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GREENSBOKO, N. C: 
PUBLISHEJP BY STERLING, CAMPBELL & ALBRIGHT. 

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PERKINS LIBRARY 

Uuke University 



Kare Dooks 



TRINITY COLLEGE LIBRARY 

DURHAM, N. C. 
1903 



Gift of Dr. and Mrs. Drcd Peacock 



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O XT H O "W M 



SCHOOL ARITHMETIC. 



B Y 



S. LA-jSTDER, a., m. 



)0 *•* 

kO GREENSBORO, N. C: 

18 6 3 



PUBLISHED BY STERLING, CAMPBELL & ALBRIGHT. 

Richmond, Va., W. Harorave White. 



Entered according to Act of Congress, 
in the year I860, 
By S. lander, 
In the Clerk's Office of tlie District Court of the Con- 
federate States, for the District of Cape Fear, 
North-Carolina. 



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P 11 E F A C E . 



In presenting to the public perhaps the first Arithmetic 
whose authorship and publication belong exclusively to the 
Confederate States, I call attention to the following as its 
leading characteristics. 

J . The pupil is furnished with a model for each class of 
operations, by which he may know precisely what kind of 
explanation is required of him as he recites. 

2. The distinction between abstract and concrete num- 
bers is carefully kept up throughout the whole book. 

3. The Tables of Relations of Concrete Numbers are un- 
usually full and convenient. 

4. The problems are designed to call into exercise the 
pupil's practical common sense, as well as to assist him in 
acquiring a correct knowledge of Arithmetic. 

5. The results of about two-thirds of the problems are 
given : those of the* remainder are omitted, and a few of 
those given contain intentional errors, to test the pupil's 
self-reliance. 

6. Progressions, and Mensuration, and the ordinary 
methods of extracting roots, are excluded entirely, because 
they lie beyond the province of Arithmetic. 

7. No space is wasted by the insertion of questions on 
the text. A teacher who can not instruct without the help 
of questions, will succeed but poorly even with them. 



IV PREFACE. 



8. The discouraging contradictions -which are so nume- 
rous in ''our best Arithmetics/' have beew sedulously 
avoided. 

For the neat appearance of the book I am much indebt- 
ed to my publishers, who have spared neither pains nor ex- 
pense to bring it out creditably ; and I am under especial 
obligations to my friend Prof. Theo. F. Wolle, of Edge- 
worth Female Seminary, without whose constant vigilance 
in revising the sheet.'?; no approximation to its present ac- 
curacy could have been attained. 

I invite my fellow-teachers to try the book by the unly 
<ure test, the test of the school-room ; and I will thank- 
fully receive any propositions of improvement which their 
examinations may suggest. 

Our Own Primary Arithmetic will follow this as soon a.< 

possible. ' 

S. LANDER. 
LiNCOLNTON, N. C, August Ij 1863, 



6 N T E N T S 



Introductiou, v i 

Arabic Notation, , -^ 

Roman Notation , 1 ♦^ 

Operations, • 

Addition of Abstract Integers, 17 

Subtraction of Abstract Integers, 2r, 

Multiplication of Abstract Integers. 31 

Division of Abstract Integers, 41 

Contraction in Addition, 54 

Contraction in Subtraction, 5:") 

Contractions in Multiplication, 5() 

Contractions in Division,. Gf) 

Oeneral Principles of Division, 72 , 

Measures and Multiples, 82 

Prime Factors, ■• ■ • • • 99 

Involution, 100 

Evolution, lOi 

Grreatest Common Measurv; 103 

Least Common Multiple, •-• 10(5 

<'ommon Fractions, lOD 

Reduction of Common Fractions, 12(' 

Addition of Common Fractions, 131 

Subtraction of Common Fractions, 134 

Multiplication of Common Fraction.^ 139 

Division of Common Fractions — 146 

Cancellation, 15r' 

Decimal Fractions, — Notation,... 154 

Addition of Decimal Fractions,.... 159 

Subtraction of Decimal Fractions, 161 

Multiplication of Decimal Fractions, 162 

Division of Decimal Fractions, 164 

Contraction in Multiplication, 167 

Contraction in Division 168 



Tl CONTENTS. 



Relations of Common and Decimal Fractions, §169 

Concrete Numbers, — Relations, 177 

Addition of Concrete Numbers, 178 

Subtraction of Concrete Num.bers, 180 

Multiplication of Concrete Numbers,, c 182 

Division of Concrete Numbers, 184 

Reductipn of Concrete Numbers, 186 

Compound Numbers, , . , 199 

Addition of Compound Numbers, 200 

Subtraction of Compound Numbers, 201 

Multiplication of Compound Numbers, 202 

Division of Compound Numbers, ., 208 

Aliquot Parts ; or, Practice, 205 

Contraction in Multiplication, 208 

Contraction in Division, 209 

Ratio, 210 

Simple Proportion, 2 1 G 

Compound Proportion, 224 

Partitive Proportion ; or. Fellowship, 229 

Conjoined Proportion ; or. The Chain Rule, 232 

Percentage, i 233 

Jnterest,. ..,......; 242 

Partial Payments, c 250 

Compound Interest,. 251 

Discount,. 252 

Bank Discount,... , 255 

Average, ^61 

Alligation Medial, 264 

Alligation Alternate,.*. _ 266 

liquation of Payments,.. 268 



ARITHMETIC. 



INTRODUCTION. 



§ 1 . Arithmetic is the science of numbers. 

^ 2. A unit is any single tiling f as, one, one dollar. 

§ 8. A number is a collection of units ; as, three, two 
<lollars, four men. 

§ 4. An abstract number is one whose unit is not speci- 
tied ; as, two, forty, seventy-one, eight. 

§ 5. A concrete number is one whose unit is specified : 
us, ten dollars, forty men, seventy-one bales, eight books. 

§ 6. Two or more numbers are similar when they have 
the same unit ; as, two, five, and seventy ; three men and 
six men. 

§ 7. Two or more numbers are dissimilar when they have 
different units ; as, two, five dollarst, seventy men, three 
books. 

iVb^e.— All abstract numbers are similar. 

§ 8. A compound number is a concrete number expressed 
in two or more denominations;, as, three dollars, fifty cents; 
ten hogsheads, forty gallons, three gills ; ten miles, seven 
furlongs, seventeen rods. 



§9 ARABIC NOTATION, 



ARABIC NOTATION. 



^ 9. Notation is the luethod of expressing numbers \>y 
figures. The Ar'abic system, which is the one in common 
use, is called also the decimal system, partly because it 
employs ten figures. These figures are : naught or zero, 
1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 
9 nine. 

The figure is used to fill vacant places^ and is omitted 
in reading. » 

§ 10. A ten is a collueiioAi often units, and is called :i 
unit of the second order. 

A hundred is ten tens,»or one hundred units, and is called 
a unit of the third order. 

A thousand is ten hundreds, or one thousand units ; and 
is called a unit of ihQfourtli order. 

So, ten units of any order jiiake one of the next higher. 

§ 11. A single figure denotes units; as, 3, three units. 
5, five units, 7, seven units. 

§ 12. When two figures are written together, the one on 
the right denotes units, and the rthertens; as, 23, two tcn.^- 
and three units, that is, twenty-three units ; 34, three t^wi^ 
and four units, that is, thirty-four units, or, simply, thirty- 
four. 

Read 27, 66, 73, 37, 84, 48, 99, ^6, 43, 55, 79, 80, KS. 

§ 13. When three figures are written together, the one 
on the right denotes units, the next tens, and the other 
hundreds ; as, 123, one hundred, two tens, and three units, 
or, one hundred and tw^enty- three ; 321, three hundred and 
twenty -one ; 132, one hundred and fliirty-two ; 402, four 
hundred and two. 



ARABIC NOTATION. §15 

Read 647, 864, 420, 301, 753, 587. 357, 735, 608, 740, 
047, 306, 700, 609, 069, 009, 290, 391, 001. 

§ 14. When more than three figures are written together, 
they arc separated into periods of three figures each, begin- 
ning at the right ; find^ in each..period, the three figures de- 
note respectively units, tens, and hundreds, of that period. 

§ 15. The names of the periods and their order from 
right to left arc given in the following 



3 a o o 



■X3 


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H 


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123,004,500,060,000,000,700,089,000,897,060,000,543,210. 

Thfe above number is read, — one hundred and twenty- 
three duodecillions, four undeeillions, five hundred decil- 
iious, sixty nonillions, seven hundred scxti-llions, eighty- 
nine quintillions, eight hundred and ninetj -seven trillions, 
.-ixty billions, five hundred and forty-three thousand, two 
hundred and ten. 

BuLE FOii READING i»UMBERS.—>S'<rjDamfe the jicjiivei^ into 
periods of three figures each, beginning at the right; then, 
heginning at the left, read each period as if it stood alone, 
•md pronounce the 7i(ime of the period after reading it, 

Kead the following : 12^ 21, 37, 86; 793, 842, 209, 319; 
1^346, 7907, 5432, 8642, 4C04, 1861, 1775 ; 24608, 13579, 
10724, 40047, 78009; 475213, 570903, 400101, 300003; 
1230456,2040608,7901035,8000005 ; 70083790,245000542, 
5430102046, 146070080009, 9000800706040, 600005000 r 





§15 ABSTRACT NUMBERS. 

Write tbe following numbers in figures : 

35. Seventy-four. 

36. Four hundred and forty-eight. 

37. Five thousand, three hundred and ninety-seven. 

38. Sixty thousand, and seventeen. ^ 

39.. Seven hundred and forty thousand, eight hundred 
and forty-one. 

40. Eight millions, sev-enty thousand, and seventy-nine. 

41. Ninety-four millions, sixteen thousand, four hundred 
and fourteen. 

42. Thre« hundred millions, and three. 

43. Two billions, two millions, two thousand, and two. 

44. Ten billions, ten millions, ten thousand, and ten. 

45. Nine hundred and twenty-five billions, eight thou- 
sand, and sixteen. 

46. Eight trillions, seven billions, sixty millions, five 
thousand, and four. 

47. Seventy trillions, eighty -nine millions, and twenty- 
one. 

48. Six hundred and forty-two trillions, three hundred 
thousand. 

49. Fifty-three quadrillions, eleven billions, and seventy- 
three. 

50. Four hundred and four quintillions, two hundred 
and two millions. 

51. Thirty sextillions, forty quintillions, fifty trillions, 
six hundred and two. 

52. Two octillions, four quadrillions, six hundred and 
eight thousand. 

53. Ten decillions, twelve nonillions, fourteen millions, 

and ninety-nine. 

54. Nine ijonillions, ten* millions, and twenty-seven. 

10 



ROMAN NOTATION. §16 



ROMAN NOTATION. 



yj) IG. The Koman Notation employs the following seven 
letters : I one, V five, X ten, L fifty, C one hundred, D five 
hundred, and M one thousand. 

All integral numbers may be denoted by combining these 
letters accordiug to the following rules : 

1. Any letter doubled denotes twice its simple value ; 
tripled denotes three times, and so on. Thus, 11=2, 
XX=20, CCC = 300. 



'J 



2. If a letter of less value is placed after one of greater 
value, tlie less is to be added to the greater. Thus, ¥1=6, 
XV=15, CGL=250. 

3. If a letter of less value is placed he fore one of greater 
value, the less is to be subtracted from the greater. Thus, 
iy==4, XC=90, CD = 400. 

4. If a letter of less value is placed hciwetni^o of greater 
value, the less is to be subtracted from the sum of the other 
two. Thus, XIX= 19, Xiy = U, XCIX=99. 

5. A dash placed over a letter multiplies its value by 
1000. Thus, "1=50000/0 = 100000. 

The above rules are sufficiently exemplified in the fal- 
lowing 







T.4lI5LiJEC. 




I--=l 


XI-=11 


XXI=21 


C=100 


II----2 


Xll-^] ■-. 


XXII=22 


CC=200 


111=3 


xiri=i3 


:vXIII=2.3 


CD=400 


IV=^1 


XlVc^li 


XXX=30 


D=500 


V=5 


XV=15 


XL=;:40 


DC=600 


VI=6 


XVI^IG 


L=50 


M=1000 ^ 


VIIr=7 


XVII=17 


LX=60 


MC=1100 ' 


vni=8 


XVIII=18 


LXX=70 


MM=2000 


IX=9 


XIX=19 


LXXX=80 


11=1000000 


X=^10 


XX=::20 


XC=90 

U 


MDCCCLXIII:^1808 



§1' 



ABSTRACT NUMBERS. 



OPEKATIONS. 



There are four operations in Arithmetic ; Addition, Sub- 
traction, Multiplication, and Division. We will explain 
these operations in succession, first with reference to ab- 
stract numbers, and afterwards with reference to concrete 
numbers. 



ADDITION OF ABSTRACT NUMBERS. 



§ 17. Addition is the operation of finding one number 
equal to several other numbers put together. 

§ 18. The result of addition is called the sum of the num- 
bers added. Thus, 10 is the sum of 6 and 4. 

§ 19. The sign of addition^ -,'-, is read jo/t^s. When pla- 
ced before a number, it denotes that it is to be added to any 
other additive number with which it is connected. Thus, 
()-f4, 6 plus 4, denotes four added to six. 

§ 20. l^hQ sign of equality^ =, is read i^ equal to. When, 
placed between two expressions it denotes that they are equal 
to each other. Thus, 6+4=10 Also, 7+4 + 3 = 8 + 6. 



1 aud are 2 


3 and are 3 


4 and are 4 


5 aud are ;' 


'1 and 1 are 3 


3 and 1 are 4 


4 and 1 are 5 


5 and 1 are G 


li and 2 are 4 


3 and 2 are 5 


4 and 2 are 6 


5 and 2 are 7 


'J and 3 are 5 


3 and 3 are 6 


4 and 3 are 7 


5 and 3 are 8 


2 and 4- are 


3 and 4 are 7 


4 and 4 are 8 


5 and 4 are 9 


li and 5 are 7 


3 and o f.re 8 


4 and 5 are 9 


5 and 5 are 10 


il and 6 are 8 


3 and C are 9 


J 4 and 6 are 10 


5 and 6 are 11 


2 and 7 are 9 


3 and 7 are 10 


1 4 and 7 are 11 


5 and 7 are 12 


2 and 8 are 10 


3 and 8 are 11 


4 and 8 are 12 


5 and 8 are IS 


2 and are 11 


3 and are 12 


4 and 9 are 13 


5 and 9 are 14 



12 



ADDITION OF INTEGERS 



§22 



G and arc G 
G and 1 nre 7 
6 and 2 nre 8 
G and 3 are 9 
G and 4 are 10 
G and 5 are 11 
G and G are 12 
G and 7 are 13 
G and 8 are 14 
<i and 9 are 15 



7 and 
7 and 
7 and 
7 and 
7 and 
7 and 
7 and 
7 and 
7 and 
7 and 



are 7 

1 are 8 

2 are 9 

3 arc 10 

4 are 11 

5 are 12 
G are 13 

7 are 14 

8 are 15 

9 arc 16 



8 and 
8 and 
8 and 
8 and 
8 and 
8 and 
8 and 
8 and 
8 and 
Sand 



are ^ 
T are 9 

2 are 10 

3 arc 1 1 

4 are 12 

5 are 13 

6 are 14 

7 are 15 

8 are IG 

9 are 17 



9 andO 
9 a4id 1 
9 and 2 
9 and 3 
9 and 4 
9 and 5 
9 and 6 
9 and 7 
9 and 8 
9 and 9 



are U 
are 10 
are 1 1 
are 12 
are 13 
are 14 
are 15 
are IG 
are 17 
are 18 



Note. — Let the above table bo thoroughly memorized before the 
pupil advances farther. 

Ex. 1. Add togetkcr 102741, 42102, and 3050. 

102741 ^ 21. Model.— 2 and 1 are 3; 5 and 4 arc 

42102 9 ^ I ^j^(2 7 arc 8 ; 3 and 2 are 5, and 2 are 
3050 7 4 i_ The sum is 147893. 



147893 



Note. — Let the teacher see to it that the pupil recites precisely 
according to the model ttoth here and -wherever a model is given. 

Explanation. — First, the numbers are arranged with 
units of the same order in the same column. Then, begin- 
ning at the right, the numbers in each column are added to- 
gether, and the sum is placed underneath in the same column . 

2. Addtogetlier23456, 10203, and 56030. Sum, 89689. 

3. Find the sum of 120242, 334124, and 224612. 

Note. — Let the pupil first say, -'Add the numbers together/" 
and then proceed as in the model. 

4. What is the sum of 2400, 1505, and 3074 ? Ans. 6979. 

5. Add 270, 102, 314, and 301 together. Sum, 987. 

6. Add together 94085, 16275, and 3367. 

94085 § 22. Model.— 7 and 5 are 12, and 5 

1627S are 17, set down 7 ; 1 and 6 are 7, and 7 

3367 are 14, and 8 are 22, set down 2 ; 2 and 3 

Sum, 113727 are 5, and 2 are 7 ; 3 and 6 are 9, and 4 

are 13, set down 3 ; 1 and 1 are 2, and 9 

are 11, set down 11. The sum is 113727. 

13 



§22 ABSTRACT NITMBERS. 



, Explanation. — After arranging tlie numbers as in § 21, 
the sum of tlie column of units is found to be 17 units, that 
is, 1 ten and 7 units ; hence, the 7 is placed under the col- 
umn of units, and the 1 is afterwards added in with the 
column of tens. The sum of the column of tens are 22 tens, 
that is, 2 hundreds and 2 tens ; hence, the right hand 2 is 
placed under the column of tens, and the other 2 is added 
in with the column of hundreds. The sum of the column 
of hundreds is 7 hundreds, and the 7 is placed underneath 
in that column. The sum of the column of thousands is 
13 thousands, that is, 1 ten-thousand and 3 thousands ; 
hence, the 3 is placed in the column of thousands, and the 
1 is added in with the column of ten-thousands. The sum 
of the column of ten* thousands is 11 ten-thousands, that is, 
1 hundred-thousand and 1 ten-thousand ; hence, the right 
hand 1 is placed in the column of ten-thousands, and the 
other 1 in the place of hundred-thousands. 

Rule. — Arrange the numbers icith units of the s<^nie. or- 
der in the same column. 

Beginning at the right, find the sum <?/ each column ; if 
this sum is expressed hy one figure, set it down under the col- 
umn; but if it is expressed by more than one figure, set th-e 
right hand figure under this column, and add the remaining 
figure or figures in with the next column. 

Set down the whole sum of the last column. 

Pnoor. — 1, Add as before, but begin at the top of each 
column. 

Or, 2. Find the sum of all the numbers but one, and to 
this sum add the number excepted. 

Ex. 7. Add together 234, 15G, 987, and 358. Sum, 1735. 
8. Add together 1020, 304, 66, and 9. Sum, 1389, 

14 



ADDITION' OF INTEGERS. §22 



9. Add together 2739, 9647, 271, 17, and 2U50. 

10. Add together 169078, 270189, and 928608. 

Sum, 1367875. 

11. Add together 27090, 2709, 2^905, 27, 2709050, and 
2r0. Sum, 3010051. 

12. Find tha sum of 369764, 275863, 10794, 273, 102469, 
and 1861. 

13. Find the sum of 173594, 240680, 10305, 678, and 
976531. Sum, 1401788. 

14. Find the sum of 97347S25, 89734782, 28973478, 
828973478, and 98289734. Sum, 1143319297. 

15. Find the sum of 1928374560, 192837456, 1928, 
19283745, 1928374, 192837, and 19283. 

16. 907050301 4-80604020-1- 123123123-=what ? 

Ans. 1110777444. 

17. 146-f 1375-M3795-f 246820-f 24682=what ? 

Ans. 2S681b. 

18. 2620-h6202-f7593-4-3694-f 1735=what? 

19. "What is the sum of 3426, 9120634, 52714, 9987, 
1137, and 97579? 

20. What is the sum of 26322, 50555, 37684, 898955, 
and 9024 ? Ans. 1022540. 

21. WhatisiAie sum of 41084, 293347, 9139919, and 
46552? Ans. 9520902. 

22. What is the sum of 245301, 586642, 51407, 1752, 
71283, and 42061 ? 

23. What is the sum of 10, 105, 1057, 10572, 105723, 
1057234, 10572349, 105723496, and 1057234968? 

Ans. 1174705532. 

24. What is the sum of 135792468,246813579,159483726, 
372684951, 123456789, 896745321, 896453217, arid 
400500746 ? Ans. 3231930797. 

15 



§23 



ABSTaACT NUMBERS. 



SUBTRACTION OP ABSTRACT NUMBERS. 



§ 23. Subtraction is the operation of finding the differ- 
ence between two iiurabers, by taking the less from the 
greater. 

§ 24. The number to he aubtracted is called the subtra- 
hend. 

§ 25. The number to he diminislied is called the imnn- 
tmd. 

§ 2%. The residt of subtraction is called the remainder 
or the difference. 

§ 27. The sign of subtraction, — , is read minus. When 
placed before a number, it denotes that it is to be subtracted 
from the number with which it is connected. Thus, 
(>— 4, 6 minus 4, denotes 4 taken from 6. Also, 7—3=4. 

§ 28. The remainder is not changed by increasing the 
minuend and the subtrahend equally. Thus, 
Min. 27 27 + 15=42 274-240=267 27 + 306;=333 
Sub. 16 16+15=31 16+240=256 16 + 306=322 

Kern. 11 n nn ~Y\ 





from 1 


leaves 


2 from 2 


leaves 




from 2 


leaves 1 


2 from S 


leaves 1 




from 3 


leaves 2 


2 from 4 


leaves 2 




from 4 


leaves 3 


2 from 5 


leaves 3 




from § 


leaves 4 


2 from 6 


leaves 4 




from 6 


leaves 5 


2 from 7 


leaves 5 




from 7 


leaves 6 


2 from 8 


leaves G 




from 8 


leaves 7 


2 from 9 


leaves 7 




from 9 


leaves 8 


2 from 10 leaves 8 




from 10 leaves 9 


2 from 11 leaves 9 



from 
from 
from 
from 
3 from 
from 
from 
from 
from 
from 



leaves 
leaves 1 
leaves 2 
leaves 3 

7 leaves 4 

8 leaves 5 

9 leaves 6 

10 leaves 7 

11 leaves 8 

12 leaves 9 



16 



SUBTRACTION OF INTEGERS. 



§29 



4 from 


4 lenves 


i 5 from 6 leaves 


i ^ 


from 


6 leaves 


4 from 


5 leaves 1 


j 5 from 6 leaves 1 


i 6 


from 


7 leavea 1 


I from 


leaves 2 


! 5 from 7 leaves 2 


1 ^ 


from 


8 leaves 2 


i from 


7 leaves 3 


5 from 8 leaves 3 


i 6 


from 


9 leaves 8 


4 from 


8 leaves 4 


5 from leaves 4 


' 6 


from 


10 leaves 4 


It from 


leaves 5 


5 froru 10 leaves 5 


I 6 


from 


11 leaves 6 


1 from 


10 leaves G 


5 from 11 leaves 6 


; 6 


from 


12 leavei G 


4 from 


11 leaves 7 


i» from 12 leaves 7 


: 6 


from 


13 leaves 7 


I from 


12 leaves 8 


from 13 leaves 8 


: G 


from 


14 leaves 8 


'{ from 


18 leaves 


5 from 14 leaves 9 


G 


from 


15 leaves 9 


7 from 


7 leavea 


8 from 8 leaves ' 


9 


from 


9 leaves 


7 from 


8 leaves 1 


8 from 9 leaves 1 \ 


9 


from 


10 leaves 1 


7 from 


1> leaves 2 


8 from 10 leaves 2 j 


9 


from 


1 1 leaves 2 


7 from 


10 leave." 3 


8 from 11 leavob 3 ' 


9 


from 


12 leaves 3 


7 from 


1 1 leaves 4 


8 from 12 leaves 4 


9 


from 


13 leaves 4 


7 from 


12 leaves o 


8 from 13 leave? 5 i 


9 


from 


14 leaves 5 


7 from 


13 leaves G 


8 from 14 leaves G ; 


9 


from 


15 leaves G 


7 from 


14 leaves 7 


8 from 15 leaves 7 1 


9 


from 


IG leaves 7 


7 from 


15 loaves 8 


8 from 16 leaves 8 J 


9 


from 


17 leaves 8 


7 from 


IG leaves 9 


8 from 17 leaves 9 ! 


9 


from 


18 leaves 



Mill. 

8ub. 

Rem. 



I. From 976348 subtract 35127. 

§ 29. Model.— 7 from 8 leaves 1 ; 2 



976348 
35127 

941221 



from 4 leaves 2 ; 1 from 3 leaves 2 ; 5 
from leaves 1 ; 3 from 7 leaves 4 ; 
from 9 leaves 9. The remainder is 
941221. 

Explanation— The subtrahend is placed under the min- 
uend, with units of the same order in the same column. 
Then, beginning at the right, each figure of the subtrahend 
is taken from the corresponding figmre of the minuend, and 
the remainder is set underneath in the same column. 

2. From 127936 subtract 14312. Rem. 113624. 



3. From 96898 subtract 13456. 

4. Subtract 864231 from 9557654. 

5. Subtract 1024370 from 12357799. 



B 



17 



Rem. 123423. 
Rem. ] 1333429. 



§30 ABSTRACT NUMBERS. 



6. Subtract 327739 from 573G47. 

Min. 573647 §30. Model.— 9 from 17 leaves 8 ; 4 

Sub. 327739 from 4 leaves ; 7 from 16 leaves 9 ; 8 

Rem. 245908 from 13 leaves 5; 3 from 7 loaves 4: 3 

from 5 leaves 2. The remainder is 245908. 

Explanation. — After arranging the numbers as in § 29, 
it is required to take 9 units from 7 units : this can not be 
done ; hence, 1 ten, that is, 10 units, is added to the minu- 
end, giving 17 units, from which 9 units taken leaves 8 units. 
Then, because the minuend is increased 10 units or 1 ten, 
the subtrahend must be increased the same amount (§ 28). 
This gives 4 tens to be taken from the 4 tons of the minu- 
end, leaving tens. Again, 7 hundreds can not be taken 
from 6 hundreds ; hence, 1 thousand, that is, 10 hundreds, 
is added to the minuend, giving 16 hundred:?, from which 
7' hundreds taken leaves 9 hundreds. Then, because the 
minuend is increased 10 hundreds or 1 thousand, the sub- 
trahend must be increased the same amount. 

The same kind of reasoning will explain the rest of the 
operation. 

Rule, — Place the siihtraJiend under (lie niinucnd, wilh 
units of the same order in the same column. 

Beginning at the rights take each figure of the subtrahend 
from the corresponding figure of the minuend. 

If any figure of the minuend is less than the corresponding 
figure of thc^ subtrahend, add 10 to this minuend figure ^ and 
add 1 to the subtrahend figure in tlie next column. 

Proof. — 1. Add the remainder to the subtrahend ; the 
sum will be equal to the minuend. 

Or, 2. Subtract the remainder from the minuend ; the 

difference will be equal t© the subtrahend. 

' 18 



SUBTEACTION OF INTEGERS. 



'M 



Ex. 7. From S9G take 307. 

8. From 1842 take 961. 

9 From 2719 take 1827. 

10. From 12791 take 9872. 

11. From 24598 take 20689. 

12. From 978637 take 97863. 

13. From 1654278 take 755429. 

14. Take 678902 from 896454. 
1^. Take 1724937 from 1963869, 
16'. Take 23468579 from 60050040. 

17. Take 9879789 from 9900000. 

18. Take 7890845 from lOOOOOOO. 

19. Miiiuend = 1234567, Subtrahend 



Eem. 589. 
Eem. 881. 

Eem. 2919. 
Rem. 3909. 

Rem. 898849. 
Rem. 217552. 

Rem. 36581461. 
Rem. 20211. 



765432. 
Rem. 469135. 

Begin by saying, "Subtract tlie Subtrahend from the 



Rem. 181765. 

Rem. 7777782. 
Ans. 2887879. 

one million, and 
Ans. 999901. 



Note 
Minuend."" ^ 

20. Min.=290178, Sub. = 108405. 

21. Sub.=20499, Min.-~1900623. 

22. Sub. =987631, Min. =8765413. 

23. 12646723-975894^=what? 

24. 2468000— 970053 = what? 

25. What is the difference between 
ninety -nine 1 

Xoic. — Begin, "Subtract the less number from the greater.'" 

26. What is the difference between thirty-seven billions, 
and eleven ? Ans. 36999999989. 

27. What is the difference between nine thousand six 
hundred and thirteen, and five hundred and forty-two ? 

28. What is the difference between eight thousand and 
tw«nty-six, and eight hundred and twenty-six? Ans. 7200. 

29. What is the difference between five thousand four 
hundred and ninety, and seven hundred and sixty-two 1 

19 



|3l ABSTRACT NCMBERf. 



^MULTlPLICATrON OF ABSTKAOT NUMBER.S. 

§ ol. Multiplication is the operation of finding a num- 
Vier ■which shall contain one of two given num})ers as many 
times as there are units in the other. 

Thus, o times 6 are 18 : here 6 is multiplied by 3, be- 
'■auae 18 contains 6, 3 times. 

§ 32. The number io he multiplied is called the muUi' 
f)Hcand. 

§ 33. The inultiplying number is called the juuftiplier, 

§ 84. The result of multiplication is called the product. 

§ 35. Either the multiplicand or the multiplier is called 
a factor of the product, and thej both are called itfi factors. 

In general, one number is a factor of any other number 
which contains it on exact number of timea. 

Thus, 3 is a factor of 18 ; 4 is a factor of 12, or of 20 ; 
5 is a factor of 10, of 15, of 30, or of 45. 

§ 36. The si'^n of multiplication^ X, when placed between 
two numbers, denotes that one of them is to b« multiplied 
by the other. It is read times, when placed after the mul- 
tiplier, and multiplied by, when placed after th« multipli- 
cand. Thus, to denote that 6 ia to be multiplied by 3, we 
may Miy, 3x6, 3 times 6, or 6x3, 6 multiplied by 3. To 
denote the successive multiplication of more than two num- 
bers, periods are used. Thus, 2.3.5 = 30. 2 times 3 times 
f)=«30. 

§ 37. The product of any two abstract factors ia the 
i^ame, no, matter which is used as multiplier. Thus, 3 x6s— 
«X3==18; 4x5-=5x4=20; 10x8-=8xl0=«80. 

20 



MULTIPLICATION OF INTEGERS. 



§o7 



IVIUI-iTlJeLlCATIOlV T^VBLE. 





Oace 





is 





Twice 





are 





3 


times 


are 


It 




Once 


\ 


is 


1 


Twice 


1 


are 


i> 


3 


times 


1 are 


o 




Once 


2 


is 


2 


Twice 


o 

Ml 


are 


4 


3 


times 


2 are 


6 




Once 


3 


is 


3 


Twice 


3 


are 


G 


3 


times 


3 are 


9 




Once 


4 


is 


4 


Twice 


4 


are 


8 


3 


times 


4 are 


12 




' hice 


T) 


is 


5 


Twice 


5 


arc 


10 


3. 


times 


5 are 


15 




Once 


« 


is 


G 


Twice 


G 


are 


12 


'"> 


times 


G are 


18 




Once 


4 


is 


7 


Twice 


7 


are 


14 


8 


times 


7 are 


21 




Once 


8 


IS 


8 


Twice 


8 


are 


IG 


o 


times 


8 are 


24 




Ouce 


9 


is 


9 


' Twice 


9 


are 


18 




time3 


are 


27 




Once 


10 


is 10 


Twice 


10 


are 


20 


8 


times 


10 are 


30 




Once 


11 


is 11 


Twice 


11 


are 


22 




times 


11 are 


33 




Once 


12 

t 


is 1 


o 


Twice 


12 


are 


24 


1 3 


times 


12 are 


36 


4 


times 





are 





5 times 





are 





6 


times 


are 





4 


times 


1 


are 


4 


5 times 


1 


are 


5 


6 


times 


"1 are 


6 


4 


times 


•) 


are 


8 


5 times 


O 


are 


10 


6 


times 


2 are 


12 


4 


timei) 


3 


arQ 


12 


times 


3 


are 


15 


6 


times 


3 are 


18 


4 


times 


4 


are 


IG 


5 times 


4 


are 


20 


6 


times 


4 are 


24 


4 


times 


5 


are 


20 


5 times 


5 


are 


25 


G 


times 


5 are 


3ft 


4 


timet 


G 


jire 


24 


5 times 


6 


are 


30 





times 


G are 


36 


4 


timts 


7 


are 


28 


5 times 


7 


are 


35 


6 


times 


7 are 


42 


4 


times 


8 


are 


32 


5 times 


8 


are 


40 


G 


times 


8 arc 


48 


4 


times 


■ 9 


are 


36 


5 times 





are 


45 


6 


times 


9 are 


64 


4 


times 


10 


are 


40 


5 times 


10 


are 


50 


6 


times 


10 are 


60 


4 


times 


11 


are 


44 


5 times 


11 


are 


55 


6 


times 


11 are 


GG 


4 


times 


12 


are 


48 


5 times 


12 


are 


60 


6 


times 


12 are 


72 


7 


time:.> 





arc 





8 times 





arc 





i 


iraes 


are 





7 


times 


1 


are 


7 


8 times 


1 


arc 


8 


9 t 


imes 


1 are 


9 


y 


times 


2 


are 


14 


8 times 




are 


16 


9 t 


imes 


2 are 


18 


^ 


times 


8 


arc 


21 


8 times 


8 


are 


24 


9 t 


imes 


3 are 


27 


T 


times 


4 


are 


28 


8 times 


4 


are 


82 


9 t 


imes 


4 ar« 


36 


7 


times 


T) 


are 


35 


8 times 


5 


are 


40 


9 t 


imes 


5 are 


45 


7 


tim(;s 


G 


aio 


42 


3 times 


G 


are 


48 


9 t 


imes 


6 are 


54 


7 


times 


7 


are 


40 


8 times 


7 


are 


53 1 


9 t 


iini>8 


7 are 


{Vd 


*T 


times 


8 


jire 


oG 


8 times 


8 


are 


64 


9 t 


imes 


8 are 


7a- 


"7 


times 





are 


68 


8 times 


9 


are 


72 


9 t 


imes 


9 arc 


81 


»T 


times 


10 


are 


70 


8 times 


le 


arc 


80 


9 t 


imes 


10 are 


90 


T 


times 


11 


are 


77 


8 times 


11 


arc 


88 


9 t 


imes 


11 are 


9& 


7 


tiraos 


12 


arc 


»4 


8 times 


12 


arc 


9G 


9 t 


imes 


[2 are 


108 



21 



ABSTRACT NUMBERS. 



10 
10 



10 times 
10 times 
10 times 
10 times 
times 
times 
10 limes 
10 times 
10 times 
10 times 
10 times 
10 times 
10 times 



are 

1 are 

2 are 

3 a,re 

4 are 

5 are 
G are 

7 are 

8 are 

9 are 

10 are 

11 are 

12 are 






11 til 


10 


n ti 


20, 


11 til 


SO 


11 ti 


40 


11 ti 


60 


11 ti 


GO 


11 ti 


70 


11 ti 


8§ 


11 ti 


90 


11 ti 


100 


11 ti 


110 


11 ti 


120 


11 ti 



mes 


are 





12 


times 





are 





mes 


1 are 


11 


12 


times 


1 


are 


12 


mes 


2 are 


22 




times 


2 


are 


24 


mes 


3 are 


33 


12 


times 


3 


are 


36 


mes 


4 are 


44 


12 


times 


4 


are 


48 


mes 


5 are 


55 


12 


times 


6 


are 


60 


imes 


6 are 


66 


12 


times 


6 


are 


72 


mes 


7 are 


77 


12 


times 


7 


are 


■84 


imes 


8 are 


88 


12 


times 


8 


are 


96 


mes 


9 are 


99 


12 


times 


9 


are 


108 


mes 


10 are 


110 


12 


times 


10 


are 


120 


mes 


11 are 


121 


1 ^ 

X.JU 


times 


11 


are 


132 


imes 


12 are 


132 


12 


times 


12 


are 


144 



Ex. 1. Multiply 24307 by 3. 
Multiplicand, 24307 § 33. Model.— 3 times 7 are 

Multiplier, 3 21, set down 1 ; 3 times are 0, 

Product, 72921 and 2 are 2; 3 times 3 are 9 ; 3 

times 4 are 12, set down 2; 3 
tfimes 2 are 6, and 1 are 7. The product is 72921. 

Explanation. — Tke smaller factor is placed under the 
larger. Then, beginning at the right, each figure of the 
upper number is taken 3 times, the right hand figure of each 
product is set down, and the remaining figure, if any,, is 
added to the next product. 3 times 7 units are 21 units, 
that is, 2 tens and 1 unit ; hence, 1 unit is set in the units' 
]jlace, and 2 tens are added to the product of the tens. 

2. Multiply 24307 by 40. 
Multiplicand, 24307 § 39^ Model.— 4 times' 7 are 

Multiplier, 40 28, sef^down 8 ; 4 times are 0, 

Product, 972280 aad 2 are 2 ; 4 times 3 are 12, set 

down 2 ; 4 times 4 are 16, and 1 
are 17, set down 7 ; 4 times 2 are 8, and 1 are 9 : annex 0. 
The produ<;t is 972280. 

Explanation. — Since 10 units of any order make one of 
t4e next order on the left, any number is rflultiplied by 10 

22 



MULTIPLICATION OF INTEGERS. 



§40 



by merely moving eacli of its figures one place to the left, 
and putting a in the place of units. Hence, to multiply 
by 40, each figure of the product by 4 is set one place to 
the left, and the units' place is filled with a 0. 

3. Multiply 24307 by 43. 



Multiplicand, 
Multiplier, 

1st partial prod. 
2nd' partial prod. 

Product, 



24307 
43 

72921 

97228 

1045201 



§ 40. Model. — 3 times 
7 are 21, set down 1 ; 3 
times are 0, and 2 arc 2 ; 

3 times 3 are 9 ; 3 times 

4 are 12, set down 2 ; 3 
times 2 are 6, and 1 are 

7 :— 4 times 7 are 28, set down 8 under 2 ; 4 times are 0, 
and 2 are 2 ; 4 times 3 are 12, set down 2 : 4 times 4 are 
IG, and 1 are 17, set down 7 ; 4 times 2 are 8, and 1 are 9. 
Add the partial products : 1 ; 8 acd 2 are 10, set down ; 

1 and 2 are 3, and 9 are 12, set down 2 ; 1 and 2 are 3, and 

2 are 5; 7 and 7 are 14, set down 4 ; 1 and 9 arc 10, set 
down 10. The product is 1045201. 

Explanation. — The upper nuniLur is multiplied, first 
by 3, as in § 38, and then by 40, as in § 39, except that the 
at the right is omitted, as being unnecessary, since the 
several figures can be placed in their proper columns with- 
out it. It must be remembered, however, that the second 
partial product is not 97,228, but 972,280. 

4. Multiply 3047 by 246279. 

5. Multiply 794378 by 4608. 



Multiplier, 
Multiplicand, 



Product, 



246279 ; 
3047 

T723953 
985116 

738837 



750412113 



Multiplicand, 
Multiplier, 



! Product, 



794378 
4608 

6355024 
4766268 
3177512 

3660493824 



23 



§40 



ABSTRACT NUMBERS, 



Rule. — 1. When either factor contains hut one valuable 
figure. Set the smaller factor under the lafger. Beginning 
at the right, multiply each figure of the upper number by the 
lower number, set doicn the right hand figure of the product, 
and add the remaining figure, if any, to the next product ; 
but set doivn the whole of (he last product. 

2. When the smaller factor contains more than one valua- 
ble figure. Set it under the larger ; multiply the upper fac- 
tor by each figure of the lower, setting the firat figure of each 
partial product under tin- .'n-i'iijli^ing figure which produced 
it, and add the partial products together in thai order. 

Proof. — Multiply the lower factor by tlie upper. 



Ex. 6. Multiply 3469 by 3. 

7. Multiply 4'by 268. ' 

8. Multiply 45274 by 5. 

9. Multiply 56295 by 6. 

10. Multiply 75397 by 7. 

11. Multiply 9 by 98765. 

12. Multiply 21179 by 27. 

13. Multiply 97825 by 34. 

14. Multiply 86906 by 45. 

15. Multiply 279862 by 52. 

16. Multiply 192837 by 67. 

17. Multiply 293705 by 75. 

18. Multiply 246835 by 83. 

19. 1964326 x98=« what? Ans. 192503948. 

NoU. — Begin, " Maltiply the first number by the second.'" 

20. What is the product of 2975x375 ? Aus. 1 1 15625. 

21. What is the product of 3047x287 ? 

24 



Prod. 10407 
Prod. 1072. 

Prod. 226370. 

Prod. 527779. 
Prod. §88885. 

Prod. 3326050. 
Prod. 3910770 

Prod. 12920079. 
Prod. 22027875. 



DIVISION OF INTEGERS. §48 

22. What isthe product of 40535x403? Ans. 19983755. 

23. What isthe product of 4-027x4027? Aus. 16216729. 

24. 719x729=whiitr 

25. 92730465xl794 = wh.it > Aus. 1G635845421U. 
2(5. 81G2035x28r>45=whatr' Au3. 233801492575. 



DIVISION OF ABSTRACT NUMBERS. 



§ 41. Division is the operation of finding how many 
times one number is contained in an otlier. Thus, 4 in 20, 
"> times : her« 20 is divided hy 4, since 4 is contained 5 
times in 20. 

§ 42. Or, Division is the operation of separating a num- 
ber into some number of equal parts. Thus, if 20 is di- 
vided into 4 equal parts, each or the parts is 5. 

§ 4$. The number to he divided is called the dividend. 

i^ 44. The dividing number is called the diviHor. 

§ 45. The result of division is called the quotievl. 

§ 46. When tlie division is not complete, the undivided 
p^rt of the dividend is called the remainder. Thus, Sin 35, 
4 times, with 3 over ; here 35 is the dividend, 8 is the divi- 
sor, 4 is the quotient, and 3 is the remainder. 

§ 47. The sign of division, -T-,is read dividcdhy. When 
placed between twu numbers, it denotes that the one before 
it is to be divided l>y tlie one after it. Thus, 20-7-5::rr:4. 

§ 48. Division is sometimes denoted by placing the divi- 
dend over the divisor with a line between them. Tiius, 
?i-=r4. 

9r, 



Z') 



^48 



ABSTRACT NUMBERS. 



I3I^V^lS10I«ir TJLBLE. 



1 in 


0, 


no time | 


2 in 0, 


no time 


3 ] 


n 


0, 


no time 


1 in 


1, 


onee 


2 in 2, 


once 


o 


n 


13, 


Onee 


3 in 


2, 


twice 


2 in 4, 


twice 


3 ] 


m 


6, 


twice 


1 in 


3,' 


3 times 


2 in 6, 


3 times 


3 ] 


n 


0, 


3 times 


1 in 


4, 


4 times 


2 in 8, 


4 times 


3 i 


n 


12, 


4 times 


1 iu 


5, 


5 times 


. 2 in 10, 


5 times 


3 


in 


15, 


5 times 


1 in 


^, 


6 times 


2 in 12, 


6 times 


3 


n 


18, 


6 times 


1 in 


7, 


7 times 


2 in 14, 


7 times 


3 ] 


n 


21, 


7 times 


I in 


8, 


8 times 


2 in 10. 


8 times 


3 


LU 


24, 


8 times 


1 in 


9, 


9 times 


2 in 18, 


9 times 


3 


m 


27, 


9 time^ 


4 in 


X 


no time 


^ 5 in 0, 


no time 


6 ] 


m 


"^"o, 


no time 


4-in 


4, 


once 


5 in 5, 


once 


6 


m 


6, 


once 


4 in 


8. 


twice 


5 in 10, 


twice 


6 


m 


12, 


twice 


4 in 


12, 


3 times 


5 in 15, 


3 times 


6 


in 


18, 


3 times 


4 in 


16, 


4 times 


"^5 in 20, 


4 times 


6 


m 


24, 


4 times 


4 in 


20, 


5 times 


5 iu 25, 


5 times 


6 


n 


30, 


5 times'. 


4 in 


24, 


6 times 


5 iu 30, 


6 times 


6 


m 


36, 


6 times 


4 in 


26, 


7 times 


5 in 35, 


7 times 


6 


n 


42, 


7 times 


4 iu 


32, 


8 times 


5 in 40, 


8 times 


6 


n 


48, 


8 times 


4 iu 


36, 


9 times 


5 in 45, 


9 times 


6 : 


LU 


54, 


9 times 


7 in 


0, 


no time 


8 in 0, 


no time 


9 : 


n 


0, 


no time 


7 in 


7, 


once 


8 in 8, 


once 


,9 


in 


9, 


once 


7 in 


14, 


twice 


8 in 16, 


twice 


9 


m 


18, 


twice 


7 in 


21, 


3 times 


8 in 24, 


3 times 


9 


m 


27, 


3 times 


7 in 


28, 


4 times 


8 in 32, 


4 times 


9 


U 


36, 


4 times 


7 in 


35, 


5 times 


8 in 40, 


5 times 


9 


m 


45, 


5 times 


7 in 


42, 


6^ times 


8 in, 48, 


6 timo-: 


9 


m 


54, 


6 times 


7 in 


4VJ, 


7 times 


8 in 56, 


7 times 


9 


in 


63, 


7 times 


7 in 


66, 


8 times 


8 in 64, 


8 times 


9 


in 


72, 


8 times 


7 in 


63, 


9 (times 


8 in 72, 


9 times 


9 


in 


81, 


9 times 


10 in 


0,^ 


no time 


11 in 0, 


no time 


'12" 


in 


i. 


), no time 


10 in 


10, 


once 


11 in 11, 


once 


12 


in 


V. 


I, once 


10 in 


20, 


twice 


11 in 22 


twice 


12 


in 


2- 


:, twice 


10 in 


30, 


3 times 


11 in 83, 


3 times 


12 


n 


86 


), 8 times 


10 in 


40, 


4 times 


11 in 44 


4 times 


12 


m 


4^ 


\, 4 times 


10 in 


50, 


5 times 


11 in 55 


5 times 


12 


in 


6( 


), 5 times 


10 in 


60, 


6 times 


11 in 66, 


5 times 


12 


m 


71 


J, 6 times 


10 in 


70, 


7 times 


11 in 77 


7 times 


12 


Q 


8^ 


t, 7 times 


10 in 


80, 


8 times 


11 ia 88 


' 8 times 


12 


m 


9( 


J, 8 times 


10 in 


90. 


9 tiinos 


11 iri 99 


9 timp= 


12 


in 


10^ 


5, 9 times 



26 



DIVISION OF INTEGERS- §50 



I. SHORT DIVISION. 

Ex. 1. Divide 3096 by 3. 

J)W^<^r,r^ 3) 3096, Dividend. ^ 49^ Model.— 3 in 3, 

1032, Quotient. once ; 3 in 0, no time ; 3 in 

9, 3 times; 3 in 6, twice. 
The quotient is 1032. 

Explanation. — The divisor is placed on the left of the 
liviilotid. Then, beginning at the left, the number in each 
rder of units is divided by 3, and each quotient figure is 
et in its proper column. 

Ex. 2. Divide 806X4 by 2. Quot. 40312. 

3. Divide 8048 by 4. Quot. 2012. 

4. Divide 90369 by 3. Quot. 30123. 
5.. Divide 17120 by 8. 

8)17120 g 50 MopEL.— 8 in 17, twice, with I over, 

2140 set' down 2 ; 8 in 11, once, with 3 over, set 
down 1 ; 8 in 32, 4 times ; 8 in 0, no time. 
The quotient is 2140. 

Explanation. — 8 is not contained in 1, that is, in 1 ten- 
thousand, in its present form ; hence, 1 ten-thousand is re- 
duced to 10 thousands, and added to the 7 thousands, mak- 
ing 17 thousands. 8 is contained twice in 16 ; so that there 
i3 1 thousand still undivided. This is reduced to 10 hun- 
dreds, and added to the 1 hundred, making 11 hundreds. 
8 is contained once in 8 ; so that there are 3 hundreds still 
undivided. These are reduced to 30 tens, and added to 
the 2 tens, making 32 tens. 8 is contained in 32 just 4 
times. The of the dividend is retained in the quotient, 
to cause the several quotient figures, 2 thousands, 1 hun- 
dred, and 4 tens, to occupy their proper plares 

.27 



§51 ABSTRACT NUMBERS. 

Ex. 6. Divide 36374 by 9. 

9)36374 § 51^ Model.— 9 in 36, 4 times; 9 in 

4041... 5 3, time, with 3 over, set down ; 9 in 

37, 4 times', with 1 over, set down 4 ; 9 in 

14, once, with 5 over, set down 1 in the quotient, and 5 a? 

remainder. The quotient is 4041, and the remainder 5. 

Explanation. — The division of the 5 units might be 
denoted ^-, as in § 48. 

Rule. — -Set the divisor on the left of the dividend, unth a 
line between them, and one under the dividend. 

Beginning at the left, see how many times the divisor is 
roiitained in each figure of the dividend, and set the result 
u/nder the dividend. 

Whenever there is a remainder, prefix it to the next fgure 
of the dividend, before dividing. 

If the divisor is not contained in any figure, except tlw 
first^ set under such figure, and regard it as a remainder, 

pjROOF. — Multiply the quotient by the divisor : the prod-' 
uct, increased by the remainder, if any, will be equal to 
th© dividend. 

Ex. 7. Divide 73052 by 2. Quot. 36526. 

8. Divide 222345 by 3. Quot. 74115. 

9. Divide 123456 by 4. 

lO! Divide 790530 by 5. Quot. 158106, 

11. Divide 78920472 by C. Quot. 13153412. 

12. Divide 945 by 7. * 

13. Divide 1240128 by 8. . Quot. 155016. 

14. Divide 743200173*^ by 9. Quot. 82577797. 

15. Divide 4703750 by 10. 

16. Divide 9009 by 11. Quot. 819. 
i7. Divide 721428 bv 12. Quot. 60119. 

28 



DIVISION OP INTEGERE. §52 



U. LONG DIVISION. 



Ex. 18. Divide 2966232 by 925. 

Dividend, 29662321925, Divisor. 

2775^ 13206, Quotient , ., ^, ^ . ■ 

ig^Q 29, 3 times ; multiply 

. _-^ . the divisor by 3 ; 3' 

^7"^'^ times 5 are 15, set 

'^:^j^ down 5 ; 3 times 2 are 

682, Remainder. 6, and 1 are 7 ; 3 times 

9 are 27, 5et down 27 : 
ubtract the product from the dividend; 2; 5 from G 
leaves 1 ; 7 from 16 leaves 9; 8 from 9 leaves 1 : — 9 in 
19, twice; multiply the divisor by 2; twice 5 are 10, set 
down 0; twice 2 arc 4, and 1 are 5; twice 9 are 18, set 
down 18 : subtract the product from the previous remain- 
der ; 3 ; U from 2 leaves 2 ; 5 fi^om 11 leaves 6; 9 from 9 
leaves 0: — 9 in 6, no time ; annex 2: — 9 in 62, 6 times; 
Multiply the divisor by 6 ; 6 times 5 are 30, set down ; 
() times 2 are 12, and 3 arc 15, set down 5; 6 times 9 are 
54, and 1 are 55 : subtract the product from the previous 
remainder ; from 2 leaves 2 ; 5 from 13 leaves 8 ; 6 from 
12 leaves 6; 6 from 6 leaves 0. The quotient iso206,nnd 
the remainder 682. 

Explanation. — The divisor is placed on the right of the 
dividend, for convenience in multiplying. The number 
is used throughout as a trial divhor. As two figures ©f the 
real divisor arc thus omitted, two figures of each partial 
dividend must be omitted also. Hence, in the third step, 
we say 9 in 6, and not 9 in 62, until we have annexed an 
additional figure. The first quotient figuile stands for 
3000 ; hence the first product is really 2775000, and the 
first remainder 191232; but, as we do not need all these 

29 



>53 ABSTRACT NUMBERS. 



figures for tke next step, we begin to subtract only one 
place to the right of the last valuable figure in the prod- 
uct- The division of the remainder might be expressed 
as in § 48. 

Ex. 19. Divide 6593 by 19. 

6593119 

57 oT« § 53. Model. — 2 in 6, 3 times; multiply 

~7."q~- the divisor by 3 ; 3 times 9 are 27 ; sot 

nn. down 7 ; o times 1 are 3, and 2 are 5 : sub- 

tract the product from the dividend ; 9 ; 

^^^ 7 from 15 leaves 8; 6 from 6 leaves :-^ — 

^'^^ 2 in 8, 4 times ; multiply the divisor by 4 ; 

4 times 9 are 36, set down 6 ; 4 times 1 are 

4, and 3 are 7 : subtract this product from 

the previous remainder ; 3 ; 6 from 9 leaves 3 ; 7 from 8 

leaves 1 : — 2 in 13, 7 times; multiply the divisor by 7 ; 7 

times 9 are 63, set down 3 ; 7 times 1 are 7, and 6 are 13, 

set down 13 : subtract the product from the previous re- 

Qiainder; 0. The quotient is 347. 

Explanation. — If the second figure of the divisor is 
less than 5, the first figure is the trial divisor ; but, if the 
second figure is greater than 5, the trial divisor is one more 
than the first figure. If, on multiplying, a quotient figure 
be found to be too large or too small, let it be diminished or 
increased a unit at a time until the right result is attained. 

Rule. — Set the divisor on the right of the dividend, with 
a line between ihem, and one under the divisor. 

Beginning at the left, see how often the divisor is contained 
in the first part of the dividend : the result will he the first 
figure of the quotient. Multiply the divisor by this quotient 
figure, and subtract the product from that part of the divi- 
dend which loas used, annexing to the remainder the next 

figure of the dividend. 

30 



DIVISION OF INTEGERS. §53 



Take this remainder as a second partial dividend, and 
from it obtain the second quotient Jif/ure. Muhiphj (he divi- 
sor by thi^Jigure, and subtract the product from the previous 
remainder^ annexing to the second remainder the next figure 
of the dividend. 

Continue this process till all the figures of ili'- dividend 
have been used. 

If any partial dividend will iiot contain the divisor, set 
in the quotient^ annex an other figure of the dividend, and 
divide again. 

Proof. 1. — The same as in § 51, for short division. 

Or, 2. Subtract the remainder, if any, from the divi 
dend ; divide this remainder by the quotient, and the re- 
sult will be the divisor. 

Ex. 20 Divide 18950 by 25. « Quot. 758. 

21. Divide 17136 by 36. 

22. Divide 42581 by 49 Quot. 8t)9. 

23. Divideud=lG7'01, Div-i.-ur^57 Quot. 293. 

P^ Note. — Begin, "Divide the Dividend by tbo Divuor.'' 

24. Dividend— 265oG, Divi3or^()2. 

25. Dividend^l5076872, Divisor=:72. Quot. 209401. 

26. Dividend— 30744, Divisor=:84. Quot. 366. 

27. Divisor=97, Dividend==84002. 

" 28. Divisor=^125, Dividend-=15625. Quot. 125. 

29. Divisor=:273, Dividendr=104832. Quot. 384. 

30. Divisor=354, Dividend=94l64. 

31. Divisor=465, Dividend^2G7375. Quot. 575. 

32. Divisor=531, Dividend=340902. Quot. 642. 

33. Divisor=^685, Dividend=543205. 

34. Divisor:==721, Dividend=r2728264. Quot. 3784. 

35. DiTisor=829, Dividend=5697717. 

31 



.^63 ABSTRACT NUMBERS. 



36. r>ivisor=937, Dividendr=981976. Qu»t. 1048. 

37. 5754375-4-1125=:whatl Ans. 5116. 

38. 4515625-^2125=:what! 

39. 48284964-T-3094=what ? Ans. 15606. 

40. 24896825~-4105=what ? Ans. 6065. 

41. 27206656-T-5216=what? 

42. 45782172-^6327=what? Ans. 7236. 

43. 313201258-f-7153=what? Ans. 43T86. 

44. 293834463995 ~ 8405==what f 

45. 572473044-i-9516=what? Ans. 60159. 

46. 93939874943-- 10471=what? Ans. 8971433. 

47. 151807041— 12321==what? 

48. Dividend=l 2741 53376, Divisor=23456. 

Qwot. 54321. 

49. Dividend==18p9739176, I>iTisor=34056. 

Quot. 54Q2I. 

50. Dividend=2642079580, Divisor=40565. 

51. Dividend==:2900 124304, Divisorzz=56504. 

Quot. 513p. 

52. Divisorrr=:65405,Dmdend=6677 19645. 

Quot. 1020». 

53. Divisoi— 74316, Dividend=4734969624. 

54. DiTisor=81634, Dividend=7571 145330. 

Quot. 92745. 

55. Dmsor=:95703, Dividend=1299551037. 

Quot. 13579. 

56. Divisor=97531, Dividend=2999956029. 
.57. Divisoi— 36805, l)ividend==800655970. 

Quot. 21754, 

58. Divisor=::234282, Dividend=83596737522. 

Quot. 356821. 

59. Divisor=5276431, DiTidend^7105901 538475. 

32 



CONTRACTED ADDITION OF INTEGERS. 



§54 



CONTRACTION IN ADDITION. 



; Note. — The judicious teacher will omit this and most of the 
following contractions as his classes proceed through the book the 
first time. 

Ex. 1. Add together the following numbers : 



469375 
237924 
472437 
853214 
975318 
242326 



§54. Model.— 26 and 10 are 36, and 8 
are 44, and 10 are 54, and 4 are 58, and 30 
are 88, and 7 are 95, and 20 are 115, and 4 
are 119, and 70 are 189, and 5 are 194, set 
down 94: — 1 and 23 are 24, and 50 are 74, 

and 3 are 77, and 30 are 107, and 2 are 109, 

3250594 and 20 are 129, and 4 arc 133, and 70 are 
203, and 9 are 212, and 90 are 302, and 3 
are 305, set down 05 : — 3 and 24 are 27, and 90 are 117, 
and 7 are 124, and 80 are 204, and 5 are 209, and 40 are 
249, and 7 are 256, and 20 are 276, and 3 are 279, and 40 
are 319, and 6 are 325, set down 325. The sum is 3250594. 

Explanation. — Beginning at the right, and taking two 
columns at a time, we take in first the tens and then the 
units, as we go up the column, and set down the two right 
hand figures of each sum. 



Ex. 2. 

123456 
789012 
345678 
901234 
567890 
987654 
821098 
765432 

4801454 



o. 

1234 

5678 
9012 
3456 
7890 
1357 
9246 
8987 

46860 



4. 

235689 
124578 
135792 
468097 
531086 
420987 
654321 
555775 



5. 

14250663 
32215941 
10340285 
92341967 
82395786 
17084657 
40558476 
91623378 



6. 

819349 
720258 
630167 
541076 
452985 
363894 
274703 
185612 



7. 

120341 
989052 
878163 
767274 
656385 
545496 
432107 
321098 



3126325 380811153 



38 



§55 ABSTRACT NUMBERS. 



CONTRACTION IN SUBTRACTION. 



Ex. 1. From 970347 take the sum of 14375; 226899, 
12534, and 369708. 

?Z?!?^ § 55. Model.— 8 and 4 are \2, and 9 are 

14375 21, and 5 are 26, from 27 leaves 1 ; 2 and 3 

226899 are 5, and 9 are 14, and 7 are 21, from 24 
12534 leaves 3 ; 2 and 7 are 9, and 5 are 14, and 8 

^69708 are 22, and 3 are 25, from 33 leaves 8 ; 3 and 

346831 9 are 12, and 2 are 14, and 6 are 20, and 4 
are 24, from 30 leaves 6 ; 3 and 6 are 9, and 

1 are 10, and 2 are 12, and 1 are 13, from 17 leaves 4 ; 1 
and 3 are 4, and 2 are 6, from 9 leaves 3. The remainder 
is 346831. 

Explanation. — As 26, the sum of the subtrahend units, 
can not be take^ from 7, the units of the minuend, we add 

2 tens, that is, 20 units, to the minuend, and afterwards 
add 2 tens to the subtrahend. (§ 28.) 

yote. — Let the pupil be required to use this contraction when- 
ever it can be applied. 

Ex. 2. From 1000 take 9+98-f-176-f 254-f 289. 

Rem. 174. 

3. From 9125 take 8-f 88 +888 -f 1297+3945. 

Rem. 2899. 

4. From 10275 take 1245 + 373 5 + 298 6 J- 895. 

Rem. 1414. 

5. From 87579 take 1477+2796 + 8972 + 10896. 

Rem. 63438. 

6. From 120225 take 246+1357+97531 + 1358. 

7. From 72575 take 575+2575+4575+15575. 

8. From 4970 take 250-|-325-|-348-|-2211. 

9. From 22907 take 3916.|-2821-|-4302-|-2309. 

34 



CONTRACTED MULTIPLICATION OF INTEGERS. 



§57 



CONTRACTIONS IN MULTIPLICATION. 



Ex. 1. Multiply 7325 by 100. 

752500 ^ ^^' ^^^^^i" — Annex two naughts to the 

multiplicand. The product is 732500. 

Explanation. — We annex to the multiplicand as many ci- 
phers as there are annexed to the 1 of the multiplier. (§ 39.) 
Ex. 2. Multiply 1358 by 10. Prod. 13580. 



3. Mult 

4. Mult 

5. Mult 
5. Mult 
7. Mult 



ply 2468 by 100. 
ply 4579 by 1000. 
ply 86725 by 10000. 
ply 1020 by 100. 
ply 32500 by 1000. 



Prod. 246800. 
Prod. 4579000. 



H. Multiply 32500 by 25000. 
32500 



__25000 

1625 

650 

812500000 



§ 57. Model.— 5 times 5 are 25, set 
down 5 ; 5 times 2 are 10, and 2 are 12, 
set down 2 ; 5 times 3 are 15, and 1 are 
16, set down 16: — twice 5 are 10, set 
down under 2 ; twice 2 are 4, and 1 
are 5 ; twice 3 are 6 :— add the partial 
products : 5 ; 2 ; 5 and 6 are 11, set down 1 ; 1 and 6 are 7, 
and 1 are 8 :— annex 5 naughts. The product is 81250000o! 
Explanation. — After finding the product of the valua- 
ble figures, we annex to it as many naughts as there are in 
the right of both the factors. 

Ex. 9. Multiply 27500 by 350. 

10. Multiply 1250 by 1500. 

11. Multiply 747000 by 250. 

12. Multiply 19500 by 1400. 

13. Multiply 124750 by 3000. 

14. Multiply 2795000 by 2700. 

85 



Prod. 9625000. 

Prod. 1875000. 

Prod. 186750000. 

Prod. 27300000. 



§58 ABSTRACT NtJMBERS, 



15. Multiply 3759 by 104. 

3759x104 I 58^ Model.— 4 times 9 are 36, set 

_j_^^*^ down 6, two places to the right of 9 ; 4 

390936 ' times 5 are 20, and 3 are 23, set down 3 ; 

4 times 7 are 28, and 2 are 30, set down 

; 4 times 3 are 12, and 3 are 15, set down 15 : — add the 

partial products : — 6 ; 3 ; 9 ; 5 and 5 are 10, set down ; 

1 and i are 2, and 7 are 9 5 3. The product is 390936. 

Explanation.— If the multiplier has only two valuable 
figures, the first of which is 1, we multiply by the other 
valuable figure, and set the first figure of the product as far 
to the right of the units figure of the multiplicand as this 
figure is to the right of the 1 . 

Ex. 16. Multiply 2376 by 12. Prod. 28512. 

17. Multiply 47475 by 107. Prod. 5079825. 

18. Multiply 57875 by 10080. Prod. 583380000. 

19. Multiply 275 by 1009. Prod. 277475. 

20. Multiply 4479 by 10006. 

21. Multiply 795310 by 10500. 

22. Multiply 1025 by 7001. 

1025x7001 § 59. Model.— 7 times 5 are 35, 

7175 set down 5, three places to the left of 

7176025 5 ; 7 times 2 are 14, and 3 are 17, set 

down 7 ; 7 times are 0, and 1 is 1 5 

7 times 1 are 7 : — add the partial products : — 5 ; 2 ; ; 5 

and 1 are 6 ; 7 ; 1 ; 7. The product is 7176025. 

Explanation. — If the multiplier has only two valuable 
figures, the last of which is 1, we multiply by the other 
valuable figure, and set the first figure of the product as 
far to the left of the units figure of the multiplicand as this 
figure is to the left of the 1. 

Ex 23. Multiply 7893 by 51. Prod. 402543. 

36 



CONTRACTED MULTIPLICATION OF INTEGERS. 



m 



24. Multiply 4685 by 601. 

25. Multiply 23795 by 7010. 

26. Multiply 1375 by 8001. 

27. Multiply 20478 by 90010, 



Prod. 2815685. 
Prod. 166802950. 



28. Multiply 27346 by 99. 

"'^o-^^fi ^ ^^' ^^^^^T^-— ^^^^ex 2 naughts to the 

.J*l___ multiplicand : — subtract the multiplicand 

2707254 from the result; 6 from 10 leaves 4; 5 from 

10 leaves 5 ; 4 from 6 leaves 2 ; 7 from 14 

leaves 7 ; 3 from 3 leaves ; from 7 leaves 7 ; from 2 

leaves 2. The product is 2707254. 

Explanation. — Since 9 is 1 less than 10, we may multi- 
ply any number by 9, by subtracting the number from 10 
times itself. If therefore the multiplier consists of 9's 
alone, we annex to the multiplicand as many naughts as 
there are nines in the multiplier, and subtract the multi- 
plicand from the result. 

Ex. 29. iVIultiply 124795 by 9. 

30. Multiply 24735 by 99. 

31. Multiply 1469 by 999. 

lultiply 70095 by 99. 
53. Multiply 9999 by 256. (§ 37.) 

34. Multiply 1276538 by 999. 

35. ^Multiply 8365712 by 99. 



Prod. 1123155. 
Prod. 2448765. 
Prod. 1467531. 
Prod. 6939405. 
Prod. 2559744. 



36. Multiply 2754 by 54. 

27540 on-. ,r r. ^ . 

§61. Model. — o4=9 times 6. First, mul- 
tiply by 9 :— (§ 60.) 4 from 10 leaves 6 ; 6 
from 14 leaves 8 ; 8 from 15 leaves 7 ; 3 from 
7 leaves 4 ; from 2 leaves 2. The product 
is 24786. Multiply this product by 6 :— 6 
times 6 are 36, set down 6 ; 6 times 8 are 48, 
and 3 are 51, set down 1 ; 6 times 7 are 42, and 5 are 47, 

37 



2754 

24f86 
6 

148716 



m 



ABSTRACT NUMHKRS. 



set down 7 ; 6 tiroes 4 are 24, and 4 arc 28, set down 8 ; 6 
times 2 are 12, and 2 are 14, set down 14. The product 
is 148716. 

Explanation. — If the multiplier is the product of two 

or more numbers, we may multiply the multiplicand by 

either of those numbers, and this product by an other, and 

so on. 

Ex. 37. Multiply 3725 by 35. 

38. Multiply 17075 by 48. 

39. Multiply 473729 by 49. 

40. Multiply 279o6 by 56. 

41. Multiply 124684 by 64. 

42. Multiply 247372 by 72. 



Prod. 130375. 

Prod. 843G00. 

Prod. 23212721. 

Prod. 1564416. 



43. Multiply 21857 by 714. 
21857 



714 

149499" 
298998 

T5248898 



§62. Model. — 14 is twice 7. First, 
multiply by 7 : — 7 times 7 arc 49, set down 
9 under 7 of the multiplier ; 7 times o are 
35, and 4 are 39, set down 9 ; 7 times 3 
are 21, and 3 are 24, set down 4 ; 7 times 
1 are 7, and 2 are 9 ; 7 times 2 are 14, set 
down 14. The product is 149499. Multiply this product 
by 2 : — twice 9 are 18, set down 8 under 4 of the multi- 
plier; twice 9 are 18, and 1 are 19, set down 9; twice 4 
are 8, and 1 are 9 ; twice 9 arc 18, set down 8 ; twice 4 are 
8, and 1 are 9 ; twice 1 are 2. Add the partial pr(-ducts : 
8 ; 9 ; 9 and 9 arc 1>^, set down 8 ; 1 and ^^ arc 9, and 9 are 
18, set down 8 ; 1 and 9 are 10, and 4 are 14, set down 4 ; 
1 and 2 are 3, ami 9 are 12, set down 2 ; 1 and 4 :n\. 5 ; 1. 
The product is 15248898. 

Explanation. — If one part of the multiplier is a factor 
of an other, the work may be contracted as in the model, 
placing the first figure of each product immediately under 

88 



CONTRACTED MULTIPLICATION OP INTEGERS. 



§64 



the right hand figure of the corresponding part of the mul- 



tiplier. 

Ex. 44. Multiply 12479 by 654. 

45. Multiply 24793 by 56248. 

46. Multiply 97635 by 53545. 

47. Multiply 86436 by 497. 

48. Multiply 23047 by 488. 

49. Multiply 902756 by 366108, 



Prod. 8161266. 

Prod. 1394556664. 

Prod. 5227866075. 

Prod. 42958692. 



50. Multiply 225 by 25. 

4)22500 ^(53^ Model.— Annex 2 naughts to the 

5625 multiplicand : — divide the r-esult by 4 : — 4 

in 22, 5 times, with 2 over, set down 5 ; 4 

in 25, 6 times, with 1 over, set down 6; 4 in 10, twice, 

with 2 over, set down 2 ; 4 in 20, 5 times. The product is 

5625. 

Explanation. — Annexing 2 naughts multiplies by 100, 
(§ 56) : hence, since 100=4x25, we divide the product by 
4, to get the true product. 

Ex. 51. Multiply 10275 by 25. Prod. 256875. 

52. Multiply 28832 by 25. Prod. 720800. 

53. Multiply 72725 by 25. . Prod. 1818125. 

54. Multiply 84287 by 25. Prod. 2107175. 

55. Multiply 96248 by 25. 
5G. Multiply 8324728 by 25. 

57. Multiply 274 by 125. 

8 )274000 ^ (54 Model.— Annex 3 naughts to the 

34250 multiplicand : — divide the result by 8 : — 8 

in 27, 3 times, with 3 over, set down 3 ; 8 

in 34, 4 times with 2 over, set down 4 ; 8 in 20, twice, with 

4 over, set down 2 ; 8 in 40, 5 times ; 8 in 0, no time. The 

product is 34250. 

39 



§65 AHBTRACT MMBBRS. 



Explanation. — Annexing 3 naughts multiplies by 1000, 
(§ 56): hence, since 1000—8x125, wc divide the product 
by 8, to get the true product. 

Kx. 58. Multiply 125 Ly 125. Prod. 15620. 

59. Multiply 625 by 125. Prod. 78125. 

60. Multiply 1776 by 125. Prod. 222000. 

61. Multiply 34070 by 125. Prod. 4259875. 

62. Multiply 934478 by 125. 

63. Multiply 7840349 by 125. 



CONTRACTIONS LN Di\ iiilO.X. 

Ex. 1. Divide l25wl 

125 64 ^^ ^^ MoDKL. — Cut off two figures at thf 

' right. The quotient is 125, and the remain 

der 64. 

ExPL-\NATiON. — We cub off at the right, for remainder, 
a.s many figures as there are naughts at the right of the I 
• •f the divisor. The remaining figures on the leftconstitut« 
the quotient. 

2. Divide 34000 by 10. (]uot. 3400. 

3. Divide 74500 by 100. Quot. 745. 

4. Divide UJ740 by 100. Quot. 107; Hem. 40. 

5. Divide 24G000 by lOUO. Quot. 24(;. 
0. Divide 147375 by 1000. 

7. Divide 24680 by 100. 

8. Divide 98630 by 800. 

8,00)086,30 ^QQ MoDKL.^Out off the 2naught.s 

123 — 230 at the right of the divisor, and 2 fig- 
ures at the right of the dividend : — 
then, 8 in 0, once, with 1 over, set down 1 ; 8 in 18, twice, 
with 2 over, set down 2 ; 8 in 2(.), 3 times, with 2 over. Tho 
(quotient is 123, and th(? rcmaindo^- '^'^'^ 

40 



CONTRACTED DIVISION OF INTEGERS. §68 



Explanation. — The remainder after dividing is prefixed 
to the dividend figures cut off, to constitute the true re- 
mainder. 

Ex. 9. Divide 127569 by 270. (.^uot. 4724; Rem. 189. 

10. Divide 56000 by 700. Quot. 80. 

11. Divide 3230000 by 1700. Quot. 190U. 

12. Divide 24600 by 2400. Quot. 10 : Hem. 600. 

13. Divide 7346790 by 72900. 

14. Divide 135073 by 21800. 

15. Divide 275 by 5. 

^"^[^ ' § 67. Model.— Multiply the dividend by 2 : 

_^ twice 5 are 10, set down ; tv/ice 7 are 14, and 

55,0 1 are 15, set down 5 ; twice 2 are 4, and 1 are 
5 : — divide this product by 10. (§ 65.) The 
(quotient is 55. 

Explanation. — Since the dividend is already 5 times 
the required quotient, multiplying it by 2 gives (2x5) 10 
times the quotient. The part cut off at the right, by this 
plan, is twice the true remainder. 

Ex. 16. Divide 10024 by 5. Quot. 2004; Kern. 4. 

17. Divide 2725 by 5. Quot. 545. 

18. Divide 49720 by 5. Quot. 9944. 

19. Divide 598405 by 5. Quot. 119681. 

20. Divide 479324 by 5. 

21. Divide 2379156 by 5. 

22. Divide 329 by 25. 

^2^ i^ 68. Model.— Multiply the dividend by 4 . 

4 times 9 = 30, set down 6 ; 4 times 2=8, and 3 

i /i- =ll,set down 1; 4times3 = 12, and 1 = 13, set 

down 13 : — divide this product by 100 (J; 65.) 

The quotient is 13, and the remainder 4. 

41 



§69 ABSTRACT NUMBERS. 

Explanation. — Since the dividend is already 25 times 
the required quotient, multiplying it by 4 gives (4x25) 
100 times the quotient. The part cut off at the right, by 
this plan, is 4 times the true remainder. 

Ex. 23. Divide 293235 by 25. Quot. 11729; Rem. 10. 

24. Divide 148532 by 25. Quot. 5941 ; Rem. 7. 

25. Divide 2475 by 25. Quot. 99. 
. 26. Divide 193450 by 25. Quot. 7738. 

27. Divide 34795 by 25. 

28. Divide 107059 by 25. 

29. Divide 23725 by 125. 

23725 I Qg Model. — Multiply the dividend by 

^ 8:8 times 5 are 40, set down ; 8 times 2 



189,800 are 16, and 4 are 20, set down ; 8 times 7 

are 56, and 2 are 58, set down 8 ; 8 times 3 

are 24, and 5 are 29, set down 9 ; 8 times 2 are 16, and 2 

are 18, set down 18 : — divide this product by 1000. (§65.) 

The quotient is 189, and the remainder 100. 

Explanation. — Since the dividend is already 125 times 
the required quotient, multiplying it by 8 gives (8x125) 
1000 times the quotient. The part cut off at the right, by 
this plan, is 8 times the true remainder. 

Ex. 30. Divide 724350 by 125. Quot. 5794; Rem. 100. 

31. Divide 111000 by 125. Quot. 888. 

32. Divide 246625 by 125. Quot. 1973. 

33. Divide 57935 by 125. Quot. 463 ; Rem. 60. 

34. Divide 793575 by 125. 

35. Divide 125364 by 125. 

36. Divide 10202 by 42. 

42 



CONTRACTED DIVISION OP INTEGERS. §71 



2)10202 



3)5101 



§ 70. Model. — 42=2 times 3 times 7. 

First, divide by 2 : — 2 in 10, 5 times; 

7)1700—1 2 in 2, once ; 2 in 0, no time ; 2 in 2, 

242 — 6 once : — divide this quotient by 3 : — 3 in 

5, once, with 2 over, set down 1 ; 3 in 

21, 7 times; 3 in 0, no time ; 3 in 1, no time, with 1 over, 

set down in the quotient, and 1 as remainder : — divide 

this quotient by 7 : — 7 in 17, twice, with 3 over, set down 

2 ; 7 in 30, 4 times, with 2 over, set down 4 ; 7 in 20, twice, 

with G over, set down 2 in the quotient, and 6 as remainder. 

The quotient is 242, and the remainder 38. 

Explanation. — If the divisor is the product of two or 
more numbers, we may divide the dividend by either of 
those numbers, and the quotient by an other, and so on. The 
true remainder is found by multiplying each remainder by 
all the divisors previous to the one which produced it, and 
adding together the several products. 

Ex. 37. Divide 7346 by 56. Quot. 131 ; Hem. 10. 

38. Divide 347934 by 35. Quot. 9940 ; Rem. 34. 

39. Divide 92384 by 64. Quot. 1443; Rem. 32. 

40. Divide 83495 by 45. Quot. 1855 ; Rem. 20. 

41. Divide 745106 by 72. 

42. Divide 656215 by 96. 

43. Divide 34635 by 285. 

^1^^^|^_?5 § 71. Model.— 3 in 3, once :— once 5 is 

J^r 121 5, from 6 leaves 1; once 8 is 8, from 14 
;*^^ leaves 6 ; once 2 is 2, and 1 are 3, from 3 

^^^ leaves : annex 3 : — 3 in 6, twice : — twice 

5 are 10, from 13 leaves 3 ; twice 8 are 16, 
and 1 are 17, from 21 leaves 4 ; twice 2 are 4, and 2 are 6, 
from 6 leaves : annex 5 : — 3 in 4, once :— -once 5 is 5, from 
5 leaves 0; once 8 is 8, from 13 leaves 5 ; once 2 is 2, and 
1 arc 3, from 4 leaves 1. The quotient is 121, and the re- 
mainder 150. 

43 



§72 ABSTRACT NUMBERS. 



Explanation. — The products are not written, but are 

iui mediately substracted as in § 55. 

Note. — Let all the exercises in Long Division hereafter be per- 
formed by this plan. 

Ex. 44. Divide 136895 by 725. Qiiot. 188 ; Rem. 595. 

45. Divide 247986 by 836. Quot. 296 ; Rem. 230. 

46. Divide 358097 by 749. Quot. 478 ; Rem. 75. 

47. Divide 469108 by 5275. Quot. 88 ; Rem. 4908. 

48. Divide 5702195 by 4386. 
40. Divide 68132050 by 5295. 



GENERAL PRINCIPLES OF DIVISION. 



§ 72. If the divisor remain unchanged, and the dividend 
be multiplied by any number,the quotient will be multiplied 
by the same number. Thus, 32-i-8=4 : then, 64-^8=8. 

§ 73. If the divisor remain unchanged, and the dividend 
be divided by any number, the quotient will be divided by 
the same number. Thus, 32-f-8=4 : then, 16-^8 = 2. 

§ 74. If the dividend remain unchanged, and the divisor 
be multiplied by any number, the quotient will be divided 
by the same number. Thus, 32-^8=4 : then, 32-^16 = 2. 

§ 75. If the dividend remain unchanged, and the divisor 
be divided by any number, the quotient will be multiplied 
by the .-ame number. Thus, 32^8=4: then, 32-^4=8. 

§ 76. If the dividend and the divisor be both multiplied 
by the same number, the quotient will remain unchanged. 
Thus, 32-8.-4: then, 64-^16:zr=4. 

§ 77. If the dividend and the divisdr be both divided by 
the same number, the quotient will remain unchanged. — 
Thus, 32-T-8==l: then, lG--4=ri. 

44 



PROMISCUOUS PROBLEMS. 5J80 



PROMISCTTOTTS PROBLEMS. 



1. The subtrahend is thirty thoiisaDcl and forty-five ; the 
remainder is fortj'-six thousand eight hundred and ninety: 
what is the minuend ? Ans. 769i:j5. 

§ 78. Minuend — Subtrahend=Remainder. 
Minuend — Remainder=Subtrahend. 
Subtrahend 4- Remainder.-rrrMinuend. 

2. The minuend is three hundred thousand ; the subtra- 
hend is ninety-nine thousand three hundred and seventy- 
four : what is the remainder? Ans. 200024. 

3. The minuend is seventy thousand and twenty-nine ; 
the remainder is sixty-five thousand and forty-six : what i.* 
the subtrahend ? 

4. The multiplicand is twenty-seven thousand and four ; 
the product is seven hundred and twenty-nine millions, two 
hundred and sixteen thousand, and sixteen : what is the 
multiplier ? Ans. 27004. 

§ 79. Multiplicand X Multiplier =Product. 
Product-7-Multiplier= Multiplicand. 
Product-f- Multiplicand = Multiplier. 

5. The multiplicand is four thousand and seventy-two : 
the multiplier is one thousand one hundred and six : what 
is the product 1 Ans. 4503632. 

6. The product is ninety-three thousand three hundred 
and sixty-one ; the multiplier is eighty-nine : what is the 
multiplicand 1 

7. The divisor is one thousand and nine ; the quotient is 
nine hundred and ten : what is the dividend ? Ans. 918190. 

§ 80. Dividend-i-Divisor=Quotient. 
Divisor xQuotient=Dividend. 
(Dividend — Remainder) -^Quotient =Divisor. 
Quotient xDivisor-f- Remainder =Dividend. 
45 



§)^0 ABSTRACT NUMBERS. 



8. The dividend is nineliuudred and forty- five thousand, 
eight hundred and eighty-eight ; the divisor is two thousand 
and four : what is the quotient ? Ans. 472. 

9. The dividend is one hundred and forty-eight thou- 
sand; the quotient is three hundred and forty-two; the re- 
mainder is two hundred and fifty-six : what is the divisor ? 

10. The quotient is one thousand and three ; the divisor 
is one thousand and two : the remainder is one thousand 
and one : what is the dividend ? Ans. 1006007. 

11. Find the sum of two hundred and forty-five thou- 
sand, nine hundred and seven, seventy-four thousand and 
seventy-four, one hundred and nine thousand and nine, and 
three hundred and ninety-seven. Sum, 429387. 

12. Find the difference between two hundred thousand, 
and one hundred and eighty-seven thousand six hundred 
and fifty-four. 

13. Find the product of one million three hundred and 
seventy-five, and one thousand three hundred and seventy- 
five. Prod. 1375515625. 

14. Find the quotient of three millions divided by six 
thousand two hundred and seventy-nine. 

Quot. 477 ; Rem. 4917. 

15. What number is that from which if 2407, 4072, 724, 
and 7240 be subtracted, the remainder will be 7042 ? 

16. What number is that to which if 2407, 4072, 724, 
and 7240 be added, the sum will be 15000 ? Ans. 557. 

17. What number is that by which if 2047 be multiplied, 
the product will be 15151894 1 Ans. 7402. 

18. What number is that by which if 2025042 be di- 
vided, the quotient will be 2021 ? 

19. 247-f 1023— 9344-3720— 4142-f245=:what? 

20. (247-154)-^3-f(247-f 154)x3=what? Ans. 1234. 

46 



PROMISCUOUS PROBLEMS, §81 



§ 81. A parenthesis enclosing two or more numbers shows 
that their united value is to be subjected to the operation 
indicated immediately before or after the parenthesis. For 
example, in the preceding problem, the difference of 247 
and 154 is to be divided by 3, and the sum of 247 and 154 
is to be multiplied by 3, and the product and the quotient 
arc to be added together. 

Two numbers thus connected are called a binomial ^ three 
numbers are called a trinomial; four, a tetranomial ; five. 
a pcntanomial ; six, a hexanomial, &c. 

The 20th problem is read, '^ Binomial 247 minus 154 
divided by 3 plus binomial 247 plus 154 multiplied by 3 is 
equal to what?" 

21. 3247 + 247-47 + 7-(247— 474-7)=what? 

22. (987— 876-f333)-H(765-543)-f-210-95=what y 

Ans. 117. 

23. 27— 30-^10-f (475 — 399)H-4=:what? Ans. 43. 

24. (204-60)-^6-(90-|.10)--5-[-(76-!.12)-T-4=what '^ 

25. 204— 60-f-6— 90-l-10-^5-K76-|-12)--4:=what ? 

Ans. 128. 

26. (204— G0)-^6-j-90-l-10-5-|-(76-|-12)^4r=:what ? 

27. (204— 60)--6— (90-|-10)-f-5-|-7G— 12-^4r=what ? 

28. 204— 60--6— (90-|-10--5.!-76)— 12-^4=rwhat ? 

Ans. 23. 

29. 123-1-41— (123— 41>|-123x41—123^41=what? 

Ans. 5122. 

30. 123-1-41— (123— 4l)-|-(123x41— 123) ^41==what ? 

31. J[(742-v-2)-r-53]x27-lJ-^53=::what? Ans. 1. 

32. [(199-78)-f-ll-(199-43)--78]x(12-3)=lwhat? 

Ans. 81. 

33. [(117-43) x2]--37-K138-128) x37==^hat 1 

47 



ABSTRACT NUMBERS. 



MEASURES AND MULTIPLES. 



§ 82, An eve7i number is one which can be exactly divided 
by 2. Thus, 12, 4, 36, 58, and 70, are even numbers. 
Xote. — All even numbers end in either 2, 4, 6, 8, or 0. 

§ So. An odd number is one which can not be exactly di- 
vided by 2, Thus, 9, 17, 25, 33, and 41, are odd numbers. 
Kote, — AH odd numbers end in either 1, 3, 5, 7, or 9. 

§ 84. A prime number is one which is not the product of 
two other numbers. Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,89, and 97, 
are all the prime numbers less than 100. 

X-A^BLK of I*I?,I^i:]K INTUMCBEl^^S ixp to lOOO. 



o 


43 


103 1 


173 


241 


317 


401 


479 


571 


647 


739 


827 


1 
919 


3 


47 


107 1 


179 


251 


331 


409 


487 


577 


653 


743 


829 


929 





53 


109 


181 


257 


337 


419 


491 


687 


659 


751 


839 


9,37 


. 1 


59 


113' 


191 


263 


347 


421 


499 


593 


661 


757 


853 


941 i 


lll 


Gl 


127 


193 


269 


349 


431 


603 


599 


673 


761 


857 


947! 


jl8 


67 


131 


197 


271 


353 


433 


509 


601 


677 


769 


859 


963 


!l7 


71 


137 


199 


277 


359 


439 


521 


607 


683 


773 


863 


967 


119 


73 


139 


211 


281 


367 


443 


523 


613 


691 


787 


877 


971 


'23 


79 


149 


223 


283 


373 


449 


511 


617 


7U1 


797 


881 


977 


29 


83 


151 


227 


293 


379 


457 


547 


619 


709 


809 


883 


983 


31 


89 


157 


229 


307 


383 


461 


557 


631 


719 


811 


887 


991 


37 


97 


163 


233 


311 


389 


463 


563 


641 


727 


821 


907 


997 


41 


101 


167 


239 


313 


397 


467 


569 


643 


733 


823 


911 





§ 85. A composite number is one which is the product of 
two other numbers. Thus, 4, 6, 9, 15, 21, and 30, are 
composite numbers, because 2x2=4, 2x3=6, 3x3=9, 
3x5=15, 3x7=21, and 5x6=30. 

Is 20 prime, or composite ? 25 ? 28 ? 31 ? 34? 37 ? 40 '/ 
43? 501 57? 64? 71? 78? 85? 92? 99? 106? 217? 
328 ? 439 ? 

48 



MEASURES AND MULTIPLES. §87 

§86. Powers. — The first jyo we r of a number is the num- 
ber itself. Thus, 5 is the Jirst power of 5 ; 7, of 7 ; 10, of 10. 

The second poicer of a number is the product of the num- 
ber multiplied by itself. Thus, 8G is the second power of 
G, because 6x6=^36: 81, of 9, because 9x0—81 : 100, of 
10, because 10x10=100. 

The second power of a number is usually called its square. 

The third power of a number is the product of the num- 
ber multiplied by its square. Thus, 8 is the third power oi 
2, because 2x4=^8 : 64, of 4, because 4x16=64 : 216, of 
6, because 6x36=216: 512, of 8, because 8x64=--:5l2: 
1000, of 10, because 10x100 = 1000. 

The third power of a number is usually called its cube. 
In like manner, what is the fourth power of a number ? 
What is the sixth power 't The 7iinth power ? ka. 

§ 87. Roots. — The first root of a number is the number 
itself. 

The second root, or the square root, of a number is one of 
the tico equal factors which produce it. Thus, 5 is the square 
root of 25, because 5x5=25. 

15 has no square root, because its two factors, 3 and 5, are 
not equal. 

The third root, or the cube root, of a number is one of the 
three equal factors which produce it. Thus, 3 is the cube 
root of 27, because 3.3.3=27. 

30 has no cube root, because its three factors, 2, 3, and 5, 
are not equal. 25 has none, because it has only two equal 
factors, 5 and 5. 16 has none, because it has four e«^ual 
factors, 2, 2, 2, and 2. 

In like manner, what is the fourth root of a number ? 
What is the seventh root ? The sixteenth root ? &c. 
D 49 



§88 ABSTRACT KUiMIiEKS. 

•^ bS. Th.Q prime f (I do i's oi n composite nuitiber arc the 
prime numbers by whose continued multiplication the num- 
ber is produced. Thus, the prime factora of 9 are 3 and 3/ 
because 3x3=9 : the prime factors of GO are 2, 2, 3, and 
5, because 2.2.3.5=60. 

§ 89. A measure of a number is a number which is con- 
tiuned in it a number of times without a remainder. Thus, 
3 is a measure of 12, because 3 is contained exactly 4 time.s 
iri 12 : 4 is a measure of 30, because 4 is confeaiiied exacfli/ 
9 times in 30. 

Is 5 a measure of 10 ? 25 ? 37 ? 40 ? 63 ? Q>~j \ 80 ? 

Is 6 a measure of 7 ? 12? 20? 30? 39? 48? 54? 

Is 7 a measure of 14? 19? 28? 36? 42? 48? 63? 

§ 90. A multiple of a number is a number which containn 
it a number of times loithout a remainder. Thus, 12 is a 
multiple of 3, because 12 contains 3 exactly 4, times : 36 is 
x multiple of 4, because 36 contains 4 exactly 9 times. 

Is 40 a multiple of 2 ? 3 ? 4 ?, 5 ? 6? 7? 8? 9? 10? 20? 

Is 56 a multiple of 2 ? 4 ? 7 ? 8 ? 9 ? 10 ? 14 ? 20 ? 24 ? 

Is 60 a multiple of 2? 3? 4? 5? 6? 7? 8? 10? 12? 15? 

§91. (7o?nmo?i means belonging equally to two or more 
numbers. 

§ 92. One number is a common measure of two or more 
numbers, if it is a measure of each of them. Thus, 3 is a 
measure of 9, also of 12, also of 18 ; hence, 3 is a common 
measure of 9, 12, and 18. Also, 4 is a common measure of 
8, 24, 32, ai)d 48. 

Is 2 a common measure of 4, 6, and 10 ? 

Is 3 a common measure of 6, 10, and 15 ? 

Is 4 a common measure of 12, 16, and 20 ? 

Two or more numbers may have several common meas- 

50 



iires. Tlui., :, .id 361iave as c.......i.... ......,.^i'cs 2, o, 4, 

6, and 1*2. In this case, lU is, of course, the greatest com- 
DKjn ::!easure of 2-i and 36. 

§ 93. Ohe number is a common multiple. of two or more 
numbers, if it- is a multiple of each of them. Thus, 40. is a 
multiple of 5, also of 8, also of 10 ; hence, 40 is a common 
uiultiple of 5, 8, and 10. Also, 45 i- a o.ovir.i.nu multiple 
of 3, 5, and 9. 

Ls 10 a common multiple of 2 and 5 ? 
Is 15 a common multiple of 3 and 6 ? 
Is oO a common multiple of 2, 5, and 10 .' 

Two or more numbers alwajs have several common mul- 
tiples. Thus, 4, 3, and 6, have as common multiples 12, 
34, 3t3, 48, 60, &o. In this case, 12 is, of course, the kcial 
eommon multiple of 4, 3, and 6. 

§ 94. Two or more numbers are prime to each other, if 
they have. np common measure. Thus, 81 and 64 are prime 
to each other. Also, 20, 27, and 77 are prime to each other. 

§ 95. 2 is a measure of every number which ends in ei- 
ther 2, 4, 6, 8, or 0. (§ 82. Note.) 

o is a measure of a number, if it is a measure of the sum 
of the figures which* denote the number. ThuS; 3 is a meas- 
ure of 246, or 462, or 624, or 612, or 426, or 261, or 2064, 
or 4602, &c., because 3 is a measure of 6-f 4-f 2, that is. 
of 12. 

4 is a measure of a number, if it is a measure of the num- 
ber denoted by its two right hand figures. Thus, 4 is a 
measure of 768, or 1860, or 95372, or 1112316, because 4 
is a measure of 68, or 60, or 72, or 16. 

5 is a measure of every number which ends in either 5 

or 0. Thus, 5 is a measure of 20, or 55, or 100, or 275. 

51 



J{96 AESTE.ACT NUMBERS. 



o is a measure of erery even number of which 3 .is a 
measnre. Thus, 6 is a measure of 462^ or 4512, or 1236 : 
but not of 471, or 632L 

8 is a measure of a number, if it is a measure of the 
number denoted by its three right hand figures. Thus, 8 
is a measure of 34800, or 753064, because 8 is a measnre of 
800, or 64. 

9 is a measure of a number, if it is a measure of the sum 
of the figures which denote the number. Thus, 9 is a meas- 
ure of 891, or 1728, or 253269, because 9 is a meavsure of 
18, or 18, or 27. 

10 is a measure of every number which ends iii 0. 

100 is a measure of every number which ends in 2 naughts. 
Is 2 a measure of 3040 ? 4047? 28? 1112? 10124? 
Is 3 a measure of 258? 869 ? 12345678 ? 5169 ? 2571 '( 
Is 4 a measure of 125784 ? 24680 ? 57932 ? 14760 ? 1 1 12? 
Is 5 a measure of 245? 12450? 7824? 12570? 3457? 
Is 6 a measure of 570 ? 378 ? 45S42 ? 123456 ? 12324 ? 
Is 8 a measure of 5070120 ? 247080? 1479008? 1234? 
Is 9 a measure of 1234566 ? 68472 ? 1357 ? 1476 ? 
Is 10 a measure of 240 ? 245 ? 3795 ? 7630 ? 1460 ? 

§ 96. A measure of a number is a measure of any one of 
its multiples. Thus, 6 is a measure of 18: then it is a 
measure of 36, or 54, or 72, or 90. 

§ 97. A common measure of two or more numbers is a 
measure of their sum. Thus, 8 is a common measure of 
16, 24, and 40 : then it is a measure of 80. 

§ 98» A. common measure of tAO numbers is a measure of 
their difference. Thus, 9 is a common measure of 18 and 
.^4 : then it is a measure of 36. 

52 



PRIME FACTORS. §99 



PulME FACTORS. 



Ex. 1. Resolve 7'*^00 into its priDie factors. 

2)7800 

2)3900 ^^^- J^I^r>EL. — Divide the namber by 2. 

'MI^Qi^ (§50). Divide tlie quotient by 2. Divide 

•^-C this cjuotieut by 2. Divide this quotient by 

'^1?.I5 5. Divide this quotient by 5. Divide this 

5)195 quotient hj 3. This quotient is a prime uum- 

jj^gT^ ber. The prime f-icfnrs of 7800 arc 2, 2, 2, 

' 1. '_ 5, 5, 3, and l.S. 

o 

Explanation. — It is better to divide first by 2 as often 
as possiblcj then b}'- 5, and then by the other prime numbers 
in succession. The several divisors and the last quotient 
are evidently the prime factors, of the number. 

Rule. — Divide the given niiviber by one of it^ prime 
nimsures ; divide the quotient hy one of its prime measures ; 
continue thus dividing until a ^^ rime number is obtained fo 
a quotient: the several divisor.^ and the last quotient will be 
ihe prime factors of the given nu,'7\ber. 

Proof. — The continued product of the prime factors will 
he equal to the given number. 

Ex. 2. Resolve 524 into its prime factors. 

P. F. 2, 2, and 131. 

3. Resolve 460 into its prime factors 

4. Resolve 770 into its prime factors. 

P. F. 2, 5, 7, and 11, 

5. Resolve 880 into its prime factors. 

P. F. 2, 2, 2, 2, 5, and 11. 

6. Resolve 999 into its prime factors. 

7. Find the prime factors of 1040. 

P. F. 2, 2, 2, 2, 5, and 13, 
53 



§100 ABSTRACT NTJMBTJRS. 

8. Find the prime factors of 1160. 

P. F. 2, 2, 5, and 29. 

9. Find the prime factors of 1275. 

iO. What are the prime factors of 1300 t 

Ans. 2, 2, l), 5, and 13. 
iL What are the prime factors of 1590 ? 

An&. 2, 5, 3, and 53. 

12. What are the prime factors of 1738? 

13. What are the prime factors of 19500 ? 

Ans. 2, 2, 5, f), 5, 3, and 15. 

14. What are the prime factors of 966000 ? 

Ars 2, 2, 2, 2, 5, 5, 5, 3, 7, and 23. 
15 What are the prime factors of 825000 ? 
16. What arc the prime factors of 13572001 

Ans. 2, 2, 2, 2, 5, 5, 3. 3, 13, and 29. 



INVOLUTION. 

§ 100. Involution is the process of finding i^ power oi -^ 
number. From the definitions of the several powers in 
§86, it is evident that any power of a number is fo raid L_j 
taking the number as a factor in miiltiplication as nianj 
times as there are units in the number of the power, 

Ex. 1. What is the square of 7? Ans 49, 

2. What is the cube of 3 ? Ans. 27. 

3. Yv^hat is the fourth power of 2 't 

4. What is the fifth power of 2 ?' Ans. 32. 

5. What is the fourth power of 5 ? Ans 625. 

6. What is the cube of 9 ? 

7. What is the square of 19? Ans. 361. 

8. What is the cube of 15 ? Ans. 3375. 

9. What is the fourth power of 20? 

54 



JiVOLUTJON. §102 



EV(^LUT10X. 



^\0l EvoLU^i*iON is the process of finding, a >oo^ of ;i 
given power. The method here explained is applicabli 
only to such numbers as have precise roots. The method 
of extracting aj»proxiraate roots of imperfect powers can 
not be explained without the use of algebraic formulas, and 
consequently is not given in this treatise. 

Ex 1. What is the cube root of 21G ? 

r 2)2 16 



2)108 

2^54 



§ 102. MoDEii.-— Resolve the given nuRi- 
ber into its prime factors. (§99.) It con- 
po)'-7 tains three twos and three threes. Hence. 
3y >= cube root is 2x8=6. 



Explanation. — Since the cube root of a number is ouv; 
of the three equal factors which produce it, we separate the 
prime factors into sets of three equal prime factors each, 
and selecting one from each setjthe produc^ of those ^elected 
is evidently the cube roqt of the given nilmber. For ari v 
other root, wo separate into sets of as many prime factors 
each a? there ape units in the ri-.mber of the root. 

3' he prime factors ')e*eparattd as above, '.ho 

required rv)ot can not be exactly found, eitlun- by this, or by afly vtfK'r 
method. 

Kui. .-■ . ..:/.;. .,.:;:.„ ^ ■...,..- into ila .....,:: fac!or>> : 

separate the factors into groups of as many equal factor,'. 

each as there are units in the niimher of the root} select one 

faetor from each f^reiip, and multiply togetlter (hose selected : 

their product tcill he the root n quired. 

Proof — Raise the root to the corresponding power. Tht^ 
result will bo counl to the given number. 

55 



§10o ABSTRACT NUMBERS. 



GREATEST COMMOX MEASURE. 



Ans. 


6. 


An». 


3, 


Ans. 


10. 


Ans. 


4. 



Ex. 2. What is the square root of 100 ? Ans. 10. 

3. What is the cube root of 125 ? 

4. What is the fourth root of 1296 ? 

5. What is the fifth root of 243 ? 

6. What is the sixth root of 64 ? 

7. What is the fourth root of 10000 1 

8. What is the fifth root of 1024 ? 

9. What is the cube root of 3375 ? 

10. What is the square root of 12321 ? Ans. 111. 

11. What is the square root of 65536 ? A^a^. 256, 

12. What is the fourth root of 65536 ? 

13. What is the eighili .■ .,t of 65536? 

14. What is the sixteenth root of 65536 ? 

15. What is the square root of 390625 ? 

16. What is the fourth root of 390625?" 

17. What is the eighth root of 390625 ? 

18. Yfhat is the cube root of 10077696? 

19. What is the ninth root <'.f 10077696? 

20. What is the cube root oi 42875 ? 

21. What is the square root of 122500 ? 

22. What is the square root of 7569? 



Ans. 


4. 


Aus. 


t.) 


Ans. 


25. 


Ans. 


5. 


Ans. 


6. 


Ans. 


35. 


Ans. 


87„ 



Ex. 1, 
and 480. 

2)60, 


. Ein 
150, 


;d the 
480 


5)30, 


75, 


240 


3) 6, 


15, 


48 


2, 


5, 


16 


2. 


,5.3=oU 



grep.test common measure of 60, 150, 

§ 103. Model — Divide each of the 
given numbers by 2. (^- 50). Divide 
each of these quotients bj 5. Divide 
each of these quotients bji 3. These 
quotients are prime to each other. 
2.5.3=30. 30 is the greatest com- 
mon measure of the 2:iven numbers. 
56 



GREATEST COMMON MEASURE. 



Explain A'iiuN. — In this operaiiuu it u uul iicoe^haij n>i 
the divisors to be prime nuiubcTs. Wo might liave divided 
by 10 and by 3, or by 5 and by 6. 

lluLE. — Divide each of the given numbers hy any one cj 
their common meamrcs ; divide each of these qnoiienta by 
ne of iheir comm07i measures ; continue f hits diriding un- 
, il the quotientc, become prime to each other : the continued 
product of the divisors will be the grcated comrnnii. mt'.asurc 
q f the g iven n k m bers . 

Ex. li. Fin'"' t'"' 'greatest common measure of '^6, 12'), 
-16, and 234. 

30=2.2.3.3 § 104. Modkl.— Resolve 36 

126 = 2. 3.3.7 inj-Q ity pi-iine factors. (§ 99). 

'?16 = 2.2.3.3. 2.3 36=2.2.3.3. Resolve 126 into 

234=2^_3.3. 13 ^^^ primefactors. 126=2.3.3.7. 

2.3.3. = 18. Resolve 216 into its prime fac- 

tors. 216=2.2.3.3.2.3. Re- 
solve 234 into its prime factors. 234=2.3.3.13. 2.3.3 = 18. 
18 is the greatest common measure of the ?iven nnm];ei"». 

ExPLANATlo.v. — The prime factors are arranged Vr-ith 
equal factors in the same column, as far as possible. The 
full columns contain the factors that are common to aU the 
numbers. The product of these factors is the greatest 
common mea.sure of tlio numbers. 

Rule. — Resolve each of the given numbers info il.s pri/iw 
factors ; select those factors ichich are common to all the 
numbers: the continued product of these factors will l-r iJtc 

(jreatest common measure of the given numbers. 

57 



§105 ABSTRACT NUMBERS. 



o. Find the greatest common measure of 108 and 261 
261:108 



216 



2 



108!45 
90!^~- ■^* 105. Model.— Divide 261 by 108. 

il (S 71). Divide 108 by 45. Divide 45 

f^^^ hy'lii. Divide 18 by 9. There is do 

'-''^ 2 reoiainder, 9 is the greatest common 

>o7^ — measure of the (nven numbers. 

_____ 

Explanation. — is a measure of IS, (§ 89); hence it i.^ 
a measure of 2x18, or 36, (§ 96) ; hence, of 86-f 9, or 45, 
(§ 97) ; hence, of 2 x45, or 90, (§ 96) ; hence, of 108, {% hi) : 
hence, of 216, (§96); hence, of 261, (§ 97);.. hence it is a 
common measure of 108 and 261. 

YhULE^—r-DLvldeihe larger ?iumber h;j the smalUr ; then 
divide the smaller numher by the remainder^ and continue 
divvUnq the last divinor bv the laM. remainder. until there ?*? 
no reinuinder : tlie last divisor iriU he fitc r/reatcst co'mrn^ 
rr. :'■ ■ ,' • of the given numhers. 

To jind the' r/rentest common 'me<!^ure of mor*'. than two 
numbers^ find the greatest common measure of two of them, 
(hen find the greatest co7nnio?i measure of this mmmire and 
an other of the mnnbos^ and w on : th' last common meas- 
ure will he Ihe greateU common measwe of alt the numher; . 

Either of the above methods may be used in the follow- 
ing exercises. 

Ex. 4. Find the greatest common measure of 48, 64,iipd 

H2. ' G. 0. M..d«. 

5. Find X\iq greatest common measure of 08, 1 19, and 357. 

(I. Find the greatest conimon messuro of GO. OjQ, and 10'-. 



GREATEST COUMON MEASUr..K. >^i()5 



7. Vi d the greatest common measure of c : 

1 .:;. (1. C. M. 13. 

i!^ ' ' lie greatest commou mcasjuro of 40, 60, and 200. 

G. C. M. 2U. 
9. Find the greatest common measure of 96, lt?8, and 320, 

10. Fiud the greatest common measure o^ 164, '-87> 
and 451. ^ 0, M. 41. 

11. Find the greatest common mea«iirc of 63, 126, i^ 15, and 
441. ' G. 0.U, 68. 

12. Find the greatest common measure of 150, '376, and 
(i75. 

13. Find the greatest common measure of. 40, CO, 68, and 
204. ^^ ^\ M. 4. 

H. FiM.^ 0- . ' :; r 2, nnfl 

n:)-.. M..214. 

I :\ Find the greatest common measure of 63. 189, 315, and 

}i}. Fuid the greatefat common me.'i^u re of 152, *j80, and 
532, ,. ..^-. ' ■ r. r'. jy[, 7^. 

17. Fijid the greatest common mcasv. ■■ 1 , 7, and 

'^^IL ^ G. C. M. 17. 

i '. d the greatest comuion measure of iOQ,' IJ^O, 210. 
and 4JV~. 

19. Find til i34, 190, and 
• 1140. G. CM. 38. 

20. Find the greate.'ic common measure of 54, 108, 324, 

nnd378. G C. M. 54. 

Si. Find the greatest common measure of oQ, 84, 140, and 

106. * a. C. M. 28. 

22. Find the greatest common Rinr.sHro of 75, 1:25, 375, 

and 67^. G C. M. 75. 

Fin<] the greateist common measure of 46, 115, aad 161 . 

50 



106 ABSTRACT NUMBERS. 



LEAST COMMON MULTIPLE. 



2)40, 


60, 150 


2) JO, 


30, 75 


^3)10, 


15, 75 


3)2, 


:J, 15 


2. 


1. 5 



Ex. 1. Find the least common multiple of 40, 60,and 150. 

§ J 00 Model. — Divide each of 
the numbers by 2. (§ 50). Divide 
some of the quotients by 2. Divide 
each of these quotients by 5. Divide 
some of these quotients by 3. These 
quotients are prime to each other. 

__ 2.2.5.3.2.5=600. 600 is the least 

2.2.5.3.2.5=600 common multiple of the given num-^ 
bers. 

Explanation. — We divide two or more of the given 
numbers by any prime number that will divide them with- 
out a remainder ; and two or more of the resulting num- 
bers by any prime number that will divide them without a 
remainder : and so on, till the quotients are prime to each 
other : — remembering to repeat in the line below, such 
numbers as cannot be divided. By this means, every factor 
of each number is used, and hence the result is a commov 
multiple of the numbers; but no factor of either number is 
used more than once, and hence the result is their least 
common multiple. 

p^uij/. — Divide two or more of the given numbers hy any 
prime romraon measure; take the qiwfients and the undivid- 
ed numbers for a new set ; divide two or marc of theAii by 
(my prime common measure; and ^o on, uniil the resulting 
numbers are prims to each other; the eontiniied product oj 
the resulting numbers and all the divisors loill he the least 
common multiple of the gicen numbers. 

Ex. 2. Find tlie least common multiple of 3C, 120, and 

216. 

60 



LEAST COMMON MULTIPLE. §107 

36=2. 2. 3 J^ § l^'^- Model. — iiesolve 06 

126=2. 3.3.7 '^^^^^ ^^^ prime factors. (§99.; 

2IQ 2.2.3.3 2.0 3Sr=2.2.3.3. Resolve 12G into 

wwoonoo if^ia its prime factors. 120=-2.3.3.7. 
^.Z.6.6.i.4.6~l0l^ Resolve 216 into its prime fac- 
tors. 216=2.2.3.3.2.3.-^2.2. 3. 3. 7. 2. 8=^1512. 1512 is the 
least common multiple of the given numbers. 

Explanation. — The prime factors are arranged as in 
^ 104, and one factor is taken from each column, whether 
full or not. 

Rule. — jResolve eacliof tJic given numbeni into its prime 
factors ; multiply together all the factors of the largest nitiTi' 
bcr, and all the factors of the other numbers that arc not fourid 
hi the largest number ; the product xcill bn the lea^t common 
multiple of the given numbers. 

Either of the above methods may be used in the following 
exercises. 

Ex. 3. Find the least common multiple of 5, 6, and 7. 

4. Find tlie least common multiple of 2, 4, 6, 8, 12, 

and 16. L. C. M. 48. 

5. Find the least common multiple of 3, 6, 9, 12, and 18. 

L. C. M. 3G. 

6. Find the least common multiple of 5, 10, 12, and 15. 

7. Find the least common multiple of 6, 12, 24, and 48. 

L. C. M. 48. 
S. Find tlie least common multiple of 8, 24, and 72. 

L. C. M. 72. 
9. Find the least common multiple of 3, 9, 18, and 72. 

10. Find the least common multiple of 2, 3, 4,5, 6, 10, 12, 
15, and 20. L. C. M. 60. 

11. Find the least common multiple of 3, 5, 7, and 11. 

L. C. M. 1155. 
61 



>ji07 iT,^TR\CT NUMBER?. 

l^. I\..:\ -A.^ iy^vfnurm multiple of 2,3,4,6,8.12, 

and 24. 
13. Find the least common multiple of 3, 7, and 18. 

L. CM. '^73. 
11. Find tlie least com,mon multiple of 2, 4, 7, and 14. 

. \ L. C. M, 28. 
15. Find the least common ninltiple of 3, 5, 15, and 30. 
10. Find the least common multiple of 2,4,8, 16, and 32. 

L. C. M. 32. 

17. Find the least common multiple of 3, 4, 6, 8, and 9. 

L. CM. 72. 

18. Find. the least common multiple of 2, 3, 6, and 9. 

19. Find the least common multiple of 4, 6, 8, 12, 16, 

and 32. L. C M. 96. 

20. Fi id the least common multiple of 2, 4, 5, 10, and '^0. 

L. C M. 20. 



PROMISCUOUS PROBLEMS. 



1. Read 279301682038040. 

2. Read i207§008u40009750. 

3 . Write twenty-seven billions, three hundred and three 
millions, four hundred and seventy-five thousand, and 
eighty-nine. 

4. Write five hundred and five billions, and fifty-five. 

5. Add 3 millions 24 thousand and 17, 4 hundred thou- 
sand 7 hundred and 98, 4 millions 247 thousand and 56, 
and 724 thousand 8 hundred and 29. Sum, 8396700. 

6. Add twenty, 2 hundred and 2, 2 thousand and 27, 
20 thousand 278, 202 thousand 7 hundred and 89, and 2 
millions 27 thousand 8 hundred and 90. 

§2 



PROMISCUOUS PROBLEMS. §107 

7, UnB 9 millions and 9, subtract 5 iiu]Iiaub789 

adred and 54. 3216355. 

8. From 80 milliaiis 85 thousand and 8, subtract 65 inil- 
•liops 764 tboueand 3 hundred and "^ ' Kem. 143206o'3> 

Multiply 4 hundred and 70 i ' "^ ^ "'"■••' • r.d 

!■ thousand 8 (mndred and 1 . 

10. Multiply 90 thousaud 7 hundred aud'S, b}^ 80 tliou- 
sand Ghundred and 4. Prod. 7::nilS5J^20. 

11. Divide 2 billions 59 million' ' ' - -ousand and. 72, 
50 thousand 7 hundred and 9. Quot, 40608. 

12. ]'>ividc 8 billions 777 millions 887 thousand 5 hui:- 
dro 31, by 97 thousand 5 hundred and 31. 

13. The minuend is 4 hundred thpusand 4 hundred ; the 
subtrahend is 364 thousand 7 hundred aud 26 : what ks the 
remainder? - ' Ans. 35674. 

14 The minuend is 57 thousand and 57 ; the subtrahend 
27 thousand 5 hundred and 79 : what is the remainder 1' 

An.'i. mis. 

15. The minuend ic? 75 thousand and 6^i ; the remainder 
■ • 36 thousand and 57 : what is the subtrahend 1 

16. The subtrahend is 3 millions and 75 ; the remainder 
I'j 5 hundred thousand 7 hundred and 5 : what is the minu- 
end ? Ans. 3500780. 

17- The remainder is 777 thousand 7 hundred and 7 ; 
t-hc subtrahend is 654 thoiis.fnd 3 hundred and 25; what 
is the minuend ? Aus. 1432032. 

18. The multiplicand is 3 millions and 75 ; the multi- 
plier is 5 hundred thousand 7 hundred and 5 : what is the 
product ? 

19 The multiplier is 3 thousand 3 hundred and 3 ; the 
multiplicand is 75 thousand 4 hundred and 25: what is the 
product ? Ans. 249128775. 

63 



§107 ABSTllACT NUMBERS. 



20. The product is 670 millions 592 thousand 745 ; the 
multiplier is 12 thousand 845: what is the multiplicand T 

Ans. 54321. 

21. The multiplicand is 40 thousand 5 hundred and 6 : 
•Le product is 413 millions 282 thousand 7 hundred and 
18 : what is the multiplier ? 

22. The dividend is 1 billion 546 millions 2(')3 thousand 
5 hundred and 4; the divisor is 71 thousand 2 hundred and 
17 : what is the quotient? Ans. 21712. 

23. The dividend is 2 billions 162 millions 6 hundred 
thousand ; the remainder is 19 thousand 4 hundred and 90 ; 
the quotient is 24 thousand and 6 : what is the divisor ? 

Ans. 90085. 

24. The divisor is 14 thousand and 20 ; the quotient is 
2 thousand 3 hundred and 45 : what is the dividend ? 

25. The divisor is 7 thousand and 2 ; the quotient is 2 
thousand and 7 ; the remainder is 2 thousand and 7 : what 
is the dividend ? Ans. 14055021. 

20. Resolve 3 thousand and 80 into its prime factors. 

P. F. 2,2,2, 5, 7, 11. 

27. Resolve 5 thousand 4 hundred and GO into its prime 
factors. 

28. Resolve 4 thousand and 4 into its prime factors. 

P. R 2,2,7, U, 13. 

29. Find the greatest common measure of 58, 87, and 
2610. G. C. M. 29. 

30. Find the greatest common measure of 118, 177, and 
295. 

31. Find the greatest common measure of 4^', 80, 128, 
and 176. Q. C. M. 16. 

32. Find the least common multiple of 3, 7, 9, 12, and 
18. L. C. M. 252. 

64 



COMMON FRACTIONS. §109 

33. Find tbo least common multiple of 2, 5, 8, 11, and 14. 

84. Find the least common multiple of 2, 4, 7, 11, 16, 
and 22. i L. C. M. 1232. 

35. What number is that to which if 1231, 8912, 5G78, 
45G7, and 9123 be added, the sum will be 47275 1 

Ans. 17761. 

31. What number is that from which if 1234,8912,5678, 
45li7,and 9128 be subtracted, tlie remainder will be 47275? 

37. What number is that by which if 9876 be multiplied, 
the product will be 121919220? Ans. 12345. 

38. What number is that by which if 5483886 be divided, 
the quotient will be 2468 ? Ans. 2222. 



FRACTIONS. 



§ 108. A FRACTTON is a part of a unit. Thus, one half, 
three fourtlis, two fifths, five sixths, four sevenths, three 
eighths, five ninths, and seven tenth-i, are fractions. 

Fractions are of two kinds, Common and Decimal. 



COMMON FRACTIONS. 

§ 109. A common fraction, or, simply, a fraction, is de- 
noted by two terms!, one above, and the other below, a hori- 
zontal line. The term above is called the numerator ^ the 
term below is called the denominator. Thus, the above 
fractions are denoted, i, f , f , |^, f, f, -|, ■^. The numera- 
tors are 1, 3, 2, 5, 4, 3, 5, and 7 : the denominators are 2, 

4, 5, 6, 7, 8, 9, and 10. 

E 65 



§110 ABSTRACT NUMBERS. 

Point out the numerator and tlie denoujinator of eacli of 
the following fractions : i, -|, f, ^, f, |, |, -V» -rr» T2» tt> 

5 7_ _P_ 3 _R_ _2_ _3_ _4_ _J>_ _fi_ _7_ _8_ _n_ lH 1 I 1^ 

14" 15» 10' 17' 18' 19' 20' 2 1' 2 2' 23' 2 4' 2 3> 26» 37'26>3»> 

iA i± ^5- JL1 <^ JLS a o 3_7 4- "^ 2^JLF 

3^' 31' 32' 3 '3' 3 4' 35' 1200' 2000' 376' 9 72* 

§ 110. A fraction is rf-ad by |>ruinnniciiig after the nu- 
merator the ordinal of the denominator iti the singular or 
the plural number accorditig as the numerator is «»ne or 
mora than one. Thus, -t is read, one fifth ; |, two fit's hs; 
^, three twenty -firsts ; -g^, four thirty-.-ecoMds ; ^^^17, five 
two hundred and-ninths; -^jy^^, t^ix two thoui^and-a nd- sec- 
onds ; -g^y'og^, seven three thoiisand-one-hundred-and .-ixths. 

But, if the denominator is 2, the fraction is read half or 
halves, and not second or seconds. Thus, -}, one half; -f, 
thrre halves. 

IIj^'kI thp followinfy • ^ ^- * — " -' "_ JL. _+_ _p_ _i_ s , _ 

4 ^- _6_ __?_ _8_ _9_ 1 O. _:i_ _4_ _S_ _6_ _7_ _8_ _?i_ J D 1 1 

IT' 16' 17' 18' 19> 20' 21' 22' 2 3' 24» 25' 2 6' 2 7» 28»2 9'30> 

12 1 3 X± XJi -L<> 11. -lA JL^l 
ST' 32' 33> 34' 35' 30' 37' 3b' 

§ 111. A fraction \^ prod me d by dividing a unit or a 
iQunjber into some nuniber of e<|ual f»arts. Thus, \ is pro- 
duced by dividing the unit into four etjual parts: f, bj 
dividing 2 into 5 equal parts ; ^, by dividing 5 into 9 e(jual 
parts. 

The nun)erator is the divi<lend, the denominator is the 
divisor, and the value of the fraction is the quotient. See 
$48. 

How is I produced ? |-? |? i? -,% ? ^? A? i^? ^? 

VP ? iJ. ? u ? 

Xs • 2 4' 35* 

§ 112. Otherwise, a fraction may he produced by divid- 
ing a unit into some number of equal parts and con.-tder* 
ing either one or several of thesb parts. Each of these 
parts is called a ^//ar/ />;/<«/ 7/1* /V ; and a fraction is f^irher 
One or several fractional units. The denominator shows 

C6 



COMMON FRACTIONS. ^113 



into liow many partes the unit is divided, and the imnierator 

fihows iiow luany parts there arc iti the fraclii)ii. Thus, iit 

|, one fifth is the fractiotial unit, and the fraction containg 

three of these units ; in ^, the fractional unit is one niutli^ 

and the fraction contains seven of them. 

In this view, how is ^^^ produced ? 3^^? {r ? f? J^ ? y\f 
7 ? _S ? j< ? 2 ? _5_ ? _«_? 

« 4 • a r> • 7 • 9 • 11" 10* 

§ 113. The o t/iifi of a fraction is the quotient of its nu- 
merator divided by its denominator. Tliis value depends 
on the value of tjie fractional units?, as well as <u\ the niun- 
her of tliein. If the fractional units of sevt-ral fraeti.mt 
are ef|ii:jl, of course the greatest fraction is the one which 
has the most fractional units. Tiiat is, if the denoui'nutors 
arc e(|nal, the groate.^'t fraction is the one wisieli has tho 
greatest numerator. Aga'n, if the nuniher of frictional 
units in several fraction!* is the same, of ct)urse thegieiresl^ 
fraction is the one which has the largest fraotiMia! nu'ib^ 
But, tiie larger the number of parts into which a unit i^ 
dividt^d, the sn>aller each part must be. Therefore, if tho 
uumiM-ators of ^several fractions are equal, the greatest frac- 
tion is tlie one which has the smallest den(Mninator. 

How do I and * compare in value ? -^- and f ? y and ^t 

r- and ^ ? ^V »»<1 t*t ? A a"^ t\ ? xV »" ^ t\ ^ U ^'»"J i^« ^ 
^ and I- ? -/3 and -,\1 -,% and A? t and ^ ? -* and /j-? 
\i- a"<l M? U «nd H? li and f^? || and fi? ^and^f 
From the definition in § 1U8, the number of fractional 
units in a frae;ion must be less than the number of pjirti 
into which tlie unit is divided ; that is, the numerator must 
be less than the denominator. Larger numbers, however, 
may be expressed in a fractional form ; and such expres- 
siocs are improperly called fractions also. Ilejce the fol- 
lowing distinctions: — 

67 



§114 ABSTRACT NtJMBERS. 



§114. A proper \rnct\im h owe whose numerator is Icsm 
than its <lon»»niinator, antl wh/»se value is less ihnn a uirit. 
Tl'iis I, ly ^, -♦, J., I, A, A, _z^, _«_^ .7^.^ and |.}, arc proper 
fractious. 

§ Ho. An improper fraction is one wlioFe nunicrator if 
not /r.v.s- tlun ita (Iciioniiniitor, jind whose value is not ]e$n 
than a unit. Thus, -?, t, 2, {-, §, ^ V, V, Vj Vu',^*"^ -f^ 
^«ir« jinpr(»per fractious. 

:I.s I a proper or an improper fraction ? * ? ^ ? -| ? -^\T 

JL; ? nV 11^ 12? 17? 209 20 9 W? ft ? P? »V f»? 7^ 
* 1 ■ J,;J • ♦ ' & • 'JO' 1 > * 2 5 ' 1 5 • 1 o • w • ¥ • y • w * 

§ lJ(j, A unit is often ouilcd, for distinction, un iTif/yral 
IBnit ; and a number of integral units is called an integer, 
.er an inlegral number. 

§ 117. A mixed number is one composed of an integer 
and a fraction. Thus, 5^, read five and one half, is a 
mixed number. So al.-o, (i^, 3^, 18^, 31;, GG^. 

A fraciional unit or a fraction may be divided into anj 
nuuilMjr of equal parts. Thu'^, if I be divided into o equal 
part.", Ciich of the parts is J of ^. So, if ^ of I be divided 
into 4 equal parts, 3 of these parts arc ;^ of ^ of ^. Such 
cxpiev-ioiJB are called comjiouiid fractions. Jlencf, 

§ 118. A compound fraction is a fraction of a fraetioiL 
Thu., I of -^ i of vV, I of }, l], of 12X, and ^ of ^ of 33^ 
are a)njj)ound fractions., 

§ 119. A complex fraction is one which has a fraction for 

2 
its numerator ojr its denomitator or each. Thus, - , read 

two divided by five and one half, is a complex fraction, 

6i 12i- .} of I \ of 18J 

*^" "^''' y ' 37i* "33i ' vTf 12.i- 

68 



DEDUCTION OF COMMON FBACriONR. ^123 



re: UCTION OF COMMON FRACTIONS. 



§ 120. T() reduce any mimbcr. cither fracti)nar or inte- 
gral, i^ to cha'ige its /'ftni of expression wlthmt r/i>n>f/ing{ 
ifs Vdfiir, Thus, a unit ni>i)' be reduced to *, or to y, or tfll 
I'o" • — i '"^^^ ^'^ reduced to ®, or to -}-i, or to ^\ : — (j\ nja v be 
reduced to ^^, or to ^£ : — and y ,iiay be reduced lo 12^. 

§ 121. A fraction is in its lowest terms \vht;n its tcrrafl 
are pnuie to each other. (^ 04). Thus, -^, \^ |, -y, ;^, -J, 
and 'J', are each in its lovve-t terms. 

Ex. 1. Reduce \^ to its Newest term?. 

4^;if^l4:i^'9j^|i ^ 122. MuDKL.— Divide botk 

terms by 4 : 4 in 14 t, 3() times; 
4 in .^TG, 144 times: divide both these quofit-nts by 4: 4 
in oG, !> times; 4 in 144, oG rime-;: divide both fho.»o <juo- 
tienrs by 9 ; 9 in 9, once ; 9 in HG, 4 times. 'J'h<^vo (|U0- 
tients are prime to each other : hence, \ is the given Irao- 
tion in its lowest terms, 

]Cxri,AVAT!<>N,— By eompHririg§^ 77 and 111, it is evi- 
dent th it the value of a Iracfioii i> not changed »iy dividing, 
both its ternjs bythesame nnnjber: and by jsucci^s.sive divi- 
sions, its terms may alwa\s be made piiujc to each oiher. 

Ex. -. lied nee -^^' !- to its lowest term^. 

2v -llf^ll^ § 128. M(n.Ki,. — Find the greatest com- 

mon measnre of the terms of tln^ fiacfion. 
(§ 108i. Their greatest con)nn'ri measuie is 27. D.vide. 
both terms bv 27 : 27 in 5G7, 21 times; 27 in 67;'), 25; 
times. 21 twenty iitths is the given fraciion iii its iowest 
terms. 

Rui-R for reducing a fraction to its lowest terms. 

1. Dciilii hotk tf^.rm< hj (till/ conitiiim mnsan; dioidt 
Loth t/iKSf (jU'ffifnfs })ij ttiii/ cninniou inensnrr • (dh/ .-o itti^ 
Ultlll ihi' qii'itif.uf>i arc in'Im.f to r<irh nlhr ; (lit: lual (juuhcHfj 
will tfS the l(jioeil term.< of the ij'ocn fractiun. 



|124 ABSTRACT NUMBERS. 



Or, 2. Divide lofh ferrns It/ tlidr f/rrafcst cornnion iiiea*- 
urf : the qnofiniis will he the loua^t lams, 
Ex. 3. Reduce -^l^^^ to its lowest termis. 

4. Reduce f i^^ to its lowest terms. Value, •^. 

5. Reduce ^|-^ to its lowest terms. A'^al. |-, 
G. Reduce -*-f^- to its lowest terms. 

7. Reduce |f^-2- to its lowest terms. A'nl. -jV 

J^. Reduce \-^^\ to its lowest terms. Yal. \\^ 

P. Reduce -^—-r to its lowest terms. 

18 9 4 

10. Reduce f-|{)^ to its lowest terras. Yal. ^, 

11. Reduce -|^|-f to its lowest terms. Yal. -/-j^ 

12. Heduce ^-^-^-l to its lowest terms. 

13. Reduce -ff-^ to its lowest terms. Yal. l^^, 

14. Reduce ||-^ to its lowest terms. Yal. -^^^ 

15. Reduce ^44^4-^ to its lowest terms. 

10. Hedtice ^ffl to its lowe-t terms. Yal. ^^^ 

17. Reduce -^Vs ^^ '^^^ lowest terms. Yal. t/qV 

18. R<duce 14x1 ^^ ^^^ lowest terms. 

19. Reduce |f?-i to its lowest teims, Yal. f?-^^ 

20. Reduce -^-«ff^ to its lowest terms. Yal. |^-^ 

21. Tn Ytt* ^f'W mapy units i' 

8G8'16 § 124. MoDKL.— Divide tlie numer- 

f58 ,54tt=«'>4-} at(»r })y the deiiomiuafor. (?^ 71). Thb 

4, rpiotient is f)-4, and tlie rtniaindor 4, 

Redufo -.-V to its luwest teruiS. The 

jiren fraction is equal to 54^. 

KxriiANATiON. — Since IG sixteenths make a unit, the 
BTimher of units in SG*"^ sixteenths is ecjual to the number 
pf limes IG is contained in 8C8. See § U.S. 

Ri'LK for reducing an improper fraction to a whole or a 

nixed number. 

/ 

70 



DEDUCTION OF COMMON FRACTIONS. §125 



D'lvlde the numerator bij the ilenominafffr ; the qKotient 
will he the integral part. Place the denondnator umUr the 
rem ai inter for the fracti mul part, 

Ex. 22. [n y* how ininy units? Anf*. 25. 

23. Ill Y l^ow many units? Ans. 24.1:. 

2\. In -i--^— ^'<J^^ many units ? 

25. In y liow many units? Ans. 3-2-. 

28. Reduce J-«n. to units. Val. 20. 

27. Reduce ^-^.s> to units. 

28. Reduce \-Y ^^ units. Val. 14-*-. 

29. Reduce V/ t^> ufiits. Val. 8^. 
80. Reduce ^.-y to units. 

31. Reduce W^ to a njixed number. Val. 134. 

32. Reduce Y/ ^0 a mixed number. Val. 12^^. 

30. Reduce "Yt^ ^0 a mixed number. 

84. Reduce y^^ to a mixed number. Val. 11-^V 
S5. Reduce y^ to a mixed number. Val. 10-^-. 

36. Reduce ^-^^ to a mixed number. 

37. Reduce y/ to a mixed number. Val. S^^. 

85. Reduce ^^^ to a mixed number. Val. *7~\ 
S9. Reduce \\/ to a mixed number. 

40. Reduce VV ^^ a mixed number. Val. 69V 

41. Reduce 10 units to fourths-. 

§ 125. MiM)Kf..— Multiply 4 fourths by 10. 
10=*jp The product is 40 fourths : hence 10 uLits— 
40 fourths. 

■ ExPLAN.VTioN.— Since 4 fourths make a unit, 10 units= 
10 times 4 fourths; that is, 40 fourths. 

Rui.K for reducing a whole number to any fractional de- 
nominiition. 

Mif/tlpij/ the nu'^ber of fractional units in a unit by the 
number of units. 

71 



§126 ABSTRACT NUMBERS. 

Ex. 42. In 3 units how many fifths ? 

43. In 5 units how many sevenths? An?. ^^. 

44. In 6 units how many ninth;^ ? Ans. '^, 

45. In 7 units how many elevenths? 

46. Ueduce 8 to thirteenths. Val. VV- 

47. Reduce 9 to fifteenths. A'al. VV. 

48. Reduce 11 to seventeenths. 

49. Reduce 12 to twentieths. Val. ^*s, 

50. Reduce 13 to twent- ^ nrths. Val. W. 

51. Reduce 14 to tweni^-i.iDths. 

52. Reduce 15 to thirty-fifths. Yal. •'W' 

53. Reduce 10 to a fraction with denominator 40. 

Val. Vo^ 

54. Reduce 17 to a fraction with denominator 4H. 

55. Reduce 18 to a fraction with denominator 53. 

Val ^-^ 

56. Reduce i9 to a fraction with denominator 59. 

Val. ^}-l^, 

3 

57. Reduce 20 to a fraction with denominator 05. 

58. Reduce 21 to a fraction with denominator 71. 

Val ±±SJL, 

' 59. Reduce 22 to a fraction with denominator 77. 
60. Reduce 23 to a fraction with detiomiuator 85. 



61. Reduce 16^- to an improper fraction, 

16i = V + 2 = V § ^-^- MoDKL.— Reduce 16 units to 
halves. Add o2 halves and 1 half. — 
The sum is 33, halves: hence 16^ is equal to 33 halves. 

Rule for reducing a mixed number to an impro[>er frac- 
tion. 

Rtduce the infeger to the (JcnomI nation of the fraction ^ 
add the two numerators tojefher, and under their sum mttthe 
common denominator. 

72 



REDUCTION OF COM MOM FRACTIONS. §127 



Ex. Ci. Reduce 3| to tliiids. Val. y. 

G-J. R«^.(iuce 4.^- to fourths. 

64. Re<luce 61 to fifrlis. V;il. \K 

65. Ri-duce, 8j to sixihv. • Val. *^*. 

66. Rod wee lOy to sevenths. 

67. Ke(]uoe 12.^ to eighths. Val. JlOJ.. 

68. Reduce 14.} to ninths. Yal. -L^-i 
60. Reduce 16 tV to tenth.*. 

70. In 17^1 ^i"^v many elevenths? An^. yy. 

71. In ISfV hovf many twelfths? 
7-j. In 19yV '^*^^ many fourteenths? 
7'1. In 2)i\ how many sixreenths? 
74-. In 21/^ how many eighteenths? 
75. In 22,/ ^- how many tA^eiity firsts? 
70. Reduce 23^:^- to an improper fraction. 
77. Reduce 24.}"- to an im[>roj)er fraction. 
7H. lleduce 25/^ to an improper fraction, 
79. Rfdiice 523^3 to an improper fraction. 
8u'. Reduce 65 y^- to an improper fraction. 

81. R-jduce -•'- to twentieths. 

•J! 

^=y^; ^1-7. Model. — 5 twentieths make one fourth, 
^lultiply both terms hy 5: 5 times 3 arc 15; 5 
times 4 are 20. a js equal to 15 twentieths. 

Exp.tiANATioN. — By comparing §.!^ 76 and 111, it is evi- 
dent that the value of a fraction is not changed hy multi- 
plying both its terms by the same number. We divide the 
re(]uired denominator by the given one, and multiply both 
terms of the fraction by the (juotient. 

RiiLK for reducing a fraction (o a larger denominator. 

AInitipf(j infh ferma hj (lie qwitient of the rKjuiitd dc' 

nomijKtfor divided bj/ fhe ijlccn one. 

73 



Ans. 


2 2 3 
"1 u • 


Ana. 


3 2 5 
1 6 * 


Ans. 


3 8 S 
16 * 


Yal. 


R n n 
a * ♦ 


Yal. 


6 7 8 
U 7 * 


Yal. 1 


718 
3 3 • 


Yal. 4 


"281 
6 » ' • 



§128 ABSTRACT NUMBERS. 



Ex. ^2. Hednce | to tenths. Val. yV 

80. Heduce A to eiglitecntlis. Yal. i-5,^ 

84. Eednce -f to tbirty-fifilis. 

85. R.duce |- to forrieths, Val. J^|^ 
8(>. T^educo |- to sixty-thirds. Val. -|-|v 

87. Reduce —o ^^ ninetieths. 

88. R«*diice ^ to ninety-ninths. Val. ^. 
81). Heduce ^^ to sixtieths. Val. f^. 
90. Reduce -^3 to sixty-fifths. 

9i. Reduce yoi; to twentieths. 

§ 128. Model. — Divide both terms by 5c 
5]_^.'^_j_^ 5 in 2o, 5 limes ; 5 in 100, 20 timccs. 25 one 

bundredths=:5 twentieths. 
For explanation, see § 122. 

Rule for reducing a fraction to a lower denominator. 
Divi'le both terms by the (jiiotlent of the given denoinina- 
tor dii't led btj the required one. 

Ex. 92. Reduce -/zr to fourteenths. x Val. ,^-. 

9o. Reduce -i§ to tifteenths. 

94. Reduce f^- to sixteenths. Val. -/g~> 

95. Reduce -^^-^ to eighteenths. Val. -^^. 
9(). In -^^-^ how many twentieths? 

97. Ill f-^- how many twenty fiists ? Ans. ^^w 

98. In -ifi how many twenty-thirds? Ans. ^^. 

99. In -~V ^^^^ many twenty-fourths? 

100. In 22o- ^^'^^v "^^^"y thirtieths? Ans. -?-§.. 

101. Reduce -,}, ^:, and |, to a common denominator. (§91). 
^ , , § 129. iMoDKL. — Multiply both termn 
''^ * ''^ of the liist fiuciion by Ji2 ; 32 times 1 

•ff ^% oi ^'"^ '^-; ^- times 2 are 04: multiply 

both terms of th»'. second fraction h) 16; 

16 times are 48 ; ]G times 4 are 64 : multiply both termi 

74 



PPBUCTION OF COMMON FRACTIONS. §129 



of tlio Oiird fr.-iotion b}' 8 ; 8 times 5 nre 40 ; 8 tiiues 8 are 
64. The j?ivpn fractions are respectively cijual to 3J, 48, 
and 40 sixty- fuurtb.". 

ExrrANATiON. — The values of the fractions are not 
chart|j(H], because both terms of each fraction are multiplied 
by the f-anie iiuiuber: and the denominators are alike, be- 
cause each one is produced by mulfiplyinc together all the 
given denojiiinators. The n)ulti[)lier 82 fur the first frac- 
tion is 4 X 8, the product of the other two denominators. 
And so for the other?. 

RiJLK tor reducing fractions to a common denominator. 

Mnlnplij both ferms of euch f taction It/ the product of th* 
other dinnminotors. 

Ex. 102. Heduce -|, -^, and -^, to a common denojiiinator. 

103. lleduce -^, -^-, and J, to a common deooniinator. 

Vi* I 1 "^ 10 8 
* *"• 3 0' To' 311- 

104. lleduce J, 2, and |, to a common denominator. 

V-i I 2 4 5 a 4 

105. Ilt>duee -^, |, and -^, to a common denominaror. 
100. Reduce *, ^, and |-, to a common detioniinator. 

' "' • 2 10' a 1 » a 1 6^ 

107. Reduce -\^ f, and y, to a common denominator. 

Vi 1 -'■ « _f fi_ JL?J» 
' ' '• 3 3 (>' a 3 6' 330* 

lOH. Re<lnce i, ^, and -y, to a common denoniibutor. 

109. Reiluce ^5 I, and -j'\j, to a commoti denominator. 

V-il ^(^ 1 «<' 2JL0 

* «J ' . -7 [. o 1 Tii " ^7 a 0* 

110. Reduce -^, -^^, -^\^ and gV, to a common deiionjina* 

tof V'il _2>oo. _8 !_« Q_ _finoo_ 7 s 6 O ^ 

'"•2 16do'»216OO'2 1C0O*21G0O" 

11 '. Ri'iluee I, -j^j, and -j^g-, to a common dtMjominator. 
111.'. Roduee -), 1^, and y'j-, to a common denominator. 

' " • « 1> 3 » G J 3 ' a 9 3 * 

lis. Reduce -j^, ^, and -fo-».^^ ^ common deitoniinaioi-. 
114. Reduce ^, fj '^^d *. to a cunimon deuumiiiittor. 

75 



§130 ABSTRACT NUMBERS. 



115. Ileduee \, ~, and ^-, to a common dononiinator. 

Vj, I 5 t 3 6 4.» 

* '* • "?!¥' 2T6> TT9' 

116. lieduce ^, |, and y, to a common denoiniiuiror. 

V-il -iL'i- j^3^ iOO 

* '" • 14 0' 1 4 » 1 ¥!?• 

117. lieduee ^, |-, and |-, to a common deuominator. 

118. Reduce i, f, and f, to a common denomiitator. 

V-il -3Jl^ ^7J! _40 
'"'• ibO> ISO' 1 sU* 



119. Reduce ^-, -f-, and — , to a commovi dennmiii i or. 

Vi 1 7 SO 1 

* '"' "2 8 0' ^so* a 

120. Reduce ^, f, and y\, to a common deuominator. 



L A A 

2 4 8 



ILM. Reduce |, -|, and |^, to their least common denomi- 
nator. 

§130. MoDFL. — Find tliejenst common 
multiple of the denominators. (§ 1(J(>). 8 is 
-| I 1^ their leist Ci>ujmon njulti|tle. MuUiplj 
both terms of the first friicrion by 4 : 4 
times 1 are 4; 4 times 2 are 8 : multiply Ixirh terni>i of the • 
seco'id frdc'i )n by 2: twice 3 are 6; twice 4 are 8; th« 
third fraction is already of tlie required dcuoiiiiiiurion. — 
Tlie given fractions are respectiveiy equal to 4, 6, and 5 
eighrh.^'. 

Exp t. A NATION. — To find the proper multiplier for the 
terms of either f^yiction, we divide the least conimon mul- 
tiple by its denominator. See § 127, 

RuLK for reducing fractious to their least common de- 
noniinafor. 

Fmd /he hast common multiple of the denominalorii, and 

rtducc each fraction to the diiiomlii'ilioii expressed hij thU 

mulfipf''. E'ich fraction iniist first he. iiL its hnct^t terms, 

Ex. 122. Pteduce f, -4,, and |, to their least con m m de- 

\ n o I n i ) i a t o r . Ya 1 . y\ » fV? -]rh 

123. Reduce I, |, and ^, to their least commou duuomi- 

iiatur. 

7G 



\ 



1?ED€CT10N OF COMMON FRACTIONS §180 

124. Il(3duce J, -f-, and |, to tlieir least common dcnoiiii- 
Udior. vai. ies, ins^ lo.s* 

125. lleduce ^, I, and y^, to their least comnKHi dcMiomi- 
nator. Val. -,V ^\„ H^. 

126. lleduce ^., ^, and -^, to tlieir least comniuii dciiomi- 
h, tor. 

127. lleduce -^, |, and ^"^,^5 to their least common denomi- 
nator. Va!. -/-, /^, I-*-. 

128. Reduce .}, ^, ^, and i, to their least common de- 
nuujiiiator. Yal. -j\-, ^\) ■^•\, -^-^, 

129. Reduce -^, i, ^, ^, and -j\, to their Uaat common de- 
nominator. 

180. Reduce |, I, |, -|, and |;^, to their least common de- 
iioiuinator. ^ Yal. aa, -^I, f^, -|^.> |->. 

181. Reduce IJ, 2, >ij To> ^'^'^ "sfj *^ their least common 
deiu.minator. " Val. if, f*-, |f, -^, if, 

132. Reduce l, ^, J-^, and H, to their least common de- 
nominator. 

138. Reduce -^, -^\, ^Vj ^^^^ 2^5 to their least common de* 
nominator. Val. |, -*, |, 1- 

184. Rtuliice I, I, -j^^, and -\l, to their least cumuxm de- 
nominator. Val. -j'V, -ft-, -i«3^, -j^a^. 

135. Reduce -^\, -^\-, -^-^^ and -^\, to their lea.st common 
deiiomiuator. 

181). Reduce I, -|, |-, f, and ^, to their least common de- 
i^^i"i"ator. Val. |§, >^, -*^, -*C,-6-C-. 

187. Reduce -J, f, /^, ^f, and |-^, to their least common 
denominator. Val. -J-^, 1% -i*, |-o, if. 

138. Reduce i, a, *, |, 3^, ^'*^, and l^, to their least com- 
mon denominator. 

*139. Reduce i, f, -*, y^^^, and i^, to their least common 
dt^nominator. Val. H-, -||, |f, -*i, H* 

14U. Reduce J, «, -^y /g, and 5%, to their least common 
denominator. Val. |^, -1%, -^, /„> »V- 

77 



§131 ABSTRACT NUMBERS. 



ADDITION OF COMMON FilziCTIONS. 



Ex. I. Add^-, A,|., and|. 

§ 181. Model. — 1 and 3 are 
¥ + ¥ + ¥ + { = ¥ = 2 4,JH.d 5 are 9, ai.-l 7 are K).— 

lo eii^htlis is eijual to 2. The 
Bum 18 2. 

ExPLAMAT'loN. — Since all the fraction-! have the samo 
fractional uni^, their ntKuerators are ad led \'oi' the tiunier- 
ttor of tlie sum, and the comiuoii deuominator is taken as 
its defioiuiiiator. 

Ex. 2. Add I, I, and -|-. 

3 § 132. MoDFL. — Reduce the given 

^ * '^ fractioLs to 'heir least c Mintioa de- 

♦ ^fi-|_L=ij~2* nominator. (§ l:-^0). 4and<)arel0, 

arid 7 a?e 17. 17 eighths i.s e([aal to 

2^,. The f^um is 2^.. 

ExPi.AVATroN. — It is evidently impossible to add the 
given fr-ictions wiihout reduction. 3 fourths and 7 eighths 
make Tieither 10 fourth's nor 10 eighths. It is not essential 
to reduce to tlie Irasf common denominator; but this gen- 
erally re<|nires less labor than to reduce simply to a com- 
mon denominator. 

Ex. 3. Add 241:. 351|, 179|-, and 187. 

24.V .l:+i-i-|- §133. MooKL.— Re- 

'^ 2 I 4 . i_ ij(_iji duce the fracti.ins to 

179^ "b'ii"t*tt H H their least comm.tn de- 

1q^ nominator. (§ 130). — 

742 f 2 and 4 are (5, and 7 

are 13. 13 eighths is 
equal to 1 and 5 eighths, set down ^; 1 and 7 are H. and 9 
are 17, and I are 18, and 4 are 22, -^et down 2 : 2 and 8 are 
10, ant) 7 are 17, and 5 are 22, and 2 are 24, set down 4; 
t and 1 are 3, and 1 are 4, and 3 are 7. The sum is 742f, 

78 



ADDITION OF C 'M3I0N FRACTIONS. §133 



Ruri*^, — Reduce the fractinns to their least cotnnto.i de- 
nomlnutor : a Id the numerators, and under their .smn set 
the coni'noa deiomlnttor. Reduce the result to iis lowest 
terms or to a mixed number ^ as the case mat/ be. 

Ex. 1. Ada I, I, and f. Sum, 1^-, 

5. Add 1,1, A, and ± Sum, 2. 

G. Add i,f.A,/;,a„d^. 

7. Add f, f, f, and f. Sum, 2|-. 

8. Add-^, I, A, and |. Sum, 1a 

9. Add 1-, A, A, ^, and ^ 

10. Add -L, f, A, and >-. Sun.,2j-5. 

11. Add^, ^, A, and A Sliu., 2^^^. 
*■-• ^^'J'l a» 4> 5> To> anu -j^j. 

l;i. Add i 4-, A, and yV Suiii, l2-|. 

14. Add i, i. A, 3Z_, and -i|. Stun, 23V 

15. Add A, ^, 8, ,A, a„j i,. 

16. Add A, ^, I, X, |i., and a^. S.hu, 4}|, 

17. Add A, I, 3, -^-, ^^_., and 3^^. Smn, 2ai. 

18. Add A, I-, A, and «. 

19. Add A. i, L, and -Jj-. Sun., ^«-V^. 

20. Add 2^, 3f, 4A, and 5*. Sim, 16aa. 

21. Add 4, 8A, 9^, and llf. 

22. Find ihe suiu of IQa, 21a. 32^-, 43^*^-, and 54/^. 

Sum, 162^*0. 

23. Find the sura of 19, 23a, 16a, and 27a. Su.n, 83». 

24. Find die sum of At, 2\'^, 3», 25a, and 33a 

25. Find the sum of 12^, 18^, 33a, 87a, and 93^ 

Sum, 245A 

26. What is the sum of », 6a, 3|, 2a, and 98 ? 

Ans. 111». 

27. What is the sum of 1, 2a, 3a, 6a, and 9a? 

79 



S134 ABSTRACT NUMBERS. 



28. AVhat is the sum of 4,^, 5-«, 17|, and 18y^ ? 

An?. 46 1^. 

29. What is the sum of 2i, 251, 125^, and 325tV? 

Ans^ 478/j-. 

30. What is the sum of If, 4|, 7«, 10-H-, and 13-}-^? 



SUBTRACTION OF COMMON FRACTIONS. 



Ex. 1. From -} take f. 

7_.3_4_i §134. M->DEL.— B from 7 leaves 4. 

4 eighths is equal to -}. The reiuaiade? 



b B 2 



IS^, 



Ex. 2. From i take i 



1 



§ 135. Model. — Keduce the fractions to 
^ their least conimon denominator. (>? loO). 

-^— |^=i 2 from 3 lenves 1, that is, 1 sixch. The re- 
mainder is -}.. 



a' 



Ex. 3. From 32?- take ISf. 

32|- x_A ^ 1S(). Model.— Reduce the 

•^Q? y g_^ fractions to their least common 

14i- "B'~s — 8 denomi»)ator. (§130). G from 

7 leavt's 1, set down |-; 8 from 
12 lenves 4 ; 2 from 3 leaves 1. The rcmaiuder is 14}. 

ExiLANATiON. — Any numher of fractional units may 
evidently be subtracted from a larger number of fractional 
units of the same denomination, just as one number of sim- 
ple units is subtracted from an other. If the given frac- 
tions liave different denominators, they must first be re- 
duced to a common denominator : ^ — ^ = neither | nor ^ 
just as 7 dollars — 3 cents=neither 4 dollars nor 4 cents. 
Ex. 4. From 27 take 19}. 

27 § 137. MoDKL. — 1 from 8 leaves 7, set down 

■L^i -^r 10 from 17 leaves 7; 2 from 2 leaves 0. 
7l- The remainder is 7|. 

80 



8" 



SUBTRACTION OF COMMON FRACTIONS. §138 

Ex. 5. From 9^ take 6h 

^8 A —i § 138. Moi^EL. — Reduce the fractions 

"2 to their least common denominator. — 

2-5. 8~-¥ (§ 130). 4 from 9 leaves 5, set down ^ 

7 from 9 leaves 2. The remainder is 2| 

Explanation. — When the fraction in the minuend is 
less than that in the subtrahend, we add an integral unit to 
the minuend fraction, subtract the subtrahend fraction 
from this sum, and then add 1 to the unit^ of the subtra- 
hend before subtracting from the units of the minuend. 

Rule. — Reduce the fractions to their least common de- 
nominator ; sid>tract the numerator of the stibtraheQid from 
the numerator of the minuend; and under the remainder 
set the common denominator. 

Jf 111 suhtracting one mixed number from an other, the 
suhtrcdicnd fraction should he larger than the one in the min- 
uend, reduce an integral unit to the common denomination 
of the fractions, add if to the minuend fraction, subtract the 
subtrahend fraction from this sum, and add one to the sub- 
trahend in the column of units, 

Ex. 6. Subtract f from y^ 

7. Subtract I from ^. Rem. -^, 

8. Subtract -{j^ from -^^. Rem. i. 

9. Subtract f from -Li. 

10. Subtract -f- from |. Rem. -i-f . 

11. From \-l take «. Rem. ^. 

1 2. From -}/^ take |. 

13. From 5§- take f. Rem. 5^- 

14. From 7^ take 4^. Rem. S^^-. 

15. From 8f- take 7f , 

16. Minuend=l7-iV; Subtrahend=6i Rem. lOff. 

F 81 



§139 ABSTRACT NUMBERS. 



17. Minuend:=2003^; Subtrahend=105^. Rem. 94| 

18. Minuend =:42i; Subtrabend = 27yV 

19. Minuends 721; Subtrahend =24fV- Rem. 47| 

20. Minuend=l75; Subtrahend =:83|-. Rem. 91-1; 

21. Subtrahend =661-; Minuend=106^ 

22. Subtrahend = 171; Minuend = 27i Rem. 10/, 

23. Subtrahend=lf ; Minuend=4.f. Rem. 3| 

24. Subtrahends 7-«-; Minuend=8y%. 

25. Subtrahend 11-}^ ; Minuend=20f5-. Rem. 8f|f 

26. What is the difference between 12|i and 21ii? 

Ans. 8-}^ 

27. What is the difference between 16-^-|^ and 10-JvV? 

28. What is the difference between 100 and 33^? 

Ans. 66| 

29. What is the difference betwcc;. : 9} and 20yV? 

Ans. I-I- 

30. What is the difference between 75 and 68^-1 



MULTIPLIOATION OF COMMON FRA^CTIONS. 



Ex. 1. Multiply I by 7. 

§ 139. Model. — 7 times 3 are 21 : 
Ax 7= V=2| 21 eighths is equal to 2|, The prod- 
uct is 2-^r. 

Explanation. — Comparing §§ 72 and 111, we see that 
the value of a fraction is multiplied by a whole number by 
multiplying its numerator by the number. 

Ex. 2. Multiply |- by 3. 

§ 140. Model. — 3 in 9, 3 times : 5 
Ax3=f=l|- thirds is equal to If. The product is 



1^ 



82 



MULTI1>LIGATI0N OF COMMON FRACTIONS. §143 

- Explanation, — Comparing §§ 75 and 111^ we see that 
the value of a fraction is multiplied by a whole number by 
dividing its denominator by the number. When the mul- 
tiplier is a measure of the denominator, this method if 
preferable to the other. 

Ex.3. Multiply 47|- by 9. 

^'^4: § 141. Model.— 9 times 3 are 27 : #27 fourths 

^ is equal to 6-2-, set down a • 9 times 7 are 63, and 

429f G are 69, set down 9 ;. 9 times 4 are 36, and 6 

■ ^ are 42. Tbe product is 429-2. 

Explanation. — As in whole numbers, we begin with the 
lowest denomination, and reduce each partial product to 
the next higher denomination, setting down the remaining 
units of the denomination in question, and reserving the 
units of the next denomination to be added to the nest 
product. 

Ex. 4. Multiply ;] by f . 

3 7_2_i §142. Model.— 7 times 3 are 21: S 
4. X s — 3-z times 4 are 32.. The product is fi. 

Explanation. — To multiply by |- is the same as to mul- 
tiply by 7 and divide the product by 8. 7 times 3 fourths 
=:21 fourths, and 21 fourths-^8 = 21 thirty-seconds: since 
a fraction (or a quotient) is divided by multiplying the de- 
nominator (or the divisor). (§§ 74, 111). 

Ex. 5. Multiply 30,^ by l. 

op.1 ^ §143. Model.— Reduce 30i to 

^^^" ^ *' fourths.(§l26). It is equal to 121 

i|-i X i= VV =7-5%^ fourths. Once 121 is 121 : 4 times 

4 are 16. 121 sixteenths is equal 
to 7^. The product is 7^\. 

83 



U44 ABSTRACT NUMBERS. 



Ex. 6. Multiply 30^ by 5^ 

30i X 6i- ^ ^'^^' ^^^^^^- — I'^educe the 

* ^ mixed numbers to improper 

i|i X V = i-^/J = 166f fraction?. (§ 126). 11 times 

121 are 1831 : twice 4 are 8, 
1331 eighths is equal to 166^. The product is 166|. 

Explanation. — It is often easier to reduce a mixed 
iiumber to'Hin improper fraction before multiplying, if the 
other factor is not a whole number. 

Rule. — To multiply a simple fraction by a whole number; 

Divide the denominator of the fraction, or else multiply 
lis numerator^ by the whole number. 

To multiply a fraction by a fraction. 
Multiply each term of the one fraction by the correspond- 
ing term of the other. 

A mixed number may be reduced to an improper frac- 
tion, or its parts may be multiplied separately. 
Ex. 7. Reduce | of 4 to a simple fraction. 

„ . \ § 145. Model. — Twice 4 are 8 : 3 times 

'3 5" i'5 5 Q^pQ 1^ fj^i^Q given fraction is equal to -^^g. 

Explanation. — One third of 1 fifth is evidently 1 fif- 
teenth ; 1 third of 4 fifths is 4 times 1 fifteenth, that is, 4 
fifteenths ; and 2 thirds of 4 fifths is twice 4 fifteenths, that 
is, 8 fifteenths. 

Rule for reducing a compound fraction to a simple one. 
Multiply together the several fractions ivhich compose it. 
Ex. 8. Multiply a by 4. Prod. 1^. 

9. Multiply A by 7. 

10. Multiply -l by 8. Prod. 7. 

11. Multiply I by 12. Prod. 10. 

12. Multiply yV by 15. 



MULTIPLICATION OF COMMON TRACTIONS. §145 



13. Multiply -j^o by 5. Prod. ^. 

14. Multiply 2i by 7. Prou. 17l. 

15. Multiply 8-1 by 8. 

16. Multiply 16f by 15. Prod. 250. 

17. Multiply 19^. by 20. Prod. 397^. 

18. Multiply 207-;} by 13. 

19. What is the product of 315-^ and 19 ? Prod. 5995|. 

20. What is the product of -'■- and -•} 1 'Prod. f-. 

21. What is the product of | and a? 

22. AYhat is the product of J^ and ^5- 1 Prod. -}. 

23. Reduce ^ of §- to a simple fraction. Val. -*. 

24. Keduce jj of ;^ to a simple fraction. 

25. Reduce -^- of -f to a simple fraction. Val. -f-l. 

26. Reduce -} of ^l of ^ to a simple fraction. Val. ■^. 
'11 . Reduce -i- of -f of -^^ to a simple fraction. 

28. Reduce -^ of 7-^- to a simple fraction. Val. 2-}. 

29. Reduce \- of \ of 7^ to a simple fraction. Val. l^-. 

30. Reduce -"; of -^ of 8^ to a simple fraction. 

ol. Find the product of f of f and \: of 12? . Prod. 1-^^. 
32. Find the product of f of f and 83^-. Prod. 2^\, 

83. Find the product of -^- of 66f and -I of 100. 

34. Find the product of f of 250 and -f of 21. Prod. 900. 

35. Find the product of -\ of ^ of 210 and \ of 83|. 

Prod. 670^. 
SO. What is the product of 16| and 16^? 

37. What is the product of 30^ and 60|- ? Ans. 183a^. 

38. What is the product of 111-,^^ and 20-^ ? Ans. 2277^^. 

39. What is the product of 275 and "^ of a of 36? 

40. What is the product of 303 and |-of 20 ? Ans. 173^-. 

41. What is the product of 3|- and 4^4? 

42. What is the product of ^ of f and f of 3f ? 



§146 ABSTRACT NUMBERS. 



DIVISION OF COMMON FRACfTIONS. 



Ex.1. Divide i-^. by 3. 
jL^_i.3— _5_ § ^"^^^ Model.— 3 in 15, 5 times. The 



^'^ ' ^° quotient is 



1 6' 



Explanation. — Comparing §§ 73 and 111, we see that 
the value of a fraction is divided by a whole numbet* by 
dividing its numerator by the number. 

Ex. 2. Divide f by 5. 



K 3 

■^— To 



§ 147. Model.— 5 times 4 are 20. The 
quotient is 3 twentieths. 
Explanation. — Comparing §§ 74 and 111, we see that 
the value of a fraction is divided by a whole number by 
multiplying its denominator by the number. 
Ex. 3. Divide f^ by |. 

§148. Model. — 3 in 15, 5 times: 
-ff-f-f =:;f=li 4 in 16, 4 times. 5 fourths is equal 
to 1|:. The quotient is l-i. 
Explanation. — 15 sixteenths-^3=i5 sixteenths (§146): 
but the divisor 3 fourths is only one fourth of 3 ; hence the 
quotient is 4 times 5 sixteenths, that is, 5 fourths. (§ 140). 
Again, since division is the reverse of multiplication, the 
process for division should be the reverse of that for multi- 
plication : and since -|-x-f-=:-j^-g^, it is evident that -^^-^2^=-^. 
Ex. 4. Divide | by |. 

§ 149. Model.— 8 times 3 are 24 : 7 
.l-hf ?=ff =-f times 4 are 28. 24 twenty-eighths ia 
equal to f. The quotient is ^. 
ExPIiANATiON. — 3 fourths-^7=:3 twenty-eighths (§147): 
but the divisor 7 eighths is only one eighth of 7 ; hence, by 
§75, the quotient is 8 times 3 twenty-eighths, that is, 24 
twenty- eighths. (§ 139). 

86 



DIVISION OF COMMON FRACTIONS. §151 



Again, Multiplying both terms of the dividend by 56, 
we have a||-^|^= I4=t- ^^> Multiplying both terms by 
14, we have -;|^f -^|-=-f, the same result as before. 
Ex. 5. Divide 2731- by 5. 

§ 150. Model. — 5 in 27, 5 times with 2 
l?_i over, set down 5 ; 5 in 23, 4 times with 3 
54f over, set down 4; 5 in 10, twice, set down f. 
The quotient is 54f. 
Explanation. — We divide the integer as usual, and 
reduce the 3 remaining units to thirds, making 9 thirds, 
which added to the given 1 third makes 10 thirds, and this 
divided by 5 gives 2 thirds. If the numerator of \£ had 
not been divisible by 5, we would have multiplied its de- 
nominator by the divisor, as in § 147. 
Ex. G. Divide 3^- by 12f . 

^^ .193 §151- Model. — Reduce the 

3"" *-'3 given mixed numbers to improp- 

i_o_^.3.s__3^__5_ QY fractions. (§126). 3 times 

10 are 30 : 38 times 3 are 114, 
30 one-hundred-and-fourteenths is equal to 5 nineteenths. 
The quotient is 5 nineteenths, 

KuLE.— -To divide a simple fraction by a whole number. 
Divide the numerator of the fraction^ or else multiply its 
denominator J by the tchole number. 
To divide a fraction by a fraction. 

Divide each term of the dividend by the correspo/iding 
term of the divisor. Or, Multiply each term of the divi- 
dend by the other term of the divisor. 

To divide a whole number by a fraction. 
Divide the dividend by the denominator of the divisor, 
(t.nd multiply the quotient by the numerator . 

A mixed number will mostly better be reduced to an im- 
proper fraction. 

87 



§162 



ABSTRACT NUMBERS. 



Ex. 7. Reduce 



2i 



of 



to a simple fraction. 



2i-^iof A 



40. 

6 ' 






§ 152. Model. — Reduce the terms 
to simple fractions. Divide |- hy ^. 
8 times 5 are 40 : 3 times 2 are 6. — 
40 sixths is equal to 6f . The 
fraction is equal to 6|. 

Rule for reducing a complex fraction to a simple one. 
Divide its numerator by its denominator. 
Ex. 8. Divide f by 5. 
9. Divide -^ by 8. 

10. Divide if by 3 

11. Divide if by 6 

12. Divide 40 by f 

13. Divide 200 by 

14. Divide 175 by 

15. Divide \% by | 
10. Divide |-^- by f. 

17. Divide i| by f . 

18. Dividend=-^' 

19. Dividend^! 

20. Dividend=f 

21. Dividend=:y^ 

22. Divisor =f 

23. Divisor^l 

24. Divisor={r 

25. Divisors ^ 

26. Divisorz^i off: 

27. Divisors I of f : 

28. Divisor=:f of f : 

29. Divisor=f of l : 

30. Divisor=i of 12^ 



divisor =-|. 
4 : divisor=|-. 
divisor=:f. 
divisor=i. 



dividend =-^. 
dividend=^|. 
dividend ={'. 
dividend =.^. 

dividend = |. 

dividend=-f-. 

dividend = *- of ^. 

dividend=f of ^0-.. 

: dividend = 1 of y\ 

88 



given 



Quot. ^V 

Quot. -V 
Quot. -^-^. 

Quot. 466|-. 
Quot. 49. 

Quot. ia. 
Quot. |. 

Quot. 2^Q, 
Quot. i'A. 

Quot. li 
Quot. ^^, 

Quot. ,«. 
Quot. Iff. 

Quot. ^. 
Quot; 1\%. 



DIVIvSION or COMMON FRACTIONS. §152 



:;l. Dividend=:12A-: clivisor=4. Quot. 3^. 

:i2. Dividend:rz207i : divisor^G. Quot. S^-^V 

:;3. DivideDd=i45f : divisor^lS^. 

34. Dividends 70^: divisor=68^. Quot. l-^-f^. 

35. Dividends 27-^: divisor=:55i ' Quot. |-f|. 

36. Dividend^:^ of 28^ : divisori=:| of 43^}. 

37. Dividend=:|- of a ; divisor=:i of 275. Quot. ■^^^. 
:]8. Dividends 2 of i of I- : divisor=| of 17-^-. Quot. ■^. 

39. Dividend=| of 27.V: divisor=^ of 38^. 

40. Pteduce — "- to a, simple fraction. Val. 3-|. 

4." 

41. Reduce -^ to a simple fraction. Val. --^y\. 

4r2. Reduce — ^ to a simple fraction. 
41 

^r of ? 

43. Reduce " " to a simple fraction. Yal. ■^\. 

2, of 4^ 

44. Reduce ~ — — -^ to a simple fraction. Yal. i|-. 

-} ot 19 

'}. of ?- 

45. Reduce --—7-"- to a simple fraction.. 

I oi 7i ^ 

27 

46. Reduce — -.— — - to a simple fraction, Val. 2tV- 

■I of 30 ^ ' "" 

1- of 20 

47. Reduce — — — - - to a simple fraction. Val. -;;%• 

-|ofl7-i- ^ 

? of 2^- 

48. Reduce to a : iinpio fraction, 

3461 

2i- 
40. Reduce ~ to a simple fraction. Val. 1- 

1 of 27j 
50. Reduce - — rr~ to a simple fraction. Val. WA\« 

f-of72f " '''^^' 

89 



^153 ABSTRACT NUMBEHS. 



CANCELLATION. 



§ 153. In multiplication of fractions, and in some other 
similar operations, the labor may be often diminished by 
canceling all the factors common to the numerators and the 
denominators, and afterwards multiplying together the re- 
maining factors of each. This is simply reducing the re- 
sult to its lowest terms in advance. 

It is customary to draw a line through a number that has 
"been canceled. 

Ex. 1. Multiply U by Ti^. 

^ 7__^ Model. — 45 in 45, once; 45 in 90, 
f^' pfp"~14 twice: 7 in 7, once; 7 in 49, 7 times :- 
7 2 the numerator is 1 ; the denominator is 

7x2 = 14. The product is ^L. 

Ex. 2. Divide I- of j- of f by f of if of ^V- 

i of f of i---f of i| of ^\ Model.— 3 

^ in 3, once ; 3 

- of 2 of '? X ^ of ?^ of — =.. «J- = 10-^- ^^ ^' ^^^^® • ^ 
27^^^/^ 1 « ^ in7, once;7 

2 2 in 21, 3 times: 

9 in 9, once ; 
9 in 9, once : 5 in 5, once ; 5 in 10, twice : the numerator 
is 3 X 27=81 ; the denominator is 2.2.2=8. The quotient 
isV = 10i. 

Ex. 3. Divide the product of 77 and 96 by the product 
of 22 and 24. 

2 Model.— 11 in 77, 7 times ; J 1 in 22^ 

^ /¥ twice : 2 in 2, once ; 2 in 96, 48 times • 

Ilj^.^14, 24 in 24, once; 24 in 48, twice. The 

^^ X ^4' ^ quotient is 7 x 2= 14. 



? 



90 



PROMISCUOUS PP.OBLEMS. SI 53 



Ex. 4. Divide 11 x 21 x 26 by 3 x 13 x 14. 

^^^Jy^^li Model.— 3 in 3, once; 3 in 21, 7 
11 X // ^xjg times : 7 in 7, once ; 7 in 14, twice : 13 
^x/^x/^ in 26, twice; 13 in 13, once: 2 in 2, 
^ once ; 2 in 2, once. The quotient is 11 
Es. 5. Multiply A of i-f- by -J^ of f ^^ of if. Prod. ^ 

6. Multiply -V of I- of if by I of -y-. 

7. Multiply f of V by a of if. " Prod. ^'^ 

8. Divide ^ of AA by V of V of -^V- Quot. i 

9. Divide f-t of if by -Jl. of ^. 

10. Divide A of A of I by"A of aa. Quot. 2i 

11. Divide the product of 22 and 56 by the product of 44, 

28, and 16. Quot. -,„ 

12. Divide the product of 72 and 96 by the product of 60 

and 64. 

13. Divide the product of 27, 2S, and 29 by the product of 

35, 86, and 37. Quot. -/a.. 

14. Divide 10 X 11 x 12 by 22 x 24 x 30. Quot.' J-^. 

15. Divide 25 x 27 x 32 x 36 by 15 x 18 x 24 x 28. 



PROMISCUOUS PROBLEMS. 



1. What is the sum of 275, 386, 497, and 608 ? 

2. What is the difference between 275386 and 497608 ? 

Ans. 222222. 

3. What is the product of 275386497 and 608 ? 

4. What is the quotient of 275386497 by 608 ? 

Ans. 452938J-AA. 
-J. Add the difference between 395 and 4:22 to the sum of 
39, 54, and 202. . Sum, 3 

91 



§163 ABSTRACT NUMBERS. 



(>. Subtract the sum of 25 and 19 from their product. 

7. Multiply ibe difference of 25 and 19 by tlieir sum. 

Prod. 264.. 

8. Divide the product of 36 and 45 by their difference. 

Quot. 180. 

9. Resolve 7050 into its prime factors. 

10. What is the greatest common measure of 25, 250, and 

375 ? " Ans. 25. 

il. What is the least common multiple of 5, G, 10, and 12 ? 

Ans. 60. 
1-.'. Reduce -rV-^V to its lowest terms. 

i z o u 

13. In "^"^ how many units ? Ans. 10-^^. 

14. In 19 units how many nineteenths ? Ans. ^^. 

15. In 15|- how many fifths ? 

16. In f how many forty-fifths ? Ans. -|-|.. 

17. In -^q\ how many twenty-fifths ? Ans. -^V- 

18. Reduce -^, -^j and {- to a common denominator. 

19. Reduce ^, |, and {^ to their least common denominator. 

Vol 2 7 .TO 2_5. 

20. Add -5^, Jj,-, and i}. Sum, 1. 

21. What is the sum of } of *- and ^ of i ? 

2^ 

22. What is the sum of f of 10-^ and — -^— , ? Ans. 7/,%. 

•" "* i of 17 

23. What is the difference between 19^- and 26^ ? 

Ans. 6-1-^. 

24. What is the difference between a of 27 and -^- of 24? 

25. What is the product of 27i.and -v of 77 ? Ans. 706|-. 

26. What is the product of '^-—^rr- — - and — --- '/ Ans. 1. 

^ 22^ .} or 11 

27. What is the quotient of I of 47 by 25^ ? 

28. What is tlie product of 5- of 47 and ^ of 25 ? 

Ans. 43-1-*-. 
93 



I'LlOiVllSCUO'tJS PROBLEMS. §l5o 

-I- of 27.^ ^ of 19 , 
29. Wlhit h the quotient of ^—xz — - by -— ^ ■ ? 

Aids. 3,W.V 
SO. Add the product of -^ of 27 and ~l of ^ to their difference. 

31. Subtract the quotient of -f- of 45 by -} of 24 from their 
sum. Eem. lO^'j. 

A of 5- 

32. Multiply the sum of ^" " and ? of 7'Sl by their dit- 

ference. Prod. 262-jyVc- 

33. Divide the product of 25^- and 17^,- by their sum. 

34. What number is that to which if 3-§- , 5^^-, 6f, and 10-/^, 
be added, the sum will be 30-]j- ? Ans. 3-^-. 

35. What number is that tVoni which if of, of, 6^^, and 10^%, 
be subtracted, the remainder will be 30-^ ? Ans. 57-|-. 

36. What number is that by which if the sum of 3f, 5|^ 
6{?, and 10y%, be multiplied, the product will be 30-|-? 

37. What number is that by which if the sum of 3f, 5f, 

6|, and lOy'^;-, be divided, the quotient will be 30-1- ? 

Ans -^-i- 
88. What is the sum of -}, ,|, 13, and IS^V ? 

39. What is the difference between -} and -^\1 Ans. y%. 

40. What is the product of 5^^ and ^ ? 

41. What is the quotient of -ff ff by 19 ? 

42. 3^-5-7-^5 + 16-^7-15'^6=what? Ans. 6|-^. 

43. (3 + 5-7)~5 + 16-^7-15-=-6=what? 

44. 3 + (5 + 7)~5-fl6-^7-15-J-6=what? Ans. 5-^-§.. 

45. 3 + 5-7-^6-fl6 + 7-15-^6=what? Ans. 27yV- 

46. (3 + 5-7)-^-5 + (16 + 15-7)-f-6=what? 

47. 3 + (5 + 7)-^5 + 16+(15-7)-^6=:what? Ans. 22H-. 

48. 3 + (7-5)-^(5 + 16) + (15-7)-^6=what? Ans. 4f 

49. 34.7_5^(5_l_16 + 15)-7-f-6=what? 

93 



§154 ABSTRACT NUMBERS. 



DECIMAL FRACTIONS. 



§ 154. A decimal fraction is one whose denominator is 
some poivcr of ten and is not expressed in writing. 

§ 155. In the Arabic or decimal system of notation (§10), 
we observed that, in passing from the units' place to the 
left, a unit of any order is ten times a unit of the preceding 
order ; or that, in passing from left to right, a unit of any 
order is 07ie tenth of a unit of the preceding order. If this 
law be extended to the right of units, the next order will 
be tenths, the next hundredths, the next thousandths, &c., 
as in the following 



■ . ^ ti -a 

^ » I t2 1 S r 2 S S ^ i ^ ^^ 

23 45.234576595875 

As 100 is 3-V of 1000, 10 is 3-V of 100, and 1 is J^ of 10, 
so one tenth is -^ of 1, one hundredth is -=^^ of 1 tenth, one 
thousandth is J^ of one hundredth, &c. 

§ 156. To write any number of tenths, then, we simply 
put the proper figure one place to the right of units ; for 
hundredths, we put the figure two places to the right, &c. 
To determine the position of units and the relative posi- 
tions of the fractional orders, we place a period, called the 

94 



NOTATION OF DECIMAL FRACTIONS. §158 



units^ point, between units and tenths ; or to the left of 
tenths, if the expression is entirely fractional. Thus, 2.3, 
two and three tenths ; 3.02, three and two hundredths ; 
5.32, five and three tenths and two hundredths ; .005, five 
thousandths ; .0006, six ten-thousandths ; .00004, four 
hundred-thousandths; .000008, eight milliontht-. 

In integral numbers, this point, being unnecessary, is 
never written : but in fractional or mixed expressions, it 
must never be omitted. 

§ 157. It will be observed that the number of places oc- 
cupied by the numerator of a decimal fraction is equal to 
the number of naughts in its denominator. If the ordinary 
exprei?sion cf the numerator does not require so many places, 
each place intervening between the units' point and the left 
hand figure of the numerator must be filled with a naught. 
Thus, .002, 2 thousandths; .023, 23 thousandths ; ,0203, 
203 ten-thousandths ; .0023, 23 ten-thousandths ; .0004, 4 
ten-thousandths ; .002034, 2034 millionths. 

§ 158. A decimal fraction is read, like a common frac- 
tion, by pronouncing after the numerator the ordinal of the 
denominator. Sometimes, in reading a mixed number, to 
prevent ambiguity, it is necessary to pronounce the word 
"units" after the integer. Thus, three hundred and fif- 
teen thousandths is written .315 ; but 300.015 is three hun- 
dred units and fifteen thousandths : so, 7000.0275 is read 
seven thousand units and two hundred and seventy-five ten- 
thousandths. 

Read the following decimal fractions: — .1, .3, .5, .7, .8 • 
.01, .05, .09, .11, .25, .34, .47, .51, .63, .75, .87, .99; .001^ 
.005, .015, .025, .075, .125, .219, .375, .487, .567, .605, .777, 
.808, .999; .0001, .0012, .0125, .1275, .3525, .6225, .7203, 

95 



§158 ABSTRACT mTMBERS. 



.8007, .9883, .9999 ; .00001, .00014, .00225, .03275, .33125, 
.42075, .53003, .70007, .87078, .99999 ; .000001, .000017, 
.000175, .003175, .063175, .475327, .796305, .634008, 
.320075, .200017, .200325; .0000001, .0000025, .0000275, 
.0020705, .0357675, .7500786; .00027625, .02700625, 
.23450275, .00073513, .23570025, .125346798, .000000125, 
.0007600025, .27340709025, .70030005345, .000257025702. 



Read the following mixed numbers : — 3.3, 70.5, 35.7, 
2.02, 3.25, 75.75, 24.05, 7.07, 30.003, 400.025, 25.125, 
375.375, 1.001, 2.325, 2.0275, 300.0025, 17.0017, 1.0005, 
2000.0002, 21.2125, 325.03725, 9180.20025, 1000.02207, 
7025.00025, 6278.374375, 2000.0002325, 3375.00000765, 
27.0000027, 3200.000000075, 2500,0000036975. 

Write tlie following in figures : 

110. Seventy-four hundredths. 

111. Four hundred and forty-eight thousandths. 

112. Five hundred units and three hundredths^. 

113. Seventy-five thousandths. 

114. Five hundred and three thousandths. 

115. Five hundred units and three thousandths. 

116. Three hundred and twenty-seven ten-thousandths. 

117. Three hundred units and twenty-seven ten-thousandths. 

118. Seventeen and seventeen hundred-thousandths. 

119. One thousand units and two thousand two hundred 
and seven hundred-thousandths. 

120. Three thousand two hundred units and seventy-five 
millionths. 

121. Six hundred and three ten-thousandths. 

122. Two thousand four hundred and sixty-one and three 
hundred and nineteen millionths. 

96 



ADDITION OF DECIMAL FRACTIONS. §160 



ADDITION OF DECIMAL FRACTIONS. 



Ex. 1. Add .3, .23, .175, and .025. 

3 

*23 § 159. Model. — 5 and 5 are ^0 ; 1 and 2 

175 ^'^^ ^' ^^^ ^ ^^^ ^^' ^^^ ^ ^^^ ^^' ^^^ down 3 ; 

025 ^ ^"^ ^ ^^® "' ^^'^ ^ ^^'^ ^> ^^^ '^ ^^^ ^' — 
'-— Point before 7. The sum is .73. 

Explanation. — Beginning at the right, we find the sum 
of the first column to be 10 thousandths, equal to 1 hun- 
dredth exactly. We do not set down the naught here, be- 
cause a naught at the right of a decimal fraction does not 
issist in determining the orders of the other figures. The 
1 hundredth is added in with the column of hundredths, 
which amounts to 13 hundredths, equal to 1 tenth and 3 
hundredths. Setting o under the column of hundredths, 
we add the 1 tenth in with the column of tenths. We then 
place the units' point at the left of the tenths. (§ 156). 

Ex. 2. Add .3, 3.5, 3.15, 35.25, and 171.275. 

§ 160. Model.— 5 ; 7 and 5 are 12, and 

•'^ 5 are 17, set down 7 ; 1 and 2 are 3, and 2 

'z'^ are 5, and 1 are 6, and 5 are 11, and 3 are 

oi'ot -^-^J ^^^ down 4 ; 1 and 1 are 2, and 5 are 7, 

171 07f^ and 3 ar6 10, and 3 are 13, set down 3 ; 1 

^'^'^'^ and 7 are 8, and 3 are 11, set down 1 ; 1 

213.475 and 1 are 2. Point before 4. The sum is 

213.475. 

Explanation. — The sum of the column of tenths being 
14, that is, 1 unit and 4 tenths, we set 4 under the column 
of tenths, and add 1 to the column of units. We place the 
units' point between the units and the tenths. (§ 156). 

a 97 



§160 ABSTRACT NUMBERS. 



Rule. — Arrange the numbers with units of the same order 
in the same column ; and add as in luholc numbers. (§ 22). 

Place the units^ point on the left of the tenths figure in 
the sum. 

Proof. — The same as in whole numbers, (§ 22). 

Ex. 3. Add 1.2, 3.56, 45.67, and 56.789. 

4. Add 1.3, 5.79, 24.68, and 90.275. Sum, 122.045. 

5. Add 27.72, 365.9, 125.008, and 236.115. 

Sum, 754.743. 

6. Add 135.709, 246.008, 145.008, and 236.709. 

7. Add 1.35795, 135.795, and 13579.5. 

8. Find the sum of 2.465, 25.G09, 100.206, and 146.27. 

Sum, 333.950. 

9. Find the sum of 100.0001, L- 1.4012, 412.5124, and 

421.5214. 

10. Find the sum of 1234.58,78.9012,3456.789, 10.234567, 

and 890.13575. Sum, 5670.020517. 

11. Find the sum of 907.0503, 890.7054, 785.4321, and 

25.457. Sum, 2608.6448. 

12. Find the sum of 12.012575, 120.125725, 1201.257725, 

and .270825. 

13. Find the sum of .760027, .000176, .012012, and 

.027945. Sum, .800160. 

14. What is the sum of .230495, .341507, .452618, and 

.563729 ? Ans. 1.588349. 

15. What is the sum of 2.30495, 34.1507, 452.618, and 
5637.029 ? 

16. What is the sum of 12.000012, 250.0025, 75.075, and 

175.0175 1 Ans. 512.095012. 

17. 175 + 6.115 + 123.1341 + 172.21275 + 5637.175:^ what? 

Ans. 6113.63685. 
98 



SUBTRACTION OF DECIMAL FRACTIONS. §161 



18. 52.8G72 + 549.72-f927.365 + 57.10715 + 13.575=wliat? 

19. 79.105 + 131.187 + 19.4201 +2643.13 + 34.8H6-t = whai? 

Ans. 2907.6785. 

50. 3844.04 + .444584 + 6.14644 + 6847.34 + 77.9899= 

what ? Ans. 10775.960924. 



SUBTRACTION OP DECIMAL FRACTIONS. 



Ex. 1. From 275.075 take 87.1275. 

^P'^I-- § 1^1- Model.— 5 from 10 leaves 5 ; 8 

"'• ^-'^^ from 15 leaves 7 ; 3 from 7 leaves 4 ; 1 

187.9175 from 10 leaves 9 ; 8 from 15 leaves 7; 9t 

from 17 leaves 8 ; 1 from 2 leaves 1. Poiut 

before 9. The remainder is 187.9475. 

Explanation. — After placing the subtrahend under the 
minuend with units of the same order in the same column, 
we find 5 ten-thousandths in the subtrahend and no ten- 
thousandths in the minuend. Adding 1 thousandth, that 
is, 10 ten-thousandths, to the minuend, we subtract from 
this the 5 ten-thousandths of the subtrahend. Then, be- 
cause the minuen'd is increased 10 ten-thousandths or 1 
thousandth, the subtrahend must be increased the same 
amount. (§28). The same kind of reasoning will explain 
the rest of the operation. We place the units' point be- 
tween the units and the tenths. (§ 156). 

Rule. — I^lace ike subtrahend under (he minuend, with 
*units of the same order in the same column, and subtract as 
In whole numbers. (§30). 

Place the unita^ point on the left of the tenths figure in the 
remainder. (|^ 156). 

Proof. — The same as in whole numbers. (§ 30). 

99 



S161 ABSTRACT JSUMBER.^. 



Ex. 2. From 8.96 take 8.07. Rem. 5.89. 

3. From 2.719 4:ake 1.827. 

4. From 97.8637 take 9.7863. Rem. 88.0774. 

5. Take 67.8902 from 896.454. Hem. 828.5638. 

6. Take 17.24937 from 1963.869. 

7. Take 234.68579 from 6005.004. Rem. 5770.31821, 

8. Take 98.79789 from 99.000099. Rem. .202209. 

9. Minuend is 1284.567 ; Subtrahend is .76542„ 

10. Minuend is 29017.05 ; Subtrahend is 10.8405. 

Rem. 29006.2095. 

11. Minuend is 2098.76 ; Subtrahend is 454.698. 

Rem. 1644.062. 

12. Minuend is 1201.257725 ; Subtrahend is 120.125575. 

13. Subtrahend is .012095; Minuend is .027945. 

Rem. .01585. 

14. Subtrahend is 2.30495 ; Minuend is 34.1507. 

Rem. 31.84575. 

15. Subtrahend is 12.000012 ; Minuend is 250.0025. 

16. Subtrahend is 75.075 ; Minuend is 175.0175. 

Rem. 99.9425. 

17. 5637.175-I72.2l275=:what? Ans. 5464.96225, 

18. 927.305-57.190715=what? ' 

19. What is the difference between one millionth, and 
ninety-nine thousandths ? Ans. .098999. 

20. What is the difference between thirty-seven billionths, 

and one hundred and eleven thousandths ? 

Ans. .110999963. 

21. What is the difference between six billionths, and nine 
hundred and ninety-nine thousandths ? 

22. What is the difference between three millionths, and 
three hundred and six thousandths ? Ans. .305997. 

100 



MULTIPLICATION OF DECIMAL FRACTIONS. §16S 



MULTIPLICATION OF DECIMAL FRACTIONS. 



Ex. 1. Multiply 5.3 by 6.25. 

6 25 
CO § 162. Model. — 3 times 5 are 15, set down 

— -— 5 ; 3 times 2 are 6, and 1 are 7 ; 3 times 6 
^^1^ are 18: — 5 times 5 are 25, set down 5 under 
^^^^ 7 ; 5 times 2 are 10, and 2 are 12, set down 2 ; 
o3.125 5 times 6 are 30, and 1 are 31: — add the par- 
tial products : 5 ; 5 and 7 are 12, set down 2; 
1 and 2 are 3, and 8 are 11, set down 1 ; 1 and 1 are 2, 
and 1 are 8 ; 3. Point before 1. The product is 33.125. 
ExrLANATiON. — Reducing both factors to improper frac- 
tions, and multiplying as in § 142, we have -^-f^ x {-^= 'VoW? 
and this product reduced to a mixed number becomes 
S3. 125. as in the model. If any decimal mixed number be 
reduced to an improper fraction, the numerator will con- 
sist of the same figures as the given mixed number. Hence 
we multiply as in whole numbers. The location of the 
units' point in the product is found by observing that the 
number of naughts in the denominator of either factor is 
the same as the number of figures in the numerator, aijd 
that the product of any two powers of ten is obtained by 
annexing to 1 as many naughts as there are in both factors 
togetiier. There are, therefore, as many fractional figures 
in the product as in both factors together. 

Ex. 2. Multiply ,15 by ,3, 

•^^ § 163. Model. — 3 times 5 are 15, set down 5 ; 

3 times 1 are 3, and 1 are 4. Prefix one naught. 

.015 Point before 0, The product is .045. 
Explanation. — When the product does not contain 
onough figures to express its proper denomination, we pre- 
fix one or more naughts to supply this deficiency. 

101 



>164 ABSTRACT NUMBERS. 



Rule. — Midt^i-ily as in loJioIe numhers, and ^Doint off ai> 
many fractional Jigiires in the product an there arc in hotU 
the facfon^y prc/?x?*wr7 navghts lohen necessary to mahe up 
the number. 

Proof. — The same as in whole numbers, (§ 40). 

Ex. 3. Multiply 12.42 by 3.2. 

4. Multiply 25.25 by 2.5. ProJ. 63.125. 

5. Multiply .25 by .25. Prod. .0625. 

6. Multiply 5.5 by 5.5. 

7. Multiply 211.79 by 2.7. ' Prod. 571.833. 

8. Multiply 97.825 by .34. Prod. 33.2605. 

9. Multiply 275.005 by 5.005, 

10. Multiply 869.06 by .045. Prod. 39.1077. 

11. Multiply 27.9362 by .0052. Prod. .14526824. 

12. 192.837x6.7== what? 

13. 293.705 X .075=what ? Ans. 22.027875. 

14. 3.047 x2.87=what? Ans. 8.74489. 

15. 2.975 x.375=what? 

16. 4.027 X 402.7== what ? Ans. 1621.6729- 

17. What is the product of 247.742 and 10.035 ? 

18. What is the product of 307.0005 and .000375? 

19. What is the product of 175.025 and 25.0175 ? 

Ans. 4378.6879375.* 
.20, What is the product of 1200.375 and 162.625 ? 



DIVISION OF DECIMAL FRACTIONS. 



Ex, 1. Divide 2.25 by .3. 

.3)2^ § 164. Model.— 3 in 22, 7 'times, with 1 
7.5 over, set down 7 ; 3 in 15, 5 times. Point 
before 5. The quotient is 7.5. 
102 



DIVISION OF DECIMAL FRACTIONS. 



§166 



Explanation. — Since the divisor and the quotient arc 
factors of the dividend, there must be as many fractional 
figures in the dividend as there are in both the factors. 
(§ 162). Hence, to find the number of fractional figures in 
the quotient, we subtract the number in the divisor from 
the number in the dividend. 



26.40 
2475 



3.2 



1650 
1650 





Ex. 2. Divide 26.4 by 8.25. 

on-' §165. Model. — Annex one naught to 
the dividend : 8 in 26, 3 times ; multiply 
the divisor by 3 ; 15, 7, 24 ; subtract the 
product from the dividend ; 5, 6, 1 ; annex 
: 8 in 16, twice ; multiply the divisor by 
2 ; 10, 5, 16 ; subtract the product from 
the previous remainder ; 0. Point before 
2. The quotient is 3.2. 

Explanation. — As the number of fractional figures in 
the divisor exceeds the number in the dividend, we annex 
a naught to the dividend to make them equal. We after- 
wards find it necessary to annex an other naught to com- 
plete the division. This makes 3 fractional figures in the 
dividend ; and, as there are 2 in the divisor, there must be 
one in the quotient. 

Ex. 3. Divide 4 by 15. 



4.00 
30 



15 



.266 + 



100 
90 



100 
_90 

10 



§ 166. Model. — Annex 2 naughts to 
the dividend ; 15 in 40, twice ; multiply 
the divisor by 2 ; 10, 3 ; subtract the 
product from the dividend ; 0, 0, 1 ; 15 
in 100, 6 times; multiply the divisor by 
6 ; 30, 9 ; subtract the product from the 
previous remainder; 0, 1; annex 1: 15 
in 100, 6 times ; &c. Point before 2, — 
The quotient is .266 + . 



103 



§166 ABSTRACT NUMBERS. 



Explanation. — Since the dividend can be extended only 
by annexing naughts, it is evident that, if the same remain- 
der should occur twice in succession, the same quotient 
figure will occur and will give rise to the same remainder 
again ; so that the same circuit of operations will occur per- 
petually. In such cases the quotient can not be obtained 
exactly, but we can always make an approximation suffi- 
ciently near for any practical purpose. 

Rule. — Divide as in ?'-7'o^/^ numbers, and point off as 
tnamj fractional figures in 6-.... (quotient as the numher in the 
dividend exceeds the numher in the divisor, ijrefixtng naughts 
when necessary to make up the numher. 

If the nmnher of fractioiial figures in the divisor exceeds 
the numher in the dividend, annex to the dividend as many 
naughts as may he necessary to make the numher in the divi- 
dend at least equal to the number in the divisor. 

Note. — When the division can not be exactly performed, 
we put the sign + at the right of the quotient. 

Proof. — The same as in whole numbers. (§ 53). 

Ex. 4. Divide 1728 by .12.' 

5. Divide 1728 by 1.2. 

6. Divide 172.8 by 12. 

7. Divide 17.28 by 12. 

8. Divide 13 by 245. 

9. Divide 2.7 by 900. 

10. Divide 189.75 by .759. Quot. 250. 

11. Divide 84.099 by .097. Quot. 867. 

12. Dividend is 4435.2, divisor is .84. 

13. Dividend is .8928, divisor is 1.24. Quot. .72. 

14. Dividend is 7049.754, divisor is 8.7034. Quot. 810. 

15. Dividend is 2.4416, divisor is 43.6. 

104 



CONTRACTED DIVISION OF DECIMAL FRACTIONS. §168 



16. Divisor is 47, divideDd is 22.09. 

17. Divisor is 18.07, dividend i^ .12649. 

18. Divisor is 180.7, dividend is .012649. 

19. Divisor is .125, dividend is 2.25. 
L'O. Divisor is 18, dividend is 19. 



Quot. .47. 
Quot. .007. 

Quot. 18. 



CONTRACTION IN MULTIPLICATION. 



Ex. 1. Multiply 23.25 by 10. • 

§ 167. Model. — Remove the 
23.25 X 10rr232.5 point one place to the right. 

The product is 232.5. 

Explanation. — To multiply by any power of ten, we 
simply remove the units' point as many places to the right 
as there are naughts in the multiplier, annexing naughts 
when necessary. See § 155. 

Ex. 2. Multiply 232.5 by 100. 

3. Multiply 10.25 by 1000. 

4. Multiply 246.25 by 100. 

5. Multiply 875.275 by 10. 

6. Multiply 96.0025 by 10000. 

7. Multiply .0025 by 1000. 

8. Multiply .0007 by 100000. 

9. Multiply .05 by 1000000. 
10. Multiply .0065 by 10000. 



Prod. 23250. 

Prod. 24625. 
Prod. 8752.75. 

Prod. 2.5. 
Prod. 70. 



Prod. ()0' 



CONTIUOTION IN DIVISION. 

^■ 

Ex. 1. Divide 23.25 by 10. 

§168. Model. — liemove the 
23.25 -f-lOmz 2.325 point one place to the left. The 

(juotient is 2.325. 
105 



^169 ABSTRACT NUMBERS. 



Explanation.^— To divide by any power of ten, we sim- 
ply remove the units' point as many places to the left as 
there are naughts in the divisor, prefixing naughts when 
necessary. See §155. 

Ex. 2. Divide 2.325 by 100. Quot. .02325. 

3. Divide 10.25 by 1000. 

4. Divide 246.25 by 100. Quot. 2.4625. 

5. Divide 875.275 by 10. Quot. 87.5275. 

6. Divide 9G.0025 by 10000. 

7. Divide 2500 by 1000. Quot. 2.5. 

8. Divide 7000 by 100000. Quot. .07. 

9. Divide .05 by 1000000. 

10. Divide .0065 bv 10000. Quot. .00000065. 



■<©>- 



RELATIONS OF COMMON AND DECIMAL 
FRACTIONS. 



§ 169. Every decimal fraction may be expressed in the 
form of a common fraction by simply removing the units' 
point, writing the denominator under the numerator, and re- 
ducing, if necessary, to its lowest terms. Thus, .5=-!%-=:-^. 
Also, .25=-^\\=l:. 

Ex. 1. Reduce .375 to a common fraction. Val. |. 

2. Reduce .625 to a common fraction. Val. ^. 

3. Reduce .1875 to a common fraction. 

4. Reduce .3125 to a common fraction. Val. -j^, 

5. Reduce .05 to a common fraction. Val. -^^. 

6. Reduce .0015 to a common fraction. 

7. Reduce 00025 to a common fraction; Val. ^o^. 

106 



COMMON AND DECIMAL FRACTIONS. . §171 



8. Reduco .004375 to a common fraction. Val. xoVo- 

9. Reduce ,08125 to a common fraction. 

10. Heduce .0175 to a common fraction, Yal. ^^-^^j. 

§170. If the denominator of a common fraction has no 
other prime factor than 2 or 5, it may be reduced to a dec- 
imal form by multiplyiog both its terms by such a number 
as will make the denominator a power of ten, removing the 
denominator, and putting the units' point at its proper place 
in the numerator. Thus, multiplying both terms of -I by 
25, we have -jY^j which may be written, .25. 

Ex. 11. Heduce | to a decimal fraction. Yal. .4. 

12. Reduce {: to a decimal fraction, 
18. Reduce f to a decimal fraction. Val. .625. 

14. Reduce ./^ to a decimal fraction, Val. .35. 

15. Reduce ^}| to a decimal fraction. 

16. Reduce l^ to a decimal fraction. Val. .475. 

17. Reduce ,.^^y to a decimal fraction. Val. .0375, 

18. Reduce '^-Jj to a decimal fraction. 

19. Reduce -^\ to a decimal fraction. Val. .5625. 

20. Reduce -^^ to a decimal fraction. Val. .09375. 

§ 171. If the denominator of a common fraction has nei- 
ther 2 nor 5 as a prime factor, it cannot be reduced to a 
decimal form. We can make an approximation, however, 
sufficiently near for all practical purposes, by the following 
plan. Taking the example of last section, if we multiply 
both terms of l by 100, we have i^, and then dividing 
both terms by 4, we have -^^^q, that is, .25. In other words, 
A common fraction is reduced to a decimal form hi/ dividing 
its numerator hy its denominator, (§ 166). This is the gen- 
eral rule, and is but a repetition of what we learned in 
§ 113. But let us attempt to apply this rule to the frac- 

107 



§172 ABSTRACT NUMBERS. 



tlon |-. I>ividing, we have 3 in 20, 6 times, with 2 over ; 
again, annexing an other naught, we have o in 20, 6 times, 
with 2 over ; and so on, evidently forever. Again, reduce 
-i\ to a decimal form. Dividing, we have, ll^n 20, once, 

with 9 over ; 11 in 90, 8 times, with 2 
11)2.0000 over; 11 in 20, once, with 9 over, 

.1818+ again ; and 11 in 90, 8 times, with 2 
over, again; and so, evidently, these 
two quotient figures might be repeated to the end of time. 
§ 172. Such expressions as these are called ^wre repctends, 
and they are denoted by placing a dot over the repeating 
figure when there is but one, or by placing dots over the 
first and last repeating figures when there are several. 
Thus, 1^.6; xV-.iS; fff=.275. 
Ex. 21. Reduce ^ to a repetend. 

22. Reduce f to a repetend. Yal. .285714. 

23. Reduce y\- to a repetend. Yal. .2*7. 

24. Reduce -^-^ to a repetend. 

25. Reduce xV ^o ^ repetend. Val. .2941176470588235. 
§ 173. If the denominator of a common fraction has ei- 
ther 2 or 5 or both, and other prime factors, the quotient 
of its numerator by its denominator will be partly a deci- 
mal fraction and partly a repetend. 

Thuo', |=:.8333 + , or .83. Also, ^^j=A1666 + , or .416. 
Also, -,\-=.2083 ; and -^^=.104<16. 

These expressions are called Tnixed rcpetends. 

Ex. 26. Reduce -^^ to a mixed repetend. Val. .585. 

27. Reduce ^ to a mixed repetend. 

28. Reduce -^^ to a mixed repetend. yal.'.2i4285t. 

29. Reduce -^-^ to a mixed repetend. Val. .46. 

80. Reduce |J- to a mixed repetend. 

108 



COMMON AND DECIMAL J-RACTIOJSS. §176 



§174. To reduce a jJiif'c^ rrpetcnd to a common fraction, 
toe remove the units' point, vjrite for denomhiator as m'jny 
nines as there are repeating figures, and reduce the result to 
lis lowest terms. 

Again, ,,V=-Oi. -^=.05, U=AO, ^ = .25, f^-.50: 
Also, -,J^:= .001, WV=.010, ^=.075, ^^i]=:.275, &c. 

§ 175. From these facts we learn that a pure rcpeteiid is 
read by pronouncing after its numerator the ordinal of the 
number formed of as many nines as there are figures in the 
repetend. Thus, .1=1-, .5^'7=|-^, &c. 

Ex. 31. Reduce .27 to a common fraction. ^. 

82. Reduce .72 to a common fraction. . -=& 



IT* 



33. Reduce .36 to a common fraction. 

34. Reduce .185 to a common fraction. -^. 

35. Reduce .270 to a common fraction. -^j-, 

36. Reduce .7^2 to a common fraction. 

37. Reduce .801 to a common fraction. -^^-^^-^ 

38. Reduce .9001 to a common fraction. uf-p* 

39. Reduce .8877 to a common fraction. 

40. Reduce .9765 to a common fraction. i-i-fi- 
§ 176. A mixed repetend is a complex fraction, having 

for its denominator some power of ten, and for its numera- 
tor a mixed number: the fractional part of the mixed num- 
ber having for its denominator a saries of nines. 

83 41 « '2083- 

Thus,.85is--;.416is-^;.2083is^». 

To reduce a mixed repetend to a simple common fraction, 

we must first reduce the numerator to an improper fraction. 

This makes it necessary to multiply the integral part by 9 

or by a series of nines ; and this multiplication can be most 

109 



§176 ABSTRACT NU31BERS. 



readily accomplished by §60. Take tLe second of tbe 
above examples, for instauce. Annexing one naught to 41, 

and subtracting 41 from 

410 369 ^^100=^"^^ *^^ result, we have 369 

_il __^ ~0~ ' ~"900 as the product of the in- 

369 375 tegral partby the denom- 

9 inator. To this product 

adding the numerator 6, 
we have 375 as the numerator of the improper fraction. 
Dividing 2.I-A by 100, we find -|-^, which should then be re- 
duced to its lowest terms. 

This result could be more easily obtained by f^uhtracting 
the decimal ])art from, the whole repctend for the numerator , 
and hij taking for the denominator as many nines as there 
are repeating' fgiircs, followed hy as many naughts as. there 
are decimal figures. 

Thus, .83r=i^=:f ; .2083=(2083-208)=i§^^=:^ 
Ex. 41. Pteduca .123 to a common fraction. t^- 

42. Keduce .50t5 to a common fraction. 

43. Reduce .779t to a common fraction. -^-|-|. 

44. Reduce .1*76 to a common fraction. -^^. 

45. Keduce .4554 to a common fraction. 



PROMISCUOUS PROBLEMS. 



1. What is the sum of 247 millionths, 26 ten-thousandths, 
163 hundred-thousandths, 3 thousandths, and 19 hun- 
dredths ? Ans. .197477. 

2. What is the difi'erence between 19 units and 19 mil- 
lionths? Ans. 18.999981. 

110 



PROMISCUOUS PROBLEMS. §176 



3. What is the product of 273 thousandths and 11*7 ten- 
thousandths? 

4. What is the quotient of 17 ten-thousandths by 16 hun- 
dredths ? Ans. .10625. 

5. What is the Bum of the product of 5 tenths and 5 hun- 
dredths, and the quotient of 5 tenths bj 5 hundredths ? 

Ans. 10.025. 
G, What is the difference between the sum of C hundredths 
and 6 units, and the product of 6 hundredths and 6 
units? 

7. What is the product of the sum of 12 thousandths and 
84 hundredths, and their difference? Ans. .115456. 

8. What is the quotient of the product of 506 thousandths 
and 78 hundredths by their sum? Ans. .306905 + . 

9. Add 27 hundredths, 538 thousandths, C4 ten-thou- 
sandths, and 9768 ni'llionths. 

10. Subtract the product of 39 hundredths and 54 thou- 
sandths from their sum. E-em. .42294. 

11. Multiply the quotient of 36 hundredths by 45 ten- 
thousandths by their difference. Prod. 28.44. 

12. Divide the sum of 497 thousandths and 608 ten-thou- 

sandihs by their difference. 

13. What number is that to which if 13 hundredths, 13 
thousandths, 13 ten-thousandths, and 13 millionthsbe 
added, the sum will be 13 units? Ans. 12.855687. 

14. What number is that from which if 11 hundredths, 12 
thousandths, 13 ten-thousandths, and 14 hundred-thou- 
sandths be subtracted, the remainder will be 15 mil- 
lionths? Ans. 123455. 

15. What number is that by which if 79 thousandths be 
multiplied, the product will be 54115 billionths ? 

Ill 



^176 ABSTRACT NUMBERS. 



IG. What number is that bj whiich if 6375 millionths be 
divided, the quotient will be 5 thousandths ? 

Ans. 1.275. 

17. The subtrahend is 25 ten-thousandths, the minuend is 
2 tenths; what is the remainder ? Ans. .1975. 

18. The subtrahend is 25 thousandths, the remainder is 2 
hundredths ; what is the minuend ? 

19. The remainder is 13 millionths, the minuend is 13 
thousandths;" what is the subtrahend ? Ans. .012987. 

20. The multiplicand is 75 thousandths; the multiplier is 
25 ten-thousandths ; what is the product ? 

Ans. .0001875. 

21. The multiplier is 18 thousandths, the product is 369 
millionths; what is the multiplicand ? 

22. The product is 1482 ten-millionths, the multiplicand is 
95 hundredths; what is the multiplier? 

Ans. .000156. 

23. The divisor is 19 huitdredths, the quotient is 21 thou- 
sandths ; what is the dividend ? Ans. .00399. 

21. The dividend is 65 and 12 hundredths, the divisor is 
17 and 6 tenths ; what is the quotient ? 

25. The quotient is 14 hundredths, the dividend is 322 
thousandths ; what is the divisor t Ans. 2.3. 

26. What are the prime factors of 3500 ] 

Ans. 2, 2, 5, 5, 5, and 7. 

27. What are the prime factors of 756 ? 

28. What different prime numbers will exactly divide 700 '^ 

Ans. 2, 5, and 7. 

29. What different prime numbers will exactly divide 850 ? 

Ans. 2, 5, and 17. 

30. What is the least common multiple of 7, 8, 10, and 14? 

112 



PROMISCUOUS PROBLEMS. §176 



31. What is the smallest number that may be exactly di- 
vided by either 9, 10, 12, or 15 ? Ans. 180. 

32. What is the smallest number that may be exactly di- 

vided by either 21, 36, 48, or 72 ? Ans. 141. 

33. What is the greatest common measure of 45, 54, and 

108? 

34. What is the largest number that will exactly divide 
either 75, 100, or 150 ? Ans. 25. 

35. What is the largest number that will exactly .divide 
either 96, 192, or 240 ? Ans. 48. 

36. What is the sum of 1-, l, -^, |, and 3%? 

37. What is the difference between f and |^? Ans. ^\. 

38. What is the product of | and j-^- 1 Ans. |-|. 

39. What is the quotient of ^ divided by -^? 

7i 

40. What is the value of -^ ? Ans. |^. 

15i 

41. What is the value of f of -f- of 7^- ? Ans. If^. 

42. What is the value of .^25 + . 025 + .715 + .225 ? 

43. What is the value of .0237-.002375 ? Ans. .021325. 

44. What is the value of .027 x .0027 ? Ans. .0000729. 

45. What is the value of .0144-^3.6 ? 

46. What is the sum of -^,-i\, yV? and ^ 1 Ans. l-i-|. 

47. What is the difference between -^\ and -^ ? 

48. What is the sum of 216 thousandths, 37 hundredths, 

15 ten-thousandths, and 10 units? Ans. 10.5875. 

49. What is the difference between 206 ten-thousandths, 

and 27 millionths ? 

50. What is the value of .211 + 3.07 + 29.6 + .0735 ? 

Ans. 32.9545. 

51. What is the value of .6501 x .736089 ? 

52. What is the value of .4396^9.3 ? 

n 113 



§177 



CONCRETE NUMBERS. 



CONCRETE NUMBERS. 



§ 177. The relations of the concrete numbers in most 
common use are set forth in the following 

I. United States Money. 





1 


mill. 


(m.) 


— 1 

— lO 


of a 


cent; 


10 mills 


= • 1 cent, 


(ct.) 


— 1 

10 


<i a 


dime ; 


10 cents 


= 1 dime. 


(d.) 


— 1 

— 10 


a ic 


dollar 


10 dimes 


=z 1 dollar, 


m 


— 1 

— 1 o 


U il 


eagle ; 


10 dollars 


= 1 


eagle, 


(E.) 








m. 


ct. 




d. 


$ 




E. 


II II II II 

rH O O O 
rH O O 


1 

1 o 
1 

10 
100 


= 


1 

10 

1 

10 

1 

10 


^^ 1000 

1 

— 1 oo 

— 1 

10 

= 1 


= 


1 


1 oooo 
1 

1000 

1 

100 

1 

1 


10000 = 


1000 


= 


100 


= 10 


— 


1. 



The denominations dime and eagle are very little used in 
calculation. In stead of 14E. 5$, 7d, 5ct., we usually write 

$145.75. 



II. £n^lisli Currency; or. Sterling Mouej 

1 farthing, (qr.) - 

4 farthings =: 1 penny, (d.) = 

12 pence = 1 shilling, (s.) - 

20 shillings = 1 pound, {£). 

114 



i of a penny; 
i-V " " shilling; 
^ " " pound ; 









RELATIONS. 




§17' 


qr. 




d. 


s. 




£ 


1 


= 


1 

4.- 


— 1 

48 


* = 


9 60 


4 





1 


1 


, 


1 


— 




12 


~^ — 


2 40 


48 


z=: 


VA 


:rr 1 


= 


1 
20 


960 


— 


240 


=: 20 


— 


1 



The pound sterling is represented by a gold coin, called 
a sovereign, valued at $4.84, U. S. currency. 

Farthings are usually written as fractions of a penny. 



III. FreificU Currency. 

1 centime, (cent.) = J- of a decimej 

10 centimes = 1 decime, (dec.) = /^ '^ " franc ; 
10 decimes = 1 franc, (fr.)- 

cent. dec. fr. 

^ 10 — 100 

10 = 1 = iV 

100 = 10 = 1 

Accounts are kept in francs and centimes. 

The franc is valued at 18ct. 6m., U. S. currency. 



--«yp~ 



IV. Troy IRTeiglit. 

USED FOR WEIGHING GOLD, SILVER, JEWELS, <fec. 

1 grain, (gr.) = ^l of a pennyweight ^ 
24 grains = 1 pennyweight, (dwt.)= -^ of an ounce.; 
20 penny weights^ 1 ounce, (oz.)= -^^ of a pound .; 
12 ounces = 1 pound, (lb.) 

gr. dwt. oz. lb. 

1 = -1- = -1- = 1 

24 480 5760 

24 — 1 _ _i_ — 1 

^^ ^ 20 240 

480 ::=: 20 = 1 = JL. 

5760 = 240 = 12 = 1 
115 



§177 



CONCRETE NUMBEHS. 





¥. Apotliecaries' W' 


elglit. 






USED IN MlJrjJVO MEDICINES. 






1 grain, (gr.) 


- ^of 


a scruple 


20 grains 


~ 1 scruple, (so.) or 9 


= i ^' 


" dram ; 


3 scruples 


=r 1 dram, (dr.) or o 


6 


^' ounce • 


8 drams 


= 1 ounce, (oz.) or 5 


1 <6 

12 


" pound ^ 


12 ounces 


= 1 pound, (lb.) or lb 






g^' 


sc. dr. 


oz. 


lb. 


1 


^^ "2 0" "6"0 

= 24 = 8 = 


1 — 

4 S 

1 — 

2 4 

,1 = 


1 


-20 
60 

480 


5 7 6 
1 

2 8 & 

1 

e 

1 
T2" 


:5760 


= 288 =96 = 


12 = 


1 


The poun 


d Apothecaries' is the same as the pound Tro}^ 



¥1. Avoirdupois 'Weaglat. 

USED FOE, WEIGHING ALL AllTICLES EXCEPT THOSE MENTIONED 

ABOVE. 



• 16 drams 
16 ounces 
25 pounds = 

4 quarters = 

20 hundredweight: 

dr. 

1 

16 

256 

6400 

25600 

512000 



1 dram, (dr..)= -^ of an ounce ; 
= 1 ounce, (oz.)= J^- of a pound ; 
= 1 pound, (lb.)= J-5 of a quarter ; 
= 1 quarter,(qr,)= ^of ahundredweight' 
= 1 hundred weight,(cwt,) = -^V of a ton; 



oz. 

1 

1 6 
1 

16 

400 

1600 

32000 



lion, (T.). 

lb. qr. 



1 . 

2 5 6' 
Ji._ . 
1 6 ■ 

1 : 

25 : 

100 : 

2000 : 

110 



1 

6 40 

1 

4 00 

1 

2 5 



cwt. 



2 5 6 00 

1 

16 

1 
100 



T. 



4 
80 



4 — 
1 = 

20 = 



5 12 000 

1 

3 2 00 

1 

2 

_1_ 

8 

1 
2 O 

1 



RELATIONS. 



§177 



144 pounds Avoirdupois = 175 pounds Troy or Apothe- 
: 7000 gr. Troy ; 1 oz. Avoir. = 437.5 gr. 



cariess 

1 lb. Avoir. 
Troy. 

The following denominations also belong here : 



28 pounds 
112 " 
2240 " 
14 " 
21^ stone 
8 pigs 

50 pounds of salt 

56 

60 

56 
100 
]96 
200 



1 long quarter ; 

1 long hundredweight ; 

1 long ton ; 

1 stone; 

1 pig; 

1 fother. 






" corn 

'^ wheat 

'' butter 

" salt fish 

" flour 

" beef, pork, or fish; 



1 bushel, 
1 bushel. 
1 bushel. 
1 firkin. 
1 quintal, 
1 barrel. 
1 barrel. 



Til. Frencli Wciglits. 

1 milligramme = -^-^ of a centigramme; 
10 milligrammes=l centigramme 
10 centigrammes=l decigramme = iV 
10 decigrammes =1 gramme == -^ 

10 grammes =1 decagramme = -^-^ 

10 decagrammes =1 hectogramme^= -^ 
10 hectogrammes =1 kilogramme = y^^ 
100 kilogrammes =1 quintal = y^^ " " millier; 

10 quintals =1 millier, or 1 ton of sea water. 



yV " " decigramme; 
^ '' gramme ; 
^ " decagramme ; 
^ " hectogramme; 
^ " kilogramme ; 



i a 



quin 



tal 



1 gramme = 15.433 grains Troy. 
117 



§177 



CONCRETE NUMBERS. 



Till, ftiong- Measure 


^ or, I^inear Measure. 


USED IN MEASURING LINES, OR DISTANCES. 




1 inch, 


(in.) = 


= yV 0^ ^ ^00* J 


12 inches — 


1 foot, 


(ft.) = 


= i " " yard; 


3 feet = 


1 yard, 


(yd.) - 


- ^\ " " rod ; 


5i yards 


1 rod, 


(rd.) = 


= ^^ " " furlong ; 


40 rods — 


1 furlong 


, (fur.) = 


= i " " mile; 


8 furlongs — 


1 mile, 


(mi.) 




in, ft. 


yJ- 


rd. 


fur. mi. 


1 — '^ -- 


— 1 


1 


— 1 — 1 


-^ 12 


3 6 


19 6 


7 9 20 63360 


12 = 1 : 


] 


2 


. 1 1 




3 


3 3 


6 6 5 2 8 


36 = 8 = 


= 1 


2 


1 1 






1 1 


2 2 1 7 60 


198 =r 16i= 


H 


= 1 


1 1 

4 ■3T0 


7920 — 660 : 


= 220 


= 40 


1 i 


63360 = 5280 : 


= 1760 


= 320 


8—1 


The following denominations are sometimes used : 


3 barley 


corns = 


1 inch ; 




6 points 


— 


1 line ; 




12 lines 




1 inch ; 




4 inches 


= 


1 hand ; 




9 " 


— 


1 span ; 




18 " 


:zjL 


1 cubit; 




21.9" 




1 sacred cubit ; 


3 feet 


: — : 


1 pace; 




6 feet 


z= 


1 fathom ; 


1 


69i miles 


1 


1 degree < 


of latitude. 


IX. Swrveyor's I<oiig Measure. 


7.92 inches 


1 link. 


00- 


2V of a rod ; 


25 links = 


1 rod. 


(rd.) = 


i " " chain ; 


4 rods = 


1 chain, 


(ch.) = 


-jV" "furlong; 


10 chains = 


1 furlong, 


(fur.) - 


i « " mile ; 


8 furlongs = 


1 mile, 


(mi.) 






118 







RELATIONS. 


§177 


in. 

7.92 — 

198 — 

792 = 

7920 = 

63360 = 


1. 

1 — 

25 — 

100 = 
1000 = 
8000 = 


rd. ch. 

"2 5" TcTo 

1 _ i ^ 

4—1 

40—10 — 

320 — 80 — 


fur. mi. 

1 — 1 


1000 SOOO 

1 — 1 

40 3 2 

1 1 

10 »o 

1 = i 
8—1 



X. Square Measure. 

USED FOR MEASURING SURFACES OF LAND, PAINTING, PLASTERING. 

PAVING, &c. 

1 squareinchj (sq.in.)=y^-^ofasquarefoot; 

144 square inches=l " foot, (sq.ft. )= I '^ " " yard; 
9 « feet =1 '' yard,(sq.yd.)=j-^y" '^ perch; 
30-i « yards =1 perch, (P.) = ^V " " ^^^^ ; 

40 perches =1 rood, (R.) = |- " " acre ; 

4 roods =1 acre, (A.) =^^0" "square mile 

640 acres =1 square mile, (sq.mi.). 

sq.in.sq.ft.sq.yd. P. R. A. sq.mi. 

^ ^~~ 1 4 -i ~~ Ta ITe"" 3 8 2 o ¥ "~~ 1 5 6 8 "i~6 o" — "e "aT 2~6 4 0" To 1 4 4 8 9 eoU 
144 — 1 — 1 — 4_ — \ — 1 — 1- 

^^^ -^ V 10«9 10890 43560 '>. 7 k 7 s 

l9Qfi — q _ 1 _ t__ — 1 — !„ 

39204 = 272:1 = 30^ z= 1 = -,V = 

1568160 = 10890 = 1210 = 40 =.: 

6272640 = 43560 =: 4840 =. 160 :.= 4 = . _ ^-^^ 

4014489600 = 27878400 =z 3097600 = 102400=2560=640=1 




XI. Cubic Measure. 

USED FOR MEASURING THE CONTENTS OF SOLIDS. 

1 cubic inch, (cu.in.)=yyi^-g-of acubicft.:; 
1728 cubic inches=l " foot, (cu.ft.) = ^V *' " " J^-'^ 
27 '' feet =1 '' yard,(cu.yd.). 

cu. in. cu. ft. cu. yd. 

1 — _3_. — L 

■^ 1728 4 6050 

1728 =z 1 = ^\ 

46656 =27 = 1 

119 



§177 



CONCRETE NUMBERS. 



Also, 40 cubic feet of round timber = 1 ton ; 
50 " " '' hewn timber = 1 ton ; 

42 " " '• shipping = 1 ton ; 

16 " '' " wood = 1 cord foot; 

128 " " ^' wood = 1 cord. 

Also, 231 cubic inches=l gallon Liquid or Wine Measure; 
268|- " " =1 '^ Dry Measure ; 
282 « " =1 " x\le Measure, (out of use.) 
537-?- " " =1 ponk; 

2150i " " =1 Lu.liel. 



XII. £iiq3£ld Measure 5 or, Wisae Measiia'e. 

USED IN MEASURING LIQUIDS ; AS, MOLASSES, SPIRITS. 
WINE, WATER, &c. 

1 gill, (gi.) = I: of a pint ; 

4 gills = 1 pint, (pt.) — 1- " " quart ; 

2 pints = 1 quart, (qt.) = -^ " " gallon ; 

4 quarts =z 1 gallon, (gal.) = -^-3 " " barrel ; 

31-|- gallons = 1 barrel, (bbl.) =: -} " " hogshead; 

2 barrels — 1 hogshead(hhd.) == ^ " " 

2 hogsheads = 1 pipe, (pi.) = ^ 



pipe 
« " tunt 



2 pipes 


r= 1 tun, 


(tun). 




gi. pt. 


qt. 


gal. bbl. hhd. 


pi. tun. 


1 - i 
4 = 1 


— Y — 

Z 126 Z 


1 — 1 — 1 — 

32 lOOb 3016 

s ^^ "2T5'"2 ^^ "sinr 

4. 126 2 52 

•*- 3 G 3 

3U-= 1 = i = 


1 — 1 

4 3 2 a 6 i 
1 1 


8 = 2 

32 = 8 

1008 = 252 


1 s 2 1 g: 
1 — 1 

504 100 tt- 

_1 — 1 

12 6 2 5 2 


2016 == 504 


= 252 = 


63^= 2 = i = 


1 1 

2 4 


4032 :=1008 


= 504 = 


126 = 4 = 2 = 


1 = i 


8064 =2016 


= 1008 = 


252 = 8 = 4 = 


2 = 1 


Also, 


42 gallons 


= 1 tierce ; 






2 tierces 


= 1 puncheon. 
120 





RELATIONS. §177 



XIII. Ale Measaire. 

FORMERLY USED FOil MEASURING MALT LIQUORS AND ?IILK, WHTCli 

NOW. HOWEVER, ARE GENERALLY MEASURED BY 

LIQUID MEASURE. 

1 pint, (pt.) = ^ of a quart ; 

2 pints = 1 quart, (qt.) = -} " " gallon; 

4 quarts = 1 gallon, (gal.) = ^fo " " barrel; 

:>6 gallons =: 1 barrel, (bbl.) - | " " hogshead ; 
li barrels =: 1 hogshead, (hhd.) 

Also, 9 gallons = 1 firkin, 

2 firkins = 1 kilderkin. 



(qt.) 

(gal.) 
(pk.) 


- 1 

i 

1. 

1 


ii 


*' gallon ; 
" peek ; 
" bushel; 


, (bu.) 








gal. 


pk. 




bu. 


1 — 


1 

1 6 

1 




— 1 



XIV. itry Measure. 

USED FOR MEASURING GRAIN, FRUITS, VEGETABLES, SALT, Ic. 

1 pint, (pt.) =: \- of a quart ; 

2 pints = 1 quart, 
4 quarts — 1 gallon, 
2 gallons — 1 peck, 
4 pecks = 1 bushel, 
pt. qt. 

1 - i = 

8 =r. 4 ^ i 

16:=.Sz::.2rrrl^A 

Also, 5 bushels zzi 1 barrel, of corn ; 
8 bushels r::: 1 quarter ; 
36 bushels — 1 chaldron. 
In the Confederate States, corn is usually bought and 
sold by the barrel. A barrel of corn should contain 280 
pounds. 

121 



y, 3 2 

1 — 1 

a' — V 



§177 



CONCRETE NUMBERS. 



V ^ — 


XV. 


Time. 












1 second. 


(sec.) 


1 
GO 


of 


a 


minute ; 


60 seconds 


— 1 minute, 


(min.) 


T 
6 




an 


L hour ; 


60 minutes 


= 1 hour, 


(hr.) 


1 
•J. 4 




a 


day; 


24 hours 


z=z 1 day, 


(da.) 


4. 
146 1 




a 


year ; 


365^^ days 


= 1 year, 


(yr.) 


1 
1 




a 


decade; 


10 years 


= 1 decade. 


(dec.) 


1 
10 




ii 


century ; 


10 decades 


= 1 century, 


(cent.) 










sec. min. hr. 


da. yr. 




dec. 






cent. 


I— 1 _ 1 _ 


_ 1 1 




1 


0"^ 


_ 


1 


•^ 6 3 e 00 


S64;06 3155760O 3155760 


"yi 5 5 7 6 OOOO 


60 = 1 = -i-. 


1 


1 


1 







1 


1 14 40 " 525960 


5 2 5 6 00 


52596000 


3600= 60 : 


== 1 = ^v - 


1 


1 

8 7 6 6 




— 1 


S 7 6 6 


S 7 6 6 00 


86400 = 1440 = 24 = 1 


146 1 


2 




1 


7 30 5 


3 6 5 2 5 


31557600 = 


525960 = 8766 = 365^ = 


1 ^ 


= 


1 1 

10 100 


315576000 = 


= 5259600= 87660 = 3652^ = 


= 10 = 


= 1 = tV 



3155760000 = 52596000 = 876600 = 36525 = 100 = 10 = 1 

x\lso, 7 days = 1 week, (wk.) ; 

30 or 31 days = 1 month, (mo.) ; 
12 months = 1 year. 

According to the table, 3651 days make a year. To ob- 
viate the difficulty arising from the fraction, we reckon 
three years of 385 days each, and one of 866 days. This 
long year is called leap year. The leap years are those 
whose numbers are exactly divisible by 4 ; except that the 
centennial years are not leap years unless their numbers 
are exactly divisible by 400. Thus, 1860 and 1848 were 
leap years ; but 1900 will not be leap year, because it is 
not divisible by 400. 

The year is also divided into four seasons; Spring, Sum- 
mer, Autumn, and Winter. These consist of the following 

months : 

122 



RELATIONS, 



§17' 



r 3. March, 
Spring, < 4. April, 
I 5. May, 

6. June, 

7. July, 

8. Aucust, 






Summer, 



(Mar.) lias 31 clays. 

(Apr.) " 30 " 

(May) " 31 

(Jun.) " 30 

(Jul.) " 31 '' 

.^..., (Aug.) " 31 " 

( 9. September,(Sept.) " 30 " 

Autumn, MO. October, (Oct.) " 31 '^ 

( 11. November,(Nov.) " 30 " 

r 12. December, (Dec.) " 31 " 

Winter, < 1. January, (Jan.) '^ 31 '^ 

{ 2. February, (Feb.) " 28 " leap year, 29. 

In most business transactions 30 days are considered a 
month. 



XVI. Circular Measure. 

USED IN SURTEYING, GEOGRAPHY, AND ASTRONOMY. 

1 second, ('' or sec.)=g^^ of a minute; 
60 seconds =1 minute, ('or min.)^:-^^ " " degree; 
60 minutes =1 degree, (° or deg.)= ^^j " "• sign ; 
30 degrees =1 sign, (S.)=y\- " " circumference; 

12 signs =1 circiimference,(C.) 

fr 

1 = 

60 = 1 

3600 = 60 

108000 = 1800 

1^96000 = 21600 

Also, 60 degrees =: 1 sextant =-1- of a circumference ; 
And, 90 " =1 quadrant =a. « « " 

123 



6 O 



o 




S. 




c. 


1 


= 


1 


= 


1 


36 OO 


10 8 000 


12 9 6000 


1 


= 


1 
1 S 00 


= 


1 


GO 


2 16 


1 


= 


1 

30 


= 


1 
360 


30 


= 


1 


z=z 


I 
1 2 


360 


=z 


12 


:;:;::: 


1 



§178 



CONCRETE NUME"£:RS. 



24 sheets 
20 quires 
2 reams 
5 bundles 



XYII. Faper. 
1 sheet, (sh.) 

1 quire, (q^-) 
1 ream, (rm.) 

1 bundle, (bdie.) 
1 bale. 



^13: of a quire ; 
2V ** " ream; 
i " " bundle ; 
i " " bale: 



-•♦^ 



1 unit, = tV of a dozen ; 

12 units =z 1 dozen, (doz.) = ^V " " gi'oss; 

12 dozen = 1 gross, (gr.) — -,V " " g^^^* S^°^^ 5 

12 gross := 1 great gross. 

Also, 20 units = 1 score. 



OPERATIONS ON CONCRETE NUMBERS. 



The numerical processes are the same for concrete num- 
bers as for abstract. In this place^ therefore, we are to dis- 
cuss only the denominations of the several results. 



ADDITION OF CONCRETE NUMBERS. 



§178. Dissimilar numbers can not be added together. 
Thus, 3 dollars and 5 cents malre neither 8 dollars nor 8 
cents. 

§ 179. The sum of several similar numbers is similar to 
the numbers added. Thus, 3 dollars and 5 dollars make S 
dollars ; 3 cents and 5 cents make 8 cents. 

Ex. 1. Add $1075, $2157, $3779, and $4209, 

124 



ADDITION'. !^i79 



2. Add ^47, £bd, £29, nnd £.63. Sum, £192. 

". Add 27fr., 3Gfr., 297fr., and 365fr. 

4. Add 291b., "STlb., 491b., and 581b. Sum, 1731b. 

5. Add 45sc., 2Si?c., 143sc., and 2S7sc. Sum, 5l.)3se. 
G. Add iOOcwt., 205cwt., 177cwt,, and 329c wt. 

7. Add 2479 grammes, 147 grammes, and 986 grummeis. 

Sum, 3GI2graui. 

8. Add 276yd., 299jd., 4G9yd., and 357yd. 

Sum, 140] yd. 

9. Add 79mi., 227mi., 37mi., and 475mi. 

10. Add 306cu. ft., 279cu. ft., and 520cu. ft. 

Sum, 1105cu.ft. 

11. Add 575A., 209A., 105A., and 258A. Sum, 1147A. 

12. Add 27gal., 72ga]., 298gal., and 143gal. 

13. Add 15bbl., 28bbl., 19bbl., 247bbl., and SGbbl. 

Sum, 395bbl. 

14. Add 47bu., 475bu., 407bu., and 4750bii. 

Sum, 5679bu. 

15. Add 27da., 38da., 52cla., and 93da. 

16. Add 12°, 26°, 37°, and 45°. Sum, 120°. 

17. Add lOrui., 14rm., 7rm., and 22rm. Sum, 53rm. 

18. Add 6doz., 27doz., 14doz., and 97doz. 

19. Add 12^-lb., 33Ub., 37ilb., and 83-^lb. Sum, 166pb. 

20. Add 3|mi., I6|mi., 18:^ mi., 62irai., and 42|mi. 

Sum, 143i{:mi. 

21. Add 19^qt., 20^qt., 7^.qt., and 28^qt. 

22. Add 3.251ir., 6.5hr., .275hr., and 700.075hr. 

Sum, 710.1hr. 

23. Add 47.5pt., 57.75pt., .375pt., and .0625pt. 

Sum, 105.6875pt, 

24. Add 9.73pk., lO.Olpk., 17.75pk., and .1775pk. 

25. Add lO.llpk., 7.369pk., and 1.002pk. 

125 



§180 CONCRETE NUMBERS. 



SUBTRACTION OF CONCRETE NUMBERS. 



§180. Subtraction can not be performed upon dissimilar 
numbers. Thus, 3 cents from 5 dollars leaves neither 2 
cents nor 2 dollars. 

§ 181. The difference of two similar numbers is similar to 
those numbers. Thus, 3 dollars from 5 dollars leaves 2 
dollars ; 3 cents from 5 cents leaves 2 cents. 

Ex. 1. From £245 take .£196. Rem. £49. 

2. From 25cwt. take 6c wt. Kem. 19cwt. 

3. From 793rd. take 546rd. 

4. From 17246sq. ft. take 8472sq. ft. Rem. 8774sq. ft. 

5. From 635cu. yd. take 473cu. yd. Rem. 162cu. yd. 
(5. From 47 decigrammes take 29 decigrammes. 

7. From 479bhd. take 3981ihd. Rem. 81hhd. 

8. From 272pt. take 199pt. Rem. 73pt. 

9. From 365da. take 175da. 

10. From 360° take 275° Rem. 85°. 

11. From 27fs. take 19is. Rem. 8is. 

12. From $75^ take $59i. 

13. From $19.75 take $ .99. Rem. $18.76. 

14. From 270^fr. take 197|fr. Rem. 72ifr. 

15. From 77in. take 17.75in. 

16. From 3706sq. yd. take 897isq.yd. Rem. 2808^sq.yd. 

17. From ^ of 246cu. ft. take -\ of 317cu. ft. 

18. From 525iqt. take 252^qt. 

19. From 27bu. take I7.25bu. Rem. 9.75bu. 

20. From SS^cu. in. take 31icu. in. Rem. 2y\cu. in. 

21. From 725dwt. take 339.17dwt. 

22. From .2468d. take .08642d. Rem. .16038d. 

23. From .1751b. take .0171b. Rem. .1581b. 

126 



MULTIPLICATION. 



§183 



MULTIPLICATION OF CONCRETE NUMBERS. 



§ 182. Every muUipUer must be an abstract number. — 
Thus, if we wish to find the cost of o yards at 25 cents a 
yard, it is evidently absurd to say, "3 yards times 25 
cents," or, '' 25 cents multiplied by 3 yards." "We multi- 
ply 25 cents by 3, because 3 yards cost 3 times the price of 
1 yard, that is, 3 times 25 cents. 

§ 183. The product is always similar to the multiplicand. 
Thus, 3 times 25 cents are evidently 75 cents ; 6 x 7 ab- 
stract units=:i42 abstract units ; 4 x $10=$40 ; 5 x 6-yards 
=30 yards. 

Ex. 1. Multiply $3179 by 27. 

2. Multiply 2764bu. by 4G. 

3. Multiply 3S5da. by 19. 

4. Multiply 347oz. by 83. 

5. Multiply 2047cwt. by 109. 

6. Multiply 347fr. by 201. 

7. Multiply 467A. by 5297. 

8. Multiply 6386pi. by 578. 

9. Multiply 7475pk. by 689. 

10. Multiply £69 by 4234. 

11. Multiply 224 by 4759. 

12. Multiply 8564wk. by 790. 

13. Multiply 9563 by 801. 

14. Multiply 10742doz. by 912. 

15. Multiply 20s. by 16750. 

16. Multiply 5}yd. by 746. 

17. Multiply 16.5ft. by 165. 

18. Multiply 30.25sq. yd. by 3.025. 

19. Multiply 7.92in. by 198. 

127 



Prod. $85833. 
Prod. 127144bu. 

Prod. 28801OZ. 
Prod. 223123cwt. 

Prod. 2473699A. 
Prod. 3691108pi. 

Prod. .£292146. 
Prod. 1066016. 

Prod. 7659963. 
Prod, 9769704doz. 

Prod. 4103yd. 
Prod. 2722.5ft. 

Prod. 1568.16in. 



i 



^184 CONCRETE NUxMBERS. 



20. Multiply 31igal. by 1008. Prod. 31752gal. 

21. Multiply SGo^da. by 365^. 

22. Multiply $29.75 by 29.75. Prod. $885.0625. 

23. Multiply $100,375 by 37.5. Prod. $3764.0625. 

24. Multiply 279.5 by 27.95. 



DIVISION OF CONCLIETE xXUMBERS. 



{M84. Division ia the reverse of multiplication. In 
multiplication, the two factors are given, to find tbe prod- 
uct ; in division, the product and one of its factors are 
given, to find the other factor. The dividend corresponds 
to the product; the divisor may correspond to either the 
multiplicand or the multiplier, and the quotient corre- 
sponds to the other. 
' Thus, 6 X 25gal.=150gal. 

Conversely, 150gal.-^6=:25gal. 
Or, 150gal.-^25gal.=G. 

§ 185. Either the divisor or the quotient must be ab- 
stract, and the other must be similar to the dividend. 

In other words, if the dividend and the divisor are simi- 
lar, the quotient is abstract : if the divisor is abstract, the 
(juotient is similar to the dividend. 

The remainder is always similar to the dividend. (§46). 

Ex. 1. Dividend=:45ct., divisor=^-3. Quot. 15ct. 

2. I)ividend==:$750, divisor=$25. Quot, 30. 

3. Dividend=::=1000bu., divisor=40. 

4. Dividend- 2451b., divisor=5. Quot. 49^b. 

5. Dividend=3003, divisor=ll. Quot. 273. 

6. Dividendz=]728cu. in., dmsor=48cu. in. 

7. Dividend=:7007yd., divisor=13yd. Quot. 539. 

128 



REDUCTION. §188 



8. Divisor=l7mi., dividend=2S9mi. Quot. 17. 

9. Divisor=25, dividends 1175gi. 

10. Divisor=:27, dividend=:2971ir. Quot. llbr. 

11. Divisor=109, dividend =:2398qt. Quot. 22qt. 

12. Divisors 245cu. ft., dividend = 5880cu. ft. 

13. Divide 642780 dozen by 36 dozen. Quot. 17855. 

14. Divide 79008oz. by 96. Quot. 823o2. 

15. Divide 847665qr. by 345qr. 

16. Divide 3475cwt. by 296. Quot. 11.739 + cwt. 

17. Divide 1001s. by 27s. Quot. 37.074. 

18. Divide SS^^ft. by 172ft. 

19. Divide 372.25sq. yd. by 250sq. yd. Quot. 1.489. 

20. Divide iA. by 13 1-. Quot. .03tA. 

21. Divide 243flb. by 19f. 

22. Divide 799.6T. by 87.'5T. Quot. 9.104 + • 

23. Divide 34.1.5 grammes by 19.25 grammes. 

24. Divide 177pt. by 771. Quot. .2295 + pt. 



KEDUCTION OF CONCRETE NUMBERS. 



§ 186. A compound number may be reduced to a simple 
one, or a simple concrete number to a-compound one by tbe 
application of the following rule according to tbe circum- 
stances of the case. 

§187. Rule. — Find from the proper table the value of 
one of the given units in terms of the required denomination ; 
and multiply this value by the number of the given units. 

Ex. 1. Reduce 6 gallons to pints. 

§188. Model. 6gal.=r6x4qt. = 24qt. 

24qt. = 24x2pt.=48pt. 

Hence, 6gal.=48pt. 
I 129 



§189 CONCRETE NUMBERS. 



Explanation. —Since 4qt.=:lgal.,6gal.j tliat is, 6 times 
lgal.=6 times 4(it. And since 2pt. = lqt., 24qt., or 24 
times lqt.=24 times 2pt. 

Observe that in each instance the product is similar to 
the multiplicand. (§183.) 

Otherwise^ 6gal. = 6 x 8pt.=:48pt. 
Ex. 2. Reduce 6gal, 3qt. Ipt. to pt. 
§189. Model. 6gal. = 6x4qt.r=24qt. 

24 + 3^:27 27qt.r=27 x 2pt. = 54pt. 

54 + 1=55 Hence, 6gal. 3qt. lpt. = 55pt. 

Explanation. — After reducing the 6gal. to qt., the giv- 
en 3qt. may be added to the result. (§ 179.) And after 
reducing the 27qt. to pt., the giv s Ipt. may be added to 
this result. 

OtlieriDise, 6gal.=6 x 4qt. = 24qt.=24 x 2pt.=:48pt. 

3 '^' = 3x2^^= 6 " 

Hence, 6gal. 3qt. lpt.=:55pt. 
Otherwise, 6gal.=6 x 8pt.r::48pt. 
3qt. =3x2" = 6" 

Ipt. = 1_^ 

Hence, 6gal. 3qt. lpt. = 55pt. 
Evidently the final result is not affected by the order in 
which the several reductions are performed. 
Ex. 3. Eeduce i^lb. to oz., dwt., &c. 
§ 190. Model. TVlb.=yV of 12oz.=l-^oz. 
^^oz.=|: of 20dwt. = 4dwt. 
Hence, -Jylb. = loz. 4dwt. 
Explanation. — This example differs from the first only 
in the fact that here each multiplier is a fraction. 

130 



REDUCTION. §193 



Ex. 4. Reduce 3795P. to A., R., &c. 
§191. Model. 3795P. = 3795x-J,tR.=H^R. = 94R, 35P. 
94R.=: 94 X 1:A.= V A. = 23A. 2R. 
Henoe, 3795P. = 23A. 2R. 35P. 
Explanation. — This example differs from the preceding 
only in the fact that here each multiplicand is a fraction. 
Ex. 5. Reduce 6gal. 3qt. Ipt. 3gi. to hhd. 
§192. Model. 3gi. =3 x ipt. = a pt, 

3^qt. =3^- X igal. = U gal. 
6||gal. = 6|ixJ3hhd.=^^hhd. 
Explanation." — Here both factors are fractional. 

Ex. 6. Reduce 30bu. Ipk. 3qt. Ipt. to pk. 
§ 193. Model. 30bu. =r 30 x 4pk. = 120pk. 
Ipk. = 1 « 

3qt.= 3x ipk.= .375 

lpt.=: t\V^'= •0625_ 

Hence, 30bu. Ipk. 3qt. Ipt."^21^43f5pk. 
Explanation. — This example is but a combination of 
two of the preceding ones, and seems to require no addi- 
tional explanation. 

Ex. 7. In $14, how many mills ? Ans. 14000m. 

8. In jS15, how many pence ? Ans. 3600d. 

9. In 19fr., how many centimes 1 

10. In 221b., Troy, how many dwt. ? Ans. 5280dwt. 

11. In 251b., Apothecaries', how many scruples ? 

Ans. 7200sc. 

12. In 261b., Avoirdupois, how many drams ? 

13. In 31 hectogrammes, how many decigrammes ? 

Ans. SlOOOdec. 

14. In 45 miles, how many feet ? Ans. 237600ft. 

131 



S193 



CONCRETE NUMBERS. 



Ans. 2048qt. 
Ans. 1728000sec. 

Ans. 8160sh. 
Ans. 720doz. 



15. In 49fur., ho^ many chains? 

16. In 50 A., how many square yards? Ans. 242000sq. yd. 

17. In 65cu. yd., how many cu. in. ? Ans. 2566080cii. in. 

18. In 72gal., how many gi. ? 

19. In 64bu., how many qt. ? 

20. In 20da., how many sec. 1 

21. In 29°j how many seconds ? 

22. In 17rm., how many sheets ? 

23. In 5gr. gross, how many doz.? 

24. In M, 3s. 2d., how many qr. ? 

25. In 5fr. 7dec. Scent., how many centimes ? 

Ans. 578cent. 

26. In 61b. 5oz. 3dwt., how many gr. ? Ans. 37032gr. 

27. In 31b. 6oz. 5dr. 2sc., how many sc. ? 

28. In 28T. lOcwt. 3qr., how many lb. ? Ans. 570751b. 

29. In 1 millier, 5 quintals, how many grammes ? 

Ans. 1500000gr. 

30. In 2rd. 3yd. 2ft., how many in. ? 

31. In 10 chains, 1 rod, how many links ? Ans. 10251k. 

32. In 4sq. yd. 6sq. ft., how many sq. in. ? Ans. 6048sq.in. 

33. In lOcu. ft. 400cu. in., how many cu. in.? 

34. In 2 tuns, Ipi. Ihhd., how many gal. ? Ans. 693fral. 

35. In 5bu. 2pk. Igal. 3qt., how many pt. ? 

36. In Icent. 6dec. 5yr., how many yr. ? 

37. In 2S. 25° 45', how many seconds ? 

38. In 2rm. lOqr. 12sh., how many sh. ? 

39. In 3gr. 4doz,, how many units? 

40. Reduce -}£ to s. and d. 
41.- Reduce ffr. to decimes. 

42. Reduce fib. Troy, to oz., dwt., &c. 

43. Reduce ^ lb. Apothecaries', to oz., &c. 

Yal. 2oz. 3dr. 12ffr. 
132 



Ans. 366pt. 

Ans. 308700^ 
Ans. 1212sh. 

Val. 6s. 8d. 
Val. 6fdec. 



REDUCTION. 



§193 



44. 
45. 

46. 
47. 
48. 
49. 
50. 
51. 
52. 
53. 
54. 
55. 
56. 
57. 
58. 
59. 
60. 
61. 

62. 
63. 
64. 
65. 

66. 
67. 

68. 

69. 
70. 
71. 

72. 



Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
R.educe 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 

Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 
Reduce 

Reduce 
Reduce 
Reduce 
Reduce 



Val. 3cwt. Iqr. S^lb. 



Val. 22fP. 



^T. to cwt,, qr., &c. 

^-mi. to fur., &c. 

f A. to R., P., &c. 

;^rcu. yd. to cu. ft., &c. Val. 4cu. ft. 864cu. in. 

f-gal. to qt., pt.j and gi. 

^jjhu. to pk., gal., &c. 

J da. to lir., min., &c. 



Yal.2pk. Igal. 2|qt. 
Val. 4lir. 48mm. 



Val. 2qr. 14.4sh. 
Val. Idoz. 9.6un. 

Val. lOoz. 8dwt. 8gr. 
Val. lib. 4dr. 



.375° to min. and sec. 

.I3rra. to qr. and sh. 

.15gr. to doz. and units 

975qr. to £. 

oOOOgr. to lb. 

300sc. to lb. Apothecaries' 

600000dr. to T. 

llOOOin. to mi. Val. Ifur. 15rd. 3yd. 2in. 

600001k. to mi. Val. 7mi. 40ch 

4000000sq. yd. to sq. mi. 

60000cu. in. to cu.yd. 

Val. leu. yd. 7cu. ft. 1248cu. in, 
lOOOOgi. to tuns. Val. 1 tun, 60gal. 2qt 

lOOOpt. to lihd., A\q Measure. 
250pt. to bu. Val. 3bu. 3pk. 5qt, 

eOOOOOmin. to yr. Val. lyr. 51da. 16hr 

2000000" to circumferences. 
27000sh. to rra. Val. 56rm. 5qr 

19000 units to gr. gross. 

Val. lOgr. gr. llgr. lldoz. 4 units 
lid. 3qr. to £. 

9oz. 9dwt. 9gr. to lb. Val. .789 -fib 

6dr. 2sc. 15gr. to lb. Val. .07204 + lb, 

Iqr. 151b. to T. 



S19o CONCRETE NUMBERS. 



73. Reduce 20rd. Syd. to fur. Val. .52!>'7fur. 

74. Reduce 2rd. 201k. to ch. Val. .7ch. 

75. Reduce IR. lOP. to A. 

76. Reduce leu. ft. lOcu. in. to cu. yd. 

Val. .03725 + cu. yd. 

77. Reduce Ipt. Igi. to gal. Val. .15625gal. 

78. Reduce Ihhd. Ibbl. to tuns. 

79. Reduce 3pk. Igal. 3qt. to bu. Val. .96875bu. 

80. Reduce lOhr. 15min. BOsec. to da. Val. .4274 + da. 

81. Reduce 1*^ 10' 30" to S. 

82. Reduce 2qr. 12sli. to rm. Val. .125rm. 

83. Reduce Igr. lOdoz. 10 units, to gr. gross. 

Val. .1585648igr. gr. 

84. Reduce J12, 10s. 6d. 3qr. to s. 

85. Reduce iOlb. 9oz. 9dv/t. 9gr. to oz. Val. 129.46875oz. 

86. Reduce 31b. 5oz. 5dr. Isc. lOgr. to dr. Val. 333. 5dr. 

87. Reduce IT. lOcwt. Iqr. 20ib. to cwt. 

88. Reduce Irai. 7fur. 20rd. 3yd. to rd. Val. 620.o4rd. 

89. Reduce 3ch. 2rd. 101k. to rd. Val. 14.4rd. 

90. Reduce lOA. 3R. 20P. to R. 

91. Reduce 2cu. yd. 6cu. ft. 75cu. in. to cu. ft. 

Val. 60.0434 + CU. ft. 
'92. Reduce lOgal. Iqt. Ipt. 3gi. to pt. Val. 83.75pt. 

.93. Reduce 2bu. Ipk. 3qt. to pk. 

94. Reduce Ida. Ihr. Imin. Isec. to min. 

Val. 1501.0l6min. 

95. Reduce 1*= 10^ 30'' to minutes. Val. 70^'. 

96. Reduce 2rm. 3qr. 5sb. to qr. 

97. Reduce IT. Icwt. Iqr. lib. loz. to lb. 

Val. 2126.06251b. 

98. Reduce Isq. yd. Isq. ft. Isq. in. to sq. ft. 

Val. 10.00694sq. ft. 
*- 134 



PROMISCUOUS PROBLEMS. §195 



PROMISCUOUS PROBLEMS. 



1. Bought a dress for $12, a cloak for $15, a bonnet for 
$7, and a pair of gloves for $1 : what did they all cost ? 

' 15 § ^^^' Model. — The whole cost is the sum of 
y the several prices : hence, add $12, $15, $7, and 
I $1. (§ 179). The sum is $35 : hence, they all 
cost $35. 



2. A owns $10475 in real estate, $3850 in slaves, $4095 
in good notes, and $1415 in cash ; what is the value of 
his whole estate ? A. $19835. 

« 

3. Three men form a partnership : A invests $2445 ; B, 
$2890 ; G, $1959 : what is the whole investment ? . 

4. A miller bought from one man 147 bushels of wheat, 

from an other 98 bushels, and from a third 273 bush- 
els ; how mueh wheat did he buy from the three ? 

5. A farmer raised on one farm 415 bushels of wheat, 548" 
bushels of corn, 827 hundred weight of hay ; on the 
other, 293 bushels of wheat, 487 bushels of corn, 2SQ 
hundred weight of hay : how much did he raise on 
both farms ? 

A. 708bu. wheat, 1035bu. corn, 613cwt. hay. 
(). Bought a farm for $2875, and sold it for $3225 ; what 
did I gain t 

^""225 ^ ^^^' ^^^^^'^- — ^^^^ g^^^^ '^^ *^® difference 

9R7^ between what I gave and what I received : 

-fiir hence, subtract $2875 from $3225. (§181.) 

$350 The difference is $350 : hence, I gained $350. 

7. A farmer owning 725 acres, sells 375 acres ; how much 
land has he remaining? A. 350A. 

185 



§190 CONCRETE NUMBERS. 



8. A man divides $3000 among three sons^ giving A ^985, 

and B $1235 : how much does he give C ? A. $780. 

9. Burnt a kiln of 100000 bricks ; sold at different times 

3475, 2800, 40150, and 35000 ; how many are still 
unsold ? 

10. The distance from Charlotte to Groldshoro', via High 

Point, is 223mi., from Charlotte to High Point is 79mi.; 
how far is High Point fj;om Groldshoro' ? A. I4lmi. 

11. What cost 2471h. of hn^ -n. at 19ct. per lb.? 

9A7 IQ + §186. xiioDEL. — 2471b. cost 247 times 

ooL^ the cost of lib.: hence, multiply 19ct.,by 

-^ 247. (§183.) The product is 4693ct.: 

4693ct. hence, the bacon cost 4693ct. 

12. How many cents are in 25 dollars ? 

13. How many gallons in 14hhcl.? A. 882gal. 

14. What will 94bbl. of flour cost at $8 per bbl.? $752. 

15. How many pages in 475 volumes of 296 pages each ? 
IG. A father divides $5460 equally among his 4 sons; what 

does each son receive ? 

§ 197. Model. — Each son's share is one 
4)$5460 fourth of the whole : hence, divide $5460 
$1365 by 4. (§185.) The quotient is $1365: hence, 
each son's share is $1365. 

17. If 755A. of land cost $12835, what will one acre cost? 

A. $17. 

18. If 125 slaves sell for $75125, what is their average 

value ? 

19. If 85 bales of cotton weigh 386751b., what does each 

bale weigh '/ A. 4551b. 

20. On 475A. of land I raised 15675bu. of wheat ; how 
much per acre ? A. o3bu. 

21. In 478241b. flour, how many bbl. 

136 



PROMISCUOUS PROBLEMS. §198 



^woci^ii, 1(^n^^. ^198. MoDEL.—As 1961b. Blake a 

478241b. 1961b. i ^ , xi i r i,vi • i x 

QP„ bbl., the number oi bbl. is equal to 

'^r'0 4 '^^^' the number of times 1961b. are con- 

f.r.r. tained in 478241b. : hence, divide 

^^^ 478241b. by 1961b. (§185.) The 

quotient is 244 : hence, there are 244bbl. 

22. How many cu. yd. iu 13122cu. ft.? A. 486cu. y^. 

23. In 11823s., how many G.I A. 563G-. 

24. How many xVcres can be bought for $5658 at $23 per 

Acre ? 

25. If a vessel make 376mi. per day, how long will she be 
in making 7144mi.? A. 19da. 

26. Find the sum of two thousand and forty-seven, three 
thousand six hundred and fifty, sixty-three thousand 
and five, ten thousand four hundred and three, and 
four hundred and seven. Sum, 79512. 

27. Find the diftereuce between ten thousand and forty- 

two, and eight thousand seven hundred and ninety- 
nine. 

28. What is the product of seven thousand three hundred 
and seventy-five, and one hundred and twenty-five? 

A. 921875. 

29. IVhat is the quotient of eight thousand six hundred and 

twenty-five, by one hundred and twenty-five ? A. 69- 
80. How many days in 4wk.? 

31. How many hours in 28da.? A. 672hr. 

32. Hovv^ many minutes in 672hr.? A, 40320min. 

33. How many seconds in 40320min.?" 

34. The minuend is 91 thousand 8 hundred and 75, the sub- 
trahend 8 thousand 9 hundi-ed and G9 ; what is the 
remainder ? A. 82906. 

35. The subtrahend is 4 thousand 2 hundred and 96, the 

137 



§198 CONCRETE NUMBERS. 



remainder 6 thousand 2 hundred and 84 : what is the 
minuend ? * A. 10580. 

26. The remainder is 7 hundred thousand and 94, the minu- 
end 2 millions 3 thousand; what is the subtrahend? 

37. How many hours in 40320min.'? A. 672hr 

38. How many days in 672hr.? A. 2Sda. 

39. How many weeks in 28da.? 

40. How many min. in 2419200sec.? A. 40320min. 

41. The multiplicand is 37 millions 43 thousand and 25, 
the multiplier 8 thousand and 64 ; what is the produef? 

A. 298714953600. 

42. The multiplicand is 7 hundred and 25, the product 593 
thousand 7 hundred and 75 ; what is the multiplier ? 

43. The multiplier is 4 thousand 9 hundred and 7, the prod- 

uct 42 millions 813 thousand 575 ; what is the mul- 
tiplicand ? A. 8725. 

44. What cost 347yd. of rope at 9ct. per foot 1 A. $93.69. 

45. How many qt. in 7gal. 2qt.? 

46. How many qt. in 8gal. Iqt.? 

47. How many pt. in 8gal, Iqt. Ipt.? 

48. How many gi. in 7gal. 3qt. Ipt. Sgi.l 

49. The dividend is 11 millions 210 thousand 202, the di- 
visor 7 thousand and 2 ; what is the quotient ? 

A. 161. 

50. The divisor is 8 thousand and 4, the quotient 5 thou- 
sand and 90 ; what is the dividend ? A. 40740360. 

51. The quotient is 1 million 2 thousand and 3, the divi- 
dend 1 trillion 4 billions 10 millions 12 thousand and 
9 ; what is the divisor 1 

52. How many sq. mi. in 228S88P.? 

A. 2Rq. mi. 150A. 211. 8P. 

53. How many R. in 1728P.? A. 43R. 8P. 

138 



PROrvilSCUOUS PROBLEMS. §198 



54. How many sq. mi. in 1895A.? 

55. How many A, in 1806P.? A. IIA. IK. 6P. 

56. What is the sum of 7 thousand, 83 thousand and 40, 9 
hundred and 70, and 17 times 5 hundred and 79 1 

A. 100853. 

57. What is the difference between the product of 85 and 
307, and the quotient of 999875 by 125 ? 

58. How many lb. in 7qr.? A. 1751b. 

59. How mary oz. in 251b. Avoir.? A, 400oz. 

60. The distance from High Point to Greensboro' is 15mi., 
from Greensboro' to Shops 22mi., from Shops to Fta- 

4 leigh 53mi.; how far is it from High Point to Raleigh, 
via Greensboro' and Shops ? 

61. The distance from Charlotte to High Point is 79mi., 
from High Point to Ilalcigh 95mi., from Kaleigh to 
Goldsboro' 49mi.; hov/ far is it from Charlotte to 
Goldsboro', via High Point and Raleigh ? A. 223mi. 

62. Bought a pair of horses for $375, a set of harness for 

$55, and a buggy for $187 ; what did the whole cost ? 

A. $617. 

83. Paid $789 for a lot of tobacco, and sold it for $910 ; 

gained how much ? 

64. How many units in 14doz. and 7 ? A. 175. 

65. How many units in 3 score and 10 ? A. 70. 

66. How many doz. in 12 gross ? 

67. How many units in 10 great gross ? A. 17280. 

68. Bought 3 stone of potatoes at 2ct. per lb.; what did 

they cost ? A. 84ct. 

69. Bought 10001b. of fish at $9 per quintal ; what did I 

pay ? 

70. What cost 6161b. of butter at $15 a firkin ? A. $175. 

71. What cost 247bbl. of flour at $5 per bbl.? A. $1235. 

139 



§198 CONCRETE NUMBERS. 



72. How far will a traiu of cars go in 3 dajs, at 16 miles 
per hour ? 

73. Bought 16yd. of calico at 15cfc., 7yd. of gingham at 

25ct., 9yd. of flannel at GSct., and 25yd. of domestics at 
lOct.; paid 16bu. of corn at GSct.; how much is still 
due ? A. $1.89. 

74. If a hook of 155 pages has 29 lines on each page, and 

39 letters in each line, how many letters are in the 
book ? A. 175305 letters. 

75. I deposited in bank $10050 : having drawn out $15, 

$175, $237, $375, $4165, $394, and $3968, how much 
have I still on deposit ? ^ 

76. The Bible contains 31173 verses ; how many verses 

must I read each day, to finish it in one year 1 

A. 85 verses a day, and 148 verses over. 

77. How many sheets of paper in 20 quires ? A. 480sh. 

78. How many sheets in 14 reams ? 

79. How many reams in 180 quires ? A. 9rm. 

80. How many quires in 19 reams ? A. 380qr. 

81. A stock-dealer bought 4'7 cows at $19, 29 horses at 

$135,53 mules at $97, and 155 sheep at $3: he received 
for them 347 acres of land at $26^ and $4125 in money; 
how much did he gain ? 

82. What will 574bbl. of pork cost at $13 per bbl.'? 

A. $7462. 

83. How far will a man travel in 6da. at 29mi. per da.? 

A. 174mi. 

84. A planter who worked 57 hands, raised 399 bales of cot- 

ton : how m.any bales did he raise to the hand ? 

85. In $45, how many ct.? A. 4500ct. 

86. In M, 5s. 6d., how many d.? A. 1026d. 

140 



PROMISCUOUS PROBLEMS. ijlDb 



87. In 240dwt., boyr raaDj oz.? 

88. In 39sc.j liow many dr.? A. I3dr. 

89. In 3T. 3qr. 201b. 12oz., how many oz.? A. 97532oz. 

90. In 7920in., how many yd.? 

91. In 4mi., how many ch.? A. 320ch. 

92. In 1568160sq. in., how many sq. yd.? A. 1210sq. yd. 

93. In 4sq. mi., how many A.? 

94. In 4cu.yd. 12cn.ft., how many cu.in.? A. 207360cu.in. 

95. In 3025gi., how many hhd.? ' A. Ihhd. 31gal. 2qt. Igi. 

96. In Sbu.j how many pt.? 

97. In 3da. lOhr. 15mija., how many sec? A. 296l00scc. 

98. In 3S. 3° 3' 3'^, how many seconds ? A. 334983''. 

99. In 2gr. gr, 3gr. 4doz. and 5, how many units ? 

100. In 6rm. 7qr. 8sh., how many sh.? A. 3056sh. 

101. In 3gal. 3qt. 8gi., how many qt,? A. 15.375qt. 

102. In lObu'. Ipk. Igal. Ipt., how many pk.? 

103. In ^6, 6s. 6d. 3qr., how many s.? A. 126.5625s. 

104. Add 4:1b. 5 -^-oz., -idwt., and -Vgr., in gr. 

Sum, 1688.2gr. 

105. Add 3.5hr.5 7.75min., and .15sec., in min. 

106. Add iA., A.R., and -^j^V., in P. . Sum, 48.1P. 

107. Add .25cu. yd., .375cu. ft., and .625cu. in., in cu. ft. 

Sum, 7.12536 + CU. ft. 

108. From .9cwt., take .251b., in oz. 

109. From .751b., take .5dwt., in oz. Rem. 8.975oz. 

110. From 10.875s., take 9.15^., in qr. Rem. 485.4qr. 

111. From 5.5da., take 5.5min., in min. 

112. Multiply .75gal. by 7.5, in pt. Prod. 45pt. 
118. Multiply 2.25A. by .125, in P. * Prod. 45P. 

114. Divide 4.5mi. by 5.4, in rd. 

115. Divide 1.55s. by 2.3, in d. Quot, 8.08695d. 

141 



§199 COMPOUND NUMBERS. 



OPERATIONS ON COMPOUND NUMBERS. 



§ 199. The operations on compound numbers are analo- 
jxous to the corresDondino; ones on abstract -iiumbers. 



ADDITION OF COMPOUND NUMBERS. 



Ex. 1. Add together 4hhd. 25gal. 3qt., 5hhd. 20gal. 2qt., 
7hhd. 17ga]. 2qt. 

§ 200. MoDEL.~2 and 2 are 4, 
4hhd. 25gal. oqt. ^nd 3 are 7, 7qt., equal to Igal. 
5 '' 20 " 2 '^ 3(^1^^^ gg. (lo^jj 3 . 1 and 17 are 18, 
7 " 17 '' 2 '- ^^^ 20 are 38, and 25 are 63, 63 



17 " " 3 " gal., equal to Ihhd., set down 

1 and 7 are 8, and 5 are 13, and 4 
are 17, i7hhd. The sum is 17hhd. 3qt. 

Explanation. — In simple numbers ten units of any de- 
nomination make one of the next higher., In compound 
numbers this uniformity of relation does not exist. Thus 
in the example above, 4qt. make Igal., but 63gal. make 1 
hhd. With this exception, the explanation in § 22 will 
suffice for this case. 

Ex. 2. Add ^10, 14s. 9d. 3qr., J^5, 16s. 6d. 2qr.,^7,10d. 
Iqr., .212, 9s. 9d. 3qr. 

3. Add ^4, 10s. lid., £J, 8s. 9d. 3qr., ^8, 10d.,and 16s. 
3qr. Sum, =£20, 16s. 7d. 2qr. 

4. Add 101b. lOoz. lOdwt. lOgr., 121b. 9oz. 6dwt. 3gr., 
91b. lloz. I3dwt. 15gr., and 24ib. Boz. 15dwt. 20gr. 

Sum, 581b. 4oz. 6dwt. 

5. Add 31b. 6oz. 9dwt. 12gr., 61b. 8oz. lOdwt. 12gr., 81b. 
lloz. 14dwt. 17gr., and 141b. lloz. 8dwt. 5gr. 

142 



SUBTRACTION. §201 



6. Add 101b. 9ok.7dr.2sc.l5gr.,10oz.6dr. Isc. lOgr:, 15 

lb. lloz. 7dr. 2gc. 19gr., and 31b. 4oz. 5dr. 6gr. 

Sum, 3 lib. loz. 3dr. Isc. lOgr. 

7. Add lOT. lOcwt,. lOlb. lOoz. lOdr., 14T. 15cwt. 3qf. 

151b. 13oz. 15dr., and 25T. 7cwt. Iqr. 201b. 8oz. €dr. 
Sam, SOT. IScwt. Iqr. 221b. loz. Idr. 

8. Add Ssq.mi. 300A. 2Fi.. 25P., 7sq.mi. 525A. 311. lOR, 

19sq.mi. 285A. 3R. 19P., and 250A. 25P. 

Sum, 31sq.mi. 82A. IR. 39P. 

9. Add 19ou.yd. 19cu.ft. 19cu.in., 25cu.yd. 25cu.ft. 250cu. 
in., and lOOcu.jd. IScu.ft. 1555cu.in. 

10. Add 4bhd. 40gal. 2qt. Ipt. 3gi., lOhbd. lOgal. Iqt. Ipt. 

Igi., and 20bhd. 43gal. 3qt. Ipt. 3gi. 

IBum, 35hhd. 32gal. 3gi. 

11. Add lObu. Spk. 7qt. Ipt., 9bu. 2pk. 6qt. Ipt., 16bu. 

3pk. Gqt., and 15bu. Ipk. 5qt. Ipt. Sum, 53bu. Iqt. Ipt. 

12. Add 30da. ;LOhr. 30min. SOsec, 15da. 15hr, 15min. 15 
sec, and lOda. 20hr. 45rain. 15s8c. 

13. Add 25<^ 15' 25'', 75° 24' 50", and 15° 50' 45'^ 

Sum, 116° Sr. 

14. Add 2rm. lOqr. 12sh., 4rm. 15qr. ISsh., and 3rm. 9qr. 
lOsh. 

15. Add 2gr. gross, 10 gross, 7doz. 5 units, 4gr. gross, 8 
gross, 6doz. 7 units, and 5 gross, 8doz. 6 units. 



SUBTRACTION OF COMPOUND NUMBERS. 



Ex. 1. From o-ei7, 5s. 6d. 3qr., take ^8, 10s. 9d. 2qr. 
o-i" - PA ^ §201. Model. — 2 from 3 leaves 

q' io^; o a Sir 1 ; ^ ^^'01^ 18 ^e^ves 9 ; 11 from 25 
. .^ilLj__l_ leaves 14 ; 9 from 17 leaves 8. The 

8, 14" 9 " 1 « remainder is ^8, 14s. 9d. Iqr. 

143 



§201 COMPOUND NUMBERS. 



Explanation. — As 9d. can not "be taken from 6d., we add 
Is., that is 12d., to the minuend, and subtract 9d. from 18d. 
We then add Is. to the subtrahend, and proceed. See 
^§28, 30. 

Ex. 2. From 501b. 6oz. 15dwt. 19gr., take 101b. 17dwt. 

Rem. 401b. 5oz. ISdwt. 19gr. 

3. From 151b. logr., take 121b. 9oz. lOdwt. 12gr. 

4. From lOT. lOcwt. lOoz., take 5T. 15cwt. 20ib. 12oz. 
lOdr. Rem. 4T. 14cwt. 3qr. 41b. 13oz. 6dr. 

5. From 6sq.mi. 2R., take 375A. 25P. 

Rem. Ssq.mi. 265A. IR. 15P. 

6. From 250cu.yd. 20cu.ft. 875cu.in.j take 79cu.yd. 25cu. 

ft. G95cu.in. 

7. From 15T. 15cwt. 3qr. 151b., take lOT. 19cwt. 3qr. 
191b. Rem. 4T. 15cwt. Sqr. 211b. 

8. From lOhhd. lOgal. Iqt. Ipt., take 9hhd. 33gal. 3qt. 

ogi. Rem. 39gal. 2qt, ]gi. 

9. From 4 tuns, Ipi. Ihhd. 5gal. 2qt. 3gi., take 2 tuns, 60 
gal. oqt. 3gi. 

10. From 175bu. Ipk. 3qt. Ipt., take 54bu. 3pk. 2qt. 

. Rem. 120bu. 2pk. Iqt. Ipt. 

11. From 27bu. 2pk. Ipt., take ISbu. 5qt. 

Rem. 14bu. Ipk. oqt. Ipt. 

12. From 30da. lOhr. 15min., take 17da. 15hr. 15sec. 

13. From 180°, take 74° 14' 45". Rem. 105° 45' 15". 

14. From 90°, take 35° 41' 15". Rem. 55° 18' 45". 

15. From 100° 17' 30", take 90° 25' 45". 

16. From 22T. Scwt. 2qr. 201b., take 12T. 18cwt. 221b. 

Rem. 9T. lOcwt. Iqr. 231b. 

17. From 16hhd. 24gal. 3qt. 2pt., take I4hhd. 37gal. 3qt. 

18. From 236bu. 2pk. 5qt. Ipt., take 17bu. 2pk. 7qt. 2pt. 

144 



MULTIPLICATION. §202 



MULTIPLICATION OF COMPOUND NUMBERS. 



Ex. 1. Multiply £11, 5s. 6d. 3qr. by 15. 

i^l7, 5s. 6d 3qr. §202. Model.— 15 x 3=45. 45qr.= 

^^ lid. Iqr.; sefc down 1: 15x6=90, 

259, 3 " 5 " 1 " nnd 11 = 101. 101d.=8s.5d.; set down 
5: 15x5=75, and 8 = 83. 88s. = ie4, 

OS. ; set down 3 : 15 x 17=255, and 4=259. The product 

is £2bd, 3s. 5d. Iqr. 

Explanation.— See §§ 38, 200. 

Ex. 2. Multiply £bS, 10s. 9d. 2qr. by 4. 

Prod. ^214, 3s. 2d. 

3. Multiply 13° 15' 45" by 7. 

4. Multiply 25° 30' 45" by 10. Prod. 255° T 30". 

5. Multiply 501b. 6oz. 15dwt. 19gr. by 13. 

Prod. 6571b. 4oz. 5dwt. 7gr. 

6. Multiply 121b. 9oz. lOdwt. 12gr. by 16. 

7. Multiply 5T. 15cwt. 201b. l2oz. lOdr. by 19. 

Prod. 109T. 8cwt. 3qr. 171b. 9oz. 14dr. 

8. Multiply 2sq. mi. 200A. 2R. 20P. by 22. 

Prod. 50sq. mi. 573A. 3R. 

9. Multiply 3cu. yd. 25cu. ft. 750cu. in. by 25. 

Prod. 98cu.yd. 14c u. ft. 1470cu. in. 

10. Multiply 9hlid. 33gal. 3qt. 3gi. by 21. 

11. Multiply 2 tuns, 60gal. 3qt. 3gi. by 24. 

Prod. 53tijps, Ipi. Ihhd. llgal. Iqt. 

12. Multiply 25bu. 3pk. Iqt. Ipt. by 29. 

13. Multiply lObu. Ipk. 4qt. by 35. Prod. 363bu. 4qt. 

14. Multiply lOda. lOhr. lOmin. lOsec. by 41. 

Prod. lyr. 62da. 8hr. 56min. 50sec. 

15. Multiply 17da. 15min. 15sec. by 50. 

J 145 



§203 COMPOUND NUMBERS. 



DIVISION OF COMPOUND NUMBERS. 



Ex. 1. Divide 15° 15' 50'' by 10. 

10) 15° 15' 50" §203. Model.— 10 in 15, once, 
1° 31' 35" with 5 over, set down 1 ; 10 in 315, 
31 times, with 5 over, set down 31 : 
10 in 350, 35 times. The quotient is 1° 31' 35". 

Explanation. — 10 is contained once in 10; so that there 
are 5° undivided. These 5° are reduced to 300', and added 
to the 15', making 315'. In like manner, the 5' undivided 
are reduced to 300", and added to the 50", making 350". 

Ex. 2. Divide £^0, 16s. 2d. Iqr. by 3. 

3. Divide £Q0, Is. 5d. by 4. 

4. Divide 291b. 2oz. 2dwt. 2gr. b;, :». 

Quot. 51b. lOoz. lOgr. 

5. Divide 2421b. 5oz. lldwt. 16gr. by 8. 

Quot. 301b. 3oz. 13dwt. 23gr. 

6. Divide 4481b. lOoz. I4dr. by 11. 

7. Divide 32T. 2qr. by 15. Quot. 2T. 2cwt. 2qr. 201b. 

8. Divide 52sq. yd. 5sq. ft. 128sq. in. by 20. 

Quot. 2sq. yd. 5sq. ft. lOOsq. in. 

9. Divide 97cu. yd. 22cu. ft. SOcu. in. by 26. 

10. Divide 91gal. Iqt. Ipt. by 34. 

Quot. 2gal. 2qt. Ipt. 2gi. 

11. Divide 79 tuns, Iqt. Igi. by 45. 

Quot. Ftun, Ipi. Ihhd. Igal. Iqt. Ipt. Igi. 

12. Divide 600bu. 3pk. 6qt. by 60. 

13. Divide 8wk. 3da. 71ir. 43min. 20sec. by 70. 

Quot. 20hr. 20min. 20sec. 

14. Divide 1150° 31' 15" by 75. Quot. 15° 20' 25". 

15. Divide 1° 41' 40" by 100. 

146 



I'ROMISCUOUS PROBLEMS. §201 



J 6'. Divide 57T. lOcwt. Iqr. 171b. 14oz. by 9cwt. Iqr.lTlb. 
lOoz. 

57T. lOcwt. Iqr. 171b. 14oz. = 1855086oz. 
Ocwfc. Iqr. 171b. 10oz. = 15082oz. 
. 1855086oz.-^15082oz. = 123. 

§ 204. MoDEL.—Recluce the dividend to oz. (§ 189). Ke- 
duce the divisor to oz. (§ 189). Divide the dividend by the 
divisor. (§ 185). The quotient is 123. 

Ex. 17. Divide 294lbii. by 45bu. 3pk. 6qt. Ipt. 

Quot. G4. 

18. Divide 97T. llcwt. 3qr. 111b. lOoz. by IT. 6cwt. 2qr. 

261b. lOoz. 

19. Divide 17bu. Ipk. 6qt. by 2bu. Spk. 5qt. Quot. 6. 

20. Divide 51A. IR. IIP. by lA. IP. Quot. 51. 

21. Divide 10 tuns, 2hhd. ITgal. 2pt. by 39gal. 6pt. 

22. Divide ^£27, 2s. 6d. by 15s. Gd. Quot. 35, 



PROMISCUOUS PROBLEMS. 



1. What is the least common multiple of 15, 24, and 27 ? 

2. What is the least common multiple of 9, 25, and 45 ? 

L. C. M. 225. 

3. What is the greatest common measure of 505, 1111, 

and 3434? 

4. What is the greatest common measure of 1015, 1260, 
and 1330? G. C. M. 35. 

5. What are the prime factors of 6105 1 

6. What are the prime factors of 4060 ? 

7. Divide ^£113, 13s. 4d. by 31. Quot. £S, 13s. 4d. 

8. Divide 10 tuns, Ipi. 17gal. 2pt. by 67. 

9. Divide 50T. 4cwt. 2qr. I4lb. by 23cwt. 3qr. 171b. 

147 



§204 PROMISCUOUS PROBLEMS. 



10. Divide 1572yd. by 32yd. 3qr. Quot. 48. 

11. Multiply 25oz. 8dwt. 17gr. by 100. 

Prod. 2111b. lloz. lOdwt. 20gr. 

12. Multiply 21da. 181ir. 42niin. by 75. 

13. Subtract 40A. 3R. 25P. from 79A. 15P. 

Rem. 38A. 30P. 

14. Subtract 4 tuns, Ipi. llihd. 25gal. 3qt. from 5 tuns, Iqt. 

15. Add 60mi., 40mi. 7fur. 39rd., and 19mi. Ird. 

16. Add 13° 14' 15", 16° 17' 25", 25° 19' 47", and 3° 15". 

17. Divide $1521808938 by 234. Quot. $6503457. 

18. Divide I42651ihd. by 45hhd. 

19. Multiply 4327bu. by 102. Prod. 44l354bu. 

20. Multiply 47935gal. by 275. Prod, 13182l25gal. 

21. Subtract 2598328fur. from 3002575fur. 

22. Subtract 187564329gi. from 923465781gi. ' 

Rem. 73590l452gi. 

23. Add 2479A., 3580A., 1358A., and 9745A. 

24. Add ^613575, £2Udb, £9475, and £31525. 

25. Divide 82960332 by 84. Quot. 987265 ; Rem. 72. 

26. Divide 82071 by 99. Quot. 829. 

27. Multiply 24068 by 13579. 

28. Multiply 1020908 by 8979091. Prod. 9166825834628. 

29. Subtract 3987456002 from 4567398745. 

30. Subtract 246 + 357 + 1298 + 982 from 3120. 

31. Add 20030405, 910285, 5821090, and 9706845. 

Sum, 36468625. 

32. Add 123, 1234, 12345, 123456, 1234567, 12345678, 

and 123456789. Sum, 137174192. 

33. A. raised 125 bales of cotton, 517bu. corn, 629bu. wheat, 

and 119bu. rye ; B., 217 bales of cotton, 865bu. corn, 

798bu. wheat, and 143bu. rye ; C, 94 bales of cotton, 

424bu. corn, 517bu. wheat, and 77bu. rye; and D., 

148 



PROMISCUOUS PROBLEMS. §204 

111 bales of cotton, 512bu. corn, 558bu. wheat, and 

98bu. rye. How much of each article did they all 
raise ? 

84. A farmer went to town with $100, and spent $9 for 
molasses, $13 for sugar, ^11 for coffee, $8 for rice^ 
$17 for dry goods, and $25 for leather. Horv much 
money had he left ? Ans. $17^ 

ii5. Bought 47 acres of land at $19, 5 horses at $125, 10 
head of cattle at $21, 14 sheep at $3, and a two-horse 
wagon for $65 ; what did they all cost ? Ans. $1835. 

36. Sold 75 firkins of butter for $1350 ; how much was 
that a firkin ? 

37. What number is that, to which if 245, 379, 124, 212, 
and 399 be added, the sum will be 1525 ? Ans. 166. 

38. What number is that, from which if the sum of 245, 
379, 124, 212, and 399 be subtracted, the remainder 
will be 1525? Ans. 2884. 

39. AVhat number is that, by which if twice 19 be multi- 
plied, the product will be the difference between 4127 
and 2759 ? 

40. What number is that, by which if 4235 bo divided, the 

quotient will be 77? Ans. 55. 

41. A miller has 5 bins, one of which holds 43bu. 3pk. 5qt.; 

the second, 39bu. Ipk. 3qt. ; the third, 45bu. Iqt. Ipt.; 
the fourth, 53bu. 2pk. ; the fifth, 34bu. 3pk. Ipt, i 
what is their united capacity? Ans. 216bu. 2pk, 2qt. 

42. How much time elapsed between Jan. 20th, 1833, and 

May 25th, 1861 ? 

43. Bought 4 lots of land, containing 3B. 27P, each ; how 

many A. did I buy ? Ans. 3A. 2B. 28P. 

44. A wine merchant has 269gal. 2gi. of wine in 30 equal 

vessels; how much wine is there in each vessel ? 

149 



P04 



PROMISCUOUS PROBLEMS. 



45. 

46. 

47. 

48. 

49. 

50. 

51. 

52. 

53. 

54. 

55. 

50. 

57. 

58. 

59. 

60. 

61. 

62. 

63. 

64. 

65. 

66. 

67. 

68. 

69. 

70. 

71. 

72. 

73. 

74. 

75. 

76. 



Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

,Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 

R.educe 

Reduce 

Reduce 

Reduce 

Reduce 

Reduce 



Val. 69119gr. 



£40, 19s. lid. 3qr. to qr. 

111b. lloz. 19dwt. 23gr. to gr. 

aib. II5. 75. 29. lOgr. to gr. Val. 2B039gr, 

2T. 19cwt. 3qr. 24lb. 15oz. 15dr. to dr. 

4L. 2mi. 7fur. 35rd. to rd. Val. 4795rd. 

12yd. 2ft. lliu. to in. Val. 467in, 

2sq. mi. 600A. 3R. 35P. to P. 

25sq. yd. 8sq. ft. lOOsq. in. to sq. in. 

21sq. mi. 250A. 2R. to R. Val. 54762R 

5cu. in. 20cu. ft. 1600cu. in. to cu. in. 



3 tuns, Ipi. llibd. to bhd. 
ohbd. 60gal. 3qt. Ipt. to pt 
2gal. Iqt. Ipt. 3gi. to gi. 
5bu. 3pk. 7qt. Ipt. to pt. 
30. 75yr. 300da. to da. 
4da. lObr. 25min. to sec. 
25° 10' J 5" to seconds. 
2rm. 15qr. ]2sb. to sh. 

4 score and 5 to units. 
lOOOqr. to £. 
6000gr. to lb. Troy. 
eOOOOOdr. to T. 
lOOOrd. to mi. 
2000sq. in. to sq. yd. 
200000sq. rd. to sq. mi. 
60000cu. in. to cu. yd. 
lOOgi. to gal. 
500qt. to bhd. 
lOOpt. to bu. 
4000sec. to hr. 
200br. to wk. 
4000" to degrees. 

150 



Val. 15bbd. 
Val. 1999pt, 

Val. 383pt. 
Val. 137175da. 

Val. 90615". 
Val. 1332sh. 

Val. £1, lOd. 
Val. lib. lOdwt. 



Val. 



om. 



Ifur. 



Val. 3gal. Ipt. 

Val. Ibu. 2pk. 2qt. 
Val. Ibr. 6min. 40sec. 

Val. 1° 6' 40". 



PROMISCUOUS PROBLEMS. §204 



77. Bought 8 firkins of butter at 20ct. per lb., 20qt. mo- 
lasses at $1 per gal., 3 stone of potatoes at 3ct. per lb., 
and 9801b. flour at $S per bbl.; what did they all cost ? 

78. Bought 5doz. Arithmetics at 30ct. apiece ; sold them 

all for $27 : how much apiece did I gain or lose ? 

79. A owes B for 5001b. of salt fish at $8 a quintal ; B 

owes A for 8bbl. flour a^ $7 a bbl., lObu. corn at 70ct. 
per bu., and 5bu. rye at 80ct. per bu. : how does their 
account stand ? A owes B $18. 

80. From 500 subtract the sum of 225, 120, and 75 ; divide 

the remainder by the diff"erence between 1000 and 
960 ; multiply the quotient by 17 ; and add 16 to the 
product. Sum, 50. 

81. Find the sum of the product of 88 and 11, and the quo- 

tient of 88 by 8. 

82. A man has 1184bu. of wheat and 468bu. of corn, which 
he wishes to pack in equal bags as large as possible. 
How many bushels will each bag hold ; and how many 
bags will be required ? Ans. 4bu., and 413 bags. 

83. What is the value of 85 + 77-64 + 6 x 19-132-^-4 ? 

84. What is the value of 15498-^54 + 41 x 63-27 x 55 ? 

85. Find the least common multiple of 4, 44, 132, and 792. 

86. Find the greatest common measure of 4, 44, 132, and 

792. G. C. M. 4. 

87. Two men travel in the same direction from the same 
place, one 40mi. a day, the other 33mi. a day ; how far 
apart are they in 7 days ? 

88. Two men travel in contrary directions from the same 

place, one 40mi. per day, the other 33mi. per day ; 
how far apart are they in 7 days? Ans. 511mi. 

89. If 10 persons use a barrel of flour in 57 days, how long 
will a barrel last one person ? Ans. 570da. 

151 



§204 PROMISCUOUS PROBLEMS. 



90. "What ig the sum of 3 numbers, of which the first is 28, 
the second 8 times the first, and the third one seventh 
of the second ? 

01. The difference is one hundred thousand four hundred 
and seventy-six, the minuend is one million ; what is 
the subtrahend ? Ans. 899524. 

92. The minuend is one hundred thousand, the subtrahend 
is sixty-seven thousand seven hundred and forty-four; 
what is the remainder ? Ans. 32256. 

93. The subtrahend is f:. ■> i Iiundred thousand and forty- 
nine, the remainder is ninety-nine thousand two hun- 
dred and seventy-eight ; what is the minuend? 

94. The multiplicand is thirty-six thousand seven hundred 

and seven, the multiplier is eighty thousand and one ; 
what is the product? Ans. 2936596707. 

95. The multiplier is eight h-T^idred and four, the product 

is sixty-one thousand nine hundred and eight ; what is 
the multiplicand ? Ans. 77. 

96. The product is eighteen billions two hundred and twen- 
ty thousand, the multiplicnnd is two thousand two 
hundred ; what is the multiplier? 

97. The divisor is one hundred and twenty-five, the divi- 

dend is nine hundred and eighty-seven thousand six 

hundred and twenty-five; what is the quotient? 

Ans. 7901. 

98. The dividend is thirty-four thousand eight hundred 
and forty-eight, the quotient is one hundred and thir- 
ty-two ; wliat is the divisor ? Ans. 264. 

99. The quotient is thirty thousand and seventy, the divi- 
sor is seven hundred and eight ; what is the dividend? 

100. What cost ISrm. of paper at $4, 3doz. Arithmetics at 
$6 a dozen, and 24 Algebras at 812 a dozen ? 

152 



ALIQUOT PARTS. §20G 

ALIQUOT PARTS. 



§ 205. An aliquot fraction is a simple fraction whose nu- 
merator is 1. Thus, ^V, jf, -}, To> -^Vj ^^® aliquot fractionj^!. 

An aliquot part of a number is a part denoted by an ali- 
(|uot fraction. Thus, ?> is an aliquot part of 12, 10s. is an 
aliquot part of ^61, 20da. is an aliquot part of 2mo. 

Ex. 1. What is the cost of 5A. 3R. 25P. of land at $45. 

50 per A.? 

2) $45.50=1A. , §206. M0DEL-5A cost 

^ - 5 times as much as lA. ; — 







hence, multiply the cost of 
$227.50r:r5A. YK ^ 5 (§183). 2R. is 

^) ""*"•/ ^ "tS" ^ Slf- ^^^ ^^^ *^^' \k.\ hence, di- 

2) 11.375=rlR. S ' vide the cost of lA. by 2. 

4) 5.687=:20P. ^ o^p (§185). IR. is one half of 

_J^421=r 5P. j • 2E.; hence, divide the cost 

$268,733 of 2R. by 2. 20P. is one 

half of III. ; hence, divide 
the cost of IR. by 2. 5P. is odc fourth of 20P.; hence, di- 
vide the cost of 20P. by 4. Add the several costs together. 
The sum is $268,733 : hence, 5A. 3R. 25P. cost $268,733. 

Ex. 2. What is the cost of 17T. 15cwt. 3qr. 101b. of iron 
at $36 per T.? Ans. $040.53. 

3. What is the cost of lObu. 3pk. 4qt. Ipt. of grass seed 
at $8 per bu.? 

4. What is the yield of 45A. 3R. 15P. of wheat land at 

20bu. per A.? Ans. 916.875bu. 

5. What is the value of 21b. 9oz. lOdwt. 6gr, of plate at 

$15 per lb.? Anso'?4l.88. 

6. What is the value of 5T. lOcwt. Iqr. 51b. of hay at $25 
per T.? 

153 



§207 ALIQUOT TARTS, 



7. What is the cost of 15bu. Ipk. 5qt. of dried peaches at 
80 per bu.? Ans. 692.4375. 

8. What is the cost of 4gal. Iqt. Ipt. of wine at $4 per 
gal.? Ans. 817.50. 

9. What is the co.st of 37rd. 2^jd. of fencing at $3.50 
per rd.? 

10. What is the cost of 7yd. 2ft. Gin. of cloth at $7.25 per 
yd.? Ans. s856.791 + . 

1 1. AVhat is the cost of SOsq.yd. 4sq.ft. 72sq.iu. of painting 
at $.75 per sq.yd.? ' Ans. $22,875. 

12. What is the cost of lOcd. SOcu.ft. of wood at $2.50 
per cd.? 

13. What will a man earn in 9mo. lOda. at $25 per mo.? 

Ans. $233.a3-J. 

14. What will lOcd. of wood cost at $2.62^ per cd.? 

2)$10 =cost of lOcd. at $1.00 per cd. 



$20 


— 


a 


a 


a 


2.00 


(( 


i( 


4) 5 


— 


a 


u 


a 


.50 


i( 


iC 


1. 


'2d= 








.12.^ 
2.02^ 




a 


$20. 


25= 


(( 



§ 207. Model,— 10 cords at $1 would cost $10. The 
cost at $2 is twice the cost at $1 ; hence, multiply the cost 
at $1 by 2. The cost at 50ct. is one half of the cost at $1; 
hence, divide the cost at $1 by 2. The costatl2ict. is one 
fourth of the cost at 50ct.; hence, divide the cost at 50ct. 
by 4. Add the several costs together. The sum is $20.25: 
hence, 10 cords at $2.G2| cost $20.25. 

15. What is the cost of 3G0bu. of wheat at $1.37-^- per bu.? 

16. What is the cost of 15yd. of cloth at £1, 4s. 9d. per 
yd.? Ans. ^18, lis. 3d. 

154 



CONTRACTED MULTIPLICATION AND DIVISION. §209 



CONTRACTION IN MULTIPLICATION. 



Ex. 1. Multiply 279 by 33^. 

q^97qnn ^^^^' ^-foDEL.— Annex 2 nauahts to the 
o;_wJuu multiplicand :— divide the result'by o The 
9300 product is 9300. 

Explanation.— See §63. When the multiplier is an ali- 
quot part of any power of ten, we may abridge the work by 
multiplying by this power of ten (§56) and dividing the re- 
sult by the denominator of the aliquot fraction. 

Ex. 2. Multiply 72 by 12i Prod. 900. 

.3. Multiply 72 by 16|-. 

4. Multiply 77 by 14f . Prod. 1100. 

6. Multiply as by li Protl. 70. 

6. Multiply 684 by 166§. 

7. Multiply 273 by 333-}. Prod. 91000. 



CONTRACTION IN DIVISION. 



Ex. 1. Divide 2000 by 142^. 

2000 §209. Model.— Multiply the dividend by 

7 7 :— divide the product by 1000. (§ 65.) The 

14 000 quotient is 14. 

Explanation.— See § 68. When the divisor is an aliquot 
part of any power of ten, we may abridge the work by mul- 
tiplying the dividend by the denominator of the aliquot 
fraction and dividing the product by the power of ten. {^65.) 

Ex. 2. Divide 150 by 33^. Quot. 4.5. 

3. Divide 250 by 14|-. 

4. Divide 1500 by 166|-. Quot. 9. 

155 



^209 rROMiscuous phoblems. 



5. Divide 245 by 12 i. gaot. 19.G. 

C. Divide 1375 by 11^-. 

7. Divide 2468 by 333^. Quct. 7.404. 

PROMISCUOUS PPtOBLEMS. 



1. Find 246 + 2468 + 24680 + 20468 + 24068 + 24608. 

Sura, 96538. 

2. Find the sum of one half, three fourths, two fifths, and 
one sixth. Sura, 1|^. 

3. Add 315, 31.57, 3157, 3.157, and .3157. 

4. A farraer raised $357 worth of corn, $475.50 worth of 
wheat, $123.75 worth of rye, and $446.37-^ worth of 
other products ; what was the total value of the prod- 
ucts of his farm ? Ans. $1402.62-v. 

5. A's gold mine yielded -j^lb. in one week, B's yielded 7 
oz., and C's 125.5dwt.; how many ounces did the three 
mines yield ? Ans. 19.275oz. 

6. A tobacco planter raised from one field 7cwt. Iqr. 20 

lb., from an other lOcwt. oqr. 101b., and from an other 
15cwt. 151b.; what amount did ho raise in all? 

7. A man sold 14 horses at $125.75 a head, and 25 head 
of cattle at $19.87^ a head ; what did he receive for 
them all '/ " Ans. $2257.375. 

8. Find the value of 135-1357 + 13570-3571-3157. 

Yal. 5620. 

9. What is the difference between thirteen fourteenths and 
fourteen fifteenths? 

10. From 500.05 take 65.556. Rem. 434.494. 

11. A dealer sold a lot of bacon for $2425.10^, which cost 
him $2177.4';^ ; wliat did he gain by the trade ? 

156 



PROMISCUOUS PROBLEMS. S200 



12. Find the difference in K. betvreeii ^A, and 27.5P. 
18. Wasliiugton was born Feb. 22, 1732, and died Dec. 14, 
1799 ; what was his age? Ans. OTyr. 9mo. 22da. 

14. A gave B 2501b. of beef at 8ct. for o?Ah. of leather at 

60ct.; how does their account stand '/ 

Ans. B owes A 20ct. 

15. What is the product of 7903 x 3907 ? 

16. Multiply three eighths by two and two thirds. 

Prod. 1. 

17. Find the value of 19.275 x 21.125. Yal. 407.184375. 

18. A planter who works 47 hands raises to each hand ten 

bales of cotton averaging 4461b.; how much does his 
cotton jaeld him at 9ct. per lb.? 

19. A farmer owns 4 farms containing each 47A. 25P.; what 

do the four contain ? ^ Ans. 188A. 2R. 20P. 

20. A jeweller has 25 gold rings weighing S^-dwt. each; how 
many oz. do they all weigh ? Ans. 4|oz. 

21. Bought 22gal. of molasses at 75ct., 2471b. of sugar at 

16|ct., 1751b. of rice at 6}ct., and 57-^-lb. of coffee at 
18 ^ct.; what was my bill ? 

22. Find the value of 25000-4-125 + 1475^25. Yal. 259. 

23. What is the quotient of three fourths by seventeen 
thirty-thirds? Ans. l|-t. 

24. What is the value of 17.375-^2.5-9.63-^3.3 ? 

25. In 392981b. of iiour, how many bbl.? Ans. 200-^bbl. 

26. A father divided 778A. 3R. 2 IP. equally among his 

seven children ; what was each child's share ? 

Ans. IIIA. IR. 3P. 

27. What is the cost of 273bu. of wheat at $1.66f per bn.? 

28. What is the cost of 29rra. of paper at $2.75 per rm.? 

Ans. $79.75. 

29. What part of lOda. is 7hr. 15min.? Ans. ■.%%. 

157 



§209 PROMISCUOUS PROBLEM ij. 



30. What part of 5gal. Iqt. is 3qt. Sgi.l 

31. Bought I of 17T. 3qr. of iron at 5ct. per lb.; what did 
I pay ? Ans. $1363. 

32. In 31b. Avoirdupois, how many oz. Troy ? 

Ans. 43.75oz. 

33. What part of lib. Avoir, is lib. Troy? 

34. How many cubic inches in 40qt. Wine Measure ? 

Ans. 2310cu.in. 

35. What cost 3doz. Arithmetics at $2.75 per doz.jl7 slates 

at I4ct., 5 gross of steel pens at 93^}ct. per gross, and 
300 slate pencils at 31|ct. per hundred ? 

36. Bought 156bbl. of flour for $936, and sold the same at 
^8.45 per bbl.; what did I gain ? 

37. In Ibu., how many qt. Wine Measure ? 

Ans. 37.236qt. 

38. How much heavier is a pound of feathers than a pound 
of gold ? Ans. 1240gr. 

39. Virginia contains 6 I352sq.mi.; North Carolina, 55500; 
South Carolina, 28000; Georgia, 58000; Florida, 
59268; Alabama, 50722 ; Mississippi,47151; Louisiana, 
41346; Texas, 325520; Arkansas, 52198; Missouri, 
65037; Tennessee, 44000; and Kentucky, 37680. 
What is the area of the Confederate States of America ? 

40. In lib. Troy, how many oz. Avoir.? 

Ans. 13.0281 -foz. 

41. What will 33J-yd. of cloth cost at $4.75 per yd.? 

Ans. $159,125. 

42. What will 17bbl. of flour cost at $7,875 per bbl.? 

43. What will 60bu. of wheat cost at $1,125 per bu.? 

44. A man borrowed $189.75, and paid at one time $37,375, 

at an other $23,625, and at an other $19.4375 ; how 
much does he still owe ? Ans. $109.3125. 

158 



PROMISCUOUS PROBLEMS. §209 



45. A lady bought a silk dress for $21,875, a lace mantle 
for $15.50, a pair of cloth gaiters for $3.25, and a 
bonnet for $9,625 ; what did they all cost ? 

4Q. If 25yd. of cloth cost $85.50, what does 1yd. cost ? 

* Ans. $3.42. 

47. What will 3651b. of flour cost at $4 per hundred ? 

Ans. $14.60. 

48. How many working days are there in a common year ? 

49. If a man receives $2000 a year, how much is that a day? 

Ans. $5,479. 

50. Bought 5bu. at $1.37-^- per bu., and sold them at 5ct. per 

qt. Wine Measure ; how much did I gain ? 

Ans. $2,434 + . 

51. How many steps of 28in. each, does a soldier take in 

marching 5 miles? 

o2. How many bottles containing Iqt. igi. each, can be filled 

from a hogshead of French Brandy ? 

Ans. 237.295 bottles. 

53. If a familj^ use 15bb]. of flour in a year, how much is 

that a day ? Ans. 8.051b. 

54. If a man travel 29mi. Tfur. I5rd. per day, how far will 
he travel in 5wk. if he rest on Sunday ? 

55. A lady went shopping with £>^, and spent -} of 14s.; 

how much had she left 1 Ans. 6|s. 

50. How many days are there from Jan. 17 to April 6 ? 

Ans. 79da. 

57. Sold one load of hay weighing I.IT., an other weighing 

1|^T., andathird weighing 17.3c wt.; what did the three 
loads weigh ? 

58. Bought } of ^ of an acre in one lot, 49P. in an other, 

and -} of lOR. in an other ; what did the three lots cost 
at $69.6875 per A.? Ans. $91,029 + . 

159 



^209 I'RO.MlbCUOUS PEOBLEMli. 

5*>. ^V iiat wiil ^o.ailb. of corn cost at 40ct. per bu.ir 

Ads. $18.21. 
GO. What will 10001b. of wheat cost at $1.37 J- per bu.? 
<il. If Ibu. of wheat will make 451b. of flour, how many> 

bl>l. will 15001b. make ? Ans. 5.74bbl. 

C>'2. How many secouds were there in the winter of 1850—60? 

Ans. 7S62400sec. 
G3. How many minutes were there in the summer of 1860 ? 
G4. How many acres of land at $1 per sq.yd. can be bought 

for S15000 ? Ans. 3.099A. 

()5. What will 200mi. of Tclegrapii wire cost at lOct. per 

yd.? An.s. $35200. 

66. How many pounds of flour in 75bbl.? 

67. What is the diiference in height between a man 5ft. 
llin. high, and a horse 16 hands high ? Ans. 7in. 

Os. Bought 101b. of rhubarb at $6.50 per lb. Avoir., and 
sold it at 50ct. pnr oz. Troy ; what did I gain ? 

Acs. 87.916. 

69. What cost 2127ft. of lumber at $ .8375 per hundred ? 

70. What cost 37560 bricks at 1^7.75 per thousand ? 

Ans. $291.09. 

71. What cotit 17 firkins of butter at 18^ct. per lb.? 

Ans. $178.50. 

72. What cost 5.5 stone of potatoes at 1.5ct. per lb.? 

73. What decimal fraction is equal to 02^-^129 ? 

" Ann. .484496 + . 

74. A man dying left $27000 to be divided so that his widow 
should have one third of it, each one of 4 sons one 
seventh of the remainder, and each one of 5 daughters 
one fifth of what was left ; what was each daughter's 
share ? Ans. $1542.857. 

160 



PROMISCUOUS PROBLEMS. §209 

75. A druggist having bought 60gal. of oil for $97.50, lost 

6.25gal. by leakage, and sold the remainder at $2,125 
per gal.: what did he gain? 

76. A merchant bought two bales of domestic, containing 
each 20 bolts of 38yd. at 13.25ct. per yd.: what did he 
pay ? Ans. $174.90. 

77. Bought SOObbl. flour at $6.75 per bbl.; sold one third 

of it at $7,375 per bbl., one half of the remainder at 
$7.9375 per bbl., and the rest at $8.50 per bbl. : how 
much did I gain ? Ans. $356.25. 

78. What will 2250bu. corn cost at |- of a dollar per bu.? 

79. A gentleman's house cost him four times as much as his 
furniture, and both together cost $4435 ; what did his 
furniture cost ? Ans. $887. 

80. A grocer had 7cwt. 3qr. of sugar, and sold at different 
times 3icwt., 3iqr., and 1271b.: how many lb. has he 
remaining ? Ans. 23l^lb. 

81. A planter sold 15 bales of cotton averaging 445.51b. at 
9ct. per lb., and with the proceeds bought land at 
$21.25 per A.: how much did he buy ? 

82. If 4f yd. of cloth cost $12|, what will 1yd. cost ? 

83. What cost 29A. IR. 18P. of land at $45,625 per A.? 

Ans. $1339.664 -f. 

84. What cost 9T. IGcwt. 151b. of iron at $37.75 per T.? 

85. A man left -} of his estate to his wife, -} of the remain- 
der to his son, and the remaining $2500 to his daugh- 
ter : what was his estate ? Ans. $7500. 

86. If 16^ days' work cost $19.75, what will 3:^ days' work 

cost'/ Ans. $3.89 + . 

S7. What must I pay for 6:2lb. of butter at 35ct. per lb., , 

12vdoz. eggs at 15ct. per doz., 10 chickens at 18fct. 

apiece, and 30 cucumbers at lOct. per doz.? 
K IGl 



§209 pROMiscuoiJS problems'. 



SS. If Ibbl. of tar cost $3,875, what will l7bM. cost ? 

Ans. $65,875. 

89. What cost 2471b. dried blackberries, at 15ct. per lb.? 

Ans. $37.05.- 

90. What will 1001b. of coffee cost at 61b. to the dollar? 

91. How many dollars will pay for 15 pieces of French cal- 

ico^ each containing 27yd., at 1.2fr. per yd.? 

Ans. $90,396. 

'92. "Howm&ny dollars will pay for 75 gross of G-illott's 

pens at ^So 6d. per gross? Ans. $63,526, 

93. What will 45bu. corn cost at 5}^ dimes per bu.? 

94. What will 7271b. salt cost at $1.25 per bu.? 

Ans. $18,175. 

95. What will Sbbl. flour cost at $.05 per lb.? Ans. $29.40. 

96. How many bu. of corn can be hauled by a team which 
can haul just 60bu. of wheat t 

.97. John's height is 3 cubits and a span ; his pony is 14 
hands high ; what is the difference of their heights ? 

Ans. 7in. 

98. How many ft. of water is drawn by a vessel which can 
not sail in less than 3 fathoms 2 feet ? Ans. 20ft. 

99. What wiiri280cu.ft. of wood cost at $1.75 per cord? 

100. What should be paid for 570bu. of corn, at $2.50 per 
bbl.? Ans. $285. 

101. A merchant bought 21 pieces of cloth, each containing 
41 yards, for which he paid $1260 ; he sold the cloth 
at $1.75 per yd.: did he gain or lose by the bargain ? 

Ans. He gained $246.75. 

102. A man receives } of his income, and finds it equal to 
$8724.16 : how much is his whole income ? 

103. If 322 books cost $371.91, what will 248 books cost at 
the same rate 1 Ans. $286.44 

162 



KATIO. §215 



EATIO. 

§ 210. The ratio of one number to an other of the same 
denomination is the quotient of the second divided hy the 
first. Thus, the ratio of 3 to 12 is 4 ; the ratio of 5ft. to 
15ft. is 3 ; the ratio of $17 to $8 is -V- 

§ 211. Since the two numbers compared are necessarily 
of the same denomination, every ratio is an abstract num- 
ber. (§ 185.) 

§ 212. Of two numbers compared, the first is called the 
antecedent, the second is called the consequent, and both to- 
fcether are called the terms of the ratio. Thus, in the first 
ratio above, 3 is the antecedent, 12 is the consequent, and 3 
and 12 are the terms. 

§213. A ratio is usually denoted by a colon placed be- 
tween the two terms. Thus, 3 : 12 is the ratio of 3 to 12 ; 
so also, 5ft. : 15ft.=:3 ; $17 : $8=-jV- 

§ 214, The ratio of two numbers ef the same nature but 
of different denominations may be found by first reducing 
them to the same denomination. Thus, 3ft. : 5yd. = 5; 
Oct. : $1 = 20. 

§ 215. The ratio of two numbers of different natures can 
not be found. Thus, 3ft. has no ratio to Sgal. 

Ex. 1. What is the ratio of 3 to 6 ? Ans. 2. 

2. What is the ratio of 10 to 75 ? Ans. 7.5. 

3. What is the ratio of 27 to 9 ? 

4. What is the ratio of 446 to 1338 ? Ans. 3. 

5. What is the ratio of $97 to $485 ? Ans. 5. 

6. What is the ratio of 27qt. to 9qt.? 

7. What is the ratio of 3qt. to 5gal.? Ans. 6.6. 

163 



§216 PROPORTION. 



8. What is the ratio of 7fur. to llmi.? Ans. 12.57142. 

9. What is the ratio of ^£.5 to 15s.? 

10. What is the ratio of -} to {-1 Ans. 1.75. 

11. What is the ratio of 3.75 to 11.25 ? Ans. 3. 

12. What is the ratio of 5:^ to 17^ ? 

13. What is the ratio of 3ioz. to 1-Vlb. Aroir.? Ans. 6-^. 

14. What is the ratio of 45min. to ihr.? Ans. -i^. 

15. What is the ratio pi 1.25cu.ft. to 2.5cu.in.? 

16. What is the ratio of f A. to 15P.? Ans. .125. 

17. What is the ratio of Ihhd. to 25gal.? Ans. .3668 + . 

18. What is the ratio of 1.5 cubits to 65 inches ? 



SIMPLE PEOPORTION. 



§216. A 2'>t'oportion is an equality of two ratios. Thus, 
the two ratios, 5 : 10, and Sin. : 6in., form a proportion. 

§ 217. A proportion is denoted by a double colon between 
the two ratios, or by a sign of equality between them. — 
Thus, 5 : 10 :: 3in. : 6in., read, 5 is to 10 as Sin. is to 6in. 
Or, 5 : 10z=:3in. : 6in., read, the ratio of 5 to 10 is equal 
to the ratio of Sin. to 6in. 

§ 218. The first two terms of a proportion are called the 
first couplet^ the last two are called the second couplet: — 
the first and third terms are called the antecedents; the 
second and fourth are called the consequents : — the first and 
fourth terms are called the extremes ; the second and third 
are called the means. 

Also, the fourth term is called a fourth proportional to 

the other three : and, when the means are equal to each 

other, either mean is called a mean proportional between 

the two extremes. 

164 



SIMPLE PROPORTION. §220 



§ 219. In every proportion, the product of the extremes 
is equal to the product of the means. 

For, if 3 : 6 :: 7 : 14, 6-3 = 14-7, or 2= V, multiply- 
ing both terms of the first fraction by 7, and both terms of 

the second by 3, we have ~~^ ^^, or 6 x 7=3 x 14. 

§ 220. This property enables us to find any term of a 
proportion when the other terms are given. 

If the two means and either extreme are given, to find 
the other extreme, we divide the product of the means by 
the given extreme. 

If the two extremes and either mean are given, to find 
the other mean, we divide the product of the extremes by 
the given mean. 

Ex. 1. 1st Term : 6 :: 5 : 15, what is the first term '/ 

Ans. 2. 
2. 7 : 2nd Term :: 14 : 70, what is the second term ? 

Ans. 35. 

o. 5-^- : 22 :: 3rd Term : 40, what is the third term ? ■ 

4. 8 : 1.6 :: 50 : 4th Term, what is the fourth term? 

Ans. 10. 

5. 15 : 1.875 :: 3rd Term : 5, what is the third term ? 

Ans. 40. 

6. 2i : 2nd Term :: 7-} : 12 1-, what is the second term ? 

7. 1st Term : 1a :: -| of a : 17, what is the first term ? 

Ans. ^. 
b. 29 : 2nd Term :: 17 : 49, what is the second term? 

Ans. 83-}-^. 
^. lo : 18 :: 3rd Term : 24, what is the third term? 
10. 2i : 31 :: 4^ : 4th Term, what is the fourth term ? 

Ans, 6.3. 
165 



§221 PROPORTION. 



§ 221. Whichever term is required, however, the given 
terms may always be arranged in the first, second, and third 
places, so that the work shall consist in finding a fourth 
proportional to the three given terms. Thus, 

The first question would become, 15 : 5 :: G : 4th Term; 

The second, 14 : 70 :: 7 : 4th Term ; 

The third, 12 : 5^ :: 40 : 4th Term. 

In finding a fourth proportional to three concrete num- 
bers, the first two terms must be of the same denomination, 
and this common denomination must be canceled before the 
operation is performed! In speaking, hereafter, of the first 
qr the second term, we will always mean the number of ab- 
stract units in such term. 

Ex. 11. Find a fourth proportional to 3in., 7in., and $12. 
3 : 7 :: $12 : 



p^2. Model.— Multiply the 

'- third term by the second. (§183). 

3)$84 Divide the product by the first 

$28 term. (^185). The fourth term 

is $28. 

Explanation. — The necessity of considering the first 
two terms abstract, is evident from the fact that $12 can 
not be multiplied by Tin., neither can $84 be divided by Bin. 

liULE. — MuIfqyJj/ the third term hy the second, and divide 
the 2^'^oduct hy the first. 

Or, Multiply the third term hy the ratio of the first to the 
second. 

Ex. 12. Find a fourth proportional to 15yd., 25yd., and 
lOda. 

13. Find a fourth proportional to $5, $75, and $40. 

14. Find a fourth proportional to 7da., 15da., and £2, 63. 
9d. 4th Term, £5, 2}d. 

16G 



SIMPLE PROPORTION, §22o 



15. Find a fourth proportional to 5|-qt., 14qt., and $17.75. 

16. Find a fourtla proportional to 3pk., 2bu., and $7.50. 

4th Term, S20. 

17. Find a fourth proportional to 1,1^ 43, and 14s. 

4th Term, 7s. 9d. S-^^qr, 

18. Find a fourth proportional to 3f, 7f, and lOgal. Sqt, 

19. Find a fourth proportional to lOgal., 3qt., and 5.5da. 

4th Term, .4125da. 

20. Find a fourth proportional to 3s., lOda., and 5yd. 

4th Term, 1.3$yd. 

21. If 15bu. of corn cost $10, what will 27bu. cost? 

15 : 27 :: $10 : $18 

27 §223. Model.— 110 is the 

%^1^ 15 third term. Since the cost of 
27bu. is 2;reater than that of 



120 



-l$18 i5bu.^ 15 is the first term, and 
27 the second. Multiply the 
third term by the second. (§ 183,) Divide the product by 
the first term. (§ 185.) Hence, 27bu. cost $18. 

Explanation. — The ratio of 15bu. to 27bu. is evidently 
the same as the ratio of the value of 15bu. to the value of 
27bu., that is, the ratio of $10 to the required amount. 
Hence, the propriety of the proportion. The first and sec- 
ond terms are considered abstract from the necessity of the 
case. 

KuLE. — Take for the third term the given number ivhich 
is of the same nature with the required term : and, if the re- 
quired term is evidently greater than this third term, take the 
greater of the remaining terms for the second and the less for 
the first ; hut, if the required term' is less than the third term, 
take the less of the remaining terms for the second and the 
grtater for the first. 

Find a fourth proportional to the three terms thus arranged, 

167 



$223 PROPORTION. 



Ex. 22. If 101b. of sugar cost $1.25, what will 17.51b. 
cost? Ans. $2.1875. 

23. If 27bbl. of flour cost $150.75, what will 94.5bbl. cost 'i 

Ans. $527,625. 

24. What cost llgal. of molasses, if 49gal. cost $34.47? 

25. What cost 3.75A. of land, if 16.375A. cost $400 ? 

Ans. $91.60 + . 

26. If 12 horses eat a load of hay in 10 days, how long 
would a load last 5 horses ? Ans. 24da. 

27. If 12 horses eat a load in 10 days, how many horses 
would eat it in 5 days ? 

28. If 91b. of tea cost $10, what will 111b. cost ? 

29. If a 6 penny loaf of bread weigh 5.5oz. when flour is 

$4.50 per bbl., what should it weigh when flour is $7 
per bbl.? Ans. 3.535oz. 

30. If -| of a yard of cloth cost $2.75, what will -^-^ of a 
yard cost? 

31. What cost 1751b. of cofi'ee, at 5-i lb. for a dollar ? 

Ans. $31.81 + . 

32. What cost 45 pr. of shoes, if 14 pr. cost $35.50 ? 

Ans. $114,107 + . 

33. If $100 gain $6 in lyr., in how many years will it gain 
$100? 

34. If 12 men build 20rd. of masonry in a week, how many 
men could build 75rd. in the same time ? Ans. 45 men. 

35. If lObbl. of flour will last a company of soldiers ISda., 
how long will lOOOIb. last them ? Ans. 7.653da. 

36. If I4lb. of butter cost $4.25, what will 1.75 firkins cost ? 

37. If a boat travels 75mi. in 6hr., how far does it go in 
25min.? Ans. 5mi. Ifur. 26.6rd. 

38. If 87.51b. of coflfee cost M, 12g. 6d., what will 7.51b. 
cost? Ans. 7s. lid. 

168 



COMPOUND PROPORTION. §224 

39. If a man can walk lOmi. in 3hr., how far can he walk 

in 5da. of 8hr. each ? 

40. If 5.51b. of sugar cost $1.00, what will lib. cost? 

Ans. $ .18. 

41. If 5ibu. of wheat make Ibbl. of flour, how much flour 
will 25001b. of wheat make ? Ans. 7.5'7bbl. 

42. If l^gal. of molasses cost $1.29, what will l-^hhd. cost? 

43. If lOijd. of cloth cost $11,625, what will 16|-yd. cost? 

Ans. $18,452 + . 

44. Iff of a ship cost ^500, 7s. 3d., what will -j^^- of her 

co«t? Ans. £260, 12s. 1.3125d. 

45. If 3 reams of paper cost $7.75, how many reams can be 

bought for $17.75 ? 

46. How many lb. Avoir, are equal to 500 lb. Troy? 

47. A grocer was detected in using as a gallon measure a 

vessel containing 3qt. Ipt. 2^gi. : how many true gal- 
lons were in 47.5 of his false gallons? A. 45.273 + gal. 

48. The same man used for his purchases a vessel contain- 
ing 4qt. 2gi. : how many true gallons were in 47.5 of 
these false gallons ? 

49. How many of his selling gallons were in 47.5 of his 
buying gallons? Ans. 52.95 + . 

50. If 1001b. of gunpowder require 751b. of saltpetre, how 

much saltpetre will 22.251b. of gunpowder require ? 



COMPOUND PROPOKTION. 



§ 224. A compound ratio is the product of two or more 

simple ratios. 

3 ' 5 ) 
Thus, . '. ^, > :: 12 : 35, is a compound ratio. 

169 



§225 PROPORTION. 



§ 225. A compound proportion is an expression of equal- 
ity between a compound and a simple ratio. 

§ 226. A compound ratio is reduced to a simple one, by 
multiplying together its corresponding terms. Thus, in the 
above instance, the antecedent 12 is the product of the an- 
tecedents 3 and 4, and the consequent 35 is the product of 
the consequents 5 and 7. 

§ 227. If the first three terms of a compound proportion 
are given, the fourth term may be found by multiplying the 
third term by the product of the second terms, and dividing 
this product by the product of the first terms : the first and 
second terms being reduced to the same denomination in 
each simple ratio, and then considered abstract. 

Ex. 1. If 5 hands can hoe 24A. of cotton in 4da., how 
many Acres can 17 hands hoe in llda.? 

^ • ^^^ •• 24A • 224^ \ ^^^^- Model.— 24A. 
4 : 11 ^ " * * "* 5- • is the third term. Since 

20 137 17 hands can hoc more 

2^^^ than 5 hands, 5 is a first 



748 
374 



term, and- 17 a second. 
Since more land can be 
hoed in llda. than in 4 
2,-0 )448,8A . (la^^ 4 jg ^ first term and 

224|-A. 11 a second. 4 times 5 

are 20. 11 times 17 are 
187. Multiply the third term by 187. (§ 183.) Divide the 
product by 20. (§185.) Hence the required number is 
224|A. 

Explanation.— As in simple proportion, we take for the 
third term that which is of the same nature with the re- 
quired term. The remaining numbers are in pairs of sim- 
ilar terms, and each pair is arranged ^s if the question de- 
pended upon it alone. 

170 



COMPOUND PROPORTION. §^8 



Rule. — Take for the third term the given number which 
is of the same natvre with the required term. Take the re- 
maining numbers in pairs of the same nature, and arrange 
each pair as in simple proportion. 

Multiply the third term by the prodiict of the second termSy 
and divide this pj'oduct by the product of the first terms. 

Ex. 2. If 60 men can do a piece of work in 40 days of 8 
hours each, how many men can do three times the work in 
90 days of 10 hours each % Ans. 04 men. 

3. If 10 horses eat 88bu. of corn in 45 days, how many 
horses will eat 120bu. in 50 days? 

4. If I can travel 75 miles in 2ida. of 7-J-hr. each, how 

many da. of 10 lir. each would it take me to travel 
225mi.? Ans. 5|da. 

5. If 5 white men can do as much work as 7 negroes, how 
many days of lOhr. each will be required for 25 negroes 
to do a piece of work which 30 white men can do in 10 
days of 9hr. each r* Ans. 15,12da: 

6. If by traveling 7hr. per da. at 4nii. per hr. I go 280mi. 

in lOda., how far will I go in 12da. by traveling 8hr. 
per da. afc 4^mi. per hr.? 

7. If 12 men build 9 rods of wall in 10 days, how many 

men can build 27 rods in 5 days ? Ans. 72 men. 

8. If $100 gain SfjO in 12mo., what will ^375 gain in 9mo.? 

Alls. $16,875. 

9. If 1001b. be carried lOOini. for 35ct., what will be the 
freight on 100001b. carried 75mi.? 

10. If 9 men, in 10 days, can build a v.all 25rd. long, 3yd. 

higl), and 5ft. thick, in what time can 20 men build a 
wall 30rd. long, 4yd. high, and 4ft. thick ? A. 5.76da. 

11. If 100 men, in 5da of lOlir. each, can dig a ditch 150 
yd. long, .3yd. wide, and 5ft, dec;', how many men, in 

171 



,^2^9 PROPORTION. 



8da. of 9hr, each, can dig a ditch 200yd. long, S^-yd. 
wide, and 2yd. deep? A. omitting fraction, 123 men. 

12. If $100 gain $7 in 12ino., in how many months will 
• $700 gain $100? 

13. If $100 gain $8 in 12mo., what sum of money will gain 
$160 in ISmo. ? Ans. 1333.33i 

14. If $900 gain $135 in 18mo., what will $100 gain in 

12mo. ? Ans. $10. 

15. If 6 school-girls spend $72 pocket-money in 4wk., what 

will 10 girls spend in 6wk. ? 

16. If 900 soldiers eat 70bbl. of flour in 20da., how many 
days will 200bbl. last 3000 soldiers at half rations ? 

Ans. 34-2-da. 



PARTITIVE PROPORTION ; OR, FELLOWSHIP. 



§ 229. Partitive Proportion is the division of a num- 
ber into two or more parts which shall have to each other a 
given ratio. 

The terms of the ratio are called the proportional terms, 
and in the operation they must be regarded abstract. 

Ex. 1. Divide 450 into three parts, which shall be to 
each other as 2, 3, and 4. 

9 : 2 :: 450 : 100 § 230. Model.--2 and 3 are 5, 
9:3:: 450 : 150 and 4 are 9. 9 is to 2 as 450 is 
9 : 4 :: 450 : 200 to the first part. Multiply 450 

by 2. Divide the product by 9. 
The first part is 100. 9 is to 3 as 450 is to the second part. 
Multiply 450 by 3. Divide the product by 9. The second 
part is 150. 9 is to 4 as 450 is to the third part. Multiply 
450 by 4. Divide the product by 9. The third part is 200. 
Hence, 100, 150, and 200, are the three parts required. 

172 



TARTITIYE rROPORTION. §230 



Explanation. — One half of the first part is evidently 
equal to one third of the second or one fourth of the third : 
so that if the whole number be divided into 9 equal parts, 
the first will contain 2, the second 3, and the third 4, of 
those parts. Hence the truth of the proportions. 

Rule. — As the sum of the proportional terms is to cither 
term, so is the whole number to be divided to the part corre- 
sponding to that term. 

Proof. — Add the several parts together : their sum is 
equal to the whole number divided. 

Ex. 2. Divide $1000 into three parts which shall be to 
each other as, 6, 1, and 3. $600, $100, and $300- 

3. Divide 141A. into three parts in the proportion of 1^, 

3i, and 4}. 

4. Divide 226gal. into four parts in the proportion of 3-^., 

2-|-, 2i,and 2-^-. 42gal., 56gal., GOgal., and 68gal. 

5. Divide 992.95 into four parts in the proportion of 1.25, 

3.2, 4.73, and 5.005. 87.5, 224, 331.1, 350.35. 

6. Divide $43.20 into four parts in the proportion of 1, 3, 

5, and 7. 

7. Two men, A and B, engage in business with a joint 
capital of $6000, of which A furnishes $2500 and B 
the remainder. What is each one's share of a gain of 
$1200 ? Ans. $500, $700. 

8. Two men, C and I), gain $1275 on a capital of which 

C's share is double D's ; what is each one's share of 
this gain? Ans. $850, $425. 

9. If E invests $2375, and F $3225, and the firm loses 
$700, how much must each partner lose ? 

10. A invests $3000, and B 83500, in a certain business, 
in which the first year they lose $325. After paying 

178 



^230 PROPORTION. 



this loss from the funds of the firm, they take in C us 
a partner with a capital of $4000, and the second year 
the new firm gains ^2035. Hov/ much of this gain is 
due to each partner ? Ans. A, $570; B, $665; C, 1800. 

11. D and E form a partnership for two years, I) contribut- 

ing $5000, and E $1750. The first year they gain 
$1350. D spends his share of the gain, but E leaves 
his share among the funds of the firm. The second 
year they gain $2130. What is each partner's share 
of this last gain '/ Ans. D, $1500 ; E, $630. 

12. Messrs. Jones, Smith, and Brown gained $5000; what 
is each man's share of this gain, if Smith owns twice 
as much of the capital as Brown, and Jones as much 
as Smitli* and Brown together ? 

13. In a certain firm, A owns 1^^, times as much stock as B, 
C owns 1^- times as much as A, and D owns 1-^ tii^ea 
as much as C: A's gain on a year's transactions is 
8500 ; what is the gain of each of the other partners ? 

Ans. B, $400 ; C, $600; D, $700. 

14. A merchant owes one creditor $2000, and an other 
$3500 : having failed, he can pay them both only 
$4015 : how much should* each creditor receive 1 

Ans. $2555, and $1460. 

15. A man dying wills to one son $2000, to an other son 
' $1500, and to his daughter $1250 ; but after paying 

his debts his executor has in his hands only $3000. 
How much should he pay to each legatee ? 



16. Three partners, A, B, and C, invest as follows : — A in- 
vests $500 for 2 months ; B, $400 for 3 months ; and C, 
$300 for 5 months. They gain $740. What ought 

each partner to receive ? 

174 



PARTITIVE PROPORTION. §;:;U 



500 X 2 == 1000 8700 : 1000 
400 X 3 rr: 1200 8700 : 1200 



$740 : $200 
$740 : $240 
$740 ; $300 



.300 X 5^:1500 8700 : 1500 
3700 
§231. Model.— Twice 500 are 1000. 3 times 400 nrc 
1200. 5 times 300 are 1500. The sum of tbese propor- 
tional parts is 3700. [Proceed as in § 230.] Hence A 
ought to receiye $200, B, $240, and C, ^300. 

Explanation. — A's investment of $500 for 2 months is 
equal to an investment of twice $500, or $1000, for 1 month ; 
so B's $400 for 3 months is equal to 3 times $400, or $1200, 
for 1 month ; and C's ^300 for 5 months is equal to 5 times 
$300, or $1500, for 1 month. The several investments, 
being thus referred to the same unit of time, evidently fur- 
nish equitjxble proportional terms. 

This is an example of what is called Compound Fellow- 

SHIV. 

Ex. 17. A firm of two partners gained $1750 : what was 
each partner's share of the gain, if A contributed $3000 for 
10 months, and B $2500 for 1 year ? 

Ans. A's, $875 ; B's, $875. 

18. A, B, C, and D, rented a pasture for $100. A kept 20 
head of cattle in it 6 months; B kept 25 head 5 months; 
C kept 30 head 5^- months ; and D kept 50 head 3 
months. What part of the rent ought each man to pay ? 

19. In a certain partnership A contributed $3000 Jan. 1st; 
B contributed $2500 Feb. 1st; and C contributed 
$4000 May 1st. On the 1st of August, they lost by 
fire $4000. What part of the loss did each partner 
sustain 1 Ans. A, $1750 ; B, $1250 ; C, $1000. 

20. Three partners in trade gained $3008 after 15 months' 

business. A put in $1000 at first, and $2000 3 months 

175 



§232 PROMISCDOUS PROBLEMS. 



afterwards ; B put in at first $4000, but took out $2000 
6 months afterwards ; and C put in $2000 at the end 
of 5 months, and $2000 5 months afterwards. What 
was each partner's share of the gain ? 

Ans. A's, $1092 ; B's, $1176; C's, $740. 



■ « »> 



PROMISCUOUS PPtOBLEMS. 



8bu. 
5 
4 
45 



J. If 8bu. of wheat are worth as much as 15bu. of corn, 
and 5bu. of corn as much as 2cwt. of hay, and 4cwt. of hay 
are. worth $6; how many bu, of wheat can be bought for 

S45 ? 

15 45.4.5.8bu.=7200bu. ^ 232., Model.— 
2 6.2.15r=180 Set 8bu. on the left, 

6 7200bu.^l80 = 40bu. and 15 on the right; 

5 on the left, and 2 
on the right ; 4 on 
the left, and 6 on the right, and 45 on the left. Multiply 
together the terms on the left. Multiply together the 
terms on the right. Divide 7200bu. by 180. The quo- 
tient is 40bu. Hence $45 will pay for 40bu. of wheat, 

Explanation. — This question, commonly referred to a 
distinct head, called Conjoined Proportion, or the Chain 
Rule, is merely a complicated case of simple proportion, as 
will be seen by stating it thus : — 

1. If 4cwt. hay cost $6, how many cwt. will cost $45 1 

6 : 45 :: 4cwt. : 30cwt. Hence, SOcwt. hay =$45. 

2. If 2cwt. hay=z5bu. corn, how many bu. corn = 80cwt. hay? 
2 : 30 :: 5bu. : 75bu. Hence, 75bu. corn=$45. 

3. If Sbu. wheat=15bu. corn, how many bu. wheat=75bu. 

corn? 
15 : 75 :: Sbu. : 40bu. Hence, 40bu. wheat=$45. 

176 



CONJOINED PROPORTION. §232 



Comparing this work with the model, we see that the 
me<ans in the proportions are 45, 4, (30,) 5, (75,) and 8 ; 
and the extremes, except the last, are 6, (30,) 2, (75,) and 
15 ; and that, omitting the two terms, 30 and 75, common 
to both, we have left in the one case the terms on the left 
in the model, and in the other the terms on the right. And 
since the product of the means ia equal to the product of 
the extremes, the product of the terms on the left divided 
by the product of the terms on the right will give the re- 
«^uired term. 

The term similar to the required term is called the odd 
terra, and the one equivalent to the required term is called 
the term of demand. Both of these must be placed on the 
left of the vertical line, and the other terms must be arrang- 
ed so that equivalents shall be opposite each other, and no 
two similar terms on the same side. In the operation, all 
but the odd^term must be regarded abstract. 
t. If 2bbl. of flour are worth as much as 26fbu. of corn, 
and 3bu, of corn as much as 7-^-lb. of bacon, how many 
lb. of bacon are equivalent to 3bbl. of flour? 

Ans. 1001b. 
S. If £93 are cqurJ to 2420fr., and 166|fr. are equal to 
$31, and 87 are C(]ual to 4bu. of wheat, how many bu. 
of wheat are equal to j£15 ? 

4. If A can do as much work in 5 days as B can do in 6, 
B can do as much in 7 days as C can do in 8, and C 
can do as much in 9 days as D can do in 10 ; in how 
many .days can A do as much as D can do in 15 ? 

Ans. 9.84da. 

5. If 10 Ells Flemish are equal to 6 Ells English, and 4 

Ells English to 5 yards, and 12 yards to 8 Ells French ; 

L 177 



§282 PROMISCUOUS PROBLEMS. 

how many Ells French are equal to 16 Ells Flemish? 

Ans. 8 E. Fr. 

6. If 191b. of butter are worth 301b. of cheese, and 191b. 
of cheese are worth $3 ; how raacy lb. of butter are 
worth S7.50 ? 

7. If a train of cars travel a mile in 2.5min., how long will 

it be in going 45 miles ? Ans. Ihr. 52.5min. 

$. If 8 men can mow a meadow in 10 days of I3hr. each, 
in how many days of llhr. each can 12 men mow it? 

Ans. 17.'72da. 

9. A, B, and C formed a partnership for two years : th« 
'first year, they lost $500 ; the second year, they gained 
$750 ; how much is each partner entitled to at the 
end of the second year, if A contributed $4000, and B 
and C $3000 each to the funds of the firm ? 

10. A and B formed a partnership for two years from Jan. 

1, 1860. On that day, A contributed $1000, and B 
$500 : July I, 1860, A added $500 to his investment, 
and Oct. 1, 1860, B added $500 to his : Jan. 1, 1861, 
A withdrew $250 from the funds of the firm, and Mar. 
1, 1861, B contributed $500 more. They gain $1090. 
How much of this ought each partner to receive ? 

Ans. A, $600 ; B, $490. 

11. If 5bbl. of cider are worth 8bu. of wheat, and 11 bu. of 

wheat are worth 2T. of coal, and 3T. of coal are worth 
501b. of tea, and 41b. of tea are worth 5oz. of quinine, 
and 7oz. of quinine are worth $6.50; how many dol- 
lars are lObbl. of cider worth ? Ans. $56,277. 

12. If the transportation of 1001b. lOOmi. cost $2.15, what 

will it cost to transport 25001b. 25mi.? 

13. What is the smallest numbeu. that can be exactly di- 

vided by either .12, 13, or 14 I Ans. 1092. 

178 



PROMISCUOUS PROBLEMS. §232 



14. What is the largest number that will exactly divide 
either 240, 720, or 840 1 Ans. 120. 

1 '). What is the total cost of 4yd. of silk at $1,875 per yd., 
Syd.'of berege at ^.625 per yd., 3doz. buttons at $.75 
per doz., and 7.5yd. of calico at $.25 per yd.? 

16. What is the total cost of 51b. of tea at 5s. 6d., 7bu. of 
corn at 4s. 4d., 8bu. of wheat at lis. 9d., and llgal. of 
molasses at 7s. 3d.? Ans. j^ll, Us. 7d. 

17. What interval elapsed between Dec. 5, 1813, and Mar. 
17, 1842 ? Ans. 28yr. 3mo. 12da. 

18. What interval elapsed between Jan. 30, 1833, and 

Sept. 3, 1862 ? 

19. John Jones was born Mar. 9, 1827, and was married 

when he was 22yr. 6mo. lOda, old ; when was he mar- 
ried ? Ans. Sept. 28, 1849. 

20. If 5 men can plough 47-^1 acres in 7|- days, in how 
many days can G men plough 31 acres ? Ans. 4 days. 

21. If o^cwt. of hemp cost ^27.50, how much hemp will 
cost $33.33i ? 

22. A merchant bought SOOObu. of salt : after having sold 

to A lOO.Sbu., to B 477.75bu., to C 329.8375bu., and 
to D 1200. 25bu., bow much has he left ? 

Ans. 891.6625bu. 

23. What is the produce of 15.375A. of corn, at 8bbl. 3bu. 

3pk. to the acre ? Ans. 134bbl. 2bu. 2pk. 5qt. 

24. A field of 25 acres produced 637.5bu. of wheat ; how 

much was that per acre ? 

25. A sum of money divided equally among 17 men gives 

to each ^17.765 ; if divided equally among 11 men, 
how much would each get? Ans. S27.455. 

26. If 20 men in 35da. earn $320, how many men will earn 
$480 in 70da.? Ans. 15 men. 

179 



§233 PROPORTION. 



27. If 18 horses eat lObu. of oats in 20(]a., how many horses 
will eat 60bu. in 36cla.? 

28. What is the greatest common measure of 75, 825, and 
1575 ? Ans. 75. 

29. What is the least common multiple of 46, 230, and 
115? Ans. 230. 

30. What are the different prime factors of 24400 ? 



PERCENTAGE. 



§233. Percentage includes :''• cases of proportion in 
whichthe first term is one liund. ■■'. 

The phrase, »er centum, that is, ptr hundred, is usually 
written, and often pronounced, ^er cent. Thus, in stead of 
"six dollars per hundred," we usually say "6 per cent." 

Ex. 1. A lawyer collected $3725 ; what is his commis- 
sion at 3 per cent.? 

100 : 3725 :: $3 : $112.75 This proportion is evi- 
dently correct : and all 
similar problems may be solved in the same manner. But, 
inasmuch as the three given terms have always the same 
unit, the same result will be obtained by regarding the sec- 
ond term concrete and the first and third abstract, by di- 
viding the third by the first and multiplying the second by 
this quotient. This method, being a little less troublesome, 
is the one usually adopted. To explain it more fully, we 
must give the following definitions. 

§234. The price or amount per hundred is called the 
rate per cent. Thus, in the above example, 3 is the rate 
per cent. 

180 



PERCENTAGE. §238 

§ 235. If the rate per cent, be divided by 100, the quo- 
tient is called the rate per unit. In all operations, this is 
regarded as an ab.'^tract number. Thus, .03 is the rate per 
unit in the example above. 

What is the rate per unit for 6 per cent.? for 10 per cent.? 
for 50 per cent.? for 75 per cent.? for li per cent.] for -J- 
per cent.? for 100 per cent.? for 33^^ per cent.? for \2\ per 
cent.? for 18 ^^ per cent.? for \ per cent ? for -^ per cent.? 

§ 236. The number on which percentage is calculated, is 
called the basis of percentage. Thus, above, $3725 is the 
basis. 

§ 237. The result of the operation is called the percent' 
age. Thus, above, $112.75 is the percentage. 

Ex. 2. Find 5 per cent, of $5750. 

$5750 § 238. Model.— Multiply the basis by the 

1^ rate per unit. (§ 183.) The product is ,^287.- 

8287.50 50. Hence the percentage is ^287.50. 

Explanation. — 5 per cent, of any number is evidently 
5 hundredths of that number, and this is found by multi- 
plying the number by .05. Observe that the rate per unit 
is simply one hundredth of the rate per cent., and is most 
conveniently expressed as a decimal fraction. 

KuLE. — Multiply the basis by the rate per unit. The 
product vnll be the percentage, 

Ex. 3. What is 1 per cent, of 7500 ? 



4. What 

5. What 

6. What 

7. What 

8. What 

9. What 



s 2 per cent, of 250 ? Ans. 5. 

s 3 per cent, of 275 ? Ans. 8.25. 

s 4 per cent, of 775 ? 

s 6 per cent, of $325 ? Ans. $19.50. 
s 7 per cent, of 89250 ? Ans. $647.50. 
s 8 per cent, of 725 men? 
181 



§239 PROPORTION. 



10. What is 9 per cent, of 1700 men ? Ans. 153 men. 

11. What is -,V per cent, of $1000 ? Ans. 61. 

12. What is 1 J- per cent, of 8175 ? 

13. What is 2^^ per cent, of 827.75 ? Ans. S.7284375. 

14. What is 31 per cent, of 8630 ? Ans. 821. 

15. What is 4f per cent, of 795 ? 

16. What is 7.} per cent, of 2775.25 ? Ans. 208.14375. 

17. What is 9a per cent, of 473.75 ? Ans. 46.190625. 

18. What is lO-j-^ per cent, of 275 ? 

19. What is 16f per ce^r. of 1500 ? Ans. 250. 

20. What is 66f per cent, of 8750 ? Ans. $500. 

21. What per cent, of 690 is 115 ? 

§ 239. Model. — Divide the percent- 
age by the basis. (§ 52.) The quotient 
is . 16|. IMultiply this quotient by 100. 
The product is 16|. -Hence, 115 is 
23o' I- 1C| per cent, of 690. 

l690!3 

ExPLA-NATiON. — Since the percentage is equal to the basis 
multiplied by the rate per unit, conversely the rate per unit 
is equal to the percentage divided by the basis. And, since 
the rate per unit is one hundredth of the rate per cent., con- 
versely the rate per cent, is found by multiplying the rate 
per unit by 100. 

KuLE, — Divide the percentage by the basis. Tlie quo- 
tient will be the rate per writ. Midtiplij the rate per unit 
by 100. The product ivill be the rate per cent. 

Ex. 22. What'per cent, of 700 is 70 ? Ans. 10 per cent. 

23. What per cent, of 375 is 125!;' Ans. 33} per cent. 

24. What per cent, of 1000 is 125 ? 

25. What per cent, of 550 is 110? Ans. 20 per cent. 

182 



115.00 


690 


690 


.16-1 


4600 100 


4140 


16|- 



I>EaC£NTACJE. §239 



26. What per cent, of S675 is $:^37.59 ? Ans. 50 per cent. 

27. What per cent, of SIOOO is ^875 ? 

28. What per cent, of $5000 is ^250 ? Ans. 5 per cent. 

29. What per cent, of $10000 is S50 ? Ans. I per cent. 

30. What per cent, of $150 is $300 ? 

31. What per cent, of 3 JOO is 4000 ? Ans. 133a- per cent. 

32. What per cent, of 275 is 302.5 ? x\.ns. 110 per cent, 

33. What per cent, of 245 is 735 ? 

34. What per cent, of 200 is 500? Ans. 250 per cent. 

35. What per cent, of 325 is 2925 ? Ans. 900 per cent. 

36. What per cent, of 81.25 is $1.50 ? 

37. What per cent, of $1.00 is $.375 ? Ans. 37^ per cent. 

38. What per cent, of $.875 is $.50 ? Ans. 57^ per cent. 
S9. AVhat per cent, of $.66f is $.22| ? 

40. What per cent, of $.125 is $.0625 ? Ans. 50 per cent. 

41. A commission merchant purchases articles amounting 

to $247.75 ; what is his commission, at 2^- per cent.? 

42. What is the commission on $312, at 12 per cent.? 

43. A merchant insured a vessel and cargo, valued at 

$75000, at 7f per cent. ; what did he pay ? 

44. What premium must I pay for the insurance of hlj 

lifo5*the policy being $5000, and the rate 2.o5 per 
cent.? Ans. $117.50. 

45. What is the premium for insuring $9450, atf per cent.? 

46. What is the insurance on a dwelling and furniture val- 

ued at $25550, at 1,} per cent.? Ans $319,375-. 

47. What is the duty, at 40 per cent., on French broafll- 

cloths valued at $15375 : Ans. $6150. 

48. What is the duty, at 20 per cent., on $6250 worth of 

Italian silk ? 
4.9. At 7,} per cent., what is the duty on an invoice of Ge- 
neva watches, valued at $7475? Ans. $560,635. 

183 



§239 PROPORTION. 



50. At 50 per cent., what is the duty on a ease of Leghorn 
hats, worth $1500 ? Ans. S750. 

51. What tax should be paid on S17725 worth of real es- 
tate, at i per cent.? 

52. What is the tax on $261000, at .15 per cent.? 

Anp. $391.50. 

53. What is the tax on $17150, at 60ct. on $100? 

Ans. $102.90. 

54. What is the amount of a dividend of 3 per cent., on 

$4200 of bank stock ? 

55. The North Carolina Bailroad company declared a div- 
idend of 2|- per cent. : what did I receive on 14 shares 
of $100 each ? Ans. $35. 

56. A merchant bought broadcloth at $3.50 per yard; at 
what price must he sell it, to gain 40 per cent.? 

Ans. $4.90. 
The percentage must he added to the basis. 

57. A grocer bought candles at 25ct. per lb.: how must he 
sell them, to gain 30 per' cent.? 

58. If broadcloth cost $4.00 per yd., how much will it bring 

at a loss of 35 per cent.? Ans. $2.60. 

The percentage must he suhtracted from the basis. 

59. A dealer bought 50bbl. of flour at $12 per bbl., but.was 
forced to sell it at a decline of 20 per cent. : what did 
he get for it all ? Ans. $480. 

60. A speculator bought $35000 worth of cotton, and sold 
it at a loss of 15 per cent. : what did he receive for it ? 

61. A man pays $406.25 for the insurance of his dwelling, 
valued at $32500 : what is the rate per cent.? 

Ans. 1-^: per cent. 

62. A vessel worth -SI 5400 was insured for $539 ; what was 

the rate per cent.? Ans. 3-^- per cent. 

184 



PERCENTAGE. §239 



63. At what rate per cent, will the insurance on $11500 

cost $172.50 ? 

64. A man bad his life insured for $277.50': what was the 
rate per cent., the policy being $10000 ? 

Ans. 2.775 per cent. 

65. If the duty on ^3457 worth of goods is $1037.10, what 

is the rate per cent.? Ans. oO per cent. 

66. What is the rate of duty, when $12657 worth of cloth- 
ing pays $6828.50 ? 

67. If $15000 worth of property pays a tax of $229.50, what 
is the tax on $100 ? Ans. $ .51. 

66. I paid a hroker $21,125 for investing $8450 in Govern- 
ment stocks; what was his rate of brokerage? 

Ans. \: per cent. 

69. My attorney charged me $260.73f for collecting $3476. 

50 : what was his rate of commission ? 

70. I bought a farm for $4000, and sold it for $5000 ; what 

did I gain per cent.? Ans. 25 per cent. 

The Jirst cost subtracted /roin the selling price leaves the 
total gain. 

71. I bought a farm for $5000, and sold it for $4000 ; what 

did I lose per cent.? Ans. 20 per cent. 

The selling 'price subtracted from the jirst cost leaves the 

total loss. Observe that in each case the first cost is the basis. 

Hence the difference in the ansiucrs of the last two questions. 

72. If I buy calico at lOct., and sell it at 12^-ct., what do I 
gain per cent.? 

73. If I buy calico at 12^ct., and sell it at 15ct., what do I 
gain per cent.? Ans. 20 per cent. 

74. A man bought a house for $7625, and sold it for $8387. 
50; what did he gain per cent.? x\ns. 10 per cent. 

1S5 



5; 240 PR ^PORTION. 



75. A man having paid S7625 for his house, was compelled 
to !^eH it for $6862.50 : how much per cent, did he lose ? 



70. By selling an article for ^1300, I gain 30 per cent, on 
it : what did it cost me ? 

$1300.00j 1.30 ^,240. Model.— Divide the sell- 

■^^^ i SlOOO ing price by 1 + the rate per unit. 
OUOU (§ 165.) . The quotient §1000 is the 

first cost. 

Explanation. — It is evident that l + thc gain per unit 
: 1 :: the first cost + the whole gain : the first cost. But 
the selling price, §1800j is evidently the first cost + the 
whole gain. Then since the second term of the proportion 
is always 1, it is easy to see the truth of the 

Rule. — To find the first cost, lohen the selling price and 
the rate 'per cent, o/ gain are given. Divide the selling price 
hy 1 -j- the rate per unit. The quotient loill be the first cost. 

Ex. 77. By selling a piece of muslin for ^50, I gain 100 
per cent. ; what did I give for it ? Ans. $25. 

78. What did I pay for eggs, if I gain 33^^ per cent, by sell- 

ing them for IGct. per doz.? 

79. A grocer sold a lot of sugar for ^1058, gaining thereby 
15 per cent. : what did the sugar cost him ? A. §920. 

80. A merchant sells some flour for $924, and gains 12 per 
cent, on it : what did he pay for it? Ans. $825. 

81. The selling price is S1800, the gain 20 per cent.: what 
is the first cost ? 



82. A merchant sold a quantity of cloth for S1410, and thus 
sustained a loss of 6 per cent. : what did the cloth cost 
him ? 

186 



PERCENTAGE. §'241 



94 '^'50(3 §241. Model. — Divide the selling 
470 price by 1 — the rate per unit. (§165.) 

^>7Q The quotient $1500 is the first cost. 

~"~m) 

Explanation. — Evidently, 1 — the loss per unit : 1 :: 

tiic first cost — the whole loss : the first cost. But the first 

cost — the whole loss is evidently the selling price, ^1410. 

The second term of this proportion is always 1, and hence 

the following 

EuLE. — To Jin J the Ji rat coiif, when the sellhig price and 

the. rrtfe j)er cent, of loss are given. Divide the selling pric^ 

hj/ \—thc rate per unit. The quotient will he the fir.st co?t. 

Ex. 83. The selling price is ?$8000 ; the loss 20 percent.: 

what is the first cost ? Ans. $10000. 

'<4. By selling flour at $12.25 per bbl.,I lost 12^ per cent, 
of wliat it cost me ; what did it cost? 

ST). I remitted ^3150 to my commission merchants to lay 
out in groceries after retaining 5 per cent, of what he 
spent: how much did he spend for me? Ans. ^3000. 
This is precisely similar in principle to the foregoing. 

86. How much sugar at lOct. per lb. can I get by remit- 
ting $864,871- to a merchant who cliarges 5 per cent. 
commission 1' Ans. 34751b. 

S7. How much stock at ^5 per cent, advance can I buy for 
S1265 ? 

.S8. How much stock at 15 per cent, below par can I buy 
farC935? Ans. §1100. 

.Si). A father settled his sou with property worth SIOOOO : 
the first year he lost 20 per cent, of it, and the second 
year he gained 25 per cent, of what he had left ; how 
much had he then 't Ans. ^lOOOO. 

187 



^242 PB.OPORTION — PERCENTAGE. 

90. A merchant sold some sugar for 31402.50, and lost 
thereby 15 per cent. Wh.it did it cost him? 

91. How much stock at :i discount of 3^- per cent, caii I. 
bought for f-5790 ? Aus. S6000. 

SIMPLE INTEREST. 

§242. Interest is the prict paid bj the borrower for 
the use of money loaned. 

§ 243. The sum of money on which interest is calculated 
is called the principal. 

§ 244. The sum of the principal and interest is called the 
amount. 

§245. The price paid for the use of one hundred doUarpi 
one year is called the rale per cent, per annum. 

Ex. 1. What is the interest of ^875, for 2yr. lOmo. 20da.. 

at 7 per cent, per annum ? 

^375 2yr. lOmo. 20da. 7n. c. 
.07 ' § 246. Model.— 

i26^=lyr. Multiply^ the priu. 
2 ...... iU.wK.. 



cinai by tlie rale per 

unit. (§!«3.) The 

^52.50=2yr. product, S26.25, is 

13.125=:6mo. -j tj,e interest for i 

6.562=3mo. -lOmo. y^,,^^ Multiply the 

2.187= Imo. j interest for lyr. by 

•^■o^']"'^^^- \ 20da. 2. The product i^ 

.364= 5da.. | !$52.50, the interest 



$75.831 = 2yr. lOmo. 20dn. for 2yr. 6mo. is one 

half of lyr. Divide 
the interest for lyr. by 2. The quotient is $13,125, the 
interest for 6mo. 3mo. is one half of 6mo. Divide the in- 
terest for 6mo. by 2. The quotient li 36,562, the iuterest 

188 



SIMPLE INTEREST. §246 



for oiiiO. Imo. is one third of 3ino. Divide the iuterest 
for Smo. by 3. The quotient is 82.187, the interest for 1 
mo. loda. is one half of Imo. Divide the interest for 1 
mo. by 2. The quotient is SI. 098, the interest f\;r 15da. 
5da. is one third of 15da. Divide the iuterest for 15da. by 
o. The quotient is S .oG4, the inleresi for 5da. Add the 
partial interests together. The sum is S75.831, the inter- 
im est for the whole time. 

ExPLA-NATiOM. — Since the rate is 7 percent, per annum, 
the interest of the given principal for 1 year is found by 
multiplying the principal by the rate per unit. Thus far 
the work is simple percentage. Far longer or shorter pe- 
riods of time the interest is proportional to the time : hence 
we take such aliquot parts of the interest for 1 year, &c., 
as the periods in question severally require. 

In the calculation of interest, a month is considered equal 

■ 30 days, and a year to 360 days. , 

Rule. — Multij^ly the principal hi/ the rate per unit. The 
product will he the inf.erefi.t for 1 year. 

Alidtiplij the 'interest for 1 year hy ihe number of years , 

and take aliquot parts for periods of time less than a year. 

To ftriii the amount, add the interest to the principal. 

Ex. 2. What is the interest of 850 for 2yr. at 6 per cent. 

per annum ? Ans. ^G. 

3. What is the interest of S75 for 6mo. at 7 per cent, per 

annum 't 

•i. What is the amount of SlOO for 9mo. at 8 per cent, peu 

annum ? Ans. ^106. 

5. What is the interest of 3125 for 3yr. at 9 per cent, per 

annum '/ Ans. $33.75. 

G. What is the iuterest of %22b for 2yr. Gmo. at 10 per 

cent, per annum ? 

189 



^24G riioroiiTiON — rELCEM..^ui:. 



7. What IS tlie iiitere.st of ^150!75 for lyr. omo. at 5 per 
cent, per annum 't Ans. $10.67. 

8. What is the amount of 3176.50 for 2yr. 9mo. at G per 

cent, per annum ? Ans. S204.45. 

9. What is the interest of s^305.50 for %r. 5mo. 15<3a. at 

6 per cent, per :iiinnni? 

10. What is the interest of S574.05 for -^jr. 7iiio. 25da. at 
5 per cent, per annum? Ans, 81oS.75. 

11. What is the intere^.t of 8615.49 for fiyr. llmo. 22da. at 

7 per cent, per annum ? Ans. $257.47. 

12. Find the amount of ?-777.75 for !>yr. 2mo. 20da. at 8 
per cent., per annum. 

13. Find the interest of ^'1225 for ojr. 5n)0. oda. at 5 per 
cent, per annum. Ans. ?332.G2. 

14. Find the intere.s: of S1525.25 for lyr. 2mo. at 8 per 

cent, per annum. Ans. S 142.356. 

15. Find the interest of 62790 for 23'r. 7mo. at 9 per ccrit. 

per annum. 

16. Find the amount of $1724.25 for 12yr. Omo. at 8 per 

cent, per annum. Ans. ^'3448.50. 

17. Find the interest of $3500 for 7yr. omo. IGda. at 7 per 

cent, per annum. Ans. 81783. 0^'5. 

18. Find the interest of -^4275 for 16yr. Smo. at 6 per cent, 

per annum. 

19. Find the interest of S5550 for 15yi'. llmo. 27da. at 

per cent, per annum. Ana. v?79S7.83{,. 

20. Find the amount of $2c;95 for tS) r. 4mo. at 12 per cent. 
per annum. x\ns. $5990. 

21. Find the interest of $3827 for 17yr. 3mo. 15da. at 10 

per cent, per annum. Ans. $6617.52. 

190 



SIMPLE INTi-UFST. §247 

CONCISE METHOD FOR 6 PER CJ^NT. PER AKNU.\L 



Ex. 22. What is the interest of SMT.OO fur 2yr. Cmc^ IS 
da. at G per cent, per annum ? 

2yr. 6mo. 18da. = 30.Grao. {^ 247. M(*DKr .— Rcdnre 

200)30.6 ^247.50 ^^^ ^'^^" ^^"^^ ^'' nsonth^,. 

~YFb 1 ^S (§1^^.) Divide tho number 

* h.'okI «^^ months by 20U. (^ 165. ) 

lOQ^'^A Multiply iho principal by 

l^d/^0 this quotient, (i^ 162.) The 

J_Z^iL product is ?P>7.8(1A, the in- 

$37.86750 terest required. 

Explanation. — Since the rate is 6 per cent, per annum, 
or for 12 mouths, one half of the number of montlis is the 
rate per cent, for any length of time : and this rate per cent, 
divided hy 100, gives the corresponding rat(3 per unit, by 
which the principal must be multiplied, to find the interest. 

For any other rate, we may find the interest at G per 
cent., and increase or diminish it, as the case may require. 
For instance, for 7 per cent., add to the interest found by 
this method, one sixth of its?lf : for 5 per cent., from the 
interest thus found subtract one sixth of itself; &c. Or, 
generally, find the interest at 6 per cent., divide it by 6, 
and multiply the quotient by the given rate. 

Rule. — Divide the number of monthly in the given time 
hy 200, and multiplij the jprincipal hy the quotient. The 
product ivill he the interest. 

Or, 3Iidtiply the number of years hy 6, and divide the 
product hy 100 ; Divide the number of months hy 2, and 
divide the quotient hy 100 .- Divide the number of days by 

191 



§248 PROPOllTION — PERCENTAGE. 

6, and diclile (he quotient hy 1000 ; Add these three results 
fo<jctherj and multiply the principal hy their sum. 

After Jill din f/ the interest at 6 per cent., as above, to find 
the interest at any other rate ; Divide the interest at 6 per 
rent, hy G, and multiply the quotient hy the required rate. 



SECOND METHOD FOR 6 PER CENT. 



Ex. 23. Find the interest of 8275.75 for Syr. lOmo. 21 
(la. at 6 per cent, per annum. 

Syr. lCmo.=46mo. §248, Model. — Reduce the 

3)21 v^275.75 years and months to months. — 

— 7 .467 Divide the number of days by 

T AOAoT 3. Annex the quotient to the 

Ifi'S^^O number 01 months. Divide this 

^^,^oAa icsult bv 1000. Multiply the 

principal by this quotient. Di- 

2)812877525 ^.iae this product by 2. The 

$64.3876 quotient is sj61.38£, the inter-. 
est required. 

Explanation, — This method is evidently the same in 
principle as the preceding, and is preferable to the other 
only on account of its greater freedom from liability to 
fractions. Of course, the multiplier iu each of these lueth- 
•ods must be considered abstract. 

Rule. — To th6 number of months annex one third of the 
humber of day^s. Divide the number thus produced by 1000. 
Multiply one half of the principal hy this quotient. 

Or, Multiply the whole principal by this quotient, and 
divide the product by 2. 

The interest for any other rate may be found as in § 247. 

192 



SIMPLE INTEREST. §249 



CONCISE METHOD FOR A^Y S1.4TE PER CENT. 



Ex. 24. Find the interest of ^^330. 60 for 6mo. 15da. at S 

per cent, per annum. 

12,00)^ 3,60.60 

"~$ .3005 

g 5 § 249. jModel. — Divide the princi- 
pal by 1200. Multiply this quotient 
by 6.5. M'lUiply this product by 8. 
The product h $15.62|-, the required 



15025 

18030 



^l,95b25 interest. 



^■15.62600 

ExPLVNATLON. — The principal -^100=:the interest for 1 
year at 1 per cent. This interest-^- I2=the interest for 1 
month at 1 per cent. This last x 6. 5 = the interest for 6.5 
months at 1 percent. And thtsx8 = the interest for 6.5 
months at 8 per cent. 

Rule. — Divide the principal by 1200. Multiply the quo- 
tient by the wumber of months in fh'i given time, and this 
product by the rate per cent. This last product will be the 
interest.- 

Either of the above methods may be used in any case. 

*Ex. 2"'. Find the interest of i?l 3 19.50 for 9 days, at 6 
per cent. |rr annum. 

26. Find ilie interest of $36")8.75 for 17 days at 6 per cent. 

per nnaum. " Int. SI 0.366. 

27. Find the interest of $5739.2,3 for 2mo. 24da. at 6 per 

cent, per annum. Int. $80,349. 

28. Find the amount of $3738^.375 for 2mo. 6da. at 6 per 

cent, per annum. Amt. $38096.88. 

M 193 



§249 , PROPORTION — PERCENTAGE. 



29. Find the amount of ^1665.25 for lyr. llmo. 9da. at C 
per cent, per annum. Amt. S1859.25. 

SO. Find the interest of ^4336.30 for 4yr. 8mo. 13da. at 6 
per cent, per annum. 

?>1.' Find the interest of ^2758.50 from July 3, 1846, to 

-May 19; 1855, at 6 per cent, per annum. 

Int. $i469.36. 

To find the interval of time, the earliest date must be 

-iQrr r to J subtractcd from the latcst. In 

Ibooyr. omo, lyda, 

1846 " 7 " 3 " • ^^^^ subtraction, the number 
^ u 10 «' IQ <' ^^ emc]! month in the calendar 

is used, and each month is 
toaken as equal to 30 days. 

Ex. 32. Find the amount of $8140.75 from Dec. 9, 1847, 
to Apr. 27, 1855, at 6 per cent, per annum. 

Amt. §11747.10. 

33. Find the interest of $3^219.15, from Apr. 8, 1850, to 
June 15, 1855, at 7 per cent, per annum. 

S4. Find the interest of $6813.45 from Mar. 5, 1855, to 
Oct. 8, 1862, at 8 per cent, per annum. 

Int. $4138.035. 

35. Find the interest of $856.85 for 6yr. 8mo. 9da. at 8 per 

cent, per annum. Int. $458,699. 

36. Find the amount 6f $742.40 from June 24, 1854, to 

Mar. 13, 1860, at 7 per cent, per annum. 

37. Find the interest of $171.80 from July 29, 1857, to 

Sept. 1, 1861, at 10 per cent, per annum. Int. $70.24. 

38. Find the interest of $670.70 from Apr. 7, 1859, to Oct. 

13, 1862, at 9 per cent, per annum. 

Int. $212,276. 

^9. Find the interest of $976.18 from Mar. 1, 1861, to Feb. 
10, 1862, at 8i per cent, per annum. 

194 



SIMPLE INTEREST. §249 



40. Find the interest of ^375.85 from Jan. 19, 1860, to 

Jan. 1, 18G2, at 11 per cent, per annum. 

Int. $80,619. 

41. Find the amount of $6.89 from June 11, 1860, to June 

1, 1862, at 9 per cent, per annum. Amt. ^8.11. 

42. What is the interest of S89.96 for 2yr. 3mo. 16da.at 8 
per cent, per annum ? 

43. What is the interest of $325 for 6jr. 7mo. 27da. at 7-^ 
per cent, per annum ? Ans. $156.88. 

44. What is the amount of $1728 from Dec. 29, 1859, to 

Oct. 9, 1852, at 10 per cent, per annum ? Ans. 152208. 

45. W^hat is the interest of $160.08 from May 1, 1851, to 
Sept. 9, 1854, at 7 per cent, per annum? 

46. What is the interest of $18.62 for 3yr. 18da. at 5 per 
cent, per annum ? Ans. $2,839. 

47. What is the interest of i£17, 6s. 9d. for 18mo. at 6 per 
cent, per annum ? 

£17, 6s. 9d.=£17.3375 The principal must first be 

^^ reduced to pounds, and then 

^1.560375 the interest may be found by 
c£l. 56=^1, lis. 2^d. any one of the preceding 

methods. 
]i]x. 48. What is the interest of ^6427, 18s. 9d. for 2 years 
at 5:} per cent, per annum ? 

49. What is the amount of ,£1096, 15s. 6d. for 4 years at 

6-1- per cent, per annum ? ^ Ans. ^61381, 18s. 8d. 

50. What is the amount of £120, 10s. for 2yr. 6mo. at 4a 
per cent, per annum ? Ans. j£134, 36s. l|,d. 

51. What is the interest of £270, 10s. >0d. for lyr. 4mo. 20 
da. at 7 per cent, per annum ? 

52. What is the interest of 1775fr. 75cent. for 3yr. 6mo. at 

6 per cent, per annum 1 Ans. 372fr. 90cent. 

195 



§250 PROPORTION — PERCENTAGE. 

53. What is the interest of 2070fr. 65cent. for 2yr. 8mo. 
20da. at 7 per cent, per annum ? Ans. 394fr. STcent. 

54. What is the amount of o29Tfr. 15cent. for oyr. 15da. 
at 8 per cent, per annum ? 

55. What is the interest of 10720fr. 25cent. for 5yr. 7mo. 

lOda. at 5 per cent, per annum ? Ans. 3007. G2fr. 

56. What is the amount of 20625fr. SOcent. for (5yr. 6mo. 

6da. at 6 per cent, per annum 1 Ans. 28689. 79fr, 



PARTIAL PAYMENTS. 



The method here given is the one enjoined by the Su- 
preme Court of North Carolina;, and used iu most, if not 
all, the States of the Confederacy. 

§ 250. Rule. — Find the amount of tlie given principal to 
tJie titne of the first p)ciymtnt^ arid if this pa 7/ men f is greater 
than the interest then due, subtract the payment from the a- 
mount. Consider the remainder as a second pr in cipaly and 
find the amount of it from the time of the first payment to 
the time of the second, and if the second payment is greater 
fhan the interest last found, subtract the second' payment 
from the second amount, and consider the remainder as a 
third principal: and so on. 

But if any payment is less than its corresponding inter- 
est, find the amount of the same principal to the time of tJie 
next payment, and if the sum of these two payments is 
greater than the interest then due, subtract their sum from 
the amount : but if the sum (fi the two payments is less than 
the interest then due, extend the time until the sum of the 
payments made shall exceed the interest due at the time of 

the last payment. 

196 



PARTIAL PAYMENTS. §250 



The principle of tlie rule is that the payment of a part 
of the debt shall not increase the debt. 

Ex. 57. $725.50. Richmot^d, Va., Jan. 1, 1858. 

One day after date, I promise to pay J.- Jones, or order, 
seven hundred and twenty-five dollars and fifty cents, for 
value received. ^ (^../^ (g^ 

On this note were the following endorsements : 
Mar. 16, 1858, $100.00 

May 16, 1859, 25.50 

. July 1, 1861, 300.00 

How much was due Oct. 8, 1862 ? 

SOLUTION. 

Original Principal, $725.50 

Interest to Mar. 16, 1858,— 2ra. 15da., 9.068 

Amount then due, ^734.568 

Amount then paid, 100. 

Second Principal, $634,568 
Interest from Mar. 16, 1858, to May 16, 1859, 

$44,419 
Amount then paid (less than interest) 25.50 
Interest from Mar. 16, 1858, to July 1, 1861, — 

3y. 3m. 15d. 125.327 

Amount then due, $759,895 

Sum of the two payments, 325.50 ^ 

Third Principal, $434,395 

Interest from July 1, 1861, to Oct. 8, 1862,— 

ly.'3m. 7d. 33.086 

Amount due Oct. 8, 1862, $467,481 

197 



§250 PROPORTION PERCENTAGE. 



')8. $3256.37. Lincolnton, N. C, Mar. 12, 1853. 

On demand I promise to pay to the order of J. Rein- 
liardtj three thousand two hundred and fifty-six dollars and 
thirty-seven cents, for value received. 

On this note were the following endorsements : 
July 12, 1855, received $654.33 
Sept. 20, 1857, " $246.50 

Jan. 5, 1859, " $945.87 

What was the "balance due Sept. 7, 1860 ? Ans. $2755.41. 




59. $108.43. Columbia, S. C, Dec. 9, 1857. 
With interest from date, for value received, I promise to 

pay J. Townsend or order oiic hundred and eight dollars 
and forty-three cents. ^^ C^^^/. ("^^ 

Endorsements. Mar. 3, 1858, received $50.04 ; Dec. 10, 
1858,113.19; May 1, 1860, S?50.11. How much was due 
Oct. 9, 1862 ? Ans. 1^55.844 

60. A note was given at Savannah, Geo., Apr, 16, 1856, 
for $450. On it the following endorsements were 
made : — Jan. 1, 1857, received ^20 ; Apr. 1, 1857, 
$14; July 16, 1857, %Z\ ; Dec. 25, 1857, ^10 ;^ July 
d:, 1858, $18. What balance was due June 1, 1859 ^ 

Note. — When r.o rate of interest is mentioned in a note, tiie 
legal rate at the place where it is given is to be used. In Louirfiana 
the legal rate is 5 per cent. : in Arkansas, Kentucky, Maryland, 
Mii-souri, North Carolina, Tennessee, and Virginia, it is 6 per cent.: 
in South Carolina it is 7 per cent. ; and in Alabama, Florida, Geor- 
gia, Mississippi, and Texas, it is 8 per cent. 

198 



COMPOUND INTEREST. |25l 



COMPOUND INTEREST. 



§ 251. Compound Interest is the interest on both princi- 
pal and interest when the interest is not paid as it falls due. 
In ordinary business transactions it is not allowed by law; 
but in a few classes of debts it is required that the inter- 
est shall be compounded annually. In such cases, the in- 
terest for one year is added to the principal; this amount 
becomes the principal for the second year ; its amount for 
the third year, and so on to the last year or part of a year. 
The original principal subtracted from the final amount 
gives the compound interest. 

Ex. Gl. What is the 'compound interest of $525.75 for 
3yr, Gmo. at 6 per cent., interest due annually ? 

SOLUTION. 

Original Principal, $525 75 

Interest for the first year, . 31.545 

Amount, — Second Principal, $557,295 

Interest on $557,295 for the second y^car, 33.437 

Amount, — Third Principal, $590.73^ 

• Interest on $590,732 for the third year, 35.443> 

Amount,— Fourth Principal, .$6267176 

Interest on $626,175 for the remaining 6mo., 18.785 

Total Amount at Compound Interest, $644,960 

Original Principal, 525.75 

Compound Interest, $119.21 

Ex. 62. What is the amount at compound interest of 
$500 at 6 per cent, for 4yr. 3ino., interest due annually ? 
63. What is the amount of $1000 for 7 years at 7 percent^, 
compounded annually ? 

199 



§252 PROPORTION — PERCENTAGE. 



64. What is the amonnt of $1000 for 6 years at 6 per cent., 
compounded semi-annually? Ans. ^1425.76. 

^. What is the interest of $1000 for 4 years at 6 per cent., 
compounded quarterly ? Ans. 1268.98. 



DISCOUNT. 



§ 252. Discount is a deduction made for the payment of 
money before it is due. 

§ 253. The present worth of a future debt is that sum 
which, at ordinary interest, will amount to the debt at the 
time it becomes due. The present worth bears the same 
relation to the debt, that the principal h^d^rs, to the amount. 

The problem to be solved, then, is, having given the 
amount, the time, and the rate, to find the principal and 
the interest. 

§ 254. Rule. — Find the amount of $1 for the given time 
at the given rate. Then, as the amount of $1 is to $1, so 
is the amount of the debt to its present worth. 

To find the discount, subtract the present ivorth from t\e 
amount of the debt. Or say, as the amount of $1 is to its 
interest^ so is the amonnt of the debt to the discount. 



Ex. 66. What is the present worth, and what is the dis- 
count, of a note due 6 months hence for $550 at 6 per cent.? 

SOLUTION. 

Amount of $1 for 6 months at 6 per cent., $1.03. 

$1.03 : $1 :: $550 : $533.98, pres,ent worth. 

$550 — $533.98=$16.02=the discount. 

Or, $1.03 : $.03 :: $550 : $16.02, the discount. 

200 



DISCOUNT. §255 



Ex. 67. What is the present worth of a note for $245, 
due 1 year hence when the rate of interest is 6 per cent.? 
68. What discount should be allowed on a note for $525, if 

paid 3mo. before it is due, interest being at 7 ner cent,? 
C9. What is the present worth of a debt of $375.50, due in 

Tnio. 15da., if interest is at 8 per cent.? 

70. What is the discount of a note for $725, due in lOmo. 

lOda., interest being 7 per cent.? 

71. In Mobile, Ala., one man gave another his note for 
$247. 50j due twelve months after date. What was the 
present worth of the note ? 

72. What discount would be allowed at New Orleans on a 

debt of $650, due 9 months hence ? 

73. What is the present worth, at Little Rock, Ark., of a 

note for $769.35, due 5mo. 18da. hence? 

74. What is the proper discount on a debt of $75.75, due 

7mo. hence at Memphis, Tenn.? 

75. What is the present worth of ^1250, due 12 months 
hence at Galveston, Texas ? 

76. What is the discount of $250, due 8mo. hence at Lex- 
ington, Ky.? 

77. What is the present worth of $55.55, due 7mo. benee 
at St. Louis, Mo.? 



BANK DISCOUNT. 



§ 255. The present worth or proceeds of a note payable in 
bank is the remainder obtained by subtracting from its face 
its interest for the time it has to run, including three addi- 
tional days — called days of grace. 

Thus, if I deposit with the Cashier of the Bank of Cape 

201 



§25(5 PROPORTION — PERCENTAGE. 



Fear my note for $1000 due in 60 chxys, lie will pay me on 
it only $1000— the interest of $1000 for 63 days, that is, 
$989.50. 

§256. The bank discount of a note not yet due is the in- 
terest of the face of the note for three days more than the 
time it has to run. 



Ex. 78. What is the present worth in bank of a note for 
$500 due in 30 days, at 6 per cent.? 

SOLUTION. 

Face of the note, ' $500. 

nterest of $500 for Soda., — hank discount, 2.75' 

Present Worth or proceeds, 

Ex. 79. What is the proceeds of a note due in bank 60 
da. hence for $250 at 6 per cent.? 

80. What is the bank discount on a note for $750 due in 
bank in 90 days, at 6 per cent.? 

81. What discount would a Kank require on a note lor 

$550.75, due 90 days hence at 8 per c(5nt.? 

82. What is the present worth of a note due in bank 90da. 
hence for $333.33 at 6 per cent.? 

83. What is the face of a note due 60da. hence, if its pres- 
ent worth in bank is $500, interest being at 6 per cent.? 

§257. The present worth of $1 : $1 :: present worth of 
the note : face of the note. In this case, $.9895 : $1 :: 
$500 : the answer. 

Ex. 84. What sum, payable in 90 days, will produce 
$750, if discounted at a bank at 6 per cent.? 
85. What sum, payable in 60 days, will produce $S000, if 
discounted at bank at 7 per cent.? 

202 



DISCOUNT. 



§257 



86. Far what amount must a note be drawn, payable in 30 
days, so that, if discounted in bank at 5 per cent., the 
proceeds will be $250 ? 

87. What must be the face of a note payable in bank in 90 

days, so that, if discounted at 6 per cent., its present 
worth may be $75.75 1 



Showing the number of days from any day of one month to 
the same day of any other month next following. 



From any 
(iay of 


To the same day of the uoxt j 


Jan. 

385 


Feb. 

31 


Mar. 
59 


Apr. 


May 

120 


June 

151 


July 
181 


Aug. 

212 


Sept. 

243 


Oct. 

273 


Nov. 

304 


Dec. 

334 


Jan. 


Feb. 


334 


36". 


28 


59 


89 


120 


150 


181 


212 


242 


273 


303 


Mar. 


306 


337 


365 


31 


61 


92 


122 


153 


184 


214 


245 


275 


Apr. 


275 


306 


334 


365 


30 


61 


91 


122 


153 


183 


214 


244 1 


May 


245 


276 


304 


335 


365 


31 


61 


92 


123 


153 


184 


214 


1 June 


^14 


245 


273 


304 


334 


365 


30 


61 


92 


122 


153 


183: 


July 


184 


215 


243 


274 


304 


335 


365 


31 


(12 


92 


123 


i53i 


Vug. 


153 


184 


212 


243 


273 


304 




365 


31 


61 


92 


122il 


: Sept. 


122 


153 


181 


2111 


242 




303 


334 


365 


30 


61 


91;' 


Oct. 


92 


123 


151 


182 


212 


243 


273 


304 


335 


365 


31 


61|i 


Nov. 


Gl 


92 


120 


151 


181 


212 


242 


273 


304 


334 


365 


30i 


Dec. 


M\ 


62 


90 


121 


151 


182 


212 


243 


274 


304 


335 


365 



To find the interval of time between Sept. 3, 1862, and 
May 19, 1863. Find Sept. in the left hand column and 
May in the upper line : then at the right of Sept. and un- 
der May, is 242, the number of days from Sept. 3 to May 3. 
To this add 16, the number of days from May 3 to May 19. 
The sum 258 is the number of days required. 

Again, from Jan. 25, to Sept. 9, is (243-16) 227 da^s, 

203 



§258 PROMISCUOUS PROBLEMS. 



PROMISCUOUS PROBLEMS. 



1. In what time will $100 amount to $200 at 6 per cen 
simple interest ? 

§258. As the interest of the given principal for 1 year 
the given interest :: 1 year : the number of years. 

In this case, as $6 : $100 :: lyr. : 16yr. Smo., the a' 
swer. 

2. In what time will $200 gain $50 interest at 6 per cent, 
per annum ? 

3. In what time will $500 gain $49 interest at 7 per cent.? 

4. In what time will $1000 gain $10 simple interest at 5 
per cent, per annum ? 

5. At what rate will $100 gain $15 interest in 2yr. 6mo.? 
§ 259. As the interest of the given principal at 1 per 

cent. : the given interest :: 1 : the rate per cent. 
In this case, as $2.50 : $15 :: 1 : 6j the answer. 

6. At what rate will $250 gain $250 interest in lOyr,? 

7. At what rate will $427.25 gain $143.60 in Syr. 4mo. 

lOda.? Ans. 10 per cent. 

8. At what rate will $746 gain $83.92 in 2yr. 3mo.? 

9. "What principal will gain $174.56 in lyr. 7mo. at 7 per 
cent.? 

§ 260. As the interest of $1 for the given time at the 
given rate : the given interest :: $1 : the principal. 

In this case, $.11081 : $174,56 :: $1 : $1575, the an- 
swer. 

10. What principal will gain $42 in Syr. 6mo. at 6 per 
cent.? Ans. $200. 

11. What principal will gain $210 in Syr. at 6 per cent.t 

204 



PROMISCUOUS PROBLEMS. §260 



• 



12. What principal will gain $400 in 4 years at 8 per cent.? 

13. What is the fourth root of 810000 ? Ans. 30. 

14. What is the value of 2.8.5.7.11 ? 

15. What are the prime factors of 1800 ? 

16. A commission merchant sold goods worth $9072; what 

was his commission at 2^- per cent.? Ans, $220.80. 

17. A capitalist sent his broker $15400 to lay out in stocks, 

after retaining i per cent, of the amount purchased.' 
How much stock did he purchase? Ans. $15801.00. 

18. A gentleman laid out 83025 in stocks which were 10 
per cent, below par. What was the nominal value of 
the stock purchased ? 

19. If I buy coffee at 30ct. per lb., and sell it for 3Gct. per 

lb., what per cent, do I gain? Ans. 20 per cent. 

20. A merchant bought 125 bushels of wheat at ^1.60 per 

bu., and sold it at a profit of 20 per cent. ; what did 
hegetfi.rit? Ans. $210. 

21. If I pay $12000 for a house and lot and sell them at 
an advance of 25 per cent., what do I gain by the 
transaction '^. 

22. A mei'chant gave $3.51 for an article which he is wil- 

ling to sell at a profit of 88^ per cent., how must he 
mark it? 

23. By selling a tract of land for $4704 I gain 12 per cent, 
on ic ; how much did it cost me ? Ans. $4:i00. 

24. If ocwt. ot sugar cost $23.40, what will lOcwt. 8qr. cost? 

25. A merehantj failing, pays only 60ct. on the dollar of his 

indebtedness ; how much will a man receive to whom 
he owes $1800 ? Ans. $1080. 

26. What cost 4G2yd. of cloth at $1.06^ per yd.? 

Ans. $490.87i. 

27. What cost 83bu. 3pk. 2qt. of clover seed at $8 per bu.? 

205 



§261 AVERAGE. 



28. What per cent, of $50 is $G? Ans. 12 pei- cent. 

29. What is 115 per cent, of $287.50 ? Ans. $330,625. 

30. At 5 per cent, commission, wkat would I receive for 
selling $240 worth of property ? 

31. A commission merchant sells property amounting to 

$550. Retaining his commission of 5 per cent., he 
lays out the balance after deducting a commission of 
2^- per cent, on the amount purchased. How much 
did he lay out ? 

32. What amount can I retain for commission at 8 per cent, 
on the amount invested, if I have received $2647.08 ? 



AVERAGE. 



§ 261. The average of two numbers is one half of their 
sum. Thus, the average of 7 and 13 is (7 + 13)-^2=10. 

The average of three numbers is one third of their sum. 

The average o^ four numbers is one fourth of their sum. 
And so on. 

§ 262. The average of two dates is a date lying half way 
between them. Thus, in any year June 23 is the average 
between June 1 and July 15. 

Ex. 1. Find the average of 2, 4.5, 5.75, 7, and 9.25. 

2. 

45 

5.75 § 263. Model. — Find the sum of the five 

7. given numbers. (§ 159.) Divide this sum by 

9.25 5. (§164.) The quotient 5.7 is their average. 

' 5)28.50 



5.7 
This needs no explanation. 



206 



ALLIGATION MEDIAL. 



^:^65 



Ex. 2. What if? the average of 2, 3, 5, and 6? 



:}. What 

4. What 

5. What 

6. What 

7. What 
S. What 
9. What 

10. What 

11. What 

12. What 

13. What 

14. What 

15. What 



s the average of 2, 5, 7, and 10 1 

s the average of 25 and 32 ? 

s the average of 34 and 19 ? 

s the average of 25, 32, and 41 ? 

s the average of 17, 29, and Go ? 

s the average of 25, 170, and 195 ? 

s the average of 2, 102, 111, and 115? 

s the average of 0, 5, 7.5, 25, and 40 ? 

s the average of 1, 7, 15, 25.25, and 37.5 ? 

s the average of 3, 7.5, 5.75, 11.75, and .625? 

s the average of 20, 47, 35, 91.5, 79.5, and 10.01? 

s the average of 13, 15, 17, 29.5, 37.5, and 63.75? 

s the average of 0, 1, 7, 9, 25, 37, and 39 ? 



ALLIGATION MEDIAL. 



§ 204. This name is given to the process of finding the 
moan value of a mixture, when the values of the substances 
composing it are known. 

Ex.16. If 41b. of sugar worth lOct. per lb. are mixed 
with 101b. worth 12ct. per lb., what is a pound of the mix- 
ture worth? • 

§ 265. Model. — Multiply 
lOct. by4. (§183.) Multi- 
ply 12ct. by 10. (§ 183.)— 
Add the products together. 
(§179.) Divide the sum by 
14. (§ 185.) The quotient, 
llf ct., is the average price 
per lb. 



4xl0ct.= 40ct. 
10x12" =1 20" 

14 i60ct. 

14 

"20 
14 

6 



14 



llfct. 



207 



§266 AViCRAaE. 



ExPL\?fATio?^. — The whole mlstare weighs 141b., whick 
evidently cost 160ct. : and 1 fourteenth of this amount is 
the average price per lb. 

Rule. — Divide the whole cost by the number of articles ; 
the quotient will be the average cost per unit. 

This rule applies to several things not embraced in the 
definition. 

Ex. 17. During 24 hours the thermometer stood for 2hi. 
at 55°, for 3hr. at 60°, for 4hr. at 65^ for 5hr. at 70°, for 
6hr, at 75°, and for 4hr. at 80^. What was the mean tena- 
perature of the day ? 

18. A goldsmith mixes lOoz. of goid 16 cirats fine with 6oa. 

17 ci]*ats fine and 8az. 19 carats fine ; what is^the fina- 
nes^ of the mixture ? 

19. A grocer mixed 4gal. of wine worth $1 i gallon, 5gal.' 
worth $1.25 a gallon. ani'lOgal. worth $1.50 a gallon ; 
wh (t s^as the mixture worth per gal.? 

20. If 30gal. of molasses at lOct., 40gii. at oOct., 70gal, at 
60ct., and SOgal. at SOjt., be mixed to^^erjier, wnat ia 
a gallon of the mixture worth '/ Ans. 32Y\ct. 

21. A farmer has 10 sheep worth $4 eaoh, 12 worth $5 
eacli, and 8 worth $10 each ; what is their averag« 
value ? 



ALLIGATION ALTERNATE. 



§286. This consists in finding the proportional quantities 
of several simple substances which shall make a compound 
of a given mean value. It is, therefore,'the converse of the 
preceding. • 

208 



ALT.IOATION ALTFRNATE. 



§26T 



Ex. 22. In wbnt, proportions inir«t sugars worth lOct., 
llct., 13ct., and ISot., be mixed, that the compuund uiaj 
be worth 14ot.? 

riO— 1 § 2C7. Model.— Connect 10 



1 with 15, 11 wi'h 15, and 13 
1 .with 15. 10 from 14 h-avf8 4; 



^^1 ^3.^ , 

L15ir_ .4 + 3 + 1 = 8 set 4 opposite 15: 11 from 14 

leaves 8; ^et 3 opposie 15 : 13 

from 14 leaves 1 ; set 1 opposite 15: — 14 tr<*iij 15 Uaves 1; 

pet 1 opposite 10, 11, an«l 13. ll« nee there muft he lib. ai 

lOct., lib. at Hot., and lib. at 13ot., to 81b. at 15cf. 

Explanation. — After arranging the several prices as in 
the model, atid placitig the mean price on the left, we con- 
nect each price below the mean with ont> above it, and t^ach 
price above the mean with one below it. Then taking the 
dififorence between each price and the mean, we set this dif- 
ference opposite the price with which this price is connect- 
ed ; observing during the operation to consider all the prices 
as abstract numbers. The reason for all this is evident 
when we consider that tach pound at lOct. falls 4ct. below 
the mean, while each pound at 15ct. is only let. above it. 
To average these two values, therefore, we mu.«t have 41b, 
of the sugar at Inct, to every one at l^cf. For a siniilar 
reason, it requires 31b. at 15ct. to counterbalance lib. afc 
llct. And as the mean price is equidistant between 13ct. 
and 15et., these two qualities must be taken in equal quan- 
tities. So that to bring the three i ferior qualities up to 
the required average, it is neces^ary to take 4-|-3-f 1, /. <?., 
8lb. of the superior quality to lib. of each of the inferior 
qualities. 

Rule. I. — JJrrange the several prices in a vertical coi- 
umriy and place the mean price on the left, 

N 209 



§267 AVERAGE. 



Connect each price below the mean with one above it, and 
each price above the mean with one below it. 

Find the difference between each price and the mean, and 
set it opposite the price luith which it is connected. If only 
one difference stands opposite any price, it denotes the j)^o- 
portion of that value ; but if several differences stand oppo- 
site any price, their sum denotes the proportion of that value. 

II. If it is required to have a specified quantity 
OF ANT VALUE. — Find the proportions as above. Then say, 
As the proportion found for this value : the quantity reqiiir- 
ed for it : : the proportion for any other value : the quanti- 
ty required for it. 

III. If the whole quantity oi' the mixture is speci- 
fied. — Find the proportions as ab JVC. Then take the sum 
of the proportional numbers, and say, As the sum of the 
proportional numbers : the required quantity of the mixture 

:: the proportion for any value '. the quantity required for 
that value. 

Proof. — By Alligation Medial. 

Ex. 23. In what proportions may gold of 10, 13, 14, and 
22 carats fine, be mixed so that the compound may be 17 
carats fine ? 

24. A grocer having brandy worth $1 a gallon, wishes to 
mix it with water so that he can sell the mixture at 
80ct. a gallon. In what proportions must he mix them ? 

25. In what proportions may liquors worth respectively $1, 
$1.20, $1.40, and $1.50 be mixed, that the mixture 
may be worth $1.25 ? 

26. A farmer wishes to mix 14bu. of wheat worth $1 per bu. 
with such a quantity worth $1.24 as will make the mix- 
ture worth $1.03 ; how much must he take ? 

210 



EQUATION OF PAYMENTS. §269 



27. How much tea at 80ct., 70ct,j and 60c t., respectively, 
should be mixed with 901b. at 90ct., so that the mix- 
ture may be worth 75ct. per lb.? 

28. A merchant having 1001b. of sugar worth lOct. per lb., 
mixed it with other sugar worth respectively 5, 8, and 
9ct., and sold the mixture at S^ct. How much of each 
quality was there in the mixture ? 

29. How much sugar at lOct., and how much at 15ct. per 
lb., must be taken to make 601b. worjth $7.20 ? 

Ans. 361b. at lOct., and 241b. at 15ct. 
W, A grocer mixes 1441b. of sugars worth respectively 12, 
10, 0, and 4ct. per lb., and sells the mixture at 8ot. 
per lb. ,• how much of each quality does he take ? 
31. A man paid S165 to 55 persons — men, women, and boys; 
to each man he paid $5, to each woman ^1, to each 
boy 50ct. ; how many were there of each ? 

Ans. 30 men, 5 women, 20 boys. 



EQUATION OF PAYMENTS. 



§ 268. This consists in finding the average date at which 
several amounts due at different times may all be paid, so 
that no interest shall be either gained or lost. 

Ex. 32. A owes B S25 due in 4mo., $50 due in 6mo., and 
$75 due in 8mo. ; what is the mean time of payment? 

25x4mo. = 100mo. §269. Model.— Multiply 4mo. 

50x6 " =300 '^ by 25. (§183.) Multiply 6mo. 

75x8 " =600 '- by 50. Multiply 8mo. by 75.— 

150 15,0)100,0mo. ^^^ ^^^ products together. Add 

— Wf- — the multipliers together. Divide 

Ogmo. iQQQ^^Q^ Ijy i^Q rp^g quotient 

6|mo. is the mean time of payment. 

211 



§269 AVERAGJ5. 

Explanation. — The irifere^t of 625 dollars for 4 months 
IS e(jn-il to the interest of 1 dollar for 100 months: the in- 
fere.-st of 350 for 6iiio. = the interest of SI forSOOuio.: the in- 
ter es of $75 tor Sin!t.=: the interest of ^ I for GJOoOo Hence 
the interest, of the several am >airth for their respectiv;e times 
Ln equal to the interest of $1 for l0(»0ii!o., and this i» equal 
to the interest of $150 for Cfmo. Oence it is fair that tb« 
whole amount .should be paid in 6|nio. 

IvUi.K. — Mnlfiplf^ e<i(}i teiDi of cn'dlt hij the number oj 
Wtita ill ih> correypititdiuij juiijmeht, iinil divide the .sum oj 
thf pmdnrts hy the aiim of (kr niu(i'plitr» : the quotient wilf 
%e ihe medv tim" of pai/mcnr, ^ 

WjS^^ '"I-^. a Qian owes an other $500 due in 8 mo., 6400 
.dn»^ ii» 6mo., and SGOO due in 9ino. ; what is the averagj* 
term of er<t;dit for the three debt?*? 
84. liou^ht goods a ■< follows : ^h)0 on a credit of 6rao.; 

,$200 on o|tiio.; and $560 on 6mo. ; what average credit 

f^hould lie allowed me on the «hole? 
B5. B'U|ihi $iOOO worth of goods lo be paid for as follows; 

$200 on the day of purchase, $400 in 5mo., and $400 

in I5m(>. What average credit &hould be allowed in* 

ot) the whole ? 

86. Ill what time should the f(*llowing amounts be paid ali 

at. once : $1000 due in 5mo., $1200 in Gmo., and $1200 
in 8mo.? 

87. I owe $100 to be paid Jan. 15 $200 due Feb. 15, and 

$300 due Mar. 9 ; on what day may the whole debt be 

paid at once ? 

jV'o/*? — Felec' }• owe day from wh'ch the periods of credit nifty 
be supi 08^•d to • omnieiicts In this i.saiKe. Jan 15 in the mosi 
conv.uJtiit Kind ihe interval ehi|s:iiK t>. tweeti tUis date wnd each 
0*" th** o»h» r«. }0(d then proceed according to the rule. Consider 
<a*ch manth 80 days. 

212 



FQUATrOV np PATMFNTJ3. §270 

ZH. A Tnai) Awe^ liis jieiii'hho'* $12f)0 duo in 8tn<>. : but at 
the <vi<l of 3m'). be pfi^ys $250, and \\\ 'Iwak more be 
pa)' $'5''; whut oxteiisioti of credit sboiild be allowed 
on the remainder ? 

250x5nio. = l 50aM). 270. M(.nKL. — Multiply 5 

150x3 '* = 450 " mo. b> 2:)!). (^ 1>^3 ) xMulti'pIj 

850)17007n7r. ^'>»''. *'> ^^'^' Add tbe pind- 

ucrs t<tfri;rhor. l)ivide ITOOnio, 

'^"**'- by 850. The quotient 2mo. i« 
the extension of cre<^it:. 

EXPL>NAT(()N.— The debtor, having pqid $2^0 (S — 3) 
5mo before it was due, is entitled to a credit of l25!)ino. 
on $1 : and, having paid SI 50, 8'no. before it was due, ii 
therefore entitb'd to a credit of 45'hiio. on 8'. For both 
prepayments be is entirliMl to a credit eipiivMlent to $1 for 
ITOOmo. Twe remainder unpaid is 81250 — ($25!) + 3l r>0) 
=:$'*'50 : nnd a credit of ITOOuio. on $1 is equal t<j a credit 
of 2mo ofi S85'>. 

Ex. 811. I owe $1000 due in 12mo. Tf T pny $100 at the 
end of Bnin., ar.d $100 at the end of 4nio.. bow long be- 
yond the r2ino. should my creditor wait for tb« jiayment 
of the 'balance ? 

40. I owe $2 100 due in 6.no. If I pny 8500 down, ?3"0 i^ 
.2mo., and S200 in omo., in bow many mOnth-i from thfc 

contraction of the debt should I pay the bjilnnce? 

41. A niercbaiit owes SI 200, of which §200 is to be paid in 
4 mon'hs, 8400 in 10 mo-nths, and the remainder in Id 
months: if he pays the whole at ouce^ at what time 
rDu.-'t he ni'tke the payment? 

42. A merchant owes $ISOh to be paid in 1.2 months, 8240O 
to be [>aid in 6 mou'hs, and 82700 to. be p.iid iu 0^- 
mouths: what is the averaj^e tuue of; payment ? 

2ia 



PROMISCUOUS PROBLEMS. 



PROMISCUOUS PROBLEMS. 



1. Reduce ,£19, 8|s. to pence. 

2. Reduce 9oz. 16|-dwt. to grains. 

3. Reduce - of ~ of IG-J- to its simplest form. 

■]r 12 ^ ^ . 

Ql. Qi. 4i 

4. Reduce 4-^, -zrrr-, tt-^ j and I of -:^ to their least com- 

'^ 20f 26 -" 9 

mon denominator. 

5. Add 900.01, 450.037, and 696.9 together. 

6. Add 2y\, 6|-, and 12if together. 

7. Add -^ of -*- of 20, -f- of -f- of 24^, and ^,- of 2-i- together. 

6a 2i 

8. From -^ take -^. 

9. From -f- of f of 3i take f of f. 

10. From $49f take i4.75 + $5^4-$9.30. 

8-1- 17 

11. Multiply -4 by -7-. 

5 

17 

12. What is the product of -fi by --- ? 

50 J 2Q_:2. 

5 

13. Multiply ^g-^^ by -^V of if of 5f 

14. Divide f of 3-| by -f- of 6i. 

15. Divide 1301|- by 161.3. 

16. Divide $1843i by 368f, 

17. What is the insurance on $3125 at 6i per cent.? 

18. A commission merchant sold 19 firkins at 45ct. per lb., 

and retained 5 per cent., commission ; how much did 
he return to the owner ? ^ 

.19. What is the par value of two certificates of stock; one ^ 

214 



PROMISCUOUS PROBLEMS. 



for S350 at 2} per cent, discount, the other for $527-.- 
60 at 5 per cent, advance ? 

20. What is the amount of ^1054, 10s. 9d. for 2yr. 9mo. at 
4 per cent, per annum ? 

21. If the interest of a certain amount of money at 6 per 
cent, is $241.80, what is the interest of the same sum 

, for the same time at 7^- per cent.? 

22. At what rate per cent, per annum will jei8295 10s. 
amount to £1898, 2s. l^d. in 9 months] 

23. What is the greatest common measure of 560, 880, 
1028, and 1296? 

24. What is the least common multiple of 36, 18, 33, 11, 
and 6 ? 

25. What is the greatest common measure of 56, 154, and 

182? 

26. What is the least common multiple of 2, 4, 10, 7, 14, 

15, and 21? 

27. Resolve 528 into its prime factors. 

28. What prime factors are common to 360, 420, and 840 ? 

29. I sold 125A. 2R. 20P. of land for ^2050 ; how much 
did I gain or lose, if I gave $15.50 per A. for the land? 

30. I bought a lot of English paper for .£698, 10s. 6d.,and 
sold it at a proJGit of 75 per cent. ; how much did I re- 
ceive for it in Federal currency ? 

31. What is the amount of £300, 10s. for 2yr. 3mo. at in- 
terest compounded semi-annually, at 8 per cent, per 
annum ? 

32. What is the square root of 509796? 

33. What is the cube root of 16003008 ? 

34. Find the greatest common measure of 1538, 2307, and 

3845. 

215 



PROMISCUOUS PROBLEMS. 



35. I sold i of my land to A, ^ of ifc to B, and retained 
20i.'A. for myself; how much had I at first? 

ii6. A, B, and C trade in p'^rtnersliip. A invests SIOOO 
'for 12 mon'hs; B, fj^l 500 for 10 months ; and C, ?2000 
for months. How bhali their profit of SlUOO be di- 
vided ? 

37. How many barrels of potatoes at S2.50 per bbl. should 
be exchanged for a Intgshead of sngnr weighing IST-^^lb. 
groBP, worth 115.00 a hundred pounds net, tare being 
8 per cent.? 

8?. How many firkins of butter, at 25ct. per lb., can bo 
bought fur 9iiiO. interest of §800 at 7 per cent, per 
annum? 

89. Having been engaged in merchandise with a capital of 
$19500, I realized a profit of 33^^ per cen^t., which I 
immediately inve>ted in land at $16.50 per A.; how 
many acres did I buy 1 

40. If [ owe three notes, one for ^630 due 3mo. hence, an 
other for ^800 due 6mo. h'nce, and the o-her for Si 000 
due 15rao. hence, in what timjc might I fairly pay the 
three notes together? 

41. If i2lb. of tea @81.->0, 151b.@$1.44, and 18ib.@S1.80, 
be mixed together, what irs liae value of 111b. of the 
ntixture ? 

42. What i.s the 4th power of 7^? 

43. What is the cube of 3.5? 

44. What is the c.ibo root of ]9.Gi»3? 
45 What is the 8(iuare root of 7l.)3^g ? 

46*. Required to fill a hogshead with two kinds of wine 
worth $1.20 and $1.05 fier gal. respectively, so that the 
mixture wil' be worth ?I.15 per gal.; how mariy gal- 
lons of each kind will be rcijuired ? 

216 



pnOMISrrOUS P'CtBLFMS. 

47. The total Mock in n UnilnKaJ is SIOOOoOO tb'^ net m- 
cctme for a veur is $501'00 ; wlia dividend will I re- 
ceive f r SlOOOn woith of .-tuck ? 

18 I exebaficed a houf^e jmd lot worth £500, 15^ fnr land 

* « • ? 

worth SIU.50 per A.; how tuuch l;iud did I r^'C'•ue • 

49. A pedlar exclxintred a pit-ce of calico, rated at 2 let. 

per yd., ff^r a fii kin of butter worih '2'2}ji5t. per lb; 
how many yards of calico were tb re? 

50. I imported V):")?. of iron wor-b S82 per T, ; what was 
th« duty o!i it at 3*3^ |>er c<Mit.? 

51. ^Vlnjt is il»e net weight ot* 275 bags of coffee, weighing 
each 78tb. irross, tnre being 4 per cent.? 

.Vi. What cost 7UA. nn. 25P. of land at $.\5.75 per A.? 
33. What co.-t 51'. IGcwf. 3.|r. o iron at $1:.125 per cwt.? 
i4. If, by se'ling a tract of land for St)450, I lo>=e 4 per 

cent, of what it cost nie, for what would I have had to 

sell it, to jjuin C^\ f>^»r cent.? 
55. Bought 2')T. IGcwr. of iron a^ =£1 1, 1 Gs. per T. ; and oM 

the whole for 32 JOO , what did [ gaiu or lose p«r T? 
^iiy^ What IS the present worth of $-00n, due in 2yr. Huio. 

1.') Ia.» ijitere:-t beiiig at fi per cetit. p^r annum r* 

57. What is the discount on £iXO^ due in Syr. Omo., inter- 

est hfiiig at 8 per cent, per annuu) ? 

58. I wish to borrow 81 150 iti bank : interest being at 6 

percent, per tinnum, what ujust be the fa<^e of t!) ; 
proper note at 90 days? 

59. In what tiute will £4o2, 15s., at C percent, per annum, 

amount to £502, 1 1 -. '^d.? 
6D. If the insurance of :B25ajO is S400, wb-it is the rate per 

C'-'Iit.? 

($1. What per cent, of 00 is 1,25? 
Q2. Wluit per cent, of 75 is 125 ? 

217 



PROMISCUOUS PROBLSMS, 



63. What is S-^\r per cent, of S11755 ? 

64. If 5|-A. of land cost $144.50, what will 17A. 3E. 19.- 
375P. cost ? 

65. If 1.37gal. of sorghum molasses cost $1.4375, what will 
I3.7gal. cost? 

66. Divide 17mi. 5fur. 25rd. by 1.5. 

67. Multiply 3deg. 17min. 45sec. by 2.03. 

68. Dividend is £1, 18s. 9.5d., divisor is 4.9, what is the 
quotient? 

69. Dividend is Ibu. 3pk. 4.5qt., e^uotient is 75bu. 2pk, 
4qt., what is the divisor ? 

70. From 1.475T. take 17cwt. Iqr. 19.29lb. 

71. Add together 4.75gal., 3.07qt., 7.45pt., and 6.19g). 

72. What cost 17bbl. flour at $10 per 1001b., 3bu. salt at 

$1.25 per bu., and 677.51b. pork at $0,065 per lb.? 

73. If 75 persons eat 800bu. corn in 1 year, how long will 

600bu. last 90 persons ? 

74. If 150 copies of a book of 200 pages require 6rm. 4qr, 
of paper, how many reams will 15000 copies of a book 
of 224 pages require ? 

75. If Irra. of paper weigh 301b. and cost 30ct. per lb., 
what will the paper cost for an edition of 1000 copies 
of a book which requires 5rm. lOqr. for 96 copies? 

76. If 83iT. of coal cost $405.50, what will 17T. 3cwt. Iqr. 
bring at 16|- per cent, advance ? 

77. The second, third, and fourth terms of a proportion are 
-|, 1^, and 2.5, respectively ; what is the first term ? 

78. If the first, third, and fourth terms of a proportion are 
$64.96, 7cwt. Iqr., and 4cwt. 2qr., respectively, what 
is the second term ? 

79. Multiplicand is 94; product is .66; what is the multi- 
plier ? 

218 



PROMISCUOUS PROBLEMS. 



80. I bought 6251b. of cheese for $62.50, and sold it at 

12ict. per lb. ; how much per cent, did I gain? 

81. I own I of a ship worth 820000, and have insured it at 
2.375 per cent. ; what insurance do I pay ? 

82. What is the amount of $2169.845 for lyr. lOmo. 17da. 
at 7 per cent, per annum ? 

83. What are the prime factors of 7825? 

84. What are the common prime factors of 875 and 1750 .'' 

85. How many hours will there be in the year 11)00 ? 

86. The Mecklenburg Declaration of Independence was 

made May 20, 1775; North Carolina unanimously 
seceded from the United States May 20, 1861 ; how 
many days elapsed between these two great events 1 

87. What cost 30001b. of corn at S3.00 per bbl.] 

88. What cost 5.25bbl. of flour at $ .04 per lb.? 

89. An officer, in pursuit of a criminal, goes lOmi. per hr.; 
the criminal, who has 36mi. the start, goes 7mi. per hr.; 
how far must the officer go, to catch the criminal ? 

90. Bought 40gal. wine at $2.50 per gal. ; lost 5gal. by 
leakage : how must I sell the remainder per gal. so as 
to gain 25 per cent, on the whole ? 

91. A vessel laden with 3000bu. wheat, found it necessary 
to throw 25 per cent, of her cargo overboard ; what 
was her loss at $1.25 per bushel 1 

92. What is the value in Avoirdupois weight of 161b. 5oz. 

lOdwt. 12gr. Troy? 

93. How many sheets in 7 reams of paper? 

94. If 7 silver spoons weigh lib. 2oz. 3dwt., what will each 
spoon weigh ? 

95. If 2A. produce 45bu. 3pk. 6c][t. Ipt, of corn, how much 

will 32A. produce ? 

219 



PR.OM.'SCUOUS TRrELEMS. 



96. A.li together i of i of an Acrp, 75^^?. -fll., and |A.r 

97. What part f.f u futhom is Sifr.V 

98. What is the anionnt of $30U0 for Gmo. 24Ja. at 7^- per 
ceiif. per arirnim ? 

99. A ari.l H [)iirclj;iPe(l a house for $3000, of which A paid 
31800, how shall they divide a rent of $350 ? 

iOO. Wliat it^ the scjuare root of 57(5? 
101. What is the 4rh root of GoGl ? 
I'v2. WhMt is the cube root of yf-^iouo ^ 

103. How much stock at 7 perceut. advance niaj be bought 
fur Sf)350? 

104. Ij.'uglit lOrm. of paper at ?3.50 per rm., and sold it 
at $ .25 per quire, h -vv luuch did I gain or lose on it all? 

105. Bought 30ubhl. of flour for $2250, sold I of it at $0 
per hbl., jiiid the remainder at $8 per bbh, how much 
did T receive for the whole? 

106. Reduce 26^ to a decimal form. 

107. Miihiply four rhousjindths by five hundredth!*. 

108. Multiply fdur hundred ati({ fifty by two hundredths. 

109. Diviile -even tenths by one hundredth. 

liO. \\''hai is the difference between thirty-five hundredths^ 
and thirty-five thouHa'jdrh.s ? 

111. What is the 2nd term of h proportion whcse T.-t, 3rd^ 
atid 4(h terms are 7, 1.3, and 19, respectively ? 

112. If one jicre of lar»d costs jC2, 15s. 4d., what will be th» 
cost (.f 173 A. 2R. 14P. at the same rate ? 

113. A geijtlemah's estate is worth dC42 !5, Is. a year : what 
may he spend per day and yet save i^lOOO per annum? 

114. A father left his son a foitune, -^ of which he ran 
through in 8 months, -^ of the remainder lasted him 12 
months longer, when he had barely £820 left: what 
hum did his father leave htm ? 

220 



PROMlSCUOye PROBLEMS. 



115. Ther? are 1000 men be>ie«^ed in a to'-vn with j)rovi" 
einiis for 5 weeks, allowing each uian 10 ounces a. day. 
If rh-y art> reinf.rrced by 500 more and no relief can 
be afforded til! the end of 8 wi-ek.s, Low many ouiKces 
n\\ii\t be given daily to each man '{ 

116. A fjitlier pave -{^ of his eMate to one Hon, and ^'^ of 
the remainder to another, leaving the rest to his widow. 
The difference of the childjen'w h-gaeies was £514, 6s. 
Si\. : what was the widow's portion ? 

117. If Hcwt. 2(jr. of sugar cost S12l\92, what will be tlie 
price of Ocwt.? 

118. If the freight of 80 tierces of Kiigar, each weighing 
Sicwt.. 150. miles, cost S8[, what must be paid fur the 
freight of 30hhd of sugar, each weighing 12cwl., 50 
miles ? 

119. If one pound of tea be equal in value to 50 ornnges, 
nnd 70 oranges be worth 8* lemons, what is the value 
of a pound of tea when a lemon is worth 2 cents? 

120. If 60 bushels of oats wilbserve 2( horses f.r 40 dajR, 
how long will^iO bushels swerve 48 horses at the same 
rate ? 

131. W!,at will be the cost of 2hhd. 5ga]. 3qt. 2gi. of mo- 
Jasse^, at 12} cents p^r (jnart? 

122. Wliat is the interest of 53153.82 for years, at 4} 
per cent, per annum ? 

123. What is the interest of 831573.25 for 10 months at 6 
per cent, per annum ? 

124. What will be the amount of $9537.15 for 11 years, 2 
months, and 18 days at 7 per cent, per annum ? 

125. What will be the amount of 83758.56 for 3 years at 

7 per cent., th© interest being compounded semi-annu- 
ally ? 

221 



PROMISCUOUS PROBLEMS. 



126. If I buy 895 gallons of molasses and lose it per cent, 
by leakage, how much have I left ? 

127. Bought a piece of cloth containing 150 yards for $650: 
what must it be sold for per yard, in order to gain $300? 

128. What is the bank discount on a note of $556.27 pay- 
able in 60 days, discounted at 6 per cent, per annum ? 

129. The sum of two numbers is 5330, their difference 
is 1999 : what are the two numbers ? 

130. How many scholars are there in a class, to which if 11 
be added the number will be augmented one-sixteenth? 

132. Sound travels about 1142 feet in a second. If then 
the flash of a cannon be seen at the moment it is fired, 
and the report heard 45 seconds after, what distance 
would the observer be from the gun ? • 

133. What number is that which being augmented by 85, 
and this sum divided by 9, will give 25 for the quotient? 

134. One -fifth of an army was killed in battle, i part was 
taken prisoners, and J^ died by sickness : if 4000 m.en 
were left, how many men^did the army at first consist of? 

135. The greatest of two numbers is 15. and the sum of 
their squares is 346 : what are the two numbers ? 

136. At what rate per cent, will $1720.75 amount to $2325.- 
86 in 7 years ? 

137. In what time will $2377.50 amount to $2852.42 at 4 
per cent, per annum ? 

138. What principal put at interest for 7 years, at 5 per 
cent, per annum, will amount^to $2327.89? 

139. What is the greatest common measure of 945, 1660, 
and 22683 ? 

140. What is the greatest common measure of 204, 1190, 
1445, and 2006 ? 

141. Find the least common multiple of 6, 9, 4, 14, and 16. 

222 



PROMISCUOUS PROBLEMS. 



142. What is the least common multiple of 11, 17, 19, 23, 
and 7? 

143. What is the least common multiple of 7, 15, 21, 28, 
35,100,125? 

144. Reduce ^-^^*- to a mixed number. 

145. Reduce 149|^ to an improper fraction. 

146. Reduce 375|^ to an improper fraction. 

147. Reduce 174947^^^ f|-g to an improper fraction. 

148. Reduce fig- to its lowest terms. 

149. Reduce -tt-Vt *^ ^^^ lowest terms. 

150. Reduce 14-1^- to its lowest terms. 

151. Reduce %, -^, and -j\- to their least common denomi- 
nator. 

152. Pteduce ^V? ^\y ^^^ v *^ their lea.<^t common denomi- 
nator. 

153. Find the least common denominator and add the frac- 
tions, yV? h h and t, 

154. Find the least common denominator and add /^, }f ^-, 
and -V- 

155. Multiply 5^- by ■}. 

156. Multiply f5 by | of 9. 

157. If 80 yards of cloth cost $340, what will 050 yards 
cost ? 

158. If 120 sheep yield 330 pounds of wool, how many 
pounds will be obtained from 1200 sheep 1 

159. If 6 gallons of molasses cost SI. 95, what will 6 hogs- 
heads cost ? 

160. If *- of a yard of cloth cost ^1|, what will 7^ yards 
cost ? 

161. What is the cost of 28|- yards of cloth, at $4f per 
yard ? i 

223 



PUOMISrUOUS PROBLTfMS. 



162. Whnt. is the interest of $1914.16 for 18 years at 3^ 

per c#Tit. per animni ?' 
ICS. What is the a»n>uni:, of $7958.70 for 9 raonths at (^ 

por ctMit per unniuij ? 

164. A Fnerch-mt h-d^ 1-00 barrels of fl )ur ; he shipped 64 
per c^uf. of it and sold the remainder: how much did 
lie sell? 

165. Two men had eacli $240. Que of them Rpend8l4per 
cent., and the otiier 1<Sl per cent. : how ir)a[»y dollar* 
more ditJ one >pend than the n<her? 

166. Wh;)t is the diife'vnce between f)^- per cent, of $800 
and 6*- per cent of Sl<'50 ? 

167. What is the square root of 1.519:^^9!? 

168. What is the Hj.nn-e root, of :^'^:i7296l ? 
1C>9. What is the cube r .ot of 18 28541 ? 

170. What is the cnhe root of 2;<>5408-6008 ? 

171. if a portion re(;='ivt\s ?1 iur ± of a djy's work, how 
innrh is that a d^y i 

112. \V\u\i nmnher \<i rh»f of whicii i, -\, and |- added to- 
gether, will muke 6-3? 



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