Horatio N. (Horatio Nelson) Robinson.
Key to the Progressive higher arithmetic : for teachers and private learners online
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LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
OR
Received
Accession No. (9 7j~ J / . Class No.
ROBINSON'S MATHEMATICAL SERIES.
KEY
PROGRESSIVE
HIGHER AEITHMETIC.
FOE TEACHEKS AND PRIVATE LEARNERS.
NEW YORK:
IVISON, PHINNEY, BLAKEMAN & CO.
CHICAGO: S. C. GRIGGS & Co.
1866.
R O B I N S O N'S
The 'most COMPLETE, most PRACTICAL, and most SCIENTIFIC SEE ES of
MATHEMATICAL TEXT-BOOKS ever issued in this country.
I. Robinson's Progressive Table Book. .....
II. Robinson's Progressive Primary Arithmetic, -
III. Robinson's Progressive Intellectual Arithmetic, -
IV. Robinson's Rudiments of Written Arithmetic, -
V. Hobiiison's Progressive Practical Arithmetic, -
VI. Robinson's Key to Practical Arithmetic. .....
vii. Robinson's Progresses Si^ii^r arithmetic, -
vm. Robinson's Key to Higher Arithmetic, -
IX. Robinson's New .Elementary Algebra, -
X. Robinson's Key to Elementary Algebra, -
XI. Robinson's University A Igebra, - - -
XII. Robinson's Key to University Algebra, .....
XIII. Robinson's New University Algebra, .....
XIV. Robinson's Key to ]Mew University Algebra, - ...
AV . Robinson's New Geometry and Trigonometry, -
XVI "rZoblnson's Surveying and navigation, .....
XVII. Hobinson's Analyt. Geometry and Conic Sections,
XVIII. Robinson's Differen. and Int. Calculus, (in preparation ,)-
XIX. Robinson's Elementary Astronomy, ......
XX Kobinson'3 University Astronomy. ......
XXI. Robinson's Mathematical Operations,
XXIi. Robinson's Key to Geometry and Trigonometry, Conic
Sections and Analytical Geometry, - - - .
Entered, according to Act of Congress, in the year 1SfiA w
DANIEL W. FISH & J, H. FRENCH,
an* I ngnin i:i the year l>63, by
DANIEL W. FISH. A.M.,
In the Clerk's Office of the District Court ,>f the TnHetl States for tbe Na them
District of tlio New York.
PROMINENT CHARACTERISTICS
OF
ROBINSON'S MATHEMATICAL SKRIES,
The books of this Series, although many of them have
so recently been published, have been recommended and
adopted by hundreds of the most critical and su^vssful
teachers, for the following reasons :
1. For the philosophical and scientific arrangement of
the subjects.
2. For the conciseness of the rules and the brevity and
accuracy of the definitions.
3. For the rigid and logical, yet full and comprehen-
sive Analysis.
4. For the new, original, and improved methods of
operations, not found in most other works of the kind.
5. For the very largo mnitbe.r and variety of practical
examples practical, because adapted to the ordinary
transactions of business life.
6. For their typographical execution, substantial bind"
ing, and general attractiveness.
7. For the easy gradation and progressiveness, not
only in the several books that compose the series, but in
the arrangement and treatment of the subjects of each
book.
8. For their adaptation to the various grades of schol-
arship in all our schools.
9. For the general unity of their plan, and the clear-
ness and perspicuity of their style.
10. For, scientific accuracy, combined with practical
utility, throughout the whole.
KEY.
ADDITION.
(67, page 25.)
Ex. 3 An*. 1982738.
Ex. 5. An*. 3189.
Ex. 7. An*. 415184.
Ex. 13. An*. 16977.
Ex. 17. An*. 6076510.
Ex. 4. Ans. 435058.
Ex. 6. An*. 289142.
Ex. 12. An*. 3001623.
Ex. 16. An*. 1881.
Ex. 20. An*. 21184000
Ex. 22. Ans. $1924950.
ADDING TWO OR MORE COLUMNS AT ONE OPERATION.
(68, page 28.)
Ex. 5. Number of churches, 35887;
" " persons accommodated, 13847902;
Value of church property, $85774659.
Ex. 6. Pounds of butter, 312625306;
cheese, 105735893;
" " wool, 52516961;
Bushels of wheat, 100485844.
SUBTEACTION.
(75, page 31.)
Ex. 5. Ans. 174333815. Ex. 6. Ans. 2361650877.
Ex. 7. An*. 86602389426. Ex. 8. Ans. 9000989311.
(25- 31)
[5]
6 SIMPLE NUMBERS.
Ex. 10. An*. 86 years. Ex. 13. Ans. $44656513
Ex. 15. Ans. 2121108 square miles; 316636286 population,
Ex. 20. Ans. 2657043.
TWO OB MORE SUBTRAHENDS.
(76, page 34.)
Ex. 5. 4568 Ex. 6. 4756+575+1404-84=5555
1320 1200
275 750
320 96
2653 Ans. 3509 Ans.
Ex. 7. $15760 Ex. 8. $75860
2175 45640
3794 25175
4587
$5045 An?
$5204 Ans.
Ex. 9. 20000 Ex. 10. 398470
11000 157548
7000 143429
Ans. 2000 square miles. 97493 Ans.
Ex. 11. $61307088 Ex. 12. $5760+$3575=$9335
52889800 2746
234000 4632
$8233288 Ans. $1957 Ans.
(31-34)
Ex. 13. 643J66
MULTIPLICATION. 7
Ex. 14. $8186793
65038
114624
Ans. 463504
Ex. 15. $12722470
7821556
424497
2355016
Ans. $2121401
5700314
904299
$1582180 Am.
MULTIPLICATION.
(85, page 38.)
Ex. 7. Ans. 43506216.
Ex. 16.^. 24500.
Ex. 20. Ans. $909000.
Ex. 8. Ans. 48288058.
Ex. 19. Ans. $31647000
POWERS OF NUMBERS
(91, page 39.)
Ex. 1. 72X72=5184.
Ex.2. 12X12X12X12X12=248832.
Ex. 3. 25X25X25=15625.
Ex.4. 7X7X7X7X7X7X7=823543.
Ex. 5. 19X19X19X19=130321.
Ex. 6. 3X3X3X3X3X3=729
(34-40)
8 SIMPLE NUMBERS.
Ex.7. Ans. 9 5 =59049; 11 3 =1331; 18 2 =g24;
244140625; 786 2 =617796; 94^=689869781056; 100 4
=100000000; 17 3 =4913; 251 5 =996250626251.
Ex.8. 8 3 =512; 15*=225; and 512x225=115200, Ans.
Ex. 9. 25 2 =625; 3 4 =81; and 625x81=50625, Ans.
Ex. 10. 7 3 X200=68600; 4 4 xll*=30976; and 68600
30976=37624, Ans.
CONTRACTIONS IN MULTIPLICATION.
(98, page 42.)
Ex. 1. 736X6X4=17664, Ans.
Ex. 2. 538X8X7=30128, Ans.
Ex. 3. 27865X7X3X4, or 27865X7X12,=2340660,^/U
Ex.4. 7856X4X4X3X3, or 7856X12X12=
1131264, Ans.
Ex. 5. $185X8X7=$10360, Ans.
Ex. 6. 17740872X8X12=1703123712 cubic feet, Ans.
(99.) -
Ex. 3. Ans. 50000 dollars. Ex. 4. Ans. 100000000000
(100, page 43.)
Ex. 3. Ans. 10350000. Ex. 5. Ans. 192128000
(102, page 41 )
Ex. 1. 5784 Ex. 2. 3785
246 721
34704 26495
138816 79485
1422864 Ans. 2728985 Ans.
(40-44)
MULTIPLICATION.
Ex. 3. 472856
54918
4255704
8511408
25534224
25968305808 Ans.
Ex. 5. 573042
24816
4584336
9168672
13753008
14220610272 Ans.
Ex. 7. 43725652
5187914
393530868
787061736
306079564
612159128
218628260
226847922169928 Ans.
Ex. 9. 2703605
4249784
18925235
132476645
113551410
227102820
11489737271320 Ans.
Ex. 4. 43785
7153
131355
656775
306495
313194105
Ex. 6. 78563721
127369
707073489
2828293956
2121220467
78563721
10006582580049 An* A
Ex. 8. 3578426785
64532164
14313707140
57254828560
114509657120
17892133925
229019314240
230923624151612740 An*
Ex. 10. 9462108
16824
75696864
227090592
151393728
159190504992 Ans.
(44, 45)
10 SIMPLE NUMBERS.
EXAMPLES COMBINING THE PRECEDING RULES.
(Page 45.)
Ex. 1. #28 X 175=44900; $37X320=$11840 5 $4900+
$11840=$16740, Ans.
Ex. 2. $1200 ($364+$275+$150+$187)=$224; and
$224X5=81120, Ans.
Ex. 3. 29+32=61; 61X17=1037 miles, Ans.
Ex.4. 834X127=84318; $47X97=$4559; and $4318+
$4559=18877, cost; 127+97=224; $40x224=88960,
sold for ; $8960 $8877=$83, profit, Ans.
Ex.5. 77+56=133; 675 133 = 542, multiplicand. 3X
156=468; 21428=186; 468 186=282, multiplier.
542X282=152844, Ans.
Ex.6. 37+50=87; 87X6=522; 98+522=620, multipli-
plicand. 6450=14; 14x5=70; 7010=60, multi-
plier. 620X60=37200, Ans.
Ex. 7. 14X25=350; 9x36 = 324; 350324 + 4324=
4350, multiplicand. 280112=168; 376 + 42 = 418;
418X4=1672; 168+1672=1840, multiplier. 4350X
1840=8004000, Ans.
Ex. 8. $2751X29967= $82439217
$5030x23905=$120242150
$37802933 Ans.
Ex. 9. 1449075X203=294162225 acres cultivated;
1922890880294162225=1628728655 acres, Ans
Ex. 10. $2258+$105=$2363, valuation per farm;
$2363x1449075=13424164225, Ans.
Ex.11. 2 4 X5 5 =50000; 7 3 =343;
50000343=49657, Ans.
(45, 46)
M ULTIPLICATION. i 1
V
Ex.12. 15=3375; 3 2 X2 5 =288; 208^=43264; 9x2 4
=144. 3375+43264=46639 ; 288+144=432 ; 46639
432=46207, Ans.
Ex. 13. 4+27+256+3125+46656=50068, Ans.
Ex. 14. 1200000X400=480000000 pounds, Ans.
Ex. 15. $2450, value of house ;
$2450X6 $500=14200 r farm;
$2450X2= 4900, stock;
s. $21550, total value.
Ex.16. 1500 X $7=$10500 ; 800x*10=$8000; 700
X$6=$4200; $8000 + $4200=$12200; $12200
$1050C=$1700, Ans.
Ex. 17. ($450+$780+$1250+$2275)X3=$14265,^t*.
Ex. 18. $115X35000=$4025000, Ans.
Ex. 19. $485X2500 =$1212500
$1450X10 == 14500
$1250X25 = 31250
$1258250 Ans.
Ex. 20. 1401944X$20=$28038880, value of double eagles;
62990 X $10= 629900, eagles;
154555 X $5= 772775, half eagles;
22059 X $3= 66177, " " $3 pieces.
Ans. $29507732, total value.
DIVISION.
(Ullage 49.)
Ex. 1. Am. 78972. Ex. 2. Ans. 121562.
Ex. 3. Ans. 152329. ^x. 4. ^ns. 6086847.
(46-49)
12 SIMPLE NUMBERS.
Ex. 9. Am. 7198. Ex. 10. Ans. 7071.
Ex. 11. Ans. 15607. Ex. 12. Ans. 48340 2 f 2 .
Ex. 13. Ans. 1253974? |. Ex. 14. Ans. 5479f| jf.
Ex. 15. Ans. 2084768|f ff . Ex. 16. Ans. 24781.
Ex. 17. Ans. 5851fff. Ex. 18. Ans. 591862f{}f
Ex 19. Ans. 15395919^f fii. Ex. 20. Ans.
Ex. 21. $147675^365^404f J | .4ns.
Ex. 22. $30732518 - 556= $55274i|| Ans.
Ex. 23. $5572470-v-287=$19416^ Ans.
Ex. 24. $8186793 - 27977=4292^f of
ABBREVIATED LONG DIVISION.
(112 page 51.)
Ex. 1. 204)77112(378 Ans.
159
163
Ex.2. 72)65664(912 Ans.
8
14
Ex. 3. 209)7913576(37864 Ans.
164
180
133
83
Ex. 4. 698)6636584(9508 Ans.
354
55
Ex. 5. 8903)4024156(452 Ans.
4625
. 1780
(49-51)
DIVISION.
Ex. 6. 6791)760592(112 Am.
814
1358
Ex. 7. 25203)101443929(4025^^ Am.
631
12786
1854
Ex. 8. 269181)1246038849(4629 An*.
169314
78062
242262
Ex. 9. 56240)2318922(41if |>f Ant.
6932
13082
Ex. 10. 17300)1454900(84^^% An*.
7090
1700
CONTRACTIONS IN DIVISION.
(121 page 57.)
Ex.1. 3(435 Ex.2. 7)4256
5)145 8)608
29 An*. 76 Am
Ex. -3. 9)17856 Ex.4. 2)15288
8)1984 3)7644
248 Am. 7)2548
364 Ana.
(51-57)
14 SIMPLE NUMBERS.
Ex. 5, 8)972552 Ex. 6. 9)526050
7)121569 7)58450
3)17367 2)8350
5789 Ans. 4175 Am
Ex. 7. 7)612360
5)87480
3)17496
5832 Ans.
Ex.8. 3)553
5)184 1
Quotient, 36 - - 4x3=12
13, remainder.
Ex.9. 3)10183
5)3394 - 1
7)678 - - - 4X3=12
Quotient, 96 - 6X^X3=90
103, remainder.
Ex. 10. 2)10197
3)5098 1
4)1699 1X2= 2
5)424 - - -3X3X2=18
Quotient, 84 - 4x4x3x2=96
^ 117, remainder.
(57)
DIVISION. 15
Ex. 11. 3)29792
8)9930 2
6)1241 2X3= 6
Quotient, 206 - - -5x8X3 =120
128, remainder
Ex. 12. 4)73522
6)18380 - 2
7)3063 2X4= 8
Quotient, 437 - - - 4x6x4= 96
106, remainder
Ex. 13. 3)63844
5)21281 1
9)4256 .... 1X3= 3
Quotient, 472 . - - 8x5X3=120
124, remainder.
Ex. 14. 2)386639
3)193319 1
4)64439 2X2= 4
5)16109 - . - 3XBX2= 18
6)3221 - - 4X4X3X2= 96
Quotient, 536 5X^X4X3X2=600
719, remainder
(57)
16 SIMPLE NUMBERS.
Ex. 15. 4)734514
6)183628 2
7)30604 4X4= 16
Quotient, 4372 18, remainder.
Ex. 16. 9)636388
9)70709 7
9)7856 5X9= 45
Quotient, 872 8x9X9=648
700, remainder,
Ex. 17. 5)4619
5)923 4
5)184 3X5= 15
Quotient, 36 4x5x5=100
119, remainder,
Ex. 18. 3)116423
7)38807 - - 2
7)5543 6X3= 18
8)791 - 6X7X3= 126
9)98 7X7X7X3=1029
Quotient, 10 - - - 8x8x7X7x3=9408
10583, remainder.
(57)
DIVISION. 17
Ex. 19. 5)79500
5)15900
5)3180 .
7)636
7)90 6X5X5X5= 750
Quotient, 12 - - - - 6x7X5x5x5=5250
6000, remainder.
(122, page 58.)
Ex. 2. AM. 79-&V Ex. 4. Ans. 230 T V$ft.
(123.)
Ex. 2. Ans. 27f3. Ex. 6. Ans. 8206|f$j|.
Ex. 7. ^ws. 3005.
EXAMPLES COMBINING THE PRECEDING RULES.
(Page 59.)
Ex.1. $4X25=$100; $3x36=$108; $100+$108=
$208 ; 2088=26, Ans.
Ex. 2. $10x12=8192; $13xl7=$221; $192+$221=
$413, cost; $18 X (12+17) = $522; $522 $413=
$109, Ans.
Ex. 3. $2X300+$750=$1350, value of produce t
$3X120+ $90= $450, stock;
$900 - 25=$36, AM.
(57-59)
18 SIMPLE NUMBERS.
Ex.4. 450+(24 12) X 5=510;
(90-^-6)+ (8 X 11) 18=30 ;
510-r-30=17, Am.
Ex.5. 648 x (3^X23)319 = 5184; 2910-f-15=194 ; 5184
194=4990, dividend; 4375-^-175=25 ; 25x4 2 +
3 2 =409 ; 2863 ~ 409 = 7, divisor. Hence, 4990-*-
7=712f , Ans.
Ex. 6. 42X34=1428; 107100-^-1428=75, Ans.
Ex. 7. Reversing the fifth operation, 12x24=288;
reversing the fourth operation, 288-7-6=48 ;
reversing the third operation, 48 +(28 16)=60;
reversing the second operation, 60 (7 2 +l)=10;
reversing the first operation, 10x45=450, Ans.
Ex.8. $60 $42=$18; $36x50=81800;
1800-^18=100 months, Ans.
Ex. 9. 251104-j-472=532, Ans.
Ex. 10. 30422=9253764, Ans.
Ex. 11. 453x307+109=139180, Ans.
Ex. 12. $4+$7=$ll ; $1276 - $11=116, number of each
kind; 116x2=232, whole number purchased;
$7 $4=$3; $3XH6=$348, difference in cost.
Ex. 13. $950+$7500=$8450;
$13982686-=-$8450=1654, and a remainder i
of $6386.
Ex. 14. 854x4300000-5-860=3870000 tons, Ans.
Ex. 15. ^3191876~400=57979f flf. Hence, by this est*
mate, 57979 persons died.
Ex. 16. 508464-f-10593=48, Ans.
(59, 60)
PKOBLEMS. 19
Jlx 11, $7680-r-$64=120, number sold;
$960-f-120=$8, gained per head;
$64 $8=$56, cost per head ;
$9800-^456=175, number bought.
Ex.18. $95X6+$1200=$1770; $1770-:-30=$59, Am.
Ex. 19. 36X16=576, number of days' work required;
576-r-24=24, number of days 24 men will require.
Ex. 20. $1650H-275=$6, cost per barrel;
($9 $6) X 186=$558, gain.
Ex. 21. 840-5-(5+10+15)=28, of each kind; hence, 28X
3=84, whole number.
Ex. 22. $965 ($5X160) = $165; 165 - 3 = 55 tons, un-
sold; 160+55=215 tons bought.
Ex. 23. $3825-s-$85=45, number sold;
$7560-v-($85-f-$5)=84, whole number of horses;
($7560+$945) $3825=$4680, to be realized on
the remainder.
Hence, $4680-K84 45)=$120, Am.
Ex. 24. $22360+$1742=$24102, total cost;
$15480 - $18=860; 860x2=1720, No. acres;
$22360~-1720=$13, original cost per acre.
PROBLEMS IN SIMPLE INTEGRAL NUMBERS.
(127, page 62.)
The following are the general? solutions :
Prob 1. Add the several numbers.
Prob 2. Subtract the sum of the given numbers from the
sum of all.
Prob. 3. Add the parts.
Prob. 4. Subtract the sum of the given parts from the whole
(60-63)
20 SIMPLE NUMBERS.
Prob. 5, Subtract the less from the greater.
Prob. 6. Subtract the difference from the greater.
Prob. 7. Add the difference to the less.
Prob. 8. Subtract the subtrahend from the minuend.
Prob. 9. Subtract the remainder from the minuend.
Prob. 10. Add the subtrahend and remainder.
Prob. 11. Multiply the numbers together.
Prob. 12. Divide the product by the given factor.
Prob. 13. Divide the continued product by the product of
the given factors.
.Prob. 14. Multiply the factors together in continued multi-
plication.
Prob. 15. Multiply the multiplicand by the multiplier.
Prob. 16. Divide the product by the multiplicand.
Prob. 17. Divide the product by the multiplier.
Prob. 18. Divide each number by the other.
Prob. 19. Divide the dividend by the divisor.
Prob. 20. Multiply the divisor and quotient together.
Prob. 21. Divide the dividend by the quotient.
Prob. 22. Multiply the divisor by the quotient, and to the
product add the remainder.
Prob. 23. Subtract the remainder from the dividend, and di-
vide the result by the quotient.
Prob. 24. Multiply the final quotient and the several divisors
together.
Prob. 25. Divide the first dividend by the continued product:
of the final quotient into all the given divisors.
Prob. 26. Divide the dividend by the several divisors sue-
cessively.
Prob. 27. Add together the numbers comprising each set,
and subtract the less sum from the greater.
(63, 64)
FACTORING. 21
Piob 28. Multiply together the factors comprising each set,
and add the several products.
Prob. 29. Multiply together the factors comprising each set,
and then add the products and given numbers.
Prob. 30. Multiply together the factors comprising each set,
and subtract the less product from the greater.
Prob. 31. Add the product of the given set or sets of fac-
tors and the given number or numbers.
Prob. 32. Subtract the sum of the products of the set or
sets of factors which form the less number from
the sum of the products of the set or sets of fac-
tors which form the greater number.
PROPERTIES OF NUMBERS.
FACTORING.
(142, page 71.)
Ex. 1. Ans. 2, 5, 5, 43. Ex. 2. Ans. 3, 5, 163.
Ex. 3. Ans. 2, 2, 3, 3, 5, 5, 7.
Ex. 4. Ans. 2, 2, 2, 2, 2, 2, 2, 2 ,2, 2, 3, 7.
Ex, 5. Ans. 2, 7, 13, 13. Ex. 6. 2, 2, 2, 5, 5, 5.
Ex. 7. Ans. 5, 5, 5, 5, 5, 5, 5, 5.
Ex. 8. Ans. 3, 3, 3, 7, 11, 13, 37.
(144, page 74.)
Ex. 2. Am. 2, 3, 7, 17, 17, 29. Ex. 3. Ans. 13, 17, 31
Ex, 4. Ans. 17, 19, 29. Ex. 5. Ans. 2, 11, 19, 487.
Ex. 6. Ans. 7, 83, 103. Ex. 7. Ans. 97, 103.
*f Ex. 8. Ans. 3, 5, 59, 139.
Ex. 9. Ans. 3, 5, 7, 47, 181.
(64-74)
22 PROPERTIES OF NUMBERS.
Ex. 10. Am. 2, 2, 2, 2, 2, 2, 41, 149.
Ex. 11. Ans. 7, 11, 37, 79.
Ex. 12. Ans. 2, 5, 13, 17, 37.
Ex. 13. Ans. 13, 17, 29.
Ex. 14. Ans. 2, 2, 2, 3, 17, 19, 23.
Ex. 15. Ans. 2, 3, 5, 7, 19, 179.
(145, page 76.)
Ex. 1. 120=1X2X2X2X3X5
1, 2, 4, 8 Combinations of 1 and 2.
3, 6, 12, 24 1, 2, and 3.
5, 10, 20, 40 ) "123 and 5
15, 30< 60, 120 J 1,4*,M
Ans. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Ex, 2. 84=1X2X2X3X7
1, 2, 4 Combinations of 1 and 2.
3, 6, 12 1,2, and 3.
7, 14, 28 } u u - o o fln j 7
21, 42, 84} 1, Z,d,and7.
. 4tw. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Ex. 3. 100=1X2X2X5X5
1, 2, 4 Combinations of 1 and 2.
5 > 10 > 20 l 1 2 and 5
25, 50, 100 } 1*3, and
^rcs. 1, 2, 4, 5, 10, 20, 25, 50, 100.
Ex. 4. 420=1X2X2X3X5X7
1, 2, 4 Combinations of 1 and 2.
3, 6, 12 1,2, and 3.
if; $ 6?} " "1,2, 3, and 5.
7, 14, 28)
21 > 42 > 84 I 1 2 3 5 and 7
35, 70, 140 f *' ^ 6 > *> and 7 '
105, 210, 420 J
. 1 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28,
" {30, 35, 42, 60, 70, 84, 105, 140, 210, 420.
(74-76)
GREATEST COMMON DIVISOR. 23
Ex. 5. 1050=1X2X5X5X3X7
1, 5, 25 Combinations of 1 and 5.
2, 10, 50 " 1, 5, and 2.
8 > 16 > 75 1 "152 and 3
6, 30, 150 j I, 0, ^, ai
7, 35, 175 1
14, 70, 350 I K
21* 105* 525 i ' > ; ; a
42^ 210 ? , 1050 J
( 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42,
I 50, 70, 75, 105, 150, 175, 210, 350, .525, 1050.
GREATEST COMMON DIVISOR.
(149, page 77.)
Ex. 2. 2X3=6, An*. Ex. 5. 6x7=42, Am
Ex.8. 3X3X7=63, ^ras. Ex.9. 91, Ans.
Ex. 11. 4X3X7=84, Ans.
(150, page 81.)
Ex. 4. Ans. 11. Ex. 5. ^is. 1.
Ex. 7. -4ns. 17. Ex. 8. Ans. 337
Ex. 10. In the operation under this rule, the quotient figure
may always be so taken that the product shall be either
greater, or less, than the dividend; in either case, the new
divisor will be the difference between the dividend and
product. It will always be found advantageous to use that
quotient figure which will give the least number for a new
divisor. In the first operation below, the second quotient
figure is 1, and the next divisor is 413690 ; in the second
operation, the second quotient figure is 2, which gives 178593"
for the next divisor, and abbreviates the subsequent work
(76-81)
PROPERTIES OF NUMBERS.
FIRST OPERATION.
L005973
4
4616175
4023892
592283
1
592283
413690
357186
1
2
3
413690
178593
169512
56504
54486
6
9081
2018
4
8072
2018
2
1009
SECOND OPERATION.
4616175
4023892
1005973
1184566
4
2
178593
8
169512
3
9081
6
8072
4
Am. 1009
2
592283
535779
56504
54486
2018
2018
Ex. 13. An*. 47.
Ex. 15. In order that the bins may be equal, the number
of bushels contained in one bin must be a common divisor of
the two quantities. And in order that the number of bins
may be the least possible, each must contain the greatest com-
mon divisor of the two quantities. Ans. 91 bushels.
Ex. 16. The pannels, to be of uniform length, must be a
common measure or divisor of the three sides ; and to be of
the greatest possible length, they must be the greatest com-
mon divisor of the three sides. Ans. 11 feet.
Ex. 17. The price to be paid by each is the greatest com-
mon divisor of the three sums, $620, $1116, aud 81488,
which is $124. Hence, B can purchase $620n-$124=5 ;
C can purchase $1116 H- $124 = 9; and D can purchase
$1488-:-$124=12.
Ex. 18. The greatest common divisor of 14599 feet and
10361 feet is 13 feet, the length of 1 joint in the fence,
(14599+10361)X2=49920 feet, the entire length of the
(81)
LEAST COMMON MULTIPLE.
25
fence. 49920 H- 13 = 3840, the number of joints in the
fence; and 3840x7=26880, the number of rails, Ans.
LEAST COMMON MULTIPLE.
(155, page 83.)
Ex. 1. 2X2X3X11X7X5=4620, Ans.
Ex.2. 7X3X3X2X2X11X5=13860, Ans.
Ex. 3. 5X3X2X2X2=120, Ans.
Ex. 4. 7X5X3X2X2X2X2=1680, Ans.
Ex. 5. 7X5X5X3=525, Ans.
Ex. 6. 19X3X7X2=798, Ans.
Ex. 7. 2X2X2X2X2X2X3X3X5=2880, Ans.
Ex. 1. 5, 3, 2
2, 2, 3, 7, 5
(156, page 85.)
15.. 18.. 21.. 24.. 35.. 36.. 42.. 50.. 60
3.. 7.. 4.. 7.. 6.. 7.. 5.. 2
5X3X2X2X2X3X7X5=12600, Ans.
, Ex. 2. 2, 2, 3
2,3,5
6. .8. .10. .15. .18. .20. .24
2.. 5.. 5.. 3.. 5.. 2
Ex. 3, 3, 5, 2
3, 5, 7, 2
2X2X3X2X3X5=360,
9. .15. .25. .35. .45. .100
3.. 5.. 7.. 3.. 10
3X5X2X3X5X7X2=6300, Ans.
Ex. 4. 3, 3, 2
2, 2, 5, 3
18.. 27.. 36.. 40
3.. 2. .20
3X3X2X2X2X5X3=1080,
(83-85)
26 P.
Ex. 5. 3, 3
2,13
ROPEKTIES
12. .26. .52
2. .13. .26
2X2X3X13=156, Am.
Ex. 6. 2, 2, 17
8,9
32.. 34.. 36
8.. 9
2X2X17X8X9=4896, Ans.
NOTE. When numbers are prime to each other, as 8 and 9 in
the above operation, their product will be their least common
multiple.
Ex. 7. 2, 2, 3
2,3,3
8. .12. .18. .24. .27. .36
2..
3.. 2.. 9.. 3
2X2X2X3X3X3=216, Ans.
Ex.
2,11
2,3,5
22. .33. .44. .55. .66
3.. 2.. 5.. 3
2X11X2X3X^=660, Ans.
NOTE. The first three numbers in Ex. 7 and the first two ia
Ex. 8, above, are factors of remaining numbers in the exam-
ples respectively. They might, therefore, have been omitted in
the operations.
Ex.9.
2, 2, 3
64.. 84.
.96.. 216
2, 2, 2, 2
16.. 7.
. 8.. 18
3,3,7
7
9
2X2X2X2X2X2X3X3X3X7=12098, Ans.
Ex. 10. The number of rods that will furnish whole days'
work to each one, must be some common multiple of 14, 25,
8 and 20 ; and the least number of rods that will furnish
(85)
CANCELLATION. 27
whole days' work to each of the men, must therefore be the
least common multiple of 14, 25, 8 and 20.
Ans. 1400 rods.
Ex. 11. The least common multiple of the prices, $4, $21,
$49, and $72, which is $3528, Ans.
Ex. 12. When all the men work together, they will dig
4 -f- 8 +6=1 8 rods per day. The ditch must therefore be
the least common multiple of 4 rods, 8 rods, 6 rods, and 18
rods, which is 72 rods, Ans.
Ex. 13. The least common multiple of 11 feet and 15
feet is 15 X 11=165 feet, the distance the carriage must
move to bring the rivets up together. Hence, 165x575=
94875 feet, the entire distance traveled ; and 94875 feet-4-
5280=17 miles 5115 feet, Ans.
CANCELLATION.
(159, page 87.)
K f>
Ex. 2.
-=80, Ans.
Jfrv/ i oiv 9f'V # V 3f
Af /\/Lfc> /\/> /\ y> XN/'
O
Ex. 3
-=32, ^Irw.
61
Ex.4,
61, Ans.
(86-88)
28 PROPERTIES OF NUMBERS.
71 11
<^X190X^
-=14839, .
Ex.5.
Ex.6.
Ex.7.
Ex.9.
=403, Ans.
9,
16
13
13,
Ex. 10. 84+56=140 cents.
240
32 cents,
240, Ans.
~\ cost of 2 yards of the
Ex. 11. 75X2+90=240 cents, [-first kind, and 1 yard
j of the second.
11 yards of the second kind ;
11X2=22 " " first "
Ex.12.
=60 cents, Am.
(88)
NOTATION AND NUMERATION. 29
FRACTIONS.
NOTATION AND NUMERATION.
(169, page 90.)
Ex. 3. Ans. |f. Ex. 4. Ans. 7 ^.
Ex. 5. Ans. |f. Ex. 6. Ans. f|f
Ex. 7. Ans. 25 H <y><>. Ex. 8. Ans. T1 /y> 52 .
Ex. 9. Ans. T WzjoW
Ex. 10. Four ninths; seven twelfths; seventeen thirty
eighths; forty-five one hundredths; seventy -two three hundred
seventy-fifths; forty-eight one thousand ninths; eighty-four
seven thousand eight hundred sixty-thirds ; four hundred fifty-
six five hundred thirty-sevenths.
Ex.11. Twenty fourths; eighty-seven thirtieths ; ninety-
five one hundredths; forty-eight twelfths; seventy-five four
hundred thirty-sevenths ; one hundred seventy-five halves ;
four hundred thirty-six fiftieths ; seven hundred sixty-six
four thousand eight hundred seventy-ninths.
Ex. 12. Four hundred sixty-seven nine hundred thirty-
sixths; five hundred thirty-six two hundred forty-eighths ; ten
thousand seventy-fifths; seventy-five ten thousandths; five
thousand seven three thousand sevenths.
Ex. 13. One hundred fifty Jive hundred thirty-sevenths ;
four hundred thirty-six nine hundred seventy seconds ; thir-
OF THE
UNIVERSITY OF CALIFORNIA.
OR
Received
Accession No. (9 7j~ J / . Class No.
ROBINSON'S MATHEMATICAL SERIES.
KEY
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1866.
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The 'most COMPLETE, most PRACTICAL, and most SCIENTIFIC SEE ES of
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VI. Robinson's Key to Practical Arithmetic. .....
vii. Robinson's Progresses Si^ii^r arithmetic, -
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XII. Robinson's Key to University Algebra, .....
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District of tlio New York.
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OF
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The books of this Series, although many of them have
so recently been published, have been recommended and
adopted by hundreds of the most critical and su^vssful
teachers, for the following reasons :
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3. For the rigid and logical, yet full and comprehen-
sive Analysis.
4. For the new, original, and improved methods of
operations, not found in most other works of the kind.
5. For the very largo mnitbe.r and variety of practical
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ness and perspicuity of their style.
10. For, scientific accuracy, combined with practical
utility, throughout the whole.
KEY.
ADDITION.
(67, page 25.)
Ex. 3 An*. 1982738.
Ex. 5. An*. 3189.
Ex. 7. An*. 415184.
Ex. 13. An*. 16977.
Ex. 17. An*. 6076510.
Ex. 4. Ans. 435058.
Ex. 6. An*. 289142.
Ex. 12. An*. 3001623.
Ex. 16. An*. 1881.
Ex. 20. An*. 21184000
Ex. 22. Ans. $1924950.
ADDING TWO OR MORE COLUMNS AT ONE OPERATION.
(68, page 28.)
Ex. 5. Number of churches, 35887;
" " persons accommodated, 13847902;
Value of church property, $85774659.
Ex. 6. Pounds of butter, 312625306;
cheese, 105735893;
" " wool, 52516961;
Bushels of wheat, 100485844.
SUBTEACTION.
(75, page 31.)
Ex. 5. Ans. 174333815. Ex. 6. Ans. 2361650877.
Ex. 7. An*. 86602389426. Ex. 8. Ans. 9000989311.
(25- 31)
[5]
6 SIMPLE NUMBERS.
Ex. 10. An*. 86 years. Ex. 13. Ans. $44656513
Ex. 15. Ans. 2121108 square miles; 316636286 population,
Ex. 20. Ans. 2657043.
TWO OB MORE SUBTRAHENDS.
(76, page 34.)
Ex. 5. 4568 Ex. 6. 4756+575+1404-84=5555
1320 1200
275 750
320 96
2653 Ans. 3509 Ans.
Ex. 7. $15760 Ex. 8. $75860
2175 45640
3794 25175
4587
$5045 An?
$5204 Ans.
Ex. 9. 20000 Ex. 10. 398470
11000 157548
7000 143429
Ans. 2000 square miles. 97493 Ans.
Ex. 11. $61307088 Ex. 12. $5760+$3575=$9335
52889800 2746
234000 4632
$8233288 Ans. $1957 Ans.
(31-34)
Ex. 13. 643J66
MULTIPLICATION. 7
Ex. 14. $8186793
65038
114624
Ans. 463504
Ex. 15. $12722470
7821556
424497
2355016
Ans. $2121401
5700314
904299
$1582180 Am.
MULTIPLICATION.
(85, page 38.)
Ex. 7. Ans. 43506216.
Ex. 16.^. 24500.
Ex. 20. Ans. $909000.
Ex. 8. Ans. 48288058.
Ex. 19. Ans. $31647000
POWERS OF NUMBERS
(91, page 39.)
Ex. 1. 72X72=5184.
Ex.2. 12X12X12X12X12=248832.
Ex. 3. 25X25X25=15625.
Ex.4. 7X7X7X7X7X7X7=823543.
Ex. 5. 19X19X19X19=130321.
Ex. 6. 3X3X3X3X3X3=729
(34-40)
8 SIMPLE NUMBERS.
Ex.7. Ans. 9 5 =59049; 11 3 =1331; 18 2 =g24;
244140625; 786 2 =617796; 94^=689869781056; 100 4
=100000000; 17 3 =4913; 251 5 =996250626251.
Ex.8. 8 3 =512; 15*=225; and 512x225=115200, Ans.
Ex. 9. 25 2 =625; 3 4 =81; and 625x81=50625, Ans.
Ex. 10. 7 3 X200=68600; 4 4 xll*=30976; and 68600
30976=37624, Ans.
CONTRACTIONS IN MULTIPLICATION.
(98, page 42.)
Ex. 1. 736X6X4=17664, Ans.
Ex. 2. 538X8X7=30128, Ans.
Ex. 3. 27865X7X3X4, or 27865X7X12,=2340660,^/U
Ex.4. 7856X4X4X3X3, or 7856X12X12=
1131264, Ans.
Ex. 5. $185X8X7=$10360, Ans.
Ex. 6. 17740872X8X12=1703123712 cubic feet, Ans.
(99.) -
Ex. 3. Ans. 50000 dollars. Ex. 4. Ans. 100000000000
(100, page 43.)
Ex. 3. Ans. 10350000. Ex. 5. Ans. 192128000
(102, page 41 )
Ex. 1. 5784 Ex. 2. 3785
246 721
34704 26495
138816 79485
1422864 Ans. 2728985 Ans.
(40-44)
MULTIPLICATION.
Ex. 3. 472856
54918
4255704
8511408
25534224
25968305808 Ans.
Ex. 5. 573042
24816
4584336
9168672
13753008
14220610272 Ans.
Ex. 7. 43725652
5187914
393530868
787061736
306079564
612159128
218628260
226847922169928 Ans.
Ex. 9. 2703605
4249784
18925235
132476645
113551410
227102820
11489737271320 Ans.
Ex. 4. 43785
7153
131355
656775
306495
313194105
Ex. 6. 78563721
127369
707073489
2828293956
2121220467
78563721
10006582580049 An* A
Ex. 8. 3578426785
64532164
14313707140
57254828560
114509657120
17892133925
229019314240
230923624151612740 An*
Ex. 10. 9462108
16824
75696864
227090592
151393728
159190504992 Ans.
(44, 45)
10 SIMPLE NUMBERS.
EXAMPLES COMBINING THE PRECEDING RULES.
(Page 45.)
Ex. 1. #28 X 175=44900; $37X320=$11840 5 $4900+
$11840=$16740, Ans.
Ex. 2. $1200 ($364+$275+$150+$187)=$224; and
$224X5=81120, Ans.
Ex. 3. 29+32=61; 61X17=1037 miles, Ans.
Ex.4. 834X127=84318; $47X97=$4559; and $4318+
$4559=18877, cost; 127+97=224; $40x224=88960,
sold for ; $8960 $8877=$83, profit, Ans.
Ex.5. 77+56=133; 675 133 = 542, multiplicand. 3X
156=468; 21428=186; 468 186=282, multiplier.
542X282=152844, Ans.
Ex.6. 37+50=87; 87X6=522; 98+522=620, multipli-
plicand. 6450=14; 14x5=70; 7010=60, multi-
plier. 620X60=37200, Ans.
Ex. 7. 14X25=350; 9x36 = 324; 350324 + 4324=
4350, multiplicand. 280112=168; 376 + 42 = 418;
418X4=1672; 168+1672=1840, multiplier. 4350X
1840=8004000, Ans.
Ex. 8. $2751X29967= $82439217
$5030x23905=$120242150
$37802933 Ans.
Ex. 9. 1449075X203=294162225 acres cultivated;
1922890880294162225=1628728655 acres, Ans
Ex. 10. $2258+$105=$2363, valuation per farm;
$2363x1449075=13424164225, Ans.
Ex.11. 2 4 X5 5 =50000; 7 3 =343;
50000343=49657, Ans.
(45, 46)
M ULTIPLICATION. i 1
V
Ex.12. 15=3375; 3 2 X2 5 =288; 208^=43264; 9x2 4
=144. 3375+43264=46639 ; 288+144=432 ; 46639
432=46207, Ans.
Ex. 13. 4+27+256+3125+46656=50068, Ans.
Ex. 14. 1200000X400=480000000 pounds, Ans.
Ex. 15. $2450, value of house ;
$2450X6 $500=14200 r farm;
$2450X2= 4900, stock;
s. $21550, total value.
Ex.16. 1500 X $7=$10500 ; 800x*10=$8000; 700
X$6=$4200; $8000 + $4200=$12200; $12200
$1050C=$1700, Ans.
Ex. 17. ($450+$780+$1250+$2275)X3=$14265,^t*.
Ex. 18. $115X35000=$4025000, Ans.
Ex. 19. $485X2500 =$1212500
$1450X10 == 14500
$1250X25 = 31250
$1258250 Ans.
Ex. 20. 1401944X$20=$28038880, value of double eagles;
62990 X $10= 629900, eagles;
154555 X $5= 772775, half eagles;
22059 X $3= 66177, " " $3 pieces.
Ans. $29507732, total value.
DIVISION.
(Ullage 49.)
Ex. 1. Am. 78972. Ex. 2. Ans. 121562.
Ex. 3. Ans. 152329. ^x. 4. ^ns. 6086847.
(46-49)
12 SIMPLE NUMBERS.
Ex. 9. Am. 7198. Ex. 10. Ans. 7071.
Ex. 11. Ans. 15607. Ex. 12. Ans. 48340 2 f 2 .
Ex. 13. Ans. 1253974? |. Ex. 14. Ans. 5479f| jf.
Ex. 15. Ans. 2084768|f ff . Ex. 16. Ans. 24781.
Ex. 17. Ans. 5851fff. Ex. 18. Ans. 591862f{}f
Ex 19. Ans. 15395919^f fii. Ex. 20. Ans.
Ex. 21. $147675^365^404f J | .4ns.
Ex. 22. $30732518 - 556= $55274i|| Ans.
Ex. 23. $5572470-v-287=$19416^ Ans.
Ex. 24. $8186793 - 27977=4292^f of
ABBREVIATED LONG DIVISION.
(112 page 51.)
Ex. 1. 204)77112(378 Ans.
159
163
Ex.2. 72)65664(912 Ans.
8
14
Ex. 3. 209)7913576(37864 Ans.
164
180
133
83
Ex. 4. 698)6636584(9508 Ans.
354
55
Ex. 5. 8903)4024156(452 Ans.
4625
. 1780
(49-51)
DIVISION.
Ex. 6. 6791)760592(112 Am.
814
1358
Ex. 7. 25203)101443929(4025^^ Am.
631
12786
1854
Ex. 8. 269181)1246038849(4629 An*.
169314
78062
242262
Ex. 9. 56240)2318922(41if |>f Ant.
6932
13082
Ex. 10. 17300)1454900(84^^% An*.
7090
1700
CONTRACTIONS IN DIVISION.
(121 page 57.)
Ex.1. 3(435 Ex.2. 7)4256
5)145 8)608
29 An*. 76 Am
Ex. -3. 9)17856 Ex.4. 2)15288
8)1984 3)7644
248 Am. 7)2548
364 Ana.
(51-57)
14 SIMPLE NUMBERS.
Ex. 5, 8)972552 Ex. 6. 9)526050
7)121569 7)58450
3)17367 2)8350
5789 Ans. 4175 Am
Ex. 7. 7)612360
5)87480
3)17496
5832 Ans.
Ex.8. 3)553
5)184 1
Quotient, 36 - - 4x3=12
13, remainder.
Ex.9. 3)10183
5)3394 - 1
7)678 - - - 4X3=12
Quotient, 96 - 6X^X3=90
103, remainder.
Ex. 10. 2)10197
3)5098 1
4)1699 1X2= 2
5)424 - - -3X3X2=18
Quotient, 84 - 4x4x3x2=96
^ 117, remainder.
(57)
DIVISION. 15
Ex. 11. 3)29792
8)9930 2
6)1241 2X3= 6
Quotient, 206 - - -5x8X3 =120
128, remainder
Ex. 12. 4)73522
6)18380 - 2
7)3063 2X4= 8
Quotient, 437 - - - 4x6x4= 96
106, remainder
Ex. 13. 3)63844
5)21281 1
9)4256 .... 1X3= 3
Quotient, 472 . - - 8x5X3=120
124, remainder.
Ex. 14. 2)386639
3)193319 1
4)64439 2X2= 4
5)16109 - . - 3XBX2= 18
6)3221 - - 4X4X3X2= 96
Quotient, 536 5X^X4X3X2=600
719, remainder
(57)
16 SIMPLE NUMBERS.
Ex. 15. 4)734514
6)183628 2
7)30604 4X4= 16
Quotient, 4372 18, remainder.
Ex. 16. 9)636388
9)70709 7
9)7856 5X9= 45
Quotient, 872 8x9X9=648
700, remainder,
Ex. 17. 5)4619
5)923 4
5)184 3X5= 15
Quotient, 36 4x5x5=100
119, remainder,
Ex. 18. 3)116423
7)38807 - - 2
7)5543 6X3= 18
8)791 - 6X7X3= 126
9)98 7X7X7X3=1029
Quotient, 10 - - - 8x8x7X7x3=9408
10583, remainder.
(57)
DIVISION. 17
Ex. 19. 5)79500
5)15900
5)3180 .
7)636
7)90 6X5X5X5= 750
Quotient, 12 - - - - 6x7X5x5x5=5250
6000, remainder.
(122, page 58.)
Ex. 2. AM. 79-&V Ex. 4. Ans. 230 T V$ft.
(123.)
Ex. 2. Ans. 27f3. Ex. 6. Ans. 8206|f$j|.
Ex. 7. ^ws. 3005.
EXAMPLES COMBINING THE PRECEDING RULES.
(Page 59.)
Ex.1. $4X25=$100; $3x36=$108; $100+$108=
$208 ; 2088=26, Ans.
Ex. 2. $10x12=8192; $13xl7=$221; $192+$221=
$413, cost; $18 X (12+17) = $522; $522 $413=
$109, Ans.
Ex. 3. $2X300+$750=$1350, value of produce t
$3X120+ $90= $450, stock;
$900 - 25=$36, AM.
(57-59)
18 SIMPLE NUMBERS.
Ex.4. 450+(24 12) X 5=510;
(90-^-6)+ (8 X 11) 18=30 ;
510-r-30=17, Am.
Ex.5. 648 x (3^X23)319 = 5184; 2910-f-15=194 ; 5184
194=4990, dividend; 4375-^-175=25 ; 25x4 2 +
3 2 =409 ; 2863 ~ 409 = 7, divisor. Hence, 4990-*-
7=712f , Ans.
Ex. 6. 42X34=1428; 107100-^-1428=75, Ans.
Ex. 7. Reversing the fifth operation, 12x24=288;
reversing the fourth operation, 288-7-6=48 ;
reversing the third operation, 48 +(28 16)=60;
reversing the second operation, 60 (7 2 +l)=10;
reversing the first operation, 10x45=450, Ans.
Ex.8. $60 $42=$18; $36x50=81800;
1800-^18=100 months, Ans.
Ex. 9. 251104-j-472=532, Ans.
Ex. 10. 30422=9253764, Ans.
Ex. 11. 453x307+109=139180, Ans.
Ex. 12. $4+$7=$ll ; $1276 - $11=116, number of each
kind; 116x2=232, whole number purchased;
$7 $4=$3; $3XH6=$348, difference in cost.
Ex. 13. $950+$7500=$8450;
$13982686-=-$8450=1654, and a remainder i
of $6386.
Ex. 14. 854x4300000-5-860=3870000 tons, Ans.
Ex. 15. ^3191876~400=57979f flf. Hence, by this est*
mate, 57979 persons died.
Ex. 16. 508464-f-10593=48, Ans.
(59, 60)
PKOBLEMS. 19
Jlx 11, $7680-r-$64=120, number sold;
$960-f-120=$8, gained per head;
$64 $8=$56, cost per head ;
$9800-^456=175, number bought.
Ex.18. $95X6+$1200=$1770; $1770-:-30=$59, Am.
Ex. 19. 36X16=576, number of days' work required;
576-r-24=24, number of days 24 men will require.
Ex. 20. $1650H-275=$6, cost per barrel;
($9 $6) X 186=$558, gain.
Ex. 21. 840-5-(5+10+15)=28, of each kind; hence, 28X
3=84, whole number.
Ex. 22. $965 ($5X160) = $165; 165 - 3 = 55 tons, un-
sold; 160+55=215 tons bought.
Ex. 23. $3825-s-$85=45, number sold;
$7560-v-($85-f-$5)=84, whole number of horses;
($7560+$945) $3825=$4680, to be realized on
the remainder.
Hence, $4680-K84 45)=$120, Am.
Ex. 24. $22360+$1742=$24102, total cost;
$15480 - $18=860; 860x2=1720, No. acres;
$22360~-1720=$13, original cost per acre.
PROBLEMS IN SIMPLE INTEGRAL NUMBERS.
(127, page 62.)
The following are the general? solutions :
Prob 1. Add the several numbers.
Prob 2. Subtract the sum of the given numbers from the
sum of all.
Prob. 3. Add the parts.
Prob. 4. Subtract the sum of the given parts from the whole
(60-63)
20 SIMPLE NUMBERS.
Prob. 5, Subtract the less from the greater.
Prob. 6. Subtract the difference from the greater.
Prob. 7. Add the difference to the less.
Prob. 8. Subtract the subtrahend from the minuend.
Prob. 9. Subtract the remainder from the minuend.
Prob. 10. Add the subtrahend and remainder.
Prob. 11. Multiply the numbers together.
Prob. 12. Divide the product by the given factor.
Prob. 13. Divide the continued product by the product of
the given factors.
.Prob. 14. Multiply the factors together in continued multi-
plication.
Prob. 15. Multiply the multiplicand by the multiplier.
Prob. 16. Divide the product by the multiplicand.
Prob. 17. Divide the product by the multiplier.
Prob. 18. Divide each number by the other.
Prob. 19. Divide the dividend by the divisor.
Prob. 20. Multiply the divisor and quotient together.
Prob. 21. Divide the dividend by the quotient.
Prob. 22. Multiply the divisor by the quotient, and to the
product add the remainder.
Prob. 23. Subtract the remainder from the dividend, and di-
vide the result by the quotient.
Prob. 24. Multiply the final quotient and the several divisors
together.
Prob. 25. Divide the first dividend by the continued product:
of the final quotient into all the given divisors.
Prob. 26. Divide the dividend by the several divisors sue-
cessively.
Prob. 27. Add together the numbers comprising each set,
and subtract the less sum from the greater.
(63, 64)
FACTORING. 21
Piob 28. Multiply together the factors comprising each set,
and add the several products.
Prob. 29. Multiply together the factors comprising each set,
and then add the products and given numbers.
Prob. 30. Multiply together the factors comprising each set,
and subtract the less product from the greater.
Prob. 31. Add the product of the given set or sets of fac-
tors and the given number or numbers.
Prob. 32. Subtract the sum of the products of the set or
sets of factors which form the less number from
the sum of the products of the set or sets of fac-
tors which form the greater number.
PROPERTIES OF NUMBERS.
FACTORING.
(142, page 71.)
Ex. 1. Ans. 2, 5, 5, 43. Ex. 2. Ans. 3, 5, 163.
Ex. 3. Ans. 2, 2, 3, 3, 5, 5, 7.
Ex. 4. Ans. 2, 2, 2, 2, 2, 2, 2, 2 ,2, 2, 3, 7.
Ex, 5. Ans. 2, 7, 13, 13. Ex. 6. 2, 2, 2, 5, 5, 5.
Ex. 7. Ans. 5, 5, 5, 5, 5, 5, 5, 5.
Ex. 8. Ans. 3, 3, 3, 7, 11, 13, 37.
(144, page 74.)
Ex. 2. Am. 2, 3, 7, 17, 17, 29. Ex. 3. Ans. 13, 17, 31
Ex, 4. Ans. 17, 19, 29. Ex. 5. Ans. 2, 11, 19, 487.
Ex. 6. Ans. 7, 83, 103. Ex. 7. Ans. 97, 103.
*f Ex. 8. Ans. 3, 5, 59, 139.
Ex. 9. Ans. 3, 5, 7, 47, 181.
(64-74)
22 PROPERTIES OF NUMBERS.
Ex. 10. Am. 2, 2, 2, 2, 2, 2, 41, 149.
Ex. 11. Ans. 7, 11, 37, 79.
Ex. 12. Ans. 2, 5, 13, 17, 37.
Ex. 13. Ans. 13, 17, 29.
Ex. 14. Ans. 2, 2, 2, 3, 17, 19, 23.
Ex. 15. Ans. 2, 3, 5, 7, 19, 179.
(145, page 76.)
Ex. 1. 120=1X2X2X2X3X5
1, 2, 4, 8 Combinations of 1 and 2.
3, 6, 12, 24 1, 2, and 3.
5, 10, 20, 40 ) "123 and 5
15, 30< 60, 120 J 1,4*,M
Ans. 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Ex, 2. 84=1X2X2X3X7
1, 2, 4 Combinations of 1 and 2.
3, 6, 12 1,2, and 3.
7, 14, 28 } u u - o o fln j 7
21, 42, 84} 1, Z,d,and7.
. 4tw. 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
Ex. 3. 100=1X2X2X5X5
1, 2, 4 Combinations of 1 and 2.
5 > 10 > 20 l 1 2 and 5
25, 50, 100 } 1*3, and
^rcs. 1, 2, 4, 5, 10, 20, 25, 50, 100.
Ex. 4. 420=1X2X2X3X5X7
1, 2, 4 Combinations of 1 and 2.
3, 6, 12 1,2, and 3.
if; $ 6?} " "1,2, 3, and 5.
7, 14, 28)
21 > 42 > 84 I 1 2 3 5 and 7
35, 70, 140 f *' ^ 6 > *> and 7 '
105, 210, 420 J
. 1 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28,
" {30, 35, 42, 60, 70, 84, 105, 140, 210, 420.
(74-76)
GREATEST COMMON DIVISOR. 23
Ex. 5. 1050=1X2X5X5X3X7
1, 5, 25 Combinations of 1 and 5.
2, 10, 50 " 1, 5, and 2.
8 > 16 > 75 1 "152 and 3
6, 30, 150 j I, 0, ^, ai
7, 35, 175 1
14, 70, 350 I K
21* 105* 525 i ' > ; ; a
42^ 210 ? , 1050 J
( 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42,
I 50, 70, 75, 105, 150, 175, 210, 350, .525, 1050.
GREATEST COMMON DIVISOR.
(149, page 77.)
Ex. 2. 2X3=6, An*. Ex. 5. 6x7=42, Am
Ex.8. 3X3X7=63, ^ras. Ex.9. 91, Ans.
Ex. 11. 4X3X7=84, Ans.
(150, page 81.)
Ex. 4. Ans. 11. Ex. 5. ^is. 1.
Ex. 7. -4ns. 17. Ex. 8. Ans. 337
Ex. 10. In the operation under this rule, the quotient figure
may always be so taken that the product shall be either
greater, or less, than the dividend; in either case, the new
divisor will be the difference between the dividend and
product. It will always be found advantageous to use that
quotient figure which will give the least number for a new
divisor. In the first operation below, the second quotient
figure is 1, and the next divisor is 413690 ; in the second
operation, the second quotient figure is 2, which gives 178593"
for the next divisor, and abbreviates the subsequent work
(76-81)
PROPERTIES OF NUMBERS.
FIRST OPERATION.
L005973
4
4616175
4023892
592283
1
592283
413690
357186
1
2
3
413690
178593
169512
56504
54486
6
9081
2018
4
8072
2018
2
1009
SECOND OPERATION.
4616175
4023892
1005973
1184566
4
2
178593
8
169512
3
9081
6
8072
4
Am. 1009
2
592283
535779
56504
54486
2018
2018
Ex. 13. An*. 47.
Ex. 15. In order that the bins may be equal, the number
of bushels contained in one bin must be a common divisor of
the two quantities. And in order that the number of bins
may be the least possible, each must contain the greatest com-
mon divisor of the two quantities. Ans. 91 bushels.
Ex. 16. The pannels, to be of uniform length, must be a
common measure or divisor of the three sides ; and to be of
the greatest possible length, they must be the greatest com-
mon divisor of the three sides. Ans. 11 feet.
Ex. 17. The price to be paid by each is the greatest com-
mon divisor of the three sums, $620, $1116, aud 81488,
which is $124. Hence, B can purchase $620n-$124=5 ;
C can purchase $1116 H- $124 = 9; and D can purchase
$1488-:-$124=12.
Ex. 18. The greatest common divisor of 14599 feet and
10361 feet is 13 feet, the length of 1 joint in the fence,
(14599+10361)X2=49920 feet, the entire length of the
(81)
LEAST COMMON MULTIPLE.
25
fence. 49920 H- 13 = 3840, the number of joints in the
fence; and 3840x7=26880, the number of rails, Ans.
LEAST COMMON MULTIPLE.
(155, page 83.)
Ex. 1. 2X2X3X11X7X5=4620, Ans.
Ex.2. 7X3X3X2X2X11X5=13860, Ans.
Ex. 3. 5X3X2X2X2=120, Ans.
Ex. 4. 7X5X3X2X2X2X2=1680, Ans.
Ex. 5. 7X5X5X3=525, Ans.
Ex. 6. 19X3X7X2=798, Ans.
Ex. 7. 2X2X2X2X2X2X3X3X5=2880, Ans.
Ex. 1. 5, 3, 2
2, 2, 3, 7, 5
(156, page 85.)
15.. 18.. 21.. 24.. 35.. 36.. 42.. 50.. 60
3.. 7.. 4.. 7.. 6.. 7.. 5.. 2
5X3X2X2X2X3X7X5=12600, Ans.
, Ex. 2. 2, 2, 3
2,3,5
6. .8. .10. .15. .18. .20. .24
2.. 5.. 5.. 3.. 5.. 2
Ex. 3, 3, 5, 2
3, 5, 7, 2
2X2X3X2X3X5=360,
9. .15. .25. .35. .45. .100
3.. 5.. 7.. 3.. 10
3X5X2X3X5X7X2=6300, Ans.
Ex. 4. 3, 3, 2
2, 2, 5, 3
18.. 27.. 36.. 40
3.. 2. .20
3X3X2X2X2X5X3=1080,
(83-85)
26 P.
Ex. 5. 3, 3
2,13
ROPEKTIES
12. .26. .52
2. .13. .26
2X2X3X13=156, Am.
Ex. 6. 2, 2, 17
8,9
32.. 34.. 36
8.. 9
2X2X17X8X9=4896, Ans.
NOTE. When numbers are prime to each other, as 8 and 9 in
the above operation, their product will be their least common
multiple.
Ex. 7. 2, 2, 3
2,3,3
8. .12. .18. .24. .27. .36
2..
3.. 2.. 9.. 3
2X2X2X3X3X3=216, Ans.
Ex.
2,11
2,3,5
22. .33. .44. .55. .66
3.. 2.. 5.. 3
2X11X2X3X^=660, Ans.
NOTE. The first three numbers in Ex. 7 and the first two ia
Ex. 8, above, are factors of remaining numbers in the exam-
ples respectively. They might, therefore, have been omitted in
the operations.
Ex.9.
2, 2, 3
64.. 84.
.96.. 216
2, 2, 2, 2
16.. 7.
. 8.. 18
3,3,7
7
9
2X2X2X2X2X2X3X3X3X7=12098, Ans.
Ex. 10. The number of rods that will furnish whole days'
work to each one, must be some common multiple of 14, 25,
8 and 20 ; and the least number of rods that will furnish
(85)
CANCELLATION. 27
whole days' work to each of the men, must therefore be the
least common multiple of 14, 25, 8 and 20.
Ans. 1400 rods.
Ex. 11. The least common multiple of the prices, $4, $21,
$49, and $72, which is $3528, Ans.
Ex. 12. When all the men work together, they will dig
4 -f- 8 +6=1 8 rods per day. The ditch must therefore be
the least common multiple of 4 rods, 8 rods, 6 rods, and 18
rods, which is 72 rods, Ans.
Ex. 13. The least common multiple of 11 feet and 15
feet is 15 X 11=165 feet, the distance the carriage must
move to bring the rivets up together. Hence, 165x575=
94875 feet, the entire distance traveled ; and 94875 feet-4-
5280=17 miles 5115 feet, Ans.
CANCELLATION.
(159, page 87.)
K f>
Ex. 2.
-=80, Ans.
Jfrv/ i oiv 9f'V # V 3f
Af /\/Lfc> /\/> /\ y> XN/'
O
Ex. 3
-=32, ^Irw.
61
Ex.4,
61, Ans.
(86-88)
28 PROPERTIES OF NUMBERS.
71 11
<^X190X^
-=14839, .
Ex.5.
Ex.6.
Ex.7.
Ex.9.
=403, Ans.
9,
16
13
13,
Ex. 10. 84+56=140 cents.
240
32 cents,
240, Ans.
~\ cost of 2 yards of the
Ex. 11. 75X2+90=240 cents, [-first kind, and 1 yard
j of the second.
11 yards of the second kind ;
11X2=22 " " first "
Ex.12.
=60 cents, Am.
(88)
NOTATION AND NUMERATION. 29
FRACTIONS.
NOTATION AND NUMERATION.
(169, page 90.)
Ex. 3. Ans. |f. Ex. 4. Ans. 7 ^.
Ex. 5. Ans. |f. Ex. 6. Ans. f|f
Ex. 7. Ans. 25 H <y><>. Ex. 8. Ans. T1 /y> 52 .
Ex. 9. Ans. T WzjoW
Ex. 10. Four ninths; seven twelfths; seventeen thirty
eighths; forty-five one hundredths; seventy -two three hundred
seventy-fifths; forty-eight one thousand ninths; eighty-four
seven thousand eight hundred sixty-thirds ; four hundred fifty-
six five hundred thirty-sevenths.
Ex.11. Twenty fourths; eighty-seven thirtieths ; ninety-
five one hundredths; forty-eight twelfths; seventy-five four
hundred thirty-sevenths ; one hundred seventy-five halves ;
four hundred thirty-six fiftieths ; seven hundred sixty-six
four thousand eight hundred seventy-ninths.
Ex. 12. Four hundred sixty-seven nine hundred thirty-
sixths; five hundred thirty-six two hundred forty-eighths ; ten
thousand seventy-fifths; seventy-five ten thousandths; five
thousand seven three thousand sevenths.
Ex. 13. One hundred fifty Jive hundred thirty-sevenths ;
four hundred thirty-six nine hundred seventy seconds ; thir-
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