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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-


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