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Full text of "The Elementary Arithmetic: Oral and Written"

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,:^^^^^NW1^:; 



'RlilCIPALOFTHC' jPeNN'- 

State Normal Sci^coi-- 

: 



FW/Z A DELPH: '\ - PORTER j> COA ^f 



J 




v^voi-li 



HARVARD COLLEGE 
LIBRARY 




GIFT OF 

JAMES STURGIS PRAY 

Chalks Eliot Professor of Landscape Architecture 

To be kept in the main coUedion of the 
College Library 





I 



« 



3 2044 096 999 370 



Z.:?u>^ 



THE 



ELEMENTARY 



AEITHMETIC, 



ORAL AND WEITTEI^. 



BT 

ALBERT N. RAXJB, A.M., 

PRINCIPAL OF THE CENTRAL PENNSTLYANIA STATE NORMAL SCHOOL^ 
AND AUTHOR OF "THE COMPLETE ARITHMETIC." 




PHILADELPHIA : 

PORTER & COATES, 

822 CHESTNUT STREET. 




u.v.ni.i 










Copyright, 1877, by Porter A Ck>ATB8. 



PREFACE. 



Believing that Oral and Written Arithmetic form 
properiy but a single study, and that they are best taught 
together, the author has in this, as in his Complete Arith- 
metic, combined both oral and written exercises in the 
same book. He claims that by this arrangement much 
valuable time may be saved to both teacher and pupil, and 
the science of Arithmetic be even better taught than by 
the plan of teaching the two from separate books. 

In the composition of this book two main objects have 
been steadily kept in view : first, that of teaching the 
elements of Arithmetic thoroughly ; and, secondly, that 
of training the pupil to think for himself and apply his 
knowledge of the science as he acquires it to the actual 
business operations of life. To accomplish the first of 
these objects, the author has adopted a natural and sys- 
tematic gradation, not only of subjects, but also the divis- 
ions of these subjects and the various operations Under 
these divisions. To this end also the problems have been 
systematically graded throughout the book. 

To accomplish the second object, simple, concise solu- 
tions have been given as a basis where it was thought 
they could be of advantage ; but in many cases the pupil 
is left to depend upon'his own knowledge, gained from his 
previous study of the book, to frame his own solutions. 
He is required to give his explanation of many of the 
written solutions of the problems as given by the author. 



4 PREFACE. 

It is believed that this will, under the guidance of a ju- 
dicious teacher, give him valuable training, not only in 
reasoning, but also in the use of language. He is re- 
quired also, in many cases, to derive his own rules from 
the preceding solutions and principles. 

In order that he may understand the practical appli- 
cation of Arithmetic, and at the same time acquire a 
knowledge of the business use of the science as he pro- 
ceeds, the problems given are largely, almost wholly, 
drawn from the actual business operations of life. 

The fact that a large number of pupils never study 
Arithmetic for any great length of time, and the neces- 
sity of their studying that only which will be of practical 
utility to them in after life, has been kept constantly in 
view in adapting the book to their wants. 

Extended exercises are given in the application of the 
Amdamental rules, because these are the basis of all the 
operations of Arithmetic. 

The definitions given are brief and accurate, and it is 
thought they are all clothed in such language as the 
pupils can readily comprehend. 

Every effort has been put forth to make the book a 
desirable one, not only for the pupil in learning the ele- 
ments of Arithmetic, but also for the teacher in present- 
ing the principles to the mind of the learner. It is 
believed, and confidently hoped, that progressive teachers 
everywhere will find it well adapted to their wants. 

A. N. R. 

Central State Normal School, 

Lock Haven, Pa., July 19, 1877. 



SUGGESTIONS TO TEACHERS. 



The author neither desires nor aims to give here a 
detailed method of teaching the whole subject of Arith- 
metic, but rather to suggest a few hints which he hopes 
may be of practical service to those who use the book. 

Objects. — Since children acquire knowledge most 
readily through their perceptive faculties, the teacher 
should illustrate largely by the .use of objects. This 
means of illustration should be used from the beginning, 
for the reason that the child thinks concretely — that is, in 
connection with objects. The idea of ten ones, or a ten, 
can be best taught by putting together ten objects and 
calling the collection Siten; in a similar manner ten tens, 
forming a hundred, and so on. So also the idea of a 
fraction and of fractional parts can be taught best in 
connection with objects. Objects themselves are much 
superior in this respect to pictures. 

In the selection of objects care should be taken, espe- 
cially in teaching the idea of a fraction, that such objects 
be used as cannot be divided without destroying their 
unity. Thus, an apple is much better for illustration 
than a stick or a line ; for if an apple be divided into any 
number of equal parts, none of these parts are units ; but 
a stick may be divided into other sticks, or a line into 
other lines, each of which is still a unit. 

Numeral Frame. — Teachers will find the numeral 

frame a great aid in teaching all the frmdamental rules. 

It has the advantage of being compact and convenient, 

aild is always at hand. It is among the most serviceable 

articles of school fiimiture. 

1* ^ 



SUGGESTIONS TO TEACHERS. 

« 

Counting. — Give much attention to the exercises in 
counting following each of the tables. It is believed that 
if these exercises are judiciously used, pupils will perform 
the operations of the fundamental rules with much greater 
rapidity and exactness. Don't continue these exercises, 
however, till they become tedious or monotonous. 

Tests. — Whenever possible give your pupils experience 
in the actual application of measurements. Thus, let 
them in the absence of a foot-rule or a yardstick mark a 
lath into feet and inches, and by the use of this determine 
the height and length of their desks, the blackboard, the 
recitation seats, the doors, etc, and of the schoolroom 
itself. If no set of weights can be secured, let paper or 
muslin bags filled with sand or corn, representing the 
ounce, the pound, the quarter-pound, etc., be used. The 
pupil is thus more readily taught, because he gains his 
knowledge largely by observation,, and the. judgment is 
trained and cultivated while the knowledge is being 
acquired. 

Solutions. — ^The solutions given in this book are sim- 
ple and concise. Teachers are not expected, however, to 
confine themselves strictly to those given by the author. 
Many problems will admit of several solutions. It will 
be well, therefore, for the teacher to encourage the pupils 
to give solutions of their own. He should be careful to 
see, however, that no unnecessary words are used, and 
that the solutions are given in good language. This will 
train the pupils to think and reason for themselves. 
In the written work everything should be done neatly, 
whether on the slate or on the blackboard. Pupils should 
be taught how to make figures. They frequently do not 
make neat figures, because they have never been taught. 

Thoroughness. — Do not hurry. Let your pupils mas* 



SUGGESTIONS TO TEACHERS. 7 

ter everything thoroughly as they pass over it. It is be- 
lieved that the exercises in this book are sufficiently 
extended and well graded to enable the learner to master 
the elements of the different subjects presented. But 
should the teacher deem it necessary to have more ex- 
tended exercises, both teacher and pupils will find it ad- 
vantageous often to originate problems •for^themselves. 
The teacher should be sure that the pupil understands 
the principles as he proceeds. It is too often the case 
that he is hurried along from one subject to another with- 
out having a clear comprehension of each, and therefore 
spends years in doing the work of as many months. 

Help. — Let the teacher give no help where the pupil 
can overcome any difficulty by his own efforts. Every 
obstacle the learner surmounts unaided, and every dif- 
ficulty he overcomes, makes him stronger and better pre- 
pared to contend successfully with the next. If help is 
needed, let it come indirectly in the shape of hints or 
suggestions. A single suggestion will often put a pupil 
on the right track or start a train of thought which will 
lead to success, and the knowledge thus gained without 
actual aid will be doubly valuable to him. 

Oral Arithmetic. — The exercises in Oral Arithmetic 
are not designed to be comprehensive. In many cases the 
teacher will find it advantageous to have his pupils solve 
some of the written problems orally. A portion of the 
class may be solving their problems on the blackboard 
while another part of the class may be solving by the oral 
process. Thus much valuable time may be economized, 
and the solutions and principles, being discussed by both 
methods in the same recitation, will be more fully com- 
prehended and more definitely understood. 



CONTENTS. 



PAOB 



Preface 3 

Suggestions to Teachers 5 



BBCnON CHAPTER I. PAOK 

Integers 9" 

I. Notation and Numeration. 9 
II. Addition 19 

III. Subtraction 30 

IV. Multiplication 41 

V. Division 63 

CHAPTER II. 

United States Money 66 

I. Definitions and Principles. 66 
II. Reductionof U.S. Money. 68 

III. Addition of U. S. Money. 69 

IV. Subtraction of U.S. Money 71 
V. Multiplication of U. S. 

Money 72 

VI. Division of U. S. Money. 74 
Bills 77 

CHAPTER III. 

Properties op Numbers 79 

I. Factors and Multiples 79 

Cancellation 82 

II. Greatest Common Divisor. 83 

III. Least Common Multiple... 85 

CHAPTER IV. 

Fractions 87 

I. Definitions 87 

II. Reduction of Fractions... 89 
Common Denominator.. 96 
Least Common Denom- 
inator 97 

III. Addition of Fractions 98 

IV. Subtraction of Fractions..! 01 
V. Multiplication of Frac- 
tions 104 

VI. Division of Fractions 110 

Complex Fractions 115 

8 



BBCTIOIf CHAPTER V. PA«B 

Decimal Fractions 118 

I. Notation and Numeration. 118 
II. Reduction of Decimals... 121 

III. Addition of Decimals 123 

IV. Subtraction of Decimals. 124 
V. Multiplication of Deci- 
mals 125 

VI. Division of Decimals 126 

CHAPTER VL 

Denominate Numbers 128 

I. Definitions 128 

II. Tables and Measures 129 

Value 129 

Weight 130 

Extension 134 

Volume 137 

Capacity 139 

Time 141 

Angles 143 

Compound Numbers 146 

1. Addition of Com- 

pound Numbers... 146 

2. Subtraction of Com- 

pound Numbers... 148 

3. Multiplication of 

Comp'd Numbers. 149 

4. Division of Com- 

pound Numbers... 150 

CHAPTER VIL 

Percentage 152 

I. Definitions and Principles 152 

II. Interest 159 

Simple Interest 159 

1. General Method... 159 

2. Decimal Method.. 161 
Review Problems 162 



Elementary Arithmetic. 



CHAPTER I. 
II^TEGEES, 



SECTION I. 

NOTATION AJfD XVMEBATIOK. 
1. In writing numbers, ten figures are used. Theyare-^ 
0123456789 

called naughty one, two, three, four, five, six, severe, eight., nine, 

Natighi, 0, is sometimes called zero, or a cipher. The 
other nine figures are sometimes called digits. 

In writing numbers greater than nine, two or more of 
these figures are taken together. 

Repeating or naming the numbers in order is called 
Coimtillg ; as, one, two, three, four, five, etc. 



One 1 

Two 2 

Three 3 

Four 4 

Five 5 

Six 6 

Seven 7 

Eight 8 

Nine 9 

Ten 10 

Eleven 11 

Twelve 12 

Thirteen 13 



NUMBERS TO 100. 

Fourteen 14 

Fifteen 15 

Sixteen 16 

Seventeen 17 

Eighteen 18 

Nineteen 19 

Twenty 20 

Twenty-one 21 

Twenty-two 22 

Twenty-three 23 

Twenty-four 24 

Twenty-five 25 

Twenty-six 26 



Twenty-seven 27 

Twenty-eight 28 

Twenty-nine 29 

Thirty 30 

Thirty-one 31 

Thirty-two 32 

Thirty-three 33 

Thirty-four 34 

Thirty-five 35 

Thirty-six 36 

Thirty-seven 37 

Thirty-eight 38 

ThittY-ivvaa ^<^ 



10 



NOTATION AND NUMERATION. 



Forty 40 

Forty-one 41 

Forty-two 42 

Forty-three 43 

Forty-four 44 

Forty-five 45 

Forty-six 46 

Forty-seven 47 

Forty-eight 48 

Forty-nine 49 

Fifty 50 

Fifty-one 51 

Fifty-two 52 

Fifty-three 53 

Fifty-four 54 

Fifty-five 55 

Fifty-six 56 

Fifty-seven 57 

Fifty-eight 58 

Fifty-nine 59 

Sixty 60 



Sixty-one 61 

Sixty-two 62 

Sixty-three 63 

Sixty-four 64 

Sixty-five 65 

Sixty-six 66 

Sixty-seven 67 

Sixty-eijght 68 

Sixty-nine 69 

Seventy 70 

Seventy-one 71 

Seventy-two 72 

Seventy-three 73 

Seventy-four 74 

Seventy-five 75 

Seventy-six 76 

Seventy-seven 77 

Seventy-eight 78 

Seventy-nine 79 

Eighty 80 



Eighty-one 81 

Eighty-two 82 

Eighty-three 83 

Eighty-four 84 

Eighty-five 85 

Eighty-six 86 

Eighty-seven .... 87 

Eighty-eight 88 

Eighty-nine 89 

Ninety 90 

Ninety-one 91 

Ninety-two 92 

Ninety-three 93 

Ninety-four 94 

Ninety-five 95 

Ninety-six 96 

Ninety-seven .... 97 

Ninety-eight 98 

Ninety-nine 99 

One hundred 100 



EXERCISE IN COUNTING. 



1. Count from 1 to 10. 

2. Count from 10 to 30. 

3. Count from 30 to 60. 



4. Count from 60 to 100. 

5. Count from 1 to 99. 

6. Count from 99 to 1» 



With what figure does 17 begin? 18? 35? 39? 
47? 46? 43? AU the forties? All the sixties? 
All the eighties ? 

2. When we write numbers, ten ones are called a ten, 

which is written 10 

Two tens, or twenty , is written 20 



Three tens, or thirty, " 
Four tens, ot forty y 
Five tens, or ffti/, 
Six tens, or sixty, 
Seven tens, or seventy, " 
'Eight tens, or eighty, " 
Nine tens, or ninety, " 



a 



u 



(t 



u 



« 



« 



« 



(( 



(( 



« 



30 
40 
60 
60 
70 
80 
90 



NOTATION AND NUMERATION. 



11 



One ten and one is called el^m, and is written. 

One ten and two, or twelve. 

One ten and three, or thirteen, 

One ten and four, or fourteen, 

One ten and ^ve, or fifteen. 

One ten and six, or sixteen. 

One ten and seven, or seventeen. 

One ten and eight, or eighteen, 

One ten and nine, or nineteen. 

Two tens and one, or twenty-one, 

Two tens and two, or twenty-two, 

Three tens and four, or thirty-four. 

Five tens and seven, or fifty-seven, 

Six tens and two, or sixty-two, 

Eight tens and five, or eighty-five. 

Nine tens and nine, or ninety-nine. 



a 



tC 



U 



it 



n 



« 



(( 



« 



i< 



tt 



u 



iC 



fC 



u 



It 



tten... 


, 11 


t( 


. 12 


u 


. 13 


it 


. 14 


(« 


, 15 


tt 


. 16 


tt 


. 17 


tt 


. 18 


tt 


, 19 


tt 


. 21 


It 


. 22 


tt 


. 34 


tt 


. 57 


tt 


62 


It 


, 85 


(i 


. 99 



EXERCISE. 

1. Write in figures thirteen, nineteen, fourteen, twenty- 
four, thirty-six, forty*three, fifty-two, sixty-one, seventy- 
seven, eighty-five, ninety-eight. 

2. Write in. words 12, 18, 27, 32, 86, 72, 64, 92, 

26, 75. 

3. Write in figures seventeen, twenty-nine, thirty, for- 
ty-eight, sixty-five, eighty-one, forty-five, seventy-eight, 
eighty-three. 

4. Write in words 39, 96, 62, 74, 87, 34, 72, 

27, 98. 

5. Write in figures thirty-one, twenty-three, fifty-eight, 
sixty, seventy-nine, forty-four, thirty-three. 

6. Write in words 22, 54, 27, 99, 28, 26, 68, 
98, 77. 

3. Ten tens taken together are called a hundred. 



12 NOTATION AND NUMERATION. 

One hundred is written 100 

Two hundred " " 200 

Three hundred " " 300 

Four hundred " " 400 

Five hundred " " 500 

Six hundred " " 600 

Seven hundred *' " 700 

Eight hundred " " 800 

Nine hundred " " 900 

4. When numbers are expressed by figures, the first 
place on the right is called ones or units; the second 
place, tens; the third place, hundreds. Thus, 326 is 
three hundreds, two tens and six ones, and is read three 
hundred and twenty-six. 

EXERCISE. 

1. Write in figures one hundred and twenty-five, two 
hundred and seventy-three, three hundred and seventy- 
two, four hundred and twenty-six, eight hundred and 
ninety-three, three hundred and ninety-eight, nine hun- 
dred and eighty-three. 

2. Write in words 126, 472, 264, 746, 827, 722, 468. 

3. Write in figures eight hundred and twenty-one, three 
hundred and forty-three, nine hundred and seventy-four, 
two hundred and seventy-six, three hundred and forty-five, 
five hundred and forty-three, four hundred and thirty- 
five. 

4. Write in words 263, 348, 482, 576, 647, 446, 644. 

5. Write in figures five hundred and forty-one, one 
hundred and fourteen, one hundred and nineteen, two 
hundred and one, seven hundred and twenty-three, eight 
hundred and four, six hundred and ^ye, 

6. Write in words 468, 144, 987, 207, 763, 875, 445. 



NOTATION AND NUMERATION. 



13 



7. Write in figures four hundred and fifty-two, eight 
hundred and thirteen, nine hundred and eleven, seven 
hundred and ninety-eight, two hundred and twenty-one, 
eight hundred and eighty-seven, four hundred and sixty. 

8. Write in words 567, 348, 843, 654, 388, 777. 

9. Write in figures seven hundred and twenty-six, three 
hundred and ninety-nine, eight hundred and seventy-two, 
three hundred and eighty-seven, six hundred and twenty- 
three, seven hundred and eighty-nine, seven hundred and 
sixty-four. 

10. Write in words 111, 406, 460, 723, 327, 273, 732. 

5. Ten hundreds taken together are called a thousand. 

One thousand is written 1000 

Twenty hundreds, or two thousand, is written... 2000 

Thirty hundreds, or three thousand. 

Forty hundreds, or four thousand, 

Fifty hundreds, or five thousand. 

Sixty hundreds, or six thousand, 

Seventy hundreds, or seven thousand, 

Eighty hundreds, or eight thousand, 

Ninety hundreds, or nine thousand, 

6. When numbers of more than three figures are writ- 
ten, the fourth place to the right is called thousands. 
Thus, 6723 is read six thousand seven hundred and 
twenty-three. 

7. The fifth place to the right is called ten-thousands. 
Thus, 86423 is 8 ten-thousands, 6 thousands, 4 hundreds, 
2 tens and 3 ones, or eighty-six thousand four hundred 
and twenty-three. 

8« The sixth place to the right is called hundred-thour 
sands. Thus, 768357 is 7 hundred-thousands, .6 ten-thou- 
sands, 8 thousands, 3 hundreds, 5 tens and 7 ones, or seven 

2 



Ma Tf J &vi/v/XAi 


>.. .WV/V/Vf 

... 3000 




...4000 




... 5000 




... 6000 




... 7000 




... 8000 




... 9000 



14 NOTATION AND NUMERATION. 

hundred and sixty-eight thousand three hundred and fifty- 
seven. 

9. In numbers of more than three figures, ev^y 
three figures counting from the right are called a 
Period. These periods are separated from each other 
by commas. 

10. The first period at the right consists of ones, tens 
and hundreds. The second consists of thousands, ten- 
thousands and hundred-thousands. Thus, in 845,675, the 
first right-hand period consists of 5 ones, 7 tens and 6 
hundreds ; and the second period, of 5 thousands, 4 ten- 
thousands and 8 hundred-thousands. 

11. The first right-hand period is also called the uniUt^ 
period, and the second to the right the thousands' period. 
Thus, 643,876 is 643 thousands and 876 units or ones. It 
is read six hundred and forty-three thousand eight hun- 
dred and seventy-six. 

12. If there are no figures for any of the places, naughts 
must be supplied. Thus, write — 

Eighteen thousand 18,000 

Six hundred and three thousand 603,000 

Four hundred and eighty thousand 480,000 

Seven thousand ^ye hundred 7,500 

Twenty thousand nine hundred 20,900 

Sixteen thousand seven hundred and eight.. 16,708 
Five hundred and seven thousand two hun* 

dred 507,200 

Five hundred and seven thousand two hun- 
dred and six 507,206 

Sixty thousand and thirteen 60,013 

One hundred and seventy-three thousand 
and six 178,006 



NOTATION AND NUMERATION. 15 

EXERCISE. 

1. Write in figures seven hundred and eighty-four, six 
thousand, eight thousand, five thousand five hundred. 

2. Write in figures one hundred and seventeen thou- 
sand, one hundred and seventy thousand, eight hundred 
and fifty thousand, one hundred and seven thousand. 

3. Write in words 1193 ; 12,700; 97,800; 20,600. 

4. Write in words 21,500 ; 180,000 ; 180,700 ; 60,500. 

5. Write in figures fift;een thousand four hundred and 
sixty, twenty-seven thousand six hundred and four, 
twenty thousand one hundred and seventeen, ninety-five 
thousand one hundred and ninety-five. 

6. Writ-e in figures eighty thousand six hundred and 
seventy, eighty-five thousand and sixty-seven, seventy-two 
thousand and five, sixty thousand and seventy-six. 

7. Write in words 16,750 ; 25,702 ; 11,720 ; 19,559. 

8. Write in words 86,700; 56,708; 52,007; 70,067. 

9. Write in figures one hundred and seventeen thou- 
sand six hundred and fifty, one hundred and forty thou- 
sand and forty, eight hundred and five thousand five 
hundred and five, six hundred and seven thousand seven 
hundred and seventy. 

10. Write in figures five hundred and forty-three 
thousand two hundred and two, one hundred and 
ninety-fiix thousand six hundred and nine, five hun- 
dred thousand and sixty-seven, four hundred thousand 
and six. 

11. Write in words 711,560 ; 440,010 ; 550,805 ; 770,076. 

12. Write in words 647,032 ; 546,607 ; 607,006 ; 600,003. 

MILLIONS. 

13. The third period of figures is c&Hedmilliom, 

14. A miUion is a thousand thousand. 



16 NOTATION AND NUMERATION. 

15. The third period consists of millions, ten-milliona 
and hundred-millions. Thus we write — 

3 million 600 thousand and 65 3,600,065 

17 million 420 thousand 642 17,420,642 

45 million 316 thousand 718 45,316,718 

264 million 376 thousand 945 264,376,945 

740 million 360 thousand and 80 740,360,080 

700 million 300 thousand and 6 700,300,006 

EXERCISE. 

1. Write in figures one hundred and seventy-eight 
thousand nine hundred and sixteen, one hundred and 
eighty-seven thousand five hundred and sixty. 

2. Write in figures six million seven hundred and 
sixty-three thousand four hundred and twenty-one, eigh- 
teen million seven hundred and sixteen thousand seven 
hundred and fourteen. 

3. Write in words 967,478 ; 875,642. 

4. Write in words 7,843,617 ; 15,673,424. 

5. Write in figures one hundred and eighty-seven mil- 
lion six hundred and forty-five thousand six hundred and 
six, seven hundred and forty-three million eight hundred 
thousand six hundred and five. 

6. Write in figures six hundred million seven hundred 
and fifteen thousand and fifteen, eight hundred and sev- 
enty-seven million four hundred and three thousand six 
hundred and six. 

7. Write in words 244,707,644 ; 873,600,707. 

8. Write in words 705,604,403 ; 700,600,006. 

9. Name the periods in order from the right. 
10. Name the places in order fix)m the right. 



NOTATION AND NUMERATION. 17 

TABLE. 

• 

m a 

pkmods, J i a 

^ ^ 
y y ' ^ , ' ^ y ' 

74 6, 68 3, 726 

§ S 

Places. I J ^ I ^ jf 

1 1 f "i I 1 "S 

WhSWhhWhP 

10 units are 1 ten. 

10 tens. " 1 hundred. 

10 hundreds " 1 thousand. 

10 thousands " 1 ten-thousand. 

1 ten-thousands " 1 hundred-thousand. 

10 hundred-thousands. " 1 million. 

10 millions " 1 ten-million. 

10 ten-millions " 1 hundred-million. 

16. A Unit is one or a single thing. 

17. A Number is one or more units. 

18. A whole number — as, 1, 3, 4, etc.-^is called an 

Integer, 

19. Arithmetic is the science of numbers and the art 
of computing by them. 

20. Notation is the art of writing or expressing num- 
bers by figures or letters. 

21. Numeration is the art of reading numbers. 

22. Figures have two values, simple and local. 

23. The simple value of a figure is the value when it 
stands alone. 

2* B 



18 NOTATION AND NUMERATION. 

24. The local value of a figure is the value given it by 
the place it occupies. 

Thus, 7, 6 and 4, when taken alone, mean 7 ones, 6 ones 
and 4 ones ; but when combined, as in 764, the value of 
7 is 7 hundreds ; of 6, 6 tens ; and of 4, 4 ones. 

REVIEW EXERCISE. 

1. Express by figures seven hundred and nine, seven 
thousand six hundred and nine, forty-three thousand 
two hundred and seventy-five, sixteen thousand and 
six. 

2. Express by figures one hundred and fifteen thousand 
and fifteen, five hundred thousand and five, eighteen 
thousand and six, eighteen thousand and eighteen. 

3. Express by figures six hundred and fourteen million 
seven hundred and eighty-three thousand and sixty-three, 
one hundred and forty-nine million six hundred thou- 
sand, eighteen million, one hundred and seventy million 
and five. 

4. Express by figures nine hundred million and nine, 
seven hundred and forty-three million seven hundred and 
forty-three, seven hundred and forty million seven thou- 
sand four hundred and seventy-four, two hundred and 
four million six hundred and seventy thousand and sixty- 
seven. 

5. Express by figures seventeen million and seventeen, 
nine hundred and sixty-two million nine hundred and 
sixty-two thousand nine hundred and sixty-two, four hun- 
dred and twenty-eight million four himdred and twenty- 
eight thousand, nine hundred and eight million seventy 
thousand six hundred and five. 

6. Express by words 1,865 ; 6,006 ; 18,014 ; 73,642. 

7. Express by words 114,000 ; 14,006 ; 128,128 ; 900,019. 



ADDITION. 19 

8. Express by words 462,352; 109,109,109; 16,000,000 
160,000,000. 

9. Express by words 1,600,000; 160,000; 4,004,004 
45,045,045. 

10. Express by words 87,500,875; 731,640,005 
62,062,062; 405,060,708. 



SECTION II. 

ABBITIOJf. 

1. A boy has one right hand and one left hand : how 
many hands has he ? 

2. A horse has two front feet and two hind feet : how 
many feet has he ? 

3. A girl has two apples in one hand and one in the 
other : how many apples has she? 

4. John had three cents, and his mother gave him two 
more : how many had he then ? 

5. Mary has three cakes and George has three : how 
many have they both ? 

25. Uniting two or more numbers of the same kind, so 
as to find how much they all equal, is called Addition. 

26« The number found by adding two or more num- 
bers together is called the Sfum. 

27. The %\^n of addition, +, is called jo^iw, and when 
placed between two numbers shows that they are to be 
added. 

28. The following sign, = , is called the dgn of equality y 
and when placed between two numbers shows that they 
are equal. Thus, 2 -f 3 « 5, and is read ^ ^\3& '^ ^njM^^ Xi, 



20 ADDITION. 

Principle. — Only similar numbers can be added. 
Thus, 3 boys and 2 boys, 4 cents and 7 cents. 

ADDITION TABLE. 



8 



9 



1234567 8 9 
1111111 1 1 

2345678 9 10 



1234567 8 9 

2222222 2 2 



6 7 8 9 10 11 



1234567 8 9 

3333333 3 3 



6 7 8 9 10 11 12 



1234567 8 9 
4444444 4 4 



6 7 8 9 10 11 12 13 



1234567 8 9 

5555555 5 5 



6 7 8 9 10 11 12 13 14 



01234567 8 9 

eJ 6 6 6 6 6 6 6 6 6 6 



8 9 10 11 12 13 14 15 



1234567 8 9 

77777777 7 



8 9 10 11 12 13 14 15 16 



01234567 8 9 

88888888 88 

8 9 10 11 12 13 14 15 16 17 



1234667 8 9 

9999999 9 9 



9 10 11 12 13 14 15 16 17 18 



1234567 8 9 

10 10 10 10 10 10 10 10 10 



10 11 12 13 14 15 16 17 18 19 



ADDITION. 



21 



1. Count 
2 and 1 are 

2. Count 
4 and 2 are 

Count by 

3. Count — 

By3's 
By3'8 

4. Count — 

By4's 
By4's 
By4'8 

5. Count — 

By5'8 
By5's 
By5's 
By5's 

6. Count — 

By6'8 
By6's 
By6'8 
By6'8 
By6's 



ORAL EXERCISE, 
by I's from 1 to 100 ; thus, 1 and 1 are 2, 
3, 3 and 1 are 4, and so on. 
by 2's from 2 to 100; thus, 2 and 2 are 4, 
6, and so on. 
2's from 4 to 60. 



from 3 to 60. 
from 4 to 100. 

from 1 to 49. 
from 2 to 74. 
from 3 to 99. 

from 1 to 51. 
from 2 to 77. 
from 3 to 93. 
from 4 to 119. 

from 1 to 73. 
from 2 to 98. 
from 3 to 117. 
from 5 to 155. 
from to 180. 



7. Count- 

By 7's from 2 to 51. 
By 7's from 4 to 95. 
By 7'8 from 5 to 110. 
By 7's from to 133. 

8. Count — 

By 8'8 from 3 to 83. 
By 8'8 from 5 to 125. 
By 8's from 7 to 119. 
By 8'8 from 1 to 145. 

9. Count — 

By 9'8 from 1 to 82. ' 
By 9's from 2 to 128. 
By 9'8 from 8 to 188. 

10. Count- 

By lO's from 2 to 102. 
By lO's from 5 to 135. 
By lO's from 9 to 199. 



CASE I. 

To Add any Colnmn of Fignres whose Sam does not exceed 

Nine^ 

ORAL EXERCISE. 

1. How many are 3 pins and 4 pms? 

2. How many are 6 pins and 2 pins ? 

3. How many are 4 boys and 5 boys ? . - 

4. How many are 5 boys and 4 boys ? 

5. How many are 7 men and 2 men ? 



22 ADDITION. 

6. How many are 6 men and 2 men ? 
7- How many are 4 cups and 4 cups? 

8. How many are 3 spoons and 5 spoons ? 

9. How many are 3 apples, 2 apples and 1 apple? 
10. How many are 2 cows, 3 cows and 4 cows ? 

WRITTEN EXERCISE. 

Ex. How many are — 
3 horses, JEJrpfcMMrftbn. — ^We write the numbers under one an- 
3 horses'? ^^^t ^^^ ^d thus : 3 and 2 are 5, and 3 are 8. Henoe, 
8 horses. ^® ^^^"^ ^ ^ horses. 

Find the sum of — 

(1.) (2.) (3.) (4.) 

6 books, 6 men, 3 pens, 4 boys, 

3 " 2 " 4 " 2 



2 " 2 






(5.) (6.) (7.) (8.) (9.) (10.) (11.) (12.) (13.) 
673425 2 4 6 
004212112 
1223626 11 

14. How many are 21 cents, 15 cents and 12 cents? 

Solution. ExplancUum, — ^We write the numbers so that units 

21 cents, stand under units and tens under tens, and begin 

15 cents, ^.t the right to add. Thus, 2 and 5 are 7, and 1 is 8, 

— 06° • which we write in the place of units ; adding the tens, 

cents, ^g have 1 and 1 are 2, ahd'2 are 4, which we write 

in the tens' place. 

Find the sum of— 

(15.) (16.) (17.) (18.) (19.) 

16 horses, 18 men, 12 books, 10 cents, 71 boxes, 
21 " 20 " 14 " 22 " 6 " 

10 " 60 " 13 " 34 ** 12 " 







ADDITION. 






(20.) 


(21.) 


(22.) (23.) 


(24.) 


(25.) 


23 


12 


20 ■ 18 


23 


123 


44 


14 


30 20 


42 


322 


22 


13 


46 40 


20 


400 


(26.) 


(27.) 


(28.) 


(29.) 


(30.) 


422 


221 


165 


141 


6 


320 


143 


200 


23 


21 


150 


20 


30 


32 


432 



23 



31. A man travelled 21 miles on Monday and 16 on 
Tuesday : how fiir did he travel in both days ? 

32. There are 33 boys and 25 girls in school: how 
many pupils are there in the school ? 

33. A horse cost 105 dollars, a cow 43 dollars and a 
sheep 10 dollars : how much did they all cost ? 

34. A boy bought two hens for 50 cents, a duck for 25 
cents and some feed for 12 cents : how much did he pay 
for all ? 

35. If I pay 122 dollars for a buggy, 45 dollars for 
harness and 200 dollars for a horse, how much do I pay 
for all ? 

How much ifr- 



36. 6 + 2+1? 

37. 4+12+3? 

38. 21 + 22+43? 

39. 10+25+60? 

40. 23+42+122? 



41. 130+200+304? 

42. 125+320+201? 

43. 172 + 205 + 602? 

44. 232 + 300+107? 

45. 455 + 203 + 340? 



CASE II. 

To Add wben the Sum of a Colnmn exceeds Nine Units of 

that place. 

ORAL EXERCISE. 

1. How many are 5 apples and 5 apples ? 3 pens and 
8 pens? 



24 ADDITION. 

2. How many are 4 dollars and 6 dollars? 

3. A boy has 5 fingers on each hand : how many has 
he on both ? 

4. How many are 5 cents, 6 cents and 7 cents ? 

5. A hog cost 10 dollars, a sheep 5 dollars and some 
hens 7 dollars : how much did they all cost ? 

6. A pair of boots cost 12 dollars, a hat 7 dollars 
and some socks 2 dollars: how much did they all 
cost? 

7. How many are 7 books, 9 books and 22 books ? 

8. Mary has 10 cents, John has 12 cents and James has 
15 cents : how many cents have they together ? 

9. A boy has 12 hens, 6 turkeys and 16 ducks: how 
many fowls has he? 

10. A boy learns 6 letters one day, 6 letters the next, 
4 letters the next, and 2 letters the next : how many does 
he learn in the four days? 

WRITTEN EXERCISE. 

Ex. How many are 6 horses, 8 horses, 4 horses and 6 
horses? 

BoLTJTiON'. Explanation, — We write the numbers under one 

6 horses, another, and add, beginning at the bottom : 6 and 4 

^ « are 9, and 8 are 17, and 6 are 23, which is 2 tens and 

5 « 3 ones. We write the 3 in the units' place, and the 2 

23 horses, i^^ the tens' place. 

Add— 

(1.) (2.) (3.) (4.) (5.) 

5 cows. 8 sheep. 6 boys. 2 cents. 6 dollars. 

6 " 4 " 7 " 6 " 7 " 
4 " 3 " 4 " 9 " 6 " 
3 " 5 " 2 " 4 " 9 " 



ADDITION. 25 

(6.) (7.) (8.) (9.) (10.) 

6 7 6 8 8 

4 8 4 7 9 

5 3 7 5 7 
9 2 3 6 2 

(11.) (12.) (13.) (14.) (15.) 

5 7 6 5 9 

6 8 7 5 7 
8 5 8 6 6 
3 4 3 4 8 
2 2 i ^ ^ 

Add. Explafuition. — We write the numbers so that units are 
423 under units, tens under tens, and hundreds under hundreds, 
347 and begin to add at the right : 9 and 7 are 16, and 3 are 
19 ones, or 1 ten and 9 ones. We write the 9 in the units' 
place, and add the 1 ten to the tens : 1 and 4 are 5, and 4 
are 9, and 2 are 11 tens, or 1 hundred and 1 ten. We write the 1 
ten in the tens' place, and add the 1 hundred to the hundreds : 1 and 
8 are 9, and 3 are 12, and 4 are 16 hundreds, or 1 thousand and 6 
hundreds. We write the 6 hundreds in the hundreds' place and 
the 1 thousand in the thousands' place. The result is, therefore, 
1619. 



849 
1619 



Add— 
















(1.) 




(2.) 




(3.) 




(4.) 


42 dollars. 


18 cents. 


55 ducks. 


48 books. 


28 


(( 




16 « 


13 


U 




25 « 


43 


it 




44 " 


84 


« 




72 « 


(5.) 




(6.) 


(7.) 


(8.) 




(9.) 


(10.) 


45 




84 


16 


46 




84 


95 


69 




72 


61 


64 




46 


50 


32 




91 


85 


51 




87 


68 



26 



\ 




ADDITION. 






(11.) 


(12.) 


(13.) 


(14.) 


(15.) 


(16.) 


642 


272 


615 


465 


956 


925 


347 


447 


421 


641 


608 


675 


872 


638 


879 


848 


467 


259 


(17.) 


(18.) 


(19.) 


(20.) 


(21.) 


(22.) 


752 


342 


253 


897 


156 


851 


423 


426 


541 


111 


481 


318 


709 


151 


422 


343 


423 


806 


820 


737 


735 


625 


782 


167 


(23.) 


(24.) 


(25.) 


(26.) 


(27.) 


(28.) 


4813 


1122 


2291 


3574 


4449 


1357 


5914 


7914 


5723 


3333 


2575 


2468 


6115 


1234 


2102 


4680 


4404 


6555 


7036 


8024 


6838 


3391 


3685 


6666 


(29.) 


(30.) 


(31.) 


(32.) 


(33.) 


(34.) 


5788 


3455 


2729 


4044 


3282 


1185 


2693 


6521 


8272 


5260 


6341 


5073 


1112 


6817 


3228 


3788 


3161 


9962 


6762 


7773 


9561 


5473 


2827 


9467 


8104 


6839 


5587 


2667 


7214 


3478 


(35.) 




(36.) 


(37.) 




(380 


43474 




73422 


77823 




13536 


38242 




75638 


21684 




71882 


67891 




18208 


18516 




81385 


84870 




32378 


33902 




80246 


22171 




27226 


. 14656 


• 


91257 



ADDITION. 


(39.) 


(40.) 


433827 


28613534 


663725 


47224466 


434958 


31821746 


367624 


18714924 


233647 


73684627 



27 



Rnd the sum — 

41. Of 6472+8733+4633+4854. 

42. Of 2762+8756 + 9783+4578. 

43. Of 1617+8743+7284+9621. 

44. Of 2650+4062+8705 + 9030. 

45. Of 5005+6007 + 7583+4783. 

46. Of 27845 + 67832 + 74281 + 68432. 

47. Of 47823 + 68421 + 70070 + 60504. 

48. Of 1 27 + 6434 + 7805 + 60007. 

49. Of 10+8756+405 + 66782. 
60. Of 7560+804+7854+87400. 
51. Of 1525 + 960+820+16 + 37800. 

PROBLEMS. 

• 

1. A merchant sells 44 yd. of cloth on Monday, 62 yd. 
on Tuesday and 30 yd. on Wednesday : how much does 
he sell in the three days ? Ans. 136 yd. 

2. A farmer raises 650 bu. of wheat in one field, 420 bu. 
in another and 725 bu. in another: how many bushels 
does he raise in the three fields ? Ana. 1795 bu. 

3. A grocer sold 850 lb. of sugar in March, 927 lb. in 
April and 1640 lb. in May: how many pounds did he 
sell in the three months? Ans. 3417 lb. 

4. A knife cost 62 cts., a slate 44 cts., a reader 95 cts. 
and a grammar 65 cts. : how much did they all cost? 

Am. 266 cts. 



28 ADDITION. 

5. A gentleman owns a farm worth 16000 dollars, a 
store worth 12500 dollars, and a house worth 25000 dol- 
lars : how much are they all worth ? Ans, 53500 dollars. 

6. If a horse cost 157 dollars, a carriage 225 dollars 
and a set of harness 120 dollars, what do they all cost ? 

AnSn 502 dollars. 

7. In my orchard I have 124 apple trees, 63 peach 
trees and 27 plum trees : how many trees are there in the 
orchard? Ans, 214. 

8. A merchant has three pieces of carpet ; the first con- 
tains 127 yd., the second 145 yd., and the third 162 yd. : 
how many yards do they all contain ? Ans. 434 yd. 

9. A man pays 1200 dollars for rent, 2750 dollars for 
clerk-hire and 175 dollars for insurance : how much does 
he pay altogether? Ans, 4125 dollars. 

10. A man was born in 1813, and died when he was 
62 years old: when did he die? Ana, In 1875. 

11. A man receives as rent from five buildings the fol- 
lowing amounts : from the first 125 dollars a year, from 
the second 320 dollars a year, from the third 650 dollars 
a year, from the fourth 240 dollars a year, and from the 
fifth 175 dollars a year: how much does he receive from 
all ? Ana, 1510 dollars. 

12. A merchant has two pieces of muslin of 45 yd. 
each, and three of 46 yd. each : how much muslin has he? 

Ans, 228 yd. 

13. What is the cost of three horses at 125 dollars each, 
and four cows at 43 dollars each ? Ans, 547 dollars. 

14. A drover bought 120 sheep for 625 dollars, 243 
sheep for 763 dollars and 571 sheep for 2855 dollars : how 
many sheep did he buy, and how much did they cost bim ? 

Ans, 934 sheep for 4243 dollars. 

15. A farmer has four hogs which weigh as follows : 



ADDITION. 29 

327 lb., 405 lb., 377 lb. and 396 lb. : how much do they 
all weigh? Ans. 15051b. 

16. The three stories of a building rent as follows : Ist 
story, 1560 dollars ; 2d, 750 dollars ; and 3d, 425 dollars : 
^hat amount of rent was received for the building? 

Am, 2735 dollars. 

17. Five timber-rafts contain the following amounts, 
respectively: 1260 ft., 1340 ft., 973 ft., 1122 ft. and 1065 
ft;. : how many feet do they all contain ? Ana. 5760 ft. 

18. A builder receives 1800 dollars for building a 
house, 743 dollars for building a barn and 2255 dollars 
for building a store property : how much does he receive 
for building all ? Ans. 4798 dollars. 

19. Independence was declared in 1776, and the Con- 
stitution was adopted 11 yr. later: in what year was the 
Constitution adopted ? Ans. In 1787. 

20. America was discovered in 1492, and the Puritans 
settled at Plymouth 128 yr. after: when did the Puritans 
settle at Plymouth? Ans. In 1620. 

21. New York was settled by the Dutch in 1613, and 
Pennsylvania was settled 69 yr. later: when was Pennsyl- 
vania settled? Ans. In 1682. 

22. The Puritans landed at Plymouth in 1620, and the 
battle of Lexington was fought 155 yr. later: when was 
the battle of Lexington fought? Ans. In 1775. 

23. Washington was inaugurated as the first President 
in 1789, and gold was discovered in California 59 yr. 
later : when was gold discovered in California ? 

Ans. In 1848. 

24. I bought a house for 2750 dollars, and soM it so as 
to gain 1225 dollars: how much did I get for it? 

Ans. 3975 dollars. 

25. I paid 175 dollars for a horse, 150 dollars for a 

3* 



30 



SUBTRACTION. 



buggy and 35 dollars for harness, and sold all at a gain 
of 13 dollars : how much did I get? Ans. 373 dollars. 

26. A merchant's cash sales were as follows : Monday, 
127 dollars; Tuesday, 67 dollars; Wednesday, 173 dol- 
lars ; Thursday, 187 dollars ; Friday, 25 dollars ; and Sat- 
urday, 316 dollars : what was the amount of his sales for 
the week ? Ans, 895 dollars. 

27. The distance from Buffalo to Erie is 88 mi., from 
Erie to Toledo 207 mi., and fi^m Toledo to Chicago 
243 mi. : how far is it from Buffalo to Chicago ? 

Ana, 538 mi. 



(28.) 


(29.) 


(30.) 


27506 


463 


32556 


3741 


117642 


897634 


82 


75 


7347 


640 


7065 


86424 


8714 


82436 


17 


618 


4782 


1700 


44106 


954 


476 


27 


8 


16964275 


8742 


12645 


8740 


196 


7253 


60006 



94372 



SECTION III. 

SUBTBACTIOJT. 

1. A boy had 2 cents, and lost 1 : how many cents had 
he then ? 

2. A girl had 3 cakes, and gave her sister 1 : how many 
had she then ? 



SUBTRACTION. 81 

3. Four birds sat on a bush ; 2 flew away : how many 
remained ? 

4. John had 5 cents, and bought an orange for 2 cents : 
how many cents had he left ? 

5. How many would he have left if he had paid 3 cents 
for his orange ? 

6. Mary bought 6 cups, but broke 3 : how many had 
she remaining? 

7. How many are 6 cups less 3 cups ? 

8. How many are 6 cups less 2 cups ? 

9. How many are 6 cups less 4 cups ? 

10. Charles had 8 ducks, but 3 died : how many had 
he then ? 

11. How many are 8 less 3 ? 8 less 4 ? 

12. How many are 8 less 2? 8 less 5? 

29. Finding the difference between two numbers is 
called Subtraction. 

30. The number found by taking one number from 
another is called the Difference, 

31. The number from which the other is taken is called 
the Minuend, 

33. That which is taken from the minuend is called 
the Subtrahend, 

33. The sign of subtraction, — , is called minuSy and 
when placed between two numbers shows that the one on 
the right of the sign is to be taken from the one on the 
left of it. Thus, 6 - 2 is read 6 minus 2, and means that 
2 is to be taken from 6. 

Principle. — Only similar numbers can be subtracted ; 
thus, 3 boys from 6 boys, 5 cents from 7 cents, etc. 



32 SUBTRACTION. 

SUBTRA.CTION TABLE. 




{12345678 9 10 

11111111 1 1 

01234567 8 9 



23466789 10 11 

22222222 2 2 

01234567 8 9 

3 4 6 6 7 8 9 10 U 12" 

33333333 3 3 

01234567 8 9 



6 



8 



9 



10 



{567 
5 5 5 
12 



4 5 6 7 8 9 10 11 12 13 
44444444 4 4 



6 7 8 9 



8 9 10 11 12 13 14 
5 5 5 5 5 5 5 



6 7 8 9 



6 7 8 9 10 11 12 13 14 15 
66666666 6 6 



01234567 8 9 



{7 8 9 10 11 12 13 14 15 16 

7777-7 777 7 7 

01234567 8 9 

~8 9 10 ii 12 13 14 16 16 IT" 

88888-888 8 8 



1 234567 8 9 

"9 10 11 12 13 il 15 16 17 18^ 
99999999 9 9 



J 1 234567 8 9 

10 ii i2 13 i4 is ie i7 is ig^ 

10 10 10 10 10 10 10 10 10 10 

01234567 8 9 



ORAL EXERCISE. 

1. Subtract by I's from 100 to 1 ; thus, 1 from 100 
leaves 99, 1 from 99 leaves 98, and so on. 



SUBTRACTION. 



33 



2. Subtract by 2's from 100 to 2 ; thus, 2 from 100 
eaves 98, 2 from 98 leaves 96, and so on. 

3. Subtract by 2*s from 95 to 1 ; thus, 2 from 95 leaves 
3, 2 from 93 leaves 91, and so on. 



- Subtract — 

By 3's from 100 to 1. 

By 3's from 99 to 0. 

By 3's from 98 to 2. 
. Subtract — 

By 4's from 100 to 0. 

By 4's from 99 to 3. 

By 4's from 98 to 2. 

By 4's from 97 to 1. 
^- Subtract — 

By 5's from 100 to 0. 

By 5*8 from 99 to 4. 

By 6's from 98 to 3. 

By 5's from 97 to 2. 

By 5's from 96 to 1. 
7. Subtract — 

By6'sfroml00to4. 

By 6's from 99 to 3. 

By 6*8 from 98 to 2. 

By 6*8 from 97 to 1. 

By 6's from 96 to 0. 

By 6's from 95 to 5. 
t. Subtract — 

By 7'8 from 100 to 2. 



By 7's from 98 to 0. 
By 7's from 97 to 6. 
By 7's from 96 to 6. 
By 7's from 95 to 4. 
By 7's from 94 to 3. 

9. Subtract — 

By 8's from 100 to 4. 
By 8's from 99 to 3. 
By 8's from 98 to 2. 
By 8's from 97 to 1. 
By 8's from 96 to 0. 
By 8's from 95 to 7. 
By 8's from 94 to 6. 
By 8's from 93 to 5. 

10. Subtract — 

By 9's from 100 to 1. 
By 9's from 99 to 0. 
By 9's from 98 to 8. 
By 9*8 from 97 to 7. 
By 9's from 96 to 6. 
By 9's from 95 to 5. 
By 9's from 94 to 4. 
By 9'8 from 93 to 3. 
By 9's from 92 to 2. 



By 7*8 from 99 to 1. 

11. Count by 4's from 3 to 39, and back again to 19. 

12. Count by 5's from 6 to 66, and back again to 26. 

13. Count by 7's from 18 to 53, and back again to 11. 

14. Count by 8*s from 25 to 65, and bacVi «i^«2«i \.q\- 



34 SUBTRACTION. 

CASE I. 

To Subtract where no Figure of the Subtrahend is greatei 
than the corresponding Figure of the Minuend. 

WRITTEN EXERCISE. 

Ex. 1. Subtract 3 from 8. 

Pbocess. 

8 Explanaiion, — We write 3 under 8, and say 3 ones fropc*^ 

3 8 ones leaves 5 ones. Or, 3 from 8 leaves 5. 

5 

EXAMPLES FOR PRACTICE. 

(2.) (3.) (4.) (5.) (6.) (7-> 

From 6 pens, 7 cows, 6 cents, 8 7 8 
Take 2 ** 3 " 4 " 5 ^ ^ 

(8.) (9.) (10.) (11.) (12.) (IS.) 

9 8 9 8 3 9 

4 2 7 6 2 3 

14* Subtract 4 from 8 ; 6 from 9. 

15. Subtract 5 from 11 ; 6 from 15. 

16. Subtract 4 from 13 ; 9 from 17. 

17. Subtract 35 cents from 69 cents. 

Process. ExplaTuition, — We write the numbers so that unit^ 
69 cents, stand under units, and tens under tens, and begin to 
§^ subtract at the right : 5 units from 9 units leaves 4 units t 

34 cents. 3 tens from 6 tens leaves 3 tens. The remainder is S 
tens and 4 units, or 34. 



(18.) 


(19.) 


(20.) 


(21.) 


(22.) 


(23.) 


From 65 


74 


49 


54 


65 


46 


Take 22 


33 


36 


31 


43 


25 


(24.) 


(25.) 


(26.) 


(27.) 


(28.) 


(29.) 


673 


478 


645 


897 


745 


683 


542 


135 


325 


783 


444 


560 





SUBTRACTION. 


i 


(30.) (31.) 


(32.) (33.) 


(34.) (35.) 


6434 7859 


4765 8493 


4682 1378 


3233 6637 


3542 4272 


2460 1164 


(36.) (37.) 


(38.) 


(39.) (40.) 


87643 • 97485 


68432 


22486 85434 


85223 66283 


46231 


11375 74321 


(41.) 


(42.) 


(43.) 


847562 


8764397 


64863387 


224530 


4231264 


53621105 



35 



44. A farmer had 75 sheep, and sold 32 of them : how 
many had he remaining ? Ans, 43. 

46. A merchant had a piece of muslin containing 47 
yards ; he sold 26 yards of it : how much remained ? 

Ans. 21 yd. 

46. A man earns 878 dollars in a year, and spends 642 
dollars : how much does he save ? Arts. 236 dollars. 

47. I buy a horse for 175 dollars, and sell him for 162 
dollars: how much do I lose? Ans. 13 dollars. 

48. A farmer had 6742 bu. of oats, and sold 1430 bu.: 
how much had he remaining? Am, 5312 bu. 

49. A man paid 6750 dollars for a farm, and sold it 
for 6620 dollars : how much did he lose ? 

Ans. 130 dollars. 

50. A lady bought a farm for 6320 dollars, and sold it 
for 7460 dollars : how much did she make? 

Ans, 1140 dollars. 

61. A carpenter bought 7500 shingles to put on a roof, 
and had 1200 remaining: how many did he put on the 
roof? Ans. 6300. 

52. A school had 85 pupils, but 23 left : how many re- 
mained ? Ans. 62. 



36 



SUBTRACTION. 



CASE II. 

To Subtract when a Figrnre of the Subtrahend is grr^ater, 
or expresses more, than the eorrespondingr Figrure of the 
Minnend. 

WRITTEN EXERCISES. 

Ex. 1. Subtract 8 from 17. 

Process. 
17 



8 
9 



Explxination, — ^Writing the 8 under the 17, we s»y 
8 from 17 leaves 9. 



Ex. 2. Subtract 29 from 85. 

Pbocess. Explanation. — We write the numbers so that uni*^ 
stand under units and tens under tens. Since we car^' 
not take 9 units from 5 units, we take 1 ten of the ^ 
tens, which equals 10 units, and unite it with the 5 unit^^ 

making 15 units ; 9 units from 15 units leaves 6 units, which w^ 

write in the units' place. 
Since we have used one of the 8 tens, we now have 7 tens : 2 ten^ 

from 7 tens leaves 5 tens, which we write in the tens' place. 



85 
29 

56 



(3.) (4.) 


(5.J 


1 (6.) 


(7.) 


(8.) 


From 16 14 


15 


23 


45 


63 


Take 9 7 


8 


14 


27 


29 


(9.) (10.) 


(11.) 


(12.) 


(13.) 


(14.) 


26 75 


83 


72 


65 


84 


19 46 


64 


48 


37 


26 


Find the value — 










15. Of 75-48. 




18. 


Of 96 - 77. 




16. Of 64 - 57. 




19. 


Of 85-49. 




17. Of 84 - 26. 




20. 


Of 73 - 19. 





21. In a school of 65 pupils, 27 left: how many re- 
mained ? Am. 38. 

22. A boy had 75 cents, and spent 39 of them for a 
book : how much had he left ? Arts. 36. 



SUBTRACTION. 37 

23. A box of soap contained 60 lb., but 47 lb. of it 
have been used : how much remains ? Ana. 13 lb. 

24. Mary had 72 pins, but lost 16: how many has she 
now? Ans. 56. 

25. Charles bought a cow for 69 dollars, and sold her 
£of 95 dollars : how much did he make ? Ans. 26 dols. 

26. Subtract 1657 from 8265. 

Solution. ErpUmation. — Writing the numbers bo that units 

8265 stand under units^ tens under tens, and so on, we sub- 
1^7 tract, beginning at the right. 

6608 Since 7 units cannot be taken from 5 units, we take 

one of the 6 tens, which is equal to 10 units, and unite it with the 
^ units, making 15 units; subtracting 7 units from 15 units, we have 
^ units. 

Since we have taken one of the 6 tens, there are but 5 tens ; 5 tens 
&om 5 tens leaves nothing. 

Since we cannot take 6 hundreds from 2 hundreds, we take 1 
t-Iiousand of the 8 thousands, which is equal to 10 hundreds, and 
tiniting this with the 2 hundreds, we have 12 hundreds; subtracting 
6 hundreds from 12 hundreds, we have 6 hundreds. 

Since we have taken one of the 8 thousands, there are but 7 
thousands remaining ; subtracting 1 thousand from 7 thousands, we 
liave 6 thousands. Hence, the result is 6 thous«inds 6 hundreds 
and 8 units, or 6608. 

(27.) (28.) (29.) (30.) (31.) (32.) 
From 643 724 642 486 645 863 
Take 428 617 139 395 696 469 



(33.) 

749 

376 


(34.) 
472 
318 


(35.) 
666 
197 


(36.) 
435 
264 


(37.) 

673. 

647 


(38.) 

462 

378 


(39.) 
450 
375 


(40.) 

507 

263 


(41.) 
570 
387 


(42.) 

607 

469 


(43.) 

725 

678 


(44.) 
742 
887 



38 



SUBTRACTION. 



(46.) 
6437 
2863 



(46.) 
5347 
3726 



(47.) 
8546 
7615 



(48.) 
8432 
6547 



(49.) 
8450 
6375 



(50.) 
9720 
6483 



(51.) 
6248 


(52.) 
5337 


(53.) 
8756 


(54.) 
8645 


(55.) 
8643 


(56.) 
9674 


3673 


4276 


4165 


3427 


5750 


2803 


(57.) 
86523 


(58.) 
42347 


(59.) 
83742 


(60.) 
73388 


(61.) 
61345 


(62.) 
52384 


63748 


26353 


36635 


37756 


22378 


41307 


(63.) 
74332 


(64.) 
634725 


(65.) 
7346000 




(66.) 
1000006 


38856 


378462 


4683745 




346087 



Find the value — 

67. Of 6700 - 1864. 

68. Of 7820 - 6437. 

69. Of 1877-1798. 

70. Of 6342-6007. 

71. Of 3600-2225. 

72. Of 72500-63497. 

73. 01*82000-18640. 

74. Of 60000 - 17400. 

75. Of 62000 - 18649. 

76. Of 56009-27346. 



Ans. 4836. 

Ans. 1383. 

Ans, 79. 

Ana. 335. 

Ana. 1375. 

Ana. 9003. 
Ana. 63360. 
Ana. 42600. 
Ana. 43351. 
Ana. 28663. 



PROBLEMS. 

1. A horse was bought for 125 dollars, and sold for 117 
dollars ; how much was lost by the sale ? Ana. 8 dollars. 

2. A farmer took 620 bu. of potatoes to market, and 
sold 455 bu. : how many had he remaining ? 

Ana. 165 bu. 



SUBTRACJnON. 39 

3. Washington was born in 1732, and died in 1799 : 
how old was he ? An8. 67 yr. 

4. A roll of carpet contained 156 yd., but 79 yd. have 
been sold from it : how much remains ? Ans, 11 yd. 

5. The battle of New Orleans was fought in 1815; 
Artlerica was discovered in 1492 : how long was America 
discovered before the battle of New Orleans was fought ? 

Ans, 323 yr. 

6. A house cost 5440 dollars, and was sold for 6000 
dollars : how much was the gain ? Ans. 560 dollars. 

7. From a flock containing 820 sheep 417 were sold : 
how many remain ? • Ana, 403. 

8. A merchant began business with 20000 dollars, and 
lost 1463 dollars : how much had he remaining ? 

An8. 18537 dollars. 

9. A man having 1600 dollars in bank, withdrew 977 
dollars: how much remained? Ans, 623 dollars. 

10. How many dollars are 15000 dollars minus 6743 
dollars ? Ans, 8257 dollars. 

11. I bought a lot for 450 dollars, and sold it for 525 
dollars : how much did I gain ? Ans, 75 dollars. 

12. At an election one candidate received 23204 votes, 
and his opponent 18675 votes : what was the majority ? 

Ans, 4529. 

13. A farmer had 240 acres of land ; he sold 27 acres 
to one man and 148 acres to another: how much re- 
mained ? Ans, 65 acres. 

14. A man died in 1877 at the age of 75 years : when 
was he born ? Ans, In 1802. 

COMBINATION PBOBLEMS. 

15. A merchant received cash 1125 dollars, and paid 
rent 215 dollars, and for clerk-hire 567 dollars : how much 
cash had he remaining ? Ans, 343 dollars. 



40 SUBTRACTION. 

16. Three horses cost as follows: 175 dollars, 116 dol- 
lars and 95 dollars ; the three were sold for 300 dollars : 
how much was the loss ? Ans. 86 dollars. 

17. A horse cost 125 dollars, a carriage 173 dollars 
and some harness 62 dollars : if the owner sell all fur 
400 dollars, how much will he gain ? Ans, 40 dollars. 

18. Mr Miller owed a man 1500 dollars ; he has paid 
him 650 dollars, 325 dollars and 92 dollars : how much 
does he still owe him ? Ans, 433 dollars. 

19. How long is it since the discovery of America? 

20. A farmer raised 320 bu. of corn in one field, 290 
bu. in another and 700- bu. in another; he sold 1269 bu. : 
how much has he remaining ? Ans. 41 bu. 

21. A boy had 120 apples; he found 72 more, and 
then gave away 59 : how many had he remaining ? 

Ans, 133. 

22. A gentleman paid 6000 dollars for his farm ; he 
built a house for 3976 dollars, and then sold both for 
12000 : how much did he gain ? Ans. 2024 dollars. 

23. A merchant paid 6000 dollars for his store ; he sold 
the goods for 8642 dollars, and paid his clerks 1640 dol- 
lars : how much did he gain ? Ans, 1002 dollars. 

24. Two men bought store goods to the amount of 1644 
dollars, and sold them so that each made 386 dollars : 
how much did they receive for the goods ? 

Ans, 2416 dollars. 

25. In a regiment of 900 men, 127 were killed and 
345 were wounded : how many escaped unhurt ? 

Ans. 428. 

26. What is the value of 1862 + 1744 - 673 ? 

Ans, 2933. 

27. What is the value of 1683 - 420 + 9684 - 6472 ? 

Ans, 4475. 



' MULTIPLICATION. 41 

28. What is the value of 89644+7842 - 6845+67340? 

A718, 157981. 

29. What is the value of 6400 + 7856 + 6834 - 20465 ? 

Ana. 625. 

30. What is the value of 6000-1463 + 8674-6340 
+1009-1001? ^n«. 6879. 



SECTION IV. 

MUL TIPLICA TlOJf. 

1. One boy has 2 eyes : how many eyes have 2 boys ? 

Solution. — 2 boys have 2 eyes and 2 eyes, or 2 times 2 eyes, 
which are 4 eyes. 

2. If 1 apple cost 2 cents, how much will 3 apples cost? 

Solution. — 3 apples will cost 2 cents and 2 cents and 2 cents, 
or 3 times 2 cents, which are 6 cents. 

3. A chair has 4 legs : how many legs have 2 chairs ? 
How many legs have 3 chairs ? 

4. A girl has 5 fingers on one hand : how many has she 
on both? 

5. If 1 orange cost 5 cents, how much will 4 oranges 
cost? 

6. If 1 hat cost 5 dollars, how much will 3 hats cost ? 

7. One cat has 4 feet : how many feet have 5 cats ? 

8. How many feet have 4 cats ? 

9. How many are 5 times 6 ? 

' Solution.— 5 times 6 are 6 + 6 -f 6 + 6 + 6, or 30. 

34. The process of taking one of two numbers as often 
as there are units in the other is called Multiplication. 

3B. The number to be multiplied or repeated is called 
the Multiplicand, 

4* 



42 



MULTIPLICATION. 



36. The number showing how often the multiplicand 
is repeated is called the Multiplier, 

87. The result obtained by the process of multiplying 
is called the Product. 

38. The sign of multiplication, x, is read timea or 
multiplied by; 5 x 6 is read 5 times 6, or 5 multiplied 
by 6. 

MULTIPLICATION TABLE. 



Onck 


Twice 


3 TIMK8 


4 TIMES 


5 TIMES 


6 


TIMES 


1 is 


1 


1 are 


2 


1 


are 3 


1 


are 4 


1 


are 5 


1 


are 6 


2 " 


2 


2 " 


4 


2 


« 6 


2 


" 8 


2 


« 10 


2 


" 12 


3 " 


3 


3 " 


6 


3 


" 9 


3 


" 12 


3 


" 15 


3 


" 18 


4 " 


4 


4 " 


8 


4 


" 12 


4 


" 16 


4 


" 20 


4 


" 24 


6 " 


5 


5 " 


10 


6 


" 15 


5 


" 20 


5 


" 25 


5 


" 30 


6 " 


6 


6 " 


12 


6 


" 18 


6 


" 24 


6 


" 30 


6 


" 36 


7 " 


7 


7 " 


14 


7 


" 21 


7 


" 28 


7 


" 35 


7 


" 42 


8 " 


8 


8 « 


16 


8 


" 24 


8 


" 32 


8 


" 40 


8 


" 48 


9 « 


9 


9 " 


18 


9 


" 27 


9 


" 36 


9 


" 45 


9 


" 54 


10 « 


10 


10 '' 


20 


10 


" 30 


10 


" 40 


10 


" 60 


10 


" 60 


U « 


11 


11 " 


22 


11 


" 33 


11 


u 44 


11 


" 55 


11 


" 66 


12 " 


12 


12 " 


24 


12 


« 36 


12 


" 48 


12 


" 60 


12 


" 72 


7 TIMES 


8 TIMES 


9 TIMES 


10 


TIMES 


11 


TIMES 


12 TIMES 


1 arc 


1 7 


1 are 


8 


1 


are 9 


1 


are 10 


1 


are 11 


1 


are 12 


2 " 


14 


2 « 


16 


2 


" 18 


2 


" 20 


2 


" 22 


2 


" 24 


3 " 


21 


3 « 


24 


3 


" 27 


3 


" 30 


3 


" 33 


3 


" 36 


4 " 


28 


4 " 


32 


4 


" 36 


4 


« 40 


4 


" 44 


4 


" 48 


5 " 


35 


5 " 


40 


5 


" 45 


5 


" 50 


5 


" 55 


5 


" 60 


6 « 


42 


6 " 


48 


6 


" 54 


6 


" 60 


6 


" 66 


6 


" 72 


7 " 


49 


7 " 


56 


7 


" 63 


7 


" 70 


7 


" 77 


7 


" 84 


8 " 


56 


8 ♦* 


64 


8 


" 72 


8 


" 80 


8 


" 88 


8 


" 96 


9 " 


63 


9 " 


72 


9 


" 81 


9 


« 90 


9 


" 99 


9 


"108 


10 " 


70 


10 " 


80 


10 


« 90 


10 


"100 


10 


"110 


10 


"120 


11 " 


77 


11 " 


88 


11 


" 99 


11 


"110 


11 


"121 


11 


"132 


12 " 


84 


12 " 


96 


12 


"108 


12 


"120 


12 


"132 


12 


"144 



ORAL EXERCISE. 

1. Multiply by 2 from 1 to 12 ; thus, 2 times 1 are 2, 
*l times 2 are 4, and so on. 

2. Multiply by 3 from 1 to 6. 



MULTIPLICATION. 



43 



3. Multiply by 3 from 12 to 6 ; thus, 3 times 12 are 36, 
8 times 11 are 33, and so on. 



4. Multiply — 

By 4 from 3 to 9. 

By 4 from 12 to 6. 
6. Multiply — 

By 5 from 2 to 7. 

By 5 from 12 to 4. 

6. Multiply- 

By 6 from 3 to 10. 
By 6 from 12 to 4. 

7. Multiply- 

By 7 from 2 to 8. 
By 7 from 12 to 5. 

8. • Multiply- 

By 8 from 3 to 10. 
By 8 from 12 to 2. 



9. Multiply- 
By 9 from 1 to 11. 
By 9 from 12 to 3. 

10. Multiply- 

By 10 from 3 to 8. 
By 10 from 12 to 2. 

11. Multiply- 

By 11 from 4 to 9. 
By 11 from 12 to 3. 
By 11 from 10 to 2. 

12. Multiply- 

By 12 from 1 to 7. 
By 12 from 12 to 5. 
By 12 from 2 to 9. 
By 12 from 10 to 3. 



Principles. — 1. When two numbers are multiplied, 
either one may be taken as the multiplier. Thus, 4x5 
= 20, or 5x4 = 20. 

2. The product is the same kind as the multiplicand. 
Thus, 3x3 cents are 9 cents ; 2 x 5 boys are 10 boys. 

ORAL EXERCISE. 

1. What will 5 hats cost at 7 dollars each? 

Solution. — If 1 hat cost 7 dollars, 5 hats will cost 5 times 7 
dollars, or 35 dollars. 

2. What will 4 pairs of boots cost at 6 dollars a pair ? 

3. A sheep cost 7 dollars : how much will 6 sheep cost 
at the same rate ? 

4. If a lemon cost 3 cents, how much will 11 lemons 
cost? 



44 MULTIPLICATION. 



« 



5. John is 8 years old : how much is 4 times his age? 

6. A boat cost 11 dollars : how much would 5 boats 
cost at the same rate ? 

7. If a coat cost 10 dollars, how much will 8 coats 
cost? 

8. There are seven days in a week : how many days are 
there in 8 weeks ? 

9. How much will 12 tons of coal cost at 6 dollars a 
ton? 

10. A man earns 4 dollars a day : how much can he 
earn in 9 days ? 

11. At the rate of 8 marbles for a cent, how many can 
be bought for 12 cents ? 

12. If a cow eat 12 pounds of hay, how miich will 10 
cows eat? 

13. What will 7 lead-pencils cost at 7 cents apiece? 

14. If a chicken has 8 toes, how many toes have 9 
chickens ? 

15. John is 7 years old ; his father is 9 times as old : 
how old is his father ? 

WRITTEN EXERCISE. 
CASE I. 

When the Mnltiplier is a Single Figure. > 

Ex. Multiply 355 by 5. 

Addition. Multiplication. Explanation.— We write the multi- 

355 355 plier under the multiplicand, and be- 

355 5 gin at the right to multiply : 5 times 6 

355 1775 ^jjj^g ^^ 25 units, or 2 tens and 5 

355 units. Write the 5 units in the units' 

J775 place, and add the 2 tens to the product 

of tens : 5 times 5 tens are 25 tens, and 
2 tens added are 27 tens, or 2 hundreds and 7 tens. Write the 7 
tens in the tens' column, and add the 2 hundreds to the next product: 



MULTIPLICATION. 45 

5 times 3 hundreds are 15 hundreds, and 2 hundreds added are 17 
hundreds, which write in its proper place. 

The following explanation is shorter than the pre- 
ceding : 

JExplanation 2. — 5 times 5 are 25 ; write the 5 and add the 2 to 
the next product. 

5 times 5 are 25, and 2 are 27 ; write the 7 and add the 2 to the 
next product. 

5 times 3 are 15, and 2 are 17. Hence, the product is 1775. 





(1.) 


(2.) (3.) 


(4.) 


(5.) 


(6.) 


Multiply 


26 


34 42 


63 


72 


81 


By 


2 


3 3 


4 


4 


4 


(7.) 


(8.) 


(9.) 


(10.) 


(11.) 


(12.) 


25 


18 


24 


35 


54 


63 


5 


5 


6 


6 


7 


8 


(13.) 


(14.) 


(15.) 


(16.) 


(17.) 


(18.) 


43 


64 


75 


87 


95 


19 


_7 


7 

t 


6 


8 


9 


5 


(W.) 


(20.) 


(21.) 


(22.) 


(23.) 


(24.) 


324 


645 


732 


841 


681 


375 


7 


8 


9 


5 


7 


6 


(26.) 


(26.) 


(27.) 


(28.) 


(29.) 


(30.) 


463 


572 


986 


785 


487 


604 


8 


7 


6 


9 


5 


8 



46 



MUI.TIPLICATION. 



Multiply — 

31. 315 by 6. 

32. 480 by 7. 

33. 614 by 5. 

34. 7842 by 3. 

35. 6843 by 7. 

36. 8742 by 5. 

37. 9764 by 8. 

38. 8973 by 6. 

39. 14068 by 5. 

40. 18007 by 4. 



41. 6742 by 8. 

42. 6040 by 9. 

43. 61783 by 7. 

44. 60784 by 6. 

45. 85643 by 5. 

46. 170604 by 6. 

47. 683471 by 5. 

48. 863478 by 7. 

49. 785473 by 8. 
60. 246853 by 9. 



CASE II. 



Process. 

642 
57 

4494 
3210 



When the Multiplier consists of Two or more Figures. 

Ex. Multiply 642 by 57. 

Explanation, — We write the multiplier under the 
multiplicand, as in the previous case, and begin to mul- 
tiply at the right. By Case I., multiplying 642 by 7 
gives 4494 ones. Multiplying 642 by 5 gives 3210, and 

since the multiplier is 5 tens, the result is 3210 tens. 

36594 . . . 

which, added to the previous product, 4494 ones, gives 

the correct product, 36594. 

Multiply — 

1. 32 by 43. 

2. 46 by 65. 

3. 67 by 35. 

4. 73 by 87. 
6. 122 by 73. 

6. 144 by 96, Am, 13824. 

7. 347 by 52. Am, 18044. 

8. 954 by 63. Am, 60102. 

9. 725 by 75. Am, 54375. 
10. 864 by 64. Am. 55296. 



Am, 1376. 
Aiu, 2990. 
Am, 2345. 
Am, 6351. 
Am, 8906. 



11. 723 by 88. Am, 6^24. 

12. 647 by 77. Am, 49S19. 

13. 493 by 82. ^m. 40426. 

14. 761 by 41. ^w^. 31201. 

15. 875 by 65. ^rw. 56875. 

16. 944 by 66. ^rw. 62304. 

17. 871 by 73. Am, 63583. 

18. 945 by 78. Am. 73710. 

19. 674 by 65. Am, 43810. 

20. 777 by 77. Am. 59829. 



'^ 






MULTIPLICATION. 4 

21. How many yards of muslin in 24 pieces of 43 yc 
each? Ans, 1032. 

22. K a cow cost 35 dollars, how much will 29 cow 
cost ? Ans. 1015 dollars. 

23. A railway train runs 38 mi. an hour : how far do€ 
it run in 75 hr. ? Ans. 2850 mi. 

24. How much will a farm of 46 acres cost at 95 do) 
lars an acre ? Ans. 4370 dollars. 

25; What is cost of 27 bu. of oats at 63 cts. a bushel ] 

Ans. 1701 cts. 

26. What is the cost of 64 doz. of eggs at 27 cts. 
dozen ? Ans. 1728 cts. 

27. There are 12 things in a dozen: how many ar 
there in 64 doz. ? Ans. 768. 

28. A drover had some horses worth 125 dollai 
each : what are 23 of these horses worth ? 

Ans. 2875 dollars. 

29. What is the value of 45 A. of land at 164 dollar 
an acre ? Ans. 7380 dollars. 

30. What will 55 mules cost at 185 dollars apiece? 

Ans. 10175 dollars. 

31. A clerk earns 755 dollars in a year: how muc" 
at the same rate can he earn in 15 yr. ? 

Ans. 11325 dollars. 

32. There are 63 gal. in a hogshead : how many gallon 
are there in 27 hhd. ? Ans. 1701. 

33. If 1 boy solve 227 problems in a week, how man 
at the same rate can 53 boys solve? Ajis. 12031. 

34. If a man earn 27 dollars a week, how much can h 
earn in 52 wk., or 1 yr. ? Ans. 1404 dollars. 

35. A merchant has 47 pieces of calico, each containiuj 
43 yd. : how many yards has he? Ans. 2021. j 



48 



MULTIPLICATION. 



Ex. Multiply 6741 by 475. 



Process. 

6741 
475 

33705 
47187 
26964 

3201975 



ExpUxnatio/n. — Multiplying 6741 by 5 units, we have 
33705 units; multiplying 6741 by 7 tens, we have 47187 
tens ; multiplying 6741 by 4 hundreds, we have 26964 
hundreds ; writing these products in their proper places 
and adding them, we have the true product, 3201975. 



Multiply — 

36. 744 by 635. 

37. 895 by 336. 

38. 972 by 243. 

39. 825 by 682. 

40. 973 by 745. 

41. 8462 by 781. 

42. 9643 by 683. 

43. 8532 by 763. 

44. 8984 by 133. 

45. 4659 by 886. 

56. 64374 by 78561. 

57. 648487 by 678432. 

58. 7846825 by 397. 

59. 75456593 by 6471. 

60. 96458 by 7354. 



46. 6484 by 6372. 

47. 7856 by 3475. 

48. 6748 by 6334. 

49. 4878 by 3437. 

50. 8547 by 7733. 

51. 85474 by 2547. 

52. 46887 by 3489. 

53. 56184 by 5474. 

54. 56664 by 4871. 

55. 25473 by 4487. 

Ans, 5057285814, 

Arts. 439954332384. 

Am, 3115189525. 

Ans, 488279613303. 

• ^rw. 709352132. 



61. What will 655 A. of land cost at 164 dollars an 
acre? Am. 107420 dollars. 

62. If a train go 597 mi. a day, how far will it go in 
313 d.? Am, 186861 mi. 

63. How much will 864 horses cost, if 1 cost 95 dol- 
lars? Am. 82080 dollars. 

64. If it cost 125 dollars a year to board 1 student, 
how much will it cost to board 453 students ? 

Ana. 56625 dollars. 



MULTIPLICATION. 49 

65. A drover sold 197 horses at 168 dollars each : how 
much did he receive for them ? Ana. 33096 dollars. 

66. If 1 log cut 475 feet of lumber, how much will 374 
such logs cut ? Am. 177650 ft. 

67. A man had 4623 tons of iron, which he sold at 73 
dollars a ton : how much did he get for it ? 

Am. 337479 dollars. 

68. A teamster hauls 676 bricks in one load : how 
many does he haul in 688 loads ? Ans. 464400 bricks. 

69. If a book has 723 pages, how many pages have 642 
similar books ? Ans. 464166 pages. 

70. If 1 cotton-bale weigh 397 lb., how much at the 
same rate will 256 bales weigh ? Ans. 101632 lb. 

CASE III. 

To Multiply when there are Nanghts at the Right of 
either the Mnltiplicand or the Multiplier, or both. 

Ex. 1. Multiply 643 by 700. 

Process. ErpUmation. — Multiplying 643 by 7 hundreds gives 

643 4501 hundreds, or 450100. 

700 This result is the same as is obtained by multiply- 

450100 ing by 7, and annexing as many naughts on the right 
as there are naughts at the right of the 7. 

Ex. 2. Multiply 614000 by 600. 

Process. Explanation. — Multiplying the multiplicand by 6 

614000 hundreds, gives 3684000 hundreds, or 368400000. 
. 99^ This result is the same as that obtained by multi- 

368400000 plying 614 by 6, and annexing to the right five 
naughts, which is the number of naughts to the right of both the 
multiplier, 6, and the multiplicand, 614. 

Hence, When there are naughts to the right of either 
mvMiplier or multiplicand^ multiply the other figures and 
annex as many naughts as are at the right of both numbers, 
5 D 



50 MULTIPLICATION. 



Find the value — 

1. Of 743x600. 

2. Of 847 X 700. 

3. Of 9642 X 6300. 

4. Of 1875 X 6340. 

5. Of 27 X 9000. 

6. Of 6000x43. 



7. Of 18000 X 623. 

8. Of 6400 X 640. 

9. Of 650 X 650. 

10. Of 83600x7500. 

11. Of 9230x7000. 

12. Of 8000x61000. 



13. There are 2000 lb. in a ton: how many pounds are 
there in 75 T. ? Ana, 150000. 

14. What is the weight of 20 loads of coal, if each load 
weigh 1800 lb. ? Am. 36000. 

15. A man sold 500 cows at 40 dollars apiece : how 
much did he get for them ? Ans, 20000 dollars. 

16. If a teacher receive 600 dollars a year salary, how 
much would 35 teachers receive at the same rate ? 

Am, 21000 dollars. 

17. What are 160 A. of land worth at 150 dollars an 
acre ? Am. 24000 dollars. 

18. What are 600 horses worth at 200 dollars each ? 

Am. 120000 dollars. 

CASE IV. 

To Multiply when there are Naughts in the Mnltiplier. 

Ex. Multiply 6043 by 7006. 

^^^^* JSr/jZanatwTi.— Multiplying 6043 by 6 units equals 

7006 ^^^^^ ""^*®- Multiplying 6043 by 7 thousands equals 

36258 ^2301 thousands, which we write in its proper place. 

42301 Adding these products, we have the true product^ 

42337258 42337258. 

Note. — Since times any number is 0, it is not necessary to 
multiply by the naught. 

In multiplying by any number of thousands, etc., the first figure 
of the result should be placed in the same column as the multiplier. 

/ 



MULTIPLICATION. 61 

'l^Qs, in mulUplyiDg above bj 7 thousands, we say 7 times 3 are 21, 
^d write the 1 in the thousands' column, adding the 2 to the next 

product. 

Find the value — 

1. Of 806 X 307. 

2. Of 7500 X 1406, 

3. Of 1800 X 709. 

4. Of 6742 X 604. 

5. Of 12000x709. 



6. Of 4600x4006. 

7. Of 8740 X 6040. 

8. Of 9000 X 70500. 

9. Of 6004x6004. 
10. Of 86457x300017. 



11. A lot cost 420 dollars: how much will 105 lots 
cost at the same rate ? Ans, 44100 dollars. 

12. A drover has 406 cows worth 30 dollars each : how 
much are they all worth? Arts, 12180 dollars. 

13. How much will it cost to build 307 miles of rail- 
road at 4060 dollars a mile? Am. 1246420 dollars. 

14. A contractor built 604 miles of railroad at 6500 
dollars a mile : hoW much did he get for it ? 

Am. 3926000 dollars. 

COMBINATION PROBLEMS. 

1. A merchant has 26 pieces of cloth of 54 yd. eaeb-, 
which he sells at 65 cts. a yard : how much does he re- 
ceive for all of it? Am. 91260 cts. 

2. A merchant sells 12 boxes of starch, each contain- 
ing 60 lb., at 11 cts. a pound : how much does he receive 
for it ? Am, 7920 cts. 

3. A man earns 25 dollars a week, and pays 5 dollars 
a week for board : how much does he save in 6 wk. ? 

An8. 120 dollars. 

4. A boy bought 3 ducks at 25 cts. each, and 4 hens at 
30 cts. each : how much did he pay for all ? 

Ans. 195 cts. 



62 MULTIPLICATION. 

5. What is the value of 7 horses at 125 dollars 
and 14 cows at 27 dollars each? Ans, 1253 doll 

6. A farmer has the following stock : 6 horses 
140 dollars each, 13 cows worth 32 dollars each, i 
hogs worth 16 dollars each : what is his stock wort! 

Am. 1512 dol] 

7. A builder hired 12 m^i at 2 dollars a day eac 
5 boys at 1 dollar a day each : how much did he p 
in 27 d. ? Ans. 783 dol 

8. A farmer took 10 bu. of potatoes worth 75 
bushel to market, and traded them for 12 yd. oi 
worth 60 cts. a yard, the balance to be paid in cash 
much cash did he get ? -4?w. 30 

9. A merchant bought 27 lb. of butter at 33 
pound, and gave in exchange 64 lb. of sugar at Ic 
pound, and the balance in cash : how much did he 
cash ? Ana, 59 

10. If a train travel 28 mi. an hour, how far wc 
go in 6 d. of 24 hr. each ? Ans. 4032 

11 . What is the cost of 40 horses at 106 dollars 
and 60 cows at 65 dollars each ? Ana, 8140 dol 

12. What is the value of 165+178 + 347-612? 

Ana 

13. What is the value of 17x643, 27x647 an 
745 ? Ans. 36 

14. What is the value of 18x24 and 16x2' 
30 X 28 ? Am 

15. A man earns 427 dollars a month, and spen 
dollars a month : how much does he save in 16 mo 

Ana. 4032 dol 

16. A farmer bought 17 cows at 45 dollars eat 
53 hogs at 12 dollars each : which cost the most, ai 
much? Ana, The cows, 129 dol 



DIVISION. 53 

17. A farmer took to market 12 hens at 33 cts. each, and 
15 bu. of potatoes at 75 cts. a bushel ; he bought 14 books 
at 95 cts. each : how much money had he remaining ? 

Ans, 191 cts. 

18. A merchant buys 12 geese at 85 cts. each, and 16 
turkeys at 94 cts. each ; he gives in exchange 27 yd. of 
calico at 9 cts. a yard, and 25 yd. of delaine at 34 cts. a 
yard : how much money does he have to pay ? 

Ana, 1431 cts. 

19. A man paid 1600 dollars for a house, and paid 
a man for 15 days' work at 2 dollars a day for fenc- 
^g : how much did the property cost him ? 

Am. 1630 dollars. 

20. A lady sold 27 tubs of butter, each weighing 
43 lb., at 27 cts. a pound, and bought 175 yd. of carpet 
*t 90 cts. a yard : how much money had she left? 

Ans. 15597 cts. 

21. A man had 465 dollars ; he earned 750 dollars, and 
then spent 540 dollars and lost 27 dollars : how much had 
he remaining ? Ana, 648 dollars. 



SECTION V. 

Dirisiox. 

1. Two boys have 4 eyes : how many eyes has 1 boy ? 

2. How many times 2 eyes are 4 eyes? 

3. Three dogs have 12 feet: how many feet has 1 dog? 

4. How many times 4 feet are 12 feet ? 

5. A bush has 8 roses: how many times 2 roses has it? 
How many times 4 roses ? 

6. A house has 1 2 doors : how many times 3 doors has 
it ? How many times 4 doors has it ? 



/.« 



54 DIVISION. 

7. An orchard has 20 trees : how many times 5 trees 
has it? How many times 4 trees? How many times 10 
trees? 

8. How many times 6 is 24 ? How many times 5 is 25? 

9. How many times is 5 contained in 15 ? 

Solution. — Since 3 times 5 are 15, 5 is contained 3 times in 15. 

10. How many times is 4 contained in 20 ? 

11. How many times is 6 contained in 30? 

12. How many times is 5 contained in 30 ? 

13. How many times is 3 contained in 18 ? 

14. How many times is 6 contained in 18 ? 

15. If a boy earn 24 dollars, how many times 4 dollars 
does he earn ? How many times 6 dollars ? How many 
times 8 dollars ? 

16. If a man has 30 cts., how many times 10 cts. has 
he ? How many times 6 cts. ? How many times 5 cts. ? 
How many times 3 cts. ? 

17. How many times 6 boys are 30 boys ? 

18. How many times 7 horses are 21 horses ? 

19. How many times is 7 contained in 28 ? 

39« The process of finding how often one number is 
contained in another is called Division. 

40. The number to be divided is called the Dividend. 

Ah The number which is contained in the other is 
called the Divisor. 

42. The result obtained by the division is called the 
QuoUent. 

43. The sign of division, -»- , is read divided by, and 
when placed between two numbers shows that the first is 
to be divided by the second. Thus, 165 - 15 is read 165 
divided by 15. 



DIVISION. 



55 



DIVISION TABLE. 



1 in 
If 1 time. 

2, 2 times. 

3, 3 



It 



4 
5 
6 

7 



4, 

?: 

8, 8 

9, 9 

10, 10 

11, 11 

12, 12 



u 
II 
it 
n 
tt 
it 
a 
a 
it 



2in 

2, 1 time. 

4, 2 times. 

6, 3 

8, 4 
10, 6 
12, 6 
14, 7 
16, 8 
18, 9 
20, 10 
22, 11 
24^ 12 



it 



a 
tt 
a 
it 
tt 
It 
it 
it 
it 



3 in 
3, 1 time. 
6, 2 times. 
9, 3 



12 
15 
18 
21 
24 
27 
30 
33 
36 



(( 



4 
5 

6 

7 

8 

9 

10 

11 

12 



ti 
a 
a 
tt 
tt 
a 
ti 
tt 
ti 



4 in 

4) 1 time. 

8, 2 times. 

12, 3 

16, 4 

20, 5 

24, 6 

28, 7 

32, 8 

36, 9 

40, 10 

44, 11 

48, 12 



it 



a 
tt 
ti 
it 
tt 
tt 
ti 
ti 
tt 



5 in 

5, 1 time. 

10, 2 times. 

15, 3 

20, 4 

25, 5 

30, 6 

35, 7 

40, 8 

46, 9 

50, 10 

55, 11 

60, 12 



it 
tt 
it 
it 
tt 
it 
it 
ti 
tt 
it 



6in 
6^ 1 time. 
12. 2 times. 
3 " 



18 
24; 

3o; 

36 
42 
48! 
54 

6o; 

66 
72 



4 
5 
6 

7 
8 
9 

10 
11 
12 



(( 
it 
ti 
tt 
tt 
tt 
tt 
tt 
tt 



14 
21 
28 
35 
42 
49 
56 
63 
70 
77 
84 



7 in 
7, 1 time. 
2 times. 
3 



ti 



4 
5 

6 
7 
8 
9 

10 
11 
12 



ti 
tt 
it 
ti 
tt 
ti 
tt 
tt 
it 



8 in 

8, 1 time. 

16, 2 times. 

24, 3 

32, 4 

40, 5 

48, 6 

56, 7 

64, 8 

72, 9 

80, 10 

88, 11 

96, 12 



(( 
it 
it 
it 
tt 
it 
tt 
it 
it 
it 



9 

18 
27 
36 
45 
54 
63 
72; 
81 
90 
99 
108 



9 in 

1 time. 

2 times. 

3 " 
•4 

5 

6 

7 

8 

9 
10 
11 
12 



a 
it 
a 
tt 
ti 
it 
tt 
tt 
a 






10 

20; 

30 

40 

50 

60 

70 

80 

90 

100; 

110 

120 



10 in 




1 time. 


11, 


2 times. 


22, 


3 " 


33, 


4 " 


44, 


5 " 


65, 


6 " 


66, 


7 " 


77, 


8 " 


88, 


9 " 


99, 


10 " 


110, 


11 " 


121, 


12 " 


132, 



11 in 

1 time. 

2 times. 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 



ft 
tt 
it 
tt 

a 
tt 
tt 
tt 
tt 
tt 



12 
24 
36 

48 

60 

72 

84 

96 

108 

120 

132 

144 



12 in 

1 time. 

2 times. 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 



(( 



tt 
tt 
tt 
ft 
it 
it 
it 
ft 

a 



56 



DIVISION. 



ORAL EXERCISE. 

1. Divide by 2's from 2 to 12 ; thus, 2 in 2, 1 time ; 2 
in 4, 2 tiraes, and so on. 

2. Divide by 2's from 12 to 2 ; thus, 2 in 12, 6 times ; 
2 in 10, 5 times, and so on. 



3. Divide- 

By 3's from 3 to 24. 
By 3's from 36 to 12. 

4. Divide- 

By 4's from 4 to 28. 
By 4's from 48 to 24. 

5. Divide — 

By 5's from 5 to 40. 
By 5's from 60 to 20. 

6. Divide — 

By &8 from 6 to 42. 
By 6's from 72 to 24. 

7. Divide- 

By 7's from 7 to 63. 
By 7's from 84 to 28. 



8. Divide- 

By 8's from 8 to 56. 
By 8's from 96 to 32. 

9. Divide- 

By 9's from 9 to 81. 
By 9*s from 108 to 27. 

10. Divide- 
By lO's from 10 to 80. 
By 10'sfroml20to40. 

11. Divide — 

By irs from 11 to 132. 
By irs from 132 to 11. 

12. Divide- 

By 12's from 12 to 96. 
By 12's from 144 to 48. 



1. How many oranges at 5 cents each can I buy for 30 
cents? 

Solution 1. — If 1 orange cost 5 cents, for 30 cents I can buy 6 
oranges, because 6 times 5 are 30. 

Solution 2. — If 1 orange cost 5 cents, for 30 cents I can buy as 
many oranges as 5 is contained times in 30, or 6 oranges. 

2. If a hat cost 4 dollars, how many hats can be bought 
for 20 dollars ? 

3. There are 4 pecks in a bushel : how many bushels 
are there in 24 pecks ? 

4. There are 7 days in 1 week ; how many weeks are 
there in 42 days ? 



DIVISION. 57 

5. There are 8 quarts in a peek : how many pecks in 
86 quarts? 

6. How many coats can be bought for 63 dollars if 1 
coat cost 9 dollars ? 

7. If 1 quart of milk cost 10 cents, how many quarts 
can be bought for 80 cents ? 

8. How many pair of boots at 7 dollars a pair can I 
buy for 77 dollars ? 

9. There are 12 eggs in a dozen : how many dozen are 
there in 108 eggs ? 

10. K 12 yards of calico make a dress, how many 
dresses can be made from 84 yards ? 

11. If a sheep cost 6 dollars, how many can be bought 
for 54 dollars ? 

12. K a man earn 8 dollars a week, how long will it 
take to earn 80 dollars ? 

13. A pair of shoes cost 3 dollars : how many pairs can 
be bought for 36 dollars ? 

14. How many horses will eat 60 bushels of corn if 1 
horse eat 5 bushels ? 

15. If a cord of wood is worth 6 dollars, how many 
cords can be bought for 54 dollars? 

CASE I. 

The Dirisor One Figrnre* 

Ex. Divide 984 by 4. 

Process. ExplarCation, — We write the divisor at the left of the 
4)984 dividend, separating them by a curved line, and draw 
246 a line under the dividend. 4 is contained in 9 hun- 
dred 2 hundreds times, with 1 hundred, equal to 10 tens, 
remaining ; adding this remainder to 8 tens, the next number in the 
dividend, we have 18 tens ; 4 is contained in 18 tens, 4 tens times, 
with a remainder of 2 tens, or 20 units ; adding this remainder to 4 



58 DIVISION. 

units, the next number of the dividend, we have 24 units ; 4 is con- 
tained in 24 units 6 units times. 

Note. — In practice we shorten the operation and say thus : 4 is 
contained in 9 twice, and 1 remaining ; 4 is contained in 18, 4 
times and 2 remaining ; 4 is contained in 24, 6 times. 

Divide as follows — 



(1-) 


(2.) 


(3.) 


(^> 


(fiw) 


(6.) 


2)6 


3)9 


4)12 


4)20 


5)26 


6)30 


(7.) 


(8.) 


(9.) 


(10.) 


(11.) 


(12.) 


4)28 


5)35 


3)36 


6)12 


6)36 


7)42 


(13.) 


(14.) 


(15.) 


(16.) 


(17.) 


(18.) 


6)54 


7)56 


6)60 


9)63 


5)45 


6)24 


(19.) 


(20.) 


(21.) 


(22.) 


(23.) 


(24.) 


7)49 


8)32 


8)66 


8)72 


8)96 


8)80 


(25.) 


(26.) 


(27.) 


(28.) 


(29.) 


(30.) 


6)18 


7)70 


9)72 


9)46 


9)81 


9)99 


(31.) 


(32.) 


(33.) 


(34.) 


(35.) 


(36.) 


4)48 


5)50 


6)66 


4)84 


3)963 


2)864 


(37.) 


(38.) 


(39.) 


(40.) 


(41.) 


(42.) 


5)70 


6)96 


8)96 


7)147 


6)126 


6)736 


(43.) 


(44.) 


(45.) 


(46.) 


(47.) 


(48.) 


6)738 


7)483 


8)624 


5)720 


8)944 


4)6832 


(49.) 


( 


:50.) 


(51.) 




(52.) 


6)68405 


6)68436 


9)78462 ( 


5)46824 



DIVISION. 59 

(53.) (54.) (55.) (56.) 
9) 468972 7) 468342 8 )847632 6)345678 

Note. — If the divisor is not contained in the dividend a whole 
number of times, write the remainder at the right, with the plos- 
fdgn between the answer and the remainder. Thus, 

3)644 
214+2 rem. 

57. If a ton of coal cost 4 dollars, how many tons can 
be bought for 464 dollars ? Ans. 116. 

58. There are 7 days in a week : how many weeks in 
364 days ? Am. 52. 

59. A pair of boots cost 9 dollars : how many pairs at 
the same rate can be bought for 225 dollars ? Ans. 25. 

60. If 9 horses are worth 1215 dollars, how much is 1 
horse worth? Ans. 135 dollars. 

61. If a farmer can raise 1504 bushels of potatoes on 
8 acres, how many bushels can he raise on 1 acre ? 

Ans. 188. 

62. If a man can earn 6 dollars in one week, how 
many weeks will it take to earn 690 dollars? Ans. 115. 

63. In an orchard of 240 trees there are 8 trees in a 
row : how many rows are there ? Ans. 30. 

64. A train of cars runs 280 miles in 8 hours: how 
far does it move in an hour ? Ans. 35 miles. 

65. If 7 horses cost 980 dollars, how much does 1 horse 
cost ? Ans. 140 dollars. 

44. The mode of dividing in the foregoing problems is 

called Short Division. 

The divisor, the dividend and the quotient only are 
written in Short Division. 

45. When also the different steps of the solution are 
written, the process is called Long Division. 



60 



DIVISION. 



Ex. Divide 744 by 3. 



Process 1. 

3)744 1 248 

6_ 

i4 
12 

24 
24 



Erpianaiion 1. — 3 is contained in 7 hundreds 2 hun- 
dred times; 2 hundred times 3 are 6 hundreds; 6 
hundreds from 7 hundreds, leaves 1 hundred. 1 hun- 
dred and 4 tens are 14 tens. 3 is contained in 14 
tens 4 tens times ; 4 tens times 3 is 12 tens, which 
subtracted from 14 tens leaves 2 tens. 2 tens and 4 
units are 24 units. 3 is contained in 24 units 8 units 

times ; 8 times 3 are 24, which subtracted from 24 leaves nothing 

Hence, 744 divided by 3 equals 248. 

Explanation 2. — 3 is contained in 7, 2 times ; 2 times 
3 are 6; 6 from 7 leaves 1 ; bringing down the 4, we 
have 14. 3 is contained in 14, 4 times ; 4 times 3 are 
12, which subtracted from 14 leaves 2 ; bringing down 
4, the next figure of the dividend, we have 24. 3 is 
contained in 24, 8 times ; 8 times 3 are 24, which sub- 
tracted from 24 leaves nothing. Hence, the quotient 
is 248. 



Process 2. 

£)744|248 

6_ 

14 
12 

24 
24 



1. Divide 

2. Divide 

3. Divide 

4. Divide 
6. Divide 

6. Divide 

7. Divide 

8. Divide 

9. Divide 

10. Divide 

11. Divide 

12. Divide 

13. Divide 

14. Divide 

15. Divide 

16. Divide 



WRITTEN 
32 by 2. 
48 by 3. 
60 by 4. 
75 by 5. 
96 by 6. 
112 by 8. 
324 by 3. 
316 by 4. 
432 by 3. 
612 by 3. 
846 by 3. 
780 by 4. 
840 by 4. 
932 by 4. 
825 by 5. 
960 by 5. 



EXERCISES. 

17. Divide 

18. Divide 

19. Divide 

20. Divide 

21. Divide 

22. Divide 

23. Divide 

24. Divide 

25. Divide 

26. Divide 

27. Divide 

28. Divide 

29. Divide 

30. Divide 

31. Divide 

32. Divide 



800 by 5. 
744 by 6. 
894 by 6. 
366 by 6. 
761 by 7. 
875 by 7. 
672 by 7. 
768 by 8. 
960 by 8. 
936 by 8. 
999 by 9. 
657 by 7. 
675 by 9. 
348 by 2. 
4264 by 4. 
6243 by 3. 



DIVISION. 



61 



CASE II. 



To Divide when the DiTisor consists of Two or more 

Fibres. 

Ex. Divide 4550 by 14. 



Process. 
14)4550 1 325 
42 



35 

28 



70 
70 



Explanation. — 14 is contained in 45 hun(}reds 3 
hundreds times ; 3 hundreds times 14 are 42 hun- 
dreds ; 42 hundreds from 45 hundreds leaves 3 hun- 
dreds, or 30 tens ; bringing down the 5 tens, the 
dividend is 35 tens. 14 is contained in 35 tens 2 
tens times ; 2 tens times 14 are 28 tens ; 28 tens 
from 35 tens leaves 7 tens, or 70 units, which is the 
dividend. 14 is contained in 70 units 5 units timed; 5 times 14 
equals 70, which being subtracted from 70 leaves nothing. Hence, 
the quotient is 325. 

Divide — 

1. 385 by 11. 

2. 312 by 12. 

3. 660 by 15. 

4. 923 by 13. 

5. 756 by 14. 

6. 520 by 20. 

7. 735 by 21, 

8. 608 by 16. 
• 9. 324 by 18. 

10. 1064 by 19. 



11. 1650 by 22. 

12. 1600 by 25. 

13. 432 by 24. 

14. 1680 by 30. 

15. 1134 by 27. 

16. 765 by 17. 

17. 3191 by 28. 

18. 6110 by 26. 

19. 5175 by 15. 

6 



20. 6095 by 23. 

21. 8760 by 24. Am, 365. 

22. 4125 by 33. Am, 125. 

23. 14770 by 35. Am, 422. 

24. 14625 by 45. Am, 325. 

25. 27398 by 38. Am, 721. 

26. 17316 by 39. ^rw. 444. 

27. 32928 by 42. Arts. 784. 

28. 26136 by 44. Am, 594. 

29. 37904 by 46. Am, 824. 

30. 35280 by 48. Am. 735. 

31. 34048 by 56. Am, 608. 

32. 27720 by 72. Am, 385. 

33. 33280 by 64. Am, 520. 

34. 35406 by 63. Am, 562. 

35. 21489 by 87. Am, 247, 

36. 62167 by 107. ^n«.581. 

37. 79005 by 115. Am.6S7, 

38. 53125 by 125. ^rw. 425. 



62 DIVISION. 

39. 1509534 by 234. Arts. 6451. 

40. 3160642 by 462. Ans. 6841. 

41. There are 24 hr. in a day : how many days are there 
in 1032 hr. ? Ans. 43. 

42. If a cow cost 32 dollars, how many at that rate 
can be bought for 2080 dollars? Am. 65. 

43. If some pigs are worth 12 dollars apiece, how many 
can be bought for 180 dollars? Ans. 15. 

44. If a man walk 25 mi. in a day, how long will it 
take him to walk 950 mi. ? Ans. 38 d. 

45. If a train run 33 mi. an hour, how long will it be 
in running 1386 mi. ? Ans. 42. 

46. There are 60 min. in an hour : how many hours in 
3900 min. ? Ans. 65. 

47. At 55 cts. a pair, how many pidrs of chickens can 
be bought for 1210 cts. ? Ans. 22. 

48. There are 16 oz. in a pound : how many pounds 
are there in 1968 oz. ? Ans. 123. 

49. Bought mules at 115 dollars each : how many did 
I get for 2760 dollars ? Ans. 24. 

50. A received 5250 dollars &» haoKB at 125 dollar 
each : how many did he sell ? Ans. 42. 

51. How many pineapples at 16 cts. eack may^ be 
bought for 400 cts. ? Ans. 25. 

52. How many pounds of beef at IS cts* a pound can 
be bought for 540 cts. ? .^*fc. ^^ 

53. Th€reare32qjLiiLabBEdidL^jMft%am^^ 

54. There are 64 pt. in a bushel: how many bush^i0 
2688 pt. ? . Ans. 42. 

55. There are 24 sheets in a quire : how many quires in 
864 sheets ? Ans. 36. 



DIVISION. 



63 



CASE III. 

To DirJde when there are Ciphers at the Blgrht of the 

Diyisor. 

Ex. Divide 6783 by 600. 

pg^ Expl<mation, — 600 is contained in 67 hun- 

6|00j67|83 



dreds 11 times, with a remainder of 1 hun* 
dred. 600 is not contained in 83 ; hence, the 



11+183 rem. ^^^^ remainder is 183. 

Note.— When the divisor with the ciphers cutoff is greater than 
^ <livide by Long Division. 

Find the value — 

1. Of 725 H- 30. 

2. Of 864 -^ 70. 

3. Of 892 ^ 80. 

4. Of 7642 -H 60. 
6. Of 6484-^200. 

6. Of 8645 -^ 500. 

7. Of 7887 ^ 700. 

8. Of 9484 ^ 600. 

9. Of 8642 -H 1200. 
10. Of 54224^1500. 



11. Of 3786 -H 1700. 

12. Of 25761 + 2100. 

13. Of 46483 + 2500. 

14. Of 3f 400 + 1600. 

15. Of 61380 - 3300. 

16. Of 75611 + 4000. 

17. Of 21500-^3600. 

18. Of 75643 + 4500. 

19. Of 45742 + 6000. 

20. Of 378751 + 12300. 



CASE IV. 

To find the Equal Parts of a Number. 

Ex. A man bought 15 cows for 495 dollars : how much 
did 1 cost ? 

Solution. 

Bkplanation, — If 15 cows cost 495 dollars, 1 cow 

cost as many dollars as 15 is contained times in 495, 

or 33 dollars. 



15)495133 
45 ~" 



45 
45 



WRITTEN PROBLEMS. 

1. If 16 hens lay 848 eggs in a summer, how many eggs 
will 1 hen lay, at the same rate? Ans, 53. 



64' DIVISION. 

2. If 15 horses costal 740 dollars, how much will 1 horse 
cost? Am. 116 dollars. 

3. If 23 tons of hay are sold for 552 dollars, how much 
is 1 ton worth? Ana, 24 dollars. 

4. What can 1 man earn in a month if 24 men earn 
1344 dollars ? Am. 56 dollars. 

5. If 42 men lay 14700 bricks in a half day, how many 
can 1 man lay at the same rate ? Ans. 350. 

6. A drover sells 64 cows for 2880 dollars : how much 
is that for each cow ? Ans. 45 dollars. 

7. A drover stells 344 hogs for 4816 dollars : how much 
does he get apiece ? Ans. 14 dollars. 

8. If 43 ducks lay 5332 eggs in a season, how many 
eggs does 1 duck lay? Ans. 124. 

9. A farm of 118 A. was sold for 17110 dollars: how 
much was that per acre? Ans. 145 dollars. 

10. A farmer raises 7656 bu. of potatoes on 29 A : how 
many bushels does he raise on 1 A. ? Ans. 264. 

COMBINATION PROBLEMS. 

1. Whatisthevalueof 644+584-500,-^104? Am.7. 

2. What is the value of 1182+4208 - 4030, - 85 ? 

Ans. 16. 

3. What is the value of 74400 + 63300, -^ 324 ? 

Am. 425. 

4. From 6492 subtract 3468, and divide the remainder 
by 27. Am. 112. 

5. A man bought 16 horses at 120 dollars each, and 
sold them all for 2000 dollars : what was the gain on 1 
horse ? An^. 5 dollars. 

6. A drover bought 25 cows for 1000 dollars, and sold 
them for 1200 dollar's : how much did he gain on each 
cow? -4n*. 8 dollars. 



DIVISION. 66 

7. A man earns 25 dollars a week, and spends 12 dol- 
lars a week ; he saves 195 dollars : how many weeks does 
iework? Ans. 15. 

8. If a farmer buy 16 horses at 110 dollars each, and 
gain 240 dollars on the lot, at what price each does he sell 
the horses? . J.n^. 125 dollars. 

9. I traded 16 hens at 45 cts. each for ducks at 40 cts. 
each : how many ducks did I get ? Ans, 18. 

10. Sold 6 lb. of butter at 35 cts. a pound, and 4 chickens 
at 30 cts. apiece, and took in exchange muslin at 15 cts. 
a yard : how many yards did I get ? Arts, 22. 

11. A laborer worked 16 d. at 80 cts. a day, and took 
his pay in potatoes at 40 cts. a bushel : how many bushels 
of potatoes did he get ? Aiu, 32. 

12. I sell to a merchant 3 bu. of potatoes at 80 cts. a 
bushel, and 15 lb. of butter at 25 cts. a pound ; he pays 
me cash 75 cts., and the rest in coffee at 30 cts. a pound : 
how many pounds of coffee do I get ? Ana, 18. 

13. A boy who wishes to buy some books worth 75 dol- 
lars, saves 7 dollars a week for 9 wk. : how much does he 
still need? Ans, 12 dollars. 

14. If two men have 15 horses worth 90 dollars each 
and 10 cows worth 40 dollars each, what is the value of 
each one's share of the stock ? Ana, 875 dollars. 

15. A man earns 544 dollars in 16 wk., but spends 3 
dollars a week of this amount : how much does he save 
each week ? Ana. 31 dollars. 

16. A farmer has 24 cows and 93 sheep, worth 1521 
dollars : if the sheep are worth 5 dollars each, how much 
is each cow worth ? Ana, 44 dollars. 

17. A drover bought 16 horses at 120 dollars each and 

8 horses at 150 dollars each : what was the average price ? 

Ana, 130 dollars. 
6» E 



66 UNITED STATES MONEY. 

18. If a carpenter charge 18 dollars a week for 9 wk. 
in building a barn, and 420 dollars for his lumber, what 
does the barn cost ? Ans. 582 dollars. 

19. A farmer gave his &rm of 160 acres for a store 
worth 12000 dollars: what was the land worth an acre? 

Arts. 75 dollars. 

20. If a clerk has a salary of 2500 dollars a year, and 
spends 6 dollars a day for 365 days, how much has he 
left at the close of the year ? Ana, 310 dollars. 



CHAPTER II. 
UNITED STATES MOK'ET. 



SECTION I. 

DEFIJ^ITIOJ^S AJ^D PRIJ^CIPLES. 

46. United States Honey, sometimes called Fed- 
eral Money, consists of dollars, cents and mills. 

Table. 

10 mills (m.) = 1 cent, e. 

10 cents = 1 dime, d. 

10 dimes, or 100 cents = 1 dollar, S. 
10 dollars = 1 eagle, E. 

47. In business, dollars, cents and mills only are used. 
A quarter-dollar is 25 cents, a half-dollar is 50 cents. 
The dollar is denoted by the following sign, J, called 

the dollar-sign. 

48. Dollars are separated from cents, in writing, by a 



UNITED STATES MONEY. 67 

point, called a aeparatrix. Thus, 3 dollars and 25 cents 
is written $3.25 ; four dollars and 5 cents is written $4.05. 

Cents occupy the second place at the right of the point ; 
the first place is occupied by dimes. 

When there are no dimes or no cents the vacant places 
are filled with naughts. 

49« United States money may be either paper money or 
coins. Coin is sometimes called specie; and paper money, 
paper currency. 

ORAL EXERCISE. 

1. How many cents in 3 dimes? 

2. How many cents in 2 dollars ? 

3. How many cents in 3 dollars and 16 cents? 

4. How many cents are equal to a five-dollar bill ? 

5. How many cents are equal to a dollar bill and 25 
cents? 

6. How many cents in a half-dollar and a quarter- 
dollar ? 

7. How many cents in 1 dollar and a half? 

8. How many dimes in 4 dollars ? 

9. How many cents are equal to 2 five-dollar bills ? 
10. How many dollars in 3 eagles? 

Read the following : 



11.15. 


$1426. 


$21.50. 


$.243. 


$3.24. 


$6.00. 


$107.16. 


$.803. 


811.17. 


$18.05. 


$107.60. 


$8.03. 


$19.30. 


$25.07. 


$100.70. 


$6,003. 


Write— 









1. One dollar and twelve cents. 

2. Eight dollars and twenty-five cents. 

3. Three hundred and fourteen dollars and forty-two 
cents. 



68 REDUCTION OF UNITED STATES MONEY. 

4. Ten dollars and nine cents. 

5. Six cents and six mills. 

6. One dollar and a half. 

7. Eighty dollars and eighty cents. 

8. Twenty dollars two cents and two mills. 

9. Three hundred dollars and thirty cents. 
10. Thirty dollare and three mills. 



SECTION II. 
REDUCTIOJf OF UJ^ITED STATES MOJfET, 

Note. — Since 1 cent equals 10 mills, and $1 equals 100 cents, 
or 1000 mills, we have the following 

RULES. 

1. To reduce cents to mills, multiply 6^ 10, or annex a 
cipher, 

2. To reduce dollars to cents, multiply by 100, or annex 
two ciphers, 

3. To reduce dollars to mills, multiply by 1000, or annex 
three ciphers, 

4. To reduce dollars and cents to cents, or dollars, cents 
and mills to mills, remove the dollar-sign and the sepa- 
ratrix, 

5. To reduce mills to cents, divide % 10 ; cents to dollars^ 
divide by 100 ; and mills to dollars, divide by 1000, and 
write the sign of the denomination required, 

WRITTEN EXERCISE. 

1. How many mills in 3 cents? In 45 cents? 

^. How many cents in 5 dollars? In 40 dollars? 

In $87. 



ADDITION OF UNITED STATES MONEY. 69 

3. How many mills in 42 cents ? In $4.18 ? 

4. How many cents in $41.16 ? In $84.12 ? 

5. How many mills in 4 cents and 4 mills ? 

6. How many mills in $6,066 ? 

7. Reduce 647 mills to cents and mills. 

8. Reduce 847 cents to dollars. 

9. Reduce $8,475 to mills. 
10. Reduce $400.03 to cents. 



SECTION III. 
jlDDITIOJf OF UJ^ITED STATES MOJ^EY. 

ORAL EXERCISE. 

1. A hen cost 40 cents, and a duck 30 cents: how 
much did both cost? 

Solution.— If a hen cost 40 cents, and a duck 30 cents, the two 
cost 40 cents plus 30 cents, or 70 cents. 

2. A book cost $1.25, and a slate 50 cents : how much 
did they both cost ? 

3. A pair of shoes cost $2.50, and a hat $2.25 : how 
much did they both cost ? 

4. If I pay $1.20 for a turkey, $1.15 for a goose, and 
60 cents for some butter, how much do I pay for all ? 

5. If a cow cost $25.50, and a sheep $7.75, how much 
will both cost ? 

6. If I earn 20 cents on Monday, and 75 cents on Tues- 
day, how much do I earn in both days ? 

7. If I pay $7.50 for a coat, and $2.25 for a pair of 
pants, how much do both cost ? 

8. A book cost 90 cents, a pen-holder 10 cents, and a 
slate 35 cents : how much did they all cost ? 



70 ADDITION OF UNITED STATES MONEY. 

WRITTEN PROBLEMS. 

Note. — In solying problems in Addition of United States Money, 
write the numbers, placing cents under cents and dollars under dol- 
lars, and add as in simple addition. Separate dollars and cents by 
a point and prefix the doUar-^ign. ^ 

Ex. Add «18.30, $16.25, $18.40 and $13.21. 

Pbocess. Explanation. — 1 and 5 are 6 cents ; 2 and 4 are 6, and 

$18.30 2 are 8, and 3 are 11 dimes, or 1 dollar and 1 dime; ] 

16.26 and 3 are 4, and 8 are 12, and 6 are 18, and 8 are 26 

13 21 clollars ; write the 6 dollars, and add the 2 ten-dollars to 

*gg ,g the next column. 2 and 1 are 3, and 1 are 4, and 1 are 

5, and 1 are 6 tens of dollars. Hence, the sum is $66.16. 

Add the following : 

(1.) (2.) (3.) (4.) 

$18.25 $54.34 $57.60 $105.20 

17.24 71.56 75.20 110.00 

121.43 84.93 18.00 409.05 

' 67.44 62.71 10.00 1000.65 

5. I sold 1 horse for $87.50, and another for $94.75 : 
how much did I get for both ? Ana. $182.25. 

6. I bought sugar for $4.24, coffee for $1.25 and rice 
for 63 cts. : how much did they all cost me? A'ns. $6.12. 

7. A merchant's receipts for 4 d. were $16.20, $13.40, 
$27.42 and $19.75 : how much were his receipts ? 

Ans. $76.77. 

8. In building a barn I paid the carpenter $97.40, the 
mason $25, and for lumber $367.45 : how much did my 
barn cost ? Ans. $489.85. 

9. I paid $9.75 for a coat, $1.50 for a vest, $3.50 for a 
hat and $3.75 for a pair of pants : what was the cost of 
all? ^rw. $18.50. 

10. A man's coal-bill for the year was $50.50, his rent 
$225, his groceries $460.73 and his other expenses $630.73 : 
what were his expenses for the year? -4 rw. $1366.96. 



SUBTRACTION OP UNITED STATES MONEY. 71 

SECTION IV. 
SUBTRACTIOJ^ OF U. 8. MOJfEY. 

ORAL EXERCISE. 

1. K a knife cost $1.25, and it is sold for $1.75, what 
is the gain ? 

SoiiUnoN. — If a knife cost $1.25, and it is sold for $1.75, the 
gain is the difference between $1.75 and $1.25, which is 50 cents. 

2. A merchant bought some goods at $2.62 a yard, and 
sold them at $3.89 a yard : what was the gain per yard ? 

3. A huckster bought turkeys at $1.20 apiece, paid 10 
cts. apiece for their feed, and sold them at $1.75 apiece : 
what was the gain ? 

4. Bought some rice for 60 cts., some sugar for 45 cts. 
and some tea for $1 : how much change should I get 
from a five-dollar bill ? 

5. Bought a horse for $120, a saddle for $15, and sold 
both for $150: what was my gain? 

WRITTEN PROBLEMS. 

Note. — In solving problems in Subtraction of United States 
Money, write the numbers, placing cents under cents and dollars 
under dollars, and proceed as in simple subtraction. Separate the 
dollars from the cents in the result and prefix the dollar-sign. 

Ex. Subtract $14.26 from $19.41. 

Process. E'Epfenohon.— .We cannot subtract 6 cts. from 1 ct. ; 

$19.41 hence we take 1 dime, or 10 cts., which with 1 ct. equals 

1^-26 11 cts. ; 6 cts. from 11 cts. leaves 5 cts. ; 2 dimes from 3 

$5.15 dimes leaves 1 dime ; 4 dollars from 9 dollars leaves 5 

dollars. Hence, the difference is $5.15. 

(1.) (2.) (3.) (4.) 

From $6.24 $27.62 $75.90 $60.50 

Take 3.12 19.45 27.54 18.46 



72 MULTIPLICATION OF UNITED STATES MONEY. 

5. What is the difference between $18 and $9.75 ? 

A718. $8.25. 

6. If a horse was bought for $175, and sold for $181.25, 
what was the gain ? Ans, $6.25. 

7. A carpenter's whole expense in building a house 
was $1562.35; he received $1700: how much did he 
make? Am. $137.65. 

8. Bought a house for $1875 and a lot for $525, and 
sold both for $2350 : how much did I lose ? Am. $50. 

9. My salary is $1000 a year; I pay for rent $150, for 
groceries $325.40, for butter $60.30, for dry goods $127.63, 
and for other expenses $75.60 : how much do I save ? 

Am, $261.07. 

10. A house and lot cost me $1927.50 ; I paid for re- 
pairs $127.67, and sold the property for $2500 : what waa 
my gain ? Ans. $444.83. 



' SECTION V. 
MULTIPLICATIOJf OF U. 8. MOJfEY. 

ORAL EXERCISE. 

1. What will 5 pencils cost at 6 cents each? 

Solution. — At 6 cents each 6 pencils will cost 5 times 6 cents, 
or 30 cents. 

2. What will 3 books cost at 90 cents each ? 

3. If 1 yard of muslin is worth 15 cents, how much are 
40 yards worth ? 

4. If a pair of shoes cost $2.50, how much will 12 pairs 
cost? 

5. What is the cost of 2 horses at 120 dollars eaich, and 
3 cows at 40 dollars each ? 



MULTIPLICATION OP UNITED STATES MONEY. 73 

6. If I buy 2 hats at $1.20 each, and a book for $2, 
how much change do I get from a five-dollar bill ? 

7. A man bought 3 bushels of wheat at $1.25, and sold 
the flour he made from it for $4.50 : what did he gain ? 

WRITTEN PROBLEMS. 

Note. — Multiply in United States Money as in simple nombers, 
separate dollars from cents, and prefix the dollar-sign. 

Ex. Multiply $16.25 by 7. 

Process. Explanation. — 7 times 5 cents are 35 cents, or 3 dimes 
$16.25 and 5 cents ; 7 times 2 dimes are 14 dimes, + 3 dimes are 

. Z 17 dimes, or 1 dollar and 7 dimes ; 7 times 6 dollars are 

$113.75 42 dollars, + 1 dollar are 43 dollars ; we write the 3 dol- 
lars, and add the 4 to the next product ; 7 times 1 ten-dollar are 7 
ten-dollars, +4 ten-dollars are 11 ten-dollars. Hence, the product 
is $113.75. 



1. Multiply $27 by 3. 

2. Multiply $121 by 16. 

3. Multiply $8.17 by 4. 

4. Multiply $20.96 by 14. 



5. Multiply $70.15 by 21. 

6. Multiply $18.27 by 84. 

7. Multiply $107.27 by 46. 

8. Multiply $150.05 by 705. 



9. What are 19 hogs worth at $13.75 each? 

Ana, $261.25. 

10. What will 43 yards of carpet cost at $1.45 a yard? 

Am, $62.35. 

11. What is the cost of 93 cords of wood at $3.75 a 
cord ? Am, $348.75. 

12. What is the value of 12 books at $1.50 each, and 
25 books at $1.75 each ? Am, $61.75. 

13. A farmer sells 117 bushels of wheat at $1.35 a 
bushel, and 40 bushels of oats at 62 cents a bushel : how 
much does he get for both ? Am, $182.75. 

14. A merchant sells 19 yards of cloth at $2.25 a yard, 

7 



74 DIVISION OF UNITED STATES MONEY. 

and 27 yards of carpet at 95 cents a yard : how much does 
he get for both ? Ana, $68.40. 

15. A lady goes to market with 10 dollars; she buys 6 
dozen of eggs at 27 cents, 7 pounds of meat at 16 cents, 
and 3 bushels of potatoes at $1.25 : how much money has 
she reraaming? Ans. $3.51. 

16. A drover bought 95 cows at $37.25 each, and sold 
them at $40 each ; how much did he make ? 

Ans. $261.25. 



SECTION VI. 
DiriSIOJV' OF UJTITED STATES MOJfEY, 

ORAL EXERCISE. 

1. If a melon cost 25 cents, how many can be bought 
for $1.50. 

Solution. — If a melon cost 25 cents, for $1.50 there can be as 
many bought as 25 cents is contained times in $1.50, or 150 cents, 
which is 6. 

2. If 6 sheep cost $42, how much will 1 sheep cost ? 

3. At 10 cents apiece, how many copy-books can be 
bought for $2.50 ? 

4. If 9 cords of wood cost $36, how much will 1 cord 
cost ? 

5. At 20 cents apiece, how many pineapples can I buy 
for $1.80? 

6. At the rate of 25 cents a dozen, how many dozen 
buttons can be fought for $3 ? 

7. K a man earn $40 in 8 d., how much can he earn in 
Id.? 



DIVISION OF UNITED STATES MONEY. 75 

8. If I buy 17 lb. of sugar at 10 cts. a pound, how 
many oranges at 5 cts. each can I get for the change due 
me from a five-dollar bill ? 

9. A yard of calico is worth 9 cts. : how many yards 
can I get for 10 doz. of eggs, worth 18 cts. a dozen ? 

10. If I trade 6 lb. of butter at 20 cts. a pound, and 
10 lb. of lard at 12 cts. a pound, for sugar at 12 cts. a 
pound, how many pounds of sugar do I get ? 

CASE I. 

To find how often One Sum of Money is contained in 

Another. 

WRITTEN PROBLEMS. 

Note. — When finding how often one sum of money is contained 
in another, reduce both sums to the same denomination and divide 
as in simple numbers. 

Ex. How often is 14 cts. contained in J3.22 ? 
Pbogbss. 

23 ' — Erplanation, — $3.22 is equal to 322 cts. ; 14 cts. is 
~42 contained in 322 cts. 23 times. 

42 

1. Divide $6000 by $12. 

2. Divide $3200 by $160. 

3. Divide $56 by 80 cts. 

4. At $16 each, how many coats can be bought for 
«384 ? Am. 24. 

5. At $45 an acre, how many acres of land can be 
bought for $3330 ? Ans. 74. 

6. If hats cost $2.25 apiece, how many can be bought 
for $56.25 ? Am, 25. 

7. How many bushels of oats at 55 cts. a bushel can I 
buy for $58.85 ? Am. 107. 



76 DIVISION OF UNITED STATES MONEY. 

8. How many cows at $24 each can be bought for 
$1512? Ans,m. 

9. At 27 cts. a yard, how many yards of dress goods 
can I get for 814.31 ? Am. 53. 

10. At $2.25 a yard, how many yards of cloth can be 
bought for $36 ? Ans, 16. 

CASE II. 

To Divide a Number into £qnal Parts. 

Note. — To find the equal parts of a number, divide as in simple 
numbers, and separate dollars from cents with the separatrix. 

Ex. If 5 cows cost $125.75, how much will 1 cow cost? 

Solution. Explanation. — If. 6 cows cost $125.76, or 12676 cts., 
6)1 25.76 1 cow will cost as many cents as 6 is contained in 
^5J5 12675, or 2516 cts., equal to $25.16. 

1. Divide $9.50 into 5 equal parts. 

2. Divide $5.10 into 6 equal parts. 

3. If 6 books cost $6.90, how much will 1 book cost ? 

Ana, $1.15- 

4. If 8 inkstands cost $6, how much will 1 inkstand 
cost ? Ana, $.75. 

5. If 11 T. of coal cost $63.25, how much ^vill 1 T. 
cost? Ana, $5.75. 

6. If 12 bu. of oats cost $3.60, how much will 5 bu. 
cost? Ans, $1.50. 

7. If a man earn $7.50 in 5 d., how much can he earn 
in6d.? ^m. $9. 

8. If 9 lb. of butter cost $2.43, how much will 50 lb. 
cost? ^/w. $13.50. 

9. If 42 A. of land cost $1890, how much will 65 A. 
cost ? Ana. $2925. 

10. If 23 yd. of carpet cost $28.75, how much will 
40 yd. of the same carpet codt? Ana. $50. 



BILLS. 77 

COMBINATION PBOBLEM8. 

1. A fanner brings to a grocer 10 doz. of eggs at 18 cts. 
a dozen, 26 lb. of lard at 15 cts. a pound, and takes in ex- 
change 10 lb. of coffee at 23 cts. a pound, and a set of dishes 
worth $2 : how much is due him in cash? Ana. $1.40. 

2. A farmer gave a horse worth 175 dollars and 3 cows 
worth 40 dollars each for some sheep worth $5 each : how 
many sheep did he get ? Ana. 59. 

3. A farmer exchanged 60 bu. of potatoes at 45 cts. a 
bushel, 3 bbls. of flour at $6.50, and 50 bu. of wheat at 
$1.20 a bushel, for carpet worth $1.50 a yard : how many 
yards did he get ? Ans,l\, 

4. A merchant sells the following goods : 10 yd. of 
calico at 9 cts. a yard, 21 yd. of muslin at 13 cts. a yard, 
18 yd. of delaine at 35 cts. a yard, and takes in exchange 
15 bu. of corn at 62 cts. a bushel, and the remainder in 
cash : how much cash did he get? Ana, $.63. 

5. A man buys a farm for $16000 ; he buys 6 horses at 
$130 each, 15 cows at $32 each, 1 pair of oxen for $96, 
15 pigs at $4.25 each, and 70 sheep at $3.75 each, and 
sells all f(tr $18000 : how much does he gain ? 

Ana, $317.75. 

BILLS. 

60. A Bill of goods is a written statement of the goods 
sold, giving quantity and price of each article and total 
cost, also the date of the sale, with the names of the buyer 
and the seller. 

61. The party who owes is called a Debtor, and the 
party to whom a debt is owed is called a Or editor. 

Note. — In mercantile and other business transactions if the 
fractional part of a cent is less than one-half, it is not counted ; if 
it is equal to a half cent or greater, It is counted a cent. 

7* 



78 BILLS. 

Make out the following bills : 

Lock Haven, Pa., JvJy 3, 1877. 

Mr. Wilson Kistler, 

Bought of Jacob Brown. 

15 lb. Coffee at 32/ $4.80 

16 " Lard at 15/ 2.40 
25 " Sugar at 13/ 3.25 
16 " Ham at 16/ 2.56 

Amt. $13.01 

(2.) 
Reading, Pa., Aug, 7, 1877. 

Dr. J. H. Barton, 

Bought of W. W. Rankin & Co. 

8 tons Coal at $5.75 

16 " Coal at 5.25 

4 " Coal at 5.00 

Amt $121.25. 

(3.) 
Lancaster, Pa., June 6, 1876. 
Mr. Samuel Christ, 

Bought o/S, D. Ball. 
6 tons Hay at $18 

16 bushels Rye at 1.25 

27 " Corn at M 

Amt. $145,55. 

(4.) 
Buffalo, N. Y., Jan. 1, 1877. 
Hon. J. W. Smith, 

Bought o/ Abraham Best. 

11 bbl. Flour at $6.75 

32 bu. Wheat at 1.63 

971b. Beef at .09 

17 bu. Corn at .45 

Amt. $142.79. 



FACTORS AND MULTIPLES. 79 

(5.) 

Brooklyn, N. Y., May 1, 1876. 
Mr. M. W. Herr, 

To John A. Bobb, Dr, 

To 27 yd. Muslin at 10/ 
" 16 pr. Shoes at $1.25 
" 25 yd. Carpet at 1.16 
" 27 yd. Silk at 85 

Or. 

By 46 bu. Potatoes at $.75 
" 33 1b. Butter at .25 
" 16doz.Egg8 at .18 

Bal. due, $28.77. 

6. Jan 13, 1877, L. A. Thompson bought of A. C. Har- 
rison of Harrisburg, Pa., 32 lb. sugar, at 13/; 11 lb. 
coffee, at 27/; 26 lb. soap, at 9/; 14 lb. rice, at 9/; 127 
lb. fish, at 13/, and 18 lb. crackers, at 12/. 

Make out a bill. Ans. $29.40. 



CHAPTER 
PEOPEETIES OF KUMBEES. 



SECTION I. 

FACTORS AJfB MULTIPLES. 

53. A Divisor of a number is any number that will 
exactly divide it. 

53. A Factor of a number is one of its exact divisors. 

54. A Prime Number is one that has no factors, and 
therefore no exact divisor. 



80 FACTORS AND MULTIPLES. 

55. A Composite Number is one that may be divided, 
and always is the product of two or more factors. 

2, 3, 5, 7, 11, etc. are prime numbers. 
4, 6, 8, 9, 10, etc. are composite numbers. 

56. A Prime Factor is a factor that cannot be divided. 

67, When any number is a factor of two or more num- 
bers, it is called their common factor. Thus, 2 is a com- 
mon factor of 6 and 8. 

68. An Even Number is one whose right-hand figure 
is 0, 2, 4, 6 or 8. All even numbers are divisible by 2. 

59, An Odd Number is one whose right-hand figure 
is 1, 3, 5, 7 or 9. 

60, Factoring is the process of finding the factors of 
a composite number. 

61, A factor is said to be common to two or more num- 
bers when it is found in each of them. 

ORAL EXERCISE. 

1. What are the factors of 10? 

Solution.— The factors of 10 are 2 and 5, because 10 is divisible 
bv 2 or 5. 

2. What are the factors of 6? Of 8? Of 12? 

3. What are the factors of 12 ? Of 15? Of 20 ? 

4. What are the factors of 18? Of 24? Of 30? 

5. What are the factors of 33? Of 49? Of 50? 

6. What are the prime factors of 30 ? 

Solution. — The prime factors of 30 are 3, 2 and 5, because 3, 
2 and 6 are the only prime numbers by which 30 is divisible. 

7. What are the prime factors of 12 ? Of 8 ? 

8. What are the prime factors of 15? Of 20 ? Of 22? 

9. What are the prime factors of 25 ? Of 27 ? Of 
36? 



FACTORS AND MULTIPLES. 



81 



10. What 
60? 

11. What 
80? 

12. What 
and 9? 

13. What 
and 12? 

14. What 
and 20 ? 16 

15. Name 

16. Name 

17. Name 

18. Name 
1 to 50. 



are the prime fisictors of 33 ? Of 44 ? Of 

are the prime factors of 75 ? Of 100 ? Of 

prime factors are found in 4 and 6 ? In 6 

prime factor is common to 10 and 12? 8 

prime factor is common to 15 and 20 ? 18 

and 20? 18 and 24 ? 30 and 50? 

the prime numbers from 1 to 50. 

the prime numbers from 50 to 100. 

the composite numbers from 1 to 100. 

the factors of the composite numbers from 



WRITTEN EXERCISE. 
Ex. What are the prime &ctors of 360 ? 



Solution. 

3 )360 

3)120 

2)40 

2)20 

2)^0 

5 



Expla-nation. — Dividing 360 by the prime &ctor 3, 
the result is 120; dividing 120 by the prime factor 3, 
the result is 40 ; dividing 40 by the prime &ctor 2, 
the result is 20 ; dividing 20 by the prime factor 2, 
the result is 10; dividing 10 by the prime factor 2, 
the result is 5, which is also prime. Hence, the prime 
divisors or prime factors of 360 are 3, 3, 2, 2, 2, 5. 

Find the prime factors — 
1. Of 30. Am. 2, 3, 5. 



2. Of 60. Ana. 2, 2, 3, 5. 

3. Of 48. Arts, 2, 2, 2, 2, 3. 

4. Of 125. Am. 5, 5, 5. 

5. Of 120. Ans. 3, 2, 2, 2, 5. 

6. Of 144. 

7. Of 175. 

8. Of 180. 



9. Of 420. 

10. Of 270. 

11. Of 475. 

12. Of 800. 

13. Of 1200. 

14. Of 1875. 

15. Of 4620. 

16. Of 1440. 



82 CANCELLATION. 



What prime factors are common — 



17. To 10 and 20 ? 

18. To 12 and 15? 

19. To 18 and 24? 

20. To 30 and 40 ? 

21. To 60 and 70? 



22. To 25 and 125? 

23. To 27 and 81 ? 

24. To 120 and 600? 

25. To 500 and 700? 

26. To 144 and 180? 



CANCELLATION. 

62. Cancellation is the process of shortening opera- 
tions in Division by rejecting or cancelling equal factors 
common to both dividend and divisor. 

Rejecting the same factor from both dividend and 
divisor does not affect the value of the quotient. 

Ex. Divide 4 x 60 by 3 x 8. 

Pbocess. JSrplanatum. — Since dividing both dividend 

1^ and divisor does not alter the value of the quo- 

4 flfl tient, dividing both by 3, 4 and 2 by cancelling, 

^ ^ ^''^ = 10 the quotient is 10. 



^x? 



EXAMPLES FOR PRACTICE. 
3x4x5x6 



1. Find the value of , ^ ^ 

4x3x6 

2. Divide 16x4x5 by 8x2x10. Ans. 2. 

3. Divide 7x16x6 by 14x3x8. Am. 2. 

4. Divide 108 x 10 x 12 by 6 x 9 x 20. Ans, 12. 

5. Divide 9 x 7 x 16 x 16 by 21 x 32 x 2. Ana. 12. 

6. Divide 108 by 27. 

o 108 2x2x3x3x3 . 
SoLunoK. — ^^-^^^ A. 

7. Divide 96 by 32 ; 96 by 24. Ans, 3 ; 4. 

8. Divide 225 by 15; 256 by 32. Ans. 15 ; 8. 

9. Divide 288 by 3 x 48 ; 500 by 5 x 20. Ans. 2 ; 5. 



GREATEST COMMON DIVISOR. 83 

10. Divide 64 x 12 x 28 by 18 x 7 x 4. Ans. 36. 

11. Divide 1 8 x 45 x 280 by 30 x 35 x 24. Am. 9. 

12. Divide 4 x 50 x 12 by 5 x 30. Ans. 16. 

13. How many hens at 30 cents each can be bought 
for 2 bushels of corn at 75 cents a bushel ? Ans, 5. 

14. How many cows at $25 each will cost as much as 
12 horses at *75 each ? Am. 36. 

15. Three pieces of cloth containing 30 yards each, 
worth $5 a yard, were exchanged for 5 pieces of cloth 
containing 45 yards each: what was the second kind 
worth per yard ? Arts. $2. 

16. If a farmer exchange 25 bushels of wheat at $1.20 
a bushel for delaine at 40 cents a yard, how many yards 
does he get ? Ans. 75. 



SECTION II. 

GREATEST COMMOJf DIVISOR. 

63. A Conunoxi Diyisor, or common factor of two or 
more numbers, is a number that will exactly divide each 
of them. Thus, 3 is a common divisor of 6, 9 and 12. 

64. The Greatest Common Divisor of two or more 

numl>ers is the greatest number that will exactly divide 

each of them. Thus, 6 is the greatest common divisor of 

12 and 18. 

ORAL EXERCISE. 

Name a common divisor — 

1. Of 6 and 9. Of 12 and 10. 

2. Of 4 and 6. Of 15 and 18. 

3. Of 16 and 20. Of 18 and 21. 

4. Of 25 and 30. Of 20 and 30. 
6. Of 16 and 40. Of 30 and 50. 



84 GREATEST COMMON DIVISOR. 

Name the greatest common divisor — 

6. Of 6 and 9. Of 12 and 10. 

7. Of 15 and 18. Of 12 and 24. 

8. Of 20 and 15. Of 20 and 30. 

9. Of 25 and 50. Of 25 and 40. 
10. Of 24 and 48. Of 48 and 72. 

PrincipLtE. — The product of the common fectors of 
two or more numbers is the greatest common divisor of 
those numbers. 

WRITTEN PROBLEMS. 
Ex. Find the greatest common divisor of 15, 30 and 60. 

Solution 1. Explanation 1. — By examining the so- 

15 = 3x5 lution we find the only prime factors com^ 

30 = 3x5x2 mon to 15, 30 and 60 are 3and5; hence 

60 = 3x5x2x2 .-, ._. j. . -^r 

the greatest common divisor of these niim- 

3 X 5 = 15 = G. C. D. hers is 3 x 5, or 15. 

Solution 2. ExpUmaiion 2.-5, being contained in 

5)15 30 60 all the numbers, is a common factor; 3, 

S)S 6 — 12 being contained in all the numbers, is also 

,"* — 2^ T a common factor, and since the numbers 

' ' have no other common factors, the greatest 

5 X 3 = 15 = G. C. D. common divisor is 3 x 5, or 16. 

From these solutions we derive the rule for finding the 
greatest common divisor of two or more numbers : 

RULE. 

Find the factors common to all the numbers, and take the 
product of these factors. 

Find the greatest common divisor — 

1. Of 15, 20, 30. Ans. 5. 

2. Of 16, 20, 24. . Ans, 4. 

3. Of 25, 50, 100. Ans. 25. 



LEAST COMMON MULTIPLE. 85 

4. Of 18, 36, 72. Am. 18. 

6. Of 28, 56, 42. Am, 7. 

6. Of 120, 240, 72. Am. 24. 

7. Of 210, 180, 150. Am. 30. 

8. Of 44, 110, 77. Am. 11. 

9. Of 210 miles, 90 miles, 75 miles. Ans. 15 miles. 

10. A man has 2 logs which he wishes to cut into 
boards of equal length ; one is 24 feet, and the other 16 
feet long : how long can he cut the boards ? Am. 8 ft. 

11. What is the greatest common divisor of $27, $36 
and $72 ? Am. $9, 

12. What is the greatest equal lengths into which two 
trees can be cut, one being 105 feet in length and the 
other 84 feet ? Am. 21 feet 



SECTION III. 

LEAST COMMOJf MULTIPLE. 

65. A Multiple of a number is a number which is 
exactly divisible by that number. 

66. A Common Multiple of two or more numbers is 
a number that is exactly divisible by each. Thus, 18 is 
a common multiple of 9 and 6, because it is divisible by 
each of them. 

67. The Least Common Multiple of two or more 

numbers is the least number exactly divisible by each of 
them. Thus, 30 is the least common multiple of 10 and 
15, because it is the least number exactly divisible by 
each of them. 

ORAL EXERCISES. 
What is a multiple — 

1. Of3? Of 4? Of 5? Of 10? Of 6? Of 20? 

8 



86 LEAST COMMON MULTIPLE. 

What is a common multiple — 

2. Of 3 and 4? Of 4 and 5? Of 5 and 10? 

3. Of 6 and 12? Of 5 and 15? Of 5 and 6? 

What is the least common multiple — 

4. Of 2 and 3? Of 3 and 4? Of 4 and 5? 

5. Of 2 and 5? Of 3 and 5 ? Of 5 and 6? 

6. Of 2 and 7 ? Of 3 and 6? Of 4 and 9? 

7. Of 5 and 7? Of 6 and 9 ? Of 10 and 12? 

Principles. — 1. Every multiple of a number contains 
the prime factors of that number. 

2. The least common multiple of two or more numbers 
contains all the prime flEictors of each of the numbers, and 
no other factors. 

WRITTEN PROBLEMS. 
Ex. Find the least common multiple of 10, 15 and 20. 

Solution. ExpUmation. — The least common 

10 = 2 X 5 multiple of the given numbers must 

OA o o ti contain the Actors 2 and 5 to be di- 

■r r^^,.. ^ « ^ «^ visible by 10 ; it must contain the fac- 

L. C.M. = 2x6x3x2=60 . « "^ k * k ^- • -ui u ic •* 

tors 3 and 5 to be divisible by 15 ; it 

must contain the factors 2, 2 and 5 to be divisible by 20. Since 

the number 60 contains all these &ctors and no others, it is the least 

common multiple of 10, 15, 20. 

Prom the foregoing we derive the rule for finding the 
least common multiple of two or more numbers : 

RULE. 

Find the prime factors of the numbers, and take (heprodud 
of these factors, vising each the greatest number of times U 
occurs in any of the given numbers. 



What is the least common multiph 

1. Of 15, 10 and 5 ? An^. 30. 

2. Of 20, 10 and 30? Ans. 60. 



FRACTIONS. 87 

3. Of 9, 12 and 18 ? Am. 36. 

4. Of 2, 3, 4 and 5 ? Am. 60. 

5. Of 3, 6 and 18 ? Am. 18. 

6. Of 10, 25 and 30? An^. 150. 

7. Of 15 and 24? ^tw. 120. 

8. Of 24, 30 and 36 ? Am. 360. 

9. Of $20, $30 and $40? Am. $120. 

10. Of 18 men, 16 men and 12 men ? Ans. 144 men. 

11. What is the smallest tract of land that may be cut 
into 6-acre, 5-acre or 4-acre lots ? Ana. 60 acres. 

12. What is the smallest sum for which I can hire 
workmen at 6, 8 or 9 dollars a week ? Am. $72. 

13. What is the least common multiple of 10, 15, 20, 
26 and 24 ? Am. 600. 



CHAPTER IV. 
FEAOTIOI^S. 



SECTION I. 

DEFIJriTIOJTS. 

If a unit be divided into two equal parts, each of these 
parts is called one-half; if into three equal parts, each part 
is called one-third ; if into four equal parts, each part is 
called onerfourth ; two of the four equal parts are called 
two-fourths, and so on. 

These parts of a unit — one-half, one-third, one-fourth, 
two-fourths, three-fourths, etc. — are called fractiom. 

68. A Fraction is one or more of the equal parts of a 
unit 



88 FRACTIONS. 

69. A fraction is usually expressed by writing the num- 
bers one above the other with a short horizontal line be- 
tween them. Thus, three-fourths is written f ; two-thirds, 
i, etc. 

70. The number below the line is called the Denamir 
nator, and shows into how many equal parts the unit is 
divided. 

71. The number above the line is called the Numerator] 
and shows how many of the equal parts are taken. Thus, 
in ^ the denominator, 5, shows that the unit is divided 
into 5 equal parts ; and 4, the numerator, shows that 4 of 
these parts are taken. 

72. The numerator and the denominator are called the 
Terms of the fraction. 

EXERCISE. 

Read— J, i i, i i, i f , f J, ^. 

Write one-third, two-fourths, three-ninths, six-sevenths, 
five-sevenths, three-eighths, five- ninths, seven-tenths, thir- 
teen-fifteenths, sixteen-fortieths. 

73. A Common Fraction is one in which the nu- 
merator and the denominator are both expressed by 
figures. 

74. Comtnon fractions are either proper or improper. 

75. A Proper Fraction is one whose numerator is less 
than its denominator; as, f, f, etc. 

76. An Improper Fraction is one whose numerator is 
equal to or greater than its denominator ; as, f , f , f, etc. 

77. A Mixed Number is one consisting of a whole 
number and a fraction. 



REDUCTION OF FRACTIONS. 89 

SECTION II. 
BEDUCTIOJf OF FRACTIOJ^S. 

CASE I. 

To Reduce Whole or Mixed Numbers to Fractions. 

ORAL EXERCISE. 

1. How many halves in 1 apple ? How many thirds ? 

2. How many fourths in an orange ? How many fifths ? 

3. How many halves in 3 apples ? 

Solution. — In 1 apple there are 2 halves, and in 3 apples 3 
times 2 halves, or 6 halves. 

4. How many halves in 4? In 5? In 6? In 8? 
6. How many thirds in 2? In 4? In 5 ? In 10? 

6. How many fourths in 3? In 6? In 9? In 12? 

7. How many fourths in 5f ? 

Solution. — In 1 there are J ; in 5 there are 6 times J, or ^ ; 
■^ and f are •^. Hence, in 5| there are -y. 

8. How many fifths in 3 ? In 4 ? In 5f ? 

9. How many sixths in 5 ? In 6 ? In 7| ? 

10. How many eighths in 4? In 10? In 6|? 

11. How many thirds in 7f ? In 8^ ? In 12| ? 

12. How many tenths in 6^^ ? In 5^ ? In 12^5^ ? 

WRITTEN PROBLEMS. 
Ex. Reduce 17^ to an improper firaction. 

Solution. 
171 Explanation. — In 1 there are f , and in 17 there 

J^ are 17 times }, or ^ ; -<^+ J= J^. 

62 thirds = ^. 

From the foregoing we derive the rule for reducing 
mixed numbers to improper fractions : 

8* 



90 REDUCTION OF FRACTIONS. 

RULE. 

Multiply the whole number by the denominator^ and to 
this add the num^eraior, writing the product over the given 
denominator. 



Reduce to improper fractions — 



7. lOf Ans. ^, 

8. 25f . Am. ^^ 



1. 4f. Ans. ^. 

2. 5^. Ans. V. 

3. 6J. Ans. ^. 

4. 7|. Ans. J^. , 10. 27f. Ans. ^^ 

5. Sf Ans. ^.11. 26|f. Ans. ^ 



9. 30f Ans. ifi 



6. 9f. Ans. ^. 1 12. 24Jj. Ans. ^ 



CASE II. 

To Bednce Improper Fractions to Whole or Mixed 

Numbers. 

ORAL EXERCISE. 

1. How many ones in -^ ? 

Solution. — Since one equals |, in -^ there are as many ones as 
I is contained times in J^-, or 3. 

2. How many ones in f ? In f ? In f ? 

3. How many ones in ^ ? In J^ ? In J^ ? 

4. How many ones in -^? In Y? In ^? 

5. How many ones in-^? In^? In^? 

6. Reduce f to a mixed number. 

Solution. — Since one equals |, in } there are as many ones as j 
is contained times in f, which is 2 times and } remaining, or 2}. 

7. Reduce f to a mixed number. 

8. Reduce ^ to a mixed number. 

9. Reduce ^ to a mixed number. 
10. Reduce ^ to a mixed number. 



REDUCTION OF FRACTIONS. 91 

WRITTEN PROBLEMS. 
Ex. Reduce ^ to a mixed number. 

Solution. Explanation, — Since there are } in 1, in ^ there 
8)63 are as many ones as f is contained times in ^, which 
7f is 7, and J remaining, or 7J. 

Prom the foregoing we derive the 

RULE. 

Divide the numerator of ike improper fraction by the 
denominator ; the quotient is the whole or mixed number. 

Reduce to whole or mixed numbers — '• 



1. ^. Ans, 3|. 

2. ^. Ana. 6^. 

3. ff . Ana. 5^. 

4. 4-' ^^' 12|. 
6. If. Ana. 2^. 



6. fj. An^. 3^. 

7. ^. Ana. 9f. 

8. ^. Ana. 13f. 

9. ^. Ana. 8|. 
10. ^. iln«. 10^. 



CASE III. 

To Reduce Fractions to Equiyalent Fractions haying 

Higher Terms. 

ORAL EXERCISE. 

1. How many sixths are there in -J^? 

Solution. — Since there are f in 1, in } there are J of f , or |. 

2. How many fourths in ^? 

3. How many sixths in ^ ? In -J^ ? 

4. How many eighths in ^ ? In J ? 
6. How many tenths in |^ ? In ^ ? 

6. How many twelfths in |? In J? In ^? 

7. How many twelfths in f ? 

Solution. — In ^ there are -fj^ and in f there are 3 times A, or 

A- 

8. How maoy twelfths in f ? In ^ ? 



92 REDUCTION OF FRACTIONS. 

9. How many fifteenths in | ? In | ? In | ? 
10. How many twentieths in | ? In f ? In ^ ? 

pRrNCiPLE. — Multiplying both numerator and denom- 
inator of a fraction by the same number does not change 
the value of the fraction. Thus, ^ equals f , which is the 
same as multiplying ^ by f . 

WRITTEN PROBLEMS. 

1. Reduce f to twentieths. 

Solution. Explanatimi, — To reduce fourths to twentieths it is 
J X A = 14 necessary to multiply the denominator by 5 ; but to 
preserve the value of the fraction the numerator must 
also be multiplied by the same number. Hence, f = i^. 

From the foregoing we derive the rule for reduction of 
fractions to higher terms. 

RULE. 

Multiply both numerator and denominator by that num- 
ber which will produce the required denominator. 

2. Reduce |, |, f to 12ths. Ans, ^, ^, |^. 

3. Reduce ^, ^, \ to 30ths. Ans, ^, ^, ^. 

4. Reduce f , |, ^ to 30ths. . Ans. fj, |^, ^. 

5. Reduce f , i^, f to 36th8. Ans. ff , ^, |^. 

CASE IV. 

To Rednce Fractions to their Lowest Terms. 
ORAL EXERCISE. 

1. How many thirds in |^? 

Solution. — Since J = J, there are as many thirds in f as } is 
contained times in |, or f . Hence, in f there are f . 

2. How many halves in f, ^, f ? 

3. How many thirds in i, i, ^2 



REDUCTION OF FRACTIONS. 93 

4. How many fourths in f , f , t^ ? 

5. How many fifths in -5^, r| , 1^ ? 

6. How many sixths in ^^, -^j, ^ ? 

7. Reduce 3^ to fourths ; ^ to thirds. 

8. Reduce \^ to sixths ; -^^ to fifths. 

9. Reduce ^ to fifths ; \% to thirds. 

10. Reduce ^ to halves ; ^ to fourths. 

11. Reduce ^J, ^ to their lowest terms. 

Note. — When a fraction cannot be reduced to one haying a less 
denominator, it is said to be in its lowest terms. 

12. Reduce ^, |f , ff to their lowest terms. 

13. Reduce ^, ff , |^ to their lowest terms. 

Principle. — The division of both terms of a fraction 
by the same number does not change its value. Thus, if 
both terms of the fraction y\, equal to ^, be divided by 6, 
the quotient is ^. The value is not changed. 

WRITTEN PROBLEMS. 
Ex. Reduce ]^ to its lowest terms. 

Solution. Fxplanation. — Dividing both terms of || J by 12, 

iff = tJ = I i* reduces to |f . Dividing both terms of If by 2, 

Or, it reduces to f . Since 5 and 6 do not have a com- 

}}J-5- J}-= J§ mon divisor, the lowest terms of the fraction |^|J 

H-*-i = * isi 
Prom the solution we derive the following 

RULE. 

To reduce a fraction to its lowest terms, divide both terms 
of the fraction by a common divisor, and this resuU again 
by a common, divisor, and so on till the terms have no com- 
mon divisor. 



94 



REDUCTION OF FRACTIONS. 



Reduc< 

1. 3^ to thirds. Ana, f . 

2. -j^ to fourths. Ana. f . 

3. ^ to fifths. Ana, f . 



4. If to twelfths. Ana, ^. 

5. ff to fifteenths. Ana, -J-f . 

6. f^ to sixths. Ana. ^. 



Reduce to their lowest terms — 



7. If- 
10. 



TjTJ 



Ana, f. 
Ana, f. 
Ana, f. 
Ana. -j^. 



19 _70 

13. Itt- 

14. -^j-. 



Ana, ^. 
Ana, ^. 



CASB V. 

To Rednee Compound Fractions to Simple Ones. 

78. A Compoimd Fraction is a fraction of a frac- 
tion, as ^ of f, or I of 2^. 

ORAL EXERCISE. 

1. What is ^ of i? 

Solution. — Since J equals f , J of J is i of f , or J. 

2. What is i of i? ^of i? 

3. What is I of I? -Jof i? 

4. Whatis Jof I? iof i? 

6. If I have ^ a dollar, and give ^ of it away, how 
much do I give away ? 

6. If a man own ^ of a store, and sell \ of his share, 
how much does he sell ? 

7. A boy had ^ of a dollar, and lost \ of it : what part 
of a dollar did he lose ? 

8. Bought ^ of a farm, and sold -J- of my share: what 
part of the farm did I sell? 

9. How much is f of f ? 

Solution. — J of J is yV » J of f is twice as much, or ^ ; and | 
of I is 2 times -j^, or -j^. Hence, | of f is ■^. 



REDUCTION OF FRACTIONS. 



95 



10. What is I of f? 

11. What is I of I? 

12. What is f of f? 

13. What is i of f? 

14. What is I of I? 



15. What is f of f? 

16. What is I of f ? 

17. Whatisf of I? 

18. Whatisf of I? 

19. What is I of f!^? 



20. If Charles has | of a dollar, and loses f of it, what 
part of a dollar does he lose ? 

21. A boy has f of a pound of candy, and eats f of it : 
how much does he eat ? 

11. If I have f of a bushel of peas, and give away f of 
it, what part of a bushel do I give away ? 



WRITTEN PROBLEMS. 



Ex. What is I of I? 
Solution. 



Whatis- 

1. ioff? 

2. |of ^? 

3. fof I? 



Ans, ^. 
Ans. ^^. 
Ana, ^. 

Reduce to simple fractions — 

7. I of f of i. 

8. iofioff 

9. |of ^of^of |. 

10. I of I of f of I 

11. I of I of A. 

12. f of I of ^. 

13. I of f of i of ^. 



Analysis. 

Jof i = 3x^ff = J^. 

4. fof ^? Ans.^, 

5. -f^ of I? Ans. ^f. 

6. -J of f of f ? Ans, ^, or ^. 



Ans. -J. 

-4 715. "I-. 

Ans. -j^. 

Ans. ^. 
Ans. ■^. 

Ans. -f. 
Ans. 7^. 



96 COMMON DENOMINATOR. 

COMMON DENOMINATOR. 

ORAL EXERCISE. 

1. Reduce f and f to fiftee»ths. 

Solution. — Since there are {^ in 1, in } there are J of |4, 
and in J, twice fV, or fj. In J there are i of |f , or ^, an 
twice T^j, or j%. Hence in } and f there are {^ and ^. 

2. How many twelfths in f and f ? 

3. How many tenths in ^ and ^ ? 

4. How many twentieths in ^, \ and f ? 

5. How many thirtieths in f , f and | ? 

6. Reduce ^ and J to fractions having a comm( 
nominator. 

Note. — When fractions have the same denominator they a 
to have a common denominator. 

7. Reduce f and f to fractions having a commc 
nominator. 

8. Reduce f and f to fractions having a commc 
nominator. 

9. Reduce f , f and f to fractions having a commc 
nominator. 

10. Reduce ^, f and 2 J to fractions having a coi 
denominator. 

WRITTEN PROBLEMS. 

1. Reduce f , f , ^ to fractions having a common d( 
inator. 
Solution 1 . Explanation 1 . — Multiplying any of the dene 

3x2 _ J g tors by the others, we find the common denonc 



8 



X 



^'^Ov^o'"^? 



3x2 ^* to be 24. Multiplying both terms of f by t 

2 ^^^ _ 16 nominators 3 and 2, it equals J| ; multiplyin 

4x2 terms of | by the denominators 4 and 2, it 

1 ^^^ _ 1 2 if j multiplying both terms of J by the dem 

3x4 tors 3 and 4, it equals Jf . 



CX)MMON DENOMINATOR. 97 

Solution 2. Exj^nalion 2. — ^We find the oom- 

Denominator = 4 X 3 X 2 = 24 mon denominator, 24, by multiply- 

} = J J ing together the given denominators 

} = J} 4, 3 and 2, and reduce each fraction 

J = J} to.24th8. 

Prom these solutions we derive the following rule for 
^educing fractions to a common denominator : 

RULE. 

Multiply both terms of each fraction by all the denomina- 
tors except its own. 

Ileduce to fractions having a common denominator — 

2. i, i, i- Ans, H» T^> tV 

3. i I, h Ans. il f^, H- 

4. 1, 1 1. Ans. m> IM, T%. 

5. h f i ^^^ AV tV7> t%. 

6. i A, f ^^«- Hi iM, m- 

7. i i i, i. ^n«. xVir, WV, i^TT, AV 

^' f» fj tV" • Ans. ^f ^, ^-j-J, ffiy. 

79. The Least Common Denominator of two or 

more fractions is the least denominator to which they can 
all be reduced. 

It is also the least common multiple of their denomi- 
nators. 

WRITTEN PROBLEMS. 

Ex. Reduce f , f, -J to equivalent fractions having their 
least common denominator. 
Solution. 

2) 4, 6, 8 Explanation. • — The least common 

2) 2, 3, 4 multiple of 4, 6 and 8 is 24, which is 

1, 3, 2 therefore the least common denomina- 

2x2x3x2 = 24, L. CD. tor. By analysis, i = /:f, and f = 3 x ^^, 

} = H or if; i = A,and J = 5x^ = J|; t = 

« = « A,andi = 7xA=JJ. 

9 Q 



98 ADDITION OF FRACTIONS. 

The following is the rule for reducing fractions to cqi 
alent ones having their least common denominator : 

RULE. 

Find the least common multiple of the denomdnators 
the least common denominator ; divide this common den 
inator by each denominator, and multiply both terms by 
quotient. 

Note. — ^Bedace mixed numbers and compoand fractions to i 
pie ones, and these to their lowest terms, before proceeding to 
the least common denominator. 

Reduce the following fractions to their least denominal 

1. i i |. ^ns. M, «, ^ 

2. I, A» A- ^^- H> M, f^ 

3. 1 h «• ^ns. m> m. m 

^' if T4> ITT" Ans, -f ifj-, -j-gir* Tsi 

6. i of I, 3% of f ^ Ans. il a 



SECTION III. 
ADDITIOJf OF FRACTIOJfS. 

ORAL EXERCISE. 

1. How many are ^, f and |^? 

2. How many are f , ^ and ^ ? 

3. How many are -f, f and ^t 

4. What is the sum of f , f , |^ ? 
6. What is the sum of ^ and \ ? 

Solution. — J equals ^, and J equals -^ ; ^ and ^^ are 
Hence, the sum of \ and J is ^j. 



ADDITION OF FRACTIONS. 



99 



What is the sum — 

6. Of iandi? 

7. Of iandi? 

8. Of land i? 

9. Of i and I? 



10. Of J and }? 

11. Of J and f? 

12. Of i, iandi? 

13. Of i,iandi? 



14. Of 2i and 3J ? . 

Note. — Questions of this kind may also be solved by adding the 
whole nambers and the fractions separately ; thus, 2 and 3 are 5 ; 
i and ^ equal ^ and -fj, or ^. Hence, 2| and 3^ equal 5^^. 



15. Of 2 J and 3i? 

16. Of 3i and 2^? 

17. Of 3|and3i? 

18. Of 2|and3|? 

19. Of 4iand2f? 



20. Of 3f and4f? 

21. Of 4t and 2f ? 

22. Of 6iand5f? 

23. Of 4 and 3f? 

24. Of 7|and6|? 



25. K I pay f of a dollar for a turkey, and i a dollar 
for a goose, what do I pay for both ? 

26. If I pay J of a dollar for butter, f of a dollar for 
eggs, and -^ of a dollar for cheese, how much do I pay for 
aU? 

27. A pair of boots cost 85^, a hat $3f , and a vest $2^ : 
how much did they all cost ? 

80. Addition of Fractions is the process of finding 
the sum of two or more fractions. 

81. Similar fractional units are those having the same 
name. 

Principles. — 1. Only similar fractional units can be 
added. Thus, thirds can be added to thirds, but not to 
fifths or other fractional units. 

2. Fractions to be added must have a common de- 
nominator. 



100 ADDITION OF FBACTION8. 

WRITTEN PROBLEMS. 
Ex. Find the sum of | and |. 

Solution. ^ . . ,«, , . ^, ^ 

,4 Analy8ta,—The oommon denominator of } and 

4 is 20. 

From the foregoing principles and solution we derive 
the following rules for Addition of Fractions : 

RULES. 

1. Reduce the fractions to a common denominator, add 
{he numerators and write the resuU over the common de- 
nominator. 

2. To add mixed numhere, add the whole numbers and 
{he fractions separately, and {hen add the results. 

Note. — 1. Beduce each fraction to its lowest terms before 
reducing to a common denominator. 

2. Beduce fractional results to their lowest teims, and improper 
fractions to whole or mixed numbers. 

Find the sum — 

1. Of land f. Ans. 1^. 
' 2. Of I and f * Ans. 1^. 

3. Of 1 1 f Ans. 2^. 

4. Of f , I, ^. Ans. Iff. 

5. Of 1 1 i • Ans. 2^. 

6. Of j, I, |. Ans. 1^. 

7. Of i, ^, i. Ans. ii- 

8. Of I A, H. ^^' 2H. 

9. Of 3f and 5f 

Pbocess. 



SUBTRACTION OF FRACTIONS. 101 

10. Of 2i and 3^. Ans. 5|. 

11. Of 3f, 6|, 3|. Ana. 14^. 

12. Of 7i 9j\, 2i. Ans, 18^. 

13. Of 6| and 9|. Ans. 16i|. 

14. Of 1^ of ^ and ^ of J. -irw. 3^. 

15. Of 2| and I of f iln«. 2||. 

16. What is the sum of $19 J, $16^, $27|? 

Ans. $63|^. 

17. What is the sum of $23f, $19|, $16|? 

Ans. $60^. 

18. What is the sum of 9| yd., 10} yd., 18| yd. ? 

Ans. 39i yd. 



SECTION IV. 
SVBTRACTIOJf OF FRACTIOJfS. 

ORAL EXERCISE. 

1. What is the difference between |- and f ? 

2. How much is f less J ? ^ less i^ ? 

3. How much is -^ less ^^ ? J less ^ ? 

4. How much is |^ — ^ ? 

Solution. — f equals J, and i equals i; f less f is | ; hence f — 
iis J. 

Subtract — 



5. \ from ^. 

6. -J from f . 

7. f from f . 

8. f from ^. 

9. ^ from f . 



10. f from ^. 

11. f from 1^. 

12. ^ from -J. 

13. ^5^ from f. 

14. I from f . 



15. What is the difference between 2^ and 1^? 

Suggestion. — ^Keduoe the mixed numbers to improper fractions 
before subtracting. 
9* 



102 SUBTRACTION OF FRACTIONS. 

16. Between aj and 2|? 18. Between 6^ and 6f ? 

17. Between 4J and 5|? 19. Between 10^ and 7^? 



What is the value — 

20. Of i+l-J? 

21. Of i+J-i? 

22. Ofi+i-i? 



23. Of |+f-|? 

24. Of l+l-f? 

25. Of 3i+2i-3i? 



26. If a boy has ^ a dollar, and spends \ a dollar, how 
much has he remaining ? 

27. I have 14 dollars, and owe 10^ dollars : if I pay 
what I owe, how much have I remaining ? 

28. John is 7^ years old, and Henry is 9J yr. old ; how 
much older is Henry than John ? 

29. A merchant has 10 yd. of cloth, from which he cuts 
2f yd. : how much remains ? 

30. A farmer sold ^ his corn in the fall, and fed ^ ; he 
has 60 bu. remaining : how many bushels had he ? 

82. Subtraction of Fractions is the process of find- 
ing the difference between two fractions. 

Principles. — 1. Two fractions can be subtracted only 
when their fractional units are similar. 

2. Fractions must have a common denominator before 
their difference can be found. 

Note. — Beduce compound fractions- to simple ones, and each 
fraction, as well as the resulting difference, to its lowest terms. 

WRITTEN PROBLEMS. 

1. Subtract f from |. 

Solution. 
6 _ 4 ExptaTMiion, — | equal fj, and f equal H » H lew 



SUBTRACTION OF FRACTIONS. 



103 



The following is the rule for subtraction of fractions : 



RULE. 



Reduce the fractions to a common denominaior^ subtract 
Ihe numeraiors and vrrite the resuU over the common denom- 
inator. 



How much IS — 








1. i-f? 

2. 1-A? 

3. l-t? 

4. l-f ? 

5. ii-A? 


Ans. ^. 

Ans. 1^. 
Ans.-^. 


6. A-A? 

7. i+i-i? 

8. f+l-i? 

9- H-f ? 
10. A-H ? 


Ans, ■^, 

Ans, f . 

Ans, ^. 

Ans,^, 


11. Prom 6} take 3|. 






Solution 1. 




Analysis. 


6i-3f 


tt,or3A 


6f-¥, 
3*=¥, 




SoLtmoN 2. 








6i-6M 
S*=3A 


Note.- 


—Let the pupil explain. 



12. From 6^ take 3^. 

13. From 6f take 5|. 

14. From 5J^ take 3^. 

15. From 7f take 2^. 

16. From 7^ take 4|; 

17. From 15| take lOf. 

18. From 17^ take 12f. 

19. From 18^ take lOf . 

20. From 27f take 19|. 

21. From 27^ take \^. 

What is the value — 

22. Of }of |-iof f? 



Ans, 3^. 
Ans, 1^, 
Ans, \W, 
Ans, 5^. 
Ans, 2^^. 
Ans, 4r^, 

Ans, 4f . 
Ans, 7^. 
Ans. 8^. 

Ans. 8f . 

Ans, \, 



104 MULTIPLICATION OF FBACTIONB. 

23. Of 6^ - f of 2^ ? Ana. 4^. 

24. Of -I of 7^ - ^ of 2|? Ans. 4. 

25. Of f + | + i-f of li? A718. lU- 

26. A farmer buys sheep at 5\. dollars apiece, and sells 
them at 8 dollars each : how much does he gain ? 

Am. S2|. 

27. If I buy groceries amounting to $3|, and give the 
merchant a twenty-dollar bill, how much change do I get? 

Ans, $16f 

28. From a farm of 120f acres there were sold 19^ 
acres : how many acres remained ? Ans. 100|f acres. 

29. A merchant has one piece of muslin containing 
45f yards, and another of 43^ yards ; he sells 50f yards : 
how much remains ? Ans. 38^ yards. 

30. Having 6| tons of hay, I sold 2| tons : how much 
remains ? * Ans. 3^ tons. 



SECTION V. 
MULTIPLICATIOJf OF FBACTIOJ^S. 

CASE I. 

To Mnltiply a Fraction by an Integer. 
ORAL EXERCISE. 

1. How much will 6 pairs of ducks cost at f of a dol- 
lar a pair. 

Solution. — If 1 pair of ducks cost } of a dollar, 5 pairs will 
cost 5 times } of a dollar, or ^^ dollars, equal to 3} dollars. 

2. At ^ of a dollar each, how much will 6 melons cost? 

3. At f of a dollar each, what must I pay for 10 hens? 

4. If a horse can eat ^ of a bushel of oats in a day, 
how much can 12 horses eat in the same time ? 



MULTIPLICATION OF FRACfTIONS. 105 

5. If a hat cost f of a dollar, how much will 10 hats 
cost at the same rate ? 

How many are — 

6. 3 times i^? 4 times ^ ? 6 times |? 

7. 5 times ^ ? 8 times |? 8 times |? 

8. 6 times ^ ? 4 times f ? 5 times ^ ? 

9. 3 times |? 6 times |? 8 times f ? 
10. 10 times f ? 9 times f ? 10 times ^ ? 

Since 2x-J=^, 3x|^ = |, etc., we have the following 

Principle. — A fraction may be multiplied by multi- 
plying its numerator. 

Since 2x^ = |^, or J, 3x^ = |^, or f, etc., we have the 
following 

Principle. — A fraction may be multiplied by divid* 
ing its denominator. 

WRITTEN PROBLEMS. 
1. Multiply ^ by 4. 

Solution 1. /^ x 4 = }J = f , or If. Let the pupil explain. 

Solution 2. &x^=f, or If. .Ecp/ana<ion.— Since dividing 

^ the denominator of a fraction 

multiplies the value, 4 x ^^ equals |, or If. 

RULES. 

1. To multiply a fraction by an integer, multiply the 
numerator or divide the denominator. 

2. To multiply a mixed number by an integer, multiply 
the whole number and the fraction separatdy, and add the 
resuMa. 



106 IfULTIPLICATION OF FRACTIONS. 



Multiply — 

2. ^ by 4. Ans, ^, 

3. i by 8. Ana. 6f 

4. I by 10. Ana. Sf. 

5. i by 12. Jn«. 9^. 

6. II by 6. ^/w. 5i. 



7. ^ by 9. Ana. 6. 

8. 3^ by 6. Ana. 20. 

9. 8| by 8. ^Tw. 70. 

10. 7f by 12. ^rw. 94. 

11. 18iby25. An8.^56\. 



CASH II. 

To Multiply an Integer by a Fraction. 
ORAL EXERCISE. 

1. If 1 yard of muslin cost 12 cents, how much will ^ 
of a yard cost? 

Solution.— If 1 yard cost 12 cents, J of a yard will cost J of 12 
cents, or 4 cents. 

2. A boy had 25 cents, and lost | of it : how much did 
he lose ? 

3. If a man earn $60 a month, how much will he earn 
in ^ of a month ? 

4. A cow cost $40, and a sheep \ as much : how much 
did the sheep cost ? 

5. A horse cost $120, and a pig ^ as much: how much 
did the pig cost ? 

6! If a ton of hay cost $25, how much will |- of a ton 
cost ? 

Solution.— If 1 ton of hay cost $25, i of a ton will cost J of 
$25, which is $5, and f of a ton will cost 4 times $5, or $20. 

7. John has 30 cents, and James has f as many : how 
many has James ? 

8. A house cost $800, and a barn f as much: how 
much did the barn cost? 

9. f of 14 yards of calico cost 60 cents : what is 1 yard 
worth? 



MULTIPLICATION OF FRACTIONS. 107 

10. If 2 hens cost 60 cents, how much will f of 30 
lens cost? 

11. f of $50 is 8 times the cost of a shawl : what was 
the cost of it ? 

12. f of 15 are how many times ^ of 6 ? 

13. f of 16 are how many times ^ of 20? 

14. I of 24 are how many times ^ of 30 ? 

15. What is i of 6 ? 

Solution.— J of 1 is J, and ^ of 6 is 6 times J, or {. Hence, 
iof 6i8f 

16. What is i^ of 7? iof 6? 

17. What is I of 8? i of 9? 

18. What is I of 12? -j^of 16? 

19. What is } of 9 ? 

Solution.— i of 9 is f, and f of 9 is 3 x f , or ^. Hence, } of 

20. What is I of 10? | of 10? 

21. What is I of 8? f of 8? 

22. What is I of 4? f of 5? 

23. Whatisf of 7? f of 8? 

24. What is I of 4? ^of 16? 

25. What is f of $26 ? | of 17 miles ? 

Principle. — The product of an integer by a fraction 
equals the fraction of the integer. 

WRITTEN PROBLEMS. 
1. Multiply 45 by f . 

Solution. 
45 X f Explanation, — 46 times f is the same as 

« i X 45 = i|^, or 33} } of 45, which is i}^, or 33}. 



108 



MULTIPLICATION OP FRACTIONS. 



5. 29 by f 

6. 54 by f . 

7. 75 by |. 



Ans, 12f. 

Am. 20^. 

Am. 25 



Multiply — . 

2. 27 by f Am. 18. 

3. 36 by |. Am. l^. 

4. 60 by |. Am. 22|. 

8. Multiply 65 by 4f. 

Solution. 
65 

4f £lrptona^tan. — 4 times 65 are 260, and f of 65 are 

260 i|ft, or 43J. Adding the two results, we have 303J. 

303J 

The following are the rules for the multiplication of in- 
tegers by fractions : 

RULES. 

1. Multiply the integer by the numerator, and divide the 
residt by the denominator. Or, Divide the integer by the 
denominator, and multiply the result by the numerator. 

2. When the multiplier is a mixed numher, muMiply the 
integer and thefra^dion separately, and add the resuits. 



Multiply — 

9. 25 by 3|. 

10. 20 by 6f 

11. 35 by lOf. 

12. 48 by 8f . 



Am. 93f . 

Am. 124. 
Am. 376^. 
Am. 41 2|. 



13. 60 by 15f . Am. 925^. 

14. 12 by I off Am. 7^. 

15. 42hjiof7l. Am. leS^. 

16. 125 by 18|. Am. 2348|. 



CASE III. 

To Multiply a Fraction by a Fraction. 

ORAL EXERCISE. 
1. Whatisf of I? 

Solution.— J of J is jiy ; J of | is 2 times i^, or A ; and } of f 
is 3 times ^, or ^y equal to J. Hence, f of } is i. 



MULTIPLICATION OF FRACTIONS. 109 



2. What is f of I? 

3. What is J of I? 

4. What is f of f? 



5. What is I of I? 

6. What is I of 3^? 

7. What is I of 4 ? 



8. How much will f of a bushel of potatoes cost at ^ 
of a dollar a bushel ? 

9. What is the cost of f of a peck of beans at ^ of a 
dollar a peck ? 

10.. What cost IJ doz. of eggs at 12^ cts. a dozen? 

11. A hat cost ^ of f of 16^ dollars : how much did it 
cost? 

12. If 4 pairs of shoes cost $9, how much will 2 pairs 
cost? 

Principle. — The product of a fraction by a fraction 
equals the fraction of that fraction. 

Remark, — The process of multiplying fractions may be 
shortened by cancellation. 

WRITTEN PROBLEMS. 

1. What is the product of f of 5^ by ^? 

Solution 1. Explanation. — Since 6 J equals ^, ^ of 

i of 5}xf 5Jxf equals f of ^xf, which is fjf, 

« Jx^xf = fjf, or IJ equal to J/, or If 

Solution 2. 
fofSJxf 

= jx^xJ = -V,orlf 

2 

The following is the rule for multiplying fitictions by 
fractions : 

RULE. 

Multiply the numerators together for the numerator, and 
the denominators for the denominator , of the product. 

Note 1. — ^Reduce mixed numbers first ,to improp)er fractions. 

2. Cancel common factors in the numerator and the denominator. 

10 



110 DIVISION OF FRACTIONS. 

Remark, — This case is practically the same as finding 
the fractional part of a fraction. 

Multiply — 



2. A by f . Ans, H- 

3. i by f Ans. |f 

4. li by |. Ans. f 

5. I of f by f ^n«. J|. 



6- it by f of 4. Ana. ^. 

7. f of ^ by I of ^. ^W5. j*^. 

8. I of 7 by 2^. Jlns. 13|. 

9. 8ibyiof7i. Ans. 2SU' 



SECTION VI. 
DiriSIOJf OF FRACTIOJ^S. 

CASE I. 

To Diyide a Fraction by an Integer. 
ORAL EXERCISE. 

1. K 3 ducks cost 1^ of a dollar, how much will 1 duck 

cost ? 

Solution. — If 3 ducks cost ^ of a dollar, 1 duck, which is J of 
3 ducks, will cost J of ^ of a dollar, which is ^, or ^ of a dollar. 

2. If 3 caps cost ^ of a dollar, how much will 1 cap 
cost ? 

3. If 5 melons cost -j!^ of a dollar, how much will 1 

melon cost ? 

Solution. — If 5 melons cost ^^ of a dollar, 1 melon will cost \ 
of ^ of a dollar, which is ^ of a dollar. 

4. How much will 1 boy earn in a day if 3 boys earn 
J of a dollar ? 

5. If 6 men can do f of a piece of work, what part of 
the work can 1 man do ? 

6. If 6 cows eat f of a ton of hay in a certain time, 
how much will 1 cow' eat in the same time ? 



DIVISION OF FRACTIONS. Ill 

7. K f of a farm be divided into 8 equal parts, hoW 
much will each part be ? 

8. 5 persons own f of a ship : what is each one's share ? 

9. 6 boys pick 10^ quarts of berries : how much does 
each boy pick ? 

10. What is i of 5i? i of 3t^? "f of 10^^? 

Since ^ divided by 2 is f , f divided by 3 is ^, etc., we 
have the following 

. Principle. — A fraction may be divided by dividing its 
numerator. 

Since ^ divided by 2 is J, ^ divided by 4 is ^, etc., we 
have the following 

Principle. — A fraction may be divided by multiplying 
its denominator. 

WRITTEN PROBLEMS. 

1. Divide H by 3. 

Solution. Explanation. — J} divided by 1 is ^; hence, JJ di- 
ll -«- 3 = ^ vided by 3 equals J of ||, or f^. 

2. Divide \i by 6. 

Solution. ExptancUion,- — }J divided by 1 equals |J; hence, 

H"*"6 ii divided by 6 equals J of |J, or ^, Therefore, 

=ixH = H H divided by 6 is H. 



Divide — 

4. 1^ by 9. Ans. f^. 

6. f by 6. Am. ^. 



7. A by 7. Ans. ij%. 

8. II by 10. Ans. ^. 

9. d^ by 5. Ans. -J. 
10. 7i by 7. ^n«. |f . 



112 DIVISION OF FBACTIONS. 

CASE II. 

To Diyide an Integer by a Fraction. 

ORAL EXERCISE. 

1. At f of a dollar apiece, how many turkeys can I buy 
for 3 dollars ? 

Solution. — If 1 turkey cost f of a dollar, for 3 dollars, or ^^ 
dollars, as many turkeys can be bought as } is contained times in 
^^y or 4 turkeys. Hence, if 1 turkey cost f of a dollar, for 3 dol- 
lars 4 turkeys can be bought. 

2. At f of a dollar each, how many hats can I buy for 
6 dollars? 

3. If 1 hen cost J of a dollar, how many hens can I 
buy for 2 dollars ? 

4. How many baskets holding } of a peck each will it 
take to hold 6 pk. ? 

5. If 1 doz. of fish cost ^ of a dollar, how many dozen 
can I buy for 3 dollars ? 

6. If f of a dollar buy 1 yd. of cloth, how many yards 
can I buy for 9 dollars ? 

7. How many pairs of gloves at f of a dollar a pair can 
I buy for 4 dollars ? 

8. How many times is ^ contained in 2 ? In 3 ? In 4 ? 

9. How many times is f contained in 2 ? In 4 ? In 6 ? 

10. How many times is f contained in 6 ? In 9 ? In 
12? 

WRITTEN PROBLEMS. 

1. Divide 9 by |. 

Solution. Explanation. — Since 9 divided by 1 equals 9. 9 

9 -J- f = divided by \ equals 4x9, and divided by } it equals 

9x J = -V- = 12 J of 4x9, or } of 9, which is equal to V, or 12. 

The following is the rule for dividing an integer by a 
Auction : 



DIVISION OF FRACTIONS. 113 

RULE. 

Multiply the integer hy the denominator^ and divide the 
remit by the numerator of the given fraction. Or, MvMiply 
the integer by the divisor inverted. 

Divide — 



2. 12 by f , 


Ars. 16. 


7. 30 by |. 


Ans. 34if. 


3. 15 by f 


Ans. 18. 


8. 60 by \\. 


Ans. 108. 


4. 20 by |. 


An8. 25. 


9. 40 by 3i 


Ans. llf. 


6. 18 by f 


Ana. 21. 


10. 55 by 4- 


Ans. 10. 


6. 25 by |. 


Ans. 41f . 


11. 75 by 6^. 


Ans. 11t^. 



CASE III. 

To Diyide a Fraction by a Fraction. 

ORAL EXERCISE. 

1. At "3^ of a dollar each, how many caps can I buy 

for f of a dollar ? 

SOH.UTI0N. — ^If 1 cap cost ^j of a dollar, for ^ of a dollar, which 
is equal to -^ of a dollar, I can buy as many caps as -]% is contained 
timefl in ^, or 4 caps. Hence, at -^ of a dollar each, I can buy 4 
caps for f of a dollar. 

2. How many pens at J of a cent each can I get for 
2 J cents? 

3. How many slates at ^ of a dollar each can I get for 

4.^ How many oranges at f of a dime apiece can I buy 
for 4^ dimes ? 

6 If silk cost f of a dollar a yard, how many yards 
can I buy for $5^ ? 

6. If 6 hens cost one dollar and a half, how many can 
I buy for $^ ? 

7. Three hats are sold for $4^: how many can be 
bought for $13^? 

10* H 



114 



DIVISION OF FRACTIONS. 



8. Divide j by ^. 

Solution. — Since 1 is contained in |, j times, ^j^ is contained in 
f 10 times | times, or ^ times ; and ^^ is contained in f , J of ^^ 
times, or ^ times, equal to 8. Hence, | divided by ^j^ equals 8. 



9. Divide f by ^, 

10. Divide | by ^ 

11. Divide ^ by f . 

12. Divide ^ by f. 

13. Divide f by |. 

14. Divide ^ by ^j. 



15. Divide J/ by f . 

16. Divide 2f by ^V- 

17. Divide 2| by f . 

18. Divide 9^ by i. 

19. Divide 9^ by f . 

20. Divide 9^ by 2^. 



WRITTEN PROBLEMS. 
1. Divide f by |. 

Solution. Analysis. — 1 is contained in |, | times ; \ is 

f -«- J contained in }, 6 x } times, and J is contained 

*=ixf = i4) or A i o^ 6xf times, or f x}, equal to JJ, or t^. 

Hence, } divided by f is ^^j. 

The following are the rules for the division of fractions 
by fractions: 

RULES. 

1. Divide the dividend by the numerator of the divisor ^ 
and multiply by its denominator, 

2. Invert the terms of the divisor, and multiply the 7ii»- 
merators together, also the denominators, 

3. Reduce the fractions to a common denominator, and 
divide the numerator of the dividend by the numerator of 
the divisor. 

Divide — 
2. I by |. 



3. |by I 

4. I by f 

5. i by f . 

6. I by f . 

7. A by i 



Ans, \^, 
Ans, 1\, 
Ans,\i. 
Ans, 1^. 
Ans. "I^. 
Ans, 1. 



8. ^ by \, 

9. 3f by f 

10. ^ by |. 

11. 4 by 2^. 

12. 6i by 34-. 



Ans, 10^. 

-4rw. 9f . 

Ans. 54-. 

-4n«. IJ. 
Ans, l-IJ. 



13. iofl7iby3i.^7w.2A, 



REVIEW. 



115 



CASE IV. 

Complex Fractions. 

Note. — Complex Fractions may be regarded hb being one 
form of Division of Fractums, The number or fraction in the nu- 
merator may be regarded as the dividend, and the number or frac- 
tion in the denominator as the divisor. Thus, in the fraction 

p f is to be divided by f . 

WRITTEN PROBLEMS. 
1. Seduce ^ to its simplest form. 

Solution. 

f _ « . Explanation. — | divided by J equals } rnul- 

i tiplied by }, which equals If, or J. 

Reduce the following fractions to th^ir simplest forms: 



2.1. 



f 

5 i 



Ans, \i. 
Ans. f . 
Ans, 1^. 
Ans. \\, 
Ans. "5^. 



I 

^■7f 



Ana. ^. 
Am. 1&|. 



Ana. ■^. 



10.^-^. An8.m- 



11 l±ii 



xL7l9» vLf X« 



REVIEW. 

WRITTEN PROBLEMS. 

1. Reduce 21^ to an improper fraction. 

2. Reduce 87-j^ to an improper fraction. 

3. Reduce ^^ to a mixed number. 



116 REVIEW. 

4. Reduce ^^^ to a mixed number. 

5. Reduce ^ of J of ^ of ^ to a simple fraction. 

6. Reduce -J^ of -^ of 6f to a simple fraction. 

7. Reduce f , f , f, -j^ to their least common denomi- 
nator. 

8. Reduce f , f , |f to their least common denominator. 

9. Add 4i, 3^ and 6^. Am. 143^. 

10. Add i of f , 3| and I of 6^. Am. 7|. 

11. From 44| take 16f Am. 27|. 

12. From 18J+16| take i of 27. Am. 20^. 

13. Multiply 12i by 12^. Am. 156J. 

14. Multiply I of 18f by | of 2^. Am. 231|. 

15. Divide 18| by ^. Am. 2|f 

16. Divide f of 16| by | of 16. Am. ||. 

T of * 

17. Reduce ^ | ^ to a simple fraction. Ana. 1^. 

18. Reduce ^ J. ^ \i to a simple fraction. Ans. fj J. 

19. A man earns 3| dollars one day, 6;^ dollars the 
next, and 4^ the next : how much does he earn in three 
days? Am. $14^. 

20. A boy has 6| dozen hens, 3J dozen ducks and 1^ 
dozen turkeys : how many fowls has he? Ans. 11^ doz. 

21. A farmer has 26f acres of woodland, and sells 16 J 
acres : how much has he remaining? Ans. 10^ acres. 

22. If a man have 27f dollars, and earn 9^ dollars 
more, how much will he have if he lose 16 J dollars? 

Am. 20^^ dollars. 

23. One hat costs $3f : how much will 14 hats cost at 
the same rate? Am. $52|-. 

24. George earned $17^ one week, and 4 times as much 
during the month, less $3| : how much did he earn in the 
month? Am. $65^. 



REVIEW. 117 

25. Twenty-seven carpenters earn in one day $87| : how 
inuch does each earn in a day ? Ana. <^3 J. 

26. If a store is worth $26246f , and is bought by a 
company of 16 men, how much does each one pay? 

Ana, $1640f 

27. A man owning a 40-acre &rm planted ^ in potatoes, 
f in wheat and the remainder in corn : how many acres 
did he plant in each ? 

Ana, 8 A. in potatoes, 26f A. in wheat, 5^ A. in corn. 

28. A farmer owned 127-J- A. ; he sold 16 J A. to one 
man and 45f to another : how much has he remaining ? 

Ana, 65|^ A. 

29. If 15 bu. of wheat cost $18J, how much will 16 bu. 
cost? Ana. $20. 

30. A boy who is 14 years old is /^ as old as his father 
and -^ as old as his grandmother : how old is each ? 

Ana. Father, 40 years ; grandmother, 60 years. 

31. If a man sleep 7^ hr. a day, how many hours does 
he sleep in 365 d., or a year? Ana. 2615| hr. 

32. A man bought a barrel of sugar ; he sold \, \ and 
J of it : what part of the barrel remains ? Ana. JJ. 

33. What is the cost of 13^ lb. of meat at 15 cts. a 
pound ? Ana. $1.96f 

34. A man earns $2| a day, and spends $5^ a week for 
board : how much does he save in a week ? Ana. $11. 

35. A farmer sells f of his land, and finds that 12^ A 
is ^ of the remainder : how much had he at first? 

Ana. 93f A. 

36. What is the value of a turkey weighing 14f lb., at 
12J ctB. a pound ? Ana. U .79^. 



118 NUHEBATIOM OF DECIMAL FRACTIONS. 

CHAPTER V. 
DECIMAL FEACTIONS. 

If a unit be divided into ten equal parts, each of the 
parts is called a tenth. 

If each tenth be divided into ten equal parts, each of 
the parts is called a hundredth. 

In the same manner, each of the ten equal parts of a 
hundredth is called a thousandth, 

83. A Decimal Fraction is a number of tenths, hun- 
dredths, ilwusandthsj etc. — ^that is, a number of the decimal 
divisions of a unit 



SECTION I. 

JfOTATIOJf AJfD JfUMERATIOJT. 

84. A decimal fraction is usually written without its 
denominator ; when so written, a point called the deemed 
point or separatrix is placed before the numerator. Thus, 
•^ and ^[%, when written decimally, are written .3 and .27. 

85. When decimal fractions are written without their 
denominator, they are called DecmwJs, 

86. The divisions, beginning at the decimal point and 
reading to the right, are named as follows : tenths, hun- 
dredths, thousandths, ten-thousandths, hundred-thousandths, 
miUionths, etc. 

The following table shows the relative place of decimals 
and whole numbers : 



NUMERATION OF DECIMAL FRACTIONS. 119 



00 






4444 4 4. 44.44 444 444 





s 


a 


.2 






1 





V Y _^— — 

Integers. Decimals. 

The decimal .175 is read thus: 1 tenth, 7 hundredths 
and 5 thousandths, or 175 thousandths. 

87. Decimals are read by naming the numerator as 
given, and the denominator, which is 1, with as many 
ciphers annexed as there are figures in the given nume- 
rator. 



Bead the following : 








.8 .64 


.125 


.3142 


6.23 


.6 .72 


.107 


.0641 


7.8 


.4 .83 


.604 


.0037 


19.303 


.3 .27 


.718 


.0005 


6.006 



Note. — When the fraction is expressed decimally, the units* 
figure of the numerator is placed in the order indicated by the 
decimal. Thus, 'in jiUjf the 9 is placed in the ten-thousandths' 
place, and the other figures in their proper connection with it. It 
is written .0129, a cipher being used to fill the vacant space. 



^^ Q^ 



Write the 


following decimally : 


A 


100 


A 


10 


A 


i8o 



Tinnr 

■nmr loooo 

lOOO TOOOO 



120 



NUMERATION OF DECIMAL FRACTIONS. 



142 thousandths. 

165 thousandths. 

63 hundredths. 

4 hundredths. 
107 thousandths. 
707 ten-thousandths. 

15 ten-thousandths. 

5 tenths. 

12 thousandths. 
104 thousandths. 



16 hundredths. 

19 millionths. 

14 thousandths. 
106 ten-thousandths. 
6 hundred-thousandths. 

76 hundredths. 
1 thousandth. 

32 millionths. 

16 ten-niillionths. 
106 hundred-thousandths. 



88. An integer and a fraction may be written as one 
number. Thus, 17^^ ^s written decimally 17.003. 



Write the following : 

9 and 7 tenths. 
16 and 3 hundredths. 

14 and 6 thousandths. 
16 and 18 hundredths. 
18 and 9 tenths. 

15 and 18 thousandths. 



603 and 19 ten-thousandths. 
721 and 17 thousandths. 
6 and 6 ten-thousandths. 
3 and 13 millionths. 
13 and 3 millionths. 
16 and 16 ten- thousandths. 



Principles. — 1. The denominator of a decimal frac- 
tion is 1, with as many ciphers annexed as there are 
figures in the numerator. 

2. Moving the decimal point to the right increases the 
decimal tenfold for every removal of one space. 

3. Moving the decimal point to the left decreases the 
value of the decimal tenfold for every removal of one 
space. 

4. Annexing a cipher to the right of a decimal, or re- 
moving a cipher from the right of a decimal, does not 
alter the value, as it simply multiplies or divides the nu- 
merator and the denominator by the same number. 



REDUCTION OF DECIMAL FRACTIONS. 121 

SECTION II. 
BEDUCTIOJ^ OF DECIMALS. 

CASE I. 

Redaction to Higher or Lower Terms. 
ORAL EXERCISE. 

1. How many tenths in 3 ones ? In 8 ones ? 

2. How many hundreds in 6 tenths ? In 9 tenths ? 

3. How many thousandths in 8 tenths? In 6 hun- 
dredths? In .7? 

4. How many tenths m .60 ? In .500 ? 

WRITTEN PROBLEMS. 

1. Reduce .16 to thousandths. 

Process. Explanation, — Since there are 10 thousandths in 1 
.16 = .160 hundredth, in 16 hundredths there are 16 times 10 
thousandths, or 160 thousandths. 

2. Reduce .13 to thousandths. 

3. Reduce .6 to thousandths. 

4. Reduce .17 to hundred-thousandths. 
6. Reduce .14 to thousandths. 

6. Reduce .600 to tenths. 

7. Reduce .0700 to hundredths. 

8. Reduce .6000 to tenths. 

CASE II. 

Reduction of Decimals to Common Fraettons. 

ORAL EXERCISE. 

1. In .4 how many fifths? 

2. How many fourths in t^ ? In .75 ? In .500 ? 

3. How many fifths in .20 ? In .60 ? In .0500 ? 

4. How many twentieths in .5 ? In .40 ? In .3000 ? 

11 



122 



REDUCTION OF DECIMAL FRACTIONS. 



Let the pupil explain. 



WRITTEN PROBLEMS. 
1. Seduce .16 to a common fraction. 
Process. 

.16-AS=A 

Note. — In reducing a decimal to a common fraction, omit the 
decimal point and write the denominator. 

Reduce the following decimals to common fractions : 



2. 


.4. 


Ans. ^. 


9. 


.125. 


Ans. ^. 


3. 


.15. 


Atu. ■^. 


10. 


.625. 


Ans. ^. 


4. 


.25. 


Ans. \. 


11. 


.0625. 


Ans. ■^. 


5. 


.60. 


Ans. f . 


12. 


4.0375. 


Ans. 4^. 


6. 


.08. 


Ans.-^. 


13. 


4.375. 


Ans. 4f . 


7. 


.09. 


Ans.T^. 


14. 


.035. 


Ans. 7^. 


8. 


.36. 


Ans. ^. 


15. 


.1875. 


Ans. ^. 



CASE III. 

Redaction of Common Fractions to Decimals. 

ORAL EXERCISE. 

1. How many tenths in ^ ; J ; f ? 

2. How many hundredths in J; -^i f ? 

3. How many thousandths in ^ ; ^ ; ^ ; -^ ? 

WRITTEN PROBLEMS. 
1. Reduce ^^ to a decimal. 

Process. 
40)5.000 1. 125 

42_ Explanation, — ^ equals -^ of 6; 6 equals fj, or 

^^Q m or M»; A of mi is ,VW, or .126. 

"200 
200 

Note. — To reduce common fractions to decimals, annex ciphers 
to the numerator and divide hj the denominator, and point off as 
many places in the quotient as there are ciphers annexed. 



ADDITION OF DECIMAL FRACTIONS. 



123 



Reduce the following to decimals : 



2.1. 


i47w. .5. 


12. |. 


uiw«. .625. 


3. i. 


Am. .25. 


13. 6J. 


uitw. 6.25. 


4. i. 


^rw. .125. 


14.1^. 


Ana. .26|. 


5. f 


-4.IW. .4. 


15. 18f. 


^?w. 18.75. 


6. |. 


uiiM. .75. 


16. 66J. 


Ans. 66.25. 


7. A. 


-Atw. .15. 


17. 12J 


-4n«. 12.5. 


8- A- 


Ans. .175. 


18. 16|. 


uirw. 16.6. 


9- A- 


^7W. .075. 


19. A. 


Am. .3125. 


10. i*. 


^7ia. .2. 


20. 7|. 


uirw. 7.6f. 


11- A 


-4rw. .4375. 


21. 8A. 


Am. 8.09375. 



Notes. — 1. When the denominator of the fraction contains other 
prime factors than 2 or 5, the division will not terminate. 

2. When a sufficient number of decimal places is obtained, the 
remainder may be expressed by a common fraction. 



SECTION III. 
ADDITIOJf OF DECIMALS. 

WRITTEN PROBLEMS. 

1. Add 3.7, 18.24, 163.017 and 614.16. 

Process. 

3.7 

18.24 

163.017 

614.16 

799.117 



ExplancUion. — The . decimals are so written that 
units of the same order stand in the same perpendic- 
ular column, and the addition is then performed the 
same as in simple numbers. 



RULE. 

Write the numbers so that the decimal points shall stand 
in the same perpendieula/r column, and add as in simple 
rmmbers. 



2. Add 16.16, 18.17, 9.16 and 20.13. Am. 63.61. 



124 SUBTRACTION OF DECIMAL FRACTIONS. 

3. Add 15.04, 7.7, 18.05 and 16.43. Ans, 57.22. 

4. Add 3.64, .75, 127.5 and 27.003. Am, 158.893. 

5. Add .718, 72.5, 5.316, 7.29. Ans, 85.824. 

6. Add 18.43, 17.03, 10.894, 1.707. Aiis. 48.061. 

7. Add 33.624, .006, 600, 800.08. Atu, 1433.71. 

8. Add 21.05, 26.005, 18.0005, 20.5. Am, 85.5555. 

9. Add 14.003, 1.27*5, 1.324, 16.5. Am. 33.102. 
10. Add 80.07, 645.3, 3.003, 64.00016. 

Am. 792.37316. 



SECTION IV. 
SUBTRACTIOJ^ OF DECIMALS. 

WRITTEN PROBLEMS. 
1. From 65.13 take 18.344. 

Process. Exphnation, — Since annexing a cipher to the right 

65.130 of a decimal does not alter its value, 65.13 may be 
}^'^^^ written 65.130. Writing the number so that the deci- 
46.786 mal points are immediately above each other, the sub- 
traction is performed as in simple numbers. 

RULE. 

Write the numbers one above the other, so that the decimal 
point of the minuend shall be immediately above the decimal 
point in the subtrahend^ and subtract as in simple numbers. 

Find the value — 

2. Of 87.15 - 63.24. Am. 23.91. 

3. Of 119.3 - 65.784. Am. 53.516. 

4. Of 107.07 - 6.45. Am. 100.62. 

5. Of 615.34 - 7.183. Am. 608.157. 

6. Of 7.004-4.7. ^iw. 2.304. 

7. Of 14 -.14. Am. 13.86. 

8. Of 3.3. - .033. Am. 3.267. 



MULTIPLICATION OF DECIMAL FRACTIONS. 125 

9. Of 21.5-16i. Am, 5.25, 

le. Of 307f - 194.785. Ana, 112.965. 

11. Of 12^-6|. Am. 5.75, 

12. From 9 and 7 tenths take 3 and 45 hundredths. 

Ana, 6.25. 

13. From 8 and 3 hundredths take 83 millionths. 

Am, 8.029917. 



SECTION V. 
MULTIPLICATlOJ\r OF DECIMALS. 

WRITTEN PROBLEMS. 

1. Multiply 3.12 by 5.25. 

Process. 
3.12 

• Explanation. — 3.12, or j^J, multiplied by 6.25, or 

llf H-h equals W^m or 16.38. 
1560 



16.3800 



From the foregoing we have the rule for multiplication 
of decimals: 

RULE. 

Multiply dedmala as aimple numhera, and point off from 
the right aa many decimal placea aa there are in the multi- 
plier and the multiplicand. 

Note. —If the number of figures in the product is less than the 
number in the two factors, prefix as many ciphers as may be neces- 
sary to make the number of decimal places in the product equal the 
number in both factors. 

2. Multiply 16.14 by .6. Am, 9.684. 

3. Multiply 17.21 by .15. Am, 2.5815. 

4. Multiply 27.48 by .07. Am. 1.9236. 

11* 



126 DIVISION OF DECIMAL FRACTIONS. 

5. Multiply 18.23 by 1.17. Ans. 21.3291. 

6. Multiply 27.007 by 1.006. ^jw. 27.169042. 

7. Multiply 20.02 by 4.8. Ans, 96.096. 

8. Multiply 372.006 by 4.09. Ans. 1521.50454. 

9. Multiply 264.0078 by 5.19. Ans. 1370.200482. 

10. Multiply 8.0008 by 6 tenths and 7 thousandths. 

Ans. 4.8564856. 

11. Multiply 96 and 374 thousandths by 81.18. 

Ans. 7823.64132. 



SECTION VI. 
DIVISIOJT OF DECIMALS. 

WRITTEN PROBLEMS. 

1. Divide 29.295 by 2.17. 

Process. 

2.17)29.295113.6 

217 Etplanation.— 29.29b is equal to ^^^; 2.17 

759 is equal to m ; ^^%^ divided by -f^ equals 

^- W, or 13.6. 

1085 

1086 

Note. — Since the quotient multiplied by the divisor equals the 
dividend, it is evident, according to the preceding section, that the 
number of decimal places in the divisor plus the number in the 
quotient equals the number in the dividend ; hence, the number in 
the dividend less the number in the divisor equals the number in 
the quotient. Hence the following rule for division of decimals : 

RULE. 
Divide as in simple numbers^ and point off as many de- 
cimal places in the quotient as the number of decimal places 
in the dividend exceeds the number of those in the divisor. 

Note. — If the number of decimal places in the divisor exceeds 
the number in the dividend, first annex enough ciphers to the divi- 



DIVISION OF DECIMAL FRACTIONS. 127 

dend to make the number of places equal to the number in the 
divisor. 

2. Divide 123.39 by 4.5. Am. 27.42. 

3. Divide 77.935 by 1.09. Arts. 71.5. 

4. Divide 205.2608 by 24.32. Am. 8.44. 
6. Divide 2.7306 by 15.17. Ana. .18. 

6. Divide 16644.3728 by 23.08. Am. 721.16. 

7. Divide 6.54 by 26.16. Am. .25. 

8. Divide 13.68 by 18.24. Am. .75. 

9. Divide 164.835 by 20.25. Am. 8.14. 
10. Divide .0121344 by .016. Am. .7584. 

REVIEW PROBLEMS. 

1. A horse cost $125.5, a cow $15.75 and some sheep 
$124.63 : how much did they all cost? Am. $265.88. 

2. A farm consists of 23.385 A. of meadow, 16.315 A. 
of woodland, and 45.3 A. of tillable land : how many acres 
are in the farm? Am. 85 A. 

3. If a man have 27.75 dollars, and spend 18.125 dol- 
lars, how much money has he remaining ? 

Ana. 9.625 dollars. 

4. If a house cost $1162.5, a store $3146.1875, and 
both were sold for $4400, how much was gained by the 
sale? Jns. $91.3125. 

5. A merchant has 650.5 lb. of sugar ; he buys 425 J lb. 
more, and sells 824.125 lb. : how much remains? 

Am. 251.625 lb. 

6. If a man travel 3.125 mi. an hour, how far will he 
travel in 6.2 hr. ? Am. 19.375 mi. 

7. How much will 7.1251b. of meat cost at 12.5 cts. a 
pound ? Am. 89.0625 cts. 

8. What is the cost of 93.5 lb. of lard at 12.5 cts. a 
pound? ^w«. $11.6875. 



128 DENOMINATE NUMBERS. 

9. What is the cost of 16^ A. of land at $125.5 an 
acre? Ana. $2012.70625. 

10. If a man travel at the rate of 31.15 mi. a day, how 
far will he travel in 14.375 days? 

Am. 447.78125 mi. 

11. If a man travel at the rate of 31.5 mi. a day, in 
how many days can he travel 204.75 mi. ? Atis. 6.5 d. 

12. If 18.751b. of meat cost $2.34375, how much wiU 
1 lb. cost. ? Ans. 12.5 cts. 

13. K I trade 15^ lb. of butter at 37.5 cts. a pound for 
coffee at 18.75 cts. a pound : how many pounds of coffee 
do I get? Ans. 31 lb. 

14. A man bought a farm of 45.375 A. for $7260 : how 
much did it cost him per acre ? Ans. $160. 

15. Exchanged 3.5 T. of coal at 84.80 a ton for lath at 
30 cts. a hundred : how many lath did I get ? 

Ans. 5600. 



CHAPTER VI. 
DEE"OMIE"ATE NUMBEES. 



SECTION I. 

BUFIJVITIOJfS. 

89. A Concrete Number is one which is applied to a 
particular unit ; as, 3 sheep, 2 pounds, etc. 

90. A DenominBrte Number is one whose unit is 
named ; as, 6 gallons, 3 inches, etc. 

Denominate numbers are always concrete. 



TABLES AND MEASURES. 129 

91. Numbers are of the same denomination when they 
have the same unit ; as, 2 pints, 7 pints, etc. 

93. A Compound Number is one consisting of several 
denominate numbers, but of the same measure ; as, 6 feet 
8 inches. 



SECTION II. 
TABLES AJfD MEASURES. 

VALUE. 

Note. — ^United States Money has been considered on page 66 
and the pages following. 

ENGLISH CURKENCY. 

93. EngUsh Currency is the money used in Great 
Britain and Ireland. 

Table. 

4 farthings (fer.)=l penny, d. 
12 pence = 1 shilling, s. 

20 shillings = 1 pound, £. 

21 shillings = 1 guinea, G. 

l£=^20s. = 240d. = 960far. 

ORAL EXERCISES. 

1. How many farthings in 3 d. ? In 5 d. ? 

2. How many pence in 3 s. ? In 12 s. ? 

3. How many shillings in 6 £ ? In 9 £ ? 

4. How many shillings in 60 d. ? In 120 d. ? 

5. How many pence in 12 s. 6 d. ? In 6^ s. ? 

6. How many £ in 480 d. ? In 80 s. ? 

I 



130 TABLES AND MEASURES. 

WRITTEN PROBLEMS. 

1. Reduce 2 £ 3 s. to slullings. 

SoLunoK. 

2 £ 3s 

20 * Eicjilainaiion, — Since 1 £ equals 20 s., 2 £ equal 

'^ 2 times 20s., or 40 s. ; 40 s. plus 3s. are 43. 

43 s. 

2. Reduce 975 d. to £, etc. 

j^ Evplanaiion, — Since 12 d. equal 1 s., in 975 d. 

^ . , there are as many shillings as 12 is contained 

2 0)81 s 3 d *^®* ^° ^^^' ^^ ^^ ®' ^ ^' 

. ^' * Since 20 s. equal 1 £,, there are as many £, in 

81 s. as 20 is contained times in 81, or 4 £ 1 s. 

3. Reduce 5 £ 6 s. to shillings. Ans. 106 s. 

4. Reduce 1 £ 5 s. 6 d. to pence. Ain, 306 d. 

5. Reduce 1680 d. to £, s. and d. Ana, 7 £. 

6. Reduce 16740 far. to £, etc. Am. 17 £ 8 s. 9 d. 



WEIQHT. 

AVOIRDUPOIS WEIGHT. 

94. Ayoirdnpois Weight is used in weighing pro- 
duce, coal, iron, groceries etc. 

Table. 

16 drams (dr.) = 1 ounce, oz. 

16 ounces =1 pound, lb. 

25 pounds = 1 quarter, qr. 

4 quarters = 1 hundred- weight, cwt. 

20 hundred-weight « 1 ton, T. 

1 ton = 20 cwt. - 80 qr. - 2000 lb - 32000 oz. 

1 lb. avoirdupois equals 7000 grains. 

The long ton, equal to 2240 lb., is used in collecting 



TABLES AND MEASURES. 



131 



duties at the United States custom-houses and in selling 
coal at wholesale. 

The following are also used : 



32- 
45 
48 
56 


lb. 

(t 

a 


01 oats, 

of timothy seed, 

of barley, 

of rye or Indian com. 


► -1 bushel. 


60 
56 


a 


of wheat, potatoes or clover seed, 
of butter 


«1 firkin. 


196 
200 
100 
100 


u 
it 


of flour, 

of beef or pork, 

of dry fish ^ 

of nails 


^ =»1 barrel. 

« Iquintal. 
«1 k^. 



ORAL EXERCISE. 

1. How many ounces in 3 lb. ? In 5^ lb. ? 

2. How many pounds in 64 oz. ? In 240 oz. ? 

3. How many pounds in 4 T. ? In 2 T. 6 cwt. ? 

4. I exchange 2 cwt. 3 qr. of flour at 2 cts. a pound for 
muslin at 10 cts. a yard : how many yards do I get ? 

WRITTEN PROBLEMS. 



1. Reduce 6 T. Icwt. 3 qr. 


2. Reduce 2750 lb. to tons, 


to quarters. 


etc. 


Solution. 


Solution. 


6T. Icwt. 3qr. 
20 

120 
1 

121 cwt. 
4 


25)2750 lb. 
4)110 qr. 
20)27 cwt 2 qr. 
1 T. 7 cwt. 


484 
3 




487 qr. 




Hence, 6 T. 1 cwt. 3 qr. = 487 qr. 


Hence, 2750 lb. - 1 T. 7 cwt 2 qr. 



Let the pupil explain. 



132 TABLES AND MEA8UBE8. 

3. Reduce 6 T. 3 cwt. to huDdred-weight. Ans. 123 cwt 

4. Reduce 18 T. 1 cwt. 22 lb. to pounds. 

Am. 36122 lb. 

5. Reduce 2300 lb. to tons, etc. Ans. 1 T. 3 cwt. 

6. What is the value of 1 T. 4 cwt. of flour at 2^ cts. a 
pound ? Ans. $60. 

7. What will 16 bbl. of flour cost at 2f cts. a pound? 

•Ana. $86.24. 

8. How much will 3 T. of wheat cost at $1.10 a bushel ? 

Am. $110. 

9. A farmer sold 60 bags of wheat, each weighing 183 
lb., at $1.15 a bushel : how much did he get for his wheat? 

Am. $210.45. 

10. What is the value of 40 bu. of clover-seed at 13^ 
cts. a pound? Am. $324. 

TROY WEIGHT. 

95. Troy Weight is used in weighing gold, silver, 

gems and jewels. 

Table. 

24 grains (gr.) = 1 pennyweight, dwt. 

20 pennyweights = 1 ounce, oz. 

12 ounces == 1 pound, lb. 

1 lb. Troy = 12 oz. = 240 dwt - 5760 gr. 

APOTHECARIES' WEIGHT. 

96. Apothecaries' Weight diflers from Troy weight 
in the division of the ounce. It is used in weighing med- 
icines. 

Table. 

20 grains (gr.) = 1 scruple, 3. 

3 scruples = 1 dram, 3« 

8 drams = 1 ounce, S- 

12 ounces » 1 pound, lb. 



TABLES AND MEASURES. 

llb. = 12S=965 = 2889-5760gr. 



1 ID. = 1^5 =1^05 

1 lb. avoirdupois = 7000 gr. 
1 lb. Troy = 5760 gr. 

1 lb. apothecaries' = 5760 gr. 



1 oz. avoirdupois « 
1 oz. Troy = 

1 oz. apothecaries' « 



133 



437 J gr. 

480 gr. 
480 gr. 



ORAL EXERCISE. 

1. How many ounces in 6 lb. Troy ? In 4^ lb. ? 

2. How many ounces in 31b. 11 oz. Troy? In 101b. 
14oz.? 

3. How many pounds in 49 oz. ? In 59 oz. ? In 63 oz. ? 

4. How many pennyweights in 4 oz. ? In 5 lb. ? In 2 
lb. 6 oz. ? 

5. How many ounces in 60 dwt. ? In 960 gr. ? In 
2ilb.? 

6. How many scruples in 3 5 2 3? In 1 lb. 6 S 53 ? 

7. What is the difference in weight between an ounce 
of gold and an ounce of iron ? 



WRITTEN PROBLEMS. 



1. Reduce81b.3oz.4dwt. 


2. Reduce 8600 gr. to 


to pennyweights. 


pounds. 


Solution. 


Solution. 


81b. 3 oz. 4 dwt. 


24)8600 gr. 


12 


20)358 dwt. 8 gr. 


96 


12)17 oz. 18 dwt. 


3 


1 lb. 5 oz. 


99 oz. 




20 




1980 




4 




1984 dwt. 




Hence, 81b. 3oz. 4dwt.= 


Hence, 8600 gr. = 1 lb. 5 oz. 


1984 dwt. 


18 dwt. 8 gr. 



Let the pupil explain. 



12 



134 TABLES AND MEASURES. 

3. Reduce 6 lb. 7 oz. Troy to ounces. Ans. 79. 

4. Reduce 7 lb. 8 oz. 4 dwt. to pennyweights. 

Arts. 1844 dwt 

5. Reduce 1600 dwt to pounds, etc. Ans. 6 lb. 8 oz. 

6. Reduce 9000 gr. to pounds, etc. 

Ans. 1 lb. 6 oz. 15 dwt. 

7. How much is a gold chain weighing 1 oz. 3 dwt 
worth, at 90 cts. a pennyweight ? Ans. $20.70. 

8. How many sfrains in 6^ lb. Troy ? Ana. 37440. 

9. How many grains in ^ lb. avoirdupois ? 

Ans. 45500. 

EXTENSION. 

97. Extension is that which has one or more of the 
dimensions — ^length, breadth and thickness. 

98. A lAne has length only. 

99. A Surface has length and breadth. 

100. A Volume has length, breadth and thickness. 

LONG MEASURE. 

101. Long Measure is used in measuring length and 
distance. 

Table. 

12 inches (in.) = 1 foot, ft. 

3 feet = 1 yard, yd. 

5^ yards, or 16^ feet = 1 rod, rd. 
40 rods = 1 furlong, jRir. 

8 furlongs = 1 mile, mi. 

3 miles = 1 league, lea. 

1 mi. = 8 for. = 320 rd. = 1760 yd. = 5280 ft. = 63360 in. 
The inch is usually divided into halves, quarters, etc. 



TABLES AND MEASURES. 



135 



ORAL EXERCISE. 

1. How many inches in 3 ft. ? In 5 ft. ? In iO ft. ? 

2. How many feet in 24 in. ? In 60 in. ? In 84 in. ? 

3. How many feet in 6 yd. ? In 40 yd. ? In 3 yd. 2 ft. ? 

4. How many yards in 18 ft. ? In 36 ft. ? In 72 in. ? 

5. How many yards in 3 rd. ? In 4^ rd. ? 

6. How many rods in 22 yd. ? In 49^ ft. ? 

7. How many inches in 6 ft. 3 in. ? In 3 yd. 2 ft. 9 in. ? 

8. At 40 cents a foot, how much will 20 rd. of fence 

cost? 

WRITTEN PROBLEMS. 



1. Reduce 7 yd. 2 ft. 3 in. 
to inches. 

Solution. 

7 yd. 2 ft. 3 in. 
_3 

21 

23 ft. 
12 

276 
_3 

279 m. 
Hence, 7 yd. 2 ft. 3 in. = 279 in. 



2. Reduce 940 in. to rods. 

Solution. 
12)940 in. 
3)78 ft. 4 in. 
6i)26yd. 

11 )52 

4 rd. f yd., or 4 yd. 

Note.— 5J = V, and 26 =A^. 
Hence, 940 in. = 4 rd. 4 yd. 4 in. 



3. Reduce 16 ft. 5 in. to inches. Ans, 197 in. 

4. Reduce 5 yd. 2 ft. 10 in. to inches. Ans, 214 in. 

5. Reduce 10 rd. to feet ; to inches. Ans, 165 ft. 

6. Reduce 620 in. to yards, etc. Ans. 17 yd. 8 in. 

7. Reduce 720 ft. to rods, etc. Ans. 43 rd. 3 yd. 1 ft. 6 in. 

8. Reduce 800 in. to rods, etc. Ans. 4 rd. 8 in. 



SUEFACE OR SQUARE MEASURE. 
103. Square Measure is used in measuring areas or 
surfaces. 

103* A Surface has length and breadth only. 



136 



TABLES AND MEASURES. 



104. A Rectangle is a fig- 
ure having four sides, each 
of which is perpendicular to 
two of the others. A slate, 
a book and a sheet of paper 
are examples of rectangles. 

105. A Square is a rectangle whose 
four sides are equal. 

A square inch is a square whose 
sides are each one inch long. 

106. The Area of a surface is equal 
to the product of the two numbers rep- 
resenting the two dimensions. Thus, 
in the rectangle ABCD, -^ 
5 ft. long and 3 it. wide, the 
unit, which is 1 sq. ft., is 
contained 3x5 times, or 15 
times. This may be foimd 
also by actual count of the 
small squares found in the 
rectangle. 

From this we have the rule to find the area of a rect- 
angle or similar sur&ce : 

RULE. 

MvUvply the length by the breadth. 

Table. 
144 square inches (sq. in.) = 1 square foot, sq. ft. 
9 square feet 
30 J square yards, ) 
272^ square feet I 
160 square rods 



— '^iV^^.^ -«Mi^^ii^iB ^i^_^^^_ ^^■BBBV^^ m^mtm^mm^ 

•b ' ' Jr 



640 acres 



= 1 square yard, sq. yd. 

= 1 square rod, sq. rd. 

= 1 acre, A. 

^ 1 square mile, sq. mL 



TABLES AND MEASURES. 137 

1 A. - 160 sq. rd. = 4840 sq. yd. - 43560 sq. ft. = 6272640 

sq. in. 

ORAL EXERCISE. 

1. How many square inches in 2 sq. ft. ? In 1 sq. yd. ? 

2. How many square feet in 3 sq. yd. ? In 5 sq. yd. ? 

3. How many square feet in 3 sq. rd. ? 

4. How many square yards in 5 sq. rd. ? In 90 sq. ft. ? 

5. How much will 1 sq. mi. of land cost at $50 an acre ? 

WRITTEN PROBLEMS. 

1. How many square rods in 2 A. 4 sq. rd. ? Ans, 324. 

2. Reduce 435600 sq. ft. to acres ? Ans. 10 A. 

3. How many acres in 3 sq. mi. ? Ans. 1920 A. 

4. A town-lot is 160 ft. long and 60 ft;, wide : how many 
square feet does it contain ? Ans. 9600. 

5. In a rectangular piece of land 480 ft. long and 320 ft. 
wide, how many acres are there ? 

Ans. 3 A. 84 sq. rd. 51 sq. ft. 

6. K I buy 10 A. of land at $150 an acre, and sell it at 

2 cts. a square foot, how much do I gain? Ans. $7212. 

MEASURES OF VOLUME. 

107. Cubic Measure is used in computing the con- 
tents of volumes or solids. 

108. A Volume or SolidhasthTee 
dimensions — ^length, breadth and 
thickness. 

109. A Cube is a volume bound- 
ed by six equal squares, which are 
called its faces. The three di- 
mensions of a cube are equal. 

A cubic inch is a cube each 
side of which is one inch in length. 

12* 




138 TABLES AND MEASURES. 

110. The Volume of a body is expressed by the product 
of the numbers representing ite three dimensions. 

Table. 

1728 cubic inches (cu. in.) = 1 cubic foot, cu. ft. 
27 cubic feet = 1 cubic yard, cu yd. 

1 cu. yd. = 27 cu. ft. = 46656 cu. in. 

WOOD MEASURE. 

Table. 

16 cubic feet (cu. ft.) «= 1 cord foot, cd. ft. 
8 coad feet, ) -. j j 

or 128 cubic feet } "^ ^'^' ^- 

A cord of wood is a pile 8 ft. long, 4 ft. high and 4 ft. 
wide. 

24f cubic feet = 1 perch of stone or masonry, pch. 

ORAL EXERCISE. 

1. How many cubic feet in 3 cu. yd. ? In 3456 cu. in. ? 

2. How many cubic feet in 3 cords ? In 2 cords ? In 
1^ cords ? 

3. How many cubic feet in a block 3 ft. long, 2 ft. 
thick and 2 ft. high ? 

4. How many cubic feet in 2 pch. of stone? In 4 
pch. ? 

WRITTEN PROBLEMS. 

1. How many cubic feet in 6 pch. of stone? 

Am, 148^. 

2. How many cubic inches in 10 cu. ft ? Ans. 17280. 

3. In :^ of a cord of wood how many cubic feet ? 

Am. 32. 



TABLES AND MEASURES. 139 

4. How many cords in a pile of wood 60 ft. long, 4 ft. 
wide and 6 ft. high ? Ans, 11 J. 

5. How many perch of stone in a wall 200 ft. long, 1\ 
ft. thick and 6 ft. high ? Am, 1%^^, 

6. How many cubic inches in a tank 6 ft. long, 4 ft, 
wide and 3 ft. deep? Ana. 124416. 

MEASURES OF CAPACITY. 

LIQUID MEASURE. 

111. laquid Measure is used in measuring liquids. 

Table. 
4 gills (gi.) = 1 pint, pt. 
2 pints =" 1 quart, qt. 

4 quarts = 1 gallon, gal. = 231 cu. in. 

1 gal. - 4 qt. = 8 pt. = 32 gi. = 231 cu. in. 

112. The ale gallon contains 282 cu. in., but it is now 
rarely used. 

113. The barrel and the hogshead are no longer fixed 
measures. Their contents are estimated by gallons. 

114. In estimating the contents of cisterns, reser- 
voirs, etc., 

1 barrel (bbl.) = 31^ gallons. 

1 hogshead (hhd.) = 63 gallons, or 2 barrels. 

ORAL EXERCISE. 

1. How many gills in 3 pt. ? In 3 qt. ? 

2. How many pints in 40 gi. ? In 3 gal. ? 

3. How many quarts in 20 gal. ? In 40 gi. ? 

4. How many pints in 3 gal. 2 qt. ? 

6. If a gill of oil cost 3 cents, how much will a gallon 
cost? 



140 TABLES AND MEASURES. 

6. K 2 gal. of wine cost $5.12, 'how much will a gil' 
cost? 

WRITTEN PROBLEMS. 

1. Beduce 16 gal. 3 qt. to quarts. Ans. 67 qt 

2. Reduce 73 gal. 2 qt. 1 pt. to pints. Arts, 689 pt. 

3. Reduce 1 gal. 3 qt. to gills. Ans, 56 gi. 

4. Reduce 600 gi. to gallons, etc. Ana. 18 gal. 3 qt. 

5. Reduce 900 pt. to gallons, etc. Ans, 112 gal. 2 qt. 

6. How many cuhic inches in a barrel of 31^ gal. 

Ans, 7276^ cu. in. 

DKY MEASURE. 

115. Dry Measure is used in measuring grain, fruit, 
salt, soft coal, etc. 

Table. 

2 pints (pt.) = 1 quart, qt. 

8 quarts, = 1 peck, pk. 

4 pecks -= 1 bushel, bu. - 2150.42 cu. in. 

1 bus. - 4 pk. - 32 qt. = 64 pt. = 2150.42 cu. in. 
1 pint dry measure equals nearly 1^ pints liquid mea- 
sure. 

116. In measuring grain, beans and other small seeds 
the measure must be even full. But in measuring beets, 
potatoes and other coarse vegetables it must be heaped 
measure. Four heaped measures are equal to 5 even 
measures. A heaped bushel equals J of an even bushel. 

ORAL EXERCISE. 

1. How many pints in 6 qt.? In 3 pk. ? 

2. How many quarts in 16 pt. ? In 7 pk. ? 

3. How many quarts in 2 bu. ? In 1 bu. 3 pk. ? 

4. How many pecks in 80 qt. ? In 4 bu. 3 pk. ? 

5. How many pints in 1 bu. ? In 3 bu. 2 pk. ? 



TABLES AND MEASURES. 141 

WRITTEN PROBLEMS. 

1. How many quarts in 3 bu. 2 pk. ? Ans. 112. 

2. How many bushels in 634 qt. of beans ? 

Ans. 19 bu. 3 pk. 2 qt. 

3. How many pints in 17 bu. 3 pk. 6 qt. ? Ans, 1148 pt. 

4. If I buy 2 bu. of wabiuts at $1.50 a bushel, and sell 
them at 3 cts. a pint, how much do I gain ? Ans, 84 cts. 

APOTHECARIES' FLUID MEASUBE. 

117. This measure is used in measuring liquids in pre- 
paring medicines. 

Table. 

60 minims, or drops (m.) = 1 fluid drachm, f 3. 

8 fluid drachms = 1 fluid ounce, f S. 

16 fluid ounces = 1 pint, O. 

8 pints = 1 gallon, Cong. 

1 gal. = 8 pt. = 128 f g = 1024 f 3 = 61440 m. 

ORAL EXERCISE. 

1. How many ounces in 3 pt. ? 

2. How many f 3 in 3 f S ? 

3. How many f S in 3 Cong. ? 

4. How many minims in ^ f 3 ? 

5. How many f 3 in 3 O. 5 f S. 

TIME. 

118. Time is a measured part of duration. 

Table. 
60 seconds (sec.) = 1 minute, min. 
60 minutes = 1 hour, hr. 

24 hours = 1 day, d. 

365 days = 1 year, yr. 

366 days = 1 leap year. 
100 years = 1 century, cen. 



142 TABLES AND MEASURES. 

Also, 
7 days -= 1 week, wk. 
4 weeks = 1 month, mo. 
52 weeks = 1 year, yr. 

119. In business transactions 30 days is considered a 
month, and 12 months a year. 

. The months of April, June, September and November 
have 30 days each. 

February has 28 d., except in leap year, when it has 29. 

The remaining months, January, March, May, July, 
August, October and December, have 31 days each. 

ORAL EXERCISE. 

1. How many seconds in 3 min.? In ^ min. ? 

2. How many minutes in 120 sec. ? In 4 hr. ? In 
IJhr.? 

3. How many hours in 240 min. ? In 4 d. ? 

4. How many hours in 5 d. ? In 6 d. ? 

5. How many weeks in 35 d. ? In 60 d. ? 

6. How many days in 6 wk. ? In 5 wk. 3d,? 

7. What part of a day is 12 hr. ? 4 hr. ? 

WRITTEN PROBLEMS. 

1. Reduce 6000 sec. to hours, etc. Ane. 1 hr. 40 min. 

2. How many minutes in 2 d. ? Ans. 2880. 

3. How many hours from 3 o'clock in the morning till 
9 o'clock at night ? Ans, 18 h. 

4. How many minutes from 6 o'clock A. M. to 4 o'clock 
P. M. ? Am. 600 min. 

Note. — A. M. means before noon ; M., noon ;- and P. M., after- 
noon. 



TABLES AND MEASURES. 143 

5. How much time passes from 10 minutes before 7 in 
the morning to 8 o'clock 25 min. P. M. ? 

Ana. 13 hr. 35 min. 

CIKCULAK MEASUBE. 

120. Circular or Angular Measure is used in mea- 
suring angles and arcs of circles. Also in determining 
the latitude and longitude of places. 

121. The measuring unit is called a Degree. It is -jj^ 

part of the circumference of a circle. 

122. A Circle is a plane figure 

bounded by a curved line, every part 
of which is equally distant from a 
point within called the Centre. 

123. The bounding line is called 

the Circumference. The space with- 
in the circumference is the circle. 

124. An Arc is any part of the circumference, as A B 
or AC. 

125. An Angle is the difference in the direction of two 
lines meeting at a common point called the Vertex. Thus, 
B O C and A O C are angles. 

126. A Bight Angle is one in which the two lines are 
perpendicular to each other. Thus, AOB and BOD 

are right angles. 

127. An angle is measured by the arc of the circle 
included between its two sides. Thus, the measure of the 
angle B O C is the arc B C. 

128. Any straight line extending from the centre to 
the circumference, as O B, is called a JRadma, and any 
straight line passing through the centre and limited by 
the circumference, as A D, is called a Diameter. 




144 TABLES AND MEASURES. 

129. Every circumference is divided into 360 equal 
parts, called degrees; each degree into 60 equal parts, 
called minviea; and each minute into 60 equal parts, 
called seconds. 

Table. 
60 seconds (") =1 minute, (' ). 

60 minutes =1 degree,- - (°). 

30 degrees -= 1 sign, s. 

12 signs or 360° »- 1 circumference, c. 
90° = 1 quadrant, or right angle. 

ORAL EXERCISE. 

1. How many degrees in 120' ? 

2. How many minutes in 5° ? In 240" ? 

3. How many minutes in 4° 30' ? 

4. How many degrees in 7200" ? 

WRITTEN PROBLEMS. 

1. Reduce 18900" to degrees, etc. Ans. 5° 15'. 

2. Reduce 19° 18' to minutes. Ana, 1158'. 

3. Reduce 1850' to degrees, etc. Ana. 30° 60'. 

4. Reduce 4° 6' 30" to seconds. Ana. 14790". 

5. How many degrees, etc., in 27604" ? Ana. 7° 40' 4". 

COUNTING. 

130. The following table is used in counting buttons, 
screws, etc. 

Table. 

• 

12 things - 1 dozen, doz. 

12 dozen = 1 gross, gro. 

12 gross = 1 great-gross, gr.-gro. 
20 units = 1 score, sc. 

1 gr.-gro. - 12 gro. = 144 doz. = 1728 units. 



TABLES AND MEASURES. 145 

PAPER. 

24 sheets = 1 quire, qr. 
20 quires = 1 ream, rm. 

2 reams = 1 bundle, bun. 

5 bundles = 1 bale, B. 

1 B. = 5 bun. = 10 rm. = 200 qr. = 4800 sheets. 

ORAL EXERCISE. 

1. How many are 1^ doz. ? 4\ doz ? 

2. How many dozen in 3 gro. ? In 2 gr.-gro.? 

3. How many buttons in 5 gro. ? 

4. How many gross in 432 screws ? 

6. How many sheets in 6 qr. of paper ? 

6. How many sheets of paper in ^ rm. ? 

7. How many quires in 3 rm. ? In 240 sheets ? 

8. How many sheets in 3^ rm. ? How many quires ? 

WRITTEN PROBLEMS. 

1. How many things in 3 gr.-gro. ? Ans, 5184. 

2. How many things in 6 J gro. ? Ans. 900. 

3. How many matches in 5 gr.-gro. ? Ans, 8640. 

4. How many gross in 1500? Ans. 10 gro. 5 doz. 

5. How many quires in 1700 sheets of paper? 

Ans. 70 qr. 20 sh. 

6. How many sheets of paper in 11 qr. 11 sh. ? 

Ans. 275 sheets. 

7. How many sheets in 3 bales of paper ? Ans. 14400. 

8. How many reams, etc. in 2500 sh. of paper? 

Ans. 5 rm. 4 qr. 4 sh. 

9. If a man use 12 sheets of paper in a day, how much 
does he use in 150 days ? Ans. 3 rm. 15 qr. 

10. If 1 desk require 6 doz. and 2 screws, how many 
screws do 20 desks require? Ans. 123 doz. 4 screws* 

13 K 



146 COMPOUND NUMBERS. 

COMPOUND NUMBERS. 

1. ADDITION OF COMPOUND NUMBERS. 

131. Addition of Oompound Numbers is the pro- 
cess of finding the sum of two or more similar compound 
numhers. 

WRITTEN PROBLEMS. 
1. Find the sum of — 

lb. 08. dwt gr. Explanaiion. — We write the numbers in ver- 

6 5 6 ^qq\ columns, so that units of the same denomi- 
18 3 9 4 
26 2 16 ^^^i^° stand in the same column. The sum of 

3 4 18 ^^^ ^^ right-hand column is 38 gr., or 1 dwt. 14 
51 2 14 S^' Writing the 14 gr. in the column of grains 

and adding the 1 dwt. to the column of penny- 
weights, the sum of the second column is 22 dwt., or 1 oz. 2 dwt. 
Writing the 2 dwt. in the proper column, and adding the 1 oz. to 
the column of ounces, the sum of the third colunm is 12 oz., or 1 lb., 
which we add to the column of pounds. The sum of the fourth 
column is 51 lb. Hence, the sum of the given numbers is 51 lb. 
2 dwt. 14 gr. 

From this and similar solutions we derive the following 
rules for the addition of compound numbers : 

RULES. 

1. Write the numbers so that units of the same denominor 
tion shall stand in the saime column, 

2. Beginning wiih the lowest denominationy add as in 
simple numberSy and reduce the sum to the next higher de- 
nominationy writing the remainder y if any, under the column 
addedy and adding the quotient obtained by the redwstion 
to the next column. 

3. Proceed in the savne manner uyith all the columns to 
the last. 

Note. — If any places are wanting, supply ciphers. 



COMPOUND NUMBERS. 147 

What is the sum of the following : 



6 


(2.) 

8. 

3 


d. 
4 


pk 
3 


(3.) 

qt. pt 
2 1 


1 


(4.) 

1^. qt 

3 1 


pt 
1 


8 


9 


6 


6 


4 




4 3 


1 


3 


12 


3 


Z_ 


6 1 




7 


1 


yd. 
6 


(6.) 
ft. 
2 


in. 

7 


d. 

7 


(6.) 

hr. min. 
3 15 


lb. 
3 


(7.) 

(H. dwt 

4 18 


6 


3 





8 


6 


4 12 


9 


6 12 


18 


2 


1 


6 


18 


9 20 


13 


4 


12 


8 


2 





40 


3 30 


18 


8 6 


6 


3 

4 


(8.) 

3 9 

3 2 


8 


cwt 
6 


qr 
2 


(9.) 
lb. 
15 


oz. 
8 


6 


6 


1 


12 


18 


3 


20 


12 


6 


4 





13 


15 


2 


6 


5 


8 


7 


2 


15 


16 





22 


15 



10. Add 6 mi. 3 for. 16 rd,, 18 mi. 14 rd., 16 mi. 7 for. 
and 20 mi. 6 for. 35 rd. Ans. 62 mi. 1 for. 25 rd. 

11. Add 12 bu. 2 pk. 3 qt. 1 pt, 17 bu. 3 pk., 14 bu. 1 pk. 
1 pt, 8 bu. 6 qt 1 pt Am, 52 bu. 3 pk. 2 qt 1 pt. 

12. A farmer raises in four years the following quanti- 
ties of corn : 106 bu. 3 pk. 1 qt, 125 bu. 1 pk. 1 qt. 1 pt, 
205 bu. 3 pk. 5 qt., 250 bu. 2 pk. 7 qt. 1 pt : how much 
did he raise ? Am, 688 bu. 2 pk. 7 qt 

13. How long is it from 2 hr. 6 min. 40 sec. before noon 
to 15^ min. past 4 o'clock in the afternoon ? 



148 COMPOUND NUMBERS. 

2. SUBTRACTION OF COMPOUND NUMBERS. 

132. Subtraction of Compound Numbers is the 

process of finding the difference between two similar com- 
pound numbers. 

WRITTEN PROBLEMS. 

(1.) Process. ExplancUion. — We write the 

From 63 lb. 4 oz. 18 dwt. 13 gr. numbers as in addition of corn- 
Take 27 6 4 16 pound numbers, and begin to 
35 lb. 10 oz. 13 dwt. 21 gr. subtract at the lowest denomina- 
tion. We cannot take 16 gr. fix)m 13 gr. ; we therefore take 1 dwt, 
equal to 24 gr., which added to 13 gr. equals 37 gr. ; 16 gr. from 
37 gr. leaves 21 gr.; 4 dwt. from 17 dwt. leaves 13 dwt. We can- 
not take 6 oz. from 4 oz. ; we therefore take 1 lb., equal to 12 oz., 
which added to 4 oz. equals 16 oz. ; 6 oz. from 16 oz. leaves 10 oz. ; 
27 lb. from 62 lb. leaves 35 lb. 

Note. — ^Let the pupil derive the rule. 





(2.) 






(3.) 




lb. 


oz. dwt. 




£ 


s. 


d. 


From 27 


6 15 




25 


6 


3 


Take 14 


7 8 

(4.) 




18 


4 
(5.) 


11 


yr. 


mo. wk. 


d. 


mi. 


fur. 


rd. 


From 115 


7 6 


2 


18 


4 


27 


Take 97 


8 3 

(6.) 


12 


12 


7 
(7.) 


36 


bu.. 


pk. qt. pt 




T. cwt. 


. lb. 


oz. 


From 140 


3 2 1 




18 7 


20 


6 


Take 60 


2 5 

(8.) 


• 


7 4 


26 
(9.) 


15 


yr. 


mo. d. 




jr. 


mo. 


d. 


From 1867 


6 13 




1706 


1 


6 


Take 1855 


3 26 




1608 


12 


9 



COMPOUND NUMBERS. 149 

10. From a barrel containing 31 gal. 2 qt. there leaked 
out 6 gal. 3 qt. 1 pt. : how much remained ? 

Arts. 24 gal. 2 qt. 1 pt. 

11. A farmer had 126 bu. 3 pk. of seed-wheat, and sold 
111 bu. 2 pk. 6 qt. 1 pt. : how much remained ? 

Ans. 15 bii. 2 qt. 1 pt. 

12. A pile of wood contained 7 cd. ; a teamster hauled 
away 4 cd. 6 cu. ft. : how much remained ? 

Am. 2 cd. 122 cu. ft. 

13. How old was a man who was born Jan. 3, 1832, 
and died July 1, 1877 ? Ana. 45 yr. 5 mo. 28 d. 

14. A lady was born Nov. 22, 1837 : how old was she 
Sept. 2, 1876 ? Ans. 38 yr. 9 mo. 10 d. 

16. A girl was born Jan. 13, 1862 : how old was she 
Sept. 2, 1877 ? Ans. 15 yr. 7 mo. 19 d. 

16. What is the time ft'om 15 min. past 4 P. M. to 3 min. 
before 10 P. M. Ans. 5 hr. 42 min. 

17. A note was dated Jan. 15, 1877, and was to be paid 
Aug. 10, 1877 : how long from the date of it to the time 
it was paid ? Ans. 6 mo. 25 d. 

3. MULTIPLICATION OF COMPOUND NUMBEKS. 

133. Multiplication of Compound Numbers is the 

process of finding the product of two numbers, one of 
which is compound. 

WRITTEN PROBLEMS. 

1. Multiply 12 £ 6 8. 9 d. by 7. 

Explanation. — 7 times 9 d. are 63 d., or 6 s. 3d.; 

Process. ^^ ^^^^ 6 s. are 42 s. ; 42 s. and 5 s. are 47 s., or 2 j& 

12£ 68. 9d. .^g 7time8l2£&TeS^£;S^£Bnd2£areSe£. 

oc£ rj — Q A Hence, 12 £ 6 8. 9 d. multiplied by 7 equals 86 £ 

7s. 3d. 

Note. — Let the pupil derive the rule. 



150 COMPOUND NUMBERS. 



(2.) 


(3.) 




6 cwt 3 qr. 7 lb. 


18 mi. 3 fiir. 


16 rd. 


Multiplied by 6 




7 


34cwt. Oqr. 101b. 


128 mi. 7 ftir. 


32 rd. 


(4.) 


(5.) 




14gal. 3qt Ipt 


16 yr. 3 mo. 


3wk. 


8 




12 


(6.) 


(7.) 




3d. 6 hr. 15 min. 


3£ 68. 4d. 


l&r. 


10 




14 



8. Multiply 4 gal. 3 qt. 1 pt by 20. Ans, 97 gal. 2 qt 

9. If a farmer can raise 16 bu. 3 pk. 4 qt. of wheat on 
an acre, how mucli can he raise on 40 acres ? 

Ans. 676 bu. 

10. If a horse can haul 17 cwt. 3 qr. 16 lb. 4 oz. of coal 
in a load, how much can he haul in 16 loads ? 

Ans. 14 T. 6 cwt. 1 qr. 19 lb. 

4. DIVISION OF COMPOUND NUMBERS. 

134. Division of Compound Numbers is the pro- 
cess of dividing one number by another, the dividend be- 
ing compound. 

WRITTEN PROBLEMS. 

1. Divide 40 bu. 3 pk. 6 qt. by 7. 

Process. Explanation, — } of 40 bu. is 5 bu., and 5 

7) 40 bu. 3 pk. 5 qt. bu., or 20 pk., remaining. 20 pk. and 3 pk. 

5 bu. 3 pk. 3 qt. are 23 pk. ; ^ of 23 pk. is 3 pk., and 2 pk., or 

16 qt., remaining ; 1 6 qt. and 5 qt are 21 qt ; 
f of 21qt is3qt 

Note. — Let the pupil derive a rule. 



COMPOUND NUMBERS. 151 

Perform the following divisions : 

(2.) (3.) 

8) 25 £ 15 8. 4 cL 9) 80 bu. 1 pk. 6 qt. 

3£ 4s. 5d. 8bu. 3pk. 6qt. 

(4.) (5.) 

12) 58 gal. 2 qt. 15) 244 mi. 2 fiir. 20 kL 

(6.) (7.) 

20) 125 yr. 10 mo. 24 )244 £ 4 s. 

8. K 8 hens weigh 42 lb., how much does each weigh ? 

Ana. 5 lb. 4 oz. 

9. If 12 bags contain 31 bu. 2 pk. of oats, how much 
does 1 contain ? Ans, 2 bu. 2 pk. 4 qt. 

Note. — When both divisor and dividend are concrete numbers, 
reduce both to the lowest denomination in either, and divide as in 
simple division. 

10. Divide 40 £ 4 s. by 2 £ 10 s. 3 d. 



Process. 

40 £ 4 8. =9648d. 



2£ 10s. 3d.= 603d. 



= 16. 



11. Divide 78 lb. 12 oz. avoirdupois by 3 lb. 15 oz. 

Am. 20. 

12. A man travels 410 mi 6 far. 10 rd. by travelling 
16 mi. 3 fur. 18 rd. a day: how many days does he 
travel ? An8. 25. 

1 3. How many loads of coal, each weighing 1 T. 3 cwt. 
27 lb., will weigh 46 T. 10 cwt. 80 lb. ? Am. 40. 

14. A number of boys gather 10 bu. 3 pk. 1 qt. of 
chestnuts; they divide them equally, and each receives 
3 pk. 4 qt. 1^ pt. : how many boys were there ? Ana. 12. 



152 PERCENTAGE. 

CHAPTER VII. 
PEEOENTAGE. 



SECTION I. 

DEFIJflTIOJfS AJfD PBIJfCIPLES. 

135. Percentage is the name applied to computations 
in which 100 is the unit or measure. 

136. Per cent is an abbreviation of the Latin phrase 
per centum, meaning by the hundred, 

ORAL EXERCISE. 

1. What is riir o^ 100? -^ of 100? yf ^ of 100 ? 

2. What is T-b of 200? ^ of 400? ^ of 600? 

3. How many hundredths of 6100 are 63 ? 65 ? 610 ? 

Note. — ^h^^ of any number is 1 per cent, of that number ; yj^ is 
2 per cent, of it ; xJ(^ is 3 per cent, of it ; ^ is 15 per cent, of it, 
and so on. 

4. How many hundredths of a number is 5 per cent, 
of it ? 8 per cent. ? 10 per cent. ? 20 per cent. ? 

6. How many hundredths of a number is 15 per cent, 
of it ? 75 per cent. ? 120 per cent. ? 100 per cent. ? 

6. What per cent, of a number is j^^ of it ? j^ of it ? 
^ofit? 3% of it? 

7. What per cent, of a number is .14 of it? .27 of it? 
.12iofit? 1.19 of it? 

Per cent, is usually written % ; thus, 25 per cent, is 
written 25%. 

8. How many hundredths is 5% ? 15%? 20%? 
40%? 75%? 6i%? 12i%? 135%? 



PERCENTAGE. 153 

9. What per cent, of a number is ^ of it ? 

Solution. — J equals ^j%, or 20 fo. Hence, ^ of a number is 
20 fo of it. 

10. What per cent, of a number is^ofit? J? ^? 

lAr? A? V V A? ^? V V i? V 

11. What fractional part of a number is 5% of it? 

Solution. — 5% =Thij ^^ A* Hence, 5% of a number is ^ 
of it. 

12. What fractional part of a number is 10% of it? 
20%? 25%? 50%? 30%? 60%? 24%? 48%? 
65%? 92%? 87%? 

13. What part of a number is J% of it? 

Analysis.— J ?& = i of xiu = ^Jiy • 

14. What part of a number is |% of it? |% of it? 
|%ofit? T^%ofit? 

WRITTEN PROBLEMS. 

Express decimally the following : 

1. 20% ; 25% ; 18% ; 30% ; 40% ; 60%. 

2. 75% ; 4% ; 8% ; 12^%; 93f % ; 175%. 

Write the following, with the per cent, sign (%) : 

3. .25; .18; .42; .12J ; .18^^; .16|. 

4. 1.24; 8.75; i;f;i;f 

137. The number of hundreds is called the Rate. Thus, 
in 5%, or yf^, 5 is the rate. 

138. The Bate per cent is the fraction which denotes 
how many hundredths are taken. 

Every number is ^^, or 100 per cent, of itself. 

139. The number of which the per cent, is taken is 
called the Base. 



154 PERCENTAGE. 

140. The result of taking the per cent of the base is 
called the Percentage. 

141. The base added to the percentage is called the 
Amount, 

142. The base less the percentage is called the Differ- 
ence. 

CASE I. 

The Base and the Bate Fer Cent, being giyen^ to find the 

Percentage. 

ORAL EXERCISE. 

1. What is 30% of 120? 

Solution.— 30% = ^, or ^ ; ^ of 120 is 36. 



5. What is 40% of 75? 

6. What is 65% of 39? 

7. What is 90% of 30? 



2. What is 5% of 40? 

3. What is 10% of 70? 

4. What is 15% of 50? 

8. What is 18% of $50? 

9. What is 32% of 75 cows? - 
10. What is 45% of 120 cents? 

11. A man's wages were $1.20 a day, but they were re- 
duced 10% : how much did he then get? 

12. A clerk's salary was $70 a month, but his employer 
reduced the salary 10% : how much did the clerk then 
get? 

13. A merchant who sells some goods at 40 cts. a 
yard, reduces the price 20% : what dpes he sell at 
then ? ^ 

14. A boy had 84 chickens, but 25% of them died: 
how many had he remaining ? 

15. A horse cost me $140 ; if I sell him so as to gain 
15%, what do I get for him? 



PERCENTAGE. 156 

WRITTEN PROBLEMS. 

1. What is 37% of 120 A. of land? 

Solution. 
120 A. 
.37 Explanation — Since 37% equals .37, the required 

840 percentage is .37 of 120 A., or 44.4 A. 

360 



44.40 
From the foregoing we derive the 

RULE. 

MvMply the base by the rate per cent.; the remit is the 
percentage. 

2. What is 21% of $170? 

3. What is 15% of 40 cows? 

4. What is 25% of 618? 

5. What is 12% of 50 chickens? • 

6. What is 9% of 200 mi.? 
♦ 7. What is 43% of «72? 

8. I bought a horse for $95, and sold him at a gain of 
26% : what was my gain? Ans. $24.70. 

9. If my salary is $75 a month, and it is reduced 5%, 
how much do I get a month? Ans. $71.25. 

10. A boy has a flock of 60 hens, and they increase 
in one year 33^% : how many has he at the end of the 
year? Ana.^O. 

11. K a man receive $1200 a year salary, and spend 
15% of it for board and 6% of it for books, how much 
has he remaining ? Atia. $948. 

12. A boy whose knife cost him $2.50 sold it at , 
a gain of 34% : how much did he get for it? 

Am. $3.35. 



156 PERCENTAGE. 

CASE II. 

The Base and the Fereentage being giyen, to find the Bate 

Fer Cent. 

ORAL EXERCISE. 

1. What per cent, of 35 is 7 ? 

Analysis.— 7 is J of 35 ; it is therefore i of 100^ of 35, or 20^ 
of 35. 

2. What per cent, of 20 is 10? 
What per cent.— 



6. Of $55 is $11 ? 

7. Of 40 is 5? Is 8? 

8. Of $200 is $40 ? 



3. Of 40 boys is 8 boys ? 

4. Of 60 ducks is 15 ducks ? 

5. Of $90 is $18? 
9. 16 is what per cent, of 200? 

10. 36 is what per cent, of 108 ? 

11. 14 is what per cent, of 35? 

12. $35 is what per cent, of $140? 

13. A l||y had 60 marbles, but lost 15 : what per cent, 
of his number did he lose? 

14. If I have a hat worth $5, and sell it for $4, what 
per cent, do I lose ? 

15. If from a barrel of vinegar containing 40 gal. I 
sell 15 gal., what per cent do I sell, and what per cent 
remains ? • 

WRITTEN PROBLEMS. 

1. What per cent of 150 is 21 ? 

Solution. 

i^^^)21-^lii. Explanation— 21 is ^^ of 160, or ^^ of 100^ 
^ of 150, which is Vifi? ^, equal to 14%. 

600 

RULE. 

To find (he rate per cent, divide the percentage by the 
base. 



PERCENTAGE. 157 

2. What per cent, of 600 is 150 ? 

3. What per cent, of 80 is 12 ? 

4. What per cent, of 66.40 is 16 cts. ? Am, 2^. 

5. What per cent, of 20 A. is 7 A. ? Ans, 35. 

6. What per cent, of 3 bu. is 2 pk. ? An8. 1&|. 

7. What per cent, of 5 gal. is 3 qt. ? Ans. 15. 

8. A man wishing to purchase a house for $1000 had 
only $800 in money : what per cent, could he pay cash ? 

Ans» 80. 

9. If I buy muslin at 8 cts. a yard and sell it at 11 cts., 
what per cent, do I make ? Ans. 37^. 

10. K a book cost me 90 cts., and I sell it at $2.25, what 
per cent, do I make ? Ana, 150. 

CASE III. 

The Bate Per Cent, and the Percentage being giren, to 

find the Base. 

ORAL EXERCISE. 

1. 40 is 20% of what number? 

Analysis. — 20% is ^^^V, or |; since 40 is J of a number, f, or 
the number, is 5 x 40, or 200. 

2. 20 is 10% of what number? 

3. 10 is 20% of what number? 

4. 24 is 6% of what number? 

5. 25 is 5% of what number? 

6. $40 is 20% of what a horse cost: how much did he 
cost ? 

7. $36 is 9% of the cost of a town-lot: how much did 
it cost ? 

8. If 12% of the cost of my coat is $3, how much did 
it cost ? 

9. If 28% of my money is $14, how much money 

have I? 
u 



168 PERCENTAGE. 

10. If 15% of the cost of a knife is 45 cts., how much 
did it cost? 

WRITTEN PROBLEMS. 
1. 120 is 30% of what number ? 

Solution 1. 

30^=120 
1^=^x120=4 Let the pupil explain. 

100% =100x4-400 

Solution 2. Explanation,— If .30 of a number is 

120 -I- .30 = ^ftp = 400 120, the number is as many as .30 is con- 
tained times in 120, or 400. 



RULE. 

To find (he base when the rate per cent and the per- 
centage are given^ divide the percentage by the raie per 
cent, 

2. $260 is 13% of what number? Ana. $2000. 

3. If I receive $210 rent for a house, which is 7% of 
the value, what is the value ? Ans. $3000. 

4. If 20 bu. of potatoes is 5% of my whole number, 
how many bushels have I ? Ans, 400. 

5. A merchant sold muslin at 15 cts. a yard, which 
was 25% more than he paid for it: what did it cost 
him ? Ans, 12 cts. a yd. 

6. A clerk spent 20% of his salary for board and 30% 
for clothing and books; he saved $400: what was his 
salary? ^/w. $800. 

7. If $175 is 14% of my money, how much money 
have I? -4rw. $1250. 

8. If a farmer sell 60% of his corn, and feed 25% 
of it, how much was the crop if he has 105 bu. remain- 
ing ? Ans, 700 bu. 



SIMPLE INTEREST. 159 

SECTION II. 

IJ^TEREST. 

143. Interest is a certain percentage paid for the use 
of money. It is reckoned at a certain rate per cent, for 
each year. 

144. The sum on which the interest is paid is called 
the Principal. 

145. The per cent, paid per annum is called the Rate. 

146. The sum of the principal and the interest is called 
the Amount. 

147. Simple Interest is that which is reckoned on 
the principal alone. 

SIMPLE INTEREST. 

1. GENERAL METHOD. 
ORAL EXERCISE. 

1. What is the interest of $100 for 2 yr., at 6% ? 

Solution. — ^At 6%, yj^y of the principal equals the interest for 
1 year, and for 2 years 2 times yj^, or ^^ ; ^-^ of $100 equals $12. 
Hence, the interest of $100 for 2 yr. at 6% is $12. 

2. What is the interest of $200 for 1 yr., at 5% ? 

3. What is the interest of $400 for 2 yr., at 5% ? 

4. What is the interest of $300 for 2 yr., at 6% ? 

5. What is the interest of $300 for 2^ yr., at 6% ? 

6. What is the interest of $140 for 2^ jr., at 6% ? 

7. What is the interest of $500 for 3yr. 6 mo., at 6% ? 

8. What is the interest of $120 for 5 yr., at 6% ? 

9. What is the mterest of $25 for 6 yr. 3 mo., at 8%. 
10, What is the interest of $60 for 8 yr. 4 mo., at 5% ? 



160 SIMPLE INTEREST. 

WRITTEN PROBLEMS. 

1. What is the interest of $360 for 3 yr. 8 mo., at 6% ? 

Solution. 

^ Qg Explanation.— 6 foy or .06, pf |360 is 

2i7g5 $21.60, or the interest of $360 for 1 yr., 

3f yr. — 3 yr. 8 mo. at 6%, and for 3 yr. 8 mo., or 3f yr., the 

6480 interest is 3} x $21.60, or $79.20. 
1440 



$79.20 

2. What is the interest of $500 for 7 yr. 2 mo., at 6% ? 

Ana. $215. 

3. What is the interest of $350 for 7 yr., at 6% ? 

Ana. $147. 

4. What is the interest of $200 for 6 yr. 6 mo., at 6% ? 

Ana. $78. 

5. What is the interest of $75 for 3 yr. 8 mo., at 5% ? 

Ana. $13.75. 

6. What is the interest of $12 for 2yr. 9 mo., at 10% ? 

Ana. $3.30. 

7. What is the interest of $600 for 7 yr. 3 mo., at 8% ? 

Ana. $348. 

8. What is the interest of $140 for 8 yr. 4 mo., at 3% ? 

Ana. $35. 

9. What is the amomit of $120 for 6 yr. 5 mo., at 6% ? 

Ana. $166.20. 

Note. — Add the interest to the principal to find the amount. 

10. What is the amount of $740.50 for 7 yr. 3 mo., at 
7%? ^Tw. $1116.30. 

11. What is the amount of $180 for 2 yr. 6 mo. 12 d., 
at 5% ? Ana. $202.80. 

12. What is the amount of $160 for 1 yr. 3 mo. 18 d., 
at6%? ^rw. $172.48. 



SIMPLE INTEREST. 161 

2. DECIMAL METHOD. 

Note. — By this method the years, months and days are reduced 
to the decimal part of a year. 

WRITTEN PROBLEMS. 

1. What is the interest of $130 for 6 yr. 7 mo. 24 d., 

at6%? 

Solution. 
30)24 d. 1130 

12)7.8 mo. _^ r» I ,. a ^ oa a 

^T— 1^ oQ Explanation, — oyr. 7 mo. 24 a. 

^' ' 6.65 equals 6.65 yr. The interest for 

3900 1 yr. at 6 % is $7.80, and for 6.65 yr. 

4680 it is 6.65 x $7.80, or $51.87. 
4680 

$51.8700 

2. What is the interest of $150 for 3 yr. 4 mo., at 5% ? 

Ans, $25. 

3. What is the interest of $450 for 6 yr. 3 mo. 24 d., at 
6% ? Ana. $170.55. 

4. What is the interest of $190 for 2 yr. 4 mo. 18 d., at 
7%? ^ns. $31.70. 

5. What is the interest of $225 for 1 yr. 9 mo. 18 d., at 
7% ? Ans, $28.35. 

6. What is the interest of $240.60 for 7 yr. 5 mo. 6 d., 
at6%? ^/w. $107.26 + . 

7. What is the interest of $920.14 for 5 yr. 8 mo., 12 d., 
at7%? -4rw. $367.14-. 

8. What is the interest of $742.10 for 6 yr. 5 mo. 12 d., 
at6%? Jn«. $287.19 + . 

9. What is the amount of $225 for 3yr. 2 mo. 12 d., 
at6%? ^iw. $268.20. 

10. What is the amount of $220.40 for 1 yr. 7 mo. 21 d., 
at5%? Jrw. $238.49 + . 

14* L 



162 • REVIEW PROBLEMS. 

11. What is the amount of $40.50 for 6 yr. 11 mo. 12 d., 
at4%? Am, $51.76 ~, 

12. What is the amount of $320.50 for 7 yr. 3 mo., 16 d., 
at6%? ^rw. $460.77+. 



REVIEW PROBLEMS. 

1. If a man earn $20 a week and pay $6 a week board, 
how much can he save in 16 wk. ? Ans, $210. 

2. How long will it take a boy to save $105, if his wages 
are $12 a month and he apends $5 a month ? 

An8, 15 mo. ' 

3. What is the cost of 6 cows at $27.50 each, and 4 
horses at $115 each? Ans, $625. 

4. A boy had 80 cts., but lost f of his money : how 
much has he remaining ? Ana, 48 cts. 

6. If 49 years is J of 2^ times my age, how old am I ? 

Ans. 42 yr. 

6. How many sheets of paper in S\ reams ? Ans. 1560. 

7. A man who was worth $25000 willed $1600 to the 
poor, and the remainder to his six children, to be divided 
equally : how much did each one get ? Atis. $3900. 

8. How much will 13 lb. 6 oz. of butter cost at 18 cts. 
a pound ? Ans. $2.41 - . 

9. If I trade 16 chickens worth 30 cts. each for 20 yd. 
of gingham at 22 cts. a yard, how much do I lose? 

Ans. 40 cts. 

10. If I trade 15 hens for 30 yd. of calico at 12 cts. a 
yard, how much do I get for my hens each? Ans. 24 cts. 

11. If I take to a store 6 lb. 4oz. of butter at 20 cts. a 
pound, and 16 lb. 6 oz. of lard at 16 cts. a pound, and get 
in exchange sugar at 9 cts. a pound, how much sugar do 
I get? Ans. 4Slh. 



REVIEW PROBLEMS. 163 

* 

12. What is the value of 6 bu. 2 J pk. of onions at $1.60 
a bushel ? Am, $10.50. 

13. How many dozen of eggs at 12^ cts. a dozen can I 
buy for 3 bu. potatoes worth 75 cts. a bushel? 

Ans, 18 doz. 

14. How many yards of carpet 1 yd. wide will it take 
to carpet a room 15ft. long and 12ft. wide? Ans. 20yd. 

15. A farmer sells 18 bags of wheat, each containing 
2J- bu., at $1.50 per bushel : how much does he get for his 
wheat ? Ans. $67.50. 

16. How much will it cost to paint a floor 18 ft. long 
^d 16 J ft. wide, at 5 cts. a squaire foot? Ans. $14.85. 

17. How much will it cost to dig a cellar 40 ft. long, 
30 ft. wide and 5 ft. deep, at f ct. a cubic foot ? Ans. $45. 

18. A merchant had a piece of cloth containing 42^ yd., 
from which he cut 12 coats of 2f yd. each : how much re- 
mains? Ans. d^yd. 

191 K a farmer buy 26 sheep at $5 each, but 6 of them 
die, how much should he receive for the others that he 
may not lose ? ^ Ans, $6.50 each. 

20. A party of 6 go on a fishing excursion ; their car- 
fare is $1.20 each ; their meals $10.20 for all of them, and 
they pay $1.20 expressage on their baggage : what is each 
one's share of the bill ? Ans. $3.10. 

21. If a merchant buy goods at 40 cts., how must he 
sell them to make 15% ? A71S. 46 cts. 

22. If I borrow $120 June 17, 1876, and pay it back 
July 5, 1877, how much interest do I pay at 6% ? 

Ans. $7.56. 

23. What is the interest on $140.50 from May 10, 1876, 
to Aug. 1, 1877, at 7% ? Ans. $12.05. 

24. What is your age in years, months and days ? 

25. If one boy travels 16.5 mi. a day, and another 



164 REVIEW PROBLEMS. 

travels 20.25 mi. a day, how far apart will they be in 12.6 
days if they travel the same direction? Ans, 47.25 mi. 

26. If the two boys mentioned in the preceding prob- 
lem travel in opposite directions, how far apart will they 
be at the end of 6.4 days ? Ajis. 285.2 mi. 

27. How many acres in a street 60 fl. wide and 1^ mi. 
long ? Ana, 10 A. 145 sq. rd. 13.75 sq. yd. 

28. A marketman buys a turkey weighing 16 lb. at 8 
cts. a pound, live weight, and sells it at 12§- ct?. a pound, 
dressed : how much does he make if the turkey loses 30% 
in dressing? Ans, 12 cts. 

29. A miller sold 4.5 tons of flour at $7.25 a barrel : 
how much did he get for the flour? Ana, $382.91. 

30. What is the value of 2740 lb. of hay at $12.50 a 
ton? ^n«. $17.12^. 

81. A boy sells 2 pk. 8 qt. of chestnute at $4 a bushel: 
how much does he get for them ? Ana, $2.87^^. 

32. A farmer plants 6 bu. potatoes worth $2 a bushel 
on some ground ; he pays a man for cultivating them 90 
cts. a day for 12 d., and raises 120 bu. which he sells at 
80 cts. a bushel : how much does he make ? Ana. $73.20. 

83. Into how many building lots, 140 ft. long and 20 
ft. wide, can a piece of ground containing 10 A. 1200 
sq. ft. be cut? Ana: 156. 



THE END. 



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