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EctucX * l"* .5 ^.^3)
Harvard College
Library
By Exchange
.1
3 2044 096 995 006
KEY
TO
HIGHER ARITHMETIC,
CONTAINING THE
ANSWERS TO THE EXAMPLES;
TOGETHER WITH
MANY SUGGESTIONS AND THE SOLUTION
OF THE
MORE DIFFICULT QUESTIONS.
BY JAMES B. THOMSON, A.M.,
AUTHOR Of MENTAL ARITHMETIC } EXERCISES IN ARITHMETICAL ANALYSIS J
PRACTICAL ARITHMETIC ', HIGHER ARITHMETIC J THOMSON'S
day's algebra; leoendre's GEOMETRY, ETC.
REVISED EDITION.
NEW YORK:
PUBLISHED BY MARK H. NEWMAN & CO.
No. 199 BROADWAY.
1852.
•
t.
w'i-Q 2/ ]ft^
'V^VMftaMMAWVI
Entered according to Act of Congress, in the year 1847,
BY JAMES B. THOMBOX,
in the Clerk's Office of the Northern District of New York, v
THOMAS B. SMITH, STIMOTYFBB*
816 WILLIAM STRUT, H. T.
1. D. BIOPOBD, PRINTER,
138 FULTOH 8TRKBT.
KEY.
SECTION I.
NUMERATION.
Abt. 40«
Ex. 1. Three thousand, five hundred and six.
2. Six thousand, and thirty-four.
3. Five thousand, and sixty.
4. Ninety thousand, six hundred and twenty-one.
5. Seventy-three thousand, and forty.
6. Four hundred and fifty thousand, three hundred and two.
1. Six hundred and three thousand, two hundred and sixty.
8. One hundred and thirty thousand, and seventy.
9. Two millions, twenty-one thousand, three hundred and five.
10. Four millions, five hundred and six thousand, five hundred
and eighty.
11. Seven hundred and six thousand, three hundred and five.
12. One million, six hundred and forty thousand, and thirty.
13. Eight hundred and thirty thousand, and six.
14. Seventy millions, nine hundred thousand, and thirty-eight.
15. Three millions, sixty-seven thousand, and three hundred.
16. Twelve millions, six hundred and four thousand, three hun-
dred and twenty-one.
17. Seventy millions, and three thousand.
18. One hundred and sixty-one millions, ten thousand, six
hundred and two.
19. Eighty millions, three hundred and sixty-seven thousand,
eight hundred and thirty.
20. Four hundred millions, thirty-one thousand, two hundred
and fifty-six.
4 NUMERATION. [SECT. L
21. Nine hundred and sixty-seven millions, fifty-eight thousand,
seven hundred and thirteen.
22. Thirty-two millions, one hundred thousand, and forty.
23. One hundred and six millions, three hundred and twenty
thousand.
24. Seven hundred and eighty millions, five hundred and seven
thousand, and thirty-one.
25. Four millions, sixty-three thousand, one hundred and seven.
26. Twenty-nine millions, thirty-eight thousand, four hundred
and fifty.
27. One billion, forty-six millions, three hundred and forty-
seven thousand, and twenty-five.
28. Twenty billions, three hundred and eighty millions, seven
hundred and twenty thousand.
29. Eight billions, five hundred and three millions, four hundred
and sixty-seven thousand, and thirty-nine.
30. Four hundred and fifty billions, six hundred and seventy
millions, four hundred and twelve thousand, four hundred and
mxty-three.
31. Four hundred and thirty billions, eight hundred and twelve
millions, six hundred and forty-one.
32. Five trillions, two hundred billions, two hundred and forty
millions, three hundred and one thousand.
33. Ninety-eight billions, seven hundred and sixty millions, two
hundred and sixteen.
34. Eighty-two trillions, six hundred billions, three hundred
and eighty-one millions.
35. Four hundred and three trillions, seventy billions, three
millions, four hundred and sixty-two thousand.
36. One hundred and twenty trillions, three hundred and forty
billions, seventy-eight millions, nine hundred and ten thousand,
three hundred and fifty-six.
37. Forty-three trillions, six hundred and one billions, three
hundred and forty-five thousand. ,
38. Five hundred and six trillions, three hundred and two
billions, eight hundred and seventy millions, forty-five thousand,
three hundred and eighty.
ART. 42.] NOTATION. 5
39. Forty-two quadrillions, eight trillions, one hundred and
twenty billions, five hundred and thirty-seven millions, sixty-two
thousand, and thirty-five.
40. Six hundred and fifty-three quadrillions, one hundred and
seven trillions, eight hundred and forty-three billions, six hundred
and four millions, eight hundred and ninety- three thousand, and
forty-eight.
41. Two hundred and ten septillions, two hundred and fifty-six
sextillions, thirty-one quintillions, four hundred and two quad-
* rillions, three hundred and eighty-five trillions, two hundred and
ninety billions, eight hundred and forty-five millions, three hun-
dred and eighty-one thousand, four hundred and sixty-seven*
42. Three hundred and sixty-one octillions, four hundred and
thirty-eight septillions, two hundred and one sextillions, two hun-
dred and nineteen quintillions, seven hundred and sixty-three
quadrillions, two hundred and eighty-one trillions, five hundred
and seventy-two billions, eight hundred and twenty-nine millions,
three hundred and eighteen thousand, two hundred and seventy-
eight
NOTATION.— Abt. 42.
1. 2,109.
2. 20,057.
8. 65,003.
4. 105,010.
-5. 710,301.
6. 2,063,008.
7. 14,000,056.
8. 440,000,072.
9. 6,006,006,006.
10. 45,000,340,076.
11. 556,003,264.
12. 810,010,075,000.
13. 96,700,000,000,054.
14. 349,005,007,004,000,020.
15. 19,000,000,000,000,000,000.
6 addition. [Sect. II.
16. 630,000,000,000,000,000,000,000.
17. 298,000,000,000,000,000,000,000,000.
18. 74,000,000,000,000,000,000,000,000,000.
19. 410,000,000,000,000,000,000,000,000,000,000,000.
20. 863,000,000,000,000,000,000,000,000,000,000,000,000,-
000.
21. 935,000,000,000,000,000,000,000,000,000,000,000,000,-
000,000.
22. 673,017,000,000,000,000,045.
23. 20,648,000,025,000.
SECTION II.
ADDITION.
EXAMPLES FOR PRACTICE.
Suggestion. — The beauty and excellence of a recitation depend not only on
the correctness of the answer, but also upon the manner of reciting. A scholar
with a wrong answer often obtains more credit at a public examination than
another with a correct answer, simply from the propriety and elegance with
which he explains his operation. The formation of correct and tasteful habits
of reciting are therefore deemed of great importance, and should receive par-
ticular attention both from the teacher and pupil. In the subsequent pages
of the Key, the method of explaining the operation upon the slate or black-
board is frequently given in full, which may serve as a model recitation^ varying
and adapting it to the conditions of different examples under the same rule.
MODEL RECITATION. ART. 59.
Ex. 1. Having written the given numbers under each Operation.
other, with units under units, tens under tens, &c., we draw 2468
a line below them, and proceed thus : seven, twelve, seven- 1645
teen, twenty-five. (Art. 60. Note 1.)* Write the 5 units 865
under the column of units, because they are units, and carry 467
the 2 tens to the column of tens because they are tens. Next, Ans. 5445 dolla
two, eight, fourteen, eighteen, twenty-four; set the 4 under
the column added, and carry the 2 to the next column because it denotes hcu>
* The references are made to the Arithmetic, unless otherwise slated.
A*ts. 5&-61.]
ADDITION.
dreds. Then, two, six, twelve, fourteen, twenty, twenty-four; set the 4 under
the column added, and carry the 2 to the next column because it is thousands.
Two, three, five. Since there are no more columns to add, we set down the
whole sum. The man therefore paid 5445 doHars.
2. 41,757 bushels.
*
28. 97,059,404.
3. 11,596 pounds.
29. 1,038,220,930.
4. 31,551 dollars.
5. 5,583 dollars.
Art. 60«
6. 65,440 square miles.
30, 31. See Book.
7. 102,451 square
miles.
32. 570,805 dollars.
8. 528,524 square
miles.
33. 6,460,458 yards.
9. 666,327 square
miles.
34. 6,657,039 pounds.
10. 1,362,742 square miles.
35. 9,429;190.
11. 233,890.
36. 11,178,170.
12. 828,463.
37. 10,306,156.
13. 990,240.
38. 10,662,291.
14. 96,181,521.
15. 127,215,713.
•
Art. 61.
16. 869,754,587.
'
39, 40. See Book.
17. 288,011,295.
41. 214.
18. 14,303,433.
42. 253.
19. 100,611,775.
43. 276.
20. 1,805,851,434.
i
44. 19,443.
21. 337,351.
45. 20,714.
22. 7,221.
46. 2,476,372.
23. 4,251,988.
■
47. 132,065,946 dollars.
24. 3,795.
■
.48. 107,109,740 dollars.
25. 73,464,390.
49. 2,069,857 tons.
26. 33,604,444.
50. 57,981,492 dollars.
27. 15,821,984.
'
51. See Book.
EXERCISES ON THE BLACK-BOARD.
Suggestions to Teachers.— It is desirable to give frequent exercises to the
class on the black-board and their slates, and it is convenient to be able to
ascertain at a glance, whether the class have obtained the right answer.
The following suggestions may be of service to the young teacher.
8 addition. [Sect. II.
I. If the class is unacquainted with muUipHaiiwn, the Operation.
teacher may direct them to write upon the black-board, or 4U82
their slates, any number, as 4682, and then require them 4682
to write the same number under this any number of times, 4683
as 4 times. By multiplication, the teacher will see at a 4683
glance that the answer is 18728. T8721 Ans.
II. Another convenient method is the following:
1. Set down any number at pleasure, as 5643, and under Operation.
it write any other number, as 3356. Now beginning with* 5643
the 3, the left hand figure of the second number, subtract 3356
each figure of it from 9 till you come to the 6, the right hand 7644
figure, which subtract from 10. The sum of these three 15643 Ans.
numbers is the same as the upper number with 1 prefixed to
it. If you wish to have the answer like the second number with 1 prefixed
to it, subtract the first number continually from 9 till you come to the right
hand figure, which must be taken from 10 as before.
2. Take any other numbers, as 5361, 31G3, and 4237, and Operation.
set them under each other. Now subtracting the 3d and 3d 5361
numbers successively from 9 as before, we obtain 6837 tor 3163
the 4th, and 5763 for the 5th. The sum of all the numbers 4337
is the first namber with as many units prefixed to it as yoa 6837
have performed subtractions ; that is, 35361. If you wish 5763
to have the answer like the 2d number, with 3 prefixed to 35361 Ant
it, subtract the 1st and 3d from 9 as before ; if you wish to
have it like the 3d, subtract the 1st and 3d from 9, Sus.
3. Again, take the numbers 63315, and 24G81, and set Operation,
one under the other. Now to obtain a 3d number, take the 63315
3d from 9 as before ; for the 4th number take any number 34681
at random, as 17303 ■ and for the 5th take the 4th from 9. 75319
For the 6th take any number, as 38035, and for the 7th 17203
subtract the last from 9 as before. Now having made three 83797
subtractions, the answer is the first number with 3 prefixed 38035
to it. If you wish to have the answer like the second, sub- 61965
tract the first from 9, &c. 363215 Ans.
This principle admits of a great varUly of applications, a
can of course vary it to suit his convenience.
1 the teacher
Art. 76.]
SUBTRACTION.
SECTION III.
»
SUBTRACTION.
EXAMPLES FOR PRACTICE.
Art.
76.
1. 7,095 dollars.
32. 55,352,005.
2. 28,984 bushels.
33. 19,957,466.
3. 30,954 dollars.
34. 77,919,261.
4. 46,025 dollars.
35. 70,051,563.
5. 58,000,000 miles.
36. 53,201,371.
6. 6,327,597 dollars.
37. 25,311,703.
7. 26,176,670 dollars.
38. 86,282,745.
8. 1,644,737 dollars.
39. 85,807,625.
9. 7,977,899 dollars.
40. 1,598.
10. 12,280,043 dollars.
41. 4,004.
11. 23,563,746 dollars.
42. 1,384.
12. 430,143 tons.
43. 14,061.
13. 149,237.
44. 12,494.
14. 3,393,329.
45. 11,547.
15. 54,399,581.
46. 3,295.
16. 8,825,431.
47. 1,606.
17. 4,001,722.
48. 3,707.
18. 2,601,900.
49. 2,664.
19. 5,313,439.
50. 925.
20. 543,679.
51. 1,511.
21. 2,007,984.
52. 41,845.
22. 45,103,074.
{ Wife rec'd. 46,900 dolls.
' /Hospital" 69,450 dolls.
23. 66,729,549.
24. 72,820,280.
54. Lost 2,410 dollars.
25. 55,301,760.
1 175250
26. 80,200,180. f
55. < 3425
27. 95,658,143.
( 171,825 Ana.
28. 9,000,001.
56. 1,674,737 dollars.
29. 99,899,999.
57: 97 dollars.
30. 83,128,433.
58. 3893 dollars.
31. 40,592,424.
59. See Book.
i*
MULTIPLICATION.
[Sect. IV
SECTION IV.
MULTIPLICATION.
EXAMPLES FOR PRACTICE.
MODEL RECITATION. ART. 93.
El. 1. Analysis.— If 1 a
re casta 57 dollars, 435 acrei Operation
will coat 435 limra as mc
ch ; and 435 dm
8 57 dollars 435
are 24195 dollars. Hence, 435 acres will coat 34795 dol- 57
lore. Since the product o
any two' numbers
is the same 3045
in whatever order they an
multiplied, for convenience 2175
we place the larger fur the
multiplicand. (A
rt. 83.) Am. 34795
2. 36,099 dollars.
23.
12,810,000.
3. 56,700 dollars.
24.
48,288,058.
4, 90,520 miles.
25.
3,473,567,604.
6. 74,175 pounds.
6. 372,500 days.
27.
9,313,702,853.
7. 960,000 rods.
28.
67,226,401,140.
8. 20,835.
20.
239,968,374,861.
9. 21,576.
30.
449,148,410,434.
10. 68,198.
31.
289,975,559,744.
11. 176,400.
32.
294,144,537,440.
12. 1,554,768.
33.
335,834,314,400.
13. 6,497,800.
34.
18,884,782,688.
14. 1,674,918.
35
109,588,282,650.
Ifil 3,931,476.
36
654,638,320,927.
16. 415,143,630.
37
396,890,151,372.
17. 31,884,470.
38
654,270,292,192.
18. 8,468,670.
39
2,985,984.
19. 43,506,216.
40
57,111,104,051.
20. 11,847,672.
41
60,435,595,442,394.
21. 57,380,625.
42
87,112,343,040,000.
22. 11,050,155,200.
AftTS. 93-100.] MULTIPLICATION.
11
CONTRACTIONS IN MULTIPLICATION.
1. 3 and 3 ; 5 and 2 ; 7 and 2
2. 7 and 5 ; 9 and 6 ; 7 and 8
3. 5 and 9 ; 8 and 9 ; 8 and 8
Art. 95.
11 and 2. f
7 and 9.
9 and 9 ; 8 and 12.
Arts. 96, 97.
4. 8=4X2=2X2X2;
16=8X2=4X4=4X2X2=2X2X2X2;
18=9X2=6X3=3X3X2;
20=10X2=5X4=5X2X2;
24=12X2=8X3=6X4=6X2X2=4X3X2=3X2X2X2.
6\ 27=9X3=3X3X3;
32=8X4=8X2X2=4X4X2=4X2X2X2=2X2X2X
2X2;
36=12X3=9X4=6X6=9X2X2=6X3X2=4X3X3=
3X3X2X2;
40=10X4=8X5 = 10X2X2=5X4X2=5X2X2X2;
48=12X4=8X6=12X2X2=8X3X2=6X4X2=6X2
X2X2=4X3X2X2=3X2X2X2X2.
6, 7. See Book.
8. 1776 dollar
9. 5760 dollars.
10. 8100 dollars.
11. 5782 shillings.
12. 23808 miles.
13. 11736 dollars.
14. 19845 shillings.
15. 32256 dollars.
Art. 99.
16. See Book.
17. 46,500 bushels.
18. 365,000 days.
19. 1,534,860,000.
20. 312,046,700,000.
21. 52,690,078,000,000.
22. 6,890,634,570,000,000.
23. 494,603,050,600,000,000.
24. 87,831,206,507,000,000,-
000.
25. 678,560,051,090,000,000,-
000.
Art. lOO.
26. 27. See Book.
28. 18,750 pounds.
29. 96,000 pounds.
12
MULTIPLICATION.
[Sect. IV.
80. 859,400,000.
58.
4629537.
81. 143,759,940,000.
82. 28,708,635,000,000.
Art. 106»
Art. lOl.
59 — 62. See Book.
63. 54530.
83. See Book.
64.
72819.
34. 123,240,000.
65.
346896.
85. 2,309,760,000.
86. 26,366,200,000.
66.
69412
95436
87. 144,447,000,000.
38. See Book.
89. 31,276,000,000.
624708 X6X4
3748248
2498832
40. 3,747,600,000,000.
41. 18,054,680,000,000.
Al * wvl/U mm
6624403632 Am>.
42. 664,726,500,000,000.
67.
324325
43. 1,075,635,900,000,000.
54426
Art. 103.
1945950X7X9
13621650
44. See Book.
17513550
45. 45514.
17651712450 Am.
46. 68476.
47. 400624.
68.
256721
48. 907002.
85632
2053768 X7X4
Art. 104.
14376376
49. See Book.
50. 132525.
8215072
21983532672 Jns.
51. 807664.
52. 2333616.
Art. 107.
58. 5691627.
69,
70. See Book.
Art. 105.
71.
72.
625.
2916.
54. See Book.
73.
5184.
55. 474309.
74.
4140.
56. 6027966.
75.
27986.
57. 7293699.
76.
154,250.
ASTS. 101-108.] MULTIPLICATION.
18
77. 11,348,400.
78. 34,639,552.
Art. 108.
19. 2,685,942.
80. 2,801,960.
81. 72,156,000.
82. 1,680,000,000.
83. 2,000,000.000.
84. 43,644,865.
85. 81,708,550.
86. 401,939,564.
87. 476,413,195.
88. 62,220,780.
89. 637,049,231.
90. 406,101,366.
91. 42,261,696.
92. 504,159,579.
93. 6,724,232,757.
94. 7,306,359.
95. 21,760,506.
96. 39,429,936.
97. 2,283,344,802.
98.
26397
24648
158382 X8X4
1267056
633528
650633256 Ana.
Or, 26397
24648
211176X8X3
1689408
633528
650633256 Ana.
99. 180,600,000.
100. 2,722,946,304.
101. 2,172,069,918.
102. 7,225.
103. 65,536.
104. 104,650.
105. 12,744,790.
106. 31,049,291,000.
107. 2,732,116,062,240.
108. 222,310,980,000.
109. 20,066,857,745,896.
110. 1,256,700,743,298.
111. 37,968,867,755.
112. 39,073,118,478.
113. 1,021,288,493,520.
114. 1,421,400,000,000.
115. 60,302,400,000,000.
116. 91,300,203,000,000.
117. 680,040,000,000,000.
118. 4,000,000,000,000,000.
14
DIVISION.
[Sect. V.
SECTION V.
DIVISION.
EXAMPLES FOR PRACTICE.
MODEL RECITATION.— ART. 127*
Operation*
66)2970(45
264
Ex. 1. Analysis. — Since 66 acres produced 2970 bushels,
I acre must have produced as many bushels, as 66 is con-
tained times in 2970. Haying written the divisor on the
left of the dividend, with a curve line between them, we 330
find that 66 is contained in 297, the fewest figures on the 330
left of the dividend that will contain it, 4 times, and 33
remainder. Placing the quotient figure on the right, we bring down the next
figure of the dividend to the right of the remainder, which makes 330. Now
66 is contained in 330, 5 times, and no remainder. Hence, 1 acre produced 45
bushels.
2. 85 barrels.
3. 68£H dollars.
4. 3 dollars.
5. 68-H- dollars.
6. 73972-fffr dollars.
7. 20-tf£days.
8. lVS-ftV days.
9. 2773ff
10. 1139-jfr.
11. 1443^.
12. 1489H-
13. 5697ff.
14. 3823£jh
15. 4166ft-.
16. 21276-Jf
17. 12152A-
18. 191^V..
19. 878.
20. 48.
21. 48ffH-
22. 87-iVi-.
23. 108.
24. 45.
25. 3679.
26. 4500.
27. 50830^.
28. 630.
29. 235.
30. 648.
31. 267l0iH.
32. 563.
33. 882621 lff*f.
34. 23434402^WV.
35. 82645 l-rWiVs".
36. 1387805ffBlHh
37. 9009009009009rh>
38. 900090009000-rW.
39. 90000900009TTTTT.
Arts. 127-139.1
DIVISION.
15
CONTRACTIONS IN DIVISION.
Art. 129.
1, 2. See Book.
3. 132f£ acres.
4. 672.
5. 460.
6. 205.
7. 1265.
Art. 131*
8. 20 ; 34 ; 56 dimes.
9. 650; 7650; 43200 dollars.
10. 267, and 50000 Rem.
11. 144, and 360791 Rem.
12. 5823, and 67180309 Rem.
Art. 132*
13. See Book.
14. 105 buggies.
15. 184 barrels.
16. 197++.
Note. — When the figures cut off from
the right of the dividend are ciphers,
if the remainder is placed over the
divisor and annexed to the quotient,
the ciphers cut off from the divisor
and dividend may obviously both be
omitted.
Art. 133.
17. See Book.
18. 36.
19. 68.
20. 36ff.
21. 75.
22. See Book.
Art. 134.
23. 1207.
24. 1690.
25. 65 12+.
26. 8654.
Art. 135.
27. 83++.
28. 76+£.
29. 77+£.
30. 142+£
Art. 136.
31. 94.
32. 194+f
33. 1693+f.
34. 3795A.
Art. 137.
35. 67.
36. 203<f¥y.
Art. 138.
37. 15.
38. 16-ftV
39. 17.
40. 30.
Art. 139.
41. 250 days.
42. 950 years.
43. l(hV dolls. (See Note, Art.
132. Ex. 16.)
44. 285*i dollars.
10
APPLICATIONS OF
[Sect. V.
45. 89-AV dollars.
46. 2 Hf dollars.
47. ll-^ft- dollars.
48. 54-g^- dollars.
49. 2 19*ff dollars.
50. 18$$.
51. 13529f£.
52. 12466H.
53. 12454ft.
54. 134469 13f
55. 2283781+i.
56. 941501if.
57. 478676**.
58. 59207.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
1826896.
138791ff.
65964+ffr.
6162^.
15831*£fc
914-3X&A
4134-t+f.
3966#ii.
1658fff.
7405-AV.
4362tVt.
3186^.
97«*Jfr
920 T ^Sfr jr .
CANCELATION.
Art. 150*
1. See Book.
2. 45.
3. 63.
4. 65.
5. 73.
Art. 151*
6. 7. See Book.
8. *X*X*X2)I*X#X12
Am. 6.
Nffte. — The process of canceling the
equal factors in the divisor and divi-
dend may, perhaps, be more readily
seen by arranging the factors-which
compose the divisor on the left of it
perpendicular line, and those which
compose the dividend on the right
Thus,
9.
n
2,*
*
4
7
10.
*
4
$
7=3* Ana.
1$
U
3 An*.
APPLICATIONS OP THE FUNDAMENTAL RULES.
Art. 152.
1. See Book.
2. 255 acres.
8. 925 bushels.
Art. 153.
4. See Book.
5. 190 dollars.
6. 1125 sheep.
AlTfl. 150-163.] FUNDAMENTAL RULES.
IT
Art. 154*
7. See Book.
8. 2240 dollars.
9. 3436 dollars.
Art. 155*
10. See Book.
11. 79 years, age of younger.
94 " " older.
12. 5 1 0-J d oils, price of carriage.
34 5£ dolls. " horses.
Art. 156.
13. See Book.
14. 65 years.
15. 175 rods.
Art. 157.
16. See Book.
17. 187825.
18. 1033062.
Art. 158*
19. See Book.
20. 48 beggars.
21*. 20 flocks.
Art. 159*
22. See Book.
23. 20 years.
24. 10 months.
25. 1842.
26. 1062.
27. 632.
28. 974.
29. 7124 and 5516.
30. 13000 and 12264.
31. 21151 and 20975.
132. 786.
SECTION VI.
PROPERTIES OF NUMBERS.
DIFFERENT SCALES OF NOTATION.
Art. 162.
1 — 9. See Book.
10. 20212331.
11. 2350147.
12. 1331124.
13. 2024122.
14. 1522365.
15. See Book.
Art. 163.
16, 17. See Book.
18. 707961.
19.. 1036993.
20. 9753020.
21. 360913096.
22. 1614386.
23. 118620366.
24. 3879090582.
18
ANALYSIS OF
[Sect. VL
ANALYSIS OF COMPOSITE NUMBERS.
Art. 165*
35.
3, and 17.
36.
2, 2, and 13.
1 — 3. See Book.
37.
2, 3, 3, and 3.
4. 9=3X3.
.38.
5, and 11.
5. 2, and 6.
39.
2, 2, 2, and 7.
6. 2, 2, and 3.
40.
3, and 19.
7. 2, and 7.
41.
2, and 29.
8. 3, and 5.
42.
2, 2, 3, and 5.
9. 2, 2, 2, and 2.
43.
2, and 31.
10. 2, 3, and 3.
44.
3, 3, and 7.
11. 2, 2, and 5.
45.
2, 2, 2, 2, 2, and 2.
12. 3, and 7.
46.
5, and 13.
13. 2, and 11.
47.
2, 3, and 11.
14. 2, 2, 2, and 3.
48;
2, 2, and 17.
15. 5, and 5.
49.
3, and 23.
16. 2, and 13.
50.
2, 5, and 7.
If. 3, 3, and 3.
51.
2, 2, 2, 3, and 3.
18. 2, 2, and 7.
52.
2, and 37.
19. 2, 3, and 5.
53.
3, 5, and 5.
20. 2, 2, 2, 2, and 2.
54.
2/2, and 19. .
21. 3, and 11.
55.
7, and 11.
22. 2, and 17.
56.
2, 3, and 13.
23. 5, and 7.
57.
2, 2, 2, 2, and 5.
24. 2, 2, 3, and 3.
58.
3, 3, 3, and 3.
25. 2, and 19.
59.
2, and 41.
26. 3, and 13.
60.
2, 2, 3, and 7.
27. 2, 2, 2, and 5.
61.
5, and 17.
28. 2, 3, and 7.
62.
2, and 43.
29. 2, 2, and 11.
63.
3, and 29.
30. 3, 3, and 5.
64.
2, 2, 2, and 11.
31. 2, and 23.
65.
2, 3, 3, and 5.
32. 2, 2, % 2, and 3.
66.
7, and 13.
33. 7, and 7.
67.
2, 2, and 23.
84.. 2, 5, and 5.
68.
3, and 31.
Arts. 165-171.] composite numbers.
19
69. 2, and 47.
70. 5, and 19.
71. 2, 2, 2, 2, 2, and 3.
72. 2, 7, and 7.
73. 3, 3, and 11.
74. 2, 2, 5, and 5.
75. 2, 2, 3, 3, and 3.
76. 120=2X2X2X3X5.
144=2X2X2X2X3X3
77. 180=2X2X3X3X5.
420=2X2X3X5X7.
78. 714=2X3X7X17.
836 = 2X2X11X19.
79. 574=2X7X41.
2898=2X3X3X7X23.
80. 11492=2X2X13X13X17
980=2X2X5X7X7.
82.
83.
81. 650=2X5X5X13.
1728=2X2X2X2X2X2
X3X3X3.
1492=2X2X373.
8032=2X2X2X2X2X
251.
4604=2X2X1151.
16806=2X3X2801.
84. 71640=2X2X2X3X3X5
X199.
20780=2X2X5X1039.
85. 84570=2X3X5X2819.
65480=2X2X2X5X1637,
86. 92352=2X2X2X2X2X2
X3X13X37.
81660=2 X 2 X 3X5X1361.
GREATEST COMMON DIVISOR.
Art.
168,
1.
See Book.
2.
3.
3.
7.
4.
fr.
5.
2.
Art.
170.
6,
7. See Book.
8.
15.
9.
14.
10.
111.
11.
39.
12.
1 ; that is,
they are
prime
to each other.
13.
See Book.
•
14.
3,
15.
16.
Art.
171.
16.
See Book.
17.
15.
18.
12.
19.
18.
20.
35.
21.
6.
22.
28.
REDUCTION OF
[Sect. VII
LEAST COMMON MULTIPLE.
Art. 176.
1 — 3. See Book.
4. 90.
5. 144.
6. 180.
1. 360.
8. 720.
9. 12600.
10. 604.
11. 1184.
12. 15015.
Art. 177.
18. See Book.
14. 144.
15. 600.
16. -2520.
17. 252.
18. 1134.
19. 360.
20. See Book.
21. 600.
22. 1440.
23. 13824.
24. 51000.
SECTION VII.
REDUCTION OF FRACTIONS.
Art. 195.
1, 2. See Book. ~
3. f.
4. f .
5. *.
6. f.
7. h
8. -Jr.
9. yj.
10. -fr
11. First find the greatest com-
mon divisor of the nu-
merator and denominator,
which is 23 ; then divide
them both by it. Am. -H-.
12. The greatest common divi-
sor is 522. Ans. •}.
13. -H.
14. f.
15. i.
16: f.
IV. tVt.
Art. 196.
19, 20. See Book.
21. 9.
22. 5.
23. 3f
Arts. 176-200.]
FRACTIONS.
81
24.
9+.
25.
1.
26.
60.
27.
21.
28.
52.
29.
60*.
30.
icftttf
•
Art.
197.
31,
32. See Book.
33.
*&.
34.
V.
35.
*♦*.
36.
w.
37.
HP.
38.
fi 1,4 OB
6 •
30.
*¥*.
40.
saaafl
41.
10 •
^
42.
**£*:
43.
**W°.
44.
anHoayg
Art.
198.
45,
46. See Book.
47.
T»T«
48.
TtT»
49.
iVfr'
Art.
199.
50,
51. See Book.
5
52., of f of- of -,«_
6
53. lofjoflofloff:
7 4 W 8
54. ^
55. f.
56. - of ^r of T- of 2^=—
7 20 35
iVofe. — It ib sometimes more convenien
to arrange the terms of the fraction
on each side of a perpendicular line
placing the numerators, which an-
swer to dividends, on the right, and
the denominators, which answer to
divisors, on the left of the line. (Art.
184.)
57.
58.
102
8
It
17
1
136
Operation,
17
13
l3=-iHftr Ana.
*
9
9=-rh- Ana.
21
59. The mixed number 4-J mua
be reduced to an impropei
fraction, before arranging
the terms for cancelation.
(Art. 197.) Ana. f.
60. -fY
Art. 200.
61. 62. See Book.
Oo. xTo » TsT i Tire > T*V
22
FRACTIONS.
[Sect. VIL
A A 444* JLflL • 444 • *0»
04. tTo > "BTTT > TTo > TTo •
65. TTo » 870 > 2T0J 4 7 o •
«e. mi; mi; mi; WW-
67 grgo • 44d_g • 4JL3-& . Alio
u «» 7 5 6 I' > 756©> 7 5 6 > 7560*
es. mi; mi; mi; mi-
69. 3 '} So » 924 > 924W-
70 #£444. AJMLOJQL. 134.80
i V. 6 3000 » 6 5 » 6 5 •
•!• 66250 > 56 250 » 66 25 0*
Art. 201.
15, 14. See Book.
16. Vt; A-
76. \f\ tt> •§£"> tt»
77. if-; £$■; -JJ-.
7ft 44 • 44 • 44 • 44
<0. fY* TT> YT» Ttf'
7ft 44 • 44 • 44 • JL.
80.
81.
6 >
4 U f
6 *
44-
4 f
4.
6 >
-?4.
40 i
44
4 0*
It will be seen at a glance that
40 is the least common multi-
ple of the denominators. (Ait
201. Obs.3.)
44.
6 >
4Z..
6 »
44 * 44
6 9 6 0*
82.
qq 2JL • sg . 44 . 44
OO. 43* T*> 4*1 tlf.
84.
85.
86.
87.
88.
89.
44.
6 >
44 •
6 i
ft)
44
6 0*
-444..
10 8 »
"TflOffJ 1008 i
i Wg *
urn
4444
5 t 76 755 >
*
5'
H;
6 900 » 6 TOO > "6 80 >
.8 4, . SO . Tfl . JJ8S
T44 > T44 i TTT > TTT'
-84 8- » -34 7.. . .4.0 4= .
126 > 124 » 126 >
fW-
■Aftfr
on 444* 444. 444* 444
91. TTo"» "Ho J TTT> tW*
ADDITION OP FRACTIONS.
Art. 202.
1 — 3. See Book.
4. 3ffr.
5. 2-ft.
6. 2A.
»• 2*m or 2f«.
8. lfff.
9. 2ttt-
10. 3Hf.
11. 6f.
12. 2tff
13. 1-fr.
14. Trnr*
15. 2ttV
16. 21f|.
17. 11 J*.
18. 6 T .
19. 61*.
20. 15-ftV
Art. 203.
21. See Book.
22. VW, or VA; tVW, or tVV
23. 4Wr,orrSV; WW-
94 1 s_ . _«7g _ •
^*- TtiT* 86 OSS*
25. tttJ i tYV
26. ljii -jft".
27. l-gVr; Trt-
Aitr. 204.
28. See Book.
29. M*.
80. M* 1 .
Arts. 201-208.]
FRACTIONS.
31.
M*F, or Ho
±B_2JZ.
32. First reduce the fractions to
a common denominator,
(16,) and add them toge-
ther ; then add the whole
numbers, reduce them,
<fec, according to the rule.
j rf+«+i=«+-rV+-rf=
and 635+427+1625=2687
Now 2687+^=HV*. 4ns.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
3fiQ3 3 7»8
35(j0t|98fl t
7 4,fl 8 9,0.8 5 9
1 06 5 9
ft
5 3 3 711
b o
86
Aft
2 0,9A aa
nns
9fil.4 7fl5
40 '
1 73
H*".
3 83 5.2281
6 16
-2J,
^p.
28
SUBTRACTION OP FRACTIONS.
Art.
206.
16.
69-H-
3 —3. See Book.
17.
121£J.
4. tV.
•
18.
278+*.
5. A-
6. •^r=i".
Art. 208
»• ftVfl*
19,
20. See Book.
«. -WW.
21.
125-fr.
». -ft'W-
.22.
238tV
10. |f u'«
23.
137*.
11. ++.
«
24.
466f.
12. 23.
•
25.
291*.
1 O 111
13» TSVff.
-
26.
603:ft,
27.
974tV.
Art.
207.
28.
*
14. See Book.
29.
8263f.
15. 12^.
80.
»l«w
84
FRACTIONS.
[Sect. VII.
MULTIPLICATION OF FRACTIONS.
CASE I.
MODEL RECITATION. — ART. 211*
Operation,
A X 4=-jy. dolls,
and -^=2* dolls.
Ex. 2. Analysis. — Since 1 lb. costs -J of a
dollar, 4 lbs. will cost 4 times as much. Now
4 times * dolls, are •*£■ dolls., and are equal
to 2* dolls. Hence 4 lbs. will cost 2* dolls?
Or, since dividing the denominator by any
number multiplies the value of the fraction
by that number, we may divide the denominator by 4, and the result is * .
which is equal to 2* dollars, the same as before.
Or, *X4=*> or 2* dolls.
Art. 212.
3. See Book.
4. -^,=3*.
5. Y=12*.
6. 18-ft-.
7. 37*.
8. 48**.
10. 66ff*.
11. 167t*t.
12. 776A.
13. 662***.
Art. 21 3t
14. 15. See Book.
16. 735*.
17. 633*.
18. 3715*.
19. 4448*.
20. 2264^
21. 12519*.
Art. 215.
22. 23. See Book.
S4. 108.
25. 127.
26. 240.
27. 435.
28. 560.
29. 621*.
SO. 701*.
31. 76-rY
32. 514*.
33. 305***.
Art. 216.
34. See Bpok.
35. 10396*.
36. 17460**.
37. 366-!%.
38. 2067t*tv
39. 35650ff.
40. 235554**.
Art. 217.
41. 42. See Book.
43. 375.
44. 738.
45. 1178.
46. 3450.
Arts. 21 1-221.1
FRACTIONS.
2ft
47. 6795.
48. 6897.
49. 15282.
50. 29318.
51. 1280.
52. 279.
53. 4496.
54. 8113.
55. 10413tV
56. 5672++.
57. 5086?%,
58. 4345*2.
59. 74290tV
60. 92280+f.
CASE II.
Art. 219.
1, 2. See Book.
3. Tu = n-
4. TT6 = "mr*
5. -AV
6. +.
•• TT3T«
8. +++.
v. (t .
Art. 220.
10. See Book.
11. 17.
12. 53+.
13. 242++.
*14. 329.
15. 1362*.
16. 3198*$.
17. 451+++.
18. 834+Jf.
19. 2001tVr.
20. 3232-ftV.
21. 5143+++,-.
22. o998g s s f*
23. 167+++.
24. 53tV^.
25. 24091-f.
26. 466+*.
27. 300000.
28. 700.
30. 78+.
31. 47*V
32. 86.
33. 401+.
34. 1474tV5V.
35. 972+ dollars.
36. 6323^ dollars.
37. 550 ltV miles.
CONTRACTIONS IN MULTIPLICATION OF FRACTIONS.
Art. 221.
1, 2. See Book.
8. +.
4. A.
5. +.
6. -V-
*. *.
8. A
9. \.
=2+.
9ft
FRACTION?.
[Sect. VIL
10. 26.
11. *.
12. -jSr.
13. ±.
U. A-
15. -ft--
16. *«&.
Art. 222.
17. 18. See Book.
19. 493+.
20. 8533+.
21. 57600.
22. 99000.
23. 1871*
24. 14220.
Art. 223,
25. See Book.
26. 2133+.
27. 18466-?.
20. 5580.
29. 430000.
Art. ' 224.
30, 31. See Book.
32. 4*762+.
33. 15937*.
34. 40187*.
35. 65450.
Art. 225.
36. See Book.
37. 1278*.
38. 4083$,
39. 8113*.
40. 93833*.
DIVISION OP FRACTIONS,
MODEL RECITATION. ART. 226*
Ex. 2. Analysis. — If 5 bushels cost 11 twelfths
of a dollar, 1 bushel must cost 1 fifth of 11
Operation.
twelfths of a dollar. But, since we cannot 12 12x5 60
divide the numerator by 5 without a remainder,
we multiply the denominator by it; for, multiplying the denominator by any
number, divides the value of the fraction by that number. (Art. 128.)
Art. 227 •
8. See Book.
4. jft.
6. «/b=ft.
9. fr.
8. "A*=-6*-
9. -sWr=VW-
Art. 230.
10—13. See Book.
14. l,to.
Abts» 322-241.] FRACTIONS.
15. ^16*
41. 30**.*
16. 3£f.
42. 6<H$.
1?. ttt.
43. 6 A a $ .
Art. 232.
Art. 237.
18, 19. See Book.
44, 45. See Book. -
20. *f.
46. 383-1%.
21. ^
47. 54-fiftft.
22. -V=23i.
23. ^=40^.
Art. 239.
24. £.
48. See Book.
25. -iV=i-
49. **=2A.
^ e\ ft i
^o« -tj> — "Te*
50. -H.
Art. 234.
51. -ft.
52. f.
27, 28. See Book.
53. Y=31i.
29. 135.
54. -sV.
30. 168.
55. -V^lf
31. 11-fr
56. 1.
32. llff-
33. 9 rib'
Art. 240.
34. 13£ff.
57. See Book.
Art. 235.
58. 1-if.
59. A.
35, 36. See Book.
60. ^.
37. 13-pjff .
61. ^=5f.
38. o~ioQ(f*
Art. 241.
Art. 236.
62. See Book.
39. See Book.
63. +.
40. 220*.
64. *,
n
FRACTIONS.
[Sect. YIL
APPLICATION OP FRACTIONS.
MODEL RECITATION. Art. 242*
Ex. 1. Since he bought 15$ yds. of one man, 19$ of Operation.
another, &c., it is manifest that he bought of all 15$ yds.
+19$ yds. +12$ yds.+41-ft- yds. Now reducing the
fractions to the least common denominator and adding
them together, we have $$ =1$$. The sum of the
whole yards is 87 and 1 to carry makes 88. The
merchant therefore bought 88$$ yards.
1*1 = 15**
19|=HhV
12*=UhV
4lA==41iV
Ans. 88$$ ych.
2. 162-rV pounds.
3. 54$ dollars.
4. 14$$ dollars.
5. 62$$ dollars.
6. 635tV dollars.
1. 1548| dollars.
8. 6516$ dollars.
9. 1515 dollars.
10. 5606-rV shillings.
11. 483$ dollars.
12. 100 dollars.
13. 161 Hi dollars.
14. 4045523? pounds.
15. 1806$ dollars.
16. 3612$ bushels.
17. 30968? dollars.
18. 6939-A miles.
19. 5229 miles.
20. 9 175* dollars.
21. 165$ yards.
22. 218$ pounds.
23. 626$ gallons.
24. 20^- pounds.
25. 53$ yards.
26. 27 boxes.
27. 153$ barrels.
28. 29$$ suits.
29. 38-ft-rods.
30. 3$$$ dollars.
31. 6$ dollars.
32. 584$ bushels.
33. 4$ dozen.
34. 6^r cents.
35. 9$f$ shillings.
86. 13^ dollars.
3*7. 5$$$ dollars.
38. 42-rfrtons.
39. 1-rWs- dollars.
40. 1$$«$ dollars.
41. 3$$$ dollars.
42. 266 T $$ T days.
43. 33798$$$ dollars.
Arts. 242-282.] compound numbers.
20
SECTION VIII.
REDUCTION OF COMPOUND NUMBERS.
EXAMPLES FOR PRACTICE.
Art. 282.
27.
142560 ft.
28.
8553600 in.
1.
See Book.
29.
5280000 yds.
2.
68810 far.
30.
54 m. 7 fur. 88 r. 2 yds. 2 ft.
3.
86768 far.
31.
9 L 2 m. 4 fur. 31 r. 1+yds.
4.
284079 far.
2 ft. 7 in.
5.
96615 far.
32.
5031 rods.
6.
£25, 13s. 6d. 3 far.
33.
17 m. 20 r.
7.
£433, Is. 2d. 3 far.
34.
132105600 ft.
8.
266 guineas, 18s. 8d.
35.
2560 na.
9.
1448 sixpences.
36.
5000 qrs.
10.
6050 threepences.
37.
6396 yds. 2 qrs. 1 na.
11.
170472 grs.
88.
9302 F. e. 4 qrs. 3 na.
12.
9000 pwts.
39.
10156 na.
13.
1010047 grs.
40.
7116 qts.
14.
2 lbs. 1 oz. 10 pwts. 16 grs.
41.
693 gals.
15.
177 lbs. 9 oz. 12 pwts.
42.
26528 gi.
16.
1596 lbs.
43.
48 bar. 20 gals.
IT.
564000 oz.
44.
117 pi. 1 hhd. 46 gals.
18.
104300 lbs.
3 qts. 1 pt. 2 gi.
19.
71680000 drs.
45.
102128 gi.
20.
lOcwt. 16 lbs.
46.
12960 pts.
21.
133 T. 12 cwt. 35 lbs.
47.
87 bar. 26 gals.
22.
1 T. 202 lbs. 1 oz.
48.
630 hhds. 44 gals.
23.
9120 drs.
49.
19520 pts.
24.
37440 sc.
50.
488 qts.
25.
64 lbs. 11 oz. 5 drs.
51.
24440 qts.
26.
88 lbs. 4 oz. 7 drs. 2 sc.
52.
28992 pts.
80
DEDUCTION.
[Sect. VIII
53. 1427 bu. 1 pk.
54. 130100 qts.
55. 36360 min.
56. 31557600 sec.
57. 84 wks. 6 hrs. 45 min.
58. 65 d. 2 hrs. 4 min. 40 see.
59. 31556928 sec.
60. 946728000 sec.
61. Analysis. — Since every 7 days must have 1 Sabbath, every
7 years must have 1 year of Sabbaths ; therefore 70 years
must have 10 years of Sabbaths. Ans. 10 yrs. of Sabbaths.
Note. — On the supposition that each year consists of 365 days, 52 of which
are Sabbaths, a different result will be obtained. For 52X70=3640;
and 3640h-365=9 yrs. 355 days. But in the 70 years preceding the pres-
ent, (1847,) there are 12 years which have 53 Sabbaths each; hence,
12 days must be added to the 3640 before obtained ; consequently in the
last 70 years there are 3652 Sabbaths. Now 3652+365=10 years, 2 days.
But in any 10 consecutive years, excepting those into which the centen-
nials not divisible by 4 enter, there are necessarily two leap years, which
will absorb the 2 days, and leave exactly 10 years as above.
62. 397200''.
63. 1350000".
64. 2126°, 11', 54".
65. 555555s. 16°, 40'.
66. 470660 sq. ft.
67. 4366073* sq. ft.
68. 32640858360 sq. in.
69. 582 A. 1 R. 3 r. 269-} sq. ft.
70. 259200 cu. in.
71. 4551552 cu. in.
72. 10877760 cu. in.
73. 49 cu. ft. 1 cu. in.
74. 306 C. 48 cu. ft.
75. 4492800 cu. in.
76. 52 T. 40 cu. ft. 180 cu. in.
APPLICATIONS OP REDUCTION.
Art. 283.
1. See Book.
2; 576 lbs. Avoirdupois.
8. 691 lbs. 10 oz. 5-ft^ drams.
4. 177+ lbs. Troy, or 145+W
lbs. avoirdupois.
5. 265+ lbs. Troy, or 218ffrt
lbs. avoirdupois.
Art. 284.
6. See Book.
7. 58 lbs. 4 oz. Troy.
8. 21-fl^r. lbs. Troy.
9. 271 lbs. 3 oz.
Art. 285.
10. See Book.
11. 360 sq. ft
12. 14 A. 10 sq. rds.
13. 108 sq. yds. 8 sq. ft
14. 446 A. 1 B.
15. 40 A.
Arts. 283-292.]
SEDUCTION.
81
16L 36 sq. yds.
17. The circuit of the room is
(18-H5)X2=66ft.
Then, by cancelation,
66X0-^0=60 sq. yds.
18. 111-J- sq. yds.
Art. 286.
19. See Book.
20. 56* cu. ft
21. 126 en. £t
22. 86 C. 2 cu. ft
23. 748 cu. ft
24. 756 cu. ft
25. 72 cu. yds.
26. 160 cu. ft
27. 1800 cu. ft
Art. 287.
i
28. See Book.
29. 17280 bu.
30. 3456 wine gak.
31. 86404- wine gals.
52. 846 X 1728 -7- 282 = -5154
beer gals. Ans.
33. 6912 b. gals. 2 qts.
34. 80T*rbu.
35. 100 bu.
36. 14ft=168in.
10 ft 8in.=128in.
6 ft 8 in. =80 in. Now,
168X128X80=1720320;
• and 1720320 -r 2150^ =
800 bu. Am.
37. 897ff wine gals*
S8. 1l2fU bar.
39. 902867HJ hhds.
Art. 288.
40. See Book.
41. 622-J cu. ft
42. 1244£ cu. ^
43. 8£J cu. ft
44. 21 Off cu. ft
45. 842-& cu. ft
Art. 289.
46. See Book.
47. 42^ bu.
48. 46Agals.
49. 14 hhds. 48tf gals.
Art. 290.
50. See Book.
51. 51-Hr beer gals.
52. 45-Jf wine gals.
53. 24ff w. gals.
54. 1598 w. gals.
55. 2207ff w. gals.
56. 3125ff qts.
57. 2734 ff gak.
Art. 292.
58. See Book.
59. 8 min. 36 sec.
60. 39 min.
61. 1 hr. 8 min. 40 sec
62. 33 min. 48 sec.
63. 12 hrs. 28 min. 12 sec
92
REDUCTION.
[Sect. VIII.
Art. 293,
64. See Book.
65. 4° 45'.
66. 12° 46'
67. 13° 23'.
Note.— The examples in Arts. 292, 293;
may also be solved by Compound
Multiplication and Compound Divi-
sion, when the pupil becomes ac-
quainted jwith these rules.
COMPOUND NUMBERS REDUCED TO FRACTIONS
Art. 296,
1—4. See Book.
6. £ illo .
7. £iAo-
8. -ft. lb. Troy.
9. -& lb. Troy.
10. £ }• lb. avoirdupois
11. Too* ■*••
12. i yd.
13. ffrjm,
15. t¥t. sq. r.
16. i gal.
17. i hhd.
18. tJt d.
19. T^nr
J 20.
21.
22.
123.
24.
25.
26.
27".
[28.
29.
[30.
31.
32.
33.
34.
■
35.
36.
ToTTO
See Book.
A.
wk.
*.
8i a
86 0*
■ 6 1 ...
16 0*
8 8 ?UU-»
lfl gfl 1
TTrtVff'
■ftV'
±.
A.
FRACTIONAL COMPOUND NUMBERS REDUCED TO WHOLE
NUMBERS OF LOWER DENOMINATIONS.
Art. 297.
1, 2. See Book.
3. 17s. 6d.
4. 7d. i far.
5. 5 oz. 2 pwts. 20-f- grs.
6. 12 pwts. 12 grs.
7. 10 oz. 10| drs.
8. 57 lbs. 2 oz. 4f drs.
9. 1250 lbs.
10. 2 ft. 4$ in.
11. 6 ft. 2iin.
1 12. 177 r. 12 ft. 10 in.
13. 2 qts. 1 pt. li gt»
AfcTS. 293-302.] COMPOUND ADDITION.
14. 55 gals. 1 pt.
15. 6 qts. 1-}- pts.
16. 3 pks. 1 qt. 1} pts.
17. 46 min. 40 sec.
18. 21 hrs. 36 min.
19. 22} sec.
20. 17' 8f.
Art. 298.
21. See Book.
22. -Hf d.
23. -ffr oz.
24. U r.
25. W- hr.=6 if hrs.
26. ^ft 5 ^ min.=F2688 min.
27. -W- na.=8 A na.
28. -Wf qts.=l7-Hqts.
29. -**H JL qte=l74-Bf qts.
30. -W- oz.=4ff oz.
31. WW pwts.=66 pwts.
32. {U r. =7tWt r -
33. -ft sq. ft.
34. HH* sec=70"
COMPOUND ADDITION.
Art. 300.
1, 2. See Book.
3. £106, 3s. Id.
4. £188, 13s. id.
5. 9 T. 8 cwt. 17 lbs.
6. 45 T. 4 cwt. 57 lbs. 2 oz.
7. 107 lbs. 7 oz. 8 pwts. 1 gr.
8. 330 lbs. 2 oz. 3 pwts. 5 grs.
9. 4 fur. 13 r. 13 ft. 3 in.
10. 109 1. 2 m. 6 fur. 1 ft.
11. 114 yds. 3qrs.
12. 387 yds. 1 qr.
13. 138 A. 114 sq. r. 80 sq. ft
14. 468 A. 1 R. 33 sq. r.
15. 43 sq. yds. 5 sq. ft. 125 sq. in.
16. 240 gals.
17. 181 hhds. 59 gals. 1 pt. 1 gi.
18. 115 wks. 15 hrs. 25 min.
19. 322 bu. 1 pk. 5 qts.
20. 135 qrs. 3 bu. 8 pks. 2 qts.
COMPOUND SUBTRACTION.
Art. 302.
1. See Book.
2. £9, 2s. 8d. 3 qrs.
3. £60, 4s. 7d. 3 qrs.
4. £499, 13s. 4d. 2 qrs.
5. 8 cwt. 1 qr. 6 lbs. 10 oz.
6. 24 T. 1 cwt. 71 lbs.
7. 19 m. 289 r. 2 ft.
8. 1 1. 1 m. 7 fur. 10 r. 12} ft.
9. 35 bu. 2 pks. 6 qts.
10. 19 qrs. 6 bu. 2 pks.
11. 55 yds. 2 qrs. 8 na.
12. 44 yds. 1 qr. 3 no.
2*
34
COMPOUND DIVISION.
[Sect. IX*
13. 6 gals. 2 qts. 1 pt.
14. Given.
15. 85 A. 119 r.
16. 235 A. 48 r.
17. 56 C. 90 cu. ft.
18. 339 cu. ft 26 cu. in.
19. 25° 3' 15".
20. 35° 3' 30".
21. 10° 26'.
22. 54 yrs. 2 mos. 2 wks. 6d.
2 hrs. 45 min. 6 sec.
Art. 303.
23. See Book.
24. 67 yrs. 9 mos. 22 d.
25. Ans. varies with the date,
26. 1 yr. 5 mos. lid.
27. 3 yrs. 9 mo. 22 d.
COMPOUND MULTIPLICATION.
Art. 305.
1, 2. See Book.
3. £247, 6s. Id.
4. £24, 9d.
5. 17 T. 55 lbs.
6. 403 T. 17 cwt. 55 lbs.
7. 689 lbs. 8 oz. 16 pwts
8. 6 lbs. 10 oz. 10 pwts.
9. 3039 hhds. 89 gals. 1 qt.
1 pt.
10. 5668 pi. 32 gals.
11. 2358 yds.
12. 5375 yds.
13. 14778 m. 1 fur. 82 r.
14. 2044 1. 1 m. 4 fur. 30 r.
15. 8962 bu. 16 qts. ^
16. 2968 qrs. 5 bu. 2 pks. 6 qts.
17. 7821 A. 20 r.
18. 25172 A. 1 R. 3 r.
19. 24645 cu. ft. 930 cu.in.
20. 96350 C. 50 cu. ft.
21. 12783 d. 11 hrs. 28 min.
22. 1199 yrs. 9 mos. 3 wks. 1 d.
23. 15891° 13' 30".
24. 204° 10'.
25. 4581 bu. 8 qts.
26. 2453 bu. 4 qts.
27. £5, 16s. 10-J-d.
28. £679, 3s. 4d.
29. £297.
30. £507, 16s. 3d.
31. 86 C. 74cu.ft. 944 in.
32. 865 lbs. 12 oz.
33. 25418 lbs. 12 oz.
34. 8662 gals. 2 qts.
COMPOUND DIVISION.
Art. 307.
1 — 3. See Book.
' 51 lbs. 3 oz. 10 pwts. 15$ grs.
U bu. 14} qts.
6. 25 bu. 1-J- pts.
7. £20, Is. 6d.
8. £4, 17s. 3d. i qr.
9. -10 yds. 3 qrs. If na.
10. 9 yds. 1 qr. -J+ na.
AftT. 303-316.] DECIMAL FRACTIONS.
11. 83 m. 2 fun 26 r. 11 ft.
12. 214 m. 2 fur. 27 r. 4 ft. fin.
13. 1 gal. 2 qts. 1 pt. l-fri gi.
14. 44 hhds. 29 gals. 1 pt H gi,
15. 24 d. 8 his. 42 min. 40 sec.
16. 10 yrs. 35 d. 1 hr. 13 min.
11-A-sec
17. 1° 48' 41 i".
18. 1 s. 17° 52' 21ff".
19. 9 C. 84 ca. ft. IOIGtV ™-
20. 6 C. 92 ft 850ft in-
21. 6s. lOid.
22. Vs. lid. 3f qrs.
23. 10s. lid. 2ft qrs.
SECTION IX.
DECIMAL FRACTIONS.
Art. 316.
(10
32 hundredths.
246 thousandths.
3624 ten thousandths.
82344 hundred thousandths.
13236 hundred thousandths.
(2)
46274 hundred thousandths.
3687 hundred thousandths.
368 hundred thousandths.
46 hundred thousandths.
9 hundred thousandths.
(3.)
42 and 68 thousandths.
17 and 401 thousandths.
23 and 7 hundredths.
61 and 4389 ten thousandths.
90 and 104 ten thousandths.
(4.)
2 and 463126 millionths.
6 and 4534 millionths.
1 and 100492 millionths.
9 and 28 millionths.
8 and 1249 millionths.
(5.)
12 and 683 thousandths.
20 and 64 thousandths.
35 and 72 ten thousandths.
67 and 4008 ten thousandths.
(6.)
6 and 754 hundred thousandths.
3 and 468 ten thousandths.
2 and 306843 millionths.
1 and 710386 millionths.
4 and 306702 millionths.
7006 millionths.
1 and 13004 hundred thou-
sandths.
9 and 203167 millionths.
86
DECIMAL
[Sect. IX.
(8.)
and 2000076 ten milliontks.
8 and 403842 ten millionth*.
8 ten miHionths.
4 and 3008004 ten millionths.
(9.)
25.1.
80.62.
72.80, or 72.8.
(10.)
4.07.
6.039.
7.060, or 7.06.
(11.)
43.2143.
13.0006.
41.0281.
3.312647.
8.000004.
9.7823457.
•
0.9.
0.25.
0.045.
(12.)
(13.)
(14.)
0.06.
0.007.
0.0132.
0.462.
0.2891.
0.00025.
0.000025.
(15.)
(16.)
(17.)
0.1637246.
0.00000065.
(18.)
I 0.071.
0.000007.
(19.)
0.23.
0.0019.
(20.)
0.00261.
0.65.
0.000121.
0.000000000751.
ADDITION OP DECIMAL FRACTIONS.
Art. 320.
1, 2. See Book.
8. 428.1739.
4. 103.8523.
5. 14.747274.
6. 60.149.
7. 832.1249.
8. 501.15998.
9. 857.005.
10. 1097.84143.
11. 1408.25559.
12. 127.05034.
13. 33.3182746.
14. 15674.1613.
15. 1.807.
Arts. 320-324.]
FRACTIONS.
87
16. 2.471092.
IT. 0.0711824.
18. 0.3532637.
19. 0.807711.
20. 0.1627165.
21. 0.996052.
22. 0.329773.
SUBTRACTION OP DECIMAL FRACTIONS.
Art. 322.
I, 2. See Book.
3. 1427.633782.
4. 20.987651.
5. 72.5193401.
6. 81.16877.
7. 0.066721522.
8. 0.01.
9. 9.999999.
10. 64.0317753.
11. 24680.12377.
12. 24.75.
13. 2.291.
14. 9.9999999.
15. 8.000001.
16. 4635.5346.
17. 541.787.
18. 46.43606.
19. 0.0000999.
20. 0.0000396.
21. 31.99968.
22. 44.99955.
23. 98.99999901.
24. 0.000999.
25. 699.93.
26. 28999.908. v >
27. 255999999.744.
28. 0.414.
29. 0.0041.
30. 0.000000000999.
31. 0.002873789.
32. 0.062156.
33. 0.71699.
34. 0.0000843174.
MULTIPLICATION OP DECIMALS.
Art. 324.
1. 681.45 ft.
2. 25020 miles.
3. 2055.375 gals.
4. 136.125 nails.
5. 788.0125 sq. yds.
6. 43560 sq. ft.
7. 2465.375 sq. rods.
8. 0.250325.
9. 18.93978.
10. 14.78091.
11. 0.613836.
12. 0.0320016.
13. 36.740232.
14. 919.82036.
15. 0.000000072.
16. 0.00105175.
88
DECIMAL
[Sect. IX*
17. 390.657556.
18. 275.230594.
19. 148.64244532.
20. 73.25771882.
21. 52.17977576.
22. 0.0306002448.
23. 4701.169144360.
24. 536.660075952.
25. 0.00164389993.
26. 160.86701632806.
27. 0.06288405909156.
28. 2.5067823.
29. 64.327106105314.
80. 0.0000118260069.
31. 11027.40199543710.
32. 94167471.869654039.
33. 0.00000006676542672.
CONTRACTIONS IN MULTIPLICATION OP DECIMALS.
Art. 325.
1. See Book.
2. 429302.13401.
3. 106723.50123.
4. 608340.17.
5. 304672.14067.
6. 44632140.32.
7. 2134567.82106.
8. 500.
9. 75000.
10. 6.5.
11. 48.
12. 2480.
13. 381.
14. 65.04.
15. 834000.
16. 10.
Art. 327,
17—20. See Book.
21. 0.09484.
22. 1.262643.
23. 0.0769.
24. 0.0389254.
25. 0.00876.
26. 0.002516.
27. 0.001789.
DIVISION OP DECIMAL FRACTIONS.
Art. 330.
1 — 3. See Book.
4. 13 boxes.
5. 8 suits.
6. 4.98347 +days.
7. 82.9997 +loads.
8. 27.7173+days-
9. 150.25 bales.
10. 5.9291+.
11. 6.632.
12. 79098.8235+.
13. 0.6344+.
14. 1210.2344+.
15. 0.03.
16. 134.8805+.
Arts. 825-335.]
FRACTIONS.
17. 59.4060+.
18. 24.8266+.
19. 4320.6V.
20. 0.02.
21. 83671000.
22. 255.1210+.
23. 0.000005.
24. 60.2589.
25. 211.076.
26. 400000.
27. 60000000.
28. 4000000.
29. 311.487360 +.
CONTRACTIONS IN DIVISION OP DECIMALS.
Art. 331.
Note. — Before beginning the contrac-
tion, it is advisable for the divisor to
have one or two figures more than
the number of decimals required in
the quotient; otherwise there may
be an error in the last figure of the
quotient.
1, 2. See Book.
3. 67234.567.
4. 103.42306.
5. 0.42643621.
6. 6-72300045.
7. 0.000012300456.
8. 0.0000020076346.
Art. 333*
9. See Book.
10. 0.1274.
11. 0.09471.
12. 1.611.
13. 0.04026.
Note. — In this example we must ob-
tain 1 decimal figure in the quotient
before we begin the contraction;
that is, we divide in the common
way, till the number of figures which
remain to be found in the quotient
is 2 less than the number of figures
in the divisor; otherwise we cannot
obtain the last figure in the quo*
tient. (See Note above.)
14. 0.0954776.
15. 2.0208.
16. 0.980439.
REDUCTION OP DECIMALS.
Art. 335.
1, 2. See Book.
3. +.
4. i*.
O. "go o •
6. **.
8. -ftV.
9. -nr»
10. tH.
11. ToTT*
12.
tVW*.
13. WW.
14. a Wo *
15. -ffc.
lw. 2 6 ©'•
17. joe*
COMMON FRACTIONS REDUCED TO DECIMALS.
Art. 337.
1—8. See Book.
4. 0.5.
- 6. 0.25.
6. 0.5.
1. 0.7S.
8. 0.2.
9. 0.4.
10. 0.6.
11. 0.8.
12. 0.5.
13. 0.125.
14. 0.25.
15- 0.375.'
16. 0.5.
17. 0.625.
18. 0.75.
19. 0.875.
Art. 342.
20. 21. See Book.
22. Terminate decimal.
23. Terminate '*
24. Interminate "
25. Terminate '*
26. Terminate "
27. Interminate "
28. Terminate "
Art. 344.
32. 0.6.
33. 0.16.
34. 0.3.
35. 0.8.
36. 0.83.
37. 0.i42857.
38. 0.285714.
39. 0.42857i.
40. 0.571428.
41. 0.714285.
42. 0.857142.
43. 0.i.
44. 0-2,
45. 0.8.
46. O.4.
47. 0.5.
48. 0.6.
49. 0.7.
50. oi.
51. Four, vis: .1875.
52. Sii, viz: .076923.
I. Three, .024.
64. Four, .0112.
55. Three, .276.
56. Ten, .0048828125.
57. 0.583.
0.076823.
59. 0.0104895.
60. 0.4683544303797'.
Nblt. — It may sometimes be convenient to find how many figure* the periot
trill contain, and where it will begin, without actually reducing the frac-
•on to a circulating decimal. This may be done in the following manner.
Arts. 337-346.]
FRACTIONS.
41
First reduce the given fraction to its lowest terms; then divide ike denom-
inator thus reduced, by 10, 5, or 2, as often as possible. By this quotient
divide 9999, df-c, till the remainder is 0.
The number of 9s used in this division will show the number of figures
in the period, and the period will begin after as many places as there are
10s, 55, and 2s used in dividing the denominator.
Illustration. — If we divide 1.0000, &c., by any prime number whatever, ex-
cept 2 or 5, the figures in the quotient will begin to repeat as soon as the
remainder is I. And since 9999, &c., is less than 10000, &c., by 1, there-
fore 9999, dec., divided by any number whatever, will leave for a re-
mainder, when the repeating figures are at that period. Now, whatever
number of repeating figures we have when the dividend is 1, there will be
exactly the same number when the dividend is any other number what-
ever. For, the product of any circulating number, into any other given
number, will consist of the same number of repeating figures as before.
Thus, let .507650765076, &c., be a periodical whose repeating part is 5076.
Now every repetend (5076) being equally multiplied, must produce the
same product. For, though these products will consist of more places,
yet the overplus in each, being alike, will be carried to the next, by which
means each product will be equally increased, and consequently every
four places will continue alike. And the same will hold for any other
number whatever. 4 *
Thus, in Ex. 59, which is already reduced to its lowest terms, the de-
nominator 286, can be divided by 2, only once ; and the period begins
with the second figure. Again, if we divide 999, &c., by 143, the quotient
of 286+2, it requires six 9s before the remainder is a cipher ; consequently
the period will consist of 6 figures.
COMPOUND NUMBERS REDUCED TO DECIMALS. «.
9.
0.25625 m.
Art. 346.
10.
0.2583 hr.
1, 2. See Book.
11.
0.127083 d.
3. £0.5375.
12.
0.0525 cwt.
4. £0.825.
13.
0.46875 lb.
5. £0.87916.
14.
0.875 bu.
6. 0.416 8.
15.
0.5625 pk.
7. 0.5416 s.
16.
1.125 gals.
8. 0.115625 m.
* Boonycaatle's Arithmetic.
42
CIRCULATING DECIMALS.
[Sect* X.
DECIMAL COMPOUND NUMBERS REDUCED TO
WHOLE ONES.
Art. 348*
1. See Book.
2. 14s. 6d.
3. 2s. 7d. 3.2 qrs.
4. Id. 2 qrs.
5. 9d. 3.6 qrs.
6. 12 lbs. 8 oz.
1. 6 oz. 15.36 drs.
8. 88 rods.
9. 1 ft. 0.51 in.
10. 11 gals. 1 qt. 1 pt. 3.7184
gills.
11. 1 qt. 1 pt. 3.4432 gills.
12. 10 hrs. 13 min. 9.12 sec.
13. 50 min. 42 sec.
SECTION X.
PERIODICAL OR CIRCULATING DECIMALS.
REDUCTION OP CIRCULATING DECIMALS.
Art. 355*
1, 2. See Book.
3. -H, or -ft.
4. TT»» or T3s •
5. Wk, or -H-.
6. U, or A-
7. A> or A-
8. •gift-* or TTT»
9. +.
10. tV-
Art. 358*
11—13. See Book.
14. tV
13* jtfuf or "yy*
16. -ft.
17. -fr.
18- "fro"*
19. f % % 3 .
20. sVAlliU , or Tffr,
Art. 361.
21. 22. See Book.
23. 0.277.
0.333.
0.045.
24. 4.3213.
6.4263.
0.6000.
AftTS. 348-362.] CIRCULATING DECIMALS.
48
ADDITION OF CIRCULATING DECIMALS.
Art. 362.
1. See Book.
2. 24.i32=24.1321321
2.23 = 2.2333333
85.24 =85.2424242
67.6 =67.6666666
3. 328.126
81.23 :
5.624
61.6 =
179.2745563
=328.12666
: 81.23333
: 5.62462
: 61.66666
476.65129
4. 31.62 =31.62222
7.824= 7.82444
8.392= 8.39239
0.027= 0.02777
47.86683
5. 462.34 =462.843
60.82 = 60.828
71.164= 71.164
0.35 = 0.355
594.691
f. 60.25 =60.2500
0.34 = 0.3444
6.435= 6.4355
0.45 = 0.4500
45.24 =45.2424
112.7224
7.
8.
9.814= 9.8148148
1.5
= 1.5000000
87.26
= 87.2626262
0.83
= 0.8333333
124.09
= 124.0999999
223.5107744
3.6
= 3.6666666
78.3476= 78.3476476
735.3
=735.3333333
375.0
=375.0000000
0.27
= 0.2727272
187.4
=187.4444444
:
L380. 0648193
9. 5391.357
—
5391.35700
72.38
=
72.38888
187.21
^z
187.21111
4.2965
=3
4.29655
217.8496
=:
217.84966
42.176
=
42.17666
0.523
=:
0.52333
58.30048=
58.30048
5974.10371
44
CIRCULATING DECIMALS.
[Sect. X.
10.
0.162
= 0.162162162
134.09
=134.090909090
2.93
= 2.939393939
97.26
= 97.266666666
3.769230= 3.769230769
99.083
= 99.083000000
1.5
= 1.500000000
0.814
= 0.814814814
339.626177443
SUBTRACTION OP CIRCULATING DECIMALS.
Art. 363.
1, 2. See Book.
3. 391.5524.
4. 3.81824.
5. 4.789.
6. 400.915.
7. 3.9046.
8. 218.60.
9. 0.61364073i.
10. 2451.386.
MULTIPLICATION OP CIRCULATING DECIMALS.
Art. 364.
1, 2. See Book.
3. 0.082.
4. 1.8.
5. 389.185.
6. 778.14.
Note. — The period or circulating part
in this example properly begins with
the figure 8, which is a whole num-
ber. Some however prefer, in such
cases, to consider the first decimal
figure the first figure of the period.
7. 750730.518.
8. 31.791.
9. 34998.4199008.
10. 2.297.
DIVISION OP CIRCULATING DECIMALS.
Art. 365.
1, 2. See Book.
3. 55.69.
4. 5.41463.
5. 7.72.
6. 8574.3.
7. 3.506493.
8., 3.145.
9. 62.323834196891.
10. i.422924901185770750988.
Aits. 363-374.] federal money.
48
SECTION XI.
FEDERAL MONEY.
Art. 370.
1. See Book.
2. $150,035.
3. $409,403.
4. $200,052.
5. $4050.653.
REDUCTION OF FEDERAL MONEY.
Art. 371.
1 — 3. See Book.
4. 46000 cents.
5. 95000 milk.
6. 900 mills.
7. 2515 cents.
8. 86408 cents.
9. 1265050 mills.
10. 4580100 mills.
11. 6886258 mills.
12. 85625400 mills.
Art. 372.
13—15. See Book.
16. $15.16.
17. 16 cento, 2 mills.
18. $1.
19. 236 cents.
20. $3.28.
21. $85.
22. $23.45.
23. 92 dolls. 35 cents, 5 mills.
24. 150 dolls. 23 cents, 3 mills.
25. $4503.41.
ADDITION OF FEDERAL MONEY.
Art. 374*
1. See Book.
2. $265.04.
3. $581,128.
4. $560.56.
5. $1795.34.
6. $1431.50.
7. $3531.432.
8. $12200.524.
9. $185,285.
10. $74.33.
11. $350.32.
12. $6491.05.
13. $8765.12.
14. $16989.
15. $378,383.
16. $300,166.
17. $256,213.
18. $1945.258.
19. $82110.17.
20. $71774.75.
21. $27860.74.
22. $81800.63.
46
FEDERAL MONET.
[Skct.XL
SUBTRACTION OP FEDERAL MONEY.
Art. 375.
1. See Book.
2. $12.13.
3. $84.82.
4. $247.15.
5. $918.48.
6. $183.22.
7. $323.47.
8. $373.82.
9. $10870.75.
10. $1699.49.
11. $9947.788.
12. $61119.364.
13. $18,981.
14. $88.11.
15. $189.92.
16. $2,937.
17. $32,056.
18. $10890.07.
19. $89989.90.
MULTIPLICATION OF FEDERAL MONET
Abt. 377.
1, 2. See Book.
3. $83.60.
4. $517,625.
Art. 378.
5. See Book.
6. $39.59375.
7. $1440.75.
8. $40.59375.
9. $84,875.
10. $193.75.
11. $205,625.
12. $326.25.
13. $2.0925.
14. $2.84375.
15. $909,375.
16. $2.70.
17. $14.0625.
18. $15.78375.
19. $28,125.
20. $220.50.
21. $142.50.
22. $2331.875.
23. $14084.125.
DIVISION OF FEDERAL MONEY.
Abt. 379.
1. See Book.
2. $4.50.
8. $0.06.
4. $3.13.
Art. 380.
5. See Book.
6. 8.207+ coata.
7. 7.871+ times.
Abts. 375-384.] federal money.
47
Art. 381.
8. See Book.
9. 308.035+ gals.
10. 543.518+ yards.
11. 991.421+ doz.
12. 360 skeins.
13. $3,524+.
14. $1.50.
15. $6.25..
16. $1,973+.
17. $3,615+.
18. $0,084+.
19. $0.04049+.
20. $0.02709+.
21. $1.78008+.
22. $1.5435+.
23. 1714.285+ bu.
24. 113.56377+ tons.
25. $0.595238+.
26. 245.517+ acres.
27. 500 cows.
28. 150 carriages.
APPLICATIONS OF FEDERAL MONET.
Art. 382.
1. See Book.
2. $800.
3. $511.50.
4. $780.
5. $780.
6 $1350.
7. $1020.
8. $864.50.
9. $2418.
10. $4440.
11. $1424.75.
12. $2691.875.
13. $5885.
14. $10538.625.
Note, — When the price of 1 article,
1 pound, &c., is 87i cts., 75 cts.,
66} cts., &c., from the number of ar-
ticles subtract |, i, or }, &c., of itself
and ike remainder will be the cost re-
quired.
Art. 383*
15, 16. See Book.
17. $6.33375.
18. $104.55.
19. $114,198.
20. $59.5856.
21. $505.3775.
22. $1901.75.
23. $5.40625.
24. $52,126.
25. $437,645.
Art. 384*
26. 27. See Book.
28. $0.0072.
29. $0.0064.
30. $13.4719+ per cwt.
$0.134719+ per pound.
31. $12.88506 per cwt.
$0.1288506 per pound.
48
BILLS AND ACCOUNTS.
[Sect. XL
BILLS, ACCOUNTS, &c— Art. 385*
(32.)
75 Thomson's Mental Arithmetic,
50 " Practical Arithmetic,
36 Porter's Rhetorical Reader,
25 Willson's School History,
30 M'Elligott's Young Analyzer,
75 Thomson's Day's Algebra,
50 " Legendre's Geometry,
at
n
tt
tt
tt
tt
tt
$.12*,
.81*,
.62*,
.46,
•31±,
.50,
.47*,
$9,375
15.625
22.50
11.50
9.375
37.50
28.75
Amount,
$129,625
(33.)
163 lbs. Butter,
at
$.14*,
-
$23,635
235 lbs. Coffee,
tt
.08*,
-
19.388
86 lbs. Chocolate,
a
.11,
-
9.46
685 lbs. Sugar,
u
.10*,
-
71.925
21 doz. Eggs,
*t
.13,
-
2.73
860 lbs. Lard,
tt
.09*,
(34.)
Amount,
81.70
$208,838
320 yds. Silk,
at
$1.12*,
-
$360.00
256 " Broadcloth,
tt
3.62*,
-
928.00
175 pair Cotton Hose,
tt
0.12*,
-
21.875
100 « Silk "
tt
0.87*,
-
87.50
15 doz. Gloves,
tt
0.62*,
«• •
112.50
120 Straw Hats,
u
1.87*,
(35.)
Amount,
225.00
$1734.875
15260 lbs. Pork,
at
$0.05*,
^ —
$839.30
7265 lbs. Cheese,
tt
0.08*,
-
617.525
11521 bu. Corn,
u
0.50,
-
5760.50
1560 bbls. Flour,
tt
6.12*,
•
9555.00
Amount, $16772.325
Arts, 385*388.]
PERCENT AGE.
40
1150 lbs. Cotton, at
8256 lbs. Sugar,
6450 gals. Molasses,
Cash to balance account,
tt
€€
CREDIT.
$0.06+,
0.07,
0.37+,
- $71,875
■ 577.92
- 2418.75
$13703.78
SECTION XII.
PERCENTAGE.
Art. 387.
1. 0.01; .02 ; .04 ; .06 ; .07; .08.
2. 0.11; .12; .14; .15; ,16;
.23; .65; .93.
9. 0.005; .0025; .002; .004;
.008 ;. 0075; .00+;. 00125;
.0025; .00625 ; .00+; .00f;
.0014285; .00+.
4. 0.045;, 0625;. 07125;. 092;
.125; .1625; 1,15;
4.0025.
Art. 388.
5, 6. See Book.
7. $7.6875.
8. $8.7526.
9. $3.4608.
10. $8.7078.
11. $114.1070.
12. $10.50. *
13. $219.
14. $43.13 receiv'd.
$819.43 paid over.
15. $402.05.
16. $134.
17. $32,625.
Note. — Whenever the given per cent,
contains a common fraction^ instead
of reducing it to a decimal, the oper-
ation will frequently be shorter to
multiply by the per cent., regarding
it as a mixed number. (Art 217.)
18. $34.03575.
19. $62.50.
20. $146,666+.
21. $8.771875.
22. 875 sheep.
23. $1568.
24. 187.5 boxes lost.
1312.5 " saved.
25. $8,125.
26. $6,316.
27. $84.52016.
28. $250.
29. $750.
30. $90.4824.
31. $844.08.
32. $4724.775.
33. $1250.
34. $12000.
35. The former $21900.
" latter $14600.
36. $200.
87. $0.95.
50
PERCENTAGE.
[Sect. XII.
APPLICATIONS OF PERCENTAGE.
Art. 395*
1. See Book.
2. $12,507. *
3. $58,878.
4. $73,159.
5. $115,203.
6. $615.
7. $583,842.
8. $52,834.
9. $155,875.
10. $619,887.
11. $44.32.
12. $673.75.
13. $57.29.
14. Sale/
Guaranty,
Total,
15. Sale,
Guaranty,
Total.
Remitted,
$231,017.
184.814.
$415,831.
$48,074.
58.757.
$106,831.
$2029.798.
Art. 397.
16. See Book.
17. $21078.431.
18. $3439.613.
19. $761904.761.
Aofc.— It will be peiceired that this
answer is a repetend, and that the
period begins and ends with the
whole number.
20
. $4126.55. (See Art. 39^
Obs.)
21
. $1413.975.
22
. $46.50.
23
. $22,113.
24
. $9,375.
25,
. $318,975.
26,
. 27. See Book.
28.
$3692.50.
29.
$2250.
30.
$8840.70.
31.
$7072.
32.
$3552.50.
33.
$1350 received.
$180 lost.
34.
$7490.50.
35.
Paid, $4128.
Received, $5088.
Gained, $960.
36.
Price of stock,
Brokerage, (Art.
$4400.00
395. Obs.)
27.50
Amount,
$4427.50
37.
Price of stock,
$8970.00
Brokerage,
58.50
Amount,
$9028.50
Aato. 305-409.]
J NT KB EOT.
SI
INTEREST.
FIRST METHOD.
Art. 404*
1. $29.61.
2. 143.255.
3. $40,367.
Note. — Since 5 per cent is tSt^hV'
the interest at 5 per cent, may be
found by taking -fc of the principal.
Or, the answer may be found by
multiplying the principal by 10 per
ct ; for, since the int is 5 per ct for
1 year, for 2 years it is 10 per ct
4. $51.20.
5. $60,263.
6. $44,414.
7. $194.58.
8. $17,803.
9. $28,206.
10. $8,103.
11. $6,853.
12. $19.14.
13. $60.27.
14. $89.40.
15. $958.41.
16. $657.45.
17. $1006.833.
18. $1585.018.
19. $889.44.
20. $1,135.
21. $1,409.
22. $1,898.
23. $102,125.
24. $154,216.
25. $704,083.
26. $2,975.
27. $76,131.
28. $3312.209.
29. $5278.162.
30. $16,158.
31. $206,718, allowing 360 d.
to the year.
$203,886, allowing 365 d.
to the year.
32. $66778.64.
SECOND METHOD.
Art. 409.
1 — 3. See Book.
4. $8.50.
5. $1,065.
6. $70,151.
7. $97.28.
8. $30.78.
0. $308,287.
10. $1177.50.
11. $1113.024.
12. $10.05.
13. $11.0025.
14. $988,761.
15. $82,078.
16. $39,179.
17. $320,833.
INTEREST,
[Sect. XII,
18. $9.8437.
19. $85,207.
20. $400.
21. $1638.442.
22. $144.
23. $90.
24. $21,075, Interest.
$12666.075, Amount.
25. $65,140, Int. at 6 per cent.
10.856 , i added.
$75.996=Int. at .07 perct.
$16360.996, Amount.
26. $307.65.
27. $227,994.
28. $8.
29. $0.07.
Art. 411.
80. See Book.
31. $15.60.
32. $21.09.
33. $1,272.
34. $4,778.
35. $46.35.
36. $129.15.
87. $168,552.
38. $137,288.
39. $481,016.
40. $391,062.
41. $1531.25.
42. $3425.655.
43. $16320.528.
Art. 413.
44. See Book.
45. $2,145.
46. $74,392.
47. $10,835.
48. $398,055.
49. $14,532.
APPLICATIONS OF INTEREST.
Art. 415*
1. See Book.
2. $5.25. N. Y. Int. is 7 per ct.
3. $3.15. N.E.Int. is 6 perct.
4. $17. Penn. Int. is 6 per ct.
5. $60. Ohio Int. is 6 per ct.
6. Time, 1 yr. 2 mo. 6 d.
Int. $45,014.
7. Time, 8 mo. 24 d.
Int. $86.08.
8. Time, 3 yrs. 8 mo. 12 d.
Int. $91,085.
9. Time, 10 mo. 3 d.
Int. $107,854+.
10. Time, 6 mo. 12 d.
Int. $13,867.
Amount, $533,867.
11. Time, 5 months.
Int. $729,166.
Amount, $25729.166+.
12. Time, 11 mo. 6 d.
Int. $347.20.
Amount, $6547.20.
Arts. 411-416.] INTEREST. 58
Id. See Book.
PARTIAL PAYMENTS.
Art. 416.
14. Principal, 9650.00
Interest to 1st payment, Aug. 13th, (7 mo. 12 d.) 24.05
Amount due on note, Aug. 13th, 9674.05
1st payment, (to be deducted from amount,) 100.00
Balance due, Aug. 13th, 9574.05
Int. on bal. to 2d payt. April 13th, (8 mo.) 22.961
Amount due, April 13th, 9597.012
2d payt. (to be deducted from amount,) 120.00
Balance due, April 13th, 1843, 9477.012
Int. on bal. to Jan. 20th, 1844, (9 mo. 7 d.) 22.022
Bal. due on taking up the note, Jan. 20th, 1844, 9499.034
15. Principal, 92460.00
Int. to 1st payt. Aug. 20th, 1845, (1 y. 4 mo. 10 d.) 200.900
Amount due on note, Aug. 20th, 92660.900
1st payment, (to be deducted from the amt.) 840.000
Balance due, Aug. 20th, 91820.900
Int. on bal. to 2d. payt. Dec. 26th, 1845, (4 mo. 6 d.) 38.238
Amount due, Dec. 26th, 1845, 91859.138
2d payt. (to be deducted from the amt.) 400.000
Balance due, Dec. 26th, 1845, 91459.138
Int. on bal. to 3d payt. May 2d, 1846. (4 mo. 6 d.) 30.641
Amount due, May 2d, 1846, 91489.779
3d payt. (to be deducted from the amt.) 1000.000
Balance due, May 2d, 1846, 9489.779
Int. on bal. to Aug. 20th, 1846, (3 mo. 18 d.) 8.816
Amt. due, Aug. 20th, 1846, 9498.595
16. Principal, 95000.000
Int. at 5 per ct. to 1st payt. Oct. 1st, 1845, (5 mo.) 104.167
Amt. due on note, Oct. 1st, 5104.167
1st payt. (to be deducted from the amt.) 700.000
Balance due, Oct. 1st, 1845, %WACVtfV
54
INTEREST.
[Sect. XII.
Balance brought forward,
Int. on bal. to 2d payt. Feb. 7th, 1846,
(4 mo. 6 d.) 877.072
2d payt. (less than int. then due,) 45.000
Surplus int. unpaid Feb. 7th, 1846, 982.072
Int. continued on bal. from Feb. 7th,
1846, to Sept. 13th, 1846, (7 mo. 6 d.) 182.125
Amt. due Sept. 13th, 1846,
8d payt. (to be deducted from the amt.)
Balance due Sept. 13th, 1846,
Int. on bal. to Jan. 1st, 1847, (3 mo. 18 d.)
Balance due, Jan. 1, 1847,
•4404.167
164.197
#4568.364
480.000
#4088.364
61.325
#4149.689
Art. 419.
17, 18. See Book.
19. £19, 5s. 10|d.
20. £8. 18s. 9 d.
21. £12, 10s.
22. £1898, 10s. 4|d.
23. £2900.
PROBLEMS IN INTEREST.
Art. 422.
1, 2. See Book.
8. The given int. is #24.80.
Int. on principal at 1 per ct.
for 8 mo. is #4.1 3}.
Amt #24.80-r4.13i=6.
Ans. 6 per ct.
4. 6 per cent.
5. 8 per cent.
6. 7-J- per cent.
7. 5-§- per cent.
8. 7 per cent.
9. 6 per cent.
10. 5 per cent.
21. S percent.
Art. 423.
12. See Book.
13. #1800.
14. #5400.
15. #10000.
16. #8000.
17. #14285.7143.
18. #20000.
19. #30000.
20. #20833*.
Art. 424.
21. See Book.
22. 4 years.
28. 6 months.
Arrs. 419-433.]
DISCOUNT.
«
24. 1 7. 3 mo. and 1 d. nearly.
25. 1 year, 6 months.
26. See Book.
27. 16 years, 8 months.
28- 14 yrs. 3 mo. 13 d. nearly.
29. 14 yrs. 3 mo. 13 d. nearly.
30. 10 years.
31. 8 years, 4 months.
32. 9 years, 6 months, 8 days.
33. 28 years.
COMPOUND INTEREST.
Arts. 426, 427.
1, 2. See Book.
3. $507,213.
4. 92177.420.
5. 94590.09.
6. See Book.
7. 91888.464.
8. 91551.328.
9. 9877.506.
10. 93491.395.
11. $16035.675.
12. 9149744.
DISCOUNT.
Art. 43Q.
1, 2. See Book.
8. 9934.579+.
4. 91488.687 +.
5. 988.461 +.
6. 983.52+.
"J. 94729.064 +.
8. 96208.955 +.
9. 93404.347 + .
10. 99950.248+.
11. Interest, 9560.
Discount, 9 523.364.
Difference, 936.636.
BANK DISCOUNT.
Art. 433.
12, 13. See Book.
14. 914.1825.
15. 916.605.
16. 926.98.
17. 95.495.
18. 92034.1213.
19. 92774.655.
20. 924.822.
21. 948.3237.
22. 937.595.
23. 943.694.
24. 96381.59.
25. 91495.625.
26. 980.
27. 9456.785.
28. 93002.500:
91875.977 -
91126.523 :
29. See Book.
=Bank dis.
True dis.
difference.
56
INRURANCE.
[Sect. XII.
x Art. 434*
Note.— In the following examples no
allowance is made for the three days
grace.
80. $414,507.
31. 8966.101. (Art.409. Obs.4.)
32. (1252.70.
33. 82514.247.
34. $3821.883.
35. 84355.102.
36. $63717.884.
37. 810416.666.
38. 851194.539.
39. 846638.655.
40. 88301.342.
INSURANCE.
Art. 437 i
1. See Book.
2. 820.70.
8. 894.20.
4. 863.75.
5. 8104/
6. 870.50.
7. 8900.
8. 81875.
9. 848t.50.
10. 8243.125.
11. 819278.
12. 83375.
Art. 438,
13. See Book.
14. 2-J- per cent.
15. 2+ per cent.
16., 1 per cent.
17. li per cent.
18. See Book.
Art. 439*
19. 852000.
20. 865600.
21. 865000.
22. 857333+.
23. 83416*.
Art. 440.
24. See Book.
25. 88365.482.
26. 813876.288.
27. 827027.027.
Art. 442*
28. See Book.
29. 848.60, the annual prem.
30. 8373.75.
31. The sum paid for insurance
was the larger.
$10000 X.03±=8350. And
8350, the annual prem.
multiplied by 35, the
number of yrs. is $12250.
Arts. 434-451.] peofit and loss.
57
PROFIT AND LOSS.
Art. 444*
1 — 3. See Book.
4. $218.
5. (680.
6. $5299.75.
7. $1366.75.
8. $68730.28.
9. Lost $12500.
Art. 445*
10, 11. See Book.
12. Since 20 per cent, is -ffo, or
f, to obtain 20 per cent,
on any sum, we may di-
vide by 5. $156,804. Ana.
13. $4238.50.
14. $5926.85.
15. $29504.875.
Art. 446.
16. See Book.
17. 23-^ per cent.
18. 4J- per cent.
19. 15+ per cent.
20. Find the selling price by
multiplying the number
of gills in a pipe into .12£.
Am. 100 per cent.
21. 20+H per cent.
22. 2£fr per cent.
Art. 447*
23. 24. See Book.
25. $460,869.
Note. — In such cases, the rule in busi-
ness operations is to call the cents
87, and reject the mills. (Art. 404.
Obs.3.)
26. $205,882.
27. $2622.222+.
28. $2736.
29. $13043.478+.
30. $6317.391+.
31. $17806.122+.
32. $42654.028+.
33. $42160.
DUTIES.
Art. 451*
1. See Book.
2. $370.80.
3. $163.20.
4. $1323.
5. 250 bags, each 65 lbs., make
16250 lbs. Now 4 perct.
of 16250 lbs. is 650 lbs. [ 10 - 11882.40626,
But 16250—650=15600.
And 15600 X$.03i =
$546. Ans.
6. $1235.22.
7. $3784.
8. $345,744.
9. $679.14.
^
«8
[Sect. ML
Art. 453,
11. See Boot.
12. $248.
13. $717.40.
14. $492.
15. $1051.71.
16. $715.75.
17. $1230.
18. $15884.75.
19. $12642.40.
20. $2807.10.
21. $11172.30.
22. $17328.75.
23. $15770.70.
ASSESSMENT OF TAXES.
Art. 456.
1, 2. See Book.
3. $54.15, B's tax.
4. $80.50, C's tax.
5. ioflperct.or8m.onfl,
6. $80. A's tax.
7. $121.92, B's tax.
8. $283.68, C's tax.
Art. 458.
9. See Book.
10. $8854.166.
11. $16125.654.
12. $17342.105.
13. $34051.815.
Abt. 459*
14. 15. See Book.
16. $78, G. A's tax.
17. $116, H. B's tax.
18. $451.50, W. C's tax.
19. $481.22, E. D's tax.
20. $314.50, J. F's tax.
21. $621.90, T. G's tax.
22. $526.40, W. H's tax.
23. $263.30, L. J.'s tax.
24. $631.00, W. L's tax.
25. $196.90, J. K's tax.
26. $404.90, G. L's tax*.
27. $370.50, F. M's tax.
28. $458.20, C. P's tax.
29. $480.50, J. S's tax.
30. $541, R. W's tax.
Proof. — The above, with A. B's
tax, (Ex. 15.) = $6000, the
amount to be raised.
Art. 460.
31. See Book.
32. The rate is .08 ; hence B's
taxis.08Xl67=$13.36.
33. The rate is .05 ; hence C's
tax is $3.45.
34. The rate is .10 ; hence D's
tax is $13.40.
Aets. 453-464.]
ANALYSIS.
w
SECTION XIII.
ANALYSIS.
Art. 462.
1, 2. See Book.
8. $300.
4. $320.
5. $12.33*.
6. $10.50.
7. $i.eaf.
8. $2640.
9. $24.80. *
10. $.055.
11. $.29f.
12. $.039Vr.
13. $64.
14. $1080.
15. $480.
16. $8.
17. 60 days.
18. 29-H months.
19. 1088 days.
20. $.56.
21. $3.
22. $7.98.
23. $6.03.
24. Analysis. — Since $ of an acre cost (108, it is plain i of,
an acre will cost i of $108, which is 836. Now, if f
of an acre costs 836, £ will cost 5 times as much; and
$36X5=8180.
Again, since 1 acre costs 8 180, \ of an acre will cost
■J- of 8l 80, which is (20. And if + of an acre costs $20>
i will cost 8 times as much ; and 920X8=8160. * Ans.
25. $3.70f.
26. $3430.
27. $11 9.9 Iff
28. $636.47-Hh|.
29. See Book.
80. 2f hours.
81. 2-rfdays.
Art. 46 3<
32. See Book.
83. 360 pounds.
34. 1500 pounds.
35. 95.2 cords.
36. 100 pair.
Art. 46 4 1
37. See Book.
38. $450, A's share.
$750, B's share.
39. $450, A's loss.
$600, B's loss.
$750, C's loss.
60
ANALYSIS.
[Sect. XIIL
40. $763.63-ft-, A's gain.
$654.54-ft-, B's gain.
$981.81-,*!-, C's gam.
41. $150.95+H> A's gain.
$164.53++£, B's gain.
$185.70-^, C's gain.
$123.80^, D's gain.
42. 66-f cts. on a dollar.
$266.66-} , 1st creditor rec'd.
$333.33+, 2d . " "
$400,000, 3d " "
Art. 465*
43. 70 cts. on a dollar.
44. 25 per cent.
45. $2990.00, A's share.
$4197.50, B's share.
$4312.50, C's share.
46. 66+ per cent.
47. 37+ per cent.
48. 10 per cent.
Art. 466.
49. 40 tons, A's loss.
80 tons, B's loss.
120 tons, C's loss.
50. 25 per cent.
51. 33+ per cent.
$30000, the man's loss.
Art. 467.
52. See Book.
53. 5s. per gal.
54. 5+s. per pound.
55. 9 cts. a pound,
56. 19+ cts. a pound.
57. 91+ cts. per gaL
58 — 60. See Book.
61. 1 part of 16, 1 of 18, 2+ of
23, and 1 part of 24 carats
fine.
62. See Book.
63. 100 gals, at 80 cts., 40 gals.
at 30 cts., and 40 gals, at
40 cts.
64. See Book.
65. 188} lbs. at 8d., 17+ lbs. at
12d., 17+ lbs. at 18d., and
17+ lbs. at 22d.
66. See Book.
model . jiEcrr ation. — Art. 468*
Ex. 67. Analysis. — Since 15 horses consume 40 tons, 1 horae will consume
•j^jf of 40 tons, and 40-*- 15 =2+ tons. And if 1 horse consumes 2-} tons in 30
weeks, in 1 week he will consume -fa of 2$ ; and 2+-f-30=-jSy of a ton. Now,
if "A of a ton will last 1 horse 1 week, 56 tons will last as many horses 1 week,
as -^j- is contained times in 50 ; and 56-f--^-=630 horses. And if 56 tons sup-
ply 630 horses for 1 week, for 70 weeks it will supply -fa °f 630 horses ; and
630-1-70=9. Ans. 9 horses.
68. 38+ days.
69. 278160 men.
90. Id}} months. >
71. $54.60.
72. $459.90.
AJLTS. 465-471. J * ANALYSIS.
Art. 469.
105. 8250*.
73. See Book.
74. 8600.
75. 83600.
76. 8630.
77. 72.
78. 360.
79. 120.
80. 240.
81. 68^- feet.
106. 8231.
107. 8119-ft-
108. 8186.
109. 8280.
110. 81170.
111. See Book.
112. 8378.
113. 8810.
114. 8288.
115. 350s.=843f.
Art. 470.
82—84. See Book.
116. 8814.
117. 8640.
118. 83000.
85. 8239.
119. 8200.
86. 81170.
120. See Book.
87. 8900.
121. 8625.
88. 81125.
122. 8480.
89. 8367.50.
123. 8808.
90. 8442.
124. 8420.
91. 8201.
125. 8690.
92. 8350.
126. 8877*.
93. 8240.
127. 8140.
94. 8754.
128. 81560.
95. 81080.
129. 8180.
96. 8630.
130. 8630.
97, 98. See Book.
131. 8180.
99. £206{.
132. 81281.
100. £266.
133. 8800.
101. 2625s.=£131i.
134. 812.60.
102. 7060s.=£353.
135. 845.
Art. 471.
136. 845.
137. 890.
103, 104. See Book.
138. 8150.
61
KATIO.
[SwjT.XIV
SECTION XIV.
RATIO.
Jber.
480.
30. I*.
1, 2. See Book.
3. 2.
31. ■}.
32. 240.
4. 4.
5. 9.
6. 6.
Art. 488*
33, 34. See Book.
7. 6.
35. 8; T .
8. 8.
86. 4; 8.
9. 9.
37. 4 ; *.
10. 9.
38. ■*; 9.
11. 9.
39. The ratio of 72 to 8 it
12. 9.
13. 4.
greater by 2.
40. The ratio of 45 to 72 is
14. 8-H.
15. ±.
16. *.
-
•
greater by tV
41. The two ratios are equal. .
42. The ratio of 936 to 560 is
17. *.
18. ±.
19. +.
•20. +.
21. 3.
greater by -ft-
43. Greater inequality.
44. Less inequality.
45. Equality.
46. The ratio of 5 : 3= If.
22. 7.
" " 12: 4=8.
23. 112 Avoirdupois.
24. 4.
Compound, 60 : 12=5.
47. 1.
25. 6.
48. f.
26. 120.
0^ ft* A 4*.
49. «.
27. 60.
28. A-
Note. — It may often be of adrantagt
to adopt the following arrangement
of the terms in compounding ratios.
Aits. 480-504.] simple proportion.
68
60.
?,**/0,2
M
21
147
18
190 ; -ftf Am.
51.
8,^0)0,2
U,2
8; | Am.
SIMPLE PROPORTION.
Abt.
502-
1.
12.
2.
8.
8.
16.
4.
8.
5. See Book.
6. 20.
7. 55f.
8. 120.
9. 10. See Book.
MODEL RECITATION. AbT. 503*
Ex. 11. Analysis.— Since 16 barrels cost $112,1 band will costfV of $119,
which k $7; and if 1 barrel costs $7, 129 barrels will cost 129 times as much,
which is $903. Ans.
Or thus : since the flour is all of the
same quality, the ratio of 16 bbls. to 129
bbls. is equal to the ratio of $112, to the
answer required. But when four num-
bers are proportional, the product of the extremes is equal to the product of the
means ; (Art. 498 ;) therefore, if the product of the means, or the second and
third terms of the proportion, is divided by one of the extremes, or the first
term, the quotient will be the other extreme or answer required.
Operation,
bbls. bbls. dolls.
*♦ : 129 : : 11% : Ans.
129x7=$903.
12. $1309.50.
18. 9225.
14. 775 miles.
15. 20 tons.
16. 2156 lbs.
17. 51i lbs. of ginger.
18. $1640.64.
19. $7066.40.
Axx, 504.
20, 21. See Book. *
bn. qts. £
22. 20X4X8 : 2 :: 1X20X12X4 : Ans. (Art. 504. Obs.l.)
2X1X20X12X4 txttXltX*
20X4X8
**X*X*
=3 far. Ans.
64
SIMPLE PROPORTION.
[Sect. XIV.
28- $792.
cam*.
dob.
24 *0 : 178:: /i00 : Ans.
173X16=$2768.
25. 435 miles.
26. 252 days.
Art. 505*
27. See Book.
28. $2.70
20. 3s. 3d. 2ft- qrs.
yd. yd. doll.
80. f : V : : i : -4*w-
5
4
20
7
tt, 8
3
63=$3.15
lb. lb. doll.
81. ♦ : *£ : : ± : ^n*.
3
3
$
77
1
9
77=$8.555.
82. $26.40.
Art. 506.
33. See Book.
34. 30 bu. of bats ; 70 bu. corn.
35. 1925| lbs. copper; 641£ lbs.
tin.
36. 1520 lbs. nitre, 280 lbs.
charcoal, 200 lbs. sulphur.
37. 980.5155+lbs.
38. $1350.
39. £45.
40. $3375.
41. $2562.50.
42. $16480.
43. 7 ft. 6 in.=90.in., and 100
miles =6336000 in.
in.
Ans.
*0 : 4***000 : : 1 : Ans.
Ans. 70400 times.
44. 9 ft. 2 in. =110 in.
in. in. time.
210 : 4**0000 : : 1
Ans. 57600 times.
45. 170.
46. 200.
47. 480.
48. 375 sheep.
49. Note. — The first express traveling 60 miles a day for 5 days
before the second is dispatched, is 800 miles in advance ;
but the second, going at 'the rate of 75 miles a day, gains 15
miles per day. The question then resolves itself into this :
If the second courier gains 15 miles on the first in 1 day, in
• how many days will he gain 300 miles.
m. m. d
15 : 300 : : 1 : Ans. 20 days.
50. Note. — Since the hound runs 8 rods, while the fox runs 5, in
every 8 rods he gains 8 rods. Hence,
tod*. rod*. todi.
8 : 150 : : 8 : Ans. 400 rods.
Arts. 505-509.] compound proportion.
65
51. The stack will keep the cow 20 weeks, or the horse 15 weeks.
In 1 week, therefore, the cow eats -5V of it, and the horse -fa ;
the two will eat in 1 week -jV+tV ; that is V<ftF+"Afc="flA»
or -g 2 ^. And if in 1 week they eat -fy, in how many weeks
will they eat ££ ? Ans. 8f weeks.
52. 10+9+8+7=34. Hence, (Art. 506,)
34 : 10 : : 80s. : share of first.
■. £ s. d. fat.
First received, 23-fr=l, 3, 6, ItV.
Second " 21-^=1, 1, 2, Oft.
Third " ' 18^=0, 18, 9, StV
Fourth " 16tV=0, 16, 5, 2-[f .
53. Oxygen, 888-fr oz.
Hydrogen, 111£ oz.
COMPOUND PROPORTION.
Art. 508.
:: 12h.:^?w.
1, 2. See Book.
8. 11 A.: 33 A.
18 d. : 5 d.
Ans. 10 horses.
4. 19£ days.
5. 1314 gals.
6. 27 laborers.
Art. 509.
7. 8. See Book.
9.
**m.
1*0 p.
10 hrs.
Ans.
m.
*00 p., 3.
8 hrs.
: : 20 d.
24 days.
10.
11.
$0 m.
M0 ft.
ft.
4 ft.
Ans.
$ hrs.
m ft., i0
8 ft.
ft.
: : 18 d.
Ans.
18X8=144 ds.
>0 d., 15
10 hrs., 5
: : 1*0 m., 15
1125 miles.
12. $225.
13. $140.
14.
$mo.
t,
Ans.
$t mo., 4
**00, 80
: : U4.0, $2.40
$768.00.
66
CONJOINED PROPORTION. [SECT. XIV
J*.
A* p.
mo.
Ans.
1$ p., %.
1% mo., 3
: : $200
1600.
16. According to the suppo-
sition, 10 boys =6 men.
Hence, 30 boys=18 men.
In the statement of the
question we substitute men
for boys.
$,£0 m.
I0hrs.
4,0 A.
It hrs., |
MA., 32
: : 20 d., %
Ans.
penona.
Or, %M
X0hrs.
$M A.
♦ P-
32 days.
I£hrs.
00 A., 8
10 p.
: : 20 d., 4
Ans.
8X4=32 d
CONJOINED PROPORTION.
Arts. 510, 511*
11, 18. See Book.
19. 10 yds. N. York=9 yds. Athens.
90 yds. Athens =112 yds. Canton.
How many yds. Canton=50 yds. N. York.
10
W0,00
Ans.
II&56
56 yds. Canton.
20. 50 yds. Boston =45 bbls. Philadelphia.
90 bbls. Philadelphia=127 bales New Orleans.
How many bales N. Orleans =100 yds. Boston.
127
: : m,t
W
Am.
127 bales N. Orleans.
Arts. 510-516.] duodecimals.
m
21. $18 United States =8 ducats (gold) Frankfort
•12 ducats, Frankfort =9 pistoles (gold) Geneva.
50 pistoles, Geneva = 24 rupees (gold) Bombay,
How many rupees, Bombay =$100 U. States.
Ann.
M
: : 100,2
4X2X2=16 rupees.
SECTION XV.
DUODECIMALS.
8. 105 ft. 5' 4" 5"' 5"" 4'"".
9. 154 ft. 3' 1" 5"' 4"" 6'""
10. 85 ft. 1' 11" 0'" 5"" 2 //,/ '
6""".
11. 195 ft. 4' 1" 3'" 8"" 0'""
Art. 516.
1, 2. See Book.
3. 28 sq. ft. 6' 10".
4. 59 cu. ft. 3' 8".
5. 268 cu. ft. 6' 11".
6. 235 sq. ft.
7. 734 sq. ft. 0' 9".
12. 50 ft. 6'X 8 ft. 3'X 7 ft 4'=3055 cu. ft. 3' ; and 8055 cu. ft. 8'
+ 128=23 C. Ill cu. ft. 3'.
13. 3840 cu. ft. 0' 5".
14. The room being 20 ft. 6' long, and 10 ft. high, one side con-
tains 205 sq. ft. and the two sides 410 sq. ft. The width
is 18 ft. Hence one end =180 sq. ft. ; and the two ends=
360 sq. ft. The ceiling is 18 ft. X 20 ft. 6' =369 sq. ft.
But 410+360+369=1139. And 1139+ 9= 126* sq. yds.,
which multiplied by $.12i-=$15.81+$. Ans.
15. 50XlOX2£=1250cu. , ft. But 1250 cu. ft. XI 728=2160000
cu. in., which divided by 64, (the cubical contents of one
brick,)=33750, the number of bricks in the, mlL
PARTNERSHIP.
[Sect. XVL
SECTION XVI.
EQUATION OF PAYMENTS.
Art. 521.
1, 2. See Book.
8. 6 months.
4. 6 months.
5. 500X0=
500X1= 500
500X2=1000
500X3=1500
500X4=2000
500X5=2500
500X6=3000
3500 )1O500
Ans. 3 yrs.
6. 62 days.
PARTNERSHIP.
Art. 523.
1. See Book.
2. $960X-flr=$240, A's gain.
$960 X -At =$320, B's gain.
$960X-ft=$400, C'sgain.
3. $860X-Hf =$274.21fift, A's loss.
$860Xfff=$373.40fft, B's loss.
*860X"H?=$212.37fff, C'sloss.
4. - $22800, amount of debts,
$11200, " effects.
$1120QXu¥ir=$llf8.947, A's receipt*
$11200XWV=$2259.649, B's
$11200Xifi&=$3340.351, C's
$11200xW l ff=$4421.053, D's
5. $3000x£J=$850; A's gain.
$3000 Xii= $800, B's gain.
$8000X44=$700, C'sgain.
$3000xH=$650, D's gain.
6. See Book.
a
t*
M
Arts. 521-533.]
EXCHANGE.
69
«
€€
a
€<
7. 3200X6 = 19200^1
2400x7=16600 [ Reduce the resulting fractions by
1800X9=16200 | dividing by 600.
52200J
$4500Xi? =$165^.172, X's loss.
$4500Xif=$1448.276, Y's
$4500 X -§£=$1396.552, Z's
9. $60X-A°i/ 1 r=$22.486, A's rent.
$60XtWt=I21.024, B's
$60X-fWr=*16.490, C's
9. 12000X6= 72000
8000X2= 16000
25000X3= 75000
15000X3= 45000
35000X2= 70000
25000X4=100000
378000
$15000X^s%=$3492.06, A's gain.
$15000X-Hi=$4761.91, B's
$15000XifJ=$6746.03, C's
= 88000
= 120000
= 170000
t€
it
EXCHANGE OF CURRENCIES.
MODEL RECITATION. ART. 533*
Ex. 3. Analysis. — Ten shillings reduced to a decimal,
are equal to £.5 ; (Art. 346 ;) therefore £850, 10s. =£650.5.
Now since according to law the value of £1, is $4.84, the
value of £850.5 must be 850.5 times as much, and $4.84
X 850.5 =$4116.42. Hence, £850, 10s. =$41 16.42.
Operation
$4.84
850.5
$4116.42 Ans.
4. $850.63. •
5. $414,667.
6. $969,815.
7. $2041.59+.
8. $4841.089+.
9. $7746.082+.
10. $60652.55+.
11. $208683.S19+.
12. $330661.605+.
13. $242840.369+.
14. $257791.397 + .
15. $369716.864+.
16. $284412.622+.
17. $4840000.
*o
EXCHANGE.
[Sect. XVI.
Art. 534.
18. See Book.
19. £82.
20. £90.
21. £181, 10|d.
22. £261, 8s. 7*d.
23. £446, 7s. 8-Jd.
24. £201, lis. 7|d.
25. £883, 5s. 8-}d.
26. £1095, 3s. ll|d.
27. £5220, 9fd.
28. £8568, 3s. 7£d.
29. £10384, 18s. 4d.
30. £20661, 3s. lid.
Art. 548.
1. See Book.
2. 84791.60.
8. $25391.084+
4. 9284.58.
Art. 536.
31. See Book.
32. £135.
33. £227.
34. £316, 9d.
35. £375.
Art. 537*
36. See Book.
37. $534,166.
88. $614.1875.
39. $986,083.
40. $7714.285
41. $20000.
EXCHANGE.
5.*$10152.527+.
6. $707.
7. $1881.60.
8. $15418.509.
9. $20665.20.
10. $36480.755.
ARBITRATION
Art.
1. $4.80 United States:
£1 sterling:
How many florins =
12-
2* 6,X$ cts.
25 francs
10, 200 pence
5 milrees
5, ££00 marcs
.06X25X10X5X5=
And 1200 marc8X35=
Then $420— $375=
OF EXCHANGE.
549.
=£1 sterling.
= 12 florins.
-$4.80= 2 1 flor. Ans.
1 franc.
jM0d. (£1.)
$ milrees.
Z$ marcs,
dollars ?
$375, circuitous exchange,
$420, direct exchange.
, gain. Ans.
Arts. 534-555.]
ALLIGATION.
*1
3.
£l sterling florins, 3
$ francs.
% milrees.
£420 sterling.
X$ florins
$ francs
milrees ?
Circuitous remittance J420 X 3 = 1 260 milrees.
Again, 70d. : £420 : : 1 milree : 1440 milrees.
Gain by circuitous remittance 180 milrees.
AL LIGATION.
Art. 552*
1. See Book.
2. $.87£.
8. 5s. 4d. lf| qr.
Art. 554»
i. See Book.
(1st.)
5 3 grs. at 18 car. fine.
1 gr. " 20
1 gr. " 22
3 grs. " 24
(2d.)
at 18 car. fine.
" 20 "
«
a
u
" 22
" 24
ti
u
3 grs.
3 grs.
lgr.
(3d.)
8+1=4 grs. at 18 c. fine.
1+3=4 grs. " 20
1+3=4 grs. " 22
3+1=4 grs. " 24
Art. 555*'
6. See Book.
«
u
it
1. (1st.)
10 oz. 16 carats fine.
5 oz. 18 " "
5 oz. 22 " "
(2d.)
10 oz. 16 carats fine.
20 oz. 18 «• "
20 oz. 22 « u
(3d.)
10 oz. of each kind.
(1st.)
8. 133 lbs. at 20 eta.
95 lbs. at 30 cts.
190 lbs. at 54 cts.
(2d.)
41$ lbs. at 20 cts.
66£ lbs. at 30 cts.
47£ lbs. at 54 cts.
(3d.)
10 lbs. at 20 cts.
76 lbs. at 30 eta.
95 lb*, at 54 cts.
73
ALLIGATION.
[Sect. XVI.
(4th.)
114 lbs. at 20 cts.
66£ lbs. at 30 cts.
1424 lbs. at 54 cts.
(5th.)
8 If lbs. at 20 cts.
76 lbs. at 30 cts.
3 If lbs. at 54 cts.
Art. 556*
9. See Book.
10. 40 gallons at 15s.
40 " 17s.
40 " 18s.
200 " 22s.
11. : 2 :: IH : ^ns.
2 X 14=28 gals, water.
And 126 — 28=98 gals. wine.
DIFFERENT METHOD OF ALLIGATION.
Note. — 1 The following method will be found a very easy and expeditious
mode of solving questions in Alternate Alligation.
Take any convenient quantity of each of the ingredients at pleasure, and
set them in a column under each other; on the right of each quantity
place its price per lb., gal, &c., together with- its cost. Find the amount
of the several quantities mixed, also their actual cost, at the given prices;
then find the cost of the same amount of mixture at the mean price, and
call the difference between the cost of these two amounts the error.
Finally, consider whether in order to diminish the error you should use
more of an ingredient whose price is less, or more of one whose price is
greater than the mean price ; divide the error by the difference between
the price of the ingredient whose quantity you choose to increase, and
the mean price ; the quotient added to that quantity will correct the error.
If it is denied to increase more than one quantity, divide by the sum of the
differences between their given prices and the mean price.
In Ex. 4th, let us take the following quantities of
the several ingredients. The amount mixed is
15 gals, and its cost is 158s. But according
to the conditions of the question ,15 gals, should
cost 150s., which is 8s. less than the cost of
this quantity at the given prices. Hence it is
evident that we have taken too much of one
of the quantities whose price is greater than
the mean price. To counterbalance this error
we must increase one of the quantities whose
price is less than the mean price. Now it is
manifest, for every additional gallon we take at 8s. we shall diminish the
error by 2, the difference between 8s. and 10s., the mean price; and if we
take as many additional gals, at 8s. as 2 is contained times in 8, it will be
the true correction to be made ; that is, 2-{-4=£, is the quantity at 8s. per
gai A*. 6 gals. at 8s. ; 3 gals. at 9s. ; 5 gals, at lis.; 5gab.aiife.
Operation.
gala.
price.
coat.
2
at
8s.=
=168.
3
<<
9s.=
=27s.
5
u
118.=:
=55s.
5
tt
12s.=
:60S.
15 gals, cost 158s.
But 15 at 10s.=150s.
Error 8s.
Now 8-4-2=4 correction.
Arts. 556-562.]
INVOLUTION.
w
Note. — 2. This principle may also be applied with equal advantage to cases
where several of the quantities are limited. Thus,
A man has 2 gals, of oil worth 8 shillings per gal. ; 3 gals, worth 9s. ; 5 gab.
worth lis., which he wishes to mix with oil worth 12s., so that the mix-
ture may be worth 10s per gallon: how much of the last kind must be
taken?
Operation.
price. cart.
os.= los.
9s.= 27s.
lls.= 55s.
12s. = 24s.
Since the mixture at the given prices comes to
more than the same quantity at the mean
price, it is manifest we have taken too much
at 12s. ; the correction must therefore be sub-
tracted from the quantity taken ; and 2 — 1=1.
Ans. 1 gal at 12s.
fab.
2
3
5
2
at
(<
(i
12 gals, cost 122s.
But 12 gals. X 10=1208.
Error 2s.
Now 2-e-2=l, the correction.
SECTION XVII.
INVOLUTION.
Art. 559*
15.
8294400.
1. See Book.
16.
10000.
2. 54*.
17.
3125.
3. 43*.
18.
279936.
4. 87*.
19.
117649.
5. 91 4 .
20.
65536.
6. 416*.
21.
387420489.
1. 299*.
22.
6.25.
8. 785 4 .
23.
.000001728.
0. 228 s .
24.
.0000015625.
10. 693 8 .
25.
*•
11. 999 s *.
26.
TsV
27.
TlW»
Art. 562*
28.
27000
1UUU Oil •
12. See Book.
29.
2<H-
13. 15129.
30.
6W.
14. 2460375.
31.
USOrVft.
M
EVOLUTION.
[Sect. XVIL
EVOLUTION,
Art. 566.
1. See Book. ,
2. V1I9, or 119*".
^ « JL
3. V231, or 231 4 .
4. V685, or 685 T .
5. V*
6. V*.
8. See Book.
9. 81 T#
EXTRACTION OP THE SQUARE ROOT.
Art. 674,
1, 2. See Book.
3. 51.
4. 73.
5. 28.
6. 9.327+.
7. 69.
8. 84.
9. 99.
10. 167.
11. 31.
12. 9.848+.
13. 2.6457+.
14. 13.78404+.
15. 209.
16. 217.
17. 23.8.
18. 2.71.
19. .9044+.
20. 34.2.
21. 792.
22. 1.7810+.
23. 3216.
24. *.
*°' it'
26. .79056+.
27. 4.1683+.
SA 28.181.
29. 14.4116+.
30. 186.9951 +.
31. 12345.
32. 345761.
33. 31.05671.
Art. 575.
34. 365(19.104973174X.
29 )265
261
381 )400
381
38204 ) 190000
152816
382089 )37 18400
3438801
382098 )279599 Begin to
267469 contract
12130
11463
667
382
285
268
17
15
2 Rem.
Arts. 566—585.] square koot.
75
35. 1.41421356237+.
36. In this example, after obtaining
the first ten figures of the root, viz :
1.732050807, we have 1970648751
for the remainder, and 3464101614
for the next true divisor. The oth-
er figures of the root may be found
by contraction in the following man-
ner. (Art 575.)
An*. 1.7320508075688772.
Dirtaor. Dividend. ftoet
3464l0l614)1970648751(568877»- r %
1732050807
238597944
2078460 97
30751847
277128 13
3039034
2771281
267753
242487
25266
24249
1017
324 Rem.
APPLICATIONS OP THE SQUARE ROOT.
Art. 581.
1. See Book.
2. 32 feet.
3. 166.709+ miles.
4. 240 rds. length of one side.
339.4112+rods, diagonal.
Art. 582.
5. See Book.
6. 10.
7. 18.
8. 36.
9. 40.
10. 66.
11. 168.
12. 11.2.
13. 67.5.
15.
61'
16.
-ftV
17.
TTT«
Art. 583.
18.
63 rods.
19.
160 rods.
20.
320 rods.
21.
480 rods long, and
160 rods wide.
22.
148 in rank.
74 in file.
Art. 584.
23.
See Book.
24.
25 and 40.
Art. 585*
25.
18 and 47.
76
CUBE ROOT.
[Sect. XVII
EXTRACTION OP THE CUBE ROOT.
Art. 590.
1 — 3. See Book.
4. 45.
5. 52.
6. 83.
7 136.
8. 217.
9. 22.6.
10. 2.74.
11. 0.623.
12. 3.332222+.
13. 1.817121+.
14. 7.217652+.
15. 8.315517+.
16. f.
17. -H.
18. 3.5463+.
19. 3f.
20. 1. 25992104+ .
21. 0.64365958974.
22. 68 ft.
Art. 591.
23. See Book.
24. 3.1748+yards.
25. Since similar solids are to
each other as the cubes of
their homologous sides or
like dimensions, it follows
that, 6 8 : 3 8 :: 32 lbs. : Ans.
32X3 8 -r-6 , =4 lbs. Ana.
26. 379-rilbs.
Art. 592.
27. 24 and 72.
28. 128 and 256.
29. 60 and 300.
30. 160 and 640.
31. 426 and 2556.
32. 747 and 6723.
EXTRACTION OF ROOTS OF HIGHER ORDERS.
Art. 593.
Art. 594.
1. See Book.
11. See Book.
2. 2.
12. 2.4872+.
3. 16.
•
13. 414.5 +.
4. 376.
5. 6.
Art. 595*
6. 26.
14. See Book.
7. 5.
15. 1.104089.
8. ,7.
16. 1.080059.
9. 8.
17. 1.004074.
10. 9.
•
18. 1.047128.
Arts. 590-612/]
PROGRESSION.
77
SECTION XVIII.
ARITHMETICAL PROGRESSION.
Art. 603,
1. See Book.
2. 5050.
3. 78 strokes.
Art. 604.
4. See Book.
5. 33.
Art. 605 •
6. See Book.
7. 44.
8. See Book
9. 3f.
Art. 607.
10. See Book.
11. 33$.
12. 502.
Art. 608.
13. The common difference is 7 ;
14, 21, and 28, the means ;
and 7, 14, 21, 28, 35, the
series.
14. 15, 29, 43, 57, 71, and 85.
GEOMETRICAL PROGRESSION.
Art. 610.
1. See Book.
2. 4.
3. 4374.
4. 13671875.
5. 2048 dollars.
6. $334.5563944, amount of
$250.
$750.3651759245, amount
of $500.
$1628.89461462237890625,
amount of $1000.
Art. 611*
7. See Book.
8. 1023.
9. 43774*.
10. $111111111.111.
11. See Book.
1X3
12.
= li.
Art. 612.
13. See Book.
14. 8.
78
ANNUITIES.
[Sect. XVIIL
ANNUITIES.
Art. 614.
1, 2. See Book.
3. $826,902.
4. $2298.262.
5. $4835.74.
6. $36785.59.
Art. 615.
7. See Book.
8. $1333.333.
9. See Book.
PERMUTATIONS AND COMBINATIONS.
Art. 618.
Note. — The reason of this role may
he illustrated in the following man-
ner : any one thing a, is capable of
only one position ; as a.
Any two things, a and 6, are capable
of two variations ; as, ab, ba, which
number is expressed by 1X2.
When there are three things, a, b, e,
any two of them will make 2 chan-
ges; consequently, by putting the
one left out before each of these
two permutations, we shall mani-
festly have 2 permutations repeat-
ed 3 times, which is expressed by
1 X2X3=6. Thus, abc,acb,bac,
bca,cab } and cba.
In like manner, when there are 4, or
any number of things, it may be
shown that the number of changes
which can be made with them is
expressed by 1X2X3X4, &c.
1. See Book.
2. 40320 ways.
3. 362880 ways.
4. -3628800 ways.
5. 479001600 days.
Art. 619.
6. See Book.
7. 15120 numbers.
8. 165765600 words.
Arts. 614-631.] mensuration.
70
SECTION XIX.
MENSURATION OF SURFACES.
Arts. 622—631.
1. 270 acres.
2. 722-J- acres.
3. The diagonal of a square is the hypothenuse of a right-angled
triangle, the base and perpendicular of which arc equal.
But, in the case supposed, the square described on the
hypothenuse is double the square described on either side.
(Art. 581. Obs. L) Now as the diagonal is 100 rods, the
square of which is 10000, the square described on either
side must be 10000—2, or 5000 rods. Arts. 31| A.
4. 320 rods, or 1 mile.
5. 360 sq. ft. .
6. 435 sq. ft.
?. 1100 sq. ft.
Note.— The area of a triangle may also
be found by multiplying the altitude
by half the base.
9. 290.4737 sq. ft.
10. 4 A. 52.82 rods.
11. 62.8318 ft.
12. 141.37155 rods.
13. 100 ft.
15. 12 A. 43.49375 rods.
Note. — 1. The area of a circle may also
be found by multiplying the circum-
ference by a fourth <of the diameter;
(Leg. V. 11 *,) or, by multiplying the
square of the circumference by .07958.
"2. The circumference of a circle may
be found from the area, by dividing
the area by .07958 ; the square root
of this quotient will be the circum-
ference.
16. 31415.9 sq. ft.
17. 2 ft. 9.94 in.
Note. — The reason of this process is
manifest from the fact, that the di-
ameter of the circle must be the
diagonal of the inscribed square;
but the square of the diagonal is
doable the square of the side of a
square. (Art. 578, Leg. IV. 11.)
18. 17.3205 ft.
so
MENSURATION.
[Sect. XIX.
MEASUREMENT OP SOLIDS.
Arts. 633—641.
1. 1864 cu. ft.
2. 3154 cu. ft. 11' 6" 8'".
3. 2615 cu. ft. 1080 in.
4. 115 sq. ft. 114.368 in.
5. 53333* cu. ft.
6. 8835.75 cu. ft.
7. 900 sq. feet.
8. 1739 sq. ft.
9. 76 cu. ft.
10. 176 sq.ft.
11. 2748.89125 cu. ft.
12. 119366.25 cu. ft.
13.' 78.53975 sq. yds., or,
78sq.yds. 4ft. 123.1128 in.
Art. 642*
14. 1017.87516 sq. in.; or,
7 sq. ft. 9.87516 in.
Note. — The surface of a sphere is also
equal to four times the area of a cir-
cle of the same diameter. (Leg. VIII.
9. Cor.)
15. 14684558.20796 sq. m.
16. I767.14437cu.ro.
17. 5291335807.60158 cu. m.
Note. — The solidity of a sphere may
also be found by multiplying the
cube of its diameter into .5236 ; or,
the surface into ■§■ of the diameter.
MEASUREMENT OP LUMBER.
Art. 644*
18. The mean breadth is 13 in.=l ft. 1'. And 12 ft. XI ft. 1'
=13 sq. ft. Ans,
19. The mean diameter is 6 ; . Now 6'X6'=36"=3' ; and 3'X
16 ft.=48 / =4 cu. ft. Ans.
Or, 16ft.=192in. And 6 in. X 6 in. X 192 m.=6912 in. Now
6912 in. -7- 1728=4 cu. ft.
20. 50X5Xi=*f iL =62£cu. ft.
Art. 646«
21. The mean diameter is 38 in. Now 38X38X45=64980 in.,
the cubical contents of the cask. And 64980 X. 0034=
220.932 w. gals. ; or, 220 gals. 3 qts. 1 pt. 1.824 gi.
22. The mean diameter is 50.2 in. 50.2X50.2X64=16128256 in.,
the cubical contents of the cask. And 16128256 X .0028
=451.591168 b. gals.; or 451 gals. 2 qts. 0.729344 pt.
Arts. 633-655. J mechanical powers. 81
TONNAGE OP VESSELS. Art. 647.
23. 150 ft.=length ; 35 feet = breadth. 17£ ft. = depth as found
by the rule. Then 150 ft. diminished by 21, three-fifths
the breadth,=129 ft. And 129X35Xl7£=79012.5 ;
which divided by 95=831.71526+ tons. Ans.
24. The dimensions are the same as above. 150X35Xl7£=
91875. And 91875-t- 95=967.10521 +tons. Ans.
MECHANICAL POWERS.
Arts. 648—655.
1. 500 lbs.
2. 133^ lbs.
3. 8 ft. : 3 ft. : : 256 lbs. :
weight on long arm.
Hence, 256X3-7-8=96 lbs.
carried by A, and 256
lbs.— 96 lbs.=160 lbs.
carried by B.
4. Since the whole length is to
the short arm, as the
whole weight is to the
weight on long arm, it
follows that the whole
w : w on long arm : :
whole length : short arm ;
That is, 90 lbs. : 30 lbs. : :
12 ft. : short arm.
Hence, 30 X 12-7-90 =4ft.
Ans. 4 ft. from A.
8 ft. from B.
5. 600 lbs.
6. 1066| lbs.
7. 1600 lbs.
8. 1250 lbs.
9. 1136.3636+lbs.
10. The power moves through
a circle the radius of which
is 12 ft., the length of the
lever; the diameter is
therefore 24 ft., and the
circumference 24 ft. X
3.14159 = 75.39816 ft.
Now 75.39816X1000-7-
tV=904777.92 lbs. Ans.
Note. — In the preceding calculations of the several Mechanical Powers, no
account has been taken of friction and resistance. In their practical
applications it is customary to make an allowance of about one third for
these impediments.
4*
82
MISCELLANEOUS
[Sect. XIX.
*v
MISCELLANEOUS EXAMPLES.
1. 980—62=918.
918-f- 2=459, the less.
459+62=521, the greater.
Proof. 521+459=980 sum.
2. 4410-7-63=70. Ans.
3. 28f = A f L ; and 145= JJ V LA .
Now
1015 201 1015
7
1015
""" t X
201 '"" 201 — 5 ^ r * Ans '
4. 7iX5£=39£,and39£-7-6£
=6-5^. Ans.
5. 24000Xlilbs=36000 lbs.
the daily consumption ;
and 720000-7-36000=20
days.
6. 861.32.
7. $351.65, int.
$16581.65, amt.
8. $1200 X. 0155, the rate for
93 days=$18.60. Ans.
9. $1843.003+.
10. $24390.243, amount invest-
ed. (Art. 397.)
11. $1800X.25=$450. And
$1800+$450=$2250,
• which divided by 500=
$4.50 per yd. Ans.
12. $5000X-12=$600theloss.
$5000— $600=$4400; and
$4400-7- 640=$6. 875, the
price per bbl. Ans.
13. $125, the profit,
$375, the cost; and
$125-r-$375=33i per ct.
14. $36.
15. 229-Ji dolls.
16. 487ff eagles.
17. 2000 miles.
18! 2£ m. = 158400 in.;
5ift.=66in. Now 158400
-7-66=2400 times. Ans.
19. 2880 times.
20. $1.50 per gross.
21. $132.00-7-5040=.02£fcts.
the price per qt.
22. $2400 per A.
23. 43 7£ bbls.
24. 21 months.
25. $1,328.
26. 18 ft.XlS ft. = 270 sq. ft
=30 sq. yds., and 30^-}
=40. Ans. 40 yds.
27. Is. Od. 3?V qrs.
28. Zli\ni days.
29. 34fff£f days.
30. $1.60 per gal.
31. 12 miles.
32. 12f days.
33. 52£days.
34. 136 gals. 1 qt.
35. $180.
36. $10,875.
EXAMPLES.
83
37. $156,615+.
38. 94 days, 3 hrs. 38 m, 10+H
kbk bbl. dolL
39. -ft- : * : : £ : Ans.
Ans.
$2
40. £l.
41. $|f =$.926 nearly.
42. $4800.
43. $197tW=$197.759+.
44. 228 gals.
45. $40.29f,
46. $41,095.
47. 2 yrs., 182 days, 12 hrs.
tt
it
tt
«
48. First discharges -rV per minute.
Second " ^V
Third " &
Now A+A+A=-H per min. The question therefore re-
solves into this : if to discharge +}■ of the cistern requires
them all 1 min., how long will £$ or the whole cistern re-
quire them. Ans. 5-fV min.
cis. cis. m. m.
Or, U : U :: 1 : ■H=5 1 B r min,
49. Both together drink -^r per day ; the man alone iV, the wo-
man tS — 1^=3^-5 — Hhr=-rihr per day. Ans. 120 days.
50. i+i+i a o+irV=£ih or £. Hence, i=24, and f =120.
Ans. 120 scholars.
51. £292.
52. $6000 whole estate.
53. ^r=45 ; hence - 4 V=15, and ^-$=600.
54. 112+15+6=133. Hence, +H of 6650=5600 lbs. tin.
tV^ of 6650= 750 lbs. lead.
tSt of 6650= 300 lbs, brass.
Proof, 6650 lbs.
55. 36iX5=181f miles. 36|x7=254i miles.
66. Analysis. — From the conditions of the question, one part con-
tained 2 lbs. as often as the other 3 lbs. It is therefore
evident that the smaller part must contain 2 lbs. and the
larger 3 lbs. as often as 5 lbs. (2+3) are contained timet
in 196 lbs. But 196-7-5=39*. Hence,
2 lbs.X39i= 78f lbs. the smaller part,
8 lbs.X39i=H7i lbs. the larger part
84 MISCELLANEOUS
5*1. $192.307-ft, A's gain. $2307.692-ft, B r s gain.
$2500.000 C's gain.
58. Analysis. — Since $2400 gain $950, $1 will gain ^ft-* part of
$960 ; and $960-r-2400=M0. Now, if $1 gains 40 cents*
$600 will gain 600 times as much ; and $600X-40=$240,
A's gain. But A and B's gain is $240+$280=$520 ;
therefore $960 — $520=$440, C's gain.
•Again, if $.40 gain require $1 stock, $1 gain will require as
many dollars stock as $.40 are contained times in $1 ; and
$1.00-7- .40=$2.50. Consequently,
$2.50X28O=$700, B's stock; and
$2.50 X440=$l 100, C's stock.
59. 20 per cent. 60. $1371.
61. $4*755.141. 62. $32000.
63. i — $50+i+i — $10=£, or the whole property. Hence,
| — $60=£. Then f =$60, and f =$360. Ans.
64. Analysis. — If the man can frame the house in 12 days, in 9 d
he can frame -ft of it. But by the conditions of the question,
the man and boy together can frame the whole in 9 days ;
therefore the boy must frame -ft in 9 days. Now, if he can
frame -ft- in 9 d., he can frame -ft in i of 9 d., which is 3
d, ; and if -ft require him 3 d., -^ will require him 12 times
3 d.=36 days. Ans.
65. Analysis. — Since it requires 15 hrs. to fill the cistern when
the discharging pipe is closed, and 18 hrs. when it is open,
it follows that the discharging pipe will empty ft or f in
18 hrs. Now, if to discharge i of the cistern requires
18 hrs., f or the whole cistern will require, 90 hrs. Ans.
66. Analysis.— Since A paid * ; B f =-f ; C £=i ; D £ =f ; it
follows that all paid i +-f +£+$ =J^. (Art. 201.) Now,
if ^ pay £255, i must pay -ft of £255, which is £17.
Hence, since A pays ■$ he must have paid £1^X3= £51;
B " i " " £17X2=£34;
C " i " " £17X4=£68;
D " i " " £17X6=£102
it
a
u
EXAMPLES. 85
67. Analysis. — Since A received £5 as often as B did £7, and
C £8, it is plain that A received £5 as often as the sum of
£5-f £7+£8=£20 is contained times in £640, the whole
gain; and £640-7- £20= 32. The same is true of B and
C'sgain. Therefore, £5X32=£l60, A's gain.
£7X32 =£224, B's gain.
£8X32=£256, C's gain.
Again, partnership implies that the respective shares of the
stock bear the same relation to each other as the shares of
the gain ; consequently A put in £5 as often as £20 are
contained times in £820, which is 41 times. Therefore,
£5X41 =£205, A's share of the stock.
£7X41 =£287, B's share of the stock.
£8X41 =£328, C's share of the stock.
68. Analysis. — $640+$880+$800=$2320. Now, since the gain
of each is repeated twice in this sura, it is plain £ of it, that
is $2320-r-2=$1160, is the whole gain. Hence,
$1160— $640 (A and B's gain)=$520, D's share.
$1160 — $880 (B and C's gain)=$280, A's share.
$1160— $800 (A and C's gain)=$360, B's share.
69. -ftXi=A. Now if fV=l> ^=i of 1 or i, and fJ=V or
20. Ans.
70. 25 persons. 71. 40 and 80.
72. 75 and 128. ' 73. 56.5685+ ft.
74. 7200 rods. 75. -3.535519+ft.
76. (3.5)*X.?854=9.6211,5, area of base.
(2.5) 2 X. 7854=4.90875, " top
14.52990, sum of areas.
Product of areas= 47.2278200625
V47.2278200625=6.87225
Sum of areas added 14.52990
21.40215
Multiplied by i of height, 3 If
Ans. 677.73475 cu. ft.
86 MISCELLANEOUS
77. Since the area of a circle is equal to the square of the diame-
ter multiplied by .7854, (Art. 629,) it follows, that if the
area of a circle is divided by .7854, the quotient will be
the square of the diameter. (Art. 628. Obs.)
Now 160 r. -7-. 7854=203. 71785077, the square of diameter,
and V203. 71785077=14. 2729, the diameter; hence,
7.13645 rods is the radius, or the length of the rope re-
quired.
78. 50 A. 3 R. 28.7399+rods.
79. 2471705622710 square miles.
80. 336009142264006.23104 cu. miles.
81. 5890.5 lbs. 82. 585.80357+bu.
83. 2482.272 gals. =39.401 -fhhds.
84. Since the timber is of equal size throughout, we may sup*
pose its whole weight to be concentrated in the centre of
gravity, which is manifestly 15 ft. from each end of the
stick. The question then may be resolved into this : if a
weight is suspended 15 ft. from one end of a lever sup-
ported by a prop, at what distance from the weight must
another prop be placed, so that the latter shall support -f
of the weight, and the former \ ? Now, (Art. 650.)
W. on short A. : W. on long A. : : long arm : short arm.
That is, 2 : 1 : 15 ft. Ans. ; and lXl5-r2=7£ ft
But 15 ft. — 7£ ft.=7£ ft. ; hence the lever must be placed
7+ feet from the end of the stick.
85. 403291461126605635584000000 ways.
86. 31 miles and 180 rods.
87. 662i. 88. $4294967.295.
89. From the conditions of the question, it is evident that the
difference between the number of bags which they carry i»
equal to 2. Now if the mule receives one from the ass, the
difference will be 4 ; but in that case the mule will have
double the number of bags that the ass has ; consequently
the mule will have 8, and the ass 4. If the mule then
gives one to the ass, the latter will have 5 and the former 7.
Ans. The ass had 5, and the mule 7.
1
EXAMPLES. 87
00. The number is 1440.
01. A's share is evidently $120+ $95=$215 more than B's:
therefore $1000 — ($2 1 5 -f $95) =$690; and$690+3=$230
B's share. Henoe, $230+ $95=$325, C's share ; and
$325+$120=$445, A's share.
92. By the conditions of the question the time past noon is equal
to £ of the time to midnight. But £ +#=-f ; it follows
therefore that i2 hrs. are equal to -£ of the -time to mid-
night. Now if -f-=12 hrs., -£ must be i of 12 hrs., which is
1+ hrs. ; and $ are 6-f hrs. But 12 hrs. — 6-f hrs.=5-J- hrs.
or 5 hrs. 20 min. Ans. 20 min. past 5 o'clock, P. M.
. Or thus: l+£ : -J- : : 12 hrs. : to the time past noon.
Ans. 20 min. past 5 o'clock P. M.
03. In one day A, B, and C will do tV of the whole work ; B, C,
and D, tV ; C, D, and A, iV ; and D, A, and B, -rV Let
these be added together, and the sum ft 4 6 9 u , is the part
done by all working 3 days ; since in each of the three
parts, tV, tV» and i^, of the whole work, there is one day's
work of A ; in each of the three parts -iV>.tV» and tV, one
day's work of B, &c. Dividing the sum by 3, we have
■ftHftr* * Qe part done by all four in one day. Hence, s a yVu of
the work is to 1, the whole work ; or 349 : 3780 : : 1 day :
10-ff-J days, the time in which it would be performed by all
of them working together. Now from s 3 7 V u , the part done
in one day by A, B, C, and D, take tV, the part done by
B, C, and D, and the remainder Tf?r> is the part done by
A. Then, xffo - 1, or 79 : 3780 : : 1 : 47fJ days, the
time in which A alone would perform the work. By pro-
ceeding in the same manner, we should find that B would
perform it in 38-frf" days; C in 27-j^y days; and D in
111-fr days.
04. Since B gains 4 miles each day on A, it is evident 4 miles : 73
miles : : 1 day : 18-J- days, the time in which B would
gain a round on A, or in which these two would first be
together again. Also, 6 miles, the space gained each day
by C on B : 73 miles : : 1 day : 12£ days, the time in
88 MISCELLANEOUS EXAMPLES.
which B and C will first be together. Now the least com-
mon multiple of 18-}-, and 12-J- which are both divisible by
6-fV> is readily found to be 36£ days. Ans.
95. This is the same as to find in what time after twelve, the
minute hand will have gained half a round on the hour
hand. Now it is evident, that in 12 hours the minute hand
gains eleven rounds ; and consequently one round is gained
in the eleventh part of 12 hours, and half a round in half
that time, or the eleventh part of 6 hours ; that is, in 32-ft-
minutes. The time required therefore is 32-^ minutes
after 12 o'clock.
96. According 'to the Julian calendar, a year contains 365 days
and 6 hours, for every 4th year contains 366 days ; but
the solar year contains only 365 days 5 hrs. 48 min. and
48 sec. ; consequently in 1 year the error of the Julian
calendar amounts to 11 min. 12 sec. Hence,
11 min. 12 sec. : 24 hrs. : : 1 yr. : the time required.
Am. 1284- years.
97. It appears from the question there were $50 left after th<
third robbery ; now since half was taken and half a dolla?
more, it is plain that $51 were taken the third night. For,
the difference of two numbers added to the less is equal to
the greater number ; (Art. 154;) consequently there must
have been $101 left after the second robbery.
Again, since on the second night half was taken and half a
dollar more, it follows that there were $203 left after the
first robbery. In like manner, since on the first night half
was taken and half a dollar more, it is evident there must
have been $407 in his desk before the first robbery war
committed.
Or thus : 50X2+1 =$101, the sum left after 2d robbery;
101X2+1=$203, " " 3d "
203 X 2 + 1 =8407, the sum which he had in his desk
THE END.
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Those books constitute the most valuable series ever published — a fact fully evinced
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1st. The child is taught to read by the use of intelligible words only — beginning
with the least, as those of two letters, and gradually advancing to those of greater length.
2d. All the words in the first book, or Primer, are learned by the scholar in the spelL
Ing lessons, before he meets with them in the reacting lessons. Also, the difficult words
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The Convention of State and County Superintendents of Schools in Vermont, held
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commended Sanders' Series of School Books, consisting of Sanders' Spelling
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D. M. Camp wag President
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MARK H. NEWMAN AND & CO.'s PUBLICATIONS.
SANDERS' SERIES OF READERS.
Jtom the Committee on the Subject ef Education, appointed by the Senate of the State
of Illinois.
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of Prof. McGuffey, and we have a very decided preference for those of Mr. Sanders, and
recommend that they be adopted uniformly by the schools of the State of Illinois.
Farther, we have examined Day and Thomson's Arithmetical Series, Willson's His-
tories, and Gray's Chemistry, and And them superior to any other works of the kind,
with which we are acquainted, and think that the interests of education would be ad-
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Jan. 16, 1849. W. B. PLATS,
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Subject of Education.
in extract from a communication to the Board of Trustees and Visitors of the Common
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After examining such reading books as we have had access to, we are of opinion that
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portant object of gradually progressive lessons, both in subjects and language, is more
nearly attained in that series than in any other with which we are acquainted.
Signed by SEVENTY TEACHERS, Cincinnati.
Extract from a Petition, presented to the Board of Education, by the Principals of the
Public Schools for the city of Rochester.
" We, the undersigned, teachers in the Public Schools in the city, believing as we do.
that some changes in our text-books, upon certain conditions, would give a healthful
stimulus, facilitating our onward progress, therefore unite, earnestly soliciting the Hon.
*V , Board of Education to substitute Sanders' Spelling Book and Series op Readers
^ for use in our schools, for those now in use.
C. MESERVE,
M. DOUGLASS,
J. R. VOSBURGH,
LEWIS BIXBY,
DONALD G. FRASER,
WM. WATSON,
From Principals of Public Schools in the city of Buffalo.
We have examined Sanders' Fifth Reader, edited by C. W. Sanders of New York,
and find the selections appropriate, and their moral tone of a highly elevated character
The work seems admirably calculated to make easy, natural, and intellectual readers.
A. L. BINGHAM, Prin. of Public School, No. 11.
SAMUEL SLADE, Prin. of Public School, No. 3.
D. P. LEE, Principal of Public School, No. 7. *
June 14, 1848. E. F. COOK, 3d Department, Public School, No. 10.
From A. S. Lovell, Principal of City High School, Middletowv, Conn.
Haying carefully examined Sanders' Series of School Books, I most cheerfully re»
commend their general adoption, as I believe them to excel in several respects any serial
at present before the public.
*Jii/|ftl845. A. S.L0VEI4*
B
Principal No. 1.
C. W. COLYER,
in No. 14.
" No. 9.
W. OGDEN,
« No. 6.
« No. 13.
E. S. TREAT,
« No. 5.
« No. 14.
A. W. FISHER,
« No. 10.
3ER, No. 15.
WM. DALLIS,
« No. 3.
« No. 11.
A. N. MERRIMAN,
« No. 11.
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DAT AND THOMSON'S SERIES OF MATHEMATICS.
FOE SCHOOLS AND ACADEMIES.
HHf ffidDDDUKDIPdDIBT (DdDHJIESIBo
NO. 1. PRIMARY TABLE BOOK.
NO. a.-MENTAL ARITHMETIC.
Or First Lessons in Numbers for Children, by James B. Thomson, AJM. N«w
edition, revised and enlarged. 18mo, 120 pages, half bound.
NO. 3.-SLATE AND BLACK-BOARD EXERCISES.
NO. ^.-PRACTICAL ARITHMETIC.
Uniting the Inductive with the Synthetic modes of Instruction, also illustrating the
principles of Cancellation, for Schools and Academies. By James B. Thomsok,
A.M. New edition, revised and enlarged. I2mo, 366 pages, half bound.
NO. 5.— HIGHER ARITHMETIC.
Or the Theory *nd Application ot Numbers, combining the Analytic and Synthetic
modes of instruction, adapted to Scientific and Commercial purposes. By Jambs
B. Thomson, A.M. Large l2mo, full bound in leather. Just published.
These books are introductory to President Day and James B. Thomson's Mathematical
Series ; which, when finished, will be the most complete ever published in a series.
isiii©iEtra (DdDnriKsn.
NO. 1.-ELEMENTS OF ALGEBRA.
Being a School Edition of Day's Large Algebra.
NO. «•— ELEMENTS OF GEOMETRY.
Being an Abridgment of Legendre's Geometry, with Practical Notes and LHustn-
tions. Bound in leather.
NO. 3.-ELEMENTS OF TRIGONOMETRY, MENSURATION AND
LOGARITHMS.
NO-4.-DAY AND THOMSON'S SURVEYING. (In Press.)
Adapted both to the wants of the Learner and the Practical Surveyor.
NO. 5.— DAY AND THOMSON'S LEGENDRE. (In Press.)
NO. 6.— DAY AND THOMSON'S BOURDON'S ALGEBRA. (In Press.)
The Publishers respectfully ask the attention of School Committees and Teachers to
the foregoing Series of Arithmetical and Mathematical Works. To any Teacher, or
Committee on Text-Books for schools, who have never given these works an examinv
tion, the publishers would be glad to present copies for that purpose, without charge
The following is taken from an editorial in the Brooklyn Eagle, June 6, 1849: —
Thomson's Arithmetic. — A few weeks since we brought to the notice of our read-
ers the two Arithmetics of Mr. Thomson, with such commendations as a brief examina-
tion seemed to point out as well deserved. Since that time the question of introducing
them into the public schools of this city has been agitated, and the book committee ha t
Mr. Thomson and Professor Da vies, [the authors of two rival arithmetical systems,] be-
fore them in a meeting to which the principals of all the public schools were invited,
that the explanations of the authors might be heard by those who were chiefly interested
in the work of instruction. After the fullest examination by the committee, and the ap-
proval of nearly all the teachers, the book committee agreed to recommend the Arith
taetJcs of Mr. Thomson to the Board of Education ; and at the last meeting, they '
Mnaalmously adopted by the board.
G
HARK H. NEWMAN is CO.'s PUBLICATIONS.
DAT AND THOMSON'S ARITHMETICS.
On the subject of the Merits of these works, the publishers will say nothing, but
merely submit the following reasons in favor of their adoption, — condensed from the
expressed opinions of teachers, superintendents, and committees, who have examined
them.
I.-MENTAL ARITHMETIC.
Among the numerous reasons given for the adoption of this work are the following :—
1. " That it begins and ends in just the right places and in just the right way."
2. " That it equally avoids the childishness and puerility of some works on the sab*
ject, and the c< implication and difficulty of others/'
3. " That th > lessons gradually increase in difficulty, and in a manner happily adapted
to the expanding minds of children, from six to ten years of age."
4. " That these lessons are rendered interesting to the young, by the great variety of
Sersons, incidents and circumstances embodied in them — strikingly contrasting with
lat repulsive monotony, where the same name and the same object occur throughout a
whole lesson."
5. "That pictures and marks are excluded from the book, and their places sup-
plied by the Numerical Frame, forthe use of which ample instructions are given."
6. ** That the lessons are so arranged that the regular increase of numbers is con-
tinually broken up, and thus the solution of each question requires thought and fur-
nishes direct mental exercise for the pupil."
7. " That in the progress of the l>ook, the first example involving a new principle
is carefully analyzed, and affords a model of reasoning for the solution of all similar
questions."
8. "That after the pupil has become practically acquainted with the principles of a
rule, and is able to solve oxamples under it with facility, the operation is then defined,
and its more prominent terms briefly explained."
IL-PRACTICAL ARITHMETIC.
Among the reasons given for preferring this work we select the following : —
1. u That the arrangement of the subjects is consecutive, systematic, and natural."
2. "That the language om ployed in the definitions and rules is peculiarly appropriate)
concise, and clear."
3. " That great care is taken never to anticipate a principle, and that no principle is
used in the explanation of another, until it has itself been explained or demonstrated."
4. " That each principle is explained separately, and carefully analyzed— the why and
the wherefore of eiich step in the process are cleurljr and explicitly given."
5. " That the examples are numerous and diversified ; their arrangement is gradual
and progressive ; and the work is calculated to impress upon the pupil's mind an abid-
ing knowledge of the subject."
6. "That the note*, observations, and suggestions, contained in the work, form an
admirable system of instruction for the student, and afford important aid to the teacher."
7. " That Mental Arithmetic, instead of being pursued to a tedious and unprofitable
length independent of written Arithmetic, is here immediately connected with it, and is
made introductory to every department of the subject. Thus mental operations are
connected with the use of the slate throughout the course."
8. "That it Is strictly an American Book*— arranged in exact accordance with the
existing state and national laws, and the practice of business men."
9. " That the old, obsolete and useless forms of arithmetical operations are rejected,
and their places supplied by valuable improvements."
10. "That instead of giving the pupil a rule and requiring him to understand it be-
fore he is furnished with an example, this work first gives an example, taeu tells the
pupil how to do it* and why he did tU and then gives a short, clear and comprehensive
rule for it."
11. " That in nearly every article, something is gained in the mode of presenting the
subject, perspicuity and precision being remarkable throughout."
12. " That in studying this book, the pupil's mind is thoroughly and actively exer-
cised ; not in seeking for hours to comprehend the meaning of obscure and knotty pro-
positions—the unraveling of which has no more practical bearing than the solution of
a riddle or conundrum, but is exercised upon practical and useful principles, which he
can understand and apply as occasions for their use arise in after life."
• Thomson's Practical Arithmetic has the honor of being the first school book wh'ch
published the standard unite of Weights and Measures adopted by the Government of
lie United States.
Tin
L. No. 11.
u
No. 8.
u
No. 15.
u
No. 12.
u
No. 6.
a
No. 17.
u
No. 18.
u
No. 9.
u
No. 1.
MARK H. NEWMAN fe CO.*S PUBLICATIONS.
^ — — ' . ■ ■ 1. 1— .^»—
From the Principals of the Public Schools in the City of New York.
After a careful examination of " Thomson's Practical Arithmetic," we cheerfully ex?
press our hearty approbation of it. Having used the work in our Schools, we are free
to say that we deem it better adapted to the purposes of instruction than any other text-
book of the kind with which we are acquainted.
WILLIAM BELDEN, Prin. No. 2. GEORGE MOORE,
LEONARD HAZELTINE, " No. 14. CHARLES S. PELL,
A. K. VAN VLECK, " No. 16. WILLIAM H. WOOD,
DAVID PATTERSON, " No. 3. ASA SMITH,
WILLIAM H. REUCK, " No. 7, THOMAS P. OKIE,
NATHANIEL W. STARR, « No. 10. MARVIN W. FOX,
JOHN PATTERSON, « No. 4. J. A. FERGUSON,
JOHN H. FANNING, " No. 13. B. G. BRUCE,
M. J. O'DONNELL, « No. 5. WILLIAM W. SMITH,
Hew York, Oct. 5th, 1848.
From the Commissioners and Inspectors of the Thirteenth Ward School, New York*
The undersigned, Commissioners and Inspectors of Common Schools of the Thir-
teenth Ward, take great pleasure in stating that, after a careful and prolonged ex-
amination into the relative merits of a great number of Arithmetics presented for their
consideration, which number embraced all the most popular ones in present use, they
have unanimously adopted Day and Thomson's Mental and Practical Arithmetics for
the use of Ward School No. 19, recently organized and opened under their supervision
— these books being considered, for perspicuity of arrangement and adaptation to the
comprehension of the pupil, with or in the absence of a teacher, preferable to any books
on the same subject which have come under their consideration.
WILLIAM A. WALTERS,
WM. TYLER ANDERSON,
CHARLES D. FIELD.
From William Belden, Jr. A. M., Principal of Ward School No. 3, JV*. Y.
A careful examination of Prof. Thomson's " Practical Arithmetic" has satisfied me
that it is a work of uncommon merit.
The plan of presenting examples, in order to introduce the rule by previously analyz-
ing its principles, will commend itself to every experienced teacher as the natural pro-
cess, both for imparting knowledge of this subject, and giving correct habits of mental
discipline. The language of the explanations and rules is peculiarly clear and intel-
ligible, and the amount and value of this part of the work much superior to that of any
other arithmetic with which I am acquainted. WM. BELDEN, Jr.
From Thomas Foulee, Esq., Principal of Ward School No. 14, New York.
Having examined with care Thomson's Mental, Practical, and Higher Arithmetics, I
am pleased to have it in my power to state, as my unqualified opinion, that I consider
each work excellent in its kind ; and, as a whole, the series is decidedly the most philo-
sophical in its arrangement, lucid in its illustrations, and superior in its adaptation to the
wants and purposes of the school-room, to any other with which I am acquainted.
I shall recommend the introduction of the series into the school with which I am con-
nected, at an early day. THOMAS FOULKE.
We heartily concur in the above recommendation.
WM. KENNEDY, Prin. Ward S. No. 2.
A. B. CLARK, Prin. Ward S. No. 16.
J. J. ANDERSON, Prin. Ward S. No. 1.
From W. C. Kibbe, Esq., Principal of Ward School No. 19, New York.
Having used "Thomson's Practical Arithmetic" during the past year, it affords me
much pleasure to commucicate my unqualified approval of it.
It is comprehensive without unnecessary details, its rules are simple and practical, its
elucidations clear and explicit, and its examples combine information of great practical
utility, approaching near the actual business transactions of life. It is indeed a treatise
Well adapted as a text-book for our schools. W. C. KIBBE.
From L G. Hubbs, A. M., Principal of Mount Washington Institute, New York City.
Gentlemen: — I have carefully examined Mr. Thomson's "Practical Arithmetic," and
do most heartily add my testimonial to those already given in its favor. It is indeed a
work of very great merit, comprising many excellencies in a small compass. Its value
is a practical school-book will be more apparent on a second and thorough examination.
While as on elementary work it deserves a place in our best school! ', / know of no
rtter *o toe// adapted to general use. ISAAC G. HUBBS.
S
MARK H. NEWMAN & CO.'S PUBLICATIONS.
DAY AND THOMSON'S ARITHMETICS.
Ill -HIGHER ARITHMETIC.
Among the various reasons given for the adoption of this work, we present the follow-
ing:
1. u The work is complete in itself, embracing the fundamental principles of Arfth
metic, together with the highest combinations of numbers."
2. "The mode of analysis and reasoning, so successfully pursued in the Mental LtuI
Practical Arithmetics, is admirably carried out in the Higher, and applied to tho mora
intricate departments of the science."
3. " It is formed upon the plan that there is an intelligible reason for every operation,
and that that reason can and ought to be explained to pupils."
4. " Nothing has been taken for granted which requires proof."
5. " It happily unites the Philosophy of Arithmetic with its various applications to the
practical purposes of life."
6. " The rules are arranged in strict accordance with reason and the natural order of
the science."
7. "No subject is anticipated, and no principle is used in the explanation of an-
other, until it has itself been explained or demonstrated."
8. " The principles embraced under each rule are arranged consecutively and sya
tematically ; the dependence of each on those that precede it, is pointed out by refer*
ences — thus presenting a regular and harmonious series of principles and propositions,
the study or which must necessarily exert the happiest influence in developing and
strengthening the reasoning powers of the learner."
9. " The definitions and rules are remarkably clear, comprehensive, and exact."
10. " Instead of presenting the rules as mere arbitrary propositions, or inferring them
from principles unexplained to the pupil, they are deduced from a careful analysis of
examples, and a logical course of reasoning upon principles already established, the
steps of which are plain and intelligible to every youth of common understanding."
11. "It contains much valuable information respecting business transactions, and
matters of scionce, not found in other works of the kind."
12. " It contains the fullest and most satisfactory account of the origin and present
standard en! up. of American and Foreign Weights and Measures and Moneys of Account,
of any Arithmetic extant."
The following are some of the Recommendations which the Publishers have received
of Day and Thomson's Series. Many others have been received from distinguished practi-
cal Teachers and friends of education ; but our present limits will not admit of their inser-
tion. Its circulation during the brief period since its publication, is believed to be without
a parallel, and affords the best practical evidence of its merits : —
From Hon. Ira Mayhew, Svp't of Public Instruction, State of Michigan.
Gentlemen :— "Thomson's Higher Arithmetic," (which you obligingly forwarded me
a short time since,) was duly received, and has been critically examined. Having pre-
viously examined "Thomson's Mental and Practical Arithmetics," with much care, and
with an unusual degree of satisfaction, I looked for a superior work in the " Higher
Arithmetic ;" and I am happy (in being able) to say to you, my expectations have been
more than realized.
For the last thirteen years I have given special attention to the subject of Arithmetic
— in the school-room and in the study — with reference to supplying (or seeing supplied)
deficiencies in existing works, and obtaining a series adapted to the wants of studentt
of all grades — a series scientific in theory, and practical in its applications.
In the prosecution of this investigation, I have accumulated a large number of Arith
metics. After the most careful examination, t am fully satisfied that each volume in the
series under consideration is unrivaled. Taken together, as a whole— leading the
learner on step by step, from the si nip lust combinations of numbers through the higher
departments of the science — f regard Day and Thomson's Series of Arithmetics
the best f have eve* seen. I shall recommend their introduction into the Schoolf
of this State. 1 1 *ust they will go into general use.
Respectfully youm, ISA MAYHEW.
F
HARK H. NEWMAN fc OO.'s PUBLICATIONS.
DAT AND THOMSON'S ARITHMETICS.
From the Principals of the Albany Public Schools.
Within the last few years no leas than ten different systems of Arithmetic hare been
more or less used in our Schools. About two years since, in view of this evil, we ex-
amined several of the more prominent Arithmetics, and agreed with perfect unanimity
upon Thomson's Series as the best adapted to the wants of the pupil, and the general
purposes of instruction.
We are happy to say that, after a trial of more than two years, we are confirmed as
to the excelleucy of the books, that they have grown in favor by daily use, and that wo -
havo succeeded in making better arithmeticians than by the use of any other books.
SAMUEL STEELE, A. T. BALDWIN,
J. W. BULKLEY, WM. H. HUGHES,
WM. JANES, WM. L. MARTIN,
ROBERT TRUMBULL, THOS. W. VALENTINE,
E. S. ADAMS, JOEL MARBLE.
Albany, April 20th, 1850.
Front. Hon, Judge Blackman, A.M., Chairman of the Board of School Visitors of tils
City of New Haven, Cu
James B. Thomson, Esq. — Dear Sir : — I have examined with attention your " Practical
Arithmetic," and consider it decidedly the best work for inculcating and illustrating the
principles and practice of Arithmetic which I have ever seen. Your illustrations, in the
form of problems to be solved, are drawn, in a great measure, from the familiar scenes
of early life ; and while the young learner is interested in the solution of problems which
ho feels are practicable, he is encouraged to persevero in a study which would other-
wise be dull and forbidding, and is thus imperceptibly led to acquire and understand the
rules of Arithmetic, which he now knows to be true.
I remain, dear sir, very respectfully yours,
ALFRED BLACKMAN.
At a meeting of the Board of School Visitors of the First School Society of the city
of New Havon, Ct., duly warned and convened —
Voted, That the " Practical Arithmetic," by James B-. Thomson, A.M., be prescribed
for use in each school of this society. ALFRED BLACKMAN, Chairman.
Certified by H. G. Lewis, Secretary.
From S. S. Green, A.M^ Principal of Philips 1 Grammar School, Boston, Mass,
Mr. Thomson. — Dear Sir: — I hereby acknowledge the receipt of a copy of tho
u Practical Arithmetic," to which I have given sufficient attention to be convinced that
it possesses superior merit as a text-book. I am particularly pleased with the practical
churacter of it, the systematic and natural arrangement of its parts, the exactness of the
definitions, tho clearness with which the principles are explained and illustrated, and
the concise, yet explicit language, with which the rules are stated. You havo dono a
good service by removing from the tables of weights and measures all denominations
out of use, and by introducing those adopted by the General Government. The work,
in fine, is weJJ adapted to the purposes of instruction. SAMUEL S. GREEN.
From Rev. C. Pierce, A.M., Principal of West Newton State Normal School, Mass.
To Mark H. Newman, Esq.— Dear Sir :— The copy of " Thomson's Higher Arithmetic,"
which you put into my hands, I have examined with considerable care. Mr. T. li&s
given us, If not the best, one of the best, school-books which have appeared in this de-
partment.' Besides happily setting forth and explaining the common principles of num-
bers and their applications, illustrating the same by appropriate examples both abstract
and practical, his book contains many suggestions, in regard to the nature of numbers
and modes of operation, which are ingenious and useful. C. PIERCE.
From Rev. J. D. Wickham, Principal of Burr Seminary, Manchester, Vt.
Having examined, with some care, the Practical Arithmetic and the Higher Arithmetic
of Day and Thomson's Mathematical Series, we know of no Arithmetical treatises that
appear so well adapted to meet the wants of our Common Schools and Academies.
With this belief; we purpose to adopt them for use hereafter iu the Burr Seminary.
J. D. WICKHAM.
G
i
i
1
This book should be returned to
the Library on or before the last date
stamped below.
A fine of five cents a day is incurred
by retaining it beyond the specified
time.
Please return promptly.
"J
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