is | of the distance ; and 5 times 20,
or 100 miles is the whole distance from Batavia to Corning.
12. Since he took $24 more than ^ of the whole for sheep
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 +
of the whole, +$6, + } of the whole + $2, or |{ of the
whole + $26, equal to the whole amount; hence T ' T of $26,
or $2 is ^ and 24 times $ 2 or $ 48 is wnat ne to k for all.
13. Of ^ that number of which i, of i and \ of 1 of 12 is
. i and i of i of 12 is 6, of 6 is 2, and 2 is j of 8 times 2,
or 16. Therefore, etc.
14. Since he earned f as much as he had spent, he only
lacks | of f =| 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 -, will cost \ of | of an eagle, or $2.
16. Since C is f 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 \ of it plus 6 years, plus \ of it plus 4 years, or f- 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 f of 36, or 24 is C's age.
17. Since C owns \ as much as A, he owns 6 acres more
than f as much as B; and we have what A owns, 12 more
than \ 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
i 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.
(164)
194 MISCELLANEOUS EXAMPLES IN THE
106.
2. Since the son's age is f of the father's, the 22 years tiio
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 18 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 the number
before removing ; i 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 ;
52 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.
Y. | less | is 2^, which by the condition of the question, is
8 more than | of his age; ? less | is y 1 ^, 6 is y'j of 72.
Therefore, etc.
8. Since he received { of his wages for his summer's la-
bor, i- as much, or of them in fall +$20, and $20 in spring,
$20 + $20 =$40 must be f or 1 of his wages, and 3 time*
f>4Q=$120, must be the whole amount.
(165)
PROGRESSIVE INTELLECTUAL ARITHMETIC. 195
10. Since A sold B f as much as B had, B now has J of
what he had ut first, which is of what A has left ; of \
i 7 j i g an d 4 times T 7 , or | what A has now f-f, what A
has left ; f f plus the f sold B, gives A ff of B's before the
sale; ^ 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 Y4 less 18 acres leaves 56
acres, what A has left.
11. Since f equal , 1 will equal \ of or , and f, 3 times
|- or | ; hence as the turkeys equal f of the chickens, 10 must
be the remaining { ; 4 times 10 is 40. Therefore, etc.
12. Since f of the price of the coat equaled \ of the price
of the suit, i would equal |- of \ or |, and f , 3 times | or f 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 j, and f are 8 times
$3 or $24. Therefore, etc.
13. Since f times his brother's equaled f of his, would
equal | of or y 1 ^ of his ; , which is 3 times T V, or T 3 7 and
the 14 more must be the remaining T 7 ; | 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 1^ times the number + 15,
or 15 equal to f of the number ; 1 of 15 or 5 is \ of it ; 2
times 5, or 10 is the less number; and 3 times 10, or 35 is
the larger.
15. Since f equal T 8 ^, } will equal \ of T 8 ^, or T 2 j ; and f, 5
times y* 5 or | ; and as the buggy cost | as much as the horse,
the difference, $40, must be ^ of the cost of the horse ; 3 times
$40 or $120 is the value of the horse ; and | of it, or $60, is
the value of the buggy.
16. 5 years since the mother's age was^S times Alice's, ana
(166)
1.96 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
as old as her mother. See Ex. 3, in this lesson.
17. Since Hobart has but f of his left, he lost of them
to Dwight, which, by the condition of the question, was equal
to i of Dwight's ; | must have equaled all of Dwight'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 ; 2f times this difference, or 19, is equal to
3J- times less 2f times=if times the less number; T ^ of 19
or 1 is ^ T ; 24 times 1=24, the less number ; and 24 + 16 =
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 would 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 4i times B's age then ; hence we have
A's age now equal to 4i times B's age at first ;
J^'g 44 it gJL H It It II
Q'g it it tt " ] " " " " "
Now if 4 years be added to B's age, f of the sum, or 4} times
B's age as first + 3 years, is equal to A's age, and 4 times B's
*t 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 ] times
8, or 80 years old.
(166, 167)
PROGRESSIVE INTELLECTUAL AlilTHMETIC. 197
107.
2. In as many hours as 2 miles, the number he gained in 1
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 gains
in running 9, is contained times in 28, the number to be
gained; 2 is in 28 14 times, and 14 time? 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 divided into eighths, 1 of
which equaled the distance he was ahead of A, A must have
run I 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, tho
cost of the horse ; and since B paid f 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, the 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
gained after 3 o'clock before they are opposite ; 11 is in 45 4 T *|
times, and 4^ times 12 is 49 T 1 T spaces. Therefore it would
be 49 T 1 T minutes past 3 o'clock.
9. Since 3 of the hound's leaps equal 6 of the fox's, 1 will
equal \ 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
(107, 168)
11)8 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 7 is 280. Therefore, etc.
10. Since the distance the sheep ran was divided into 5 parts,
3 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 equals 3 parts, or the distance betwee
thorn.
11. Since the interest at 5 per cent., for 2 years 7 months
and 6 days, is -ffa of the principal, the amount will be ||f ;
_i_ of $2260 is $20. and 100 times $20 = $2000, the sum at
interest ; and since B's money equaled -| of A's, the whole
equaled f c^ A's ; i of $2000 is $400 or | of what A had in,
which is ] of all, and 5 times $400 or $2000 equals A's ;
$400 equaled i- of what B had in, which is | of all, and
8 times $400 or $3200 equals B's.
12. Sincp B's fortune is 1 times A's, i of A's is equal to |
of B's, and th^ hiterest on it for 5 years at 6 per cent, would
equal ft of it | of $600, or $200, is T V of 10 times $200,
or $2000; \ 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's.
13. Since he lost 8 per cent, or -/j of the cost on the sale,
he sold for f f of the co.'t ; hence { of his calves arid f O f his
sheep cost $25, and 4 times $25, or $100 is the cost of all
the calves and f of the sheep ; this exceeds the whole cost by
$24, which must equal the cost of the f of the sheep ovei
the whole number ; -J- of $24 is $, and 5 times $8 is $40
which would buy 20 sheep at $2 ; $76 less, $40 givep $36
for calves, which would buy 12 calves at f 3 Therefore, etc
(168)
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