EPOCHS OF ANCIENT HISTORY.
^ EDITED BY TIIK ■ .
Bev. a. W. COX, taa^. al?d-by^HABLES SAij-KEY,' M.A.
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The GEEEKS and the PERSIANS. By the Rev. G W Cox M A
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' Mr, Cox's work is the first volume in
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tended for youthful students, the pubU-
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History. Mr, Cox's high reputation as
an authority on historical matters will
no doubt insure the success of The Greeks
and the Persians.'
Edinburgh Cocrakt.
The EARLY ROMAN EMPIRE, from the Assassination of Julius
?v^fTo n , ^^s^f^-^ination of Domitian, By the Rev. W. W. Cai'r^ il \
With 2 Coloured Maps, price 2s. M. '» ■ v.ai j, , ji.a.
•^^^^^o"^?-^^^' ^''°^ *^® Foundation of the City to its Destruction hv
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The ROMAN TRIUMVIRATES. By the Very Rev. Charles
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books by reason of the fact that, unlike
sclioolbooks in general, it is the outcome
ot a complete knowledge of all that
modern criticism has doue to separate
tbe fabulous in early Roman history
from that which may be accepted as
really a part of the warp and woof of the
gi-eat nation.
' Dean Merivale's volume on the
Roman Triumvirates is written with the
easy fluency and grace of a scholar who
has already travelled over the same
gi-ound in the production of a more ex-
tensive and important history. This
Jipoch differs from some others of the
series in the fact that its chief excellence
consists not so much in the unravelling
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Ihe moral to be drawn from this Epoch
IS that Rome had reached the period
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political liberty, and that the struggles
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School Board Ohbonicle.
Looking at the names which appear
upon these title-pages, it is superfluous
to say that this series of historical
epochs is no mere compilation. They
are works of flrst-rate merit, written by
masters of these fields of historic study
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evidences of old history ; and the reader
feels that he can trust himself to his
Authors as to a guide who will help him
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history as well as to make himself
acquainted with the course and causes
and consequences pf events,
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the pei-iod covered is hardly more than j
that of an ordinary lifetime, and yet it is
a period of momentous interest and of
vast developments in the history of the
human race,
' Dr. Iune's Earli/ Rome should occu-
py a place of its own among educational
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MACEDONIAN EMPIRE, its Rise and
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Assistant-Master, Murlboroiii,'h College.
ROME and CARTHAGE, the PUNIC
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By A. H. Beesly, M.A. Assistant-
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The AGE of TRAJAN and the ANTQ-
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AEITHMETIC
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V
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1876.
f o3
THE LATE MASTER OF TRINITY.
Extract from Dr. Whewell's Work on ' A Liberal
Education,' pp. 158, 159.
As the basis of all real progress in Mathematics, boys ought
to acquire a good knowledge of Arithmetic and a habit of per-
forming the common operations of Arithmetic, and of applying
the rules in a correct and intelligent manner. This acquirement
appears to be often neglected at our most eminent classical
schools. Such a neglect is much to bo regretted ; for the want
of this acquirement is a great practical misfortune, and is often
severely felt in after-life. Many persons who ara supposed to
have received the best education which the country affords, are,
in all matters of numerical calculation, ignorant and helpless, in
a manner which places them, in this respect, for below the
members of the middle class, educated as they usually are.
Arithmetic is a matter of habit, and can be learnt only by long-
continued practice. For some years of boyhood there ought to he
a daily appropriation of time to this object.
dJ
PBBFACE.
Since this book was first published, some considerable
additions have been naade to it, besides further modifica-
tions, with a view to correcting any defects which expe-
rience has from time to time detected, and bringing it up
to the reqidrements of the present day. These have been
carried out under my sanction and superintendence,
and to my entire satisfaction, by the Rev. J. Huntee,
formerly of the National Society's Training College,
Battersea, and chiefly at his suggestion ; and I consider
that the book has been much improved by them.
I have taken the opportunity, however, of my
being in England for a few weeks, to insert some addi-
tional pages on the Metric System of Weights and
Measures, the principles of which, by a rule of the
Council of Education in force in 1872, were required
to be taught to all children of Standards V. and VI.
in schools under the control of the Government. The
rule in question has, however, been since rescinded, as
requiring too much from elementary schools, while the
use of the Metric System has not yet been rendered
compulsory by Act of Parliament. But the general
adoption of that System in England is only, it seems
plain, a question of time.
J. W. NATAL*
London: December 24:, 187-i.
TABLE OF CONTENTS.
Elementary Arithmetic.
•
Page
- .1
Definitions, Notation, and Numeration
Simple Addition
- 2
„ Subtractioq -
4
„ Multiplication
- 5
5, Division
7
Answers to Examples
t
- - 9
Arithmeticai, Tables
- [9]. [10]
Compound Arithmetic.
Chap.
I. Eeduction - - -
- 11
Compound Addition
- 14
„ Subtraction
- 17
„ Multiplication -
- 10
„ Division
- 21
Square and Cubic Measure
- 25
Miscellaneous Examples
- 29
II. Greatest Common Measure
- 34
Least Common Multiple -
- 35
III, Vulgar Fractions -
- 38
Miscellaneous Examples
- 54
IV. Decimal Fractions -
- 57
Miscellaneous Examples
- 71
V. Practice -
- 74
Miscellaneous Examples
- 79
Till
TABLE OF CONTENTS.
Chap.
\I. Proportion
Single Eule of Three
Double Eule of Three
VII. Interest -
Discount -
Insurance, &c.
Stocks
Profit and Loss
Proportional Parts
concllding problems
Miscellaneous Examples
Appendix - - - " -
Standards of Money, Weight, Space, and Time
Decimal Coinage - - - -
The Metric System . - -
Notes and Examination-Paieks on Arithmetic
Examination-Papees - . -
Ans"oers to the Exan:plcs - - - ■
Page
- 81
- 84
- 94
- 99
- 105
- 107
- 109
- 112
- 114
- lis
- 126
- 141
- 143
- 149
- 156
- 167
- 178
- 200
AEITHMETIC.
Arithmetic is the science wliich treats of nujiibers — of the
mode of expressing them — of the manner of computing by
tliera — and of the various uses to which they are applied in
the practical business of life.
The number one is called unity ; and an integer^ or wliole
TiMmher^ is a collection of ones, unities, or U7iits.
The figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, denote, respeetivdy,
the numbers one, two, three, four, Jive, six, seven, eight, nine;
the figure 0, called zero or a cypher, expresses nought or
nothing ; but by means of these figures, which are called
the ten digits, or more commonly the nine digits and zero,
any number whatever can be expressed. This is effected
thus :
A figure standing by itself, or on the right hand of other
figures, has its own proper value, expressing so many units;
A figure standing in the second place from the right is
considered to express so many tens of units ;
In the third place, so many tens of tens, or hundreds of
units J
In the fourth place, so many tens of hundreds, or thou-
sands of units, &c., according to the folbwiug Table, called
the
t;u},ieration table.
1
2
8
1
4
3
5
7
1
2
3
4
1
o
3
^
£
W
B
y^
B
B
CO
R"
cc
B
02
B
w
2.
o
B
&.
a
1
S'
2,
B
&
3
2,
B
»
S
a
^
&
g
&
^
Sc'
O
o
o
o
•=^
>-*
o
I-*
g
o
5
1
s
Bf
B
i
H
1
o
1
find so pn to trillions^ quadrillions, &c. if necesEt
B
ARITHMETIC.
Notation is the art of expressing any given number by
tbeso figures ; Numeration the art of reading them, when so
expressed.
N.B. Examples in Notation and Numeration may be ob-
tained from those given in Addition and Subtraction.
The Romans used I for 1, V for 5, X for 10, L for 50, C for 100,
D or Iq for 500, M or CIo for 1000.
When any character was followed by one of less or equal value, the
expression denoted iha sum of their simple values; but when preceded
by one of less value, the difference; thus III stood for 3, IV for 4, and
VI for 6, XL for 40, and LXX for 70, &c.
Every q annexed to Iq, and every C and 3 joined to CI^, increased
its value tenfold ; thus 133 stood for 5000, CCIoo for 10,000, &c.
A line drawn over a character increased its value a thousand-fold ;
thus V stood for 5000, C for 100,000.
The following signs are also made use of in Arithmetic : .
+ (plus) shows that the number before which it stands is
to be added ;
— (minus) that the number before which it stands is to
he subtracted;
X (into) that the numbers between which it stands are to
be multiplied ;
-r- (by) that the number which stands before it is to be
divided by the one which follows ; and
= (equal) that the numbers between which it stands are
equal to each other.
Addition, — When any numbers are taken together, or
added, the resulting number is called their sum.
T? ^jj ni,/.o In order to add whole numbers together, we
Jbx. Add 94163 „ , , , ,.,.,.
21934 ^"*^*' ph'^ce them under one another, with their
7812 units-figures in the same vertical line; we then
593 add these figures thus, 5 and 7 are 12, atid 3 an
^^^"^l 15, and 2 are 17, and 4 are 21, and 3 are 24, i. c.
24 U7iits, or 2 tens and 4 units; we set the 4 under
165064 jjjg units-figures, to be the units-figure of the resi/lt,
fmd carry the 2 tens to be added to the second or tens column ; adding
ADDITION. 3
this in the same manner, beginning with the 2 carried, thus 2 and 9 are
11, ajid 4: are 15, &c,, we find the sum of the column to be 36,'i. c. 3(>
fen«, or 3 tens of tens (i. e. 3 hundreds) and 6 tens; we set the 6 under
the tens-figures, to be the tens-figure of the result, and carry the 3
hundreds to the third or hundreds column : pursuing the same course with
this, we find the sura of this column to be 40, i. e. 40 hundreds or 4 tens
of hundreds (i. e. 4 thousands) and 0 hundreds; we set the 0 under the
hundreds-figures, to be the hundreds- figure of the result, and carrt/ the 4
thousands, &c.
N.B. Any suras may be set at pleasure in Addition, and the
Answers proved by repeating the operation, beginning with the top
figure of the units column, when the result will be the same, if the sum
be worked correctly.
EXAMPLES IN ADDITION.
321413
2. 543123
3. 536123
4. 123456
452734
234512
453215
234561
130421
713145
1234
345612
3718
104234
4231
456123
24561
36142
51234
561234
341323
3451
613254
612345
761284
6. 657890
7. 692387
8. 768453
612874
278679
4956
358428
8719
5798
87958
8796
46759
67843
769.^78
54937
587999
489567
5790
495
987678
37429
87658
876578
9. Add together five hundred and ninety-seven thousand six hundred
and eighty-five, forty-nine thousand three hundred and seven, four hun-
dred and nine thousand and sixty-seven, fourteen thousand and nineteen,
seven hundred thousand and seventy-four, sixty-five thousand and nine.
10. Add together seven hundred and seven thousand four hundred and
fifty- nine, ninety-eight thousand and seventy- four, six thousand eight
hundred and seven, five hundred thousand three hundred and nine,
seven thousand nine hundred and seventy-eight, nine hundred and nine
thousand nine hundred and ninety -nine.
11. Add together ^hy-fiyQ millions seven hundred thousand and five,
seven hundred millions nine hundred and eight thousand two hundred
and five, seventy-six millions fourteen thousand and fifty -nine, eight
hundred and seventy-seven millions nine hundred and two thousand and
forty-seven, seven millions eight hundred and four thousand five hundred
and twelve, five hundred and seventy-five millions eiglit hundred and
one thousand and' ninety-nine.
b2
4 SUBTRACTION.
12. Add together three hundred and nine millions four hundred and
seventeen thousand and eighty-seven, six hundred and seventy-five
thousand and forty-nine, seven thousand and ninety-seven millions
eight hundred and fourteen thousand three hundred and five, seventy-
nine millions five hundred and four thousand and forty-nine, six thousand
and seventy-eight millions four hundred and thirty-nine thousand six
hundred and forty-seven, seven thousand millions eight hundred and
seventy-six thousand four hundred and twenty-nine.
Subtraction. — When one number is taken from another,
or subtracted, the result is calle 1 the remainder or the dif-
ference,
Ex. Fi-om 794327 In order to subtract one whole number from
Take 342814 another, we first place the number to be sub-
451513 triieted under the other, with their units-figures
in the same line ; we then take the units-figure, 4, of the lower number
from that of the other, 7, thus 4 from 7, 3, i. e. 3 units, and we place the
3 under the units-figures, to be the units-figure of the result ; then we
proceed to the tens-figures, and say, I from 2, 1, i. e. 1 ten, and we set
down 1 under the tens-figures; then to the hundreds-figures, and say 8
from 3. . . / cannot; but if we take or borrow 1 out of the 4 thousands
(leaving 3 thousands), and treat it as 1 ten of hundreds, we shall now
have 13 hundreds in the upper line; we can now say 8 from 13, 5, i.e.
5 hundreds, and we set down 5 as the hundreds-figure of the result: and
we have now to take 2 thousands from 3 thousands, or, which is just the
same, but more convenient in practice, instead of supposing the upper
figure, 4, diminished when we borrow 1, we may suppose the lower cor-
responding figure, 2, increased, i. e. we may carry one to it, and say 3
from 4, 1, i. e. 1 thousand, and so on.
N. B. Any sums may be set at pleasure in Subtraction, and the
Answers proved by adding the remainder to the lower number, when
the result will be the upper, if the sum be worked correctly.
EXAMPLES IN SUBTRACTION.
1.
765439
2.
C97438
3.
758452
4.
543625
343418
635036
418234
492708
5.
683125
6.
712345
7.
564307
8.
702306
492816
538159
479176
475429
9. From six hundred and nine thousand seven hundred and one take
three hundred and ninety-seven thousand and forty-nine.
MULTIPLICATION. 5
10. From four hundred and fifty thousand and ninety-four take nhicty-
nine thousand nine hundred and nine.
11. From seven hundred and eighteen millions fourteen thousand and
fifty -seven take ninety-seven millions eight hundred and four thousand
seven hundred and sixteen.
13, From fifty-three thousand millions eighteen thousand and ninety-
seven take forty thousand five hundred and twenty-eight millions seven
hundred and six thousand seven hundred and nine.
Multiplication is the method of finding what number would
result from adding several of the same numbers together ;
thus, if we add 6 sevens together, the result is
7-i-7 + 7-f7 + 7 + 7=42,
the same number as that given in the Multiplication-table
for the value of 6 titnes 7 : and, since the same number is
also the sum of 7 sixes, or the value of 7 times 6, it follows
that, when two numbers are multiplied together, it matters"
not which we take as multiplier.
The numbers multiplied in any case are called /acfor^, and
the result is called the product.
Ex. 1. 3467 When the multiplier, as in Ex. 1., is not higher than
? 12, we first set it with the units-figure under that of
6934 the multiplicand; then we begin to multiply, saying,
twice 7 is I A— four and carry one, i. e. we set down the 4 units under the
units- figures, and carry the 1, which means 1 ten, to be added to the
tens; we now proceed, twice ^ is 12 (i.e. 12 tens, since 6 means 6 tens),
and 1 (i. c. the one carried) zs 13. . . 3 and carry 1, i. e. we set down the 3
tens, and carry the 1, which means 1 ten of tensor 1 hundred, to be added
to the hundreds, and so on throughout the Hue.
Ex.2 3467. ...2 "When the multiplier, as in Ex. 2., is higher
692.. ..8 than 12, we first set it under the multiplicand
6934 as before, and, having multiplied the upper
31203 line \)y tiie units-figure, 2, of the lower, as in
20802 o > > 5
Ex. 1., we now multiply by the tens-figure, 9,
2399164. ...7 saying 9 times 7 is 63 (i.e. 63 tens, since 9
means 9 tens) ... 3 and carry 6; i. e. we set down the 3 tens, and carry
the 6 tens of tens or hundreds, and so on : we now multiply by the
hundreds-figure, 6, of the lower line, in the same manner; and then add
up the separate lines, when the result is the product required.. The
2, 8, and 7, on the right, will be explained presently.
6 MULTIPLICATION.
Ex. 3. 37218 Since it is immaterial which number we take as
^ multiplier, it is best always to choose that which is
22330S simplest; and if it can be separated into two or more
I factors each less than 12 (thus 42 = 6 x 7), we may
1563156 multiply separately by each, as in Ex. 3.
N.B. Any number which can be separated into factors is called a
composite number; any number which cannot be so separated, such as
7, 11, 13, 17, &c., is called a prime number.
"2700 0 If the multiplier ends with one or
— more cyphers, the sum may be worked
2268700 .V ^ i -u 1 • I
g^gr, as in the annexed example, by which
"rrTrrTr. ^ many useless cyphers are saved.
8/0O/OO 0 *' ''
N. B. Any sums may be set at pleasure in Multiplication, and the
Answers proved, either by repeating the operation with the other number
for multiplier; or by the process of casting out nines (for the proof of
which see Algebra), as follows : add up the figures in the upper number,
divide this by 9, and set down the rem''; do the same with the other
number; then do the same with the product of these rem'*, and with the
product of the two numbers; and if tlie new rem" are the same, the sum
is most probably right; but, if different, it is certainly wrong. Thus in
Ex. 2,, the first pair of rem" are 2 and 8, and their product 16; the
rem' from this is 7, the same as from the Ans': in Ex. 4., the first pair
of rem" are 1 and 0, and their product is 0; the rem' from this is 0, the
same as from the Ans'. See Note I.
It is desirable that the pupil should be made to apply one or both of
these methods to the Examples below given.
EXAMPLES IN MULTIPLICATION.
1.
4.
7.
10.
345673x2
371281 x5
378914x8
978564x11
25. 234915x123
28. 391525x861
31. 1644405x7749
34. 1389294 x 8900
2.
5.
8.
11.
457632x3
635432 X 6
476539 X 8
496782 X 12
26. 704745x615
29. 1174575x2214
32. 231549x8856
35. 926196x7896
415763x4
421375 x7
435976x9
876549x12
13.
378125x16
14.
456932 X 18
15.
712436x24
16.
543817 x27
17.
593654 X 30
18.
697128x36
19.
765438x40
20.
596437 X 45
21.
642198x60
22.
756328 X 72
23.
814765x84
24.
913748x96
27. 469830x369
30. 3523725 x 2583
33. 463098 x 7380
86. 2778588x9867
DIVISION. 7
Division is the metliod of finding how often one number is
contained in another, i. e. how often one number must be
taken to make up another. Hence Division bears the same
reference to Subtraction^ as Multiplication bears to Addition;
for we might go on subtracting the divisor from the divi-
dend, and then from the 1st rem^, then from the 2nd rem'',
and so on, until the final rem"^ is either zero, or is less than
the divisor itself; and if we counted the number of times we
had subtracted it, this would be the result required, or, as it
is called, the quotient. But the Multiplication-table will
enable us much more easily to divide one number by another;
thus, since 7 times 9 is 63, if we divide 63 by 7, we shall have
the quotient 9, or if by 9, the quotient 7 : and the method of
applying it to more difficult cases will be seen by what follows.
Ex. 1 4^)2379 When the divisor, as in Ex. 1., is not higher
177^ than 12, we first set it in a loop before the
* dividentl ; then -we take the first figure of the
dividend, 2, i. e. 2 thousands : but, since 4 will not be contained at all
in this, we take then the first two figures, 23, i. e. 23 hundreds, and say
4 is in 23 . . 5 times and 3 overj and we set down the 5, i. e. 5 hundreds,
In the quotient, and carry the 3 hundreds, or 30 tens, to the tens-figure,
7, of the dividend: we have now 37 tens, to be divided by 4; we say,
^liercforc, 4 is m 37 . . 9 times, arid 1 over, and we set down the 9, i. e.
9 lens, in the quotient, and carry the 1 ten or 10 units to the units-
figure, 9, of the dividend: we have now 19 units to be divided by 4;
we say, therefore, 4 is in 19 4 times aiid 3 over, and we set down the
4, i. e 4 units, in the quotient, and place, as is usual, the final rem*
3 over the divisor with a line between them, as |(</iree-/bMrfAs), a quantity
meaning 3-r4, and called a fraction, of which more will be said hereafter.
It appears then that 4 will be contained 594 times in 2379, with 3
over; i.e. we might subtract 4 from 2379 594 times, and have still 3
remaining. This is an example in Short Division.
Ex. 2. 42) 379543 (9036|i When the divisor, as in Ex. 2., is
378 higher than 12, we place it, as before, in
154 a loop before the dividend, and the quo-
^26 tient in a loop after it ; and we see that
283 42 will not be contained in the 3 (i. e. 3
^^^ hundreds of thousands'), nor in the 37
SI (i. e. 37 tens of thousands), but will be
8 DIVISION.
contained 9 times in the 379 (i. e. 379 thousands)', or, which is the same
thing, but more convenient in practice, we take the first figure only of
the dividend, and say 4 is in 37 . . 9 times; we set therefore the 9 (i. e. 9
thousands) in the quotient, and, multiplying 42 by 9, subtract the pro-
iluct, 378 (i. e. 378 thousands) from the dividend; and we have now the
rem', i. e. 1 thousand or 10 hundreds, to be carried to the hundreds: we
take in then the hundreds-figure, 5, of the dividend, and have now 15
hundreds to be divided by 42; we say then (42 is in 15, or) 4 is in 1 . »
I cannot; we set, therefore, 0 (i.e. 0 hundreds) in the hundreds place of
the quotient, and have now 15 hundreds, or 150 tens, to be carried to the
iens; we take in tlien the tens-figure, 4, of the dividend, and have now
154 tens to be divided by 42; we say then 4 is in 15 . . 3, and we set the
3, i. e. 3 tens, in the quotient, and so on till, at last, we have the final
rem' 31, which we set over tlie divisor, as a fraction, and have the whole
quotient 9036|^. This is an example in Long Division.
Ex. 3. But when the divisor, as in this case, is made up of two or
more factors, less than 12, it is often more convenient to divide by each
separately, as follows.
6) 379543 There is here a fraction | over in the Jirst quo-
7) 63257i tient, and a rem' 5i in the second, which, according
9036^ to our previous practice, should be written -^ ; but
such an expression may always be simplified (as will be shown here-
after) by putting the rem' 5^ in the form -^, (which we obtain by multi-
plying the 5 by the 6, and adding in the 1); and then multiplying the
6 by the 7, so making ||, the same as the fraction obtained by the other
method. See Note II.
Ex. 4. 39,00) 7134,53 (182|g§§ In this Ex. and in all others where
.^^ there are cyphers at the end of the
323 divisor, the work may be abridged
312 by marking off, with a comma, or
114 point, these cyphers, and as many
"Q figures also from the right of the
3653 dividend ; then Ave proceed, 3 is in
7 tioice; but on trial we should find that 2 would be too large for the
first figure in the quotient, (which comes of using 3 for the divisor instead
of 39, and this difficulty will sometimes occur, but not so as to embarrass
the student, when he gets accustomed to division); we set, therefore, 1 as
the first figure in the quotient, and go on, as before, till we have taken
down all the figures before the point in the dividend ; and then we com-
ANSWERS TO THE EXAMPLES. 9
plete the last rem' by taking down the two figures cut off, and put it over
the divisor as a fraction.
N. B. Any sums may be set at pleasure in Division, and the an-
swers pi'oved by either of the methods given in Multiplication ; since
the product of the divisor and quotient (if the sum be worked correctly)
will give the dividend^ diminished, however, by the remainder (or upper
number of the fraction; if any. Thus in Ex. 2., the divisor is 42 and
quoiient 9036, and the rem" from these are 6 and 0; the product of
tlicse is 0, and the dividend, diminished by the rem' 31, is 379512, and
the rem" from these are 0, 0 : in Ex. 4., the dirisor is 3900 and the
quctient 182, and the rem" from these are 3 and 2 ; the product of
these is 6, and the dividend diminished by the rem' 3653, is 709800,
and the rem" from these are 6, 6.
The pupil should be required to apply one or other of these methods
of proof in the following examples.
1. 4325l6-r2.
4. 713915-J-5.
7. 465328-7-8.
10. 457848 -r 11
EXAMPLES IN DIVISION.
2. 3517894-3.
5. 385734-7-6.
8. 395424 -T-8.
11. 716855-rl2.
3. 543750-^4.
6. 616824-7.
9. 567035-r9.
12. 936571^12.
13.
2366745-rl5.
14.
7954326 -r 18.
16.
6549372^36.
17.
4733491-^45.
19.
7825687-4-64.
20.
3795469 -^70.
22.
6598769-^84.
23.
8791605-^88.
15. 6342576-r24.
18. 5674331 -r 60.
21. 3754329 -r 80.
24. 7654325 -j-96.
25. 3765897 -^23.
28. 395437 l-f-47.
31. 34568135 -r 357.
34. 56854327-7-7323.
26. 4613578-^37. 27. 5123495-r41.
29. 3755123-7-234. 30. 5764123-^340.
32. 76549139-^543. 33. 29876533 -r- 6930.
35. 95642371-^8790. 36. 34568795-5-9879,
ANSWERS TO THE PRECEDING EXAMPLES.
ADDITION.
1.
1274170.
2.
1634607.
3.
1659291. 4.
2333331.
5.
3005313.
6.
1537206.
7.
1648127. 8.
20676S7.
9.
1835161.
10.
2230626.
11.
2294129927.
12.
20566726566.
nS
10
ANSWERS rO /IIE 1LXA5IPLES.
SUBTRACTION.
1. 422021, 2. 62402. 3. 340218.
5. 190309. 6. 174186. 7. 85131.
9. 212652. 10. 350185.
11. 620209341. 12. 12471311388.
4. 50917.
8. 226877.
MULTIPLICATION.
1. 691346. 2. 1372896. 3. 1663052. 4. 1856405.
5. 3812592. 6. 2949625. 7. 3031312. 8. 3812312.
"9. 3923784. 10. ' 10764204. 11. 5961384. 12. 10518588.
/13. 605GOOO. 14. 8224776. 15. 17098464. 16. 14683059.
17. 17809620. 18. 25096608. 19. 30617520. 20. 26839665.
21. 38531880. 22. 54455616. 23. 68440260. 24. 87719808.
25. 28894545.
28. 337103025.
31. 12742494345.
34. 12364716600.
26. 433418175.
29. 2600509050.
32. 2050597944.
35. 7313243616.
27. 173367270.
30. 9101781675.
33. 3417663240.
36. 27416327796.
DIVISION.
1. 21625i. 2, 117263. 3. 135939. 4. 142783.
5. 64289. 6. 73832. 7. 58166. 8. 49428.
9. 63003|. 10. 41622^. 11. 59737i|. 12. 78047-;^.
13. 157783.
16. 181927.
19. 122276§|.
22. 78556|^.
25. 163734l|.
31. 96829|ff.
S4. 7763II&.
14. 441907.
17. 105188fi.
20. 54220^.
23. 99904 1^.
26. 124691ii.
29. 16047l|f.
32. 140974ff|.
3.5. I0880ii|l.
15. 264274.
18. 9457211.
21. 46929^.
24. 79732|5.
30.
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111
II
CHAPTER I.
ELEMENTARY RULES.
deduction.
1. This is the name given to the method of converting
a quantity expressed in one denomination to another, as from
pounds to pence, from ounces to tons, from inches to yards,
&c.; thus £3=720c?., 2r»0880 oz.= 7 tons, 72 in.=2 yds. &c.
2. To reduce a quantity to a lower denomination.
Rule. Multiply the given quantity by the number whicli
shows how many of the next lower denomination make one
of the higher ; and so on, step by step, till we arrive at the
proposed lower denomination.
Ex. 1. Reduce £37 to pence.
^Q Here, since £\ contains SO*, we fil'St multiply the
7405. :S37 by 20, to bring them into shillings ; and then
12 since 1*. contains 12(/., we multijily these shillings by
Alls 888oZ ^^' *® bring them into pence. See Not3 ni.
If the given quantity consist of several terms of different
denominations, we must add in with each product, as we
proceed, the term (if any) of corresponding denomination.
Ex. 2. Beduce £15 7*. 0|(f. to farthings.
£15 Is. Qld.
20
307s. Here we first reduce £l5 to shillings, adding
12 in the 7s. ; then these shillings to pence ; and lastly
3684d ^hese pence to farthings, adding in the 3 farthings.
4
14739/ Ans,
12 REDUCTION.
Reduce Ex. 1.
1. £513 to farthings ; and 320 guineas to halfpence-
2. £2000 to halfcrowns ; and 2000 guineas to sixpences.
3. £27 10s. to pence ; and l7s. 6|c?. to farthings.
4. £75 10s. 6c?. to sixpences ; and 220 crowns to fourpcnny- pieces.
5. £47 105. life/, to farthings, and £85 Os. lOld. to halfpence.
6. £29 10s. O^d. to halfpence ; and 1373 halfcrowns to farthings.
7. 23 tons to pounds ; and 115 cwt. to ounces.
8. 27 lbs. to drams ; and 11 tons to ounces.
9. 3 qrs. 14 oz. to drams ; and 47 cwt. 25 lbs. to ounces.
10. 34 cwt. 3 qrs. 11 oz. to drams ; and 2 tons 3 qrs. 5 oz. to ounces.
11. 4 tons 15 cwt. 2 qrs. 12 lbs. to lbs. ; and 14 cwt. 1 qr. 8 drs. to drams.
12. 15 cwt. 2 lbs. 9 oz. to ounces ; and 3 tons 3 qrs. 3 oz. to drams.
13. 16 lbs. Troy to grains ; and 105 lbs. Troy to dwts.
14. 27 oz. 10 dwts. to grains ; and 3 lbs. 13 dwts. to dwts.
15. 9 oz. 17 dwts. 22 grs. to grains ; and 2 lbs. 11 oz. 20 grs. to grains.
16. 7 oz. 19 dwts. to grains ; and 3 lbs. 9 oz. 7 grs. to grains.
17. 23 miles 7 fur. to feet ; and 2 lea. 2 m. 7 fur. to yards.
18. 3 fur. 135 yds. 4 in. to inches ; and 5 fur. 171 yds. 2 ft. to inches.
19. 2 lea. 2 m. 2 fur. 200 yds. to feet ; and 5 m. 200 yds. 3 in. to inches.
20. 73 yds. 3 qrs. to nails ; and 35 ells 4 qrs. to nails.
21. 54 A. 3 B. to poles ; and 17 sq. yds. 8 ft. to inches.
22. 7 a. 12 p. to poles ; and 29 sq. yds. to square inches.
23. 13 cub. yds. to feet ; and 7 cub. yds. 20 ft. to inches.
24. 23 cub. yds. 1000 in. to inches ; and 12 cub. yds. 23 ft. to inches.
* 25. 137 gals, to pints ; and 13 gals. 3 qts. to gills.
26. 17 qrs. to gals. ; and 220 bushels to quarts.
27. 3 loads 3 qrs. 3 pks. to gals. ; and 2 qrs. 1 gal. to pints.
28. 3 loads 3 bus. to quarts ; and 2 qrs. 7 bus. 2 pks. to gallons.
29. 27 years to days ; and 3 yrs. 315 d. to minutes.
30. 5 mo. 3 w. 4 d. to hours ; and 27 w. 5 d. 15 hrs. to seconds.
3. To reduce a quantity to a higher denomination.
Rule. Divide the given quantity by the number which
shows how many of the lower denomination make one of the
next higher ; and so on, step by step, till we arrive at tlie
proposed higher denomination.
Ex. 1. Reduce 137 B20farthivgs to shillmys.
4) 137520/: ^^^^'^ ^^'^ ^^^^ divide the given number of far-
, ^ things by 4 to bring them into pence, and then
^ -1 -_L we divide these pence by 12 to bring them into
2865s. Ans. shillings.
REDUCTION. 1 3
If there should be a remainder after any division, we must
set it down as a term of the same denomination as the
dividend from which it came.
Ex. 2. Reduce 13799 farthmgs to pounds.
4) 13799/ Here, after dividing the given farthings by 4,
12) 3449(/ . 3f. ^® ^^^^ ^ ^^^^ ^' which means that in 13799/
.-7—7— , tlierc are 3449c/,, and 3/ over ; we set down
' — -! — - ' ' ' ' therefore the rem' as 3/., that is, as a term of the
£14 7s. 5|rf. Ans. g^^g ^^.^n ^g jj^g dividend from which it came ;
after dividing the pence by 12, we have a rem' 5, which we set down, for
a similar reason, as 5c?. ; and after dividing the shillings by 20, we have
a rem' 7, which we set down as 7s.
N.B. We have divided by 20 by the usual short method, cutting off
the last figures of the dividend and divisor.
Reduce Ex. 2.
1. 78790236s. to guineas ; and 150080 sixpences to pounds.
2. 1758960/ to crowns ; and as many halfpence to halfcrowns.
3. 480144/ to sevcnsliilling-pieces ; and 50000c/. to pounds.
4. 284061/ to pounds; and 110012(/. to pounds.
5. 101010c/. to guineas ; and 123290/: to pounds.
6. 350000/ to pounds ; and 538483 halfpence to guineas.
7. 37568 lbs. to tons ; and 108190 drs. to cwt.
8. 2345820 drs. to tons ; and 108234 oz. to cwt.
9. 100000 oz. to tons ; and 12821 drs. to qrs.
10. 229601 oz. to tons ; and 314735 drs. to cwt.
11. 156423 drs. to cwt. ; and 1008001 oz. to tons.
12. 237023 oz. to tons ; and 371283 drs. to cwt.
13. 13172 grs. to lbs. Troy; and 30066 dwts. to lbs. Troy.
14. 17073 grs. to lbs. ; and 12327 grs. to lbs.
15. 108970 grs. to lbs. ; and 189081 grs. to lbs.
16. 272821 grs. to lis. Troy ; and 127272 grs. to lbs. Troy.
17. 36090 ft. to milcG ; and 231031 yds. to leagues.
18. 120835 in. to fmlongs ; and 378135 ft. to miles.
19. 517900 in. to miles ; and 183810 ft. to leagues.
20. 13587 na. to yards ; and 181970 na. to el!s.
21. 121321 p. to acres ; and 33333 sq. inches to yards.
22. 20000 r. to acres ; and 20000 sq. inches to yards.
23. 200000 cub. in. to yards ; and 138297 cub. in. to yards.
24. 106921 cub. in. to yards ; and 180831 cub. in. to yards.
25. 18191 pts. to gallons ; and 30983 gills to gallons.
14 ■ ADDITION.
26. 28716 qts. to loads ; and 91356 pints to quarters.
27. 89765 pks. to loads ; and 56789 pts. to loads.
28. 356187 qts. to loads ; and 598712 gals, to quarters.
29. 137819 days to years ; and 3561829 sec. to weeks.
30. 235967 hrs. to weeks ; and 71871900 see. to years.
Addition.
4. Rule. Set the quantities to be added under one another,
so tliat terms of the same kind may be in the same column.
Add the numbers in the right-hand column ; divide the
result by the number of things in this column, which make
one in the next ; set the remainder, if any, under the first
column, and carry the quotient to be added to the next ; and
so on with all the columns.
£ s. d.
Ex. 1. 13 0 8 Here, adding Tip the pence in the right-hand
2 5 6 cohimn, wc have A2d. ; in order to bring this into
23 4 7 shillings, we divide by 12, wliich goes 3 times with
37 8 10 6 Qyer, so that 42r/. = 3*. 6c/. ; we set down the Gd.
^^ ^ 7 under the first column, and caiTy the 35. to the
0 13 4 .V
£89 2 6
£ s. d.
next ; and so on.
Ex. 2. 22 4 6i Here, adding up the farthings in the right-hand
0 2 6i column, wo have 7/., which=lfcf. ; we therefore
set down the ^t?., and carry \d. to the next column.
36 0 43
7 1 1^
£65 8 65
Ex. 3.
£ s. d. £ s. d. £ s. d. £ s. d.
1. 3 13 6 2. 14 13 7 3. 65 4 3^ 4. 23 13 6^
2 11 9 22 15 9 22 0 2i 35 17 Qi
3 17 8 29 11 11 46 15 7^ 35 7 72
2 5 2 82 17 7 73 12 62 67 16 8i
5. 41 16 8i 6. 36 17 6| 7. 24 16 8^ 8. 71 17 2i
21 10 7i 14 17 6 51 14 2| 41 2 9i
31 17 72 21 12 72 11 0 8 54 7 62
24 16 81 13 13 3i 27 1 3 2 11 6
9. 16 5 4 10. 11 13 32 11. 42 13 4 12. 76 15 42
35 7 9| 32 12 2i 17 6 82 32 4 10
16 10 8 13 13 32 90 9 8 21 3 7*
42 13 81 24 3 0 21 12 4l 62 18 4l
ADDITION,
15
lb. oz. dr.
qr.
lb. oz.
cwt. qr.
lb.
qr. lb. oz.
13.
7 3 13
14.
3
27 15
15. 18 2
23
16.
13 25 7
12 0 9
1
11 2
17 1
19
4 18 6
23 13 14
0
21 13
15 3
17
24 17 5
3 15 7
dr.
2
13 14
9 2
23
tons
37 9 14
qr. lb. oz.
cwt. qr.
lb. oz.
cwt. qr. lb.
17.
2 1.5 13
11
18. 27 2
13 4
19.
4
17 3 18
3 5 11
8
32 1
12 15
2
3 0 15
2 27 13
2
28 0
15 12
13
9 2 25
3 17 15
4
lb.
32 1
14 3
gr
22
18 3 15
oz. dwt. gr.
oz. dwt.
oz. dwt.
lb. oz. dwt.
20.
9 17 23
21.
23
8 14
22. 7 17
21
23,
.25 8 14
4 18 20
7
9 19
11 5
13
37 3 15
7 5 15
37
5 3
4 14
20
25 9 10
8 19 4
gr.
15
7 13
10 17
5
lb.
44 7 11
lb. oz. dwt.
lb. oz.
dwt. gr.
oz. dwt. gr.
24.
12 5 13
22
25. 35 3
4 12
26.
27
0 17 22
24 7 19
13
27 8
14 22
5
9 0 23
47 11 17
19
41 9
17 10
17
8 11 13
31 4 11
17
oz
2 3
13 21
• gr.
22
7 9 15
dr. scr. gr.
dr. scr.
dr. scr,
oz. dr. scr.
27.
5 0 13
28.
11
7 2
29. 7 1
19
30. 11 7 2
7 2 14
4
3 2
8 0
1
10 5 2
3 1 17
10
5 0
11 2
13
5 2 1
6 0 12
9
4 1
9 1
14
116 2
yds. ft. in.
fur.
po. yds.
m. fur.
yds.
lea. m. fur.
31.
12 1 11
32
. 7
31 41
33. 5 7
137
34. 7 1 6
22 2 9
3
19 21
2 4
121
8 2 4
9 0 3
8
27 3
8 6
213
1 0 5
13 1 4
4
35 5
3 5
23
9 1 7
fur. po. yds.
po.
yds. ft.
yds. ft.
in.
po. yds. in.
35.
, 5 33 4i
36,
. 27
41 2
37. 5 2
10
38
. 7 31 11
7 21 3|
35
31 1
8 1
4
9 2 10
2 13 2i
24
41 0
6 0
7
5 11 8
6 21 5
In.
13
3 1
m. fur.
9 2
5
m.
6 21 6
po. yds. ft.
po. yds.
fur. yds. in.
39
.731
11
40. 14 3
17 21
41. 3
5 137 9
12 21 2
4
23 5
33 4
7
7 77 7
9 4 0
7
37 1
24 5
9
6 203 6
2 31 1
9
43 7
31 11
5
4 156 2
It>
ADDITION.
42.
yds. qrs. na.
25 3 2 43.
37 0 3
54 1 1
49 2 3
yds. qrs. n.i.
183 3 2
297 0 1
328 2 3
169 1 2
4i.
ci:s
:-9
C7
82
98
qr?.
3
4
1
3
rn.
3
1
3
2
f-Hs q-s. na.
45. 35 2 3
42 4 5
37 2 2
25 4 3
46.
s.yds. s.ft. s.im
20 8 100 47.
31 7 85
24 5 34
37 8 113
r. s.yds. s. ft. s. in.
2 13 7 85
3 20| 8 24
5 2.5i 6 99
4 22f 8 37
c.yds. eft. c. in.
, 13 25 872
22 17 1000
34 11 1534
21 8 479
R. p. s.yds.
7 33 201 .
8 13 14i
7 25 2i
6 17 ?.l
18. 27
35
22
45
s. yds.
121
271
11
t. c.in.
856
979
787
842
R.
2
3
1
0
5
r.
31
24
17
29
R.
2. 37
21
18
25
A. R. P.
49. 27 1 31
41 2 28
51 0 19
42 1 25
50.
A. R. P.
51. 35 1 23
9 2 15
11 1 24
42 0 35
c. yds. c. f
54. 27 22
31 15
24 19
22 6
p. s. yds. s. in.
33 231 121
25 17 135
17 201 102
12 25 97
53.
c
55.
. yds. c. ft. c. in.
14 20 1431
32 3 1560
25 18 937
22 21 1364
56.
gal. qts. pts. g;il. qts. pts.
27 3 1 57. 17 3 1
31 2 0 24 2 1
54 11 35 3 0
37 0 1 25 2 1
56
pks.
!. 3
4
5
7
gal. qts.
1 3
0 2
1 1
1 3
bus. pks. gal.
59. 23 3 1
31 2 1
24 0 0
35 3 1
60.
qrs. bus. pks. Ids. qrs. bus.
13 3 2 61. 13 4 7
24 6 1 24 3 4
37 3 1 37 4 0
43 5 2 43 2 1
bus
62. 31
25
41
27
. gal. qt.-:.
1 3
0 2
1 1
1 3
bus. pks gal.
63. 29 3 1
37 2 0
53 3 1
47 2 1
64.
gal. qts. pts. gills.
, 22 3 1 3
31 2 0 1
13 3 1 2
24 3 1 1
bus. pks.
65. 13 2
42 3
51 1
47 3
gal. qts.
1 3
1 2
0 3
1 2
66
qrs. bus. pks. gal.
. 23 3 3 1
32 4 1 0
41 6 2 1
52 2 0 1
67.
d. hr?. min. sec.
5 13 39 42
4 22 19 33
6 20 29 45
4 17 59 59
mo. w.
63. 13 3
21 2
37 3
41 2
d.
5
4
6
5
hrs.
11
15
17
19
d. hrs. min. sec.
69. 4 11 39 28
2 13 10 32
5 21 40 29
7 23 19 19
70.
yrs. d. hrs. min.
6 130 23 15
7 354 10 17
8 45 22 14
9 313 13 17
yrs. w.
71. 14 13
22 47
35 39
21 44
d.
5
4
3
6
lirs.
23
3
18
15
yrs
72. 8
6
5
7
• d. hrs. min.
244 22 49
315 17 38
223 13 45
129 21 48
£ s.
d.
1. 34 17
9f
27 8
^
£1 9
^^i
£ s.
d.
2. 19 12
8^
IG 17
^\
SUBTRACTION. 17
Subtraction,
5. Rule. Set the quantity to bo subtracted under the
other, so that terms of the same kind may be in the same
coUimn.
Subtract the right-hand term of the lower line from that
of the upper, if possible ; if not, subtract it from the number
of things in this column, which make one of those in the
next, and add the upper term to the remainder ; place the
result under the first column, and carry one thing to the
lower term of the next ; and so on with all the columns.
Here, taking \d. from ^., we have left \d. to he
Ex. 1. 34 17 9| set under tlic farthings ; then takhig 4 J. from 9(/.,
we have left bd. to he set under the pence ; and so
on.
Here Vv^e cannot take yi. from \d. ; we borrow
Ex. 2. 19 12 8i therefore Irf. from the 8 rf., and convert it into far-
things, thus changing the ^\d. into Id. + lid., or
£2 15 3f 7 J. 5/; taking, then, the i<l or 2/ from 5/, we
have left 3/. or fd, to be set under the farthings, and have now to take
4d. from Id., which leaves 3d to be set under the pence.
N. B. In practice, it is best to take the |rf. at once from the \d. bor-
rowed, which leaves \d., and add in the \d. to this rem'', which gives |J.
ns before; and also, instead of t' king Ad. from Id., we may take bd. from
8f/., which will leave the sanr" rem' 3J., i. e. we need not alter the quan-
tity from which we subtract;, if we add, or carry, one to the quantity-
subtracted.
Again, as we cannot take lis. from \2s., we borrow £l from the £19,
and thus taking 17s. from £\ 12«. or 325., we have left 15s., and then,
taking £16 from £18, we have left £2. Here, too, it is best to take the
17s. at once from the £l borrowed, which leaves 3s., and add to this the
12s., which gives 15s. as before; also, to carry £l to the £16, making
£17, and take this from the original £19, which leaves £2 as before.
£ g^ ^^ Here, taking |cZ. from \d. borrowed, we have \d.
Ex. 3. 23 6 0\ left, to which we add the \d., making |J. to be set
22 18 11| down; then carrying \d. to the llc^.,we have \2d.,
£0 7 0| which we take from Is. borrowed, and have no rem"";
again, carrying Is. to the 18s., we have 19s., which we take from £l
borrowed, and have Is. left, to which we add the 6s., making 7s. to be
set down; and carrying £1 to the £22, we have £23 to subtract, and
no rem'. ^
IS SUBTRACTION.
Ex. 4.
£ s. d.
23 10 8
£ *. d.
2. 45 14 7i
3.
74
0
6^
4. 89 15 7
13 7 5
12 7 51
13
8
4i
74 11 9
5.
93
0
9
37
10
11
9.
137
13
0}
111
15
n
17.
qrs. lbs.
17 11
OZ.
3
8 27
15
21.
OZ. dwt,
11 19
. gr.
3
8 14
17
6.
24 0 5
15 12 11
10.
234 0 111
195 18 10|
14.
qrs. lbs. cz.
13 3 1
5 12 14
18.
tons cwt. qrs.
32 1 1
30 14 3
22.
OZ. dwt. gr.
32 7 21
18 9 22
7.132 11 61
129 13 41
11.317 14 01
239 18 10|
lbs. OZ. dr. qrs. lbs. cz. cwt. qrs. lbs.
27 11 3 14. 13 3 1 15. 33 0 11
13 7 1 5 12 14 12 1 24
cwt. qrs. OZ.
19. 27 1 3
13 0 7
Ib^ OZ dwt.
23. 13 7 15
6 11 18
8.
225 0
0
37 18
9|
12.
345 0
0
129 17
8|
16.
qrs. lbs.
2 23
OZ.
0
1 25
9
20.
cwt. lbs.
45 0
OZ.
3
44 6
13
24.
OZ. dwt.
11 0
gr.
0
2 18
22
OZ. dwt. gr. OZ. dwt. gr. oz. dwt. gr. oz. dwt. gr.
25. 23 0 4 26. 3J. 0 0 27. 22 2 2 28. 42 0 3
1 15 20 Oil 13 13 11 11 27 13 21
dr. scr. gr. oz. dr. scr. llig. oz. dr. dr. scr. gr.
29. 7 1 18 30. 11 0 0. 31. 37 7 1 82. 8 0 11
4 0 19 8 5 2 19 II 2 6 2 15
yds. ft. in. po. yds. ft. ' fur. po. yds.r tn. fur. vds.
£3. 13 17 34. 23 3 1 35. 6 37 2 36, 13 6 'l23
11 2 10 13 41 2 1 15 41
m. fur. po.
37. 24 0 7
11 5 18
8 5 2
34.
38.
po. yds. ft.
23 3 1
23 41 2
fur. po. yds.
6 37 4
5 18 41
42.
yds. ft. in.
23.; 0 0
15 2 7
lea.
m.
fur.
39.
37
0
5
18
0
7
yds.
qrs.
na.
43.
17
3
2
13
0
1
8
7
219
fur.
po.
yds.
40.
7
23
31
6
35
5
ells
qrs.
na.
44.
24
1
3
19
2
1
po. yds. ft.
41. 23 3 2
15 41 1
s.yds. s ft. s.in.' 4tP. s.yds. s.ft. R. p. s.vds. A. r. p.
45. 13 2 73 '^0^22 13 5 47.3 2 25 48.37 2 29
6 8 131 ^ *■ 13 201 8 2 35 281 23 3 35
-^, —
A. R. P. * R. P. s.j'ds. R. S.vds. S.ft.
49. 45 2 35 50. 2 35 20 51. 10 131 4
19 3 39 1 21 281 8 10 7
s
.yds.
s.ft.
s.in.
52.
12
2
13
8
7
130
MULTIPLICATION.
19
c. yds. eft. c.in.
53. 23 13 357
10 25 lOU
c.yds. eft. c.in.
54, 37 2 459
7 24 1532
c.yds. eft. c.in,
55. 45 24 656
12 19 999
c.vds. eft. c.in.
56. '27 13 2
13 23 731
57.
gals. qts. pts.
36 2 0
33 3 1
61.
qrs. bus. pks.
45 3 1
39 7 2
65.
hrs. m. s.
22 39 19
8 41 30
69,
yrs. d. hrs.
32 131 22
19 300 13
58.
gals. qls. pts.
35 0 1
29 3 0
62.
Ids. qrs. bus.
22 3 5
9 3 7
66.
d. hrs. m.
14 17 20
6 21 35
70.
yrs, w. d.
27 35 4
18 47 6
^.
pks. gals
.qts.
59.
23 1
0
19 1
3
bus. pks
gals
63.
57 1
0
39 3
1
60.
bus. pks. gal
47 2 0
28 3 1
64.
Id--, qrs. bus
5 1 1
2 4 5
68.
mo. \v. d.
12 2 5
8 3 6
72.
yrs. d. hrs.
26 213 11
19 231 21
\v. d. hrs.
67. 3 5 2
2 6 13
yrs. w. d.
71. 45 45 3
36 1 6
Multiplication .
6. Rule. Set the multiplier under the right-hand term
of the multiplicand ; multiply this term by it, and find, as
before, how many are to be carried to the next term, writing
the rem^ under the right-hand term : then multiply the next
term, and add in the number carried ; and so on.
Here 5d. x 4 = 20^, = 1». Sd. ; we set down 8ci.,
and carry 1*. : — 135, x 4 = 52s., and, adding the
Is. carried, we have 53s. =£2 13s.; we set down
13s. and carry £2:— £23 x 4 = £92, and, adding
the £2 carried, we have £94.
Here 2/. x 11 =22/ =5 J. 2/ or SlJ.; wc set
down Id., and carry 5d. ; and so on.
Ex. I. £23 13
Ans. £94 13 8
Ex. 2. £37 13 Si
11
^ws. £414 10 9.i
Ex. 5.
3,
5.
7.
9.
11.
13.
15.
17.
19.
£1.
23.
£ s.
23 8
59 13
78 2
99 17
171 13
134 6
1G5 14
115 7
124 5
xlO
171 13 11 xll
37 0 2| X 12
128 17 3 xl2
£
s.
d.
2.
37
13
5|x 2
4.
48
17
7ix 3
6.
96
15
6ix 4
8.
75
14
2|x 5
10.
154
11
32 X 6
12.
161
12
7|x 7
14.
173
18
51 X 8
16.
135
15
4|x 9
18.
175
4
91x10
20.
183
12
lOfxll
22.
51
10
01 X 12
24.
171
13
51x12
20
MULTIPLICATION.
7. When the multiplier is large, but is composed of two or
three factors* we may multiply separately by each of these.
Ex. 1. Multiply £23 Us. 4ft/. by 36.
■ 4 X 9, or = 3 X 12, the sum may stand thus:
£ s. d. £ s. d.
or 23 11 4| or 23 U 4?
4 3
incc 36 = 6
x6
£
."?.
d.
23
11
4f
6
141
8
?
94
70
14 2|
12
848 10 3 Ans.
848 10 3 Ans.
848 10 3 Ars.
Ex. 2. Multiply £17 3*. 0^1. by 140.
Since 140 = 4 x 5 x 7, the sum may stand thusi
£ s. d.
Ex. 6.
17
3 01
4
68
12 2
5
343
0 10
7
2401
5 10
£ s.
d.
£ s.
d.
1.
23 17
Six
15
2.
79 14
101 X 18
3.
93 8
31 X
21
4.
49 12
8 X 28
5.
68 7
4|x
35
6.
97 19
91 X 48
7.
87 4
31 X
64
8.
92 11
10 X 70
9.
37 13
21 x
81
10.
42 10
91 X 88
11.
98 18
3 X
96
It?.
43 12
5|xl32
13.
£2 10
81 X
128
14.
3 15
6 xl76
15.
10 11
8|x
270
16.
13 7
4|x275
8. When, however, the multiplier, though large, cannot bo
broken up into factors, we must proceed as in the first case.
£ s. d. Here 3/. x 37 = lll/ = 27rf. 3/, or 27^^/.; we set
Ex. 23 1 1 4f doAvn ^d., and carry 27^/. :— 4d x 37 = 148rf., and, add-
37 ing the 27 d., we have 175tf. = 14s. 7c?.; we set down
£872 1 7^ Id. and carry 145.; and so on.
* In order to find thesp, note that any no. is exactly divisible by 5, if it ends in 5 or 0;
by 1, 4, 8, if the no. formed by its last orie, two, three figs, respectively is div. by 2, 4, 8 ;
by 3 or 9, if the sum of its figures is divisible by .3 or 0, respectively ;
by 11, if the sums of its figs, in odd and even places, when div. by 11, leave the same
renir:
Thus 75 and 30 are each divisible by .5, since they end in h and 0 respectively ;
?.A bv 2, since 4 is div. by 2 ; 7ofi by 4, since .^^6 is by 4 ; 1528 bv 8, since 528 is by 8 ;
72908374 by 11, since figs, in orftf places =7+9+8 +7=31, and'in eveM = 2+0+3+4 = 9,
. and 31 and 9, when div. by 1 1, leave th? same remr, 9,
Ex. 7.
DIVISION.
£ s.
d.
£ 3,
d.
1.
43 8
6ix 19
2.
47 13
2|x 23
3.
33 15
Six 29
4.
79 16
3 X 34
5.
18 15
2ix 47
6.
24 14
31 X 62
w
19 10
Six 79
8.
15 17
45 X 93
9.
23 18
eixlOG
10.
IG 13
7|xl39
21
11, 3 qrs. 6 lbs. 13 oz. 15 dr. x 8
13. 5 tons 27 cwt. 27 lb. 5 oz. x 25
15. 17 cwt. 3 qrs. 15 oz. 7 dr. x 36
1 7. 3 lbs, Soz, 1 5 dwts. 1 3 grs. x 49
19. 5 fur. 78 yds. 2 ft. 7 in. x 56
21. 5a. 3r. 27p. X 70
23. 3sq. yds. 8 ft. 131 in. x 80
25. 87 gals. 3 qts. 1 pt x 90
27. 4 qrs. 6 bus. 2 pks. x 100
29. 5 d. 17 h. 39 m. 20 s. x 120
12. 4 tons 1 3 cwt. 171b. 10 oz. X 9
14. 9 tons 16 cwt. 1 qr. 5 oz. x 32
16. 18 tons 3 qrs. 5 lb. 13 drs. x 45
18. 2 lb. 7 oz. 9 dwts. 22 grs.
20. 7 fur. 87 yds. 1 ft. 5 in.
22. 17a. 1r. 31 p.
24. 17 cub. yds. 21ft. 57 in.
26. 37 gals. 2 qts. 1 pt.
28. 3 qrs. 5 bus. 2 pks.
30. 17yrs. nod. 17 h. 57s. x 144
X 50
X 64
X 72
X 84
X 96
X 108
Division,
9. Rule. Set the divisor in a loop to the left of the divi-
dend, and divide the left-hand term by it, setting the quo-
tient under that term : if there be any rem^, reduce it to the
next lower den", adding in that term (if any) of the div^,
which is of this lower den", and divide the result by the div^ i
and so on.
£ g^ J Here first wc have to divide £38 by 3,
Ex. 1. 3) 38 6 8i whence wc get £12 with £2 over : now, as wo
£12 15 e'^ cannot divide £2 by 3, we reduce it to 40s., and
adding in the term 6*. in the dividend, we have
now to divide 46s. by 3: — hence we get 15s. with Is. over; and since
ls. = l2d., adding in the term 8d. in the dividend, we have now to divide
20d. by 3 : — hence we get 6d. with 2d. over; and since 2J. = 8/., we
have lastly to divide 8/. + 1/, or 9/. by 3, which gives us 3/. or Id.
£ s d Here the number of pounds is exactly divisible
Ex. 2. 8) 376 2 6 by 8 ; and since we cannot divide the term, 2s.,
£47 0 3^ of the dividend by 8, we reduce it to pence, and
adding in the term 6d., we have now to divide
sod. by 8 ; whence wc get 3d. witli rem"" Gd. ; and since (jd. = 24/ we
divide 8*1/ by 8, and thus have 3/ or \d,
c
22
Ex. 8«
DIVISION,
£ .9.
d.
£ s.
d.
!.
26 15
3|-j- 2
2.
12 14
n-^ 3
3.
56 15
8^4
4.
76 17
2K o
5.
84 10
3 -T- 6
6.
90 13
SH 7
7.
75 7
6 4-8
8.
87 16
8i^ 9
9.
91 14
4i-rlO
iO.
74 17
71-Ml
11.
57 13
0 -rl2
12.
87 13
6 4-12
Ex.
10. iJivision by 10, 100, 1000, &c. is usually performed
hj pointing 0^ one, two, three, &c. figures, respectively, from
the rij^lit of tlie dividend.
Here, dividing 2315 by 100, wc have a quotient
23 with rem'^ 15 ; we may j^oint off, therefore, the
last two figures as the rem'', leaving the rest for the
quotient ; reducing now this rem' into shillings,
and adding in the term 145., we have to divide
314s. by 100 ; and sinee the quotient is 3 with
rem' 14, we may again point off the last two
fiiiures as the rem'' : and so en.
o'
£ .9.
23.15 14
_J0
3.1 4«T
12
1.7~5</.
Ex.9.
3.00/
£
1. 176
3. 329
5. 1511
7. 645
d.
84-10
3^100
2 4-1000
8 4-10000
£ s. d
2.
30 6 34-10
4.
73 12 ll-^100
6.
72 18 44 1000
8.
1062 10 04-10000
11. When the divisor islarge,but can be broken up into two
or more factors, we may divide separately by each of these.
Ex. 1. Divide £3762 35. 6f/. by 24.
Since 24 = 4 x 6, or =3 x 8, or =2 x 12, the sum may stand thus:
£ s. d. £ s. d. £ s. d.
4)3762 3 6 or 3) 3762 3 6 or 2)3762
6)940 10 10^
£156 15 \lAns.
Ex. 2. Divide £4081
8)1254 1 2
3 6
12)1881 1 9
£156 15 \lAns.
£156 15 \lAns.
15*. Od. by 1200.
Since 1200 = 12 x 100, the sum may stand thus :
£ s. d.
0 Here there is no quotient from the shillings, and
3 vfQ have the
Ans. £S4 Os. 3^.
12) 40818 15
100) 34.01 11
_20_
.31.9.
12
3. 7 'id.
4
3^0/;
N.B. In a case where one of the factors of the
divisor is 10, 100, &c., it is generally best to divido
last by that factor.
Ex. 10.
DIVISION.
£
s.
d.
£
5.
d.
I
702
6
3 -
- 20
2.
187
14
11 -
- 14
3.
275
15
n-
- 18
4.
345
13
4 -
- 40
5.
345
10
5-
- 25
6.
351
14
8 -
- a2
7.
485
17
6 -
- 120
8.
457
18
4 -
- 400
9.
203
16
9 -
- 36
10.
362
19
101-
- 42
11.
G92
10
0 -
- 800
12.
1137
10
0 -
-2400
13.
347
1
3 -
- 45
14.
457
1
6|-
- 63
15.
362
10
0 -
-6000
16.
1556
5
0 -
-3600
17.
408
0
9 -
- 54
18.
453
11
6|-
- 77
19.
363
18
n-
- 81
20.
473
14
0 -
- 96
21.
386
16
4-
- 99
22.
374
19
3 -
- 108
23.
319
2
9 -
- 132
24.
576
3
0 -
- 144
12. When, however, the divisor, though large, cannot be
broken up into factors, we must proceed as in the first case,
only setting the quotient in a loop at the right of the divi-
dend, instead of under it,
Ex. Divide £3715 18«. 9(/. by 470.
Since 470 = 47 x 10, the sum may stand thus i
£ s. d. £ s. d.
47)3715 18 9 (79 1 3
329
Here the rem' from the pounds is £2, which we
reduce into shillings, adding in the term 18s. in the
dividend : and so on.
^We have now to divide this first quotient by 10 :
£ s. d.
10) 7.9 I 3
20
425
423
2
20
58(1
47
12
141(3
141
Ans. £1 185.
l¥'
18.1
12
1.5
_4
2.0
Ex. 11.
£ s.
375 13
2.58
371
412
1375
11. 2456
d.
91-
21 -r- 370
6|-T- 123
11 -T-3G50
13
190
29
£
289
456
513
d.
11
8. 712 18
10. 2559 7
12. 2348 11
17
23
9^-7- 3100
71-r 41
6 -T- 18900
41-T- 354
13. Hitherto we have had to divide some quantity of
money, weight, &c., or, as it is called, some concrete quantity,
by a simple, or abstract^ number, that is to say, we have had
c2
24 DIVISION.
to find a certain part of sucli a quantity : thus, to divide
£3 75. 6d. by 8, is to find the eighth part of £3 75. 6d. ; and
here the quotient will also be a concrete quantity of the same
hind as the dividend — as in this case, 85. 5^d.
But if we have to divide a concrete quantity by another
of the same ki?id, this amounts to finding hoiv many times
the divisor is contained in the dividend : thus, to divide
£3 Is. 6d. by I6s. lO^d., is to find how many times I6s. lO^d,
is contained in £3 75. 6d. ; and here the quotient will be an
abstract number — as in this case, 4.
The quotient in cases of this latter kind is to be found by
reducing the two quantities to the same denomination, and
then performing the division.
Ex. 1. Divide £3 7s. Cd.bi/lGs. 10|rf.
Here £3 7 6 =1G20 halfpence 1 hence 405) 1620 (4 ^r.s.
16 10^= 405 halfpence J 1620
Ex. 2. Divide 3 tons 2 cwt. 1 qr. 21 lbs. by 2 qrs. 7 lbs.
tons cwt. qr. lbs. lbs.
Here 3 2 l 21 = 69931 hence 63) 6993 (111 ^?js.
2 7= 63J G3_
69
63
63
63
Ex. 12.
£ s. d. £ s. d £ s. d. £ s. d.
I. 11 7 9|4- 1 5 3f 2. 22 15 71-^ 3 15 111
3. 102 10 32-^11 7 93 4. 68 6 lOi-j- 2 10 7^
5. 68 6 lOi-f- 7 11 101 6. 205 0 7i-r34 3 51
7. 684 7 6 ^76 0 10 8. 171 1 10i-r57 0 71
9. 89 cwt. 22 lb. 4-3 cwt. 1 qr. 6 lb. 10. 196m. 7 fur. -r7ft. 6 n.
11. 81 cwt. 1 qr. 1 6 lb. -j- 1 cwt. 3qr. 16 lb.
1 2. 9 lb. 9 oz. 3 dwts. 12 grs. -^ 5 tlwts. 9 grs,
13. 513m.4fur. 23po.-rl7m. 5 fur. 27 po.
14. 1027 m. 1 fur. 6 po.-r 17 m. 5 fur. 27 po.
15. 244 qrs. 3 bus. 1 pk.-^3 qrs. 3 pks. 16. 2366A. 3r. 86p. +91^, 6p.
14. To this head also may be referred certain cases of
Reduction, in which we cannot pass directly, step by step,
from one den" to another, but must reduce both the given
quantity and the proposed to some common lower den", (it
will be best to take the highest den" to which they can both
DIVISION. 25
be reduced), and then find by div" what quantity of the pro-
posed den" is equivalent to the given quantity.
Ex. JReduce £96 16s. to guineas.
£06 165.
Here, since we know that 21s. hiake a guinea, we first
reduce the given sum into shillings, and then divide
by 21, to bring these shillings into guineas. The rem'
4 we set down (Art. 3., Ex, 2.) as 4s.
Ans. 92g. 4s.
Ex. 13.
835 guineas to pounds ; and 538 pounds to halfguineas.
2. 760 halfcrowns to guineas; and 670 halfguineas to halfcrowns.
3. 325 crowns to halfguineas ; and 253 guineas to crowns.
4. 18756 foarpenny-pieces to crowns ; and 3700 halfcrowns to four-
penny-pieces.
5. £36 175. 6d. to crowns ; and £27 5s. 4d. to sixpences.
6. 100 halfguineas to fourpcnny-picccs ; and £100 to seven-
shilling-pieces.
7. 1 cwt. 2!bs. Av. to Troy weight ; and 16 dwts. to Ap. Avcight.
8. 20 lbs. Av. to Trey weight ; and 5 drs. Ap. to Troy weight.
9. 478 ells to yards ; and 14 hands to feet.
10. 500 fathoms to yards ; and 5 furlongs to fathoms.
15. It must be noticed, that we can never divide a con*
Crete quantity of one kind by another of a different kind, as
ehillings by ounces, pounds by hours, &c. ; since no quantity
of shillings will contain ounces, nor of pounds, hours, &c.
Kor can -sve multiply together concrete quantities of any
kind, whether the same or different : thus, we cannot mul-
tiply either shillings by shillings, or shillings by ounces.
16. Mensuration of rectangular areas.
Suppose ABCD to represent the surface of
a table, of which the length AB is 5 feet, and
the breadth AD, 3 feet. Divide then AB into
5 equal parts, and AD into 3, as in the figure,
and through the points of division draw lines
parallel to AB and AD, By this means •\VG
F
^6 DIVISION.
shall have divided the whole surface into small figures, such
as AEFG, all equal to one another ; and since ^^=one
foot, and AG=z one foot, it is plain that the sur^Ace AEFG
measures a foot every way, a foot long and a foot broad, — i.e.
AEFG is a square footy and so are all the other small figures.
Now the number of these figures is 5x3= 15, each
horizontal row of 3 square feet (the number of feet in AD)
being repeated 5 times (the number of feet in AB) ; so that
tlie number of square feet in the surface is found by multi-
plying together the n° of feet in its length and the n° of
feet in its breadth.
17. As the same method of proof would apply in any
similar case, it appears that the n** of square feet in any
rectangular surface is found by multiplying together the n<»
of linear feet in its length and breadth; or if we express
the. length and breadth in yards, i7iches, &c., and multiply
them in this form, we shall obtain the n^ of square yards,
square inches^ &c. in the surfjice.
Ex. Find the surface of a floor 17 ft. 8 in. long by 3 yards wide.
Here 17 ft. 8 in. = 212 in.* 12) 22896
3 yds. = 108 in. 1 2) 1908
1696 9) 159
2120 "TTe
22896 sq. in. = 17 sq. yds. 6 ft. Ans.
Ex. 1ft. 1. 37 ft. 2 in. X 2 ft. 9 in. 2. 23 ft. x 3 ft. .5 in.
3. 3 yds. 2 in. X 3 fY. 4. 1 yd. 2 ft. x 1 yd. 1 in.
5. 15 ft. 7 in. X 11 ft. 1 1 in. 6. 22 ft. 5 in. x 3 yds.
• 7. What is the area of a court, 10 yds. 2 ft. long, and 5 yds. 1 ft.
broad ?
8. How many sq. yds. of carpet will it take for a room 26 ft. by 32 ft.?
9. What is the surface of a marble slab, whose length is 5 ft. 7 in., and
breadth 1 ft 10 in.?
10. Find the area of a square building, whose side is 46 ft. 8 in.
♦ It miglit at first sight appear that we are here multiplying inches by inches, con-
trary to the statement in (l.'i) ; but, in reality, it is only the numbers 212 and 108 that
we multiply, not the quantities 212 in. and 108 in. : so also the resulting product is
only the number 22896, to which we append sq. in., because we know from the above,
that this IS the number of square inches in the given area. A similar remark applies
to all such cases, and to all such expressions as multipiyinjr the length by the breadth,
&c. The Studeat's attention should be strongly drawn to this.
DIVISION.
27
1 1 How many square yards of paper will be required for a room 17 ft.
long, 12 ft. 7 in. wide, and 8 ft. 5 in. high ?
10. How much wainscoting is there in a square room, 18 ft. 3 in. long,
and 8 ft. 6 in. high ? See Note IV.
18. Since, by multipli/ing the length and breadth, we get
the square area of any rectangular surface, it follows that,
by dividing the square area by the letigth, we shall get the
breadth, or, dividing it by the breadth, we shall get the
length— ivi\i\ng care to express the quantities concerned,
before div", as quantities of the same den", as, for instance,
not dividing sq. feet by inches, but first bringing them to
sq. inches, &c.
Ex. What length of paper, that is 2 ft. wide, will be required for a
room 14 ft. square, and 10 ft. 4 in. high ?
The room being square, the united length of its four sides will b3
14x4 = 56 feet, and their height being 10 ft. 4 in., we shall find the
square area of the whole surface of the walls by multiplymg these
quantities, first reducing them to inches.
Here 56 ft. = 672 in. The surface of the walls being 83328
10 ft. 4 in. = 124 in. sq. in., we have now to divide this by
2638 2 ft. = 24 in., the width of the paper.
■104 1 in. sq. in. in.
g72 24) 83328 (3472
83328 sc^. in. rrz
1 1«5
96
Ans, 3472 in. = 96 yds. 1 ft. 4 in. 172
168
48
48
Ex. 15.
1. 5sq.yds.6ft.lMm-18ft.7in. 2. 1 Isq. yds. 3 ft 129 in-21t.9in.
i 8sq.yds.6ft.84in.-5ft.9in. 4. 17,sq. yds. 4 ft. 24 m.-23 It.
5. 17 sq. yds. 0 ft. 45 in. -18 yds. 1 ft. 9 m.
6. 42 sq. yds. I ft. 50 in. -7-23 ft. 10 in.
7 What is the length of a room, whose breadth is 11 ft. 11 in., and
which it takes 17 sq. yds. 2 ft. 131 in. of drugget to cover ?
8 One side of a rectangular building measures 26 yds. 5 in., and its
area contains 683 sq. yds. 2 ft. 25 in. ; show that it is square.
9. How many yards of carpeting, 2 ft. 4 in. broad, vAW it take to covcf
a room whose dimensions are 26 ft. by 35 ft. ?
10. It is found that 288 yds. of paper, 2 ft. 8 in. wide, will cover the walls
of a reom ; how many would be required »f paper 2 ft. 3 m. wide ?
28 DIVISION.
11. How many yards of matting, 2 ft. 3 in. wide, will be required for a
square room, whose side is 18 ft. 9 in. ?
12. If the room in (11) be 13 ft. 4 in. high, how many yards of paper
1 ft, 4 in. wide will be required for it ?
19. Mensuration of rectangular solids.
Suppose -we place upon cacli of the little squares in the
preceding figure, a solid (as, for instance, a brick) in the
form of a cubic foot, that is, measuring a foot every way —
a foot long, a foot broad and a foot high — wc shall have a
layer of such bricks one foot high, and containing as many
cubic feet as there are square feet in the base ; if upon this
We pile another similar layer, we shall have the whole solid
two feet high, and containing twice as many cubic feet as
there are square feet in the base ; and so on ; hence the
whole ToP of cubic feet in any such solid, will be found by
taking the product of the n° of feet in height by the n" of
square feet in the base, and this last, as in (17), is the pro-
duct of the n® of feet in length by the n° of feet in breadth.
Hence the n° of cubic feet in any rectangular solid or
space is found by multiplying together its length, breadth^
and height (or thickness, as the height would be called when
small, as, for instance, in the case of a beam of timber), these
quantities being all reduced first to the same den", and their
product being of the same den", but in cubic measure.
Exi rind the solid content of a team of timber, 30 ft. long, 2 ft. 3 in,
wide, and 1 ft. .5 in. thick.
in.
ere 30 ft. = 360
2 ft. 3 in. =• 27
1728) 281880 (163 cub. ft;
1728
2i520
720
9720 sq.
5 ft. 5 in. = 29
logos 27) 163 (6 (Jub. yds,
10368 162
in. c . 54dO 1 ft.
5i84
87480
19440
216 cub, ifl.
Ans. 281880 eub. in. ^6 cub. yds. 1 ft. 216 in, by Red*.
MISCEttANEOtJS EXAMPLES. 50
20. So also, as before, having given the solid content of
any space and any two of its three dimensions, we may find
the third by dividing the content by the product of these
two, reducing all to the same den".
Ex. What is the length of a room, whose width is 10 ft. 4 in. and
height 10 ft. 6 in. ; and which contains 1519 cub. ft. of air ?
and 1519 cub. ft. =2624832 cub. in.,
hence, performing tlic div", we have
sq, in, cub. in. in.
15624) 2624832 (168
15624
106243
93744
Here
10 ft.
4 in. = 124
in.
10 ft.
6 in. = 126
in.
744
1488
1.5624
sq
in.
Ans.
168 in.
= 14 ft.
124992
124992
Ex.16,
1. ISfi-. 9in. X 13ft. 4in. x8ft. 4in. 2. 3 ft. 9 in. x 6ft. Sin. x 2ft. T in.
3. 1 1 ft. 3 in. X 3 ft. 4 in. x 10 ft. 5 in.
5. 7 ft. 4 in. X 5 yds. x 8 ft. 3 in.
4. 5 yds. X 6 yds. 2 ft. x 4 ft. 2 in.
6. 9 ft. 2 in. X 2 yds. x 6 ft. 8 in.
7. How many cubic feet of water can be contained in a vessel with
square base, whose side is 3 ft. and height 2 ft. 10 in. ?
8. What quantity of timber is there in a beam, whose length is 20
feet, breadth 3 feet, and thickness 2 ft. 6 in. ?
9. Find the solid content of a cube, Avhose side is 7 ft. 5 in.
10. In making a square pond, whose side was 12 yds., there were
taken out 336 cub. yards of earth ; how deep was it made ?
11. What must be the length of a trench, 5 ft. 6 in. deep, and 10 ft.
8 in. wide, that it may contain 7040 cubic feet ?
12. The depth of a canal is 7 ft. 3 in., the width 20 ft. 4 in., and the
length 10 miles ; how many cubic feet of water will it con-
" tain?
Miscellaneous ExAMrLES. 17.
1. A sovereign weighs nearly 493 quarter grains; how many lis. will
loco sovereigns weigh?
2. In 2551443 seconds, which is the exact length of the lunar
month, how many days ?
3. What is the cost of 530 lbs. of tea at 35. 7d. per lb ?
4. Six persons on a journey spend £97 9^. 6d. ; how much is that
for each person ?
5. The circimiference of the Earth contains 131250000 feet ; express
the same in miles.
cS
30 MISCELLANEOUS EXAMPLES
6. If 81 oxen are bought for £1779 195. 6d., what is the average
price per head ?
7. How many letters, paying penny postage, require stamps to the
amount of £7947 2s. lOd. ?
8. A pint will contain 9000 barleycorns, and 3 of these, placed end
to end, would reach an inch ; how many feet would they all reach ?
9. How many days would it take to count a million of sovereigns, at
the rate of 100 a minute?
10. "What is the amount of 42 cwt. of sugar at £2 35. 7c?. per c-w-t. ?
11. Divide 3587 yds. 9 in. into 27 equal distances.
12. What sum must bo divided among 27 men, bO that each may
receive £14 6s. S^d.?
13. How many ducats, each worth 45. 9d., are contiiitted in £231 165. ?
14. Divide £1478 125. 9|o?. into 77 equal portions.
15. How many days in a solar year, which contains 31556928
seconds ?
16. A cubic foot of water weighs lOOD ounces ; what weight of water
is there in a vessel, the length, width, anddepthof which are each a yard?
17. The battering ram employed by Titus against the walls of Jeru-
salem weighed 100000 lbs. ; how many tons did it contain?
18. The Calcutta rupee is worth l5. 1]|<?. ; what is the value of a
lac, which consists of 100000 rupees?
19. Sound travels at the rate of 1140 feet a second ; how many miles
is a thunder-cloud distant, when the sound follows the flash after 7
seconds ?
20. Light travels at the rate of 186040 miles a second; if the Sun's
light takes 8 min. 13 sec. in reaching us, what is his distance from the
Earth?
21. A cannon-ball travels at the rate of 400 yards a second ; how
many miles will it go in a quarter of a minute ?
22. Find the amount of 200 tons 81 lbs. of iron railing at 7d. per lb.
23. Suppose a weekly newspaper, price 3c?., has a circulation of
11800 ; what is the sum realised by its sale in a year ?
24. If 2 cwt. I lb. cost £116 195. O^d., what is the cost of 1 lb. ?
25. How much silk at 65. 8d. a yard may be bought for 20 guineas?
26. To how many persons may £60 155. Od. be distributed, giving
£4 135. 6£?. to each?
27. An Attic drachma was worth 7^. ; what was the value of the
tiilent, which contained 6000 drachmae? and how many minge did it
contain, each worth £3 4.v. 7d. ?
28. A Jewish shekel weighed 219 troy grains, and was worth 25.
B^d. ; what was the weight of a talent, containing 3000 shekels ? and
the value of 10000 talents ?
IN ELEMENTARY RULES. 31
29. The captains of Israel, after the destruction of Midian, made a
free-will offering of 16750 shekels ; what sum did this amount to? See
Ex. 28.
30. How long would a cannon-ball, moving at the rate of 1200 foet
a second, be in passing from the Earth to the Moon, 230500 miles ?
31. How much is spent in 15 years by a person who spends £825
1 85. 9d. yearly ? and how much would he have saved in that time out
of an income of £1500 ?
32. How many pounds weight of bronze are there in a million of
pennies, each weighing one-third of an ounce avoird. ?
S3. A plate of gold cost £161 175. 6d., at £4 7s. 6d. per ounce;
what was its weight ?
di. How many patients will an hospital maintain, whoso revenue is
£5629 105., when each requires on an average £8 135. 9d. per annum?
35. If the duty on brandy, at 105. 5d. a gallon, amounted to £26357
65, lOd., on what quantity was it paid ?
30. Twenty bricklayers and ton carpenters were employed in build-
ing a house, each of the former receiving 275. per week, and each of the
latter 295. ; what was the amount of their wages in 16 weeks ?
37. Two boats start in a race, and one of them gains 5 feet upon the
other in every 55 yards ; how much ^vill it have gained at the end of
half a mile ?
38. What is the area of a playground 58 ft. 6 in. long, and 54 ft,
9 in. broad?
39. A has £100 4s. liy., and B 64393 farthings ; if A receive from
B mil fiirthings, and B from ^ £11 II5. Hid., how much will ^ have
more than B ?
40. What is the value of a beam of timber, whose length is 20 ft,
breadth 3 ft, and thickness 2 ft., at 35. B^d. per cubic foot?
41. If the length of a cubit was 22 inches, what was the cubic
content of the Ark, which was 300 cubits long, 50 broad, and 30 high ?
42. A grocer mixes 3 c-svt. 24 lbs. of sugar at 6ld. per lb. with 2 cw t,
64 lbs. at i\d. ; at what price per lb. must he sell the mixture, so as not
to lose by the sale ?
43. A person gives a five-pound note to pay for lodgings during the
month of August at 25. Sd. per night ; what sum will bo returned to
him?
44. Of the three quantities 1847 lbs. avOird., 449 shillings, and
£6286, it is required to multiply One quantity by the quotient of the
Other two.
45. What is the cost of 6 packs Of cloth, each containing 6 parcels,
each parcel 6 pieces, aild each piece 60 yards, at 2^,. per yard ?
46. A labourer's house-reut is £5 25. 1 Id. a year j what must he lay
tip -Weekly to pay it ?
32 MISCELLANEOUS EXAMPLES
47. It is estimated that the average strength of a man is equal to
raising 100 lbs. through 1 foot in a second, working 10 hours a day;
how many tons will he raise at this rate in the day ?
48. In marching, soldiers take 75 steps a minute, in quick marching
108 ; how far would a regiment advance in 3 hours, the last half-hour
at quick march, reckoning each step as 2 ft. 8 in. ?
49. If a compositor set up 8500 letters a day, and be paid o^d. for
every thousand, how much will he earn in a week ?
60. Divide £184 lis. 2ld. eqxially among 39 persons; and, sup-
posing 15 of them to have received their portions, and of the rest only
21 to appear, how much might be given to each of these?
51. A mixture is made of 9 gallons of spirits at 125. 6d. per gal.,
16 gallons at 18s. 9c?., and 90 gallons at 225. 3d. ; what is the value of
a gallon of it ?
52. A corn-factor buys 2 quarters at 395. per quarter, and 7 bushels
at Qs. per bushel ; at what price per bushel must the whole be sold, so
as to gain 235. 9d. in all ?
53. A side of Lincoln's Inn Square is 770 feet, and of Eussell Square
670 feet ; how many acres does each contain ?
64. What weight of water may be contained in a canal whose depth
is 8 feet, width 25 feet, and length 12 miles? See Ex. 16.
65. How many yards of carpet, 25 inches wide, will bo required to
coveta floor that is 19 ft. 7 in. long by 18 ft. 9 in. wide?
66. A wished to exchange 50 gallons of brandy, at 2l5. 9d. per gal-
lon, with B, for ale at I5. dd. per gallon ; how many gallons of ale
should he receive ?
67. A wall is to be built, 15 yards long, 7 feet high, and 13 inches
thick, with a doorway 6 feet high and 4 feet wide ; how many bricks
will it require, if each, including mortar, occupy 108 cubic inches?
68. Divide £115 10s. among 5 men and 6 women, giving to each
man thrice as much as to a woman.
59. An equal number of men, women, and boys earned £55 135. in
6 weeks ; each man earned 2s. 4c?. a day, each woman Is. 3c?., and each
boy 10c?. ; how many were there of each ?
60. There is a plantation in the form of a hollow square, length ex-
ternally 252 yards, and depth 16 yards; find the area of the plantation
and that of the inner square.
\ 61. Divide £39 into four equal numbers of guineas, half-guineas,
crowns, and half-crowns respectively.
63. A clergyman commutes his tithes, valued at £500, for equal
quantities of whefit, barley, and oats ; how much grain will he receive,
supposing the average price of wheat to bo 6s. Id, a bushel, of barley
3s. lid, and of oats 25. 10c?. ?
m ELEMENTARY RULES. 83
63. A and B go to bed at the same hoxir daily, but A rises at a
quarter past 6, and Z? at 8 ; how much of waking life will A hare had
more than B in 40 years, paying attention to the Leap-years ?
6i. Divide £20 among three persons, so that one may have £3 155,
more than each of the others.
65. Divide £550 35. l^d. among 4 men, 6 women, and 8 children,
giving to each man double of a woman^ and to each woman triple of a
child.
66. Divide £2 95. 2c?. among A, B, C, so that B may have Gs. 8d,
more than A, and 6"s share may be double of B's.
67. The circumference of the fore wheel of a carriage being 8 ft,
3 in., and that of the hind wheel 11 ft. 11 in., how many more revolu-
tions would be made by the fore wheel than by the hind wheel in going
from Cambridge to London, a distance of 52 miles ?
68. In new enclosures, the cost per acre of the first crop (wheat) is
£G lis. Gel., and the produce 18 busliels at 8?. ; that of the second crop
(barley) is £3 I65., and the produce 25 bushels at is. ; and that of the
third crop (potatoes) is £12 II5. 2d., and the produce 100 bags at Zs. ;
deducting one-tenth of the whole produce for tithes, find the result of
enclosing 500 acres, in one year and in three.
34
CHAPTER II.
GHEATEST COMMON MEASURE: LEAST COMMON MULTIPLE.
21. One number is said to be a measure or a, factor of
another, when it divides it exactly, without remainder.
Thus, 1, 2, 3, 4, 6, 12 are all measures or factors of 12.
Unit}/, however, is not generally named among the divisors
of a number.
22. Any number, which divides without remainder each
of two or more numbers, is said to be a common measure or
common factor of those numbers ; and, of course, the greatest
number which so divides them is their Greatest Common
Measure (g. c. m.)
Thus 2 is the only common measure of 4 and 6; 3, 5, 15 are, each of
them, common measures of 30 and 45, and 15 is their greatest common
measure; 2, 7, 14 arc, each of tliem, common measures of 14, 42, and
70, and 14 is their greatest common measure.
23. To find the Greatest Common Measure of two numbers.
Rule. Divide the greater by the less, and the preceding
divisor by the remainder, and so on continually, until there
is no remainder : the last divisor will be the G. C. m. re-
quired.
Ex. 1. Find the o. c. m. of 3575 and 125455; and of 279 and 4185.
3575) 125455 (35 279) 4185 (15
10725 279
18205 1395
17875 1395
330) 3575 (10 Ans. 279.
330
275) 330 (1
275
55) 275 (fl
275
Ans. 55.
LEAST COMMON JIULTU'LE. 35
Ex.2. Find the G, c. M. of 17 and 36.
17) 36 (2
34
2) 17(8
16
1)2(2
2
Ans. 1.; i. e. the given numbers have no common measure but unifi/.
The reason of this Rule can hardly be explained without some know-
ledge of Algebra in the Student. The Rule itself is here introduced,
because it is often useful in reducing Vulgar Fractions to simple forms.
See Note V.
18,
, Find the G. c. M. of
1.
224 and 336.
2.
348 and 1024.
3.
175 and 2042.
4.
1225 and 625.
5.
2121 and 1313.
6.
429 and 715.
7.
377 and 1131.
8.
2431 and 770.
9.
900 and 34 74.
10.
1379 and 2401.
11.
2314 and 3721.
12.
7007 and 7392.
13.
2793 and 2660.
14.
4165 and 686.
15.
5325 and 8307.
16.
3775 and 10000
17.
7056 and 7392.
18.
6327 and 23997.
19.
12321 and 54345.
20.
24720 and 4155.
24. One number is said to contain, or to be a multiple oj\
another, when it can be divided by it without remainder.
Thus 12 is a multiple of each of 1,2, 3, 4, 6, 12 ; and any number is
a multiple of each of its measures.
25. A common multiple of two or more numbers is one
which contains each of them ; and, of course, the least such
number, is their Least Common Multipte (l. c. M.).
Thus 6, 12, 18, &c., are all common multiples of 2 and 3 ; but 6 Is
their least common multiple : 12, 24, 36, 48, &c., are all common multi-
ples of 2, 3, 4, 6, and 12; but 12 is their least common multiple.
Of course, a common multiple of any given numbers may
be found, by multiplying them all together ; thus a common
multiple of 6 and 8 is 48, of 4, 6, and 9 is 216. In practice,
however, we require the least common multiple, especially
in preparing Vulgar Fractions for Addition and Subtraction
36 LEAST COMMON MULTIPLE.
26. To find the Least Common Multiple of two or more
7irimhcrs.
Rule. Set them in a line, and strike out any that are
contained in any of the others. Divide those not struck out
by any number that will exactly divide one of them; under
any which it exactly measures, place the corresponding quo-
tient ; under any which it partially measures (containing
some factor common to it, but not being itself wholly con-
tained in it), place the quotient obtained by dividing it by
the common factor; and under any which it does not mea-
sure at all, repeat the number itself.
Now treat the new line thus formed, in the same maunet
as the first; and so on, until all the numbers left in anyliiJC
have no common measure but unity.
Then the continued product of the numbers in this line
and all the divisors is the l. c. m. required of the given
numbers.
Obs. It will generally be most convenient to talie pretty
large numbers, if possible, for divisors; as fewer lines will
thus be necessary, especially if such be chosen as contain
themselves many simple factors. Thus 12 contains the fac-
tors 2, 3, 4, 6, 12, and is therefore, when possible, a xcry
good divisor to be employed.
Ex. 1. rind the l. c. m. of 24, 16, G, 20, 4, 8, 10, 30, 12, 25.
12) 24 . 16 . t^ . 20 . ^ . )^ . Xt^ . 30 . ^'i^ . 25
^^ . 4 . '^ . \ . 25
Ans. 4x25 x 12 = 1200.
The reason of this process may be thus explained.
Wc are required to find a number, which shall contain 24, 16, 6, 20, 4^
8, 10, 30, 12, and 25. Kow if wc find a number which contains 24, it
will, of course, contain 0, 4, 8, and 12, which are themselves contained
in 24. We may therefore strilvc out 6, 4, 8, and 12; and for a similar
reason, 10, which is contained in 20; and we thus reduce the question to
finding the L. C. M. of 24, 16, 20, 30, and 25.
Now wc choose for divisor, according to the Rule, the number 12y
ts-hich exactly divides one of these, viz. 24. In order, therefore, that the
t. c. M. required may contain 24, it must, of course, contain this number'
LEAST COMMON MULTIPLE. 37
12, and besides that a factor 2 ; "but we now wish to find what factors
beddes 12 and 2, the l. c. m. must contain, so as to contain all the given
numbers. "Wo see then that 12 will also supply the factors 4 of 16, 4 of
20, and 6 of 30 ; so that the only others besides 12, which must bo con-
tained in the required number, arc 2 to make up 24, 4 for 16, 5 for 20,
5 for 30, and 25, i.e. the numbers given by our process in the second lincj
— to which a similar reasoning applies.
Ex. 2. What is the least number that can be divided by each of 14,
16,40,50,25,8,64?
1 0)14.^^. 40 ^50 ._^!^J_. 6 4
7. \. V. '32
Arts. 7x5x32x 10 = 11200.
Ex. 3. Find the l. c. m. of 27, 24, 6, 15, 5, 9, 126.
9)27 . 24.16, . 15 .^. Si. 126
2)^
. 8 .
5.
14
3
. 4 .
5 .
7
Ans. 3x4x5x7x9x2 =
= 7560.
Ex. 19. Find the L.
C. M. of
1. 15, 20.
2.
14, 21.
3. 8, 4, 16.
4.
3, 9, 22.
5. 12, 15, 16.,
6.
8, 16, 20.
7. 9,15,18,20.
8.
16, 9, 12, 18.
9. 8, 12, 15, 20.
10.
34, 68, 17, 2.
. .
Jl. 6,12,16,18,24.
12.
8, 12, 18, 24, 27.
13. 2, 4, 8, 16, 10, 48.
14.
1,2,3,4,5,6,7,
8,9.
15. 7, 12, 15, 27, 35, 40,
17. 4, 9, 10, 15, 18, 20,
,45.
21.
16.
18.
9, 16,42,6.3,21,
7, 15, 21, 28, 35,
14^72.
100, 12.9.
19. 8, 9, 10, 12, 25, 32,
75, 80.
20.
15, 16, 18,20,24
, 25, 27. 30.
38
CHAPTER m.
VULGAR FRACTIONS.
27. A Fraction is SL\quantity which represents a part or
parts of an integer, or whole.
28. A Vulgar (that is, a common) Fraction, in its sim-
plest form, is expressed by means of two numbers placed one
over the other, with a line between them.
Tlie lower of these is called the Denominator, and shows
into how many equal parts the whole is divided ; the upper
is called the Numerator, and shows how many of those parts
are taken to form the fraction.
Thus f denotes that the whole is divided into four equal parts, and
that three of them are taken to form the fraction.
29. A proper fraction is one whose numerator is less than
the denominator, and which is itself therefore less than the
whole in question ; as f , f .
An improper fraction is one, whose numerator is equal to
or greater than the denominator, and which is itself, there-
fore, equal to or greater than the whole in question ; as J, -V-.
30. A mixed number is one formed of a whole number
and a fraction ; as 2 J, 5 J.
A compound fraction is a fraction of a fraction ; as J of J.
2iof|of3i.
A complex fraction is one in which either the num^, or
gi 2 li - of 3
den^ or both are fractions ; as ^^» -r^ -^^ *tt—
J, 4tj Or A-^
VULGAR FRACTIONS. 39
31. Every whole number may be considered as a fraction
whose den*" is 1 ; thus 6 is f .
32. A fraction may be considered as expressing the divi-
sion of the num"^ by the dcn^
Thus 5 expresses 3 T- 4: for wc should obtain the same, whether we
divide 07ie unit into 4 equal parts, and then take three of these parts, that
is, three-fourths of the one unit; or divide three units, each into 4 equal
parts, and then take one part out of each four, i. e. one-fourth of each
unit, and therefore one-fourth of the whole three units; so that f of 1, or
f, = lof3, or 3^4.
For instance, | of £l, which is 15.?., =| of £3, which is also 15s.
33. To reduce a whole number to a fraction ivith a given
denominator.
Rule. Multiply the number by the given dcn^, and the
result will be the num'' of the fraction required.
Ex. Reduce 5 to a fraction, with denominator 6.
Since 1 contains 6 sixth parts, .*. 5 contains 30 sixth parts ; or 5 = -^.
Ex. 20. 1. Reduce 8 and 27 to fractions with den" 5 and 27.
2. Reduce 34 and 135 to fractions with den" 11 and 17.
3. Reduce 6, 9, 12, 20, to fractions with den"" 15.
4. Reduce 25, 34, 70, 111, to fractions with den' 34.
34. To reduce a mixed number to an improper fraction*
Rule. Multiply the whole number by the den^ of the
fractional part ; add the result to the num*^ of that part for
the new num^, and retain the same den''.
Ex. 1. 7i=-f-: for 7=^^- (33); and hence, 7i=-^ + §=-f-.
Ex. 21. Reduce to improper fractions
1. 3^. 2. 10§. 3,
6. 200|^. 7. 711i. 8,
11. 200|2. 12. 125|f. 13.
16. im 17. 17§^.' 18.
Ex.
3
n
=-^.
actions
221/t-
4.
13lf.
5.
32]^.
use-
9.
128i|.^
10.
37^.
514^.
14.
1011^.
15.
71911.
io|^.
19.
llllf^
20.
851-
40 VULGAR FRACTIONS.
35. To reduce an improper fraction to a whole or mixed
number.
Rule. Divide the num^ by the den'^ : the quotient will be
a whole number, and the remainder, if any, the num'' of the
fractional part of the mixed number required.
Ex.1. -f- = 5. Ex.2. -Vf = 7^.
Obs. All improper fractions, occurring in any sum, should
(except the contrary be desired) be expressed as whole or
mixed numbers.
Ex. 22. Xletluce to v.hole or mixed numbers
1.
r!7
0 •
2.
¥v
3.
13 •
4.
W-
5.
^^.
6.
^.
7.
W'
S.
W'
9.
ILI
10.
31 3R
y.-> •
11.
^W-
12.
3 -.7 7
■ 102 •
13.
^^.
14.
221 •
15.
3V3.-.
1-22 •
16.
fiono
■ 37 3"'
17.
-w-.
18.
6.- 56
401 •
19.
12321
■ 200 •
20.
23438
36. To multiply a fraction by any whole number or integer^
either multiply the numerator, or divide the denominator
by it.
Ex.1. ^x7=i4.
For in each of the fractions, ^ and 34, the whole is divided into 1.5
equal parts, and 7 times as many of them arc taken in the latter case as
in the former.
Ex *> -^ X 4—1 — 12
For the whole being divided into 4 times as many equal parts in -^ as
it is in I, each of the parts in the latter is 4 times as great as in the
former; and the same number of parts being taken in both cases, the
latter fraction is therefore 4 times as great as the former.
Ex.3. |x9=-fL=5|. Ex.4. ^x4=f| = llf.
Ex.5. M^9=-¥- = 4l. Ex.6. 1^x7=^ = 3^.
37. Conversely — To divide a fraction by any integer^ ei ther
divide the numerator, or multiply the denominator by it.
Ex.1. 3|-6 = ^. Ex.2. M-5=^.
Ex.3. 1-5=^. Ex.4. |-h6 = f,.
VULGAR FRACTIONS. 41
Ex. 23. 1. Multiply ^^ by 9, 12, 18, 25 ; and divide it by 5, 7, 8, 12.
2. Multiply i|£ by 7, 8, 9, 16 ; and divide it by 5, 8, 12, 25.
a Multiply §f§ by the numbers 2, 3, 4, 5, 7.
4. Divide |f§ by the numbers 7, 8, 9, 10, U.
38. If the num' and den^' of a fraction be both multiplied
or both divided by the same number, its value v/ill not be
altered.
Tor if the numr be multiplied by any number, the fraction is multiplied
by it (36), and if the derC' be multiplied by any number, the fraction is
divided by it (37) ; and if any quantity be both multiplied and divided
by the same number, its value is not altered.
Similarly, when the num' or den' are both divided by the same
number.
39. To reduce a fraction to loiver terms.
Rule. Divide both the num'^ and dcn'^ by any common
factors they contain.
Pv T 5) 270_3) 54^18 -Py o 9)315_7)35_5
From (38) it appears that the value of a fraction is not altered by this
process.
When a fraction is reduced as much as possible by such
division, it is said to be in {{^lowest terms. (See p. 20, note.)
Obs. All fractions, occurring in any sum, should (except
where the contrary is desired) be expressed in their lowest
terms.
Ex. 2ft. Reduce to their lowest terms
1.
6.
11
m
2.
720
8G4'
3.
^Ul5*
4.
5U40-
5.
129«
la-io*
m-
7.
495
1210'
8.
129R
1728-
9.
1R72
2010*
10.
990 ^
r.ooo
12.
2S9a
13.
aiof/*
14.
pr,4
3()72'
15.
3300
41^35 •
C030
17.
r.544
essa-
18.
7040
73n)5-
19.
113R.T
20.
• 2217(1
23328*
49. A fraction may be reduced immediately to its lowest
terms by dividing both its num^ and den^' by their G. c. m.
42 VULGAR FRACTIONS.
This process is generally longer than the other, and is
therefore, if possible, avoided in practice. It is, however,
sometimes, the only way of reducing a fraction, when we are
unable to detect by inspection the common factors of the
num^ and den*". Thus we should not see, perhaps, that the
fraction |-f|^ may be reduced to -J^ by dividing both its
terms by 113, their g. c. m.
Fv 1 179) 4117 _ 23 X'v 9 564)13636 — 24
riX. i. 0487—53- l^X. J. T7484 — 3l*
Ex. 25. Reduce to their lowest terms
1 321 g 510 o 209 A 1407
K 1905 /? 1715 7 6509 n 1589
''• 3l7a* "• iJBS)5* '• 7«8a* "* 2270*
9 8251 l n 3575 i l 12fil l Q 10759
I47ia* *"^'* 471b' ^*^' 4423;j* *— 20405*
41. We shall now give examples of the application of the
foregoing rules to the mult" and div" of concrete quantities.
£ s. d.
Ex. 1. 23 13 9| X 35 jj^^j.^ | ^ 5^^^43 . ^g get down |J., and
carry Ad. :
so also i X 7 = -^ = 2§ ; we s^WownJf/., and
carry 2d.
118 9
7
829 3
£
7) 37
9|
s. d.
14 8-f
4) 5
7 9f
litre, in the first div°, there are 5d.
Ex.2. ?)37u 8-28 7'-' 'v*-^ <':"'''?,,^^^';:'f \"« ^^'
down as ft/., since (32) f of 5J. = f of 1 J.
We might have brought these 6d. to 20/.
* " ^'^ and then, dividing by 7, should have had
2f/ ; but as a farthing itself is only & fraction of a penny, it is usual,
when the result does not come out a clear number of farthings, to ex-
press the whole below the pence as a fraction of a penny.
In the second div", there is Ifrf. over, or ^c?., which, divided by 4,
gives fc?.
Here, in the first div", Ihere is \\d.
^Q over = |^., which, divided by 8, gives
■^d. ; in the second div", there is 4^^.
over = -^^., which, divided by 5, gives
£ s. d.
Ex. 3.
8) 175 19 53h
5) 21 19 11^
4 7 Ufi
VULGAR FRACTIONS. 43
£ s. d. £ 8. d.
Ex. 4. 13) 54 10 5^ (4 3 IQl"-
52
2
20
50 (3s. _
39 Here thsre is 1\d. over = ^W , which, divided
11 by 13, gives |[c/.
12
137J(10J.
130
7.1
£ s. d.
Ex. 5. 3) 1115 17 8|-^300
100) 3.71 19 2^
20
14.39
12
/
Qere there is 70|J. over=^|^., which,
divided by 100, gives \lld. = '^^d.
4.70§ Ans. £3 14s. ^^d.
Ex. 26.
£ s. d. £ s. d. f 8. d.
1. 8 17 4| X 5. 2. 5 11 2i X 7. 3. 4 0 5| x 9
4. 7 8 llg xll. 5. 6 1 7^ xl5.
Q"^.
7. 6 17 4f x32. 8. 2 19 9| x 44. 9. 4 13 0| x 29.
12
10. 5 3 4l§x31. 11. 7 14 9^ x 37. 12. 6 18 0/^x41
£ s.
d.
13.
2 0
1 -^3.
15.
29 17
8 -fS.
17.
8 13
0 -r9.
19.
73 0
5i-i-8.
21.
69 17
5^4-9.
£3.
124 15
6 -7-15.
2.5.
135 14
10 -7-40.
27.
1275 3
8 -^200.
£
s.
d.
14.
9
7
31-. 4.
16.
72
13
5 -^6.
18.
37
6
2 --10.
20.
29
7
01-^7.
22.
53
4
0l-rl2.
24.
131
11
81^-18.
26,
Hi
11
iii-^co.
28.
675
13
6i-.500.
29. 1134 15 10 -rlOOO. 30. 4332 13 7|^3000.
44 VULGAR FRACTIONS.
42. To reduce a compound fraction to a simple one.
Rule. Multiply together all the num" for a new num',
and all the dene's for a new den^.
Ex. 1. iof| = i.
For one-thinl off is ^ (37) ; therefore two-thirds, whidi must be twico
ns great, is ^^ (36).
By similar reasoning, f of | = ^ = g of ^.
Ex.2. iof5=foffl^.
Mixed numbers must be reduced to improper fractions,
before the rule can be applied.
Ex.3. 2fof5of3l=-U.offof| = aM = 48l.
Compound fractions may often be reduced by striking out
factors common to one of the num^^ and one of the den^'^
5
Ex. 27. Express as simple fractions
1. §of§of4. 2. foffofe. 3. foffof3.
4. I of! of 31 5. I off off. 6. fof3iof9i.
7. foffof^. 8. f off of 31 9. 4lof3|ofl0.
10. 2loffof7^ 11. foffof?!. 12. 3loflfof3§.
13. f of 11 of 9 of 61. 14. l|of2|oflioflil.
15. ^ of ^ off of 7. 16. i of 61 of If of ^.
17. ^of IfofSlofl. 18. llof2|of33of4f.
19. 3^ of 21 of I of ^. 20. ^ of 21 off of 101.
43. To reduce fractions to a common denominator.
Rule. Find the l. c. m. of all the den^^, and take this for
the common den^ : for the new num*'^ multiply each num*" by
the number obtained by dividing the common don^ by its
own den^.
Ex. Reduce f, ll, ^, to their least common denominator.
The L. c. M. of 8, 12, 18, being 72, we have
55 x9_45 11_11 X 6_66 7 _7x4_S8
8 72 72' 12 72 72' 18 72 "72*
VULGAR FRACTIONS. 45
where the factors 9, C, 4, in the new nuni" arc obtained by dividing the
connnon den' 72 by the original den" 8, 12, 18, respectively.
For, in any one of these fractions, it is plain that its num' and den'
have both been multiplied by the same number, viz. that which makes its
den' = 72.
Ex. 28. Reduce to their least common den'
1.
\^v\.
2.
%
1'
h
^.
3. hhhl^
4.
8» 'h Tg» T5'
5.
I
h
ii'M.
6. §,f'f'M-
7.
1^» Ta» 24» gi' 4l'
8.
2 4 16 ^ J.q.
3' U' Jff 81' 5243'
9.
3 7 6 11 13 2.1
5' 10' 25' 30» 45' UU*
10,
12.
27' -14' h 15' 9' it-
5 11 2 8 9 17
Y» 12' l3' 27' 35' 40*
44. Addition of Fractions,
RuLR. Reduce them (if necessary) to their least commoii
clen^; and take the sum of the num^^ retaining the common
Ex.1. f + i=|.
For the whole being divided into 5 equal parts, 3 of those parts, to-
gctlicr with 1 of those parts, must make 4 such parts.
Tv O 2 - 3 , 4_12±i£±i^_Jl33._2l3
IjX. J.. 3 + 4+5— 60 ~" 60 ~~^G0'
If any of the given quantities are whole or mixed numbers,
it is best to take separately the sum of the integral and
fractional parts, and then add the two results togeth«^.r.
Ex.3. 25 + 3j"j+5^ + 4.
TTpvn 3 . JL + _5_ — 2!it^t??_ JJJ5 _155_ ill.
XltlL 5 + j^+ 1^— 60 (TcT — ^SU— 'T2»
.*. 2 + 3+5 + 4+lll=15l|.
Improper fractions should be reduced to mixed numbers,
and compound fractions to simple ones, before the application
of this rule.
Ex.4. ii3 + |of^ + 2|of 2^of f + 5 = 14| + |-»-3|' + 5.
'"•^"'. _37_T13.
24>
13_OQl3
.-. 14 + 3 + 5+ li| = 23i|,
D
46 VULGAR FRACTIONS,
Ex. 29. Find the value of
1. i+f+?ff+2. 2. i+ifi+A. 3. n^+e+i.
4. \^ + ^^lh^E' 5. i + lfi + ll. 6. | + J, + X + ^.
7. tli + ^? + l.^. 8. ^ + A+l + n. 9. 21 + 31 + 41 + 5.
10. 3i + 2i + ^ + 3^. 11. 22 + 2 + 4 + 50. 12. l| + l + A + 2i.
16. 17;^ + ? + A+l^. 17. f of l8 + 2ofl^.
18. 11+ l--_4. J. +011+ X 10 111: + 02D + .<:*M + 41D
12 ^ ' I.-, ^ K, ^ -18 ^ 20' *''• ^T(i + -24^''25 ^ *30'
20. 52 + 2 of 71 + 8^. 21. 2 + 7^ + 4 of I of 101.
22. 2;^ of 3^- + JjLL + 21 of 41 of 1 2 + 4| of ^ of 2l of 1 ?-.
£
s.
d.
£
*.
d.
£
s.
d.
£
s.
d.
23.
3
5
n
24.
7
5
Si
'25,
. 3
15
H
26. 7
11
8i
4
10
H
2
13
^t
5
14
n
2
9
7^
5
6
5§
5
11
41
7
G
lOf
6
5
4^
6
12
n
2
8
5^
8
1
ni^
3
18
7^
7
5
2^
7
17
3^
2
4
6^
4
5
61
2
8
3
5
7^
28.
0
10
4^
29.
1
4
5f^
3
19
2^
27.
7
13
i;i
17
13
5i
30.23
2
6i
6
1
2i
2
17
4f
32
6
lU
14
1
5^
5
17
8i
5
2
H
12
10
9f
7
8
H§
6
4
2^
6
11
2M
7
0
8|
4
9
5M
5
1
7!
4
5
Of
11
5
4|
16
4
2|
7
12
6M
6
3
4|
6
16
5^
5
4
3i
45. Subtraction of Fractions.
EuLE. Reduce them (if necessary) to their least common
den^, and take the difference of the num", retaining the
common den^.
Ex 1 *-l-5
For the whole being divided into 5 equal parts, and 1 of those parts
being taken from 4 of those parts, there will remain 3 such parts.
*^^' ^' XO 15 ~ 3U ~"S0*
VULGAR FRACTIONS. 47
If the given quantities are both mixed numbers, or con-
sist of a whole and a mixed number, it is best to taice sepa-
rately the difference of the integral and the fractional parts,
and then add the two results together.
Ex. 3. 5| - 2|.
Here i-i=^V* = ^; .-.5-2 + 1 = 31.
Ex.4. 5^-21.
Hero i-i=^«-^=-i; .-.5-2-1 = 3-1=21.
Ex.5. 6-42 = 2-? = li.
Improper fractions should be reduced to mixed numbers,
and compound fractions to simple ones, before the application
of this rule.
Ex. 6. lof2lof 16-lf of5l = 8-7f=-.7|-7| = i.
Ex. 30. Find the value of
] 13, 8^. 13 _7_. _8 9_. 1 1
'• 15 15» 20~20» 15 20 » 2 3'
2cS il. o3_95. r._<)G. inS 11
S. 1^-1; 9-3^; 97l-48i; 5^-2l^.
4. 13^-3,4; i^-3^; 31-^; 24^-21^,.
5. 1^-f; 17i-^; 4f-lof|; ^-^of^.
6. l|of2l-3l|; 51 of 41-31 of 31.
7. 3l + 4|-5l+16|-71i+10-14§.
8. 5l-.2iJ-3^ + -f— 16l+3i+8l.
9.
12.
46. 3Iultiplicatio7i of Fractions.
Rule. Multiply the num^'s together for the new num^, and
£ s.
d.
£ s.
d.
£ s.
d.
13 0
H
10.
4 17
Ill
11.
9 0
oi
4 17
6^
13.
3 19
4^
14.
8 17
n
15 0
3f
7 17
7^
8 13
6f
§ 19
m
6 19
9^
4 19
n
the den^s for the new den^
Tv- 1 2 V 4^ 8
d2
48 VULGAR FRACTIONS.
This method is the same as that we should have used to
find the value of the compound fraction f of f, or | of §,
(42) ; and we must here observe that the same word 'Multi-
plication's used to signify, not merely, in its original sense,
and as we have hitherto employed it (wlien the multiplier
Avas a whole number), the taking a inuUiple of a quantity,
i. e. repeating it some number of times, but also (wlien the
multiplier, as here, is fractional) the taking any part or
parts of it ; so that ' to mulliplij 5. by J ' is only another way
of-saying 'to take \ of -J'; and hence the Rule for the oper-
ation is the same in the two cases.
It will be seen, however, that this Rule includes the case of
Mult" by whole numbers ; thus if we had to find the value of
f X 5, we might say, J x 5 = J x f = J^5 = J^i-, obviously the
same result as we should have obtained by the common rule
of Mult" by whole numbers (36) : and it is on this account,
viz. that the general method of taking any part or parts of
a quantity includes the particular case of taking any multiple
of it, that mathematicians have adopted the name, properly
belonging to the latter case only, and applied it also to the
former, calling the operation in both cases multiplication.
The method, therefore, of Mult" of Fractions is the same
as that for reducing a compound fraction to a simple one ;
and (as in that case) mixed numbers must be reduced to
improper fractions before applying the rule, and the result
may be simplified by striking out factors common to num^
and den^.
Ex. 2. 25 X 3i X If of I of 10 = -^^- X I X I X I X ^^^ =5.03 ^ 102|.
Ex. 31. Find the value of
1. -^yJLvO^. OJLvJLvl-'-' 2-5-v9iy-iL
*• ja^'iu^-ii» -KJ^iT'^^u' ^TI^'^5^31'
2. H X 21 X 100; 1.31 X 3| X 1^^^; Gf x 2§ of 21.
21 of 3| X 4| of 11; 21 x If of l^Sj ^ 3I of 1^..
lof ^of fx^ofSl; liof^x tkof2iof8.
f X If of 121 X 21 of ^; i of 11 X 2§ of 43 of 2|.
VULGAR FRACTIONS. 49
47. Division of Fractions,
Rule. Invert the divisor, and multiply.
i.X. 1. 4 — 7-4^5— 20- '20*
Here also the word * Division * is used in a more general
sense than heretofore, to denote the finding that quantity,
which, multiplied by the divisor, will produce the dividend
— the word multiplied^ being here used in the enlarged sense
explained in (46). Hence, in the above Example, where the
div'' is ^ and the div^ J, we must have quotient x ^ = -J :
multiply each of these equals by the same quantity f , and the
products must be equal; .*. quotient x -J x ff=f X f : but
^ X |- — 1 ; hence the quotient— \ x | = 1t,V> as above.
The quotient thus obtained will have its usual meaning,
when the div^'is an integer, i. e. will express how many times
the div^ contains the div'', or what multiple the div^ is of the
div"* ; thus J-j-o =|-h ^ =^ x 1 =^^5 and hence f contains -j^
five times or =5 x -^^ : but when the div^' is a fraction the
quotient will express what part ov parts the div<^ is of the
div*" ; thus :f-J-y= (as above) l^V? and hence f =l^'jy of f.
Mixed numbers must be reduced to improper fractions, and
compound fractions to simple ones, before applying this rule
Ex.2. 21-33 = 1^ --=«,_i = 3^.
Ex.3. (2§of3^)^(4loff of ^) = 10-f = 10x1 = 35.
Ex.4. t2£l = i=2x^=l^
Hence it follows that a complex fraction, in which both
the num'^ and den^* may appear as fractions, may be simplified
by multiplying together the outside numbers, or extremes, for
the num'', and the middle numbers, or means, for the den'.
-, ^ f 15 I I 2 2 2 ]o 2^ 2j G3
• |~28' 3~f~9' 2^ V~ll'3i '3"~80*
So also, in a complex fraction, common factors that appear
in either one of the extremes and also in one of the means,
may be struck out of both.
50 VULGAR FRACTIONS.
V, 3 5 XL
2| 8 _3, 5 ^.^4. 1^^.112^1
X.X. 6. 7-— -8' 3f-\^ 3 3' lOi ^J^ 6*
1 4 ^
Ex. 32. Find the value of
1. 2-§; l-a; 2I-MI; 2^-31; 161-121; ||-^.
2. lli^^f; 5-14; (I of e)-(f off); (4|of ^)-(5f of If).
3. 209-1 of 20; (f of |)-(| of 1 of 5); (4i of 3i)-(2i of 6i).
KO Q^ 14 11 Q"!^ *; ^ fi3 1 c3
«^4 •* ^25 'is -^27 ■*]I «^8 '5
" 23 ?^ofU. 3f of2ll 21+ 1| 4^of2|
°* 02 4. 2 ' fl of 11' -i- of 8—' 3^ — '?!' 5i_4l ■
We shall here give examples of the application of the pre-
ceding rules to the Mult" and Div" of concrete quantities.
Ex. 1. Find the value of | of £4.
Since (32) f of £4 is the same as 1 of £4 x 3, we first multiply £4 by
3. and then divide the result by 8.
£4
3
8) 12 0 0
£l 10 0 Ans.
This is the same (46) as to mulUphj £4 by |.
Ex. 2. Divide 1 ton 13 cwt. 15 lbs. by ll.
Since 15=5, we have here (47) to multiply by f. We may do this as
ill Ex. 1, or (which is often more convenient) by first dividing by 2,
which gives 1 of the quantity, and then dividing this half by 2, which
gives 1 of it; and adding the two results together, we shall have f of it.
ton cwt. qrs. lbs.
1 13 0 15
3
ton
1
cwt.
13
qrs.
0
lbs.
15
for 1
fori
0
0
16
8
2
1
^i
1
4
3
lU
4) 4 19 1 17
1 4 3 lll^?is.
Compound fractions must be reduced to simple ones before
the application of this rule ; but, in the case of mixed numbers,
VULGAR FRACTIONS. 51
it is best to multiply separately for the integral part, and add
the result to that obtained by the Rule for the fractional part.
Ex. 3. Multiply £2 10*. 4d. by 3^.
£2 10 4
3A
12)1-2 11 8
7 11 0
£S 11 11§
Sometimes it is convenient to reduce the given quantity
to one denomination, before applying the Rule.
Ex. 4. Divide 7*. ly. by ^-f,.
Here 7*. l^d. =342 forthings, which we have to multiply by ^^-= 12|.
342/:
12"^
9)^2394
2C6
4104
4)4370
I2)l09-2ld.
9 Is. 01'/. =£4 lis. Ojd.
Ex. 33. Find the value of
1. I of £1 ; 15 of £0; Gs. 8cl. X f ; 3^ of 2s. U ; 2§ of 21s.
£5 4 6l-M|.
£10 11 21x35.
£2 10 6|x35.
5. £30 14 61^-4; £7 13 4--if; £4 7 35-^.
6. f of a ton ; | of a lb. Troy ; 3 cwt. 1 qr. -4- 1^ ; 11| of 6s. 1 llrf.
7. 2 wk. 3 d. 4-^ ; 3a. 3r. 3p. x 1 0^ ; 2s. 9}d. x 1 of 51
8. 1 of I8I9.; 1 cwt. 2 qrs. 13 lbs. x 3II ; 13|f of £7 5s. lOd.
9. £1 lls.6d-^^f; ^ of £8 8s. 51c/. ;^ of ^ of 5^ of 27s.
10. 1 m. 5 fur. 91 yds. 2 ft. -r 2| of 1^ ; £.3| + 9^s. + 5|c/.
11. £3 + ^s. + 1 of 2 Is. ; f cwt. + 8§ lbs. + 3^ oz. ; 4 d. 5 h. x I/5.
12. l|of 10s. 6rf.-|of 2s. 6(/. + £^-^of 21s.
2.
£3 6 8x^;
£3
7 5^11;
3.
£7 6 Slxlf;
£8
0 75x2|
4.
£13 15 4x4f ;
£18
17 0x4|;
52 VULGAR FRACTIONS.
11. f of 2 15. -^ ^ of 5s. 4 g of 75. 6(/. - ^ of 2(1.
15. 2| of 1 1 of 8f^i. + 3§ of 1|^ of ^i of 4W.
16. |of£l5 + 3fof£l +loff of2of£r+|off«.
48. Tg reduce a given quantity to the fraction of another
given quantity.
Rule. Reduce both to the same denomination ; and lake
the result of the former for the num^, and of the latter for
the den^, of the fraction required.
Ex. 1. Rcdiicc Is. Id. to the fraction of £l.
Since Is. 7(1. — Old., and £l =24 Or/., tlic fraction required is ^.
Tor Id. is ^^ cf £1 ; and therefore 7s. 7d., Avliich = 9k/., is ^ of £l.
Any common denomination, to which the two quantities
may be reduced, would answer the purpose of expressing one
of them as the fraction of tlie other ; but if the higliest, of
which they both admit, be taken, the fraction will be ex-
pressed in lower terms.
Ex. 2. Reduce half-a-cro^n to the fraction of half-a-guinca.
Reducing them to pence, we have tlic required fraction = ^"^7-, Lut
reducing to sixpences we liavc tlie same fiuct on in lower terms, = ~.
Note, rfx expresses what is called the Raiio of 2*. 6rf. to 10*. 6^/. ( 79 ).
Ex. 34. Reduce
1. 35. 4r/. to the fr. of £l ; 2.9. 61^/. to the fr. of 6fZ.
2. £7 9.V. Gf/. to tlic fr. of ^^13 4.?. G(/. ; 65. 8^. to the fr. of l^d.
3. 3 qrs. 1 4 lbs. to the fr. of 3 cwt. 1 qr. ; 1 ton 4 CAvt. to the fr. of
15 cwt. 1 qr. 20 lbs.
4. 3.S. 71/1 to the fr. of £l 3.9. ^d. ; £4 75. G^J. to the fr. of 275.
5. 3 cwt. 2 qrs. 3 lbs. to the fr. of a ton ; 14 h. 15 m. to the fr. of
31 days.
6. 2r. 13p. to the fr. of 3 acres ; 14 half-crowns to the fr. of ^s. 8d.
7. A ton to the fr. of 3 cwt. 3 qrs. 21 lbs. ; 30r. 5 yds. to the fr. of
1 fur. 2Sp.
8. 3 w. 16 m. to the fr. of half-an-hour ; 3 qrs. 2 qts. to the fr. of
4 qrs. 3 bus.
9. 8a. 3r. to the fr. of 2a. 32p. ; 1 ft. 2f in. to the fr. of a yard.
10. 7 h. 12 m. to the fr. of a day ; £4 1 25. 1 Id. to the fr. of £l 9.9. SU.
VULGAR FRACTIONS. 53
11. 17 lbs. to the fr. of I qr. 14^ lbs. ; 1 m. 4 fur. to the fr. of 3 yds.
1 ft.
12. 2 sq. yds. 2 ft. 120 hi. to the fr. of 3p. 13^ yds. 1 ft. 72 in. ; 3 cwt.
14 lbs. to the fr. of 2 ton 2 cwt. 2 qrs.
13. £22 13s. 8ld. to the fr. of 3i gs. ; £3 16s. 6fd to the fr. of
£l 3s. 51(1.
14. 3000 in. to the fr. of 1 fur. 5r.; £2 Os. S^d. to tlie fr. of £l 4s. 2lr/.
15. 1| guineas to the fr. of £l| ; £ll 6s. 5c/. to the fr. of £10 5s. 4d.
16. 3| crowns to the fr. of £l 12s. 9f(/. ; 2| half-guineas to the fr. of
10s. Hid.
49. To reduce a fraction of one given quantilj to a frac*
lion of another.
Rule. Express by (48) the first quantity as a fraction of
the second ; and the fraction required will then be found by
reducing the resulting compound fraction to a simple one.
Ex.1. Reduce |s. to the fraction of £1.
ls. = iof£l: .M9.=§of£^^ = £^.
Ex. 2. Reduce l^h. to the fraction of 10 min.
1 h. =^ of 10 m. = Q of 10 m. ; .*. 1 ^^ h. = 1^. of ^ of 10 m. = 6f of 10 ni.
Ex. 3. Reduce 3% of £l Os. ^d. to tha fraction of £l 10s. \Qd.
£1 Os. 9|c/.=999/:, and £l 10s. 10(/.= 1480/. ;
hence the required fraction =3| of ^[^ = 2|.
Ex. 35. Reduce
1. £^ to the fr. of a guinea ; l^s. to the fr. of £l.
2. Id. to the fr. of 15s. ; 12| of 3s. G(/. to the fr. of £l.
a. f of Is. 6c/. to the fr. of Is. ; f of a sixpence to the fr. of £l.
4. 3i of £1 3s. id. to the fr. of £5 ; 2| of 17s. ^d. to the fr. of 10s.
5. 3\ of 1 cwt. 3 qrs. to the fr. of a ton ; 3f d. to the fr. of 3 wks.
6. \\ of £3 13s. 6c/. to the fr. of 10s. 6c/. ; 2| of £6 to the fr. of £ I 13s.
7. 2± of 4 cwt. to tlie fr. of 3 qrs. 4 lbs. ; 4| crowns to the fr. of 5 gs.
8. I lb. Tr. to the fr. of a lb. Av. ; § po. to the fr. of a fathom.
9. I sq. ft. to the fr. of a pole ; 12§ of 1 qr. 3l lbs. to the fr. of
1 ton 2 cwt.
10. Slof 2a. 3r. to the fr. of 2r. 2lp. ; 1-^ of £2 4s. ^d. to the fr. of 5s.
1 3
54 VULGAR FRACTIONS.
11. 3f wks. to the fr. of Id. 8|hrs. ; 2§ of 45 yds. to the fr. of 10 miles.
12. 2| of 3r. 6p. to the fr. of lA. 2r. 3p. ; | of 1| of 105. 7|(/. to the fr.
of £4 4s. 4|d
13. 33^ of 3 qrs. to the fr. of 3| tons ; 3f of 1|a. to the fr. of 2a. 2|p.
14. 7} of £2 3s. 6|J. to the fr. of 7s. 6d. ; | of 5s. + fs. to the fr. of 21s.
15. 4| of £2 13s. 7^d. to the fr. of £2 Us. Sid. ; If of £2 Os. 1^. to
the fr. of £2 2s. 2ld.
16. 6|f of £1 10s. 5|(/. to the fr. of £3 3s. O^^. ; | of £l -§ of 21s. to
the fr. of 10s. 6d.
Miscellaneous Examples. 36.
1. Which is the greatest and which the least of ^, ^, ^ ?
2. Divide the sum of 5, i, and ^ by the difference between ^ and 1.
3. What n" added to H makes If ? and what taken from Ip leaves
119
15 •
4. Which is the greater, | of 2f, or ^ of 1|, and by how much ?
5. Divide the sum of 10 and ^ by the difference, and also the
difference by the sum ; and find the sum and difference of the two
quotients.
6. Divide the sum of f of £3 7s. 6d., and | of 4| guineas, by lOf.
7. If I pay away i of my money, then ^ of what remains, and then
i of what still remains, what fraction of the whole will be left ?
8. What n" added to if, li, ^, ||, will make the sum total 3 ?
9. What must be the length of a plot of ground, if the breadth be
15| feet, that its area may contain 46 square yards ?
10. Add together the sum, difference, product, and quotient (the
greater being divided by the less) of f and ~.
1 1. Find the value of f lb. Troy + 1 oz. Troy ; and of £f-|s.
12. Express 2|ells as a fr. of a yard ; and mult. 3 ft. 7i in. by 2i in.
13. Add the sum and difference of f of 3 guineas and | of £4.
14. Divide ^ (^fo^fT^ by -?-, and find the value of
14' i 1 i.
2i'^3i'^4i
15. To ■^ of a dozen add || of three hundred, and divide this sum l)y
the difference of 3| of a hundred and 433.
Vulgar fractions. 55
16. Multiply the sum of 1, |, |, and |, by the difference of ^ and ^;
Bnd divide that product by the double of 21|.
17. Take from 1 its half, third, and twenty- fourth parts; add the
product of those parts to the rem'; and multiply this sum by 7yg.
18. Mukiply the sum of 3|, 4|, and 4i, by the difference of 7f and
5|; and divide the product by the sum of 94^ and 93,j.
19. Divide 2 by the sum of 2§, f, and 4 ; add If-^ to the quotient;
and multiply the result by the difference of 5^ and -Ji.
20. Findthcvalueof (| + |)x(l| + 2|) x(2^-ll)x(3i-f); and
of l|-i-2|+5|^3l.
21. A person had ^ of a lottery ticket, which was drawn a prize of
of £518 10s. • what was the value of his share?
22. Express the sum and difference of £^ and f of a crown as
fractions of half-a-sovcreign; and find how many times the first contains
the second.
23. Multiply U^s. by 109f, and divide £g1 4s. 7{'^d. by 2G7^.
24. IIow often is f.?. contained in half-a -crown ? and how often is
£| contained in 24 guineas?
25. If a yard of luce cost £l|^, what will 16i| yards cost ?
26. If I of a ship be worth £3740, what is the value of the whole?
27- Compare, as fractions of their highest common denomination, the
values of ^ of £l, t^j of a guinea, and -^ of a crown.
28. Find the value of ,,, ^/.t^ , i x - of3^^i%
(liof§)-10i o 13|ofo^
29. If 3 of an estate be worth £220, find the value of ^ of it.
30. Express in Tr. weight the difference between | lb. Tr. and | lb. Av.
31. Find the value of (12|-8|-1^ + ^) x 4i x (7^-6^), and of
32. Compare, as fractions of their highest common denomination, the
values of i of half-a-crown, ^ of 35. 4f/., and ^ of 4s, 2id
33. Express, as a fraction of £5, the difference between £7f and
£7 X f ; and find the value of £l4^-^lif.
34. A person owes a guinea to each of 4 creditors : to one he pays
I of his debt, to another |, to another f, and to another ^ ; what Avill he
still be owing altogether ?
35. Express in Troy weight the sum of 3| lbs. Tr., and 16^ lbs. Av.
56 VULGAR FRACTIONS.
36. Find the value of ^^~^^of ii+£M of !ilii.
37. If ^ of a ton is worth £4 10^., what is the vahie of i of it ?
38. After taking out of a purse | of its contents, § of the remainder
was found to be 13s. 5^d. ; what sum did it contain at first ?
39. The dimensions of a room are 29^ ft. by 1 15 ft. ; what length of
carpet, f yd. wide, will cover it? and what will be the expense of it, at
3^s. per yard ?
40. A ship is worth £16000, and a person, possessed of ^ of it, sells
I of his share ; what share has he remaining, and what is it worth?
41. Express 4 bus. 1 pk. 1 gal. 2 qts. as a fr. of a qr. ; and reduce
5 cwt. to lbs. Troy.
42. If I of a ship be worth £36 IO5. 7^., what share will cost
£125 5*. ?
43. Multiply 3^ by 15f, and divide -- by '*; and add together the
3j 3
sum and difference of these results.
44. A party having a bill to pay of £12 7s. l^d., one of them pays
for himself and three friends the sum of £5 9s. lOc/. ; how many were
they?
45. Express both in Tr. and Av. weight, ^Ib. Tr. +^lb. Av.
46. A pint contains 34§ cubic inches ; how many gallons of water
will fill a cistern 4 ft. 4 in. long, 2 ft. 8 in. broad, and 1 ft. l|in. deep ?
47. Add together If, 2|, and 3|; multiply this sum by the product
of these fractions; subtract from the result tlic difference of 2§ and li;
and divide the remainder by the sum of 5^ and 1| of 3^.
48^ How many yards of paper, f yd. wide, will be required for the
walls of a room that is 20g ft. long by 1 U fr. wide, and 12^ ft. high ?
and what will be the cost of it at 2^(/. a yard ?
49. A cubic foot of wood weighing 1 l^lbs., wliat is the weight of a
beam 24 f>. long, 2^ ft. wide, and 2^ ft. thie k ? and what is its value at
3||s. per cubic foot ?
50. A person dies worth £10000, and leaves i of his property to his
wife, i to his son, and the rest to his daughter. The wife at her death
leaves 3 of her legacy to the son, and the rest to the daughter ; but the
gon adds his fortune to his sister's, and gives her 1 of the whole. How
much will the sister gain by this ? and what fraction will her gain be of
the whole ?
57
CHAPTER IV.
DECIMAL FRACTIONS.
50. In common numbers, or decimal integers, the actual
value of each figure depends upon its position with respect to
the place of units, its value in any one position being one-
tenth of what it would be, if it stood one place further to the
left : thus 3045 denotes 3 thousands, 0 hundreds^ 4 tens, and
5 units, or 3000 + 0-1-40 + 5; where Ave may obtain the
actual value of any figure by multiplying it by 10, 100, 1000,
&c., according as it stands in the 1st, 2nd, 3rd, &c. place to
the left of the place of units.
Now if we continue the same method of notation to the
right of the place of units, still reckoning the value of each
figure to be one-tenth of what it would be, if it stood one
place further to the left, we obtain what are called decimal
fractions, or briefly decimals ; thus setting, as is usual, a dot,
called the decimal point, after the unit's place, the number
3.045, &c. will denote 3 units, 0 tenths, 4 hundredths, 5 thou'
sandths, &c., or 3 + y^g- + -j^--^ + yy^g^ + &c. ; where we may
obtain the actual value of any figure by dividi?ig it by 10,
100, 1000, &c., according as it stands in the 1st, 2nd, 3rd,
&c. place to the right of the place of units.
51. Hence it follows that a decimal may also be defined to
be a fraction, whose den'^ is 10, or some power* of 10, as 100,
1000, &c., which den**, however, is not set down, as in vulgar
fractions, under the num'^, but expressed by marking oiF by a
point, from the right of the num^, as many figures as there
are cyphers in the den'^, prefixing cyphers to the former, if
necessary, to make up the requisite number of figures after
the point.
* A power of a number is the product of a number multiplied by
itself once or successively. When the number is used as a factor twice,
thrice, &c., the product is called the second power, third power, &c., of
the n».
58 DECIMAL FRACTIONS,
inns joo — c5 + Y0 + Y00-J.4/ ,
i;^ 10+3_ 1 , 3 _ Qiq .
1000 ToOO ~ioo 1000 — '^"^ '
flM = 2.125, ^lie^ = .0119, -^ = .00027, &C.
52. Conversely, any decimal maybe expressed as a vulgar
fraction by setting down tlic figures which comiiose it as the
num^, and for the den^, 10, 100, 1000, &c. according as there
are one, two, three, &c. figures after the point. This, in
fact, amounts to expressing each figure separately as a vulgar
fraction with its own den'', and then bringing all these frac-
tions to one common den''.
Tlirs^n*^-*?" or O"-"^ . QTQ — ^ J. ■?■ , . n _ 300+70-1-0 _ 379 .
J.nus ^.Uc>-/j-„o or -^^^ , .cJ/ J— j-^ +• Xoo + Iooo Toco 1000 »
42.037 =42^ or .Mni^; .0029 = ^^; 15.001 = 15^^ or J^^I-.
Sometimes the resulting fractions admit of reduction to
lower terms.
Thus 13.75 = 13^=131; 23.0625 = 23^i§^ = 23^.
53. Any decimal is multiplied by 10, 100, 1000, &c. by
moving the point one, two, three, kc. places to the right, jind
divided by moving it similarly to the left.
Thns
3.247=f§^; licnce 3.247 xlO = =^^^ = 32.47; 3.247^ 10 = ^^ = .3247,
3.247 X 1 00 =2fn = 324.7; 3.247 -=-100 = ^^ = .03:4 7.
So .0023 X 100= .23, 2.3 -j- 100= .023,
2.3 X 1000 = 2300, 2.3 -r- 1000 = .0023, &c.
54. It should be carefully noticed, that adding cyphers to
the right of a decimal does not alter its value ; thus .3, .30,
.800, are all equal, representing each of them y\, or as in (52)
TO' Tuo> tWo5 respectively ; but prefixing cyphers to the left
of a decimal after the point is equivalent (53) to dividing it by
10, 100, &c.; thus .3, .03 .003, are respectively j\, yj^, i^'V^.
Ex. 37. Express as decimals
117
■"• 100' 10000' 10000' lOOOOOOO*
3. 2 tenths + 3 hundredths + 37 millionths.
4. 11 tenths +11 thousandths + 1 1 hundred-thousandths.
5. 13 + 3 thousandths + 5 millionths.
DECIMAL FRACTIOXS. 59
6. 101 tenths + 10 thousandths + 101 millionths.
Express as vulgar fractions
7. .037, .0002, .25, .375. 8. .0075, 1.225, .1875, 3.225.
9. .0006875, .0009375, 23.038125.
10. 15.203125, .0023437.5, 4.0078125.
Multiply and divide
11. .3 by 10 and 1000, .00125 by 100 and 10000, 538.734 by ten
thousand.
J2. 1.1 by iOOO and 1000000, 11.025 by 1000 and 100000, and
213.012 by a million.
65. Addition and Subtraction.
Rule. Set down the decimals with their points in the
same vertical line, so that units of the same kind may be
under one another, filling up the bhank places with cyphers ;
tlien add or subtract as with common integers, setting tlie
point in the result in the same line with the other points.
Ex. 1. Add together 2.8146, .0938, 8, .875, 31.2788, 4.00S7.
2.8146 Here the figures in the right-hand column represent so
.0938 many ten-thousandths ; so that we have to add together
8.0000 r)+8+o+n+8+7_ 20 _ a . 9 .
.8750 lotuj 10000 1000 "^ 10000 »
31.2788 we set down therefore the 9 under the column of icn-
^■^^^"^ thotismidths, and carry the 2 ihomaudths to the next column ;
47.0709 and so on.
Ex. 2. Eind the difi'crence of 2.418 and 1.2234.
Here we have 4 ten-thousandths in the lower line, but
L2234 ^^^^ i" ^he upper ; we therefore have to borrow one from
ryrTr the 8 in the next column, 1. c. we borrow 1 thousandlli
= 10 ten-thousandllis, from which we take the four ten-
thousandths, and have 6 remaining ; we have now only 7 thousandths
in the upper line, from which we are to take 3 thousandths, or, instead
of this (as in former cases of borroiving in Subtraction), we may take
4 thousandths from 8 thousandths; and so on.
Ex. 38. Find the value of
1. 11.275 + .34132-1-. 00414 + .0001 + 23.001.
2. 321.4 + 12 + 31.6 1 54 + .01 + 2.214 + 415.62.
3. .001213 +45.613 + 234 + .0012 + 141.00056.
4. 1.0000123 + 81.1 + 117.154 + 2343.008 +.0002.
5. 32.001-12.999 5 and 3.45-.00098.
6c 23.1415-2.008 ; and 3.412 -2.99987.
60 DECIMAL FRACTIONS.
7. 22.0001-2.9999; and 2415.6-2414.598^
8. .001 -.0009987; and 24.004-.987516.
9. 1.3742-.03742; and 3.054 -.3054.
10. .0123 -.009087; and 3.33 -2.^8765.
56. Multiplication.
Rule. Multiply the given decimals as if they were com-
mon integers, and mark off in the product as many decimal
places as there are in the multiplier and multiplicand
together.
Ex. 1. Multiply 1.002
1.002 5
2.5
5 b^
25 X
'2.5.
2.5 =
.1001
■ 100(
5012 5 For 1.00:
2 0050
ix!^=^MS^M = 2.50625.
2.5062 5 Ans.
Ex. 2. Multiply .0048
by.
000012;
and 1.005 by .005 x .0064.
.0048
.000012
1.005
.005
.0000000576
.005025
.0004
20100
30150
.0000321 600 = .0a00321G Ans.
Ex. 39. Find the value of
1. 22.5 X 32.16; and 4.41 x 33.21.
2. .0001 X .001 ; and 32.1 x 2.31.
3. .0032 X 23.45 ; and .0002 x 3.01.
4. 22.5 X .0241 x .0024; and .0003 x .01 x 500000.
5. 2.7 x .27 X .027 X 270; and .2 x .04 x .008 :. 64000.
6. 1.1 X. Oil X 1.01 X. 0101; and .013 x 1.6 x .007 x 3.05.
57. Division.
Rule. If the given divisor is not a whole number, make
it so by removing its decimal point altogether, and shift the
decimal point of the dividend as many places to the right
us there were decimal figures in the divisor ; annexing for
this purpose decimal cyphers, if necessary, to the dividend.
DECIMAL TRACTIONS. 61
Then divide as if the given decimals were common inte-
gers ; and when, in the process of division, the decimal
point of the dividend is arrived at, place a decimal point in
the quotient.
Decimal cyphers may be annexed to the dividend, to any
extent that may be wanted for carrying on the division. (54)
Ex. 1. Divide 277.53 by 12; also .27753 by 12; and 1037 by 305.
19\o'.'' conn Here tbc divisors are all integral, and the position
- — :^„- ; -v of the point in the qnoticnt is very simply dctcr-
,qv '»J,„„,. mined. In the first sum, we take the 12tli part of
' 27 tens, which is 2 tens and 3 over; then the 12th of
37 units is 3 units and 1 over; then the 12th of
^^^ 9?5^^^'^ 15 tenths is 1 tenth, &c.; so that the point in the
1^20 quotient comes exactly under that of the dividend,
1220 I" the second sum the 12th of 2 tenths is 0 tenths;
the 12th of 27 hundredths is 2 hundredths, and 3 over,
&c.; and here the student should particularly observe, that when the
divisor is a Avhole number, there will always be a quotient figure, though
eometimes, as here, a cypher, for every decimal figure of the dividend.
Ex. 2. Divide .805 by 2.3, .001029 by 1.68, and 1 by .007.
2.3). 805 1.6S).001029
23)8.05(.35 1687". 1029 (.0006125
69 1008
115 ^210
115 168^
420
' 007)1.0000 ■ 336
7)2000 "00 840
~1 4 2.857 &c. yO
In the 1st of these sums the divisor, 2.3. is mulf^ by 10,
which removes the point, and the dividend is also raulf^ by
10, by having the point shifted one place to the right. In
the 2nd sum the divisor and dividend are mult^ by 100,
nnd in the 3rd by 1000, to make the divisor integral. In
the 3rd sum the quotient will not terminate, but, by annex-
ing cyphers to the dividend, we may continue the quotient
ns far as we please.
Obs. An integral divisor ending with cyphers may be de-
prived of the cyphers, if we shift the point of the dividend
one place io the left for every cypher withdrawn : thus,
.45 --60 =.045 -6
62 DECIMAL FRACTIONS.
A little consideration will enable us often to avoid the trouble
of counting the decimal places of the dividend and divisor.
Ex. 4. Divide 15.95 bj 2.75.
2.75) 15.950 (5 8 ■^^''®' '^^^^^^^"'^ counting, we may set at once the
13 75 " point after the 5 in the quotient, because it is plain
2 200 that the divisor, which is a little greater than 2,
2 200 will go about 5 times in the dividend, which is a
little greater than 15.
Ex. 40. Find the value of
1. 15.G25-i-2.5; and .015625-^25.
2. 1562.5-^.00025; and 1.5625-^25000.
3. 18 1.3 -H. 00037; and 171.99-^27.3.
4. 9,065^.049; and .03 -v-.OOl.
5. 8-^.002; and 37.5 -^ 7.68.
6. 15^6.25; and 17.28-^.0144.
7. .00128 -=-8.192; and 1708.4592 -=-.00024.
8. .0002 -=-.0163; and 4 -^ .00255.
9. ll.l-^32.76; and.0123-=-3.21.
10. 2.1 17 -r. 0073; and .032-7-2.137. I i
— ■ N ;, ; ,
58. To reduce any fraction to a decimal.
Rule. If the den^ be 10, 100, &c. v/e may at once express
it as a decimal (ol) : in other cases, if 10, 100 &c. be [i factor
of the den^, divide the numerator by it as in (53), and then
divide the num'' as it now stands by the remaining factor as
in (57), and the result will be the decimal required.
59. Sometimes the division will not terminate, but the
same figures will be repeated over again continuallj.
T?^ 15 ^ 95 9.5 3 .03 J 4 ^ . ,
tun. Kcducc ^^'^'"g »fT^ or— -, and -, to decimals.
9) 9.50000 11) .03000000 7)4.00 00000
.57 14285 &c. = f
Decimals of this kind, in which the same figures are con-
tinually repeated without end, are called Circulating, Re*
peating, or Recurring, Decimals ; and the part repeated is
called the Period or Repetend.
DECIMAL FEACTIO.ISS. 63
It ifs usual to express any circulator by writing it down to
the end of the first period, and setting dots over the first and
last figures of the period ; which dots will, of course, be on
adjacent figures, when the period consists of only iioo figures,
and will coalesce into one dot, when the period consists of
only one figure.
Thus the above results would be written 1.05, .002?, .S71428.
A pure circulator is one in which the period begins im-
mediately after the decimal point ; all others are called
mixed.
Ex. 41. Reduce to decimals
1 2 . 13 . 42 . loon O 106 . 1 1 17 . 4000 . k3
*• 50' 250 » 8 » (:2.-. • — 125 » *'l25
*^' 'G4' 128' G400 » "^'31250* ^"
15^ . 11
125 ' '■ '1250 ' 25G > ^
l_ . 1025 . 13 . 7
12 ' 1024 ' 1600 ' 5120*
^ of ^^ • 7i of ^^ • 1 ^ of 1^ of •
16 62r' ^ «2500 » ^19 "^ ^75"^
60. Any fraction, to be expressed as a decimal, should
first be reduced to its lowest terms ; and then, if the den'*
contain only powers of 2 and 5 as factors, it maybe reduced
to ^finite or terminating decimal.
For, in reducing a fraction to a decimal, we set a point after the num%
and annex cyphers to it, until the den'^ will, if possible, exactly divide it.
Or, leaving out of consideration the point, (which, it is plain, docs not
affect the division, but only determines the place of the point in the
result), this amounts to multiplying the num^ by such a power of 10, as will
make it contain the den^. But now, since the fraction is supposed to have
been originally in its lowest terms, the den' can have no factor in
common with the original num' ; if, therefore, it be exactly contained in
the num' as it now stands, that is, with the annexed cypliers, it can only
bj by its being contained in that power of 10, by which the original
num' has been multiplied. But, since 10 contains only the factors 2 and
5, any power of 10 will contain only powers of 2 and 5 ; and, therefore,
the den% in order to be contained exactly in some power of 10, must be
made up only of powers of 2 and 5 as factors. In this case the division
would terminate, and the decimal be finite; but not so, if the den'
contain any other factors, such as 3, 7, 11, &c., since then no power of
\() whatever would contain the den', nor, therefore, would the original
num"", whatever be the number of cyphers annexed, become exactly
divisible by it.
64 DECDIAL FRACTIONS.
61. If the deii^ of a fraction, in its lowest terms, contain
any other factor than powers of 2 and 5, the fraction may be
expressed as a Circulating Decimal, where the number of
figures in the period will be less than the den**.
For since, in the division, the figures to be taken down are
always the same^ riz. cyphers, it follows that, whenever we
have any former remainder repeated, we shall also have the
same series of figures repeated in the quotient: but, if we go
far enough, we cannot help having some former remainder
repeated ; for, all the remainders must, of course, be less
than the divisor (or den'), and so the number of different
remainders must be less than the den' itself.
Ex. 1. Reduce f to a decimal.
0 ^-^ (85/142 jjpj.g y,Q haA'e had in order the remainders 6,4.
^Lr 5, 1, 3, 2, which are all there are less than the
^^ divisor, 7 ; the next remainder must therefore he
one of these again, and accordingly we find it to be
^2 6 ; now, since the same figure, 0, is taken doiim to it
as before, it is plain that the whole series of figures
^2 hi the quotient will be reproduced in exactly the
same order as before.
^2 In the above Example, all the possible remainders
have occurred, and the period, consequently, consi.'-ts
of as many figures as it possibly could, viz. one less
— than the den'' : this, however, is not usually the case.
6
JEx. 2. Reduce 1^ = 3^^ to a decimal.
3/
49
LC
7_
3(
28
22
22) 3.0 (.13(j
80
66_
140
132
Ans. 3. 136.
Sometimes a decimal of very long period may be carried
out easily to many places, as in the following example :
DECIMAL FRACTIONS.
65
Ex. 3. lied ace ^ to a decimal.
19) LOO (.05263 Hence ^ = .05263^1^, .•.j1j = .15789^ ;
95 and hence i = . 05263 15789^ ;
-^ .•.^ = .4736842101fi-.4736842105-\- ;
38_ and hence i=.03263157894736842105^1y,
120 and, by continuing this process, we obviously double at
114 every step the number of figures obtained.
60 This decimal, it will be seen, circulates after the
57 eighteenth figure ; so that
■3 i = .05263157894736842i.
Ex. 42. Reduce to decimals
1 13 . 103 . 120 . _J.7_ 9 41 . Ill .
■»• ~~u~ i Mitt > nr, > 1M7S' ^'
o 89 . 121 . T7 6401 . 4111 A JI15_ . 297 . 378. 1139
"• 999» » 21 ♦ *' 49300 » :i3300* ^* SfOO' 2900' 925 »
5.
14 » 22 » 16U5 » -"333*
LI 39
555£
62. To reduce a pure circulator to a fraction.
Since ^ = .111111 &c., it follows that § = .2222 &c., §=.5555 &c. ;
SO that any pure circulator, having one figure in the period
may be expressed as a fraction Avith that figure in the num',
and 9 in the den^.
Again,
^ = 14.11 =.010101 &c. ; hence j^=. 050505 &c.; § = .232323 &c.',
SO that any pure circulator, having two figures in the period,
may be expressed as a fraction Avith those figures in the
numf, and 99 in the den^
In like manner, since
^ = l-f 111 =.001001 &c,^V^ = l4-llll=.0001 &c.,
and so on, it will follow that any pure circulator may be ex-
pressed as a fraction with the period itself in the num«", and
in the den** as many 9's as there are circulating figures.
Thus .378 = ee = |f, .0378=^ = ^, .000378 = 5,^^ = ^.
63. To reduce a mixed circidator to a fraction.
If we had a pure circulator with any figures before the
point, we might either keep these to form a mixed number,
as 3.4=3*, 5.4S=5f| ; or we might bring the whole at
once to an improper fraction, with the same den^ as before,
36 DECIMAL FRACTIONS.
by writing for the niim'' all the figures to the end of the first
period, subtracting, however, the figures before the point ;
thus 3.4 ^5^^ = -^= 3^; 5.43=^^ = 4^ = 511; &c.
The reason of this method may be thus seen :
4 3 X 9 + 4_3 (10- l) + 4 30 + 4-3 34-3
^9" y ~ 9 ~ 9 9 '
43 5(100-l) + 43_.543-5 „
•'^99 ~ 99 99 '
Now, if the point be not immediately before the period,
as in these examples, but moved towards the left, this is
equivalent to dividing the decimal by 10, 100, &c., and we
must therefore annex to the den**, as found by the preceding
Eulo, as many cyphers as there are figures between the
point and the first period :
. 34-3 31 .J., 543-5 533 269
thus .034 = -^^=—; •5^'^=-i97r = iJ^O = 4-^-
If there should be any figures of a mixed circulator still
left before the point, it will be best to leave these as they
are, to form a mixed number :
thus 2.46 = 2^ = 2^ = 2/,, the same as ?ii=^ = |? = 2M
The above results may be thus stated, as a Rule for
reducing any circulator to a fraction :
Consider only the figures after the point ; then
For the num^, wiite the decimal to the end of the first
period, subtracting from it (if any) the figures which do not
circidate ;
For the den^, ^orite as many 'd's as there are figures
circulating^ followed by as many 0*5 as there are figures not
circidating. See Note VI.
Ex. 43. Reduce to fractions
1. .3; .05; .64; .?29. 2. .OM; .0432; .00675; 2.0432.
3. 3.4i8; .0443; 1.145; .00449. 4. 4.053i; 7.6631; 2.345; .09318.
5. 2.OO0D; .54950; 1.0428571. 6. 2.6428571; 5.19318; 11.28?,
64. It may be noticed that, according to the above rule,
the circulator 9 = f = 1. It is true, we cannot reverse this
DECIMAL FRACTIONS. 67
operation, and reduce 1 to the decimal .999 &c. ; yet it will
be evident, by repeating the period, that this decimal really
differs from 1 by a quantity so small as to be absolutely
insensible : thus
1 Q_i n_i i_QQ_i 09_i_-i_ qqq _ i . , noo _ i o,^
1 — ,v — I Y6 — 10' '■ ''''' — '■ Too — 100' ^ •iJi'i' i 1000 ~~ 1000' ""'•'
where we see that, by repeating the 9's, the difference be-
tween 1 and the corresponding decimal becomes less and less,
and thus may be made as minute as we please, and will at
length become absolutely insensible.
It is in this sense that 1 is said to be the value of the
circulator .^, and, indeed, that a?i7/ vulgar fraction is assigned
as the value o? any circulator ; so that, in fact, the equivalent
vulgar fraction for any circulating decimal is that to which
the value of the decimal will become more and more nearly
equal as we repeat its period, and from which it may; by
such continued repetition, be made to differ by a fraction as
minute as we please, and altogether insensible.
Whenever, therefore, in a decimal we find the figure 9
circulating, we may at once«get rid of the period, by adding
1 to the figure preceding it : thus .4999 &c. = .5, the same
result as we should obtain by the Rule, since
90 90 10
Q6. Arithmetical operations in which circulating decimals
are concerned, may often be performed, with sufficient accu-
racy for all practical purposes, by repeating the period as
often as shall seem upon consideration necessary to ensure the
result being correct to some given number of decimal places.
Ex. 1. Add together 13.S, 2.026, 111.0004, 3.14 15^, 2.024 correctly
to 6 decimal places.
13.55555555
2.02525252 Here, by carrying out the decimals to 8 places, we
111.00044444 ensure the accuracy of the first 6 places; for, although
3.14159159 the last two are incorrect, and would be altered, if we
2.02402402 carried on our periods farther, yet a little consideration
131.74686812 will show us that the sixth and all the preceding figures
will not be altered, however often we may repeat the periods.
68 DECIMAL FRACTIONS.
In such a case it is generally sufficient to carry out the periods to three
decimal places more than the number required to be accurate.
Ex. 2. From 1.02341 take .62S, correctly to 6 decimal places.
1.02341.3413
.G2S88S888
.394524525 Ans. .394524.
It is sometimes convenient to reduce the circulators to
vulgar fractions, especially for the purpose of multiplying or
dividing one circulator by another, in which case the fraction,
resulting from the multiplication or division, may be after-
wards reproduced in the decimal form.
Ex.3. .2ax.36 = ixM = _i_^ = .084; .I0-.OOL>?=M-I7io = ^'
61.1.
Ex. 44:. Find the value (correct to 7 places of decimals) of
1. .13S + .14285? + 2.4iS + 2.06 + 42.65 + .008497185.
2. 37.23 + .26 + 7.tS + .297 + 3.973 + 8 -r 4.75 h- 74.0367 + 32.41.
3. .3-.09; and .04-.00769238. 4. 7-6.14285? ; and .042 -.036-
5. 37,23 X .26 ; and 7.72 x .^9?.
6. 3.973 x8; and 74.0367 X 4.75.
7. .3-r09; and .04-r.?692.00. 8.* 7-f-.l42S57 j and .042-T-.036.
QQ^ To find the value of any decimal of a give?! quantity
Rule. As in common Reduction, multiply the given
decimal by the number of units of the next lower den° wliich
make one of the given den" : the integral part (if any) of
the result will be so many units of that lower den", and the
fractional part may now be reduced in the same manner to a
lower den" ; and so on.
Ex. 1. Find the value of £.36875.
.36875 or, omitting useless cyphers, .36875
20 * 20
7.37500 7.37500
12 12
4.50000 4.500
4 4
2.00000 2.0 Ans. 7s. 4^.
DECIMAL FRACTIONS. 69
If the given quantity be expressed in more than one den",
it should be reduced to one^ before applying the Rule.
Ex. 2. Find the value of .07 of £2 IO5. ; and of .7365 of 65. 8rf.
Here £2 105. = 505., and 6s. 8(/. = 80d
.07 .7365
50 80
3.50 58.9200
12 Ans. 45. I0.92rf.
Ans, 35. 6i. 6.0
Ex. 3. Find the vahic of .177083^?.
.17708333 Or thus; .1770Si
20 20 ■
3.54166660 3.54166^
12 12"
6.4999992 = 6.5 as in (64). 6.50000
Ans. 35. 6|J. 2.0 2.0
But it is often best to convert a circulator entirely to
a vulgar fraction in such a case, and so find its value.
Ex. 4. Find flic value of 3.27 of a ton.
Here 3.2? =t 3JL j and 3 j\ of a ton = 3 tons 5 cwt. 2 qvs. 6 lb. 3^ oz.
Ex. 45. Find the value of
1. .45 of £1 ; .68125 of £l ; and 2.325 of £\.
2. 32.5 of 5.'. ; 1.85 of 35. id. ; and 2.375 of 135. 4c?.
3. .13125 of £5 ; and .001953125 of £40.
4. 3,45 of 5 guineas ; and .325 of \\ ton.
5. 23.42 of a day; and 1.46875 of an acre.
6. 2.74 of 125. 6cZ.; and 22.25 of £2 25. &±
7. 3.225 of 21 fruineas ; and 22.75 of £5 IO5. 6c/.
8. 3.03 of lOs. 5d.; and .0474609375 of £lO 135. 4.d.
9. .176 of 1 fur. 3^^ p. 2 yds. 5 in. ; and .22 of 3 qvs. 1 5 lbs;
10. .2775 of 1 sq. yd. 0 ft. : 2 in. ; and 32.156 of 3 m. 330 ydi..
11. 2.441 of £32 05. Aid.', and 33.25 of £3 125. 4^^/.
12. 44.045 of U^a.; and .55, + .7 of a crown + .125£.
13. .634375£ + .025 of 255. + .325 of 305.
14. 8.71875 of Sd. + 1.146875 of 65. 8t/. -.0625 of a guinea.
15. .375 of a guinea + .18 75 of a crown + .3 of 75. 6f/.-.87o of2dc
16- 3.83 of 4s.; and 6.15 of 2*. 9|of.
E
70 DECniAL FRACTIONS.
17. 23.4o of 3 m. 5 fur.; and 13.2^5 of oa. 2r,
18. 2.20t of £3 95. 4lc/.; and 2.14o of 55. 8|a.
19. .397910 of £l ; and .40972 of a guinea.
20. .571428 of a qr.j and .285714 of a cwt.
67. To reduce a given quantity to the decimal of another
give)i quantity.
Rule. Begin with the term of lowest den" in the first
given quantity, and reduce it to a decimal of the next higher
den" ; prefix to this decimal the term (if any) of this higher
den", which is found in the first given quantity, and reduce
the result to a decimal of the next higiier den" ; and so on,
until we have thus brought it, if possible, to the decimal of
the second given quantity.
Ex. 1. Reduce £3 17*. 6|d to the decimal of £5.
4) 3.00 Here we first reduce 3/. to a decimal of a penny, by
12) 6.7500 dividing by 4; the result is .75, i. e. 3/.".75<f., and, pre-
20) 17.562500 fixing the 6c/., we have now 6.75c?., which we reduce to
5) 3.878125 the decimal of a shilling; and so on.
.775625 Ans.
Sometimes, as in common Reduction, we cannot thus pass
directly, through different successive den"^, from the first to
the second given quantity ; and then it will be necessary to
express the first as a fraction of the second, and then to
reduce this fraction to a decimal.
Ex. 2. Reduce 25. 9|d to the decimal of 7*. 9|c/.
25. 9|c?. 135 farthings 9^ 25) 9.00 (.36 Ans,
^^^^■® T^fcA ~ 375 farthings " 25 ]J_
1 50
1 50
Ex. 46. Reduce
1. 95. 6cf. to the dec. of £l ; and 25. 2\d. to the dec. of £5.
2. 55. to the dec. of 13s. 4c/.; and 175. 3c/. to the dec. of 105.
3: £\ 25. 6(/. to the dec. of £l; and 25. 1\d. to the dec. of 105.
4. 35. 3|c/. to the dec. of £\ 65. 6c/. ; and £3 45. 2d. to the dec. of 25. 4c/.
DECIMAL FRACTIONS. 71
5. 6s. 6|c?. to the dec. of a guinea ; and 7s. lO^d. to the dec. of £2.
6. 9 oz. 2 dr. to the dec. of a lb. ; and 3 fur. 33 yds. to the dec. of
a mile.
7. 2 m. 1100 yds. to the dec. of a league ; and 12 h. 55' 21" to the
dec. of a day.
8. 3 qrs. 3 lbs. 1 oz. 7 drs. to the dec. of a cwt. ; and 18| days to
the dec. of a year.
9. 15*. 6|d to the dec. of £4 ; and 1 cwt. 3 qrs. 7 lbs. to the dec. of
2h tons.
10. 3| gs. to the dec. of £100 ; and 4| lbs. to the dec. of 3 qrs. 12 lbs.
11. 135. 4d. to the dec. of a crown, and 2 tons 4| cwt. to the dec. of
1 ton 11| cwt.
12. 3i in. to the dec. of | mile ; and 22 guineas to the dec. of £25.
13. 2r. 4p. to the dec of In. 5p. ; and £2 Us. eld. to the dec. of £3.
14. 8 sq. ft. 20 in. to the dec. of 12 sq. in. ; and 7s 6|(/. to the dec.
of£l.
15. 2 w. 6id. to the dec. of 4 d. 3 hrs. ; and £6 12s. G|d to the dec.
of li guinea.
16. 3 hrs. 3' 2i" to the dec. of a day ; and £24 12*. 6^^ to the dec.
of £4.
MISCELLANEOUS EXAMPLES IN DECIMAL FRACTIONS.
1. What vulgar fraction is equivalent to the sum of 14.4 and 1.44
divided by the diflfereHce ?
2. What is the value of .0333 &c. of half-a-crown multiplied by .5 ?
3. The circumference of a circle = 3. 14 16 times the diameter; find
the radius of the Earth, whose circumference is 24857 miles.
4. If the length of the year be taken at 365^ days instead of
365. 24§ days, its true yalue, what will the error amount to in four
ceuturies ?
5. Reduce 5I5 and ^^ to decimals; 3.75 and 3.76 to vulgar frac-
tions ; and multiply .235 by .0021 and 1.2.
6. Reduce 7s. &d. to the decimal of £1 ; find the value of £2.6625 :
and, if 1 oz. cost .03125£, what wuU .0625 lbs. cost ?
7. Find the value of . 6£ + .3125s. + .2 of a guinea.
8. Reduce ^ and 4^ to decimals ; .01^3 to a vulgar fraction ; and
divide 18.073 by .0341 and 5300.
9. Find the value of .453125£ + 1.1484375s. + . 71875a.
e2
72 DECIMAL FRACTIONS.
10. Reduce .875^ to the decimal of a guinea ; and ] .25 of 3.675£ to
the decimal of 10.55.
11. Find the value of .300694 of a day ; and of .9178977^ of 2a.
1 2. Find the value of 3f + 4^ + l|i + 3^ both by vulgar fractions and
by decimals ; and show that the two results coincide.
13. rind the value of 1.875 guinea + 1.875 crown + 1.875 of 3.C25£.
14. Find the difference between 5| half-guineas and 3.125£; and
reduce the result to the decimal of half-a-crown.
1 5. Multiply Is, 7ld. by 5782.5 ; and divide £l68 55. 4^d. by 1.32.
16. The price of ^ an oz. of coffee is .45835. ; what is the value of
.0015C25 of a ton? ^
17. Find the difference between 1.6 of 3.4 of 1.125£ and | of 3.6
of 9.1125£.
18. Reduce ^^^ and ^^ to decimals ; .0675 and .0675 to vulgar frac-
tions ; and find the value of .73125 of £5.
19. If a lb. of sugar cost .0703125 of 8s., what is the value of
.0625 cwt. ?
20. Add together f, f, ^ and ^, both as vulgar fractions and as
decimals ; and show that the two results coincide.
21. Find the value of 3.55. + 2.9 of 23.3755.-g of 1 6.65.
22. Find the difference between 17.428571 sq. ft. and lOO.S sq. in.j
and between 1.76 cub. yds. and 26.66 cub. ft.
23. Multiply .0235 by 8.03; divide .0625 by 2.5; and find the
value of .843541 6 of £5.
24. Multiply 65. O^d. by 85.3125 ; and divide £10 II5. 3 J. by 29.25.
25. Find the value of 4.4 of a guinea - 3.75 of half-a-crown + .4l6£
— .357142S of a guinea.
26. How many yards of matting, 2,4 feet broad, will cover a floor
that is 27.3 feet long and 20.16 feet broad ?
27. Find the value of .375 of 5.375£, and of .06328125 of £100;
and reduce £2 7s. 9^d. to the decimal of IO5.
28. Find the values of 3.5 + 2.83 + .6 + 1.175 ; ll.73-10.9l6;
3.375 X 1.6 X 4.8 ; ^^ ; and find the product of the results.
29. Find the value of 1,2 of 3.5 of 4.375J. + 1,83 of .954 of . 428571
of 4.5d.
30. What is the quarter's rent of 22.7916 acres of land, at 3.72£
per annum per acre ?
31. Reduce ^ and /^ to decimals ; .65 and .0651 to vulgar fractions ;
and £2 3s. 3|c?. to the decimal of £4.
DECIMAL FRACTIONS. 73
32. Find the'value of .^85714 of £30 + 6.85714S£ + .6 of .? 14285 of
M+1.3 of .42857 is.
33. Ecduce 2| and ^ to decimals ; 2.05 and .20o to vulgar frac-
tions ; and £19 17s. 2icZ. to the decimal of £5.
34. Multiply 1 cwt. 2 qrs. 3 lbs. by 5.125 ; and divide £3834 05. 5^1
by 441.75.
35. If an ounce of gold be worth £4.0099, what is the value of a bar
of gold, weighing 1.683 lbs ?
36. Ivcducc .6 of £1 + .6 of 5s. 3(7. + 3.75 of a crown to the decimal
of 165.
37. Find what decimal multiplied by 175 \\ill give the sum of |, if,
|g, and 31.
33. Multiply .285 by 4.02 ; divide 2.961 by .007 ; and find the value
of 2.778125 of G5. 8d.
39. Eeduco (^} of ^4-") - (^L^ of ~i- "j to a simple quantity.
V3.16 .0620/ \7 0.625/ ^ "^
40. Multiply £2 I65. \0.75d. by 144.33 ; and divide £9753 145. 8lJ.
by 234.5.
41. Find the value of 3.275 of £10; multiply 3.275 by 12.8; and
divide .0625 by .00005.
42. Reduce ^ and ^ to decimals; 2.0325 and .340o to vulgar
fractions ; and 2 lbs. 3 oz. to the decimal of a ton.
43. llcduce 1.755. to the decimal of ill ; and 2.(5 of £.877083 to the
decimal of half-a-sovercign.
44. Find the value of 3^| of £3 12i. 6^^/.; and reduce the result to
the decimal of £35 O5. S^d.
^^ ^ . 2.8 of 2.27 4.4-2.85 „ 6.8 of 3 . ,
45. Reduce -jjr;f- + 1.6 + 2.629 "^ IIT *° ^ ''""P^' ^"^''^'^^•
46. Find the value of f of 2.625 guineas ; and the difTerencc between
26.5P. and 705 sq. yds.
47. Find the value of 6.8^ of £3.867708^ + 5.8 of £2.4114583-4.375
of £1.3.
48. Reduce to a decimal, accurate to 5 places,
1^(--;t-^, + v^.5-.-^ + &c.V.tIk. SeelsoTEYIL
\o 3 X 0' 0x0 7x5^ / 239
40. Find the sum of £1.15 + 2.0625 guineas + .0078125 of 325. ; and
reduce the result to the decimal of half-a-soA'croign.
60. Reduce to a decimal, accurate to 7 pla
aces.
1 1x2 1x2x3 1x2x3x4
74
CHAPTER V,
PRACTICE.
68. This IS an expeditious method of finding the value c^
any quantity of merchandise, &c. ivhen the value of a unit
of any denomination is given ; as of 456 cwt. at £3 13s. Qd.
per cwt., or of 3 cwt. 3 qrs. 13 oz. at £2 Qs. l^d. per lb. &c.
69. Case i. Where the given quantity is expressed in the
same denomination as the unit whose value is given.
Under this head will occur such examples as the follow-
ing : 36 cict at £3 \0s. per cwt., 25 lbs. at £2 I6s. Sd. per lb.,
37 oz. at £5 17 s, 6d. per oz., &c. ; where the unit, whose
value is given, is of the same den^- as the quantity whose
value is required. It is obviously immaterial ivhat the unit
itself is ; that is to say, the values would be the same either
of 36 cwt. at £Z \0s. per ciot, or of 36 lbs. at £3 lOs. per lb.,
or of 36 oz. at £3 \0s. per oz., or (without specifying any
unit) of 36 articles at £3 IO5. each, or, as it is briefly ex-
pressed, of 36 at £3 \0s.
Ex. 1. Find the value of 36 at £3 10s.
Here wc have, in fact, to multiply £3 10^. by 36; let us first then
multiply £3 by 36, or, which amounts to the same, multiply £36 by 3,
and we shall have £108 as the amount of 36 at £3.
£36 Now, instead of multiplying the lOs. by 36, we observe
5 that, since 10s. is £h we may take 36 times lOs. by
1 i 108 taking 36 half pounds, which is the same as taking half
10*- ill ]^ of 36 pounds = £18, which vrc add to tbc £108, and
An^. £126 thus have the whole product of £3 IO5. x 36.
PRACTICE.
75
Ex. 2, Find the value o/253 at £2 16s. 8d.
10s.
6s. 8d.
£253
2
506
126 10
84 6
A71S.
£716 16 8
£2 165.
£
S.
d.
C.48
1.
129 at
6
10
0
3.
157 at
9
5
0
5.
271 at
8
3
4
7.
289 at 11
1
8
9.
447 at
1
16
8
11.
361 at
9
11
8
Hero we find, as before, the value of £2 105.
X 253 : and then, since 6s. 8d. is £^, dividing
253 Ly 3, we have £84 65, 8d., the value of
65. Sd. X 253, which we add to the other two
lines, and thus have the whole product of
£
5.
d.
2.
343 at 4
6
8
4.
362 at 7
4
0
6.
187 at 1
2
6
8.
495 at 12
11
0
10.
555 at 4
13
4
12.
677 at 2
12
6
Ex.
105.
55.
25. 6d.
£371
5
Find the value of 371 at £5 175. 6d.
Here we find the value of £5 105. x 371 as
before : then instead of taking, as we might,
65. 8d. as £i &c., we may take 55. as \ of 105.,
and so find the value of 5s. x 371, by taking
half the value of IO5. x 371, i.e. half of £185
105., or £92 155. : in like manner, we may then
fcike 25. 6d. as \ of 55., and find the value of
25. 6d. X 371 by taking half of £92 155.
1855
185 10
92 15
46 7
6
A71S. £2179 12 6
Ex. 4. Find the value 0/ 713 at £4 85. lUc^.
£713
4
55.
35. 4'i.
7¥-
2852
178 5
118 16
8
22 5
n
Here we find, as iu Ex. 2, the value of
£4 85. 4c^. X 713 ; we then take 7ld. as i of 55.,
and so divide by 8 the line £178 os., which is
the value of 55. x 713.
Ans. £3171 7 3i
£ 5.
d.
£ 5.
d.
Ex. ft9.
1.
127 at 3 15
0
2.
235 at 5 7
6
3.
339 at 4 12
0
4.
341 at 6 17
6
6.
253 at 7 17
0
6.
457 at 1 18
6
7.
365 at 11 14
6
8.
573 at 7 15
6
9.
285 at 1 6
6
10.
389 at 8 13
6
11.
492 at 6 18
9
12.
297 at 1 16
9
%
PftACTICfl.
2d.
Ex. 5. Find the vahie of 89 at 35. life?.
Ilere there are no £" in the given raluo ; hut if
-n-e multiply 89 by 3 the result will be in sldllings;
then {id. being \ of 35. we take \ of 2675. ; and 2d.
being ^ of I5. we take l of 895. ; lastly, |i. being
^ of ^d. wo take ^ of QQs. U.
89s.
3
267
1
€6 9
1
U 10
^
6 6|
3515. 12-^. = £17 Us. l^c/. ^«5.
Ex. 6. i^'m,'? ^^e iT^j/e 0/ 111 at I85. lU.
£111
6.V. 8f7.
1
G5. 8(^.
1
65.
1
Sc/.
T
•.'0
U.
h
37
37
27 15
1 7
0 4
Ans. £103 7 4[
£111 x.9 = £99 185. Oa\
GfZ. 'i^' 2 15 6
0 13 lOJ
i^^l
Ans. £103 7 4^
Ex. SO. 5. d.
1. 227 at 2 li
4. 356 at 4 9f
7. 177 at 8 Hi
10. 193 at 13 6|
Wo might treat this as the last Ex. ; or, to
avoid the final reduction, we may begin at
once by taking 65. 9>d. as £\, &c., drawing
a lino imder the 111, before wo divide by 2,
since it is not to bo added in with the other
lines.
Otherwise : — As the u° of shillings in tho
price is even, we can conveniently change it
into the decimal of £1, viz., £.9, and miilti-
plying by .9, we may mentally double the
decimal of the product for shillings ; &c.
2. 149 at 3
5. 365 at o
8. 784 at 9
11. 395 It 14
5. d.
3. 8.54 at 4 2i
6. S73 at 7 8|
9. 480 at 11 8^
12. 499 at 17 11|
70. It is often convenitnt to suppose the given value
increased so as to become an exact number of pounds, or
shillings, &c. for which we may calculate by common
Mult" ; then, if we find by Practice the value of the part
added to the true value, and subtract it from the other
result, we shall get the required amount.
£253
3
85. U.
759
42
3 4
Thus, in Ex. 2, supposing tho given value to
be £3, we should multiply £3 by 253 ; and then
taking 35. 4id. (the part added to the given
value) as £\, and subtracting tho correspond-
ing amount, we have the sjime result as before.
Similarly, in Ex, 3, Wo may add 25. Qd. to tho given value, making it
£6 ; then multiply £6 by 371, and from the result subtract 25. Qd. x 371,
or i of £371.
Ans. £716 16 8
Ex. 51.
PRACTICE.
£ 8.
d.
£ s.
J.
1.
135 at 2 19
n
2.
217 at 4 17
n
3.
273 at 3 18
4|
4.
322 at 7 14
5^
5.
289 at 8
8|
6.
373 at 9
'4
7.
431 at 5 17
111
8.
397 at 6 15
10
9.
511 at 7
m
10.
623 at 1 1
n
XL
271 at 6 15
10|
12.
333 at 5 18
HI
Ti
71. Case it. When the given quantity is not expressed in
the same denomination as the unit ivhose value is given.
Here we shall have to find the value of 3 cwt. at £2 13^. Qd.
per lb., or of 2 cwt. 3 qrs. 16 lbs. at £Z 5s. 7^d. per cwt.,
or per qr., or per lb,, &c. In all instances, (like the first of
these,) where the given quantity can be immediatelj reduced
to the same den" as the given unit, we may do this, and shall
then have only an example under Case i. : thus 3 cwt. =
336 lbs., and the value of 336 lbs. at £2 ISs. 6d. per lb.
may be found as before. So also, if we can reduce an?/ part
of the given quantity to the same den" as the given unit,
we may find its value by mult" ; and, for the rem^, we may
take parts of the given u?iit itself, and proceed as in the
following examples.
Ex. 1. Find the value of 7 cwt. 3 qrs. 11 lbs. at £2 13s. Id. per qr.
The given unit being a qr., we reduce 7 cwt. 3 qrs. to 31 qrs., and find
the value of them by multiplying by 31; then to find the value of
11 lbs., we consider that 7 lbs. are l qr,, and 4 lbs. are } qr. ; so that,
dividing £2 13s. Id. by 4 and 7, and adding up, we have the value of
7 cwt. 3 qrs. 11 lbs.
£ s. d.
2 13 1 X 31
10
26 10 10
3
71bs. = i
4lbs. = i
'9 12 6 value of 30 qrs.
2 13 1 „ 1 qr.
13 31 „ 7 lbs.
•7_Jl „ 4 lbs.
Ans. t>iZ 6 51 „ 31 qrs. 11 lbs.
£8
7S PRACTICE.
Ex. 2. Find the rent of 8a. 3r. IOp. at £l lis. 8d. per acre.
Here we find the rent of 8a., as in the last example 5 and then calcu-
late that of 3r. IOp., and add as before.
£ s. d.
1 17 8x8
8
15 14 rent of 8a.
2R.=1
18 10 „ 2r.
1R.=1
9 5 „ IR.
10P. = i
2 4i „ lOr.
Ans. £\^ 11 \\\ „ 8a. 3r. IOp.
Ex. 3. Find the rent for 3 mo. 3 w. 5 d. at £3 13s. 6d. per month.
Here 1 mo. =4 wks. ; and we can take 3wks. = i of 3 mo. &c., as in
one of the subjoined forms, or 1 wk. 5 da. =^ of 3 mo. &c., as in the
other.
£ s.
3 13
d.
6
3
£ s.
Or thus: 3 13
11 0
Iwk. 5da.=i 1 11
2wk.=|m. 1 16
£U 8
d.
6
3
11 0
3wk. = l 2 15
4 da.=im. 0 10
1 da.=i 0 2
£14 8
6
n
6
7|
9 Ans,
6
6
9
9
Ex. 52.
1. 6 cwt. 1 qr. 11 lbs. at £2 17s. 9d. per cwt.
2. 3 cwt. 3 qrs. 5 lbs. at £4 14s. per cwt.
3. 9 cwt. 21 lbs. at £5 lis. l^d. per cwt.
4. 2 cwt. 4 lbs. 12 oz. at £3 Is. per cwt.
5. 3 qrs. 5 lbs. 9 oz. at £2 14s. 6c/. per lb.
.6. 2 qrs. 9 lbs. 13 oz. at 15s. 9c?. per lb.
7. 2 qrs. 7 oz. 9 drs. at 18s. 6d, per lb.
8. 2 cwt. 2 lbs. 2 oz. 12 drs. at £1 3s. Od. per lb.
9. 3 cwt. 3 qrs. 27 lbs. 15 oz. 12 drs. at £7 per cwt.
10. 6 oz. 18 dwts. 20 grs. at 7s. 9c?. per oz.
11. 3 lbs. 5 oz. 14 dwts. 12 grs. at 17s. 6c?. per oz.
12. 22 yds. 2 ft. 2 in. at 18s. 8f?. per yard.
13. 13 yds. 1 ft. 7 in. at 9s. 4c?. per fooi.
14. 37a. 1r. 28p. at £2 2s. per acre.
15. 17a. 3r. 1 9 p. at £5 18s. ed. per acre.
16. 21a. 2r. 12 r. at £3 15s. Sd. per acre.
PRACTICE. 79
17. 5 mo. 3 w. 4 d. at 17s. 6c?. per week.
18. 7 mo. 2 w. 5 d. at £2 85. 4d. per month.
19. 9 mo. 1 w. 6 d. at £1 25. 9c/. per week.
20. 6 mo. 3 AV. 2 d. at £3 Os. 6c/. per montli.
72. The method of Practice may be applied, as we have
said, to any case where the value of any quantity is sought,
that of a unit of any den" being given. It is not^ however,
necessary (as in the foregoing Examples) that this given value
should be the price of the unit, &c.; but, whenever any given
amount is charged for any reason upon the unit, we may
find thus the corresponding amount for the given quantity.
Ex. A bankrupt is able to pay 12s. ^\d. in the £, and his debts are
£3600 : what was his estate worth?
3600
This means that, for every £ he owes, he
1800 Qg^jj pj^y Q,^]y 125. 6^/, } here then we have to
7 10
105.
25. 6c/.
4-50
find the value of 125. 6|c/. x 3600, wdiich must
Ans. £2257 10
have been the value of his whole estate*
Miscellaneous Examples. 53.
1. What must be paid to 721 labourers for a week's service, at
17.9. 4lc/. each?
2. What would be the amount of 137 tons 12 cwt. of goods, at the
rate of £2 45. IQic?. per cwt. ?
3. Calculate the amoimt of a .salary of 21118 rupees, valued at
25. 1^. each.
4. A bankrupt's debts are £7357, and he is aide to pay 125. d^. in
the £ ; what are his effects w'ortli ?
6. To how much will a charge of £28 85. 2c^. per day amount in 365
days?
6. Lodgings at £5 IO5. 6d. per month being occupied for 8 mo.
21 days, how much mu.st be paid for them ?
7. What must be given for a gold snuff-box, weighing u oz. 9 dwts,
20 grs., at the rate of £4 35. Qd. per oz. ?
8. What is the dividend on £1710 145. 6c/., at 13s. 4ir;. in the £ ?
9. How many acres \nll supply 53 horses with hay and oats, if eacV
horse consimie annually the produce of 5a. 3k. 26p. ?
10. What is the expense of digging a ditch, of which t?he cubic con-
tent is 5755 cubic yards, at the rate of I5. 7^1. per yard?
80 PRACTICE.
11. A bankrupt owes £2468, and can pay los. 6d. in the £ ; wliat
are his effects %vorth?
12. rind tho weight of 1000 pieces of gold coin, each weighing
6 dv,t 7 gr.
13. An officer's pay is 12«. 3c?. per day ; what is that in a year?
14. A labourer's pay being 2s. 9|^. a day, what is the whole pay of
23 men for 25 days ?
15. If lodgings kt at 135. 6d. per week, how much do they let for
during 273 days?
16. A merchant bought 182 quarters of wheat at £2 Is. od. per quar-
ter, and retaihd ihe sania at £2 18s. id. per quarter; what was his
gain, and at wiiat per quarter should he have sold it to have gained
exactly 104 guineas?
17. "What sum would be required to pay the wages of 377 labourers
for a week, at 2s. bd. a day each ?
18. If a persons estate be worth £1384 16s. per ann., and the laud-
tax be assessed at 25. ^^. in the £, what is his net annual income ?
19. An iron bridge consists of 3 arches^ the centre one weighing
3046 tons, and the two others 2600 tons each ; what is the cost of tho
iron at £6 135. Qd. per ton?
20. What will a room cost in painting, at I5. 1\d. per square yard,
those height is 10 ft. 3 in., width 16 ft. 0 in., and length 18 ft. 10 in.?
21. An estate of 134a. 3r. 16p. is rented at £2 125. Qd. per acre,
and afterwards the best pasture, consisting of 51a. 2r. 12p., is let at
£3 105. per acre ; what will the first tenant still hare to make up of
his rent ?
22. A bankrupt's liabilities are estimated at £3758 175. Qd.\ what
are his assets, if ho can pay 135. 'J\d. in the £?
23. "What is the joint raluo of 5 qu. 3^ bu. of wheat at 75. 4|</. per
buslicl, and 5 qu. Z\ bu. of (»ats at 45. 2\d. per bushel ?
24. There were sold three pieces of land, containing oO^a., 76jA.,
89a. 12p. respectirely : the price of tho first piece was £12 75. lOcZ., of
the second £13 155. 9f?., and of the third £16 85. Qd. per acre; what was
given for the whole ?
25. "What will be the cost of replacing a cistern, to weigh 8 cwt.
2 qrs. 14 lbs., at the rate of £2 05. Qd. per cwt., if the plumber allows
£1 II5. 6c?. per cwt. for the lead of the old one, which weights 6 cwt.
Iqr. 10 lbs.?
81
CHAPTER VI.
PKOPORTION.
73. The Ratio of one quantity to another is the number
^vhich expresses what fraction the former is of the latter,
and is therefore obtained, as in (48), by dividing the former
by the latter.
Thus the ratio of 103 to 144, or (as it is written) of 108 : 144, Is
-}^ = |, meaning that 108 is | of 144.
The former of the two terms in any ratio is called the
antecedent^ and the latter the consequent ; and it is plain
from the above, that all ratios are equal which may be made
to have the same antecedent and consequent by striking
common factors out of their two terms.
Thus the ratios of 108 : 144, 36 : 48, 21 : 28, 15 : 20, 3 : 4, &c., are
all equal, since each of them is equivalent to the fraction |; and it will
be seen that the first of each of these pairs of quantities is £ of the
second.
74. When two ratios are equal, they are said to form a
Proportion, and the four terms coniposing them are called
proportionals, or are said to be proportional to one another.
Thus, since 15 is | of 20, and 21 is | of 28, and so (as before was
said) the ratio of 15 : 20=thc ratio of 21 : 28, these four quantities
form a proportion, which is usually expressed thus, 15 : 20:: 21 : 28,
and read as 15 is to 20 so is 21 to 28, or 15 is to 20 as 21 is to 28; and
here 15 and 21 are the two antecedents, 20 and 28 the two consequents,
of the ratios which form this proportion.
N.B. It should be well noticed that the proportion
15:20:: 21: 28 expresses that 15 is the same fraction
(proper or improper) of 20 that 21 is of 28.
75. In any proportion, the product of the 1st and 4th
tcrms=the product of the 2nd and 3rd terms, or, as it is
commonly said, the product of the extremes ^the product of
the means.
■^^=^, whence 20 : 8:: 15 : 6
f? = ^, whence 20 : 15::8 : 6
I = in, whence 6 : 8 : : 15 : 20
1^ =i, whence 6 : 15::8 : 20
82 PROPORTION.
Thus in the proportion 15 : 20:: 21 : 28, since the two ratios arc
equal, wc have |§=|^; and, if wc multiply each of these equals hy
20 X 28, wc get 15 x 28 = 20 x 21, or 1st x 4th = 2nd x 3rd.
76. Conversely, if the product of any two quantities =
the product of two others, the four are proportionals, the
factors in one product being the extremes, and those in the
other the means, of the proportion.
Thus, since 6 x 20 = 120 = 8 x 15, if wc divide each of these equals
by 6 X 8, 6 X 15, 20 x 8, 20 x 15, respectively, we get
or -ir^-¥-» whence 15 : 6::20 : 8;
01* I •=15' whence 8 : 6::20 : 15;
01" M = t» whence 15 : 20::6 : 8;
or ^ =^, whence 8 : 20::6 : 15;
in the first set of which proportions it is seen that the terms of one
product, 6 and 20, are the extremes, and those of the other product,
8 and 15, the means; and vice versa, in the other set.
77. Hence also it follows, that, if four quantities in any
given order are proportionals, they will also be proportionals
in any other order, in which the same two terms will go
together, either as extremes or meaiis.
Thus, since 6 : 9::10 : 15, it follows by (75) that 6 x 15 = 9 x 10,
and therefore by (76) we have also 6 : 10::9 : 15, 10 : 15::6 : 9, &c.,
in which 6 and 15 still go together, either as extremes or as means. We
could not have, however, 6 : 15 : : 9 : 10, &c., in which this is not the case.
78. If we have given any three of the four terms of a
proportion, we may by means of them easily find the fourth ;
for since by (7o) the 1st x 4th ^ 2nd x 3rd, we have the
, , 2nd X 3rd ,, .^. 2ncl x 3rd ,, r, i 1st x 4th
1st = r-. , the 4th = , the 2nd = — -— —
4th ' 1st 3rd
and the 3rd ^il^^ii^
2nd •
Ex. Find the numbers which shall form the 1st and 2nd terms,
respectively, of a proportion with the numbers G, 7, 8.
2nd X 3rd 6x7
Here the 1st = —j^^ — "" 8~~ ^ ^^' 5i : 6 : : 7 . 8 ;
, I:tx4th 6x8 ,
tli2 2nd - 'q^— = -f- = 65, and 6 : 6f : : 7 : 8,
PROPORTION. 83
Ex. 54. Find numbers whicli shall form the 1st, 2ncl, 3rd, 4th terms,
respectively, of a proportion with
1. 2, 3, 4. 2. 3, 4, 5. 3. 4, 5, 6. 4,,- 5, 6, 7.
5. 2, 0, 7. 6. 4, 5, 8. 7. 2, 7, 9. 8. 5, 7, 7.
79. Wc have hitherto given instances only of the ratios
of «&5^rac^ quantities, or numbers, to one another; but wo
may similarly obtain the ratios of coticrete quantities.
Thus the ratios of £\m : £144, of 9 cwt. : 12 cwt., of 15 gals.
: 20 gals., of 39 ft. : 52 ft., are (by 48) respectively i^, ^, |f, ff, each of
which reduces to £; and we say, therefore, that the ratio of £108 : £l44
is the same as that of 3 : 4, or f, meaning that £108 is f of £144; and so
with the other ratios.
Of course, however, the quantities forming such a ratio
must be of the same hiiid; for, otherwise, one of them could
not be a fraction of the other.
Thus it would be absurd to speak of the ratio of £108 .* 144 cwt., or
of 9 cwt. : 12 gals., &c.
So also, though they be of the same kind, we must besidct
reduce them, as in (48), to the same denomination, before we
can express the one as a fraction of the other, and so find
their ratio.
Thus the ratio of 7*. 6c/. : 45. 2(/. = the ratio of 90J. : 50(/. = §^=2
= 9:5.
N.B. AVhatevcr be the nature of the quantities themselves,
their ratio is always a mere abstract number, expressing, as
stated in (73), what fraction the one is of the other.
Thus in the last instance the ratio of 90J. : 50J. is, as in (48), the
number f, not §c/.; for it has no reference whatever to the fact, that the
given quantities were pence, but only to the magnitude of the one with
respect to the other, i. e. to the fact that the one is f of the other; and
it would plainly have been just the same, if the ratio had been that
of £90 : £50, or of 90 cwt. : 50 cwt,, &c.
SO. So also, when two such ratios are equal, they form a
proportion ; thus £108 t £144 : : 9 cWt. : 12 cwt. ; only here
we cannot, as in (77), change the order of the terms, except
the change be such as still leaves the two ratios possible.
84 PROPORTION.
Thus, it will be true, as before, that :^144 ; £108 : : 12 cwt. : 9 cwt , or
9 cwt. : 12 cwt.::£l08 : £144, &c. ; but we cannot say that £144 ;
12 cwt. ::£108 : 9 cwt., because the two ratios £144 : 12 cwt. and
£108 : 9 cwt. are absurd. It would, however, be true, that £144 : £I2
:: 108 cwt. : 9 cwt. &c.
81. For the like reason, we cannot exactly say of such a
proportion, that the product of the extremes = that of the
means; thus it would be absurd to speak of multiplying £144
by 9 cwt., &c. : if, however, we consider only the numerical
values of the terms, this would be still true : and, having three
terms of such a proportion given, we may, by means of their
numerical values, find as in (78) the numerical YdXue of the
fourth term, which will be of the same kind and denomination
as the other term of the ratio to which it belongs.
Thus to find a fourth proportional to £108, £100, 9 cwt., we have its
100 X 9
numerical value = "^"^ = 83 > which must be 8^ cwt., since it must be
of the same kind and denomination with 9 cwt., the other terra of the
ratio to which it belongs ; and the proportion will tlierefore be
£108 ; £100:-. 9 cwt.: 8^ cwt.
82. The method above referred to, by which we may find
the fourth proportional to three given quantities, — viz. hij
mitUiplt/ing together the 2nd and 3rd, and dividiiig the pro-
duct hij the 1st, — is commonly known by the name of the
liule of Three,
In practical applications of this Rule, the three given
quantities are generally concrete ; and a very large class of
Examples are those, where, the cost of a given quantity of
some article being given, we are required to find, either
what will be the cost of another given quantity, or else what
quantity may be bought for another given cost. For it is
plain that, in any such case, if the first cost be double,
treble, half, &c. of the second cost, the first quantity will
be double, treble, half, &c. of the second quantity, and,
generally, the first cost will be the same fraction of the
second cost that the first quantity is of the second quantity ;
PROPORTION. 85
i.e. tlie ratio of tlie two costs will be the same as tlie ratio
of the two quantities, or the four will be proportionals, so
that we may apply to them the preceding observations.
Ex. 1. 7/" 39 cwt. of sugar cost £91, what will be the cost of 18 cwt. ?
39 CTvt. : 18 cwt. ::£91 : the Ans., -vrliose numerical value we obtain by
18 multiplying £91 by 18, and dividing
728 by 39, without considering these as
91 concrete quantities, and the result 42
39)1638(£4:2 will be of the same ki?id as the 3rd
156 term, viz. £'.
78
Ans. £42. ^
Ex. 3. If £42 will hmj 18 cv:t. of sugar, what qnantitg may he had
for £91 ?
£42; £91:: 18 cwt.
18
728
Here we have multiplied the 2nd by the
qJ" 3rd (the least of the two) for convenience,
„v , „Q and the result 39 will be of the same kind
42 s as the third term, viz. csvt.
X 7)273
Ans. 39 cwt.
Ex. 55.
1. If 12 yards of cloth cost £15, what would 8 yards cost at the
same rate ?
2. If 46 bu. of wheat cost £16, how many may be bought for £72?
3. AVhat will be the cost of 00 gals, of wine, if 495 gals, cost £396 ?
4. How many acres of land may be rented for £Q>b, if the rent of 168
acres be £364 ?
5." If 63 loads of straw can be bought for £180, how many may be
had for £100?
6. How much must be given for 25 doz. of wire, at the rate of £176
for 80 doz. ?
83. Since the .I7i5i6'ef = ^^^^'^ and the value of this
1st
fraction is not altered by striking common factors out cf
its num"^ and den^, we may sometimes simplify the operation
by striking out (before we multiply and divide according
to the 'Rule) a common factor, either from the 1st and 2nd
terms, or from the 1st and 3rd terms.
86 fROPORTIO:?.
Ex. 3. 7/" 275 reams of imper cost £158 lo5., wJiat would 990 reams
275 rms. : 990 rms. : :£lo8 los. : ^^'^'^s.xQdQ
20 275
127 317o^\
§i&-?i^= 1275.x 90 = 114305. = £571 105.
1 1 Ans.
Here haying first, for douveniouce, reduced the 3rd term to shillings,
•wo have signified tlie value of the 4th terra by a fraction representing
the product of the 2nd and 3rd terms divided by the 1st. Then striking
out 25 from 3175 and 275, vre get 127 and 11 ; then dividing 990 by
the 11, -vve obtain 127 x 90.
Ex. 4. If 14 to7is of bar-iron cost £106 lis. Gd., how much may ha
had for 1^0 guineas 1
£106 ll5. Qd. \ 100guin.::14tons : ^^^-^^^00
20 42 4263
2131 4200 si^p.
2
4263 sisp.
200
14xW5_2x_200_
\^^^ ~ 29 -i^"t--^y
203
29) 400 (13§ tons. Ans.
29_
110
J7
23
7. If 385 yards of cloth cost £253, how many may be had for £138?
8. How much cambric may be bought for £45, if 714 yds. cost £85?
9. If 36a. 3r. of land are rented for £84, what should be the rent of
2lA. 3r. 20p.?
10. If I pay £18 for 7 cwt. 3 qrs. 14 lbs. of sugar, what would bo
the cost of 4 cwt. 1 qr. 14 lbs. ?
11. How much oats, at £80 15?. for 51 quarters, may be bought for
£62 145.?
12. If 172 cwt. 2 qrs. 18 lbs. of pot:itoes cost £94 175. Gd., how much
must be given for 7 cwt. 3 qrs. 11 lbs.?
84. Tlie Principles of Proportion may, however, be applied
to numberless cases, besides such as we have been hitherto
considering ; and w^e must here say a little more of the
general nature of what are called Frojjortional Quantities.
PROPORTION. 87
We have already seen what is meant by saying that/bwr
quantities are proportionals ; but it is common also to speak
of tivo quantities being proportional to each other (or varying
as each other) ; only here the quantities are used generally^
whereas the four quantities, in the former case, yvQVQ par-
tieular values of such general quantities.
Thus, for example, we say commonly that the weight of an
article is proportional to, or varies as, the price ; where the
words weight and price are used generally, without reference
to any particular weights or prices : but by saying this we
mean, that if we took any two particular weights, and the
two corresponding prices, the four would be proportionals :
and thus, having given any two weights, and one of the
corresponding prices, we might find, by the Rule of Three, the
other price ; or, having given any two prices, and one of
the corresponding weights, we might find the other weight :
and this we have been doing in the preceding examples.
85. But now other quantities, considered generally, may
be similarly proportional to each other ; and to these the
same principles may be applied, Thus, tlie rent of a house
will vary as the time it is occupied, a workman's wages will
vary as the time he labours, the distance run by a coach will
vary as the rate at which it moves, &c. ; in all which cases,
if we take any two particular values of the first quantity,
and the two corresponding values of the latter, the four
would be proportionals ; so that, if three only were given, we
could apply the Rule of Three, as before, to find the fourth.
86. Sometimes we can only know from philosophical rea-
sons, that two such general quantities are proportional ; as,
for instance, that the length of the shadow cast by a vertical
rod, at any given hour of the day, varies as the height of the
rod ; that the velocity acquired by a heavy body in falling
varies as the time of motion from rest, &c. ; but, in most cases
that occur in common practice^^itis easy to apply at once the
88 PROPORTION.
test of proportionality, as in former examples, viz. by con-
sidering whether, by taking any two particular values of one
quantity, and the corresponding two of the other, the four
would be proportional, i.e. whether the Ist of the former set
would be the same fraction of the 2ud, as the 1st of the
latter set would be of the 2nd.
This, perhaps, we may do most simply thus : consider if,
by doubling, trebling, &c. any value whatever of the one
quantity, the corresponding value of the other quantity
would also be doubled, trebled, &c. ; — in which case the
above test would be satisfied with these four quantities, and
therefore the two given general quantities would be propor-
tional to each other.
87. Hence, when any question i,: proposedj in which,
having given any two values of one quantity, and one of the
corresponding values of the other, we are required to find the
other of these values, we must first enquire whether the
case be one of Proportion. If so, we may proceed to state
and solve the sum by the Rule of Three. It will be best,
first to set down the 3rd term, which will always be the single
given term, and of the same kind as the answer (being the
antecedent of the ratio in which the answer is the con-
sequent) ; then the other two terms, (which will always be
of the same kind, being two given values of the other
quantity,) will form the other ratio — the antecedent, or 1st
term, being that value which corresponds to the antecedent
of the second ratio, or 3rd term.
Wo may now proceed as before — reducing the 1st and
2nd terms to the same den^-^and, if desirable, the 3rd, to
any den" we please — striking out common factors (if any)
from the 1st and 2nd, or 1st and 3rd— ^multiplying together
<he 2nd and 3rd, and dividing by the l?t — when tlic quo*
lient will give the Answer, in the same den" as that in
wh;o.ii we have expressed the 3rd term.
PROPORTION. 89
Ex. 1. What is Oie coachfarcfor 130 miles, if it is £l 95. id. for 85 miles?
Here 5) 85m. :5)1^0m. ::£l 9 4 Here it is plain that, if
17 26 20 we c/owi/e the distance, what-
29* ever it may be, the corre-
12 spending fare will also be
352 doubled; hence the fare va-
26 lies as the distance, and we
2112 proceed as before, setting as
704_ the 3rd term the single given
17) 9152 (538^. quantity £l 9s. 4c/., and, for
85 the terms of the first ratio,
65 the pair of given quantities
51 of the same kind, 85m. and
142 • 130m., of which we set 85m.
136 ^rsf, since it is that distance
Ans. 538-% J. =£2 4 10 ''-. 6 which corresponds to the 3rd
term.
Ex.2. The rents of a parish amount to £\750,and a poor-rate is
wanted o/£61 195. Id. ; what is that in the £ 9
£1750 : £1 ::£61 19 7 Here it is plain that, if we double the rent,
20 whatever it may be, the corresponding rate
1239 will also Ije doubled: so that the rate varies
12 as the rent. The single terai is here tlio
1750) 14875 (8i. whole rate, £61 195. Id.; the terms of the
14000 fust ratio are the two given rents, £1750 and
875 ^ £1, of which £1750, since it corresponds to
Ans. %\d. 1750~2 the 3rd term, is set first.
This sum in fact amounts merely to one in division; since if £1750
will supply a rate of £61 195. Id., we may obtain that supplied by £l,
by simply dividing this amount by 1750.
Ex. 56.
1. Afield of 18 acres is let for £24 185. 6c?.; what would be the
rent of 42 acres at the same rate?
2. If a servant's wages be £25 a year, what should he receive for 87
days' service ?
3. If the coach fare for 65 miles be £l l5. 8t?., how far ought one to
go for £2 185. 8J.?
4. If a carding-machine throw off 54 lbs. of wool in 2 hrs. 46 min.
SO sec, in what time will it throw off 24 lbs.?
5. How much land may bo rented for £70 105. 6(/., if 5 acres are
rented for £4 135. 4 J. ? -
90 PROPORTION.
6. What is the assessment on 20a., if that on 445a. be £l 4 145. 9^</.?
7. If the tax on a rent of £25 is £2 lOs., what will it be on a rent
of £10 95. 4lr/.?
8. What is the amount of poor-rates to be paid upon £95 IO5. 9^'/.,
when £39 11*. 8d. is levied upon £791 135. 4iU
9. The expenses of the poor in a parish amount to £110 75. C>d., and
the whole rent is £2000 ; how much in the £ must be levied to pay it ?
10. What is the tax on a house rented at £65 IO5. Q(/.; if that on
one rented at 25 guineas be £4 11 5. loirf.?
88. Sometimes we may have two general quantities so de-
pending on each other, that, if we double any value whatever
of the one, the corresponding value of the other, instead of
being doubled, will be halved. Thus, if any given number
of men would do a piece of work in a certain time, it is plain
that double that number would do it in half the time. In
this case the four quantities will still be proportional, but
with the terms of the second ratio in inverted order; since
the 1st value of the former quantity will be the same fraction
of the 2nd, that the 2nd of the latter quantity is of the 1st.
Tlie two general quantities are here said to be inversely
proportional to each other, whereas in the former examples
they were directly proportional : but the Rule of Three may
still be applied, if we take care to state the sum rightly, viz.
by setting last, as before, the single term, and then setting
as the second term, or consequent of the first ratio, (instead
of, as before, the first, or antecedent,) the corresponding
value of the other two given ones.
Ex. 1. A person completed a journeij in 32 days, travelling 8 hrs. 2
day ; how long would he take to do the same, travelling only 6 hrs. a day?
6 hrs. : 8 hrs. : : 32 days Here the term 8 hrs. corresponds to the
? term 32 days, and it is plain that if we
6) 256 double the n° of hrs. in each day, the n" of
Ans. 42| days, days required will be only half of what it
was before ; so that the n* of hrs. in a day
varies inversely as the n** of days required. The single or 3rd term is
32 days, and here we put the corresponding term, 8 hrs., second instead
o( first, as in the former cases.
PROPORTION. 91
Or wc might reason thus. The whole number of hrs. must be the
same in both cases; and tlicrcforc 32x8 = 6x^w5., ^Yhcnce we havo
.1 J S2xS 2.")G ,-,0 ,
the Ans. = -0- = 0" = ^^-5 ^'^J^-
Ex. 2. 1/ S4 sheep can be grazed in a field for 12 days, how long
might 112 sluep have been grazed in the same field ?
Here it is plain that, if we double the
" * ]2" " ^'^"If ^^^^ '^"^^ "1 t^i6 s^^ie field; so that
1 12)1^(9 davs/l... the no of sljeep varies inversely as the n^
lOOS " ^^ ^^y^' ^^^^ ^"^S^^ *^*'"^ ^s 12 days, and
wc set the corresponding term, 84 sheep,
second.
Or thus: 84 sheep for 12 days consum:; as much as 84 x 12 sheep in
one day; and we have 84 x 12 = 1 12 x ^^5. .*. Ans.=-.^^ = 9 days.
Ex. 57.
1. If 100 workmen can do a piece of work in 12 days, how many can
do the same in 8 days ?
2. If a besieged garrison have 4 months' provisions, at the rate of
18 oz. per man per day, how long Avould they be able to hold out, if
cacli man were allowed only 12 oz. per day ?
3. If I borrowed of a friend £300 for 8 months, for how long a
time should I lend him .€200 in return ?
4. How many men would perform in 1C8 days a piece of work,
which 108 men can perform in 266 days ?
5. If a person, travelling 12 hrs. a day, w(^ld finish his journey in
3 weeks, how many weeks would he take to do it, if he travelled only
9 hrs. a day at the same rate ?
6. If 475 shilling cakes can bo made of a quarter of wheat, what will
be the price of a cake, if 70 are made of the same quantity of flour ?
7. How much land, at 275. per acre, should be given in exchange for
480 acres, at 355. per acre ?
8. A besieged fortress has provisions for 3 weeks, at the rate of
14 oz. a day for each man; at Avhat rate per day must the provision
be distributed, so that the place may hold out 5 weeks ? ^^^
89. We must always be assured, as in the preceding
Examples, that the two general quantities concerned in any
case are proportional to one another, either directly or in-
versely, and so that the question is one which falls under the
92 PROPORTION.
Rules of Proportion. But when satisfied of this, we may
relieve ourselves of some of the care required in stating the
sum, by the following general Rule, which includes both
cases, and is that commonly given as the
RULE OF THREE.
Set last the single term, (viz. that ivhich corresponds to the
AnsweVj) and the greater or less of the other two terms
second, according as it is seen that the Ansiver will he greater
or less than the third term.
The reason of this is plain ; for, if the three quantities do form the
first three terras of a proportion, the single term must b^ set 3rcl, since it
belongs to the ratio of which the Ans. is the other term ; and then, as wg-
know that the Ans. will be found by multiplying this term by one, and
dividing by the other, of th) two remaining terms, it is obvious that, if
the Ans. is to be greater than the 3rd term, we should have to multiply
by the greater and divide by the less ot the two, i. c. we should have to
put the greater of them second; if less, the less.
This explanation, however, is only intended to show that the above
Rule will enable us to make the same statement of the sum as we should
have done by the proper considerations, and so to get the correct result.
It docs not at all profess to give the truo reason for so stating, which
depends upon the foregoing observations. See Note VIII.
Ex. 7/" IO7 lbs. of scdt cost l^.s., what will 3| cwt. cost?
Here tne single, or 3rd term, is l{^s.;
and since the A7is. will plainly be
greater than this, we set the greater 15?
the two others in the second place, viz.
3| cwt. or 3|x 112 lbs. when reduced
to the same den" as the 1st term.
Ex. 58.
1. If 69 lbs. of salt cost 9s. l^c?., what ^11 be the cost of 15 lbs.?
2. AVhat is the value of sheep per score, if 311 sell for £080 Is. 4|fZ. ?
3. A bankrupt owes £4726 10s., and his effects are worth £1181
12s. 6^. ; how much will he be able to pay in the £?
4. If 275 bushels of potatoes cost fed 4s. 6d., what quantity will cost
£25 14s. 7d. ? ' .
6. If 39 ewi:. 1 qr. 11 lbs;,. cost £59 6$, 6d., what win 13 jcwt. cost at
the same rate ? --^-
1 Of lbs.
: 32 X
112 lbs.:
:illi^.
Ans.=
11*. X
= Ms.x
7x7.
, Q
I3IXI12
= £2 19s.
10§df.
X
PROPORTION. yo
6. Wliat weight of sugar may h& bought for £3,74 Bs., when the
cost of 6 c^-\ 2,qrs./is £14: lis. Bid. ?,
7. If the tax on :^335 75. 6d. amount to £58 135. 9|<?., what is that
in the £ ?
8. How many gallons of wine, at the rate of £31 165. id. for 46 gals.,
may be bought for £117 H*. Sd. ?
9. If 17 cwt. 3 qrs. 14 lbs. of tallow cost £38 25. 8d., howm^chmay
be bought for £5 12s. 6d. at the same rate?
10. If the sixpenny loaf weighs 3 lbs. when wheat is at 6s. a bushel,
what ought it to weigh when Avheat is at 65. Od. a bushel?
11. Suppose there are 12,000,000 sheep fed in this country; what is
the value of their wool-produce yearly, if 11 sheep produce 25 lbs. of
Wool, which is sold at £8 125. per cwt. ?
12. From 3 tons 5 cwt. take 1 ton 16 cwt. 3 qrs. 12 oz., and find the
Value of the remainder at £1 75. 6d. for 1 qr, 27 lbs.
13. If a nobleman's rental be £8050 per annum, and the land-tax b»
charged at the rate of £11 55. per £100, what will be his nett income ?
14. If il yards of cloth cost £5 145. i^., what would 20 yds. cost?
15. The chain for measuring land is "^6 feet long, and divided into
100 links ; what is the length of a wall which measures 2456 links ? V
16. The rateable value of a parish amounts to £1250, and a poor-
rate of £27 IO5. 6d. is to bo raised; what will a person have to pay
whose rents are £525 ?
17. A wedge of gold, weighing 14 lbs. 3 oz. 8 dwt., is valued at
£514 45. ; what is the value of an oz. ?
18. A bankrupt has assets to the amount of £1020, and debts to tho
amount of £3225 ; what will his creditors receive in the £ ?
19. A bankrupt's effects amounted to £980, which paid his creditors
135. 6d. in the £ ; what did his debts amount to?
20. What is the income corresponding to an income-tax of £l 3 2s. Qd.,
at the rate of 7 pence in the £ ?
21. A borrowed of B £175 55. for 102 days, and afterwards would
return the favour by lending B the sum of £210 6s. ; for how long
should he lend it ?
22. What is the height of a steeple, whose shadow was 148 ft. 4 in.,
at the same time that the shadow of a staff 6 ft. 4 in. long was 5 ft. 3 in.?
23. A coach goes 'fromTO'ndon to Liverpool, at the rate of 9 miles
an hour, in 24 hours ;• in what time would the distance be performed on
the railroad, at the rate of 32 miles an hour ?
24. A besieged town, containing 22400 inhabitants, has provisions
to last 3 weeks ; how many must be sent away that they may be able to
hold out 7 weeks ? .
94 PROPORTION.
25. If a serrant receive £3^ for 20 weeks* service, how many weeks
ought he to remain in his place for 12 guineas?
26. If the carriage of 15- cwt. for 60 miles came to 7^. 9d., how far
ought 2\ cwt. to be carried for the same money ?
27. How much may a person spend in 73 days, if he wishes to lay
by every year 50 guineas out of an income of £450 ?
28. The carriage of a parcel of goods, weighing 1 ton 3 cwt 2 qrs.,
cost £2 lis. ; what will be the charge for 4 other parcels, weighing
each 17 cwt. 3 qrs. 7 lbs. ?
29. If 3| shares in a speculation are worth £27 105., what are 4f
shares worth ?
80. If If yard of cotton print cost 2s. 6d., what is the cost of 24i
yards ?
31. If 1| cwt. of sugar cost 3| guineas, what must be given for
17|lbs.?
32. At 35. 4^. for 4f lbs., what is the price of 14| lbs. ?
33. If 2| yards of cotton print cost l5. lO^d., what is the cost of 13|
yards?
34. If 6| yards be worth 27s. ^d., what quantity is worth 1 8s. 2^. ?
35. What is the value of f of f of a ship, when f of the whole is
worth £525 ?
36. If 6336 stones of 3^ ft. length complete a certain quantity of wall,
how many similar stones of 2| ft. length will raise a like quantity ?
37. If a ball falling from rest acquire a velocity of 115^ ft. in 3f
seconds, at what rate will it be moving at the end of the first second,
and at the end of 4| seconds ?
38. What will 3 cwt. 1 lb. 1| oz. of merchandise cost, if the cost of
13| tons be 500 guineas?
39. If 4| oz. Av. cost 8||s., what will 8|| lbs. cost?
40. If ^ of I of 2i of 40 lbs. of beef cost l^d., how many lbs. may
be bought at the same rate for 65. 7^. ?
90. Suppose it were asked, ' If 9 men can reap 30 acres
of wheat in 10 days of 6 hours each, how many men would
reap 40 acres in tJie same time .<" This would he an instance
of common Direct Proportion, and we should have
30a. : 40a.::9 men : ^x9 = 12 men.
But now suppose that, instead of 'm the same timey the
question had said, 'in 12 days of tJie same length.'' Here il
PROPORTION. 95
is plain that, after finding, as above, the n^ of men, 12, who
would reap 40a. in 10 days, we must still have another Pro-
portion, to find the n<* who will reap the same n^ of acres in
12 days; thus (the case being here one of Inverse Proportion),
12 days : 10 days :: 12 men : — x 12 men=10 men.
Once more, suppose that, instead of * 12 days of the same
lengthy the question had said, * 12 days of 7i hrs. each.'
Here, after having found, as above, the n^ of men, 10, who
will reap the 40a. in 12 days of 6 hrs. each, we must still
have a third Proportion, to find the n^ who will reap the
same n^ of acres in the same n^ of days of 7|^ hrs. each ;
thus (the case being here also one of Inverse Proportion).
7^ hrs. : 6 hrs. : ; 10 men : s^ X 10 men = 8 men.
91. Now the above is an instance of Compound Proportion,
whereas the preceding Examples were all instances of Simple
Proportion ; the difference between questions in Simple and
Compound Proportion being, that, in the former, we have
one general quantity proportional to another ; whereas, in
the latter, we have one general quantity proportional to
each of several others, taken separately, i. e. supposing that,
while we take the two different values of any one of them,
the others meanwhile retain the same fixed values.
Thus, in the above Proportions, the n» of men is proportional,
in the 1st, to the n" of acres (directly) when the n° of days continues
the same, and the n° of hours in each day the same —
in the 2nd, to the n" of days (inversely) when the n° of acres continues
the same, and the n" of hours the same —
in the 3rd, to the n" of hours (inversely) when the n" of acres continues
the sawje, and the n® of days the same.
/
92. We have seen that, in cases of Simple Proportion,
when a single value of one general quantity is given cor-
responding to one given value of the other, we may find that
f2
96 PKOPORTION.
corresponding to another given value of the other by the
Rule of Three. In like manner, in cases of Compound
Proportion, when a single value of the first quantity is
given, corresponding to one given set of values of the
other quantities, we may find that corresponding to another
given set of them, either, as above, by successive Propor-
tions, or by what is called the Double Eule of Three,
which arises from the following consideration. Taking
the numerical value of the 1st result in its original form,
40 10 X 40
-^ X 9, we have that of the 2nd, -^ oq x 9, and of the
^^^> ¥T To oTS ^ ^> which would, of course, reduce itself
/g- X l.i X o\j
to the final answer, 8, i. e. 8 men : but now this is the same
result as we should get, if we made only one statement,
in which we set down the single term, 9 men, as usual, last,
and, for the 1st and 2nd terms, the products, respectively, of
the numerical values of the 1st and 2nd terms of the three
Proportions.
The same will be true in other cases. It is best to set down,
one under another, the num. values of iha first ratios of these
Proportions, observing to state them by considering each
general quantity separately, with reference to that quantity
whose single value is in the 3rd term ; and then w^e may
multiply these together, (striking out, as before, common
factors from the 1st and 2nd, or 1st and 3rd,) and, finally,
multiply together the 2nd and 3rd terms of the resulting
compound statement, and divide by the first.
Ex. If 5 compositors set up a work of 6 sheets in 8 days, in what
time ■will 6 compositors set np a "O'ork of 0 sheets ?
Here 8 daj-s is the single term, to Le set last : now, if we doubled the
n** of men (supposing the same n' of sheets), the n° of days would .be
halved ; hence the n" of days varies inversely as the n° of men, and the
corresponding first ratio will be 6 men : 5 men. Again, if we doubled
the n° of sheets (supposing the sa-me n° of men), the n" of days would bo
doubled ; hence the n" of days varies directly as the n° of sheets^ and the
PEOPOETION. 97
corresponding first ratio will be 6 sheets : 9 sheets ; "we have, therefore,
setting down the numerical values of thcsa ratios),
6:5] 8x5x9 ,
). : : 8 da. : ,. — ^r- da.
6:9] ^^^
and now striking out 4 from the dividend, and 2x2 from the divisor,
we have
2
^x5x9 ^ , , ,
— - — - — -2 X 5 = 10 days. Ans,
3 3
Ex. 59.
1. If 15 pecks of wheat serve 9 persons for 22 days, how long will
20 pecks serve 6 persons ?
2. If £33 55. pay 15 labourers for 18 daj's, how many labourers
will £79 165. pay for 24 days?
3. If 27 men can dig 2^ acres in 2 days, how many men can dig 2
acres in 3 days ?
4. If 7 horses be kept 20 days for £12, how many may be kept 14
days for £18?
5. If 9 persons spend £147 in 6 months, how many will £130 135.
^d. last for 4 montlis ?
6. If 6 horses consume 375 lbs. of oats in 8 days, what quantity will
4 horses consume in 10 days?
7. How much paper is required for 5000 copies of a book of 12^
sheets, if 66 reams are required for 3000 copies of a book of 11 sheets?
8. If 8 men earn £9 wages for 5 days' work, liow much would -36
men eiirn for 24 days' Avork at the same rate ?
9. If £100 will pay the expenses of 5 persons for 22 wks. 6 da., how
long would 12 persons be supported by £150 under similar circum-
stances ?
10. If 7 men earn £9 105. Qd. in 10^ days, what sum will 28 men
earn in 31i days?
11. If the wages of 25 men amoimt to £115 in 16 days, how many
men must work 24 days to receive £155 55., the daily wages of the latter
being one-half those of the former ?
12. If 21 men mow 72 acres of grass in 5 days, how many must be
employed to mow 460a. 3r. 8p. in 6 days ?
13. If 9 persons spend £120 in 8 months, how much will servo 26
persons for 12 months?
14. If 12 horses in 41 days plough 10^ acres, how many horses would
plough 35 acres in 20 days ?
98 PROPORTION.
15. If a 3 lb. loaf costs 7c?. when wheat is at 525. 6df. per quarter,
■what should be the price of wheat when a 2 lb. loaf costs d-d. ?
16. If a man travels Go miles in 3 days, by walking 7| hours a day,
in how many days ^vill ho travel 156 miles by walking 8 hours a day?
17. What will be the wages of 15 men for 10 months, when 9 men
receive £261 155. for 8 months?
18. If 3 persons are boarded 5 weeks for £17 10;?., how long should
14 persons be boarded for 60 guineas ?
19. How far should 80 cwt. be carried for £29, if 30 cwt. be carried
17 miles for £5 85. 9d.?
20. If 6 men can reap 34 acres of corn in 5 days, hcvr many men will
"be required to reap 95a. 32p. in 10^ days?,
21. If 40 bushels of corn serve 12 horses 37 days, how many days
would 195 bushels serve 9 horses?
22. A person completes a journey of 160 miles in 3 days, travelling
11 hours a day; in how many days would he complete 1000 miles,
going 15 hours a day at the same rate ?
23. If 3 men can reap 7 acres of wheat in 2 days, how long will it
take 8 men to reap 20 acres at the same rate?
24. If a ton of turnips will last 25 sheep for a fortnight, how much
will be required to supply 40 sheep during the months of January and
February in Leap-year ?
25. If 6 men can dig a trench, 220 yards long, in 2idays, by working
8 hours a day, how many will dig a trench, 187 yards long, in 4^ days,
working 6 hours a day?
26. If 12 men build 24 rods of wall in 30 days, working 8 hours a
day, how many hours a day must 18 men work to build 64 rods in 40
days?
27. If 8 men can plough 84 acres in 12 days of 85 hours each, how
many acres can be ploughed by 20 men in 11 days of 7| hours each ?
28. If 8 men can dig a trench 100 ft. long, 3 ft. broad, and 4 ft. 6 in.
deep in 9 hours, how many will be required to dig a trench 80 ft. long,
6 ft. broad, and 2 ft. deep in 5^ hours ?
29. If 7 men can erect a certain piece of wall in 20f days of 9| hours
each, how long would it take 3 men to do 2| of the same work, reckoning
10| hours to the day ?
30. If 20 men can excavate 185 cubic yards of earth in 9 Lours, how
many men could do half the work in a fifth of the time ?
99
CHAPTER VII.
MISCELLANEOUS RULES.
93. Interest is the consideration paid for the use of money.
The Rate of Interest is the sum paid for the use of a certain
sum, generally J 100, for a certain time, generally one year :
thus, if £6 is paid for the use of iClOO for one year, the in-
terest is said to be at the rate of 5 per cent.
The sum originally lent is called the Principal; and the
principal, together with its interest for any time, is called
the Amount for that time.
When interest is only taken for the original principal, it
is called Simple Interest; but, when at the end of any
stated period, as a year, the interest accruing is added to
the previous principal, and interest reckoned upon this sum,
taken as the principal, for the next year, it is called Com*
pound Interest.
94. To find the Simple Interest on a given sum for (X
given time at a given rate per cent, per annum.
Rule. Multiply the principal by the number of years,
and by the rate of interest per cent., and divide the result
by 100 ; the quotient will be the interest required.
Ex. L Find the Simple Interest on £725 for 3 years at 5 per centi
jper annum.
£725 For the Int. will be the same, whether We
o
."^ suppose the Principal, £725, repeated three times
2175 in three successive years, or three times in one
^ and the same year ; that is, the Int. on £725
108.75 for three years is the same as the Int. on
£2175 for one year: and this we find, accord-
Ans. £10S 15s. 15.00 ing to the above definition of Int., by dividing
by 100, to see how many Cents there are in the sum, and then taking 5
for each, i. e. multiplying by 5 ; or, which is the same thing, bat more
convenient in practice, we first multiply by 5, and then divide by 100,
100
INTEREST.
Ex. 2. Find the Simple Interest on £212 105. Ad, for 2| yrs. at 2^ per
cent, per ann.
for
fori
£212 10
Here the rem'', after dividing by 100, is
425
0
8
106
5
2
53
2
7
584
8
5
2|
1168
16
10
292
4
2|
14.61
1
0^
20
12.21
12
^.
!25rf.
?lrf.;
100 200 40
and, the Int. being £14 12s. 2|ic/., we have the
whole amount £227 2s. ^Hd. But it is gene-
rally best to represent the whole procedure first
symbolically, in order to ascertain whether the
calculation may be simplified; thus we have
£212 IQj?. Ad. X 2^ x2|_£212 105. Ad.^W
100 160
SO that J^ of the given principal will be the
interest.
2.52^
Ex, 60. Find at Simple Interest,
1. Interest on £500 for 5 yrs. at 5 per cent.
2. Interest on £375 for 3 yrs. at 4 per cent.
3. Amount of £l 125 for 4 jts. at 3 per cent.
4. Amount of £2275 for 3i yrs. at 5 per cent. ^
6. Interest on £347 165. 8d. for 15 yrs. at 4| per cent. <
6. Amount of £2000 for 12i yrs. at 3^ per cent.
7. Amount of £575 for 8| yrs. at 3| per cent.
8. Interest on £325 10s. for 4 yrs. at 5^ per cent.
9. Interest on £500 135. Ad. for 2^ yrs. at 2f per cent.
10. Interest on £ 1 50 for 3^ yrs. at 4 per cent.
If parts of a year be given, they may be expressed as a
fraction of a year.
Thus the Int. for 2 yrs. 3 mo., at any given rate, would be the same
as that for 2| yrs. at the same rate.
But, in practice, more accuracy is generally required;
and we must express the given parts of a year in days, and
then, finding first the Int. for one year, we may find by a
proportion the Int. for the given portion of a year.
Ex. 3. Find the Int. on £Z2bfrom March 1, 1871, to May 31, 1874,
at 4 i^er cent, per ann.
INTEREST. 101
When interest is thus required from one date to another, the day of
the first date is to be left out, because it is not until the day folloAving
that one day's interest will have accrued. Accordingly, we have here
the whole time = 3 yrs. 9 1 da.
Now, the int. for 1 year is (£325 x 4)-r 100=£13; and for 91 days we
have by Proportion —
365 da. : 91 da. :: £13 : £3 4s. 9f|(/.
Int. for 3 yrs. = £13 x 3= 39 _0 0_
Ans. The whole int. is £42 4 9||
If the rate of Interest be given in parts of a £, they may be
expressed as ^fractio7i of a £, and the sum treated as before ;
or we may work for them by the method of Practice.
Ex. 4. Find the Tnt. on £500 for 4 yrs., at £5 7s. 6d. per cent.
£500j<4_x5|^5^21| = £l07 10*. Ans.
100 "^
Ex. 5. Find the Int. of £307 15s. 6d. for 156 datjs, at £i Us. Bd.
per cent.
Here it will be best to work throughout by decimals, and to extend
them only to so many places as will insure the accuracy "of the final
result to two or three decimals of a penny. Also we may employ the
method of Practice, not only for the rate, but also for the days, 156 da.
being = 146 + 10 da,=|yr. + 10 da.
£3.07775 =Principal-r 100.
4
12.31100
10s. = 1
4s. =-1
1.538875
.61555
6c/. =1
.076944
14.542369 =
Int.
for 1 yr.
10
365)
145.423690
.398421 =
Int.
for 10 da.
^3da.=i
2.908474 =
)j
73 da.
7ada.=i
2.908474 =
if
_73da.
£6.215369 =
}f
156 do.
20
4.307380
12
Ans. £6 4s. 3.688d
f3
102 INTEREST.
C
Ex. 61. Simple Interest \
1. Find the amt. on £500 from March 1 to Jan. 10, at 4| per cent, i
2. Find the amt. on £7500 from May 5 to Oct. 27, at 3| per cent.
3. Find the amt. on £l 1 58 1 1s.^ 6d, for 1 y r. 1 1 5 d. , at £2 1 0«. per cent.
4. Find the int. on £250 125. ^6d, from March 26, 1870, to Oct. 31,
1872, at 3 per cent.
5. Find the int. on £3996 15*. for 4yrs. 225 d. at £2 13«. 4c?. per cent.
6. Find the int. on £2755 155. for 3yrs. 110 d. at £3 25. 6c?. per cent.
95. To find the Compound Interest on a given sum, for a
given time, at a given rate per cent, per ann.
Rule. At the end of each year add the Interest for that
year to the Principal at the beginning of it, and this will be
the Principal for the next year ; and so on, till we have
found \\\Q final Principal, or whole Amount, See Note IX.
Ex. Find the Compound Interest on £750 for 3 yrs., at 4 per cent,
per ann. ; and also at 2^ per cent, per ann.
£750 First Principal
^= 30.00 Int. in 1st year.
780.00 Second Principal.
loo= 31.20 Int. in 2nd year.
811.20 Third Principal.
xUo= 32.448 Int. in 3rd year.
£843.648-750 = £93 125. ll|fc/. 1st Ans.
, £750 1st Principal.
_i2_=_i_ = 18.75 Int. in 1st year.
^^^ 768.75 2nd Principal.
i= 19.21875 Int. in 2nd year.
787.96875 3rd Principal.
• i= 19.69921875 Int. in 3rd year.
807.66796875 -750 = £57 135. 4 ^c?. 2nd Ans.
Ex. 62.
1. Find the amt. of £95 I65. 8d., for 2yrs., at 2| per cent.at comp.int.
2. Find the amt. of £50, for 3 yrs., at 5 per cent, at comp. int.
3. Find the difference between the simple and compound interest on
£41 135. 4d., for 2 years, at 5 per cent.
4. Find the difference between the simple and compound interest on
£365 45. 8|c?., for 3 years, at 4 per cent.
6. Find the comp. int. on £225, for 3 years, at 3| per cent.
6, Find the comp. int. on £300, for 3 years, at 2| per cent.
INTEREST. 103
96. There a.refotcr things to be considered in all questions
of Interest— the Pri?ieipal, the Hate of Interest^ the Time,
and the Total Interest, (the Amount being only the sura of
the first and hast of these) ; and, if any three of these be
given, we are able to obtain the fourth. Hitherto we have
only considered the case which most commonly occurs in
practice, viz. that in which the Principal, Rate, and Time
are given to find the Interest, (or the Amount) ; we shall
now give an Example of each of the other three cases which
may arise in Simple Interest — those in Compound Interest
being more difficult, and of less frequent occurrence.
I. When the Principal, Interest {or Amount), and Rah
are given to find the Time.
Ex. In what time will £91 135. 4(/. amount to £105 65. 0|f/., at 4|
per cent, per ann. ?
Subtracting the prhicipal from the amount, we have here given the
interest = &\2> 12s. ^d.', now in o/ie year £91 13s. 4cf. produces, at tho
given rate, 4?Ji^=^JiijLlI; we have, therefore,
^ '100 48
€iL^_L^ : £13 125. 8M. :: l year,
48 1 3 y
or, £U X 17 : £13 125. 8|c?. x 48 :: 1 year.
12
163 12 6
4
11)654|
17) 591
3| years. Ans^ ^
II. When the Rate, Time, and Intef-SSi (or Amount) are
given to find the Principal,
Ex. What sum of money, put out to interest for 4 yrs» at 3| per cent.,
will amount to £259 75. ?
At the given rate for the given time the interest of £100 would be
£85 X 4 = £14, and therefore its amount £114 ; We have, therefore,
£114 : £259 75.:: £100 : the Ans.,
which We obtain in the usual manner. =£227 lOs,
104 INTEKEST.
III. When the Principal, TimCy and Interest {or Amounf)
are given to find the Rate.
Ex. 1. At what rate per cent, will ^142 10«. amount to £163 13*. \\\d,
in \\ years?
The interest of £142 105. is £21 3s. \\\d, in 4| years,
.*. for 1 year it is 20349c/. -^ 1 7 = 1 197(/. ;
and £142 10s. being=34200d, we have
34200c/. : £100 :: 1197(/.
or, 38c/. : £1 :: 133c/. : £3|.
Ans. 3| per cent, per ann.
Ex. 2. At what rate per cent, per annum will £5 amount tc
5 guineas in 219 days?
In 219 da., or § of a year, the int. of £5 is 5s.
.*. in 1 year it is 5s. -r-f = 8|s»
£5 : £100 : : s^s. : £8|.
Ans, 8| per cent, per ann.
Ex. 63. Simple Interest.
1. At what rate will the int. on £102 10s. amount to £12 13s. 8|c/.
in 2\ years ?
2. What sum will amount to £45 Os. 9|c/. in i year, at 6| per cent. ?
3. In what time will the int. on £498 16s. 8c/. amount to £lO 9s. 3ic/.,
at 61 per cent.?
4. At what rate per cent, will the int. on £200, for 146 days, amount
to £4 16s.?
5. In what time will £732 lis. 10c/. amount to £1709 7s. 7|c/., at
.5i per cent. ?
6. What sum must be put out to interest at 4| per cent., to become
£49 Os. b\d. in 51 years ?
7. At what rate will the int. on £4127 10s. amount to £92 17s. 4lc/.
in a year ?
8. What principal will produce £121 15s. bd. in 2 yrs. 1 mo., interest
at 5| per cent. ?
9. In what time will £419 amount to £486 4s. 3lc/., at 4| per cent. ?
10. At what rate will £220 12s. 6c/. become £240,4s. 8|c/. in 3i yrs. ?
11. What principal in 3 years 73 days will become £10 Is. 10|</.,
interest at 6i per cent, ?
12. In what time will the interest on £812 10s. 10c/. amount to
£771 18s. 31c/., at 43 per cent. ? ,
DISCOUNT. 105
97. Discount is the sum allowed for the payment of
money before it is due.
Thus, if A has to pay to B £525 at the end of a year, and the rate
of interest is 5 per cent., he might arrange to discliarge his debt by-
paying him now £500, because this sum put out to interest would
amount to £525 at the year's end. In this case, therefore, £25 would
be the discount which B would allow to A, for paying him the debt at
the present time.
The present value of a sum, due at some future time, is,
therefore, the sum left, when the discount for that time is
deducted, (as £500 in the above instance); and may be
defined to be that sum which, put out at interest for the
time in question, would amount to the sum due at the end
of the time ; and the discount is the diifference between the
whole sum and its present value, or the interest upon the
present value.
98. The most common form in which Discount occurs is
in the prepayment of Bills or Notes of Handy which are both
documents (but differing somewhat in form and character)
by which a person engages himself to pay a certain sum, at
a certain future time, both named therein. If the credit of
the party promising payment, or of the party holding the bill,
be considered satisfactory, a banker will discount it, that is,
will pay its present value at once, deducting from the whole
amount the discount upon it for the time that must elapse
before it will become due.
99. In practice, however, it is usual to charge as dis-
count the interest on i\iQ future debt itself; by which means
the present value obtained is evidently less than it should
equitably be.
Thus, if a banker discounted at 5 per cent, a bill for £525, due at
a year's end, he would not calculate what sum (viz. £500) at interest
would produce £525 at the year's end, and so deduct the interest
(viz. £25) for this sum as discount ; but he would calculate the interest
on the debt, £525, itself (viz. £26 5s.), and, deducting this, would pay only
£498 15*. to the holder of the bill as its present value. By this means,
since £498 15«., with its own interest, would not amount to £525 in a
106 DISCOUNT.
year, the holder is a loser, and the banker gains, as we have seen, the
difference of £500 and £498 155., viz. £l 55., by the transaction— being,
in fact, the interest upon the true discount.
In practice, therefore, questions in discount are reduced
merely to questions in Simple Interest ; but we shall, here
and throughout, give examples in the more correct rule>
unless the contrary be expressed.
N.B. In Great Britain and Ireland 3 days, called Days of Grace^
are always allowed, after the time that a bill is nominally due, before it
is legally due. Thus, if a bill of £250 were drawn on July 10, at 3
months, it would be nominally due on Oct. 10, but legally on Oct, 13;
and, if a banker were to discount it on Aug. 20, he would reckon forward
to Oct. 13, (the last of these days inclusive,) and, finding the interval to
be 54 days, he would reckon the interest on £250 for that time, and,
deducting it as discount, would pay the difference as the present value
of the bill.
' It may be noticed, also, that, if a bill would fall nominally due on the
29th, 30th, or 31st of February, or on the 31st of any month which has
only 30 days, it is considered to be nominally due on the la&t day of the
month, and therefore legally on the 3rd of the following month : and, if
any fall legally due on Sunday, they are paid in Great Britain on the
Saturday, but in Ireland on the Monday.
Ex. 1. What is the discount on £396 175. b\d., due at 9 months, at
4 per cent. ?
This example ftills under (96), Case 11, in Simple Interest; since,
thereforie, £100 produces in 9 months, at 4 per cent., £3, we have £100,
the present value of £103, due at the end of 9 months ; and thus we get
the proportion,
£103 : £396 175. 5iJ.::£l00,
which, being solved as usual, gives us the present value £385 6A 3tf., and
therefore the discount, £11 lis. 2^(7.
Ex. 2. What would a banker gain by discounting on Sept. 21a bill
of £318 35., dated July 31, at 4 months, at 5 per cent. ?
This bill will be nominally due on Nov. 30, and legally on Dec. 3 ;
and, reckoning from Sept, 21 to Dec. 3, (the last inclusive), we have
79 days. We shall find the interest on £318 35. for 73 days, in the
tisUal manner, to be £3 35. I^d. ; and the present value of it, i.e. that
principal, which at 5 per cent, would become £318 35. in 73 days, we
shall find, as in Ex. 1, to be £315, and therefore we have the discount—
£3 35. i so that the banker gains upon the whole 1^.
INSURANCE. 107
Ex. 6«.
1. Find the present value of £284, due at the end of 2 years, at 3|
per cent, per annum.
2. What is the present value of £860, due at the end of 3 years, at
3| per cent. ?
3. Find the discount on £1336 lis. Bd., due at the end of 3| years,
at 5 per cent,
4. Required the present value of £151 17s. Qd., due at the end of 4
years, at 6| per cent.
6. What is the discount on £88 2s. 5d., due at the end of 6 months,
at 4| per cent. ?
6. Find the discount on £210 125. Id., due at the end of 3| years, at
4| per cent.
7. Find the present value of £598 9s. dd., due at the end of 1 year
115 days, at 21 per cent.
Find the true discount upon the following hills —
Drawn. Discounted.
March 6, at 7 months Sept. 15, at 5 per cent.
Sept. 12, at 5 months Jan. 13, at 4 per cent.
Feb. 29, at 3 months April 27, at 3| per cent.
March 17, at 3 months May 31, at 6 per cent.
Aug. 5, at 5 months Dec. 6, at 3^ per cent.
May 31, at 4 months Sept. 3, at 5 per cent.
Dec. 25, at 2 months Feb. 8, at 6 per cent.
See Note X.
& s.
d.
8.
419 12
1
0.
457 18
0
10.
637 5
2
11.
755 5
9
12.
1006 15
6
13.
1337 14
6
14.
1846 5
2
100. There are other cases of common occurrence in
which a rate per cent, is charged.
Insurance is a per centage paid for securing property from
fire, &c. The charge is regulated by the nature of the pro-
perty insured, and the hazard to which it is exposed, as laid
down in the Tables of the different Insurance Companies,
The whole annual payment is called the Premmm^ and the
legal document by which the Insurer is secured from los3
is called the Policy of Insurance.
Life Insurance is a per centage paid for securing the pay-
ment of a sum of money upon the death of a person. The
charge is regulated by the age and healthiness of the person
whose life is assured, at the time the Policy was firsfc taken
108
INSURANCE.
out, as laid down in the Tables ; and, being thus settled, it
is reckoned per cent, upon the whole sum secured — the
whole annual payment being called, as before, the Premium
upon the Policy of Assurance,
In each of the above cases the Premium, like Interest, must be
renewed every year, while the Policy is in force ; but the following charges
are, from their nature, paid only once.
Insurance from sea risk is a per centage charged upon
the value of a cargo, just as in Fire Insurance.
Commissioji is a per centage paid to an agent for buying
or selling goods.
Brokerage is a smaller per centage of the same nature,
paid usually for transacting money concerns.
101. It is usual with tradesmen to allow (what is called)
a discount of 5 per cent, for ready-money payments upon
goods purchased, or, (since 5 per cent, is the same as 1 in
20), to allow a shilling in the pound upon the account to be
paid : thus, for ready-money payment of an account of
£1 135. 6c?., most tradesmen would allow 7*. 6d, (7^. for
the ^7, and 6c?. for the 10^.,) and would be content there-
fore to receive as full payment £7 6^. This, however,
differs from the discount of which we have before been
speaking, since it takes no account of the time^ at which the
debt would otherwise be paid ; but is merely an arrange-
ment to secure to the seller the convenience of a ready-
money payment, by giving to the buyer a corresponding
advantage.
Ex. 1. What is the sum to be
paid for insuring a vessel and cargo,
worth £2225, at 3^ per cent.?
£2225
31
fori
6675
556
72.31
20
6.25
12
Ans. £12 6s, 3i. 3.00
Ex. 2. What is the premium
upon a policy of £375 upon a life
of 28, the rate being £2 8«. 1d>
per cent, for that age ?
Here £375 -3| of £100} and
the premium is 3| of
£2 8 7
3
4)7 5
1 16
£9 2 2i Ans.
STOCKS. 109
Ex. 3. What sura should be insm-ed at 4 per cent., on goods worth
£735, that the owner may receive, in case of loss, the value both of
goods and premium ?
Here, if £100 were insured, it would cover goods to the amount of
£96, together with the premium £4 ; hence we have the proportion
£96 : £735:: £100,
.whence we get, as usual, the ^n5. = £765 I2s. 6d.
Ex. 65.
1. What would be" the ready-money payment of an amount ol
'£27 135. 6d., discount being allowed at 5 per cent. ?
2. What would be the expense of insuring a vessel and cargo, whose
value is £2516 10*., at 3| per cent. ?
3. What is the premium on a policy of assurance for £2286 13s. 4d,,
upon the life of a person aged 42, at the rate of £3 10s. per cent, for that
age?
4. At 4| per cent., for what sum should goods be insured, which arc
worth £427 15s. 3d., in order that, in case of loss, the owner may recover
their value, together with the premium paid ?
. 5. What would be the cash payment of an account of £27 17s. 5^., at
5 per cent. ?
6. What is the brokerage upon a money transaction of £273 15s,, at
3s. 4c?. per cent. ?
7. For what sum should a cargo, worth £5263, be insured, at 7§ per
cent., so that the owner may recover, in case of loss, the value both of
cargo and premium ?
8. What is the commission upon £713 6s. 8d., at 2f per cent. ?
9. What is the premium of insurance upon £3208 17s. Id., at £2 1 2s.
per cent, ?
10. What is the premium on a policy of insurance for £1237 10s.,
upon a life of 21 years, at the rate of £2 2s. 4d. per cent, for that age ?
11. What is the brokerage on £768 2s. 6d., at 3s. 4d. per cent.?
12. For what sum should goods, worth £4384 Os. 3c?., be insured at
£2 6s. 8d. per cent., that the owner, may recover, in case of loss, the
value of both goods and premium ?
102. Stock is the name given to Money, lent to some
Trading Company, or, more comi?ioiily, to our own or some
foreign Government, at some given rate of Interest, which
is settled at the time the Money is first lent, according to the
circumstances then existing.
iio gtocKg.
Thus, if Government were to boiTOw to the amount of £500,000 at
4 per cent., and A had lent £100 of this sura, A would be said to have
£100, 4 -per cent, stock, and would receive a document entitling him to
receive the Interest (viz. £4) upon this stock from year to year, until
Government chose to repay the Principal, and put an end to the debt
The source from which the Interest is paid is called the
*Public Funds,' being, however, only an imaginary Property,
representing the credit of the Country itself, which is pledged
to the payment of the debts contracted by its Government ;
the Interest is paid half-yearly, and the document, entitling
the possessor to receive it, may be sold, and transferred from
one party to another, just as any other kind of property.
« If money would always bring the same amount of Interest,
the average price of £100 stock would be always the same,
(viz. £100, the price first given for it) — we say the average
price, because even then the price would evidently be some-
what less immediately after the payment of a dividend than
it would be immediately be/ore it. But not only does this
cause affect the price of Stocks, but the continual fluctuations
in the value of Money, arising from commercial or political
changes or expectations abroad and at home, are constantly
disturbing it, even two or three times in the sa?ne day,
according to the news which reach us. The price of stock,
then, will rise or fall according as it seems most likely that
Money would fetch elsewhere a higher or a loiver rate of
Interest, i. e. would be more scarce, and in demand, as in
prospect of war, or of active speculation, or be lying upon
hand and plentiful, as Avhen trade is looking dull, and there
are no means of employing capital.
Thus, if at the time A wished to sell his stock, money was elsewhere
making 5 per cent., it is plain that no one would give him £100 for the
right to receive only 4 ; but since £80 of common or sterling money
(as it is called) would now bring £4 interest, he would be able to sell
his £100 stock for £80; and the 4 per cents, would be said to be
Belling at 80.
With this explanation, the mode of treating questions on
Stocks will be easily seen from the following Examples.
STOCKS. Ill
Ex. 1. If £3500 be invested in the 3^ per cents, at 98, what is the
annual income thence derived ?
Here -^=no of cents, purchased, for each of which £3| arc paid as
interest : hence the whole income = — — x3l=£\25.
Ex. 2. The 3i per cents, are at 99| ; how much money must be in-
vested in them to produce an income of £140 ?
Here^-^ = n" of cents, required, for each of which £99| are paid ,•
hence the whole sum paid = H? x 99|=£3995.
Ex. 3. If a person were to transfer £29000 stock, from the 3i per
cents, at 99, to the 3 per cents, at 90f, what would be the difference in
his income ?
Here £29000 m the 3| per cenrs. produces 290 x£3|=£l015 Int.,
and would be sold out for 290 x 99 =£28710 ; this money, invested in
the 3 per cents, at 9 Of, wouM purchase — '-^ cents., and therefore
90g
2871 0
would produce, as Int., ^ x 3=£950 8s. ; and his income, there-
90^-
fore, would be diminished by £64 12*.
Ex. €6.
1. The 4 per cents, being at 82|, what must be given for £1000
stock? and what sum would be gained by selling out again at 86^?
2. What income should I get by laying out £1188 in the purchase
of 3 per cent, stock at 81 ?
3. If I lay out £3000 in the 3 per cents, when they are at 84|, what
mcome should I thence derive?
4. A person having £4200 invests it in the 3^ per cents, at 90; find
his income.
5. What is the price of stock per cent., when a person can purchase
£2766 13s. 4d. for £2490?
6. What sum must be invested in the 3 per cents, at 94j, to yield an
annual income of £500?
7. How much stock at 92| can be bought for £494, a commission of
I per cent, being charged on the stock purchased ?
8. What is the cost of 850 Bank Annuities at 90|, | per cent, being
paid for brokerage? And what sum would be lost by selling out
at 89i?
112 PEOFIT AND LOSS.
9. If I lay out £1000 in the 3i per cents, at 96, what should I lose
by selling out at 95?
10. If a person lays out £4650 in the 3| per ceilts. at 93, what will
be his loss of property by the stocks falling | per cent. ?
11. What would be the difference in income, made by the transfer of
£5000 stock from the 3 per cents, at 72 to the 4 per cents, at 90?
12. A person transfers £11000 from the 4 per cents, at 92 to the 5
per cents, at 110; what is the difference in his income?
13. What would be the difference in annual income from investing
£3450 in the 4 per cents, at 92, and the 3| per cents, at 69?
14. A person invests £18150 in the 3 per cents, at 90f , and, on their
rising to 91, transfers it to the 3| per cents, at 97|: what increase does
he make thereby in his annual income?
15. If I lay out £l 1 10 in the 4 per cents, at 92^, at what price should
they be sold to produce a gain of £100?
16. In which is it most advantageous to invest, in the 3 per cents, at
89i, or the 3^ per cents, at 98|?
17. A sum of £3750 was sold out of the 3 per cents, at 95, and put
at compound interest for 2 years at 4 per cent. ; the amount being laid
out in the 3| per cents, at 104, find the alteration in income.
18. A person has £1000 in the 3| per cents.; how much must he
have also in the 3 per cents, that his whole income may be £200,
and what sum would he realise by selling out at 83§ and 77| re-
spectively ?
19. A sum is laid out in the 3 per cents, at 89|, and a half-year's
dividend received upon it; the stock being then sold at 94|, and the
■whole increase of capital being £54, find the original sum laid out.
20. The sum of £1001 was laid out in the 3 per cents, at 89|, and
a whole year's dividend having been received upon it, it was sold out ;
the whole increase of capital being 72 guineas, find at what price it was
sold out.
103. Profit and Loss. — The method of treating ques-
tions of this kind -will be best learnt from the following
Examples.
Ex. 1. If tea be bought at 55. 6d. per lb., and sold at 6s. 8c?., what is
the gain per cent.?
Here the gain on the prime cost, 5s. 6d., is Is. 2d. ; hence we have
5s. 6c/. : £100 :: is. 2c?. : the Ans.
which is found by the usual method to be £21 4s. 2^.
PROFIT AND LOSS. 113
Ex. 2. If bar- iron, which cost in making £2 1*. ^d. per cwt., be sold
at a loss of 5| per cent, what price did it fetch per cwt.?
Here bar-iron, which cost £100, would only have sold for £100— ^Sf
= £94|; hence we have
£100 : £2 \s, 8 J. : : £94f : the Ans.
which is found by the usual method to be £l 195. 5\d.
Ex. 3. If 5 per cent, be gained by selling 125 yards of cloth for £95,
what was the prime cost per yard?
Here, if the cloth had sold for £105, the prime cost would have been
95 X 20
£100; therefore the selling price per yd. bemg -l— — 5., we have
£105 : ?^i^. :: £100 : ^^s.^Us. 5fJ. Ans.
125 21 ^
Ex. 4. If 4 per cent, be lost by selling linen at 2s. 9c?. a yard, at what
price must it be sold to gain 10 per cent.?
Here, cloth which would have cost £100 would have been sold for
£96 at the first price, and for £110 at the second; we have, therefore,
£96 : 2s. 9d. :: £110 : second price = 3A 1^.
Ex. 67.
1. How must nutmegs, which cost 18s. dd. per lb., be sold, so as to
gain 16 per cent.?
2. If tea be bought at 2s. lid. per lb., and sold at 3s. 7c?., what is
the gain per cent. ?
3. A merchant, by selling sugar at £1 16s. 6d. per cwt., loses 18
per cent. ; what was his prime cost ?
4. If cheese, which was bought at £3 4s. 7d. per cwt., be sold at
£3 12s. id., what is the gain per cent. ?
5. If iron, raised at an expense of £4 5s. Sj^d. per ton, be sold at
£4 19s. 9d., what is the gain per cent.?
6. If I buy 2048 yards of linen at 3s. 2ic?. per yard, and sell
the whole for £359 6s. 8t/.; required the whole gain and the gain per
cent.
7. If hemp cost £48 7s. 6d. per ton, and be sold at £43 per ton,
how much per cent, is lost, and how much is lost in the sale of 39 tons,
3 cwt.?
8. If 64 ells of lace cost £l 1 5, at what price per yard must it be sold,
so as to gain 18 per cent.?
9. A plumber sold 96 cwt. of lead for £109 2s. 6J., and gained at
the rate of 12i per cent.; what did it cost him per cwt.?
10. On the sale of 1 12 yards of silk velvet at 14s. dd. per yard, a
114 PROPORTIONAL PARTS.
merchant loses £10 145. 8d. ; find the prime cost of the whole, and the
loss per cent.
11. If teas at 25. Qd., Bs. Sd., and 25. id. be mixed in eqnal quanti-
ties, and the mixture sold at £16 165. per cwt., what will be the gain or
loss per cent. ?
12. A person has fth of a ship, worth £6600, and insured for 91|
per cent, of its real value; what damage would he sustain iu case of its
being lost?
13. What was the cost of printing 500 copies of a book, which was
sold for 55., if the expense of sale was 34 per cent., and the author's
profit £37 155. upon the whole?
14. If 5i per cent, be gained by selling butter at £5 5s. ed. percwt.,
what will be the gain per cent, by selling it at Is. 3c?. per lb.?
15. If 8 per cent, be gained by selling 218 yards of cloth for £92 135.,
at what price per yard must it be sold, so as to gain 17 per cent. ?
16. A person buys 50 reams of paper, which he thought to sell at
£1 2s. 6d. per ream, making 8 per cent, profit on the prime cost ; but,
5 reams being damaged, what did he gain or lose per cent, by selling
the remainder at the same rate ?
17. A person buys 4 cwt. of goods for £15, intending to gain 12 per
cent, by the sale; but, a guinea's worth (at this calculation) being
damaged, at what price should he sell per cwt., to gain as much upon
his whole outlay as he intended ?
18. Bought 236 yards of cambric at 7s. 10|c?. per yard, and sold one-
fourth at lOs. 3c?., one-third at 8s. 6c/., and the remainder at 7s. per
yard; what was the gain or loss per cent, upon the whole outlay?
19. If eggs be bought at the rate of 5 a penny, how many should be
sold for 7c?., to gain 40 per cent. ?
20. A person purchases pins, 18 in a row, and sells them, 11 in a
row, at the same price j how much is his gain per cent, on his outlay?
There are various examples depending upon the following Rule, the
method of treating which will be best explained in the instances below
given.
104. Proportional Parts. — To divide a given quantity
into parts ivliich shall have to each other given ratios,
Rule. Form fractions whose common den'* is the sum of
the numbers expressing the ratios, and the num*'^ the sepa-
rate numbers themselves ; and take these fractions of the
given quantity : they will be the parts required.
PROPORTIONAL PARTS. 115
Ex. 1. Divide 75 into two parts which shall have the ratio of 2 : 3.
Here the fractions are | and f, and the parts required are f of 75 = 80,
and I of 75 =45, which are plainly in the given ratio.
The reason of the Rule is evident, since the sura of the num" maked
up the den% and therefore the sum of the fractions makes up unifi/, i. e.
the sum of the parts makes up the whole of the number; while the parts
themselves, having a common den', are in the ratio of their num".
Ex. 2. Gunpowder is composed of 76 parts of nitre, 14 of charcoal,
and 10 of sulphur: how much of each of these will be required for a cwt.
of powder?
Here the fractions are ^= if, ^=^, ^=^, and the parts are
Sq. l^lbs., 15iflbs., and llilbs. respectively.
Ex. 3. Divide £1000 among A, B, C, so that A may have half as
much again as B, and B a third as much again as C.
Here, representing C's part by 1, B's is 1|, and ^'s 1| + | of ll=2;
and, therefore, the parts are to be as the numbers 2, 1^, 1, or 6, 4, 3.
Hence the fractions will be ^, i%, 1^3; and the parts required
£461 10s. 9^., £307 13s. lO^J., £230 15s. 4^.
N.B. —It will be found most convenient, where there are many frac-
tions with the same den', to find the part corresponding to that den*
•with num' unify, and then multiply this successively by the num" of the
different fractions; thus we should find ^ of £1000, and then multiply
this by 6, 4, 3, respectively.
Ex. 4. A, B, and C form a joint capital for conducting a business, of
which A contributes £500, B £650, and C £700. At the end of a year
the profits are £555; what share should each receive?
Their shares should evidently be in the ratio of their contributions of
capital, i.e. in the ratio of 500, 650, 700, or of 10, 13, 14; hence the
fractions are if, if, |f, and since gV of £555 = £15, we have the shares
required £150, £195, £210.
Ex. 5. A begins business with a capital of £800, and, at the end of
3 months, takes B into partnership, with a capital of £1000; at the end
of another 6 months they divide their profits, £330 ; what should each
receive ?
Here A contributes £800 for 9 months, and B £1000 for 6 months ;
and the interest of £800 for 9 months = interest of £800 x 9 for 1 mo.,
and the interest of £1000 for 6 months = interest of £1000x6 for
1 month ; hence the value of A's and B's outlay may be represented by
the products 800 x 9 and 1000 x 6, or 7200 and 6000 respectively, and
their shares of the profits must be in this ratio = that of 6 : 5 ; hence A's
8hare=T\of £330 = £180,
116 PROPORTIONAL PARTS.
N.B. — It appears, as in the above Ex., that the values of sums
employed in business, &c., for different times are proportional to the pro-
ducts of the suras by the times, or rather of their numerical values, the
sums being expressed in the same den°, and so also the times.
Ex. 6. A and B enter into partnership, A contributing £500 and B
£300 ; at the end of 9 months they take in C as partner, who brings into
the concern a capital of £1000. The profits, £2000, being divided at
the end of another 9 months, what shares did they each receive?
Here, as in Ex. 5, at the end of 18 months, the shares of capital
supplied by A, B, C, respectively, may be measured by the numbers
500 X 18, 300 X 18, 1000 x 9, or 5, 3, 5 respectively: hence the fractions
will be t^, ^, ^; and since ^ of £2000=£l53 16^. U^J., their
shares of profit will be £769 4s. T^d, £461 10*. 9^., £769 4s. 7^.,
respectively.
Ex. 68.
1. Divide 1065 into parts, which shall be to each other in the ratio of
3, 5, 7 ; and also into parts which shall be in the ratio of \, i, \.
2. A, B, and C engage in trade, investing capital to the amount of
£128, £176, £192 respectively: their profits amount to £279; what
were their shares of it?
3. How much copper and tin will be required to cast a cannon
weighing 16 cwt. 3 qrs. 11 lbs., gun-metal being composed of 100 parts
of copper and 11 of tin?
4. Divide £153 among five persons in the proportion of the fractions
11111
3» 4' 5> 0' Y'
5. Divide 1400 into parts, which shall have the same ratio to one
another as the cubes of the first four natural numbers.
6. Pure water is composed of 2 gases, oxygen and hydrogen, in
the proportion of 88.9 to 11.1; what weight of each is there in a cubic
foot ClOOO oz.) of water ? • V
7. Divide £300 among three persons, so that the first shall have
twice as much as the second, and the third twice as much as the other
two together.
8. A works regularly 9 hours a day ; B remains idle the first two
days of the week, and works 6^, 8|, lOf, 12 hours, respectively, on the
other four; what sum should each receive out of £11 12s. 6|(/. at the
month's end?
9. The standard silver coin of this realm is made of 37 parts of
pure silver and 3 of copper, and a lb. Troy of this metal yields 66 shil-
lings; what weight of pure silver is there in 20s.?
10. In England, gunpowder is made of 75 parts of nitre, 10 of sul-
phur, and 15 of charcoal; in France, of 77 of nitre, 9 of sulphur, and 14
b
PROPORTIONAL PARTS. 117
of charcoal: if half a ton of each be mixed, what weight of nitre, sul-
])htjr, and charcoal, will there be in the compound?
1 1. The standard gold coin of this realm is made of gold, 22 carats
fine, and a lb. Troy of this metal yields 46|2 sovereigns; what weight of
pure gold is there in 100 sovereigns?
12. If 4 oz. of gold, 17 carats fine [see Appendix'}, are mixed with
3 oz., 13 carats fine, how much fine gold will there be in a gold orna-
ment made of the compound, and weighing 3| oz.?
13. A and B engage in trade, their capitals being in the ratio of
4:5; and, at the end of three months, they withdrew respectively §
and f of their capitals: how should they divide their whole gain, £335,
at the end of the year ?
14. Ay B, C join their capitals, which are in the proportion of \, i,
and \ ; at the end of 4 months A withdraws i of his capital, and at the
end of 9 months more they divide their profits, £284; what should each
receive ?
15. A and B rent a pasture for £16', A puts in 80 sheep and B
100, but at the end of 6 months they each dispose of half their stock,
and allow C to put in 50 sheep to feed; what should A, B, C, severally
pay towards the rent at the year's end?
16. Four parcels of gold, weighing respectively 10, 4, 2, and 4 oz.,
and of 13, 12, 11, and 10 carats fineness, being mixed, what was the
fineness of the compound ?
17. If the preceding be reduced by refining to 16 oz!, what will bo
the fineness of the mass? or if its fineness, when reduced, bo 16 carats,
what will be the reduced weight?
18. If 8 oz. of gold, 10 carats fine, and 2 oz,, 11 carats fine, bo
mixed widi 6 oz. of unknown fineness, and that of the mixture be 12
carats, what was the unknown fineness?
19. A, B, C, are sent to empty a cistern, by means of two pumps of
the same bore. A and B go to work first, making 37 and 40 strokes re •
spectively a minute ; but, after 5 minutes, they make each 5 strokes less
a minute, and, after 10 minutes more, A gives way to C, who works at
the rate of 30 strokes a minute. The cistern is emptied in 22 minutes
altogether, and the men are paid 12s. Id. for their labour. What should
each receive ?
20. A and B are partners, having each embarked £500 in their
business. At the end of 3 months they gained £300, when A withdraws
£200, and B at the same time advances £200. At the end of the next
3 months, they gained £780, when A again withdraws £200, and B at
the same time advances £200. At the end of the year they separated,
dividing their property, which by losses during the last 6 months was
reduced to £400. AVhat should A and B each receive ?
G
118 CHAIN BULE.
105. Chain Rule. — When a comparison of several suc-
cessive quantities is made by stating how many of the
second are equivalent to a given number of the first, how
many of the third are equivalent to a given number of the
second, and so forth, and it is required to find how many of
the last are equivalent to a given number of the first, the
answer is conveniently found by the Ghain Bute, The fol-
lowing is an example :
What is the value of 20 lbs. of bacon, if 15 lbs. of bacon be equal iu
value to 14 lbs. of cheese, and 35 lbs. of cheese equal to 46 lbs. of pork,
if pork be worth 66*. Zd. per stone of 8 lbs. ?
In applying the Chain Eule to this question, we first set down the
direct demand — How many jpeiice = 20 lbs. of bacon ? — which may be -wTitten
briefly thus : ? pence = 20 lbs. bacon ; then we set down a given quantity
of bacon as equivalent to a given quantity of something else : thus, 15 lbs.
bacon =14: lbs. cheese; then another given quantity of cheese as equiva-
lent to something else: thus, 35 lbs. cheese=4:6 lbs. pork; then another
^iven quantity of pork as equivalent to something else : thus, 8 lbs.
pork = 75 pence. These equations should be placed in successive lines,
as follows :
? pence =20 lbs. bacon,
if 15 lbs. bacon = 14 lbs. cheese,
35 lbs. cheese = 46 lbs. pork,
8 lbs. pork = 75 pence ;
where it may be observed that the first and last quantities in the state-
ment are of like denomination, viz, pence, and that the second side of an
equation is always of the same kind and denomination as the first side of
the next equation. The answer for the term of demand {? pence) will
now be found by di^-iding the continued product of the right-hand num-
bers by that of the left-hand numbers. Thus :
!^^ xXV<.46 xXS = 5 X 46 = 230^. = 195. 2d. Ans. 1"
X ^
The reason of the equating and calculating processes will be evident
if we employ unity to express the antecedent of each condition ; thus :
? pence = 20 lbs. bacon,
if 1 lb. bacon = ^ lb. cheese,
1 lb. cheese = If lb. pork,
I lb. pork = "^-^ pence ;
CHAIIf BULE. 119
for now it is obvious that 20 lbs. bacon = i^x20 lbs. cheesG = ^xi^
X 20 lbs. pork = '/ x ^ x 1* x 20 ponco.
The most important application of tho Chain Rule belongs to •what is
called Arbitration of Exchange. — See Note XI.
' Ex. 69.
1. If 10 first-class labourers do as much work per hour as 12 second-
class, 14 second-class as much as 16 third-class, 18 third-class as much
as 21 fourth-class, what number of the first class corresponds to 8 of
the fourth ?
2. When 94i Dutch florins is the exchange for 100 Austrian florins,
and 16 sovereigns are given for 193^ Dutch florins, how many Austrian
florins should be given for 28 sovereigns ?
3. How many lbs. of tea are equivalent to 10| lbs. of butter, when 5
lbs. of tea are equivalent to 14 of coffee, 9 of coffee to 20 of sugar, 10 of
sugar to 6 of cheese, and 10 of cheese to 9 of butter?
4. If 8 sacks of flour be equal in value to 13 loads of straw, 3 sacks
of flour to 10 sacks of potatoes, 27 sacks of potatoes to 26 cwt. of rice,
and 18 bushels of oats to 5 cwt. of rice, how many loads of straw aro
worth as much as 10 bushels of oats ?
5. If 16 pears be equal in price to 25 apples, and 18 oranges equal
to 12 pears, and 20 lemons equal to 27 oranges, and lemons cost Id^. a
dozen, what is the cost of 15 apples ?
6. How many yards of velvet are equal in value to 60 of muslin,
when 25 of muslin are equal to 16 of calico, 21 of calico to 13 of flannel,
40 of flannel to 27 of linen, 68i of linen to 28 of silk, and 47 of silk to
35 of velvet?
7. How many pounds sterling will be the value of 1000 rupees, when
15 rupees aro worth 7 American dollars, 5 dollars worth 26 francs, and
101 francs worth £4?
8. If 4 quarters of oats be worth 3 quarters of barley, 14 quarters of
barley worth 11 quarters of wheat, 27 quarters of wheat worth 32 bags
of rice, 24 bags of rice worth 67 sacks of potatoes, and 2 sacks of pota-
toes weigh 3 cwt., what quantity of potatoes is equivalent to 63 bushels
of oats ?
9. When ^ of a lb. of tea is equal in value to ^ of a stone of mutton,
and § of a stone of mutton equal to 3 lbs. of coffee, and ^ of a lb. of
coffee equal to | of a lb. of beef, how many lbs. of beef are equivalent to
20 lbs. of tea?
10. If an ounce troy of standard silver, of which 37 in 40 parts of
the whole are fine, be worth 5s. 1^., and copper worth 5 guineas per
CTi't., what is the ratio of the value of fine silver to that of copper ?
q2
120 SQUARE EOOT.
106. Square Root. — The square root of a given number
is that number which, when multiplied by itself, produces
the given number. Thus, the square root of 49 is 7,
because 7x7=49.
The sign of the square root is v/, a corrupted form of the
initial letter of the Latin word radix, root ; thus we write
>v/49 = 7.
Few numbers, comparatively, are perfect squares ; as
may be seen by the intervals of the numbers 1, 4, 9, 16, 25,
36, 49, 64, 81, which are the squares, respectively, of 1, 2,
3, 4, 5, 6, 7, 8, 9, and which indicate that every perfect
square must have 1, 4, 5, 6, or 9, as its last significant
figure.
1Q7. ITow, as the square root of 49 is 7, because 7 x 7=49,
so the square root of 186624 is 432, because 432x432
=186624; but, while simply from recollection of the ordi-
nary Multiplication Table it is easy to teU what is the
square root of 49, a process somewhat complex is requisite
to extract from 186624 the square root of that high number.
We proceed to exemplify the method of extracting the
square root of a large number, referring for proof of the
method to the chapter on Involution and Evolution in
Colenso's Algebra.
Ex. 1. Extract the square roots of 186624, 77841, 9G596G4.
186624(432 'i7^n{279 ^659664(3108
16 4 2_
83)266 47)378 61) 65
249 329 61_
862)T724 549) 4941 6208) 49664
1724 4941 49664
Here we first place a dot over the last figure, and then over every
second figure, reckoning from it ; by which means the number will be
divided into periods, as they are called, consisting each of two figures,
except the first, which (when the number of figures in the given number
is odd) will evidently consist of only one figure.
We then take the nearest square n^ not greater than the first period:
this is 16 in the first of the above instances, and we set its square root, 4.
SQUARE HOOT. 121
as the first figure in the root ; we then subtract its square, 16, and bring
ilown the next period, 66.
We now set the double of the first figure in the root, 8, in a loop, as
divisor, to the left of the rem', regarding it, however, as standing for 80,
not for 8, since we shall presently have to set another figure after it.
.Dividing the rem' by this div', 80, we set the quotient, 3, as the second
figure both in the root and also in the div' : then, multiplying the 83 by
3, we subtract the product, and take down the remaining period, 24.
To form the next div', we double the last figure of the preceding
one, making 86, which (as before) we regard as 860, and proceed
exactly in the same manner : and if finally, as here, we find there is
no rem', we may conclude that we have found the exact square root.
In the 2nd instance, notice (i) that the second rem', 49, is greater
than the div, 47 j this may sometimes happen, but no difficulty can
arise from it, as it would be found that, if instead of 7 we took 8 for
the second figure, the subtrahend would be 384, which is too large :
And (ii), that the last figure, 7, of the first div, being doubled in order
to make the second diV, and thus becoming 14, causes 1 to be added to
the preceding figure, 4, which now becomes 5.
In the 3rd instance, we have an intermediate cj'pher in the root.
Ex. 2. Extract the square roots of 1000, 2, 1.6, .002.
Vl000.00 = 31.6&c. a/2 = 1.41&c. Vi.60= 1.26&C. v/.0026 = .0147&C.
9 I I 2^
61)100 24) 100 22) 60 84) 400
61 96 44 336
626)3900 281) 400 246) 1600 887) 6400
3756 281 1476 6209
144 119 124 ToT
In the 1st instance, we find there is a rem', 39, when we have made
use of the last period of the given number, 1000 ; but we may continue
the operation as long as we please in such a case, by setting a decimal
point after the given number, and annexing cyphers as decimal places ;
and for every period of two cyphers thus formed we shall obtain a
decimal figure in the root.
The same is true of the 2nd instance, except that we have not taken
the trouble to set down the extra cyphers at the end of the given number,
though we have taken them down as required, and set the decimal point
in the root.
In the 3rd instance, it is to be noticed that the first dot must always be
placed on the last figure of the Integral part of any number, i.e. on the
one next before the decimal point, and then on every second figure on
each side of it. Of course, in the 4th instance, the figure next before
G 3
122 SQUAEB ROOT.
the decimal point, though not expressed, is 0. And, in both these, we
have had to annex one cypher to the original number, to complete ite
points.
In all such cases the square root can never be exactly obtained ; but
by annexing cyphers, it may be ascertained to as many places of deci-
mals as we please. Such roots are called irrational or surds.
Ex. 3. Extract the square roots of 1^^, ^1, 1 , and - .
^ 289 64 12 7
(i ) 169_13x 13 . /169_ a/169_13
^ "^ 289 17x17' ** Ay/ 289 V289 if
Or, /?^= A/.578125 = . 760345 f.
'V 64
(iii.) /Z-= /ll = ^ = i!M^ =.76376 + .
^ '' a/ 12 V 36 6 6
Or, /I = V.583333.. = .76376 + .
'V' 12
(iv.) /^= /LS^^ = ^-^^^"»- =.845154.
35__ a/35 _ 5.91608-
49 7 7
°'Vr
A/.71428571+ =.845154.
In the 1st instance, the given fraction is a perfect square, and its root
is found by extracting separately the roots of num' and den'. Observe
that the square root of a proper fraction is always necessarily greater
than the fraction.
In the 2nd instance, the den' only is a perfect square, and we may
either proceed as in the 1st instance, or reduce the given fraction to a
decimal, and then seek the root.
In the 3rd instance, neither num' nor den' is an exact square, but if
we multiply both by 3, we shall have the latter an exact square, and
may then proceed as in the 1st instance. Otherwise, we may first re-
duce ^ to a decimal.
In the 4th instance, we proceed as in the 3rd.
Ex. 70.
Extract the square roots of —
1. 5329 and 8836.
2. 34225 and 137641.
3. 531411 and 350164.
CUBE ROOT. 123
4. 95481 and 249001.
6. 348100 and 6512490000.
6. 37491129 and 16949689.
7. 3534400 and 65561409.
8. 99960004 and 24088464.
9. 119550669121 and 368451428004.
10. 8, 20, and 363.
11. 35120 and 8837.
12. 134909.29 and 650506.7716.
13. 6663.114 and 27.773.
14. .225 and 51.12965025.
15. .012012 and .00158404.
16. .000082355625 and .021.
17 JL2JL 18 nnrl HH
18. ^, ^^> and 3^.
19. 2871, 6136i, and 367f.
20. A, '^^, and 3-jL^,
303' .155 ' ^ 6-f
21. Ifof (41 + 51), and 1+i-l + i-i.
22. Ho-w many links in length is a square field containing 8 ac. 2 ro.
9 pa?
23. Find the length of a square having the same area as a rectangle
43 ft. 5 in. long and 34 ft. 7 in. broad.
24. What sum of money must be divided among A, B, C, so that A
may have 65. and C 9s. ^\d., and that B may have as much per cent,
more than .4 as C has more than B ?
108. CcBE Root. — The cube root of a given number is
that number which, when multiplied by its square produces
the given number. Thus, using %/ as the sign of the cub6
root, we have 1/ 512 = 8, because 8 x 8 x 8=512.
The first nine numbers are the respective cube roots of
1, 8, 27, 64, 125, 216, 343, 512, and 729.
The method of extracting the cube root of a large number
is mucb more complex than that required for the square
root, as will appear from tbe following example. A proof
of the method will be found in the chapter on Involution
and Evolution in Colenso's Algebra.
12i
PROBLEMS.
Ex. Extract the cube root of 80677o68161.
A!/8067?56816i = 4321
128
1202
12061
4800
369
5169
16677
15507
551700
2584
557284
1170568
1114568
55087200
12961
56000161
56000161
56000161
Here -vre first
divide the number
into periods by
placing a dot over
the last figure, and
then over every
third figure begin-
ning from it. Then
we take the nearest
cube n** not greater
than the first peri od,
80; this is 64, and
we set its cube root, 4, as the first figure in the root; then, subtracting
its cube, 64, wc l)ring down the next period, 677. AVe now set the
triple of the first figure of the root, 12, at some distance fo the left of
the rem""; (there is 123 in the sum, but the 3 will be accounted for by
and by ;) then we multiply this triple by the first figure of the root, and
place the product, 48, between 12 and the rem'", annexing two cyphers
to it.
We now divide the rem' by this 4800, and set the quotient, 3, as the
second figure in the root, and also after the 12, making 123 : now wo
multiply this 123 by 3, the second figure in the root, set tlie product, 369,
under 4800, add them up, multiply the sum, 5160, l^y tie second figure
in the root and subtract the product, 15507. We bring down the next
period, 568, and have now to form the two quantities to the left of it.
The first is obtained by tripling the last figure, 3, of 123, which gives 129
(the final 2 in 1292 will be accounted for when the next figure in the
root is found) ; and the other quantity, 5547, is found by adding 9, the
square of the second figure in the root, to the two preceding middle
369
lines
5169*
We now add two cj-phcrs, and repeat the whole process
described in this paragraph.
The remarks made above with respect to surd square-roots apply
also to cube-roots : thus, .01, 24.1 would bo pointed for the cube-root
.010, 24.100.
Ex. 71.
Find the cube roots of—
1. 185193 and 405224.
2. 21952 and 6859000.
3. 4330747 and 35287552.
4. 94818810 i:nd 959530803000.
rROBLEMS. 125
5. 529475129 and 111123515328.
6. 261775532773 and 176369715712.
7. 357759791.299 and .050243403.
8. 60000 and 527.71.
9. 80i7 and 5678.9.
10. g and 30}.
11. A box is 3 ft. 5 in. long, 1 ft. 8 in. wide, and 14 inches deep.
Required the edge of a cubical box of the same capacity.
12. The yolumes of spheres are to one another as the cubes of their
diameters. If, therefore, the Sun be l^ million times as large as the
Earth, and the Earth's diameter be 7912 miles, hov many miles will the
Sun's diameter measure ?
126
MISCELLANEOUS EXAMPLES.
1. The circumference of a cocach-wheel being 16| ft, how often "will
it turn round bot-ween London and Oxford, a distance of 59 miles ?
2. If a person's estate produce £400 a year, and the land-tax be
assessed at 25. Qd, in the pound, what is his net annual income ?
3. Eeduce ^^ to its lowest terms, and £1 155. 6d. to the fraction
of a guinea ; find the value of £^ of half-a-guinea, and add together |,
I of 15, If, and 3 -^2§.
4. Divide 2U guineas equally among 12 men.
6. What is the rent of 145a. 1r. 32p. of land, at £10 5s. 3d. per
acre?
6. The produce of a farm one year was 150 quarters, which were sold
ftt 58s. a qu. ; in the next year the price of wheat fell to 485., but the crop,
being plentiful, produced on the sale the same amount as before : of how
many quarters did the second crop consist ?
7. A straight plank is 3^ in. thick, and 6^ in, broad ; what length
must be cut off so as to contain 65 cubic feet of timber?
8. A person holding 50 shares in the London and North-Western
Eiiilway, sells out at 170 ; what income would ho have by buying into
the 3i per cents, at 93^ ?
9. ' If 5 lbs. of tea be worth 12 lbs. of coffee, and 7 lbs. of coffee worth
20 lbs. of sugar, and 14 lbs. of sugar worth 75. l^., what is the worth
of 9 lbs. of tea?
10. A common pasture containing 54a. Zn. 35|p., another containing
39a. 13|p., and a third containing 54|a., are to be divided into 60 equal
parts, after deducting from the whole 11a^23b. for tithes ; of how much
does one part consist ?
11. Find the square root of 370881, and the side of a square con-
taining 7367 sq. ft. 52 in.
12. If the produce of wheat be tenfold of the seed, how many
quarters can be obtained from one grain in 1 0 years, supposing there to
bo 7580 grains in a pint?
13. If I lose lid. in 35. id., how much do I lose per cent. ?
14. In the centigrade thermometer the freezing point is zero, and the
boiling point is 100°; in Fahrenheit's the freezing point is 32°, and the
boiling point is 212° ; what degree C. corresponds to 68 F. ?
15. How much water must be added to a cask, containing 40 gallons
of Spirits at 135. 8c/., to reduce the price to 105. 6d. ?
MISCELLANEOUS EXAMPLES. 127
16. A bill for £100 has six months to run, and the holder has it dis-
counted at 5 per cent., and receives £97 105. ; how much less than his
due does he receive ?
17. Find the value of § of a guinea ; reduce 2s. S^d. to the fraction
of a pound, and 1 hr. 7| min. to the fraction of 1 da. 6 hrs.
18. A person invested £1000 in the 3 per cents, at 90| ; but the price
rising to 9I5, he sold out, and invested the proceeds in the 3| per cents,
at 975 : find the increase in his income.
19. Find the square root, and also the cube root, of 95951|fi.
20. A general levies a contribution of £870 on four villages, con-
taining 250, 300, 400, and 500 inhabitants respectively ; what must they
each pay ?
21. -4 can do a piece of work in 10 days, which B could do in 13 ; in
what time would they do it together ?
22. A stationer sold quills at lis. a thousand, by which he cleared |
of the money ; what would he clear per cent, by selling them at 135. 6d.
a thousand ?
9a ■Rfidnno 3872_ 1 7 J_ + jL 4. 1 4.4.II 9l3_17 3 ^.f B ^ ^ ^.f 11 nf 2 1
zd. J^.eauco y^HoJ' -^'12 + 15 ^^**2l' -^35 25' i ^^ I'^TE "^ H "^ aS'
and 6347 ■T-2|, to their simplest forms.
24. Di^nde the value of 79 florins between A and B, giving A half-
a-crown more than B.
26. Three persons rent a piece of land for £60 10s. ; A puts in 5 sheep
for 4i months, B, 8 sheep for 5 months, and C, 9 sheep for 6| months :
what must each pay of the rent ?
28. What is the present worth of £75, due 15 months hence, at 5 per
cent. ?
27. If A can do a piece of work in 10 days, and A and B can do it
together in 7 days, in what time would B alone do it ?
28. Find the cube root of 133354510.
29. Divide £16 0$, lOd. among 4 persons in the proportion of the
fractions i |, i, i.
30. Divide 1037 into two parts, which shall have to one another the
ratio of the sum of 7.625 and 5.375 to their difference.
81. A cistern has two pipes, by one of which it may be filled in 40
min., and by the other in 50 min. ; it has also a discharging pipe, by
which it may be emptied in 25 min. If all these three were open toge-
ther, in what time would the cistern be filled ?
32. There is a number which, when divided by § of f of 1|, will pro-
duce 1 ; find its square.
83. If a person lend me 1296 guineas for 125 days, how long should
1 lend him £1620 to requite the flivour?
84. Find the square roots of 9.21677 and 921677.
35. If 6 men will dig a trench, 15 yds. long and 4 broad, in 3 days
128 MISCELLANEOUS EXAMPLES.
of 0 hours each, in how many days of 8 hours each will 8 men dig a
trench 20 yds. long and 7 broad ?
36. Reduce 1 Ss. lid. to the decimal of a pound, and f of Is. 6^1. to tho
fraction of half-a-crown ; divide 1001 by 390625, .1001 by .000390625,
and 10.01 by 390.625.
37- The cost price of a book is 35. 9(f. ; if the expense of sale be 6
per cent, upon this, and the profit 24 per cent., what would bo the retail
price?
38. If the Sun moves through 360^ in 365 days 5 hrs. 48 m., how
many minutes and seconds will ho pass through in a day?
39.* Divide £15 among 10 men, 13 women, and 25 children; each
man to receive twice as much as each woman, and each child half as
much as each woman.
40. There is a fraction which, when multiplied by the cube of U, and
divided by the square root of Ij, produces | ; find it.
41. A floor, 24 ft. 4 in. broad and 96 ft. 6 in. long, is to be laid at
\yi: per square foot ; find the cost.
42. A sells to i? I of i of f of 30 sheep for ^ of jf|r of f of £210 ;
"what was tho average price of Ciich sheep ?
43. The estate of a bankrupt, £21000, is to be divided among four
creditors, whose debts are, ^'s to Fs as 2 : 3, iTs to Cs as 4 : 5, C's to
Z>'s as 6 : 7 ; what must each receive ?
44. A cubic foot of water weighs 63 lbs. ; what is the weight of water
in a vessel 1 ft. deep, 16 ft. 7 in. long, and 8 ft. 4 in. wide?
45. The profits of a mine for one year amounted to £3296 135. b\d.,
and a person holding 14 shares received for his dividend the sum of
£1025 125. lid. ; how many shares were there in all?
4G. If the price of gold be 4 guineas an oz., what is tho cost of a gold
ornament weighing 3 oz., of which 18 parts out of 24 are pure gold ;
allowing 35. 4d per oz. for the value of alloy, and 25 per cent, upon the
wliole for expense of workmanship ?
47. Find the square roots of .064 and 26.123456790.
48. What is the price of a piece of timber, of which the length,
brc'adth, and thickness are respectively 23 ft. 9 in., 2 ft. 4 in., and 2 ft.,
at ^\d. per solid foot?
49. If 90 degrees correspond to 100 French grades, how many
degrees and how many grades are there in the sum of 36.45 degrees,
and 36.45 grades?
50. A man can reap 302^ square yards in one hour j in what time
■will 3 such men reap 2| acres ?
61. A fiirmer gave for a horse a bill of £156, due 8 months he&ce, at
4^ per cent., and sold him at once for £180 ; required his gain per cent-
02. A can do a piece of work in 3 days, B can do thrice as much in
MISCELLANEOTJS EXAMPLES. 129
8 days, and C five times as much in 12 days : in what time would they
do it together?
63. If a tradesman marks his goods 20 per cent, above the cash price>
what ready money would he take for an article marked 265. ?
54. If 6 men can earn £20 in 21 days, when the days are 12 hrs.
long, how much can 4 men earn in 35 days, when the days are 10 hrs.
long?
55. If 45 bricks will pave a square yard, how many will be wanted
for a space 34 ft. long and 14 ft. wide, allowing for a path, 2 feet wide,
all round ?
56. Reduce Z\s. to the decimal of ^\ of a guinea ; and find the values
of .232 of a cwt., and 4.0m of a mile.
57. A gentleman had 5 sons, to whom he left £3750 in cash, and
two bills of £151 each, due at the end of two and three months respec-
tively; the eldest son had by the will \ of the property, and, taking
charge of the whole, he paid the others their shares, which were equal, in
cash. "What would these be, reckoning interest at 4 per cent. ?
68. Find the sq. root of 39.0625, and the cube root of 2116.874304.
59. What is the annual interest on £76978, bought into the Danish
3^ per cents, at 77 ? and what sum would be gained by selling out
at 771?
60. It is desired to cut off an acre of land from a field 1 S^p. in
breadth ; what length must be taken ?
61. . Express a degree (69j^ miles) in metres, when 32 metres are
equal to 35 yds.
62. At 9|«?. per sq. yd. what is the cost of painting a room which is
24 yds. round, and 10 ft. 4 in. in height?
63. Find the difference between V| and y\.
64. What is the alteration in income made by transferring £10000
from the 3 per cents, at 92 to the 4 per cents, at 110 ?
65. Divide 4i into two parts, one of them to be 4^ times the other.
66. A plate of gold, 3 in. square and |*in. thick, is extended by
hammering so as to cover a surface of 7 sq. yds. ; find its present
thickness.
67. I bought 171 gallons of brandy in bond for £79 35. 4(f., and on
taking it out paid duty equal to 112i per cent, of the bonded value;
what was the duty per gallon ?
68. Compare the interest on £350 at 4| per cent., with the interest
on £450 at 3| per cent., for one year.
69. The dfiy's journey in Turkey being 10 hours, of 4| English
miles each, and the proportion of an English to a Roman mile being
12 : 11 nearly, how many Eoman miles are there in 13 days' journey in
Turkey?
130 MISCELLANEOUS EXAMPLES.
70. A dramng-room, 36 ft. 10 in. long and 23 ft. 2 in. wide, is sur-
rounded with a cornice 3^ in. wide, the gilding of which cost £4 1 Is. 10i<^. ;
how much was that per square foot ?
71. A steward receives for his landlord £1987 of rent, and disburses
one-fifth ; he pays his landlord £195 125., and the renminder is invested
in an estate at 30 years' purchase : find the rent of the estate.
72. Eeduce ^ of half-a-crown to the fraction of half-a-guinoa,
and 65. S^d. to the decimal of a £ ; find also the value of f of ^ of
£6666 135. ^d.
73. What is the yearly interest on £1127 bought into the 4 per cents.
at 92?
74. Find the value of £1368 7s. 5d. sterling in dollars and cents, a
dollar being equal to 100 cents, and to 45. 4d. English money.
75. A sum of £333 35. 3ld. is to be divided among 4 persons, whoso
shares are to be in proportion as 1, 2, 3, 4 ; find the share of each.
76. The circumference of the Karth in the lat. of London is 15120
miles ; find the disti\nce between two successive meridians of longitude,
and the space passed over by the Sun in his apparent daily motion in a
minute.
77. If a person accepts £247 Is. 8d. as present payment of £252 O5. 6d.
due four months hence, at what rate per cent, does he allow discount?
78. Divide 135. l^d. into six parts, each succeeding part to be 6|c?.
more than each preceding.
79. How much stock, at 93i per cent, can be purchased for £540, a
commission of ^th per cent, being charged on the stock purchased?
80. If either 5 oxen or 7 horses will eat up the grass of a close in 87
days, in what time will 2 oxen and 3 horses eat up the same?
81. The sum of £3 135. 6d. is to be divided among 21 men, 21 women,
and 21 children, so that a woman may have as much as two children, and
a man as much as a woman and a child ; what will each man, woman,
and child receive ?
82. A sells to 5 I of f of ^ of a package of tea, which weighs f of |
of 1 cwt. 21 lbs. at 35. 6d. per lb. ; what did it come to?
83. How many revolutions will a carriage-wheel, whose diameter is
a yard, make in a mile, the ratio of the diameter to the circumference
being 1 : 3.14159?
84. A cistern can be filled by two pipes, A and B, \h 4 miii. and
6 min. respectively, and emptied by C in 2f min. A is opened for
2 min., and theft A atid B together for 1 miii. more, when C is also
opened. In what time would the cistern, 'Vvhich now contains 861 gals.,
be full ? and how many gallons would have passed through A and Z?
Respectively ?
85. What is the yeariy interest on £27225, bought into the 3i per
cents, at 97^ ?
MISCELLANEOUS EXAMPLES. 131
86. Express in its simplest form if— y| + if— li ; and add together |
of a guinea, ^ of a crown, and ^ of 75. 6d., and reduce the result to the
decimal of 165.
87. Find the simple interest on £325 165. Sd., for 5 months, at 4|
per cent.
88. If 18 men eat 165. worth of bread in 3 days, when wheat is at
645,, what value of bread will 45 men eat in 27 days, when wheat is
at 465. ?
89. "What length of paper, 22^ in. broad, will be used for a room
21 ft. 9^ in. long, 15 ft. 7 in. broad, and 8 ft. l^in. in height? and what
will it cost at l5. Zd. a yard?
90. Find the value of 36.42 tons of coal, at 175. 7\d. per ton ; and
the difiference between i x lu x H ^^ 1^^-' ^^^ ? of | of £3 II5. Od.
91. The 3 per cents, are at 85|; what price should the 3§ per
cents, bear, that an investment may be made witli equal advantage
into either stock ? And what income would be derived by eo investing
£5000?
02. A farm lets for £92 per annum : the tenant pays for 2 years'
occupation, with interest accumulating at 5 per cent. ; the landlord
pays 5 the amount for repairs of house, g of this for repairs of barn, and
£2 35. id. for other expenses : find the balance.
93. Wliat will bo the cost of painting a room at d^d. per square yard,
if the sides are each 19 ft. 10^ in., the ends 16 ft. 1| in., and the height
10 ft. 3 in. ?
94. Express 1618| Eng. miles in degrees (a degree = 69^ miles):
find the values of f of £2 75. 8^., and of {^ of £1 65. 8c/., and reduce
their difference to the decimal of £20.
95. Twenty-six wedges of gold, weighing in all 33 lb. 3 oz. 7 dwt.
4 gr., are to be coined into sovereigns : find the weight of each wedge,
and the number of sovereigns coined from the whole, at the rate of 3^^
sovereigns per oz.
96. How many feet in 150 must a road 10798 feet long rise, to bo
carried from a plain to a hill 463 feet in perpendicular height ?
97. A gentleman selling a mortgage of £4410, for which he received
6 per cent, interest, bought into the 3^ per cents. Bank Stock at 70 ;
after receiving the interest for 5 years, on the stocks rising to 76, he sold
out. What was his gain upon the whole transaction, over what he would
have received had he continued the mortgage ?
98. What is the present worth of £325 I65. 8d., due at the end of 5
Inonths, at 4i per cent. ?
99. Find the square roots of 6242^ and 1438.^37, and the cube roots
of .000328509 and 27054.036008.
100. If 40 men in 7? days cah dig 3 rectangular fields, each 160 ydi.
132 MISCELLANEOUS EXAMPLES.
by 130; how long will 37 men bo digging 5 fields, each 129| yds.
hy 90 ?
101. If 3 men, 5 women, or 8 children, could do a quantity of work
in 26i hours, in what time will 2 men, 3 women, and 4 children com-
plete it ?
102. A person, leaving Paddington at 13 minutes Lefore 2, p.m.,
travels the first 162 miles at 27 miles an hour, the next 121 miles at 9|
miles an hour, and the last 27 miles at 8 miles an hour : when will he
reach his destination, Penzance ?
103. How many square yards are therein a parade, 864 ft. 3 in. long
and 62 ft. 6 in. broad ?
104. A met two beggars, B and C, and, having in his pocket
(3^-7-4f ) of (lOf -^7^) of ^ of a moidore (275.), gave Z? i of | of that
sum, and C § of the remainder ; what did each receive ?
105. What is the present worth of £1147 105., duo 3 years hence, at
41 per cent, simple interest ?
106. A and B entered into partnership : A put into stock at first
£2000, and at the end of 8 months £1000 more ; B put in at first £750,
and at the end of 4 months £3000, but took out £1300 at the end of
3 months more. At the year's end they had gained £1635; what should
each receive ?
107. Allowing that 44| guineas weighed a lb. Troy, when 32 half-
pennies weighed a lb, Av., and observing that a lb. Av. contains 7000 gr.
Troy, what was the difference in grains between the weights of a guinea
and half-penny ?
108. How much stock must be bought at 88 per cent., in order that,
by selling out when the stocks are at 90, 20 guineas may be gained?
1 09. A bankrupt pays 3|c?. in the pound, and the total of his payments
amounts to £154; what was his debt?
110. A person has £18752, for which he is receiving 3| per cent,
but spends annually £27 more than the whole original interest ; what
has he at the end of 3 years ?
111. If £100 be placed at interest at 5 per cent, and the interest be
added to the principal every 20 years, in how many j-ears will it amount
to £1000?
112. The prime cost of a 50-gall. cask of wine is £25, and 10 gall,
are lost by leakage ; at what price per gall, must the remainder be sold,
so as to gain 10 per cent on the whole original cost?
113. To do a certain piece of work A by himself would require 16
hours, jB 18, C20. Suppose that after A and B working together for
6 hoxirs, and then B and C for 3 hours, the remainder of the work is
left for C to finish, in wb.r.t time would he finish it?
MISCELLANEOUS EXAMPLES. 133
114. If the Ccarriago of 60 CAvt. for 20 miles cost £14l what can I
have carried 30 miles for £ojo ?
115. Find the side of a square whose area equals 14 sq. ft. 11 in.
116. A and B engage in a speculation, and di-vnding the proceeds
of it, A took £57 185., and B £29 145., as their respective portions;
what sura did each layout, it heing known that A paid £7 I6s. 8d. more
than^?
117. A person had £2950 in the Danish 3 per cents., at 7o\, which
he transferred to the Russian 5 per cents., at 11 Of; required the altera-
tion in his income.
118. Extract the square root of .009059 and of 464ff, and the cube
root of .578?03.
119. If 7 oxen are worth 64 sheep, and 3 sheep cost £5 125., what
must be given for 100 oxen ?
120. A person buys teas at 35. and 45. the lb., and mixes them in
the proportion of 4 : 7 ; what will he gain per cent. l)y selling at 35. Qd.
per lb. ?
121. Find the difFt^rence between the simple and compound interest
on £150 in 3 years, at 4^ per cent.
122. If 5 men can reap a field, in length 800 ft. and breadth 700 ft.,
in 3i days of 14 hours each ; in how many days of 12 hours each will 7
men^reap a field of 1800 ft. by 960 ft. ?
123. Three soldiers, A, B, and C, divide 770 cartridges in the follow-
ing manner: as often as A takes 4, B takes 3 ; and as often as A takes
6, C takes 7 : how many will each have ?
124. If £100 in 2 years gain £12 interest, what principal will gain
£6 155. in 4i months?
125. A person desires to exchange 25 Spanish £100 bonds, and
£800, Z\ per cent. Stock, for 3 per cent. Consols ; the prices of these
securities being 48, 99, 93| respectively, what quantity of Consols can
he obtain ?
126. A person buys three estates of 56, 67, and 71 acres, and gives
£81 35. 6(f., £92 45. %d., and £109 35. 2d. an acre for them respectively;
what should they produce annually to pay 15 per cent, upon his whole
outlay ?
127. If a beam which is 10 in. wide, 8 in. deep, and 5 ft. 6 in. long,
weigh 8 cwt. 1 qr., find the length of another beam, the end of which is
a square foot, which shall weigh a ton.
128. A and B have 185. and 125. respectively; and if A give B
2|-j-4| of the difference of 2^-^ 13^ of their respective sums, and \ of
2\ of As present sum be added to ^ of i of jB's, C» w.onay will be 1| of
this sum : find it.
120. What is the expense of carpeting a room, 28 ft. long and 19 ft.
wide, with carpet ^ yd. wide, at bs. 9d. a yard ?
VS4i MiSCELLANEOtTS EXAMPLES.
130. A person transfers £2tH)0 sterling from the 3| per cents, at 99,
to the 3 per cents, at 86f ; what is the difference in his income?
131. Multiply £2 165. 10.75d. by 144.33, and divide £9753 Us. S^.
by 234.5.
132. What would be the purchase-money for an estate producing a
rental of £3228 35. 4cf., at the rate of 8| per cent. ?
133. What will be the expense of glazing a hall- window contain-
ing 60 sqxiares, each 1 ft. 3 in. long, and 11| in. wide, at Is. 10c?. per
sq. ft. ?
134. A lb. of tea and 4 lbs. of sugar cost 55. ; but if sugar were to risie
60 per cent, and tea 10 per cent., they would cost 65. 2d. Eequired
the prices of tea and sugar per lb.
135. If I buy 14 sheep for £39 65. b^d., and sell 6 of them at 365.
each, for what must the remainder be sold that I may gain 4 per cent,
on the whole ?
136. The weights of equal quantities of lead and cork are as 11.324
and .24 ; and 60 cubic inches of lead, with 54 of cork, weigh as much
as 1538|of fir: what number represents proportionally the weight of
fir?
137. By selling an article for 105., the seller loses 5 per cent. ; what
will be the loss or gain when sold for 125. ed., and what was its prime
cost ?
138. A puts out to interest £2000 at 4 per cent. ; he spends annually
£75, and adds the remainder of his dividend to his stock : what is he
worth at the end of 5 years ?
139. A country containiilg 711117 inhabitants increases to 732666;
find the increase per cenj^^
140. If 12 men can complete a piece of work in 15 days, working
6 hrs. a day, how many can do it in 85| days working 12i§ hrs. a day ?
141. A bankrupt has good debts to the amount of £456 185. Id.,
and the following bad debts, £360 75. lOd., £120 135., and £19 185., for
which he receives respectively 4, 5, and 9 shillings in the £ ; his own
liabilities amounted to £3408 125. : how much can ho pay in the £?
142. A had £2 135., and B, when he had paid A 6f-7-l§ of
£l ll5. 6d., found that he had remaining ^ of the sum which A now
had : what had B at first ?
143. Find the sq. roots of .0026009 and .0002404, and the cube root
off.
144. A rectangular cistern, of which the length is 13| ft. and the
breadth 6 ft., contains 29 4^ cubic feet of water ; what is the depth of
the cistern, and what is the weight of water when one cubic inch weighs
262.6 grains?
145. At what rate per cent, of simple interest will £1 become a
guinea in 6 years?
MISCELLANEOUS EXAMPLES. 1S5
146. How mucli will a broker, who charges 5 per cent, discount,
give for a bill for £600 due at 2 months ?
147. Hiding a journey of 27 miles into town, I meet the coach which
left town at the same moment that I started from hence (viz. 7 o'clock),
at the 15th mile-stone from town. Supposing that it travels 10 miles an
hour, find the hour when we meet, and the time when (proceeding at
the same rate as before) I shall reach London?
148. If 12 casks are carried 18 miles for £^6 when the carriage is at
Is. Sd., how far ought they to be carried for £7^N<^hen the carriage is at
lOd.?
149. Add together | of f of a guinea, ^ of a pound, and 3^\ of
145. 8d.; and reduce the sum of l-^3| of half-a-guinea and 3-5-3| of
155. 6d. to the decimal of a pound.
150. What number of lbs. of tobacco, at the same number of pence
per lb., amounts to £16 10s. dd. ?
151. A manufacturer employs 50 men and 35 boys, who work re-
spectively 12 and 8 hours a day during 5 days of the week, and half-
time the other day ; each man receives 6d., and each boy 2d., an hour.
What is the whole amount of wages for a year ?
152. A man buys 27 sheep for £30, and sells 12 of them, so that he
loses 3 per cent, in the sale ; at what price per sheep must he sell the
remainder, so that he may gain 2| per cent, on the whole purchase ?
153. Two persons buy respectively, with the same sums, into the 3
and 3| per cents., and get the same amount of interest ; the 3 per cents,
being at '7 5, at what are the 3^ per cents. ?
154. Find the present worth and discount on £226 Is. lid., due 7
months hence, at 4| per cent.
155. Three tons of merchandise cost £26 15s. 5d.; at how much
per cwt. must it be sold so as to gain 20 per cent. ?
156. Divide 3i guineas among 6 persons, so that their shares may be
in the proportion of the reciprocals of the first 6 units.
157. Divide 999 into three parts, so that 6 times the first, 7 times
the second, and 11 times the third may be equal.
158. Half the trees in an orchard are apple trees, a fourth pear trees,
a sixth plum trees, and there are besides 50 cherry trees ; how many
trees are there altogether ?
159. A banker borrows money at 3| per cent., and pays the interest
at the end of the year : he lends it out at 5 per cent., but receives the
interest half-yearly, and by this means gains £200 a year : how much
does he borrow ?
160. By selling tea at 25. 8d. a pound, a grocer clears |th of his
outlay ; he then raises the price to 35. : what does he clear per cent,
upon his outlay at the latter price?
136 MISCELLAiJEOtJS EXAMPLES.
161. How much tea, at 25. i^d., must I give for 28 lbs. of sugar, at
4|^., so as to gain 5 per cent, by the exchange?
1 62. Eeduce —^ to its lowest terms, and ~ to a decimal ; and add
together 2|, 3^, ^, and 1| ; and divide 2g of U of 1§ by 7^.
163. If 54.32 cub. in. of gold be as hea^^ as 101.36 cub. in. of silver,
how many oz. of silver are equal in bulk to 226j oz. of gold?
164. "What is the present worth of £131 12.s. 6d., payable in ^ of a
year, at 5 per cent. ?
165. The length of a street is 937 ft. 6 in., and its breadth 66 ft. 8 in.;
find the cost of paving it at 8hd. per square yard.
166. If 100 men, in 6 diys of 10 hours each, can dig a trench
200 yards long, 3 wide, and 2 deep, in how many days of 8 hours long
\rill 180 men dig a trench of 360 yards long, 4 wide, and 3 deep ?
167. A person spendin* annually £240, saves £2 ]5?. of it quarterly
by ready payment; what is the rate of discount? and if he by this
means makes an increase of 20§ per cent, upon his annual saving, what
was his annual income?
168. A certain sum of money was divided among three persons. A,
B, C. Suppose that ^'s share was £264 12s., and Cs £2 85., and that
A's share contained the value of B's as often as -B's share contained Cs ;
what must the whole amount have been ?
169. Add together 3^ of 24 of 7^ of a £, 9f of 3§ of a shilling, and
8^ of 4i of a penny, and divide the sum by ^i of ^ of | of S^d.
170. Extract the square roots of 2.054 and of 42.03361 ; and the
cube roots of 15.438249 and 629.422793.
171. If 6000 lbs. of iron are cast off at a foundry in 24 hours, how
many tons weight will be cast off in 308 days, supposing them to work
16 hours each day ? and if the price of iron be £3 3-9. per ton, what will
be the gain per cent, upon the annual expenditure, supposing it to be
£20 per week of 6 days ?
172. How must wine, which cost 16s. per gall., be sold, so as to gain
21i per cent. ? and how so as to lose the same?
173. The value of a pound of goM is 14 times that of a pound of
silver, and the weights of equal quantities of gold and silver are in the
ratio of 19 to 10 ; find the value of a bar of silver equal in bulk to £1750
worth of gold.
174. A, B, and C, together, can dig an acre of land in 7^ days. A
digs 32 perches in 6 days, and B 54 perches in 7 days. Find the three
lowest integral numbers expressing the comparative powers of these men ;
and the time in which C digs 17| perches.
175. Wliat is the prica of a silver cup weighing 1 lb. 10 oz. 12 dwt.
6 grs., worth 5*. an ounce ?
176. Divide the cube root of ^M^l^i by the square root of 260100.
MISCELLANEOUS EXAMPLES. 137
177. Eeduce 2 w. 2 d. 19^ hrs, to tho fraction of a month, and {^ of
a shilling + g of half-a-crovvn + ji of a guinea to the decimal of a £.
178. A fast train leaves Bristol for London, a distance of 120 miles,
at 2 o'clock, and travels at the rate of 25 miles per hour ; at M'hat time
must a luggage train, which travels at the rate of 15 miles in 50 minutes,
have left, so as not to be overtaken by the fast train ?
179. Find the commission on £126 at | per cent, and reduce tho
answer to tho decimal of £1 Us. 6d.
180. If, by selling lino Irish cloth at 5s. per 3'ard, I gain 8 per cent.,
what will be my rate of profit if I sell at 6s. 4:d. per ell ?
181. Add together the cube roots of .007301381 and 32768, and
multiply tho result by the sq.iare root of 72^
182. What ready money will discharge a debt of £528 Qs., due
4 months hence, at 4| per cent. ?
183. Find the least common multiple of 64,720,960 ; and find what
decimal 17 yds. 1 ft. 6 in. is of a mile, and what fraction of 3s. 6d. is §
of i| of 2s. 6d. ?
184. The 3 per cent, stock is at 98|, and the 3^ per cents, at 106^ ;
into Avhich is it most advantageous to buy ?
185. £1000 is to be divided among A, B, and C, so that for every £3
given to ^, Z? is to receive £5 and C £8 ; what sum had they each ?
186. Reduce 4^^^ lbs. Av. to Troy weight, and 3 cwt. 34 lbs. 2 oz.
to the decimal of a ton ; and .0975, .65, .5245, to their equiA'alent fr-ac-
tions.
187. The quantity of copper ore sold at Truro on a certain day was
3696 tons (of 21 cwt. each), and the produce 6| per cent. ; find the quan-
tity of fine copper obtained from it in common weight.
188. A rectangular parish, 6 fur. long and 4 fur. broad, is enclosed ;
a belt of plantation, 200 ft. wide, is carried the whole way round ; a
main road, 60 ft. wide, runs across the land in the direction of its length,
and a cross road, 41 ft. wide, in the direction of its breadth : how many
acres of field were there ?
189. If the sixpenny loaf weigh 51 lbs. when wheat is at 5|s. per
bushel, what must be paid for 52^ lbs. of bread when wheat is at 8s. 6d.
per bushel ?
190. Find the present value of £273 Os. 9^,, due 3 months henco, at
4i per cent., and tho compound interest on £105 in 3 years, at 3| per
cent.
191. A body of 7300 troops is formed of four battalions, so that ^ of
the first, I of the second, | of the third, and f of the fourth, are all com-
posed of the same number of men ; how many were there in each ?
192. Among the Jews the coin mina (or pound) was worth 50
shekels of silver, each weighing 219 grs. ; the weight mina, when of gold,
138 MISCELLANEOUS EXAMPLES.
weighed 100 shekels, when of silver, 60 ; what were the values of these
minse, rating gold at £i and silver at 5s. an ounce ?
193. A father left to the elder of his two sons if of his estate, and I?
of the remainder to the younger, and the residue to his widow ; find
their respective legacies, it being found that the elder son received
£l6d0 more than the younger.
194. Divide 240 into two parts, such that | of one added to ^ of the
other shall equal 36.
195. If 193 Eussian versts be equal to 205.9 French kilometres, and
1552.94 kilometres equal to 964.9 English miles, how many miles aro
equal to 100 versts?
196. If the rent of 2 acres for | of a year be £1 Zs. Sd., what will be
the rent of 547 acres for a half year ?
197. If I buy 3 per cents, at 78|, and ^ at 95y%, which is the best
investment? If I had invested £6962 19s. S^d. in each, and the former
rose and the latter fell ^, how much should I lose or gain ?
198. If 3 men can mow 7 acres of grass in 5 days of 9 hours each,
in how many days of 8 hours each will 5 men mow 17| acres ?
199. Add together 3^, 2j|, i and ^ ; find the difference of 3^ and
2f , and divide 3^ by 2§.
200. Five thousand copies are issued of a 6s. book : the cost of
printing is Is. per copy, of binding 4^., and of carriage, advertising,
&c., 2d. : the publisher disposes of them to the retail bookseller, charg-
ing 25 copies as 24, and 30 per cent, less than the selling price, and
upon the whole receipts takes 10 per cent, commission for himself: what
are the gains respectively of author, publisher, and bookseller on this
edition ?
201. Find the square root of ^, and the cube root of 352045.367981.
202. Find the discount on £1294 10s. for 1| year, and the interest
on the discount for the same time, at 4i per cent.
203. Divide 100 guineas into an equal number of guineas, half-
guineas, crowns, half-crowns, shillings, and sixpences, and reduce the
remainder to a fraction of a pound.
204. A person has £3500 to lay out ; the 3 per cents, are at 82|, and
the 3i at 96 : what would be his income from each ?
205. How many inches are there in the diagonal of a cub. ft., and
how many square inches in a superficies made by a plane through two
opposite edges ?
206. A merchant employs £700 in trade, and at the end of 3 years
takes another into partnership, who advances £1900. At the end of 4
years from this time they have gained £500; how ought this to be
divided between them ?
207. If 24 pioneers, in 2| daj^s of 12| hours long, can dig a trencli
MISCELLANEOUS EXAMPLES, 139
189.75 yds. long, 4| yds. wide, and 2i yds. deep, what length of trench
will 90 pioneers dig in 4^ days of 9§ hours long, the trench being 4| yds.
wide and Si yds. deep ?
208. What is the discount on £257 8s. S^^., paid 210 days before
due, at 4| per cent. ?
209. What is the cost of papering a room 15 ft. long, 12 ft. "wide,
and 10 ft. high, with paper 30 in. broad, at 7ld. per yard ?
210. The sum of £925 was so divided among A, B, C, and D, that
Z?'s portion was equal to ^ of ^'s, <7s was equal to | of Fs, and i)'s was
half as much as Bs and C"s together : what did each receive ?
211. A draper bought 5 pieces of silk, each 52 yards, at 45. Zyi. per
yard, and sold the whole so as to gain as much as 16^ yards were sold
for ; what was the selling price per yard ?
212. £100 stock, in the 3 per cents., is sold for £91 155. ; howmuch
can be bought for £540, allowing, for commission, | per cent, upon the
stock bought ?
213. A gentleman's income is £896 135. 4(?. per ann. ; he gives to
the poor quarterly £13 105., and lays up 200 guineas at the year's end :
how much does he spend in 6 days ?
214. A grocer buys 13 lbs. of tea at 25. Zd., 16 lbs. at 25. bd., and
18 lbs. at 35. Zd., and mixes them : at what rate per lb. must he sell the
mixture so as to gain on the whole 17| per cent. ?
215. AVhat is the present worth of £2035 15s., due in 2 yrs. b\ mo.,
at 4| per cent. ?
216. What is the expense of paving a rectangular court-yard, whose
length is 63 ft., and breadth 45 ft., it being paved with pebbles at Is. Qd.
per sq. yard, except a foot-path, which runs the whole length, 5 ft. 3 in.
broad, and is paved with flag-stones at 35. per square yard ?
217. A and B can do a piece of work alone in 12 and 16 days
rfespectively ; they work together at it for 3 days, when A leaves it, but
B continues, and after 2 days is joined by C, and they finish it together
in 3 days ; in what time would C do it alone ?
218. Find the value of 135II0 of 2 cwt. 2 qrs. ; and of f| of £8 85.
219. A can mow 2| acres of grass in 6| hours, and B 2\ acres in 5|
hours : they mow together a field of 10 acres ; in what time will they do
it, and how many acres will each mow ?
220. In making gold thread for embroidery, a cylinder of silver
weighing 360 oz. Av. is cased with one of gold weighing 6 oz. ; and this
mass is drawn through a series of circular holes, continually diminishing
in diameter, until it becomes so thin that 202 feet in length weigh one
dram : what is now the length of the thread ?
221. The gross weight of the Chinese silver, brought homo ia
140 MISCELLANEOUS EXAMPLES.
January 1842, was 143639 lbs., and the mint-refiner undertook to pay
all expenses of refining on being allowed 3| grs. of gold (less 10 per
cent.) on every pound weight gross of silver : what sum did this amount
to, at £4 l5. dd. per oz. ?
222. The weight of gold extracted from the above was 2530 oz.
1 dwt. 17 grs. ; what was its value at the same rate?
223. What would be the interest on £256 5^. 9d., at 4^ per cent., for
4 yrs. 5~ mo. ? and what would be the compound interest on £1040, at
4 per cent., for 3 years ?
141
APPENDIX.
The choice of the number 10, as the base or radix^ as it is called,
upon which the decimal system or scale of Notation depends,
common as it is to so many nations, barbarous as well as civilised,
may be conceived to have had its foundation in the natural practice
of counting on the fingers, whence the term digit; but we might
have taken any other number for base, and, having characters for
zero and all the figures less than the base, we might express any
number whatsoever in such a scale. {See Alg. Notation.)
Tlie admirable method of notation by the use of the nine digits
and zero is of extreme antiquity ; and though called the Arabic
method, (because first introduced into Europe through the !Moors
in Spain about the 11th century, though it was not till about
the 14th that it superseded the old Roman system,) was certainly
known to the Hindoos long before the rise of Arabian science,
and even by them ascribed, for its excellence and the remoteness
of its origin, to the direct revelation of the Divine Being. It
seems to have been traced with some probability to the regions
of Thibet.
The system of the Greeks was almost identical with that of the
Hebrews, or Phoenicians : that of the Romans, though very simple,
was singularly cumbrous and inconvenient; and it is a striking
proof of their extreme indifference to any advances in scientific
matters, that tliey so pertinaciously retained it, notwithstanding
their acquaintance with the far more perfect and comprehensive
notation of the Greeks.
The Jigures now in use are derived from the old Arabic, though
much modified and corrupted by the course of time.
When numbers are used with reference to the things numbered,
as when we say 3 apples, 4 pens, 5 shillings^ they are said to be
H
142 APPENDIX.
concrete numbers ; when used without such reference, merely to
indicate a certain number of units of the same kind, as when we
say simply 3, 4, 5, they are called abstract numbers.
The concrete quantities, required in ordinary calculations, are
those which are necessary to express Money ^ Weight, Space, and
Ti7ne. In the Tables will be found the most common of these
quantities ; but we shall here make a few additional remarks about
them, and explain the Standards, which are used in each of thes2
classes.
The standard gold coin of this realm is made of a metal, of
which 22 parts in 24 are pure gold, and 2 parts alloy, a mixture
of silver and copper. From a lb. Troy of this metal are coine(i
465§ sovereigns = £46 145. 6^/. ; so that the Mint price per oz. of
standard gold = ^V of £46 145. Qd. = £3 IT*. lO^d. ; and since there
are 11 oz. of pure gold in 12 oz. of standard, we shall have (neg-
lecting the value of the alloy) the value per oz. o? pure gold at the
Mint = Jy of £46 145. 6fZ. = £4 45. U^\d.
The standard silver coin is made of a metal, of which 37 parts
in 40 are pure silver, and 3 parts alloy (copper). From a lb. Troy
of this metal ai-e coined 665., so that the Mint price per oz. of
standard silver is 6s. 6d. : and since there are ^ J of an oz. of pure
silver in this, the value per oz. of pure silver at the Mint is |^
o£5s.6d. = 5s.Ul^d.
From a lb. Av. of copper are coined 24 pence : but this is not
a legal tender for more than 1 2d., nor is the silver coinage for more
than 405., the gold coinage being the standard of the realm.
The following coins are noticeable, occurring often in ancient
documents : —
Groat = 4d., Tester = 6d., Noble = 65. Sd., Angel = 105.,
Merk=135. 4d., Carolus = 235., Jacobus = 2o5., Moidore = 27s.
Great Inconvenience having been long felt in this country, from
the want of uniformity in the systems of weights and measures,'
which were in use in different parts of it, an Act of Parliament was
passed in 1824, and came into operation on Jan. 1, 1826, by which
certain weights and measures, therein specified, were declared to
be the only lawful ones in this realm, under the title of Imperial
Weights and Measures.
ArrENDix. 143
It was settled "by this Act —
1. That a certain yard measure made by an order of Parliament
in 1760, (by comparison with the yards then in common use,)
should be henceforward the Imperial Yard^ and the Standard of
Length for the kingdom : and that in case this Standard should be
lost or injured, it might be recovered from the knowledge of the
fact, that the length of a pendulum, oscillating in a second, in
vacuo, in the latitude of London, and at the level of the sea,
(which can always be accurately obtained by certain scientific
processes,) was 39.13929 inches (or twelfth parte) of this yard;
2. That the half of a double-pound Troy, "made at the same
time, should be the Imperial Pound Troy, and the Standard of
Weight; and that of the 5760 grains, which this lb. contains, the
lb. Av. should contain 7000 : and that in case this Standard should
be lost or injured, it might be recovered from the knowledge of
the fact, that a cubic inch of distilled water^ at the temperature
of 62° Fahrenheit, and when the barometer js at 30°, weighs
252.458 grains ; ......
3. That the Imperial Gallon, and Standard of Capacity, should
contain 277.274 cubic inches, (the inch being above defined,)
which size was selected from its being nearly that of the gallons
already in use, and from the fact that 10 lbs. Av. of distilled
water, weighed in air, at a temperature of 62°, and when thC
barometer is at 30°, will just fill this space.
The name Troy Weight has been derived from Troyes, a city
of France, where great fairs were once held, and to which it wag
introduced, about the time of the Crusades, from Cairo in Egypt;
but It has also been derived from the monkish name for London,
Troynovant, from Trinovantum. The name Avoirdupois is probably
derived from the old Norman, avoirs, goods and chattels, and pois^
weight.
It is probable that a grain of wheat was- the element of iveight
in former days, and a grain of barley (barleycorn) the element of
length.
The pennyweight was so called as being the weight of the silver
penny then in use.
The words ounce and inch are both derived from the Latin
uncia, or twelfth part, of a pound and foot respectively.
h2
lU
APPENDIX.
The following weights and measures are noticeable, besides
those given in the Tables.
Carat (of Diamond) . =3| grs.
Carat (of Gold or Silver) =240 grs.
Firkin (of Butter) . . =56 lbs.
rodder (of Lead) . .=19icwt.
Great Pound (of Silk) =24 oz.
Pack (of Wool) . .=240 lbs.
Yard (of Land)
Hide (of Land)
= 30 acres
= 100 acres
Line . .
Barleycorn
Span . .
Cubit . .
Pace . .
:iin.
:9in.
1 8 in.
5 ft.
Degree =69^miloa
Flemish Ell
French Ell
= 3 qrs.
= 6 qrs.
Firkin (of Beer)
Kilderkin . .
Barrel. . , .
Hogshead , .
Butt ....
Tun , , , .
= 9 gals.
= 18 gals.
= 36 gals.
= 54 gals.
= 108 gals
= 2 butts
Anker (Wine or Spirits) = 10 gals.
Runlet =18 gals.
Tierce =42 gals.
Hogshead .... =63 gals.
Puncheon = 2 Tierces . = 84 gals.
Pipe = 2 Hogsheads . =126 gals.
Since there are 24 carats in a lb. of gold, the fineness of gold is
often expressed by saying that it is so many carats Jine^ meaning
so many parts out of 24 ; thus our standard gold is 22 carats fine,
and jewellers' gold (as marked on the stamp of a watch) is 18 carats
fine.
In measuring land, surveyors use a chain^ called Gunter's chain,
which is 22 yards long, and divided into 100 links; and 10 square
chains, or 100,000 square links, make an acre.
In France, the standard of linear measure is the metre^ which
is one ten-millionth part of the Terrestrial arc from the Equator to
the Pole=39.37l inches; and their other measures are all decimal
parts or multiples of this : thus the decimetre=^3.9B7l in., centi-
metre=.S9S7l in., millimetre = .03937 in., &c., and so the decametre
=393.71 in., and similarly for the hectometre (hecatometre), hilo'
metre (chiliometre), myriometre., &c.
The standard of weight is the G^rfwime = weight of a cubic
centimetre of distilled water = 15.4340 grs.; and this is likewise
subdivided and multiplied into the decigramme^ centigramme,
kilogramme^ &c.
APPENDIX. 145
•j.'Iie standard of capacity is the litre = 61.028 cub. inches, that
of superficial measure, the are = 119.6046 sq. yds., that of solid
measure, the stere = So S17 cub. ft. — all of which may be sub-
divided and multiplied as before.
The Greek unit of linear measure was the roue = 12.135 inches.
The principal Attic measures of length were
Uktv\05 (iir) =1 in. nearly. irXiQpov (1 OOtt) = 101 | ft.
irrixvi (li7r) = l|ft. oriyd. araUov (6007r) = 606|ft.
dpyvia. (Gtt) = 6 ft. or a fathom. ZlavXos (1 2007r) = 12 13| ft.
It will be found that there are very nearly 8^ stadia in a mile.
The Persian parasang was 30 stadia^ rather more than a league.
The principal square measures were the square irovq and irXWpov^
which latter contained 4 dpovpat, and was a little less than a 7'ood,
The Roman unit of length was the pes = 11.6456 inches.
Their other ordinary measures Avere the digitus (^ pes), uncia
(^jP')i palmus {\p.),palmipes (1^;?.), cubitus (l^p.), gradus (2^p.),
passus (5jt?.), milliarium or mille passuum {5000p. = 1618 yds.).
Their principal square measure was the jugerum (240 J9. by 120)
= 28800 pedes quadrati, or ^ acre, nearly.
For rough calculations, the ttovq and pes may each be considered
to be equivalent to a, foot English.
The Greek and Roman systems of mojiej/ were naturally founded
upon those of weight, the denominations of money and weight
being identical.
The Attic unit of weight and money was the drachma, which,
as a weight, was equivalent to 66^- grs. ; and this weight of silver
being worth d^d., this was the value of the silver coin, drachma.
Their other coins (all in silver) were as follows —
6 ohols (SfioKoi) made 1 drachma (Bpaxp.-f))
100 drachmoe ... 1 mi}ia (iJ.ua)
60 min(B . . . , 1 talent (rdXavrop) ;
SO that the obol was worth about l^d., the mina £4 Is. Sd., the
talent £243 15*.
Besides these, there were the dioholus, triobolus, didrachm, tetra-
drachm (or stater), Sec, whose values are explained by their names.
In later times, the value of the drachma as a coin corresponded
to the Roman denarius ^Sld
146 APPENDIX.
The Roman unit of weight was the libra^ or pound, = 5204 grs.,
that is, nearly % lb. Av., or very nearly -^ lb. Troy. This weight
of the metal as or bronze (a mixture of copper and tin) formed
originally the coin as, or pound ; but the weight of the coin was
subsequently reduced in the proportion of 8 : 5.
The as or lih?'a was divided into 12 uncia, i.e. twelfth-parts;
and the following names were given to the different multiples of
an uncia.
7 unc sepiunx
8 ... (|lb.) hes
9 ... (fib.) dodrans
10 ... (fib.) dextans
11 deunx
12 , libra ov as
li line, (sesqui-uncia) sescunx
2 ... (I lb.) sextans
3 ... (i lb.) ter-uncius or quadrans
4 ... (ilb.) trlcns
5 quincunx
6 ... Q\b.'=semi-as) semis
The name hes is supposed to be formed from des (as bis from ^t'c),
and this from de~triens (dcsit triens), meaning an as wanting a triens
or third; just as dodrans, dextans, deunx, are formed from de-quad-
vans, de- sextans, de-uncia.
It should be observed that the word uncia, or ounce, means
simply a twelfth-part; and therefore the above terms sescunx,
sextans, &c. were used by the Romans, as so many fractions, for
subdivisions of other units, as well as of the as : thus, we have had
above the uncia of length = -^^ pes, and see also below among the
measures of capacity.
The uncia of weight = 434 grs. = very nearly an ounce Av.
The Romans had also a silver coinage, consisting of the denarius
Rild its parts. These were the denarius, worth 8hd., and equivalent
(as its name denotes) to 10 ancient ases or 16 later ones ; the qui'
narius (5 ancient ases^ = ^\d., called also victo?'iatus, from the image
of Victory upon it ; the sestertius (i. e. semis tertius nummus, or
a coin worth 2^-, viz. ancient a5e5)=2|c?. ; libella = -Y^^ den., sem-
hella (semi-lihella) = -^den., terunqius=:.f^Q den. ■= (a,s above) ^ an-
cient as or I later as = ^d., nearly, , . ,
For rough calculations we may reckon the as at ^d., sestertius 2d.,
denarius S^d. The sum of 1000 sestertii was called a sestertium
7= £8 175 Id., but there was no coin for this amount.
The Greek KeTrtjg = Roman sextarius. may be conveniently taken
as the unit of capacity, being equivalent to (.9911 or) just one pint
APPENDIX. 147
English. The sextarius "was so called as being ^ of tlie congius, and
contained 12 KvaOoi, cyatlii; and the multiples of the cyathus had
the same names among the Romans as those of the uncia^ or ounce
of weight : thus, 2 cyathi was a sextans^ or i of a sextarius^ &c.
The Greeks had also the kotv\i]=- \ sext. = ^ pt,, xohn^=-\\ pt.,
;^oDg = 4;to/rtKfc=3 qts., fieTpqr)jf^=9 gals., /x^ St nvog= 12 gals.; and
the cfcrot; and iijAtKTOQ were the sixth and twelfth parts of the me-
dimnus. The Romans had, beside the cyathus and sextarius, the
hemina ■=. \ pt., congius = G 5ea'^. = 3 qts., modius = 2 gals., unia =
3 gals., aw/>/iom = 6 gals.
k
A /S^Zar X)«y is the interval between two successive transits
of the Sun over the meridian of any place ; but, from several
causes, this interval is continually varying, though slightly, in
duration. If, hoAvever, we take the 7nean of many observations,
we shall get the length of the 3Ieau Solar Day, and this is the
Standard unit for the measurement of I'ime in ordinary life;
though Astronomers have another unit in common use.
The Solur Year is the interval between the Sun's leaving and
returning to a certain fixed point in his apparent orbit round the
Earth (the Ecliptic), and is accurately determined by Astronomers
to contain 3G5.242218 mean solar days = 365 days, 5 hrs., 48 min.,
47 h sec. nearly. Hence the common, or Civil, Year, which contains
only 36o days, is somewhat shorter than the Solar, or True, Year ;
and this error, being nearly J of a day, would accumulate, if not
corrected, so as to produce at length a complete confusion in the
times at which the seasons would return, and we should have
Summer, sometimes in July, sometimes in December.
Julius Cccsar first corrected this ; and, supposing, in the then
state of Science, that the Solar Year contained exactly 365 days,
6 hrs. = 365.25 days, he ordered that every fourth year should
contain 366 days instead of 365. But this correction was really
too great by .007782 of a day, since the Solar Year contained
only 365.242218 days; and in 400 years this error amounted to
400 X .007782 = 3.1128 days; and hence it happened that the
vernal equinox, which fell, in a.d. 325, at the Council of Nice,
on March 21, fell in a.d. 1582 on March 11. Pope Gregory, in
consequence, caused 10 days to be omitted in that year, making
Oct. 15 to follow Oct. 4, so that the vernal equinox fell next year
148 APPENDIX.
again on Marcli 21 ; and, to prevent the recurrence of this error,
he ordered that in every succeeding cycle of 400 years, 3 of the
leap years should be omitted, viz. those which complete a cen-
tury, when the number of hundreds is not divisible by 4 ; thus,
1600, 2000 are leap years, but not 1700, 1800, 1900, &c.
The Gregorian correction was introduced in England in 1752.
when it had become necessary to omit 1 1 days of the current year ;
and the Calendar thus rectified is called the New Sti/le, the Julian
reckoning (which is still retained in Russia) being the Old Style.
This correction is too great on the other side by .000282 of a
day, but the error only amounts to a day in 4000 years.
N.B. — Until A.D. 17o2, the New Year's day in England for all
official records was the 2oth March : hence, we often find, in works
relating to an earlier period, a double date given, as 1703-4,
wiicnever the event referred to occurred during the month of
January, February, or March, up to INIarch 25 — the former indi-
cating the year according to the old, and the latter, according to
the modern, reckoning.
uo
DECIMAL COINAGE.
1. It maybe desirable to say here a few words upon the
subject of a Decimal Coinage, which has been for sometime
under the consideration of the Government, has been recom-
mended for adoption by a Committee of the House of Com-
mons, and is lil^ely, therefore, before long, to be introduced
in England, as it has been already in France and in the
United States of America.
2. Two systems of decimal coinage have been proposed,
and each has met with warm supporters, — the one based
upon the penny or farthing, the chief coin of the poorer
classes, as the unit of reference, the other upon the pound
sterling or sovereign, the chief coin of the wealthier classes.
Each of these systems has its own peculiar advantages and
disadvantages, which we shall proceed briefly to explain.
Of the two, the advantages of the latter, based upon the
pound sterling, seem to be upon the whole the greatest;
and as it has been specially recommended by the House of
Commons' Committee, it is probably that which will be
ultimately sanctioned by Act of Parliament, perhaps, with
some modification of its details, as, for instance, in the
names at present proposed for the new coins.
3. I. One system of decimal coinage takes the farthing
for its unit of reference, and its money-table would be some-
what as follows : —
10 Farthings make 1 Doit = \0f. = 2\d.
10 Doits make 1 Florin = 100/ = 2,9. Id.
\0 Florins mako \ Pound = 1000/. = 20s. lOd. *
h3
loO DECIMAL COINAGE.
The coins required for use in this system would be the
following : —
Copper — fiirtliing, halfpenny, and penny, as now ;
Silver —doit (2|(/.)» g^oat (5J.). shilling {\2\d.), florin (25c/.) ;
Gold — half-pound {\25d.), pound (250d.).
It might also be convenient to have a dollar or double'
florin (50c/.) in silver, and a crown (621</.) in gold, so that
five dollars, or four crowns, would go to make the pound.
The difference in size between the doit and the groat, being
much greater than that existing between the present 3</.
and 4c?. pieces would allow very well of their being both
coined in silver.
4. The advantnges of this system are the following : —
(1.) All coins now in use would be still available ; and
thus, while the banks would be collecting the old coins,
and gradually withdrawing them from circulation, business
might bo carried on as usual with the old shilling, florin,
and pound. This would prevent, no doubt, much confusion
at first, especially among the poorer classes.
(2.) The farthing, halfpenny, and penny, would be per-
manently retained, and the price of food, the rate of wages,
&c., being generally fixed by the penny, much inconve-
nience would be saved by this means to the mass of the
population.
(3.) No change need be made in the penny postage,
the penny-stamp, the tolls for turnpikes, bridges, &c., nor
in any fixed payment whatever, as now existing.
5. The disadvantages of this system are the following : —
(1.) The present pound sterling, which is the usual unit
of reference in all great questions of national and com-
mercial finance, would be ultimately displaced altogether.
(2.) The accounts of bankers, merchants, &c., kept during
past years according to the old coinage, or the sums of
money mentioned in statistical or other records could not
be immediately compared with corresponding entries under
the new system, nor without the trouble of reducing them
DECIMAL COINAGE. lol
in each case to their equivalent expressions in the new
coinage.
(3.) The process of reduction from the old coinage to the
new, though easy on this system, is much more easy on
the other system of decimal coinage, as will presently
appear.
6. We may here complete what we have to say on this
system, by explaining the process of reduction from the old
coinage into the new.
To reduce a Sum of Money from the present Coinage into
the neio Decimal Coinage {Penny System),
Since one old pound contains 960 farthings,
and one new pound contains 1000 farthings,
it follows that if a denote the number of old pounds, and
b the number of new pounds, in the same given sum of
money, then
960a =10006*, or 6=^"^ a = (l-^) a = a-. 04a.
Hence we may find the number {b) of new pounds, cor-
responding to any given number (a) of old pounds, by sub-
tracting from a the quantity .04«, which we obtain by
merely multiplying a by 4, and moving the decimal point in
the result two places to the left, or otherwise by deducting
4 per cent, from the amount.
Ex. 1. Reduce £765 from the old Coinage into the new (Penny)
Coinngo.
Here a = 765.00
.G4a= 30.60
b = 734.40 = 734 Pounds 4 Florins (new Coinage).
Since 1 shilling = (^V = tSct = ) -05 of a pound, any
number of shillings in the given sum may be expressed at
once as a decimal of a pound, by merely multiplying by 5,
♦ For if we denote that sum by -S when reduced into farthings,
S S
960=«'^"^lUo5=^'
/. 6'=960a = 10006,
152 toJEClMAL COINAGE.
and setting the product to fill the two places of figures
immediately after the point.
Ex. 2. Reduce £343 17s. into the new (Penny) Coinage.
Here a = 343.850
.04a = 13.754
b = 339.096 = 330 Pounds, 0 Florins, 9 Doits, 6 /.
(new Coinage).
If there are any odd pence in the given sum, these have
only to be reduced to f\irthings, and added in as thousandths
of a pound,
Ex. 3. Redr.ce €409 lis. %\d. into the new (Penny) Coinage.
Here a = 409.550
.04a = 16.382
393.168
8irf. = 34
b = 393.202 = 393 Pounds, 2 Fl, 2/. (new Coinngc).
7. The converse process of reduction from the new
coinage into the old would be performed as usual.
Ex. 1. £734.4 (new) = 4) 734400/
12)18360Q(/. -
20)153005.
765£ (old).
Ex. 2. £330.096 (new) = 4)330096/:
12}82524</.
20)687 7s.
343£ 17s. (old).
Ex. 3. £393.202 Cnew) = 4^393202/:
12)983U0^rf.
20)8 19 Is. 8^r/.
40y£ lis. 8|(/. (old).
8. II. The other system of decimal coinage takes the
vound sterling, or sovereign, for its unit of reference, and
its money-table Avould be somewhat as follows: — the mil
being the yJ^ ^ of a pound sterling — ^-^ of a penny = |f of
a farthing.
10 Mils make 1 Cent. =j^£ = 2M.
10 Cents make 1 Florin = ^ £ = 2s.
10 Florins make 1 Pou7id sterling = 20*.
DECIMAL COINAGE. 153
The coins required for use in this system would be the
following : —
Ccppcr— »u7 (^d.), two-mils or double (^d.), five-mils or doit
Silver — cent (2f</.), two-cents or groat (4f J.), five-cents or
shilling (lid.), florin (,2s.).
Gold — half-sovereign (10s.), sovereign (205.).
It might also be convenient to have a dollar or double*
/lor in (45.) in silver, and a crown {5s.) in silver or gold.
9. The disadvantages of the system are the following : —
(1.) It would abolish the coins most in use with the poor,
namely, the farthing, halfpenny, penny, and Sd.y 4d., and 6d.
pieces, leaving them only the shilling, and coins of larger
value. The sixpence, indeed, might still be used for a time,
as it is exactly equivalent to 25 mils; but it would ulti-
mately be withdrawn from circulation.
(2.) It would be impossible to pay exactly in the new
coinage a sum in the old coinage which contained (besides
pounds and shillings) any number of pence, except it were
six-^QWQQ. For Ic?. = 4^ mils, 2c?. = 8^ mils, &c.
(3.) Hence also it would be necessary that, wherever a
rate of Ic?. is now levied for any purpose, a change should
be made, and either 4 mils or 5 mils charged instead.
Where large sums are raised by such a rate, this would
produce a very considerable difference in the amount so
obtained.
To take, for instance, the case of the penny postage : if 4
mils be charged instead of \d.=^Al mils, the loss to the
government upon every penny would be \ mil, and upon a
million of pounds 240000000 x i mils = 40,000,000 mils =
£40,000; whereas, if 5 mils be charged instead of Ic?., the
gain to the government would be f mil upon every penny,
or, upon a million of pounds, £200,000.
The same would be true of tolls taken for turnpikes,
bridges, &c., which are usually rated at Ic?., 2d., 3c?., 4c?.,
&c., and the difficulty of coming to a satisfactory arrange-
154 DECIMAL COINAGE.
ment in such cases would be much greater than in that of a
government impost. For, in the hitter case, it is the
government, that is, the nation itself, which would be the
gainer or loser by the loss or gain of the public in paying
the tax; whereas, in the former, the loss or gain of the
public would occasion a corresponding gain or loss to the
private individuals or companies who might be the pro-
prietors of the tolls.
10. Notwithstanding the above disadvantages, the re-
commendations of this system are so great, (1) from its not
abolishing the shilling, florin, crown, half-sovereign, and
sovereign; (2) from its allowing old accounts to be com-
pared at sight with those of the present day, without the
trouble of reduction ; (3) from the facility w4th which a
sum may be converted on this system from the old coinage
into the new; that there is little reason to doubt its being
ultimately adopted, if our present system is exchanged for
any other.
11. To reduce a Sum of Money from the present Coinage
into the neiv Decimal Coinage (^Pound System).
Here the number of pounds remains unchanged; the
shillings, if any, may (as before) be expressed as a decimal
of a pound by multiplying by -j-^ or .05 ; and, since \d,=A\
mils, if the pence be converted into farthings, the number
of farthings will give the number of equivalent mils, except
that 1 mil must be added whenever the number of pence is
Qd., or above it. If special accuracy be required, then 1 mil
should be added for any number of odd pence between 3c?.
and 9d., and 2 mils for any number of odd pence above 9c?. ;
by which arrangement the loss and gain upon the fractional
parts of a mil, when there are several sums of money con-
cerned, would in the long run be fairly balanced.
Ex. Reduce £409 Us. 8|<f. from the old Coinage into the new
(Pound) Coinage.
Here £409 lis. OJ. = £409.550
8l(/. 35
Ans. £409.585 =£409 5fl. (85 cents, ox) 8 Cents
5 mils.
DECIMAL COINAGE.
1,
12. The converse operation would be performed as
usual.
A.71S. £409 Us. &|i. nearly.
13. We may exemplify the application of this system in
one or two instances.
Ex. 1. Multiply £37 175. 4|f/. by 43.
Old Coinage,
£37 17 4i
10
£378 13 9
£37 17
New Coinage.
4i = £37.850
19
£1514
113
£162J
£1628.36;
l\ = £1628.356 (new coinage).
N.B. — The difference in these two results arises from the fact that in
the one we have expressed \\d. by 6 mils, instead of 6i mils, its true
value, and in the other we have expressed 4i(/. by 1 9 mils, instead of
182 mils, its true value. The second error of \ mil when multiplied
by 43 produces an error of 10| mils, which added to the first error of |
mil makes up the whole difference of 11 mils.
Ex. 2. Find the value of 5 cwt. 3 qrs. 14 lbs. at £14 9*. 8c/. per cwt.
£14 9 8 = £14.483
3
2 qrs.
1 qr.
14 lbs.
72.415
7.2415
3.62075
1.810375
£85.087625 = £85.088 (nearly) = £85 Ojl. 88/
14. It would be of little use to pursue this subject any
further at present, while the whole matter is yet under
consideration, and the details of the measure, to be here-
after proposed to Parliament, are by no means fixed.
156
THE METEIC SYSTEM.
15. Besides the Decimal Coinage, there is also a Decimal
System of Weights and Measures, commonly called the
French or Metric System, which has been adopted by nearly
all the Continental nations of Western Europe,* and will
probably at no very distant day be established also in
England. The first step indeed to such estabHshment had
been already taken, when the Council of Education required
in their Code of Regulations (1871) f that a chart of the
Metric System should be hung conspicuously on the walls of
all schools under Government inspection, and that in all
such schools children in Standards Y and YI should know
the principles of the Metric System, and be able to explain
the advantages to be gained from the uniformity in the
method of forming multiples and sub-multiples of the unit. J
* The Metric System h<as been <acIopted in France, Holland, Belgium,
Greece, Spain, Portugal, Italy, Eoumania, the North German Confede-
ration, Wurtemberg, Bararia, Baden, and also by Chili, Equator,
Uruguay, Brazil, the Argentine Confederation, New Granada, Peru,
Venezuela, and partially or in substance in Norway, Canada, British
India, and the United States ; while a Decimal System of "Weights and
Measures, differing only from the Metric System in the unit chosen as
the base of the System, exists by law in Austria and Switzerland.
t But this rule is not at present (1874) in force.
\ In 1864 the Metric Act of Parliament (27 & 28 Vict. c. 117) was
passed, which provides that, * Notwithstanding anything contained in
any Act of Parliament to the contrary, no contract or dealing shall be
deemed to be invalid or open to objection on the ground that the weights
or measures expressed or referred to in such contract or dealing are
weights or measures of the Metric System, or on the ground that
decimal subdivisions of legal weights and measures, whether Metric or
otherwise, are used in such contract or dealing.' In other words, this
Act permitted the use of the Metric System. And yet, ' by a strange
THE METRIC SYSTEM. 157
16. The advantages in question are obvious. Thus in
Avoirdupois Weight 16 drams make 1 ounce, 16 ounces
make i pound, 28 pounds make 1 quarter, 4 quarters make
1 hundred- weight, 20 hundred-weight make 1 ton, where
the numbers, indicating the multiples of the unit of the
next lower denomination which make one of the higher, are
respectively 16, 16, 28, 4, 20 ; and so in Troy Weight they
are 24, 20, 12, in Apothecaries' Weight, 20, 3, 8, 12 ; and
the same irregularity prevails in the Tables of Measures.
But in the Metric System the number is always the same,
viz. 10, so that te7i times the unit of the next lower denomina-
tion makes always one of the higher— except a slight modi-
fication in Square Measure, as shown below. By this means
all laborious multiplications and divisions are avoided, such
as are required under the old system, e.g. for reducing
ounces to tons, or miles to inches. And arithmetical opera-
tions of all kinds are so much simplified in practice by the
use of the Metric System that (to use the words of Prof.
Leone Levi, Metric System^ p. vi), ' Here is a tool which
offers facilities for saving one-half of the time in arithmeti-
cal education, and one-fourth, or one-third, of the time spent
in all the transactions which include calculations of weights
and measures.' Being, moreover, so generally employed
on the Continent, it is very desirable, with a view to inter-
national communication, that it should be as soon as prac-
ticable adopted also in England. And, in fact, it is already
used exclusively in some popular scientific class-books, and
a knowledge of it is reqnired by Examiners in Physics and
Chemistry.
inconsistency, as tho law now stands, -vrliilst the restriction is removed
against contracting in terms of the Metric System, any person using
such weights and measures for the purpose of buying and selling in
shops and other places subject to the visits of Inspectors of Weights
and Measures, or having them in his possession, is liable to have them
seized and to conviction and forfeiture.' Prof. Leone Levi, Theory
and Practice of the Metric System, p. 6.
158
THE METRIC SYSTEM.
17. The Metric System is so called from the French word
metre (derived from the Grreek metrouy ' measure '), the name
given to a line of a certain length (39*37 inches, rather more
than a yard), which was fixed upon in 1799 by the French
Legislature as the standard unit of linear measure, and
which was at that time supposed to be the
ten-millionth part of the distance from the
Equator to the Pole. It has been since
found, however, that the measurement of tho
Earth's circumference then made was not
quite correct. And, consequently, the Metre,
as originally determined by that measure-
ment, is really an arbitrary length, like the
English imperial yard.
M
n
E
to
'5-
^
&9
E
~
f^
E
—
iA
E
E
CJ
E
E
•4
~
—
=
00
E
E
to
=
zz
=:
o
—
18. The Metric System has four principal
units, all depending on the metre.
1. The Metre (39*37 indies) is the unit of
measures of length.
2. The Are (120 square yards), the square
of ten metres, is the unit of measures of
surface.
3. The Litre (61 ciihic inches), the cube of
the tenth of a metre, is the unit of measures
of capacity.
4. The Gram (ISJ grains) is the unit of
measures of weight, and is the weiglit in
vacuo of so much water at its greatest
density as would fill the cube of the hun-
dredth part of a metre.
19. The standard Metre is a platinum bar, and the
standard Kilogranr (p. 161) a platinum cylinder, which aro
preserved carefully in the Hotel des Archives at Paris.
Exact copies of them are deposited at the Conservatoire
THE METKIC SYSTEM.
159
des Arts et Metiers, and are used to verify tlie metric stan-
dards for foreign countries. But England possesses two
platinum copies of the standard Metre, deposited -with the
Eojal Society in London, and a platinum copy of the stan-
dard Kilogram, deposited at the Standard Department.
Besides these, brass copies of the Metre, Kilogram, and
Litre, have been carefully made, and presented by the
French to the British Government, and are now deposited
at the office of the Warden of the Standards.
decimal multiples and sub-
Surface
hectare
centiare
20. Each
unit has its
multiples,
as
follows : —
1000
100
10
Length
kilometre
hectometre
dekametre
1
•01 ( =
METRE
= jL) decimetre
= ioo) centimetre
"001 ( = 1^) millimetre
Capacity
Weight
kilolitre
kilogram
hectolitre
hectogram
dekalitre
dekagram
LITRE
GRAM
decilitre
decigram
centilitre
centigram
millilitre
milligram
'21. The following are the tables of measures employed in
the Metric System, with their respective tmits.
I. Measures of Length or Linear Measure.
The unit of Linear Measure is the Metre=39'S7 inches,
or 3-28 feet, or 1-09 yard (more correctly 39'3708 in.
=:3'2809//.=l-0986 7/c7s.). •
10 millimetres make
10 centimetres
10 decimetres
10 metres
10 dekametres
10 hcctomptrcs
10 kilometres
1 centimetre.
1 decimetre.
1 metre.
1 dekametre.
1 hectometre.
1 kilometre.
1 myriometre.
Hence, in order to reduce from one denomination to
another, the French arithmetician merely throws the deci-
mal point Qua or more places to the right or left as the case
160 ' THE METRIC SYSTEM.
may require. Thus 98765-4321 metres=9S7 6H^2-l milUm.
= 9*87654321 myriom. ; whsreas under the Enghsh system,
in order to reduce 987654321 inches to leagues, we should
have to divide by 12, 3, 5 J, 40, 8, 3, successively, a very
laborious process.
N.B. The deJcametre (10 m. or 100 decwi.=S2-8 ft. or
10 "9 yds.) is used as a chain in surveying, and is divided into
50 links, each containing 2 decim.
The hilometre (1000 7?i.=1093-6 yds.) is nearly 5 furlongs
(1100 yds.)y so that 8 Jdlom.=5 miles nearly.
The myriometre (10 h'lonietres or 10,000 m.) =50 furlongs,
or 6^ miles nearly (more nearly=10936 yds. or 6i miles).
II. Measures of Surface or Square Measure.
The unit of Square Measure is the Are or square
dekametre, that is a square of which t<he side is a deka-
metre=10 metres, and which therefore contains (p. 26)
100 square metress=Ell9"6 square yards.
100 centiareB (square metres) make 1 are.
100 ares ( = 10,000 square metres) „ 1 hectare.
111. Measures of Solidity or Cuhic Measure.
The unit of Cubic Measure is the Stere or cubic metre
=61027 cubic inches, or 35'3166 cubic feet, or 1-30802
cubic yard, nearly.
10 decistercs make 1 stere.
10 stercs „ 1 dekastere,
K.B. These measures are chiefly used for wood and
carpentry.
IV. Measures of WeigJd,
The unit of Weight is the Gram, which is the weight in
vacuo of 1 cubic centimetre of distilled water at its greatest
density, viz. at the temperature of 4° of the centigrade
thermometer aEl5-43234 grains or 15^ grains, nearly.
THE METRIC SYSTEM. 161
10 milligmms make 1 contigram.
10 centigrams „ 1 decigram.
10 decigrams „ 1 gram.
10 grams < . „ 1 dekagram.
10 dekagrams „ 1 hectogram.
10 hectograms ,, 1 kilogram.
10 kilograms „ 1 myriogram.
N.B. The kilogram or kilo, as it is often called,
=15432-34 grains=24- lbs. Av. (15,400 grains) nearly, is
the weight usually employed on Continental railways ; and
the half-kilo (=lnj lb. Av.) is also generally used as a
weight on the Continent.
The centner=50 kilos. = 771,6l7 grains=110:^ lbs. Av.
(771,725 grains) =1 cwt. (112 lbs.) nearly.
The quintal=10 myriogr. or 100 kilos.=220i lbs. Av.
=2 cwt. nearly.
The milHer or tonne=10 quintals or 1,000 kilos=2205
lbs., or 20 cwt., or 1 ton, nearly.
V. Measures of Capacity,
The unit of Capacity is the Litre or cubic decimetre
= 61-027 cubic inches=l-76 pint.
10 centilitres make 1 decilitre.
10 decilitres „ 1 litre.
10 litres „ 1 dekalitre.
10 dekalitres ,, 1 hectolitre.
10 hectolitres „ 1 kilolitre.
N.B, The hectolitre=100 litres=176 pints=22 gallons,
or 2| bushels, nearly.
22. Since 1 decimetre=10 centimetres, therefore (p. 28)
a cubic decimetre or litre =1000 cubic centimetres. Hence
the weight in vacuo of a litre of distilled water at its
greatest density is the weight of 1000 cubic centimetres
of such water, or 1000 grams, that is to say, the weight
of a litre of such water is 1 kilogram.
162 THE METRIC SYSTEM*
In like manner, since 1 metre=]0 decimetres, therefore
the weight of a cubic metre of such water is that of 1000
cubic decimetres, viz. 1000 kilos or 1 milHer. Thus a
mass of rock 4 metres long, 3 metres wide, and 2 metres
deep, would contain (4x3x2=) 24 cubic metres, and fill
24 kilolitres ; and as this quantity of water would weigh
24 milliers, the weight of the mass in question would be
found at once by multiplying this weight by the number
which expresses the specific gravity of the rock compared
with water o
23. A metric quantity may be read in various ways, in
terms of one denomination or of more than one, at pleasure.
Thus 35*703 metres may be read as 35 metres 7 decim.
3 millim., or as 3'5703 deJcam., or as 357 decim. 3 millim.^
or as -035703 hilom.
But, in writing a metric quantity from dictation, it is
necessary sometimes to insert cyphers, as in the following
examples :—
Thirteen kilometres, seven grams= 13*007 kilometres
or 13007 grams ;
Seven hectolitres three centilitres =7*0003 hectolitres or
700-03 litres ;
.. Seven hectares six ares five centiares^ 706-05 ares or
7-0605 hectares.
But it should be noted carefully that in Square Mea-
sure such an expression as 5*7 sc[. m. means — not 5 sg.
metres 7 sq^. decim., but— 5-7 (=5y^^) sci. metres ~h sq. metres
70 sq. decim. (since 1 sq. metre=100 sq. decim.). Similarly
in Cubic Measure 5*07 cuh. m. means 5y^^ cuh. metres
=5 cub. metres 70 cuh. decim. And conversely, since
1 sq. metre='100 sq. decim. and 1 cuh. metre=-\000 cuh.
decim., therefore 9 sq. metres 5 sq. decim.=9'05 sq. m.,
and, in like manner. 8 ciih. metres 91 cub. decim.=S'Odl
cuh. m.
.T£.E METRIC SYSTEM. 16^
24 Since the metre=l*09 yard or ly'^- yard, nearly, and
the half-kilo =ly^o lb. Av., nearly, it follows, that when
goods are sold by the metre or half-kilo, the prices should
be 10 peii cent, higher than when they are sold by the yard
or pound respectively. In like manner since the centner
(50 kilos.) = 110:^ lbs., which is less than a hundred-weight
(112 lbs.) by IJ ^^.=6? czt;^., the prices of goods, when sold
by the centner or millier (20 centners), should be -g^ less
than when sold by the hundred-weight or ton (20 civt.)
respectively, which amounts to a reduction of 2|cZ. in the £.
• 25. The metre, half-kilo, centner, and millier, might be
called the metric yard, metric jpoimd, metric hundred^u'eigJit,
metric ton, respectively. And the following names, corre-
sponding to the names of English measures, arc given' by
Prof. Levi, Metric System, p. 64.
Metric league (half-myriometre) = 3'1 miles.
„ mile (kilometre) = 1094: yards.
„ furlong (double-hectometre) = 219 „
„ chain (double-delcametre) = 21-9 „
„ pole (half-dekametro) = S'S „
„ fathom (double-metre) = 6-56 feet.
„■ cubit (half-metre) = 1"6 „
„ hand (decimetre) = 3-9 inches.
26. The following is a table of approximate equivalents
in the English and Metric Systems, where great accuracy
is not required (Prof. Galbraith, as quoted by Prof. Levi,
Metric System, p. 49).
Length,
1 metre = 3 feet 3 inches 3 eighths,
64 metres = 70 yards.
Linear, Square, and Cubic Measure.
10 metres =11 yards.
10 sq. metres = 12 sq. j-ards.
10 cuh metres = 13 cub. yards.
164 THE METRIC SYSTEM,
Land Measure.
1 are = 4 perches,
10 ares = 1 rood.
1 hectare = 2| acres.
Weiglit.
1 kilogram = 2| lbs. Av.
80 grams =17 drams Av.
Liquid and Dry Measure.
A\ litres = 1 gallon.
1 hectolitre = 22 gallons.
27. The following table gives a more accurate list of the
equivalents of the principal metric measures in terms of
English measures, and vice versa.
Measures of Length.
Millimetre = -03937 inch.
Centimetre = -3937 „
Decimetre = 3"937 inches.
or -32809 foot
Metre = 39-37079 inches.
or 3-28089 feet.
or 1-09363 yard,
Bekamctre = 10-93633 yards.
or 1-98842 pole.
Inch = '0254 metre.
Foot = -30479 „
Yard = -91438 „
Pole. = 50291 metres.
Chain (4 jx) = 20-1164
Furlong {lOp.) = 201-1644
Mile = 1609-3149 „
or 1-6093 Jcilom.
Measures of Surface.
Square decimetre = 15-50059 sqicare inch.
Square metre = 1-1960Z square yard.
or 10*76429 sqtcarefeet.
THE METRIC SYSTEM.
165
Hectare
:=
2-471 U acres.
Are
=
•02471 acre.
Square inch
=
6-45137 square ce^itimetrea.
Square foot
=
9-28997 square decimetres.
Square yard
=
•8361 sq. ')netre ( centiare).
Square pole
-
•2529 are.
Rood
=
10-11678 ares.
Acre
«
40-4671 „
or
•40467 hcktares.
Measures of Solibity.
Cubic decimetre = &l-Q2'JQb ctthio inches.
Cubic metre = 35-31658 cubic feet.
or 1 '30802 cubis yard.
Cubic inch = 16-38618 cubic centimetres.
Cubic foot = 28'3153 cubic decimetres.
Cubic yard = '7645 cubic metre.
Measures of Weight.
Gram
a=
•56438 dr.
or ■
•03527 ounce Avoirdupois.
or
15^43234 grains.
or
•64301 dwt.
Hectogram
=r
3-52739 ounces Avoirdupois,
or
3-21507 ounces Troy. ■
Kilogram
=
35-2739 ounces Avoirdupois.
or
2*2046 pounds Avoirdupois,
or
2-6792 pounds Troy.
or
•01968 hundred-weight.
MUlier
=
•98420 ton.
Grain
=
•0648 gram.
Pennyweight
=
l'o5517 „
Ounce Troy
=
31-1035 grams.
Pound Troy
=
373-24195 „
Dram
=
1-77184 gram.
Ounce Avoirdupois
■ =
28-34954 grams.
Pound Avoirdupois
; =
453-59265 „
Stone (14 lbs.)
=
6-3503 kilometres
Quarter (28 lbs,)
=
12-70059 „
I
166
THE METRIC SYSTEM.
Hundred-weight
Ton.
60-80238 kilometres.
1016-0475
10-160475 quintals.
1-0160475 millicr.
Measures of Capacity.
Centilitre ==
•07043 gill.
Pint =
•56755 litre.
Decilitre =
'17607 pint.
Quart =
113510 „
Litre =
1-7607 „
Gallon =
4-54041 litres.
or
-88038 quart.
Bushel =
Z-6S233 dekalitres.
or
•22009 gal.
or
36-3233 litres.
Belcalitre =
2-20096 „
Quarter =
2-90586 hectolitres.
Hectolitre =
22-0096 „
or
29-0586 dekalitres.
or
2-751208 i«s.
or
290-586 litres.
or
•343901 gr.
167
NOTES AND EXAMINATION-PAPERS
ON
ARITHMETia
NOTES.
Note I.
Casting out the Nines, as a method of Proof for Multiplication,
depends on the two following considerations: —
(i.) Any n" divided by 9 leaves the same remainder that would be
left if the Jura of its digits were divided by 9.
Thus, 687 -=-9 leaves 3; and (6 + 8 + 7)^9 leaves 3.
(ii.) If each of two no" be divided by any n°, say 9, and the product of
their remainders be taken, this product divided by 9 will leave the same
remainder that would be left if the product of the two no^ were divided
by 9.
Thus, 15474-9 leaves 8, and CS7-r9 leaves 3; then, (8 x 3)-f9 leaves
6, and (1547 x 687)4-9 also leaves 6.
The frst of these considerations will appear just from the following
illustration.
10, or 100, or 1000, or any other power of 10, is an exact n° of nines
+ 1 ; therefore,
80 is an exact n° of niiies + 8,
600 is ditto +6,
680 is ditto +6 + 8,
687 is ditto +6 + 8 + 7;
so that 687 -+ 9 leaves the same remainder as (6 + 8 + 7) -+9.
It is evident, then, that to ascertain what remainder would be loft
after dividing any n° by 9, we need only sum the digits of the n°, and
cast out 9 as often as it arises in the addition.
The second consideration may be illustrated by the following
example ;-=^
i3
168 NOTES.
Since 1547 = 171 nines +8,
and 687 = 76 nines + 3,
therefore, 1547 x 687 is equal to
(171 nines + 8) x 76 nines, [which gives an exact n« of nines']
+ (171 niries + 8) x 3; [which gives an exact n" of nines + 8x3];
evidently, therefore, the whole product is an exact n<* of riines + 8 x 3, oi
+ 24, or + 6; the 6 being obtained by adding the digits 2 and 4.
Note II.
When a divisor is composed of two or more factors, and the quotient
is found by using those factors successively, the remainders after the
several divisions may be converted into the full remainder in the manner
employed in the following example: —
Divide 39711 by 35, or by 5 x 7.
5)39711 Or, 7)39711
7) 7942. ..It 5x4 + 1 5)_5673...0 "4 7 x 3 + 0
11 34.. .4 J =21 rem. 1134...3 J =21 rem.
Quotient, 1134^, ov U3q.
Dividing by 5 first, the successive remainders are 1 and 4; or, divid-
ing by 7 first, they are 0 and 3; and to find the entire remainder, we
multiply the first divisor by the second remainder, and to the product
add the first remainder.
The reason of this procedure may be shown thus: —
We are required to find how many thirty-fives are contained in 39711
units. Dividing first by 5 units we find that 39711 is = 7942 fives +
1 unit; and then dividing the fives by 7 we find them = 1134 thirty-fives
+ 4 fives; so that 37911 units are equal to
1 134 thirty fives + 4 fiVes + 1 unit, = 1 134 thirty-fives + 21 units;
= 1134 thirty-fives + 1^ of 35; = 1134|l thirty-fives.
In the second form of the division we have 0 as the first remainder:
in such instances, the second remainder placed over the second divisor
gives the fractional part of the quotient in a simpler form.
Note III.
Strictly, in reducing £37 to shillings, we multiply—not £37 by 20,
which would produce £740, but 37 by 20; the reasoning is that £37
contains 20 times as many shillings as pounds.
Note IV.
The multiplication of dimensions is frequently performed by what is
called the method of Duodecimals, which subdivides both square feet
and cubic feet into denominations called prmes, seconds, thirds, Sec; 12
NOTES. ' 160
superficial primes being = a square foot, 12 cubic primes = a, cubic foot,
tmtl, in both cases, 12 seconds = a prime, 12 thirds = a second, &c.
Primes, seconds, &c., are marked thus,
15 sq. ft. 7' 10" 6'"; 15 cub. ft. 7' 10" 5'".
In the first of these expressions the seconds evidently are square
inches, for they arc ]A4ths of a square foot; and if to these we add the
7 jirimes, or twelfths of a sq. foot, = 84 one-hundi'ed -and-f or ty- fourths of
a sq. foot, we have 94 sq. inches, and the whole expression is equivalent
to 15 sq. ft. 94^ sq. in.
In the second of the expressions the thirds are evidently cubic inches,
for they are I728ths of a cubic foot, and if to these we add the 7 primes
and 10 seconds, which are = — + 12. = !^ + i^ of a cubic foot, we
12 144 1728 1728
have 1128 + 5 cubic inches, and the whole = 15 cub. ft. 1133 cub. in.
Suppose, now, it is required to find by duodecimal multiplication the
area of a rectangular surface, 37 ft. 7 in. by 5 ft. 9 in.
Here, since 37 ft. 7 in. = 37^ ft., if the rect-
angle were 1 ft. broad the area would be 37i5
sq. ft., or 37 sq. ft. 7'; then, as the breadth is
5f^ ft., we multiply 37 sq. ft. 7' by 5 units 9
twelfths, as follows: — Placing the greater di-
"ZTZ 7 y 17 mension over the less, we first multiply 37 sq.
ft. 7' by 5, then we multiply the same quantity
by 9 considered as twelfths, and by setting the remainder, arising from
a twelfth of 9 times 7, one place to the right of inches, and canying 5 to
the next product, of which in hke manner we take a twelfth, we shall
evidently have 9 twelfths of 37 sq. ft. 7' = 28 sq. ft. 2 twelfths of a sq. ft.
3 twelfths of a twelfth of a sq. ft.
The entire product is 216 sq. ft. 1 prune 3 seconds.
If I2ths of an inch, commonly called parts, occur in cither of the
factors, the duodecimal multiplication is performed in the same way.
Let it be required to multiply 28 ft. 9 in. 6 pts. by 11 in. 9 pts.
It should be observed that the
annexed proceas, which is con-
ducted in the same way as the
preceding one, is equivalent to
finding first — of the multiplicand,
12
9 1
then— or — of it, and that we do not really multiply one concrete
144 16 J i-J
quantity by another, which would be absurd.
ft.
pr.
37
7
5
9
187
11
28
2
3
ft.
pr.
sec.
28
9
6
0
11
9
6
26
4
8
1
9
7
1
6
28 sq. ft.
2'
3"
7'"
6'
170 NOTES.
Note V.
For a demonstrative aritlimctical example of the process of finding
the greatest common measure of two numbers, see Hunter's Art of
Teaching Arithmetic, p. G4. A very slight acquaintance with Algebra
will enable the student to understand the following illustration of the
general Rule for finding the g.c.m.
Let it be required to determine the g.c.m. of 1275 and 561.
The G.C.M. of 1275 and 561 evidently cannot exceed 561, and must be
= 561-:- some factor of 561. Let x denote that factor. Therefore, the
G.C.M. of the proposed no' will be — , when x has the least value that
X
561
allows '- — to measure 1275.
X
t/? -t
We have to find, then, the least A'alue of x making 1275 -r — , or
X
i^Z^ a whole no.
561
Now, l^I^=2x+ — of a: ; so that l^-- of :r is a whole n°.
561 561 561
Put
153
561
ofx =
= A;
; '.x =
561
153
of A,
, = 3a +
Ii:-'
•'•
102
153
of A =
= a wliolc
n°, w
diich
we may
calls;
•'•
A =
153
102
of
B = B +
51
102
of Bi
&imila:
^•ly, ^
^of
L02
B
= c; .
.*. B =
102
""sT
of c, = :
2 c exactly.
Now, we should get b = a whole n°, whatever whole n° we might choose
for the value of c ; but we must take c = 1, the lowest whole n°, that we
may obtain the lowest integral value of x.
Honco, lHZS£=liL5 of ^1 of 15? of 'B of 1 =i?I«,
561 561 153 102 51 51
•T 1 561 ^, ,1 .J
. . — = — , or, — = 51, the g.c.m. rcqmred.
561 51 0,'
From the above analysis, then, it appears that the g.c.m. of two no'
is obtained by dividing the greater by the less, then the less by the
remainder, and so on as prescribed by the Rule.
To determine the g.c.m. of three no% find that of two of them, and
then that of the result and the third number. Thus, the g.c.m. of
12528, 16182, and 13804, will be found = 58 ; for that of the first two
no" is 522, and that of 522 and 13804 is 58.
To find the g.c.m. of fractional quantities, as, for example, of 8^ and
NOTES. 171
19§, express tlicm as fiactions having a common denominator, then find
the G.C.M. of the numerators, and under it write the common denomina-
tor. The result for the supposed example will be — , which is con-
tained 15 times in the first n" and 34 times in the second.
Note VI.
For the conversion of a mixed circulating decimal to a vulgar frac-
tion, the following rule is self-demonstrating: — Multiply the given
decimal by 10, or 100, or 1000, &c., according as there are one, two,
three, &c., decimal places before the circulating period ; express the
result as a mixed fraction, and then divide it by the 10, or 100, &c., pre-
viously used as a multiplier, which will evidently restore the value of the
given expression.
Thus, to convert .034 and .27345 to vulgar fractions :—
(i,) .034x100 = 3.4 = 31;
and 3t X —
•" 100
_3l
900*
1000
=273.45
= 273i§
= 273A
f
and
273/^^ X
1
1000
ii
II
376
1375
(ii.) .27345 X
What is further included in the usual Rule has reference to an easy
method of multiplying by the denominators 9, 99, 9S9, &c.
Thus. 27311 X J- being=?I?ilS»-±15,
®^ 1000 99 X 1000
and 273 x 99 being = 273 x (100-1),
27300-273 + 45 27345-273
we have
99000 99000
27072 3008 376
yyooo 11000 1375
Note VII.
The series proposed for calculation in Ex. 47, 48, is one by which
the ratio of the circumference of a circle to its diameter may be
approximately computed. See Colenso's Plane Trigonometry^ Part II.
p. 7. The result signifies that the circumference of any circle is nearly
3.14159 times the diameter.
The series proposed for calculation in Ex. 47, 50, is that whereby
what is called the base of the Napierian system of Logarithms is
approximately computed. See Colenso's Plane Trigonometry, Part I.
p. 121, or Hunter's Treatise on Logarithms, p. 55. The result signifies
172 KotES.
that the Napierian Logarithm of any given number is that power of
2.71 828 1 8 which ^Yhc^ calculated produces the given number.
Note VIII.
Questions in Proportion can always be worked independently of the
artificial Kule of stating, and though sometimes not so conveniently, yet
always in a more satisfactory way as regards simplicity of demonstra-
tion. It will appear from the following examples that a knowledge of
the first principles or fundamental rules of Arithmetic is suflBlcient for
the solution of all problems in the Rule of Three.
(1) If 15 lbs. of salt cost Is. Gd., what cost 25 lbs.?
Cost of 15lbs.==18(/.
1 lb. =5\of 1 8c/.
25lb3. = f5of 18</.
l^ijL^= GJ. X 5 = 2s. 6d. Ans.
15
(2) If 25 lbs. of salt cost 2s. 6f/., what quantity cost Is. 6d.?
No. of lbs. for 30(/. = 25 lbs.
l^.=JgOf 25lb3.
„ 18c/. =1^ of 25 lbs.
25 lbs. X 18 ► 1, „ o ^r ^\.r, A
= 5 lbs. X 3 = 1 5 lbs. Ans.
30
(3) What is the coach fare for 130 miles at the rate of £l 9*. Ad.
for 85 miles?
Fare for 85 miles = 29^5.
„ 1 mile =5^ of 29^5.
„ 130 miles = iaf of 29^5.
88.. X 130^88.. X 26^^^^^ 10^^. Ans.
3x85 51
(4) If 112 sheep were grazed in a field for 9 days, how long n)ight
84 sheep have been grazed in the same field?
Time that 112 sh. were grazed «=9 da.
■ „ 1 sh. might be grazed =112 times 9 da.
,i 84 sh. „ „ =^ of 112 times 9 da.
9 da. X 1 12 rt 1 1 ift J A
, = 3 da. X 4 == 1 2 da. ^4??*.
84
(5) A person comj leted a journey in 32 days, travelling 8 hours a
day; how long would he have taken to do the same, travelling only
6 hours a day?
No. of days at 8 hrs. a day = 32 da.
i^ at 1 hr. a day =8 times 32 da.
„ at 6 hrs. a day = i of 8 times 32 da,
6 3^
NOTES. 11^3
(6) Three partners with a joint stock of £1036 lis. 6d. gain
£287 6.?.; what share of the gain falls to one of the partners whose
stock is £365 175.?
Gam on £1036 lis. 6c?. (or 41463 sixp.) = 5746s.
„ on 1 sixp.sr: of 5746s.
41463
„ on £365 I7s. Od. (or 14634 sixp.)
14634
41463
5746s. X 14634 5746s. x 1626
of 5746s.
£101 8s. Ans.
41463 4607
(7) If lOy lbs. of sug^ir cost 4\^s., what will 3§ cwt. cost?
Cosf of 10|lbs. = 4l^Js.
„ of 1 lb. =1 of L^s. = JL of 7s.
" 75 16 16
„ of 112x3§ Ibs. = iHii^'of 7s.
" -^ 16
^"•'-^^^^^^^ =12£llil = £8 19s. 8f/. Ans.
16x3 3
Note IX.
In calculating the amount of any sum of money, by compound In-
terest, for any n" of years, at 4 per cent, per annum, wc add to the
original principal of itself to obtain the 2nd principal, then to this
principal we add — of itself to obtain the 3rd principal, and so on.
Now, adding to any n° — of itself is the same as multiplying it by
1-,^, or by 1.04; and accordingly, the amount of £750 for 3 year?, at
4 per cent, per annum., comp. int. might be found thuss^.
£750x1.04x1.04x1.04,
= £750 X 1.04^ = £750 x 1.124864,
= £843.648.
Similarly, the amount of £?50 for 4 yrs. at 5 per cent. -U-ould bo
£750 X 1.05*. And, generally, to find the amount of £p, by comp.
interest, for any n" of years, at any annual rate, we may first add a
hundredth of the rate to 1, then raise the sum to that power which is
denoted by the n° of year?, and then multiply by p.
Suppose that, in this way, we have to find the compound interest of
£95 6s. 8f/., for 3 yrs., at 5 per cent, per ann., payable half-yearly: —
the rate is here intended to denote 2g per cent, per half-year, for 6 half-
years.
i3
174 NOTES.
Wc have accordingly to find the 6th power of 1.025; and this we
could obtain at once from compound interest Tables; or we could very
3asily calculate it from a Tabic of Logarithms. The simplest form of
the arithmetical process is as follows; the divisor 40 determining the
interest in each case, because 2^ is ^^ of 100.
40)1.025 Amt. of £l for I hf. yr.
.025625
40)1.050625 Do. „ 2 do.
.0262656
40)1.0768906 Do. ,, 3 do.
.0269223
40)1.1038129 Do. „ 4 do.
.0275953
40)1.1314082 Do. „ 5 do.
.0282852
1.1596934 Do. „ 6 do.
Hence the compound interest of £ I, at the end of the 3rd year, is
£.1596934 ; which multiplied by 951 gives the comp. int. of £95 6s. 8d.
= £15.2241, or £15 45. 5,78d. Ans.
Now, suppose it is required to find what principal at 2| per cent, per
annum, comp. int., will in 6yrs. amount to £110 3s. 5c/.: that is, what is
the present worth, by comp. int., of £110 35. 5d. payable in 6 yrs.: — we
have
1.025" xP= 110.170833;
/. 11 0.170833 -M.l 596934 ==£95. Ans.
Again; let it be required to find at what rate of comp. int. £95 will
amount to £110 35. 5d. in 6 yrs.: —
110.170833-^95 = 1.1596934, the 6th root of which may be found by
logarithms = 1.025 5 or, Vl. 1596934= 1.0768906, the cube root of
which is 1.025. Hence the rate is 2l per cent. Atis.
Lastly; to find in what time £95 will amount to £1 lO.l 70833, at 2l per
cent, per ann., comji. int.: — Here we should ascertain by logarithms
what power of 1.025 is equal to 1.1596934; but when the time is an
exact n° of years, as in this instance, it would be found by raising
1.025 through consecutive powers till the required amount of £l is found
equal to the 6th power, denoting the time to be 6 yrs.
Note X.
A Kule called Equation of Payments is introduced in some treatises
on Arithmetic. It teaches how to ascertain the single time at which two
or more debts, due at different times, might be discharged by one pay-
ment of the sum of the debts. It is merely a particular application of
NOTES. 1/0
the principle of Discount; and it is given in two forms, according to
true discount and mercantile discount, respectively. ;
Examp. I owe £1085; of which £651 is due 5 months hence, and
£434 is duo 8 months hence; how many months hence would one pay*
raent of £1085 discharge both debts, reckoning the use of money
worth 5 per cent, per annum?
We compare the several sums by means of their present values, con*
sidering that the discount on £651 for 5 montlis added to the discount
on £434 for 8 months, should be equal to the discount on £1085 for the
time sought.
Now, according to Mercantile Discount^ we have
— - of £3255 = int. of £651 for 5 monthsj
1200
and -A- of £3472 = int. of £434 for 8 months;
1200 — —
o
1200
of £6727 =int. of £1085 for G-2 months. Ans.
because 6727-5-1085 = 6-2.
This method is evidently independent of the rate of interest; and
hence, for equating terms of payment according to mercantile discount,
we have the following
Ordinary Ride. Multiply the several debts by their times in any uni-
form denomination, and divide the sum of the products by the sum of
the debts.
Thus, the above process is reduced to the following: —
651x5 = 3255
434x8 = 3472
1085 )_6727
6i months. Ans.
The meaning of which is, that as the int. of £651 for 5 months Is
that of £3255 for a month, and the int. of £434 for 8 months is that of
£3472 for a month, so the int. of £6727 for a month is that of £1085 for
6f months.
But secondly, according to Trite Viscount, we have
A of £5, or £2^5 = disc, on £102^ for 5 mths.
or -~ = disc, on 1 }
49
® of £5, or £31= disc, on £103| for 8 mths>
12
or - = disc. on 1;
81
176 NOTES.
/. ^^^ =^£131 is the disc, on £651 for 5 mths.
~*« 14 is the disc, on 434 for 8 mths.
31 . _=_
£27f is the disc, on £lOS5 for the time sought,
"We have to find, therefore, in what time £l057f would produce
£271 interest, or J7404 would produce £191.
7404:100-) ,^ ^,,, .
5 ; 191/ •• 12mo. : e^^ mo. Ans.
this answer, equal to about 6.19 months, is a little less than 6.2, the
Answer found according to mercantile discount; but as the method of
true discount is much more laborious than the other, and in most prac-
tical questions gives a result very little less than the other, it is generally
sufficient, as it is more convenient, to follow the ordinary rule.
The Rule for equating according to true discount may be given as
follows : —
Find for each of the debts the discount that would reduce it to its
true present value; then find the time for which the sum of the dis-
counts would be the true discount on the sum of the debts.
For a discussion of the principle of Equation of Payments, see
Hunter's Art of Teaching Arithmetic, p. ?9.
ITOTE XI.
In Paper IX. will be found a vnriety of Questions relating to the coni«
parison of the money of different countries. This subject is frequently
treated in books on Arithmetic under a special Rule called Exchange.
The Par of Exchange is the intrinsic value of the coin of one country
as compared with a fixed sum of the money of another. The Course of
Exchange is the variable sum of the money of one country actually
given for a fixed sum of the money of another.
Thus, France exchanges with England a variable number of francs,
averaging about 25.30, for the pound sterling; for the actual Course of
Exchange, being dependent on the course of trade, is in almost con-
tinual fluctuation. Moreover, as in England gold is the adopted
standard of value, and France has a silver standard; — as also the values
of gold and silver are not always in the same proportion, and each
metal has not always the same value in both countries, — the Par itself is
not invariable.
Arbitration of Exchange is the estimation of the rate of Exchange
implied in the purchase of indirect Bills of Exchange, Bullion, Coins,
&c., in one country, as compared with their sale in another.
Thus, to find what arbitrated rate of Exchange is established between
NOTES. 177
London and Paris by bills on Vienna bought in London at 10 florins
1 kreutzer per & sterling, and sold in Paris at 254 francs per 100 florins;
a florin being = 60 krcutzers : —
Here we have given £1 = 601 kreutzers, and 600 kreutzcrst=2o.4
franc.-. lkr.=?^*fr.,
and 601 kr. =25.4 fr. x 1^^=25.44 fr. per i.. AnS,
Again; to find what arbitrated rate is established between London
and Paris by the purchase of gold in London at 77s. 10|(/. per ounce
standard, and the sale of it in Paris at 4 per millc premium : an ounce
Troy being = 3 1.1 grammes, and 1000 grammes of English standard
gold being worth 3151 francs : —
Here we have 311 grammes = 10 oz., or 1 gramme = — oz.;
.*.1000 grammes = l^^ oz.,
1000 grammes bought in London for 77|5. x 2^;
oil
1000 grammes sold in Paris for 3151 frs. x 1.004;
6230000^^ . 2Q^ .. 31Q3 Q f^,g^ . 35.27 frs. nearli/. Ans.
8x311
r
178
EXAMINATION-PAPERS.
Paper I.
Questions on the Introductory Pages.
1. (a) Explain the principle by ■which the decimal system of nota-
tion is made capable of expressing any number whatever.
(6) Distinguish between the arts of Notation and Numeration.
2. Add Thirteen thousand thirteen hundred and thirteen to Scveiiteen
thousand seventeen hundred and seventeen.
3. Subtraction may be performed (a) for the purpose of diminishing
a quantity by taking away some quantity it contains, or (i) for the pur-
pose of comparing two quantities as to their absolute magnitudes.
Give properly distinctive names for the results in these two cases.
4. (a) If two numbers be equally increased, how is their difference
affected? A fathei is 3 score and 5 years old, and his son is 37; what
is the difference of their ages? and what will be the difference of their
ages 10 years hence? (i) Apply these considerations to explain the
process of borrowing ten and carrying one in subtraction.
5. (a) "What name is given to two or more numbers connected by
multiplication? (b) Show how six sevens are equal in amount to
7 sixes, (c) Show why multiplying successively by 6 and 7 gives the
same result as multiplying by 42.
6. "What are the methods commonly used for proving the accuracy
of multiplication? How might division (if the pupil understood that
process) be used as a trial of correctness in multiplication?
7. (a) Divide 27564 by 21 in two ways: — resolving 21, first, into
successive divisors 7 and 3, and secondly, into successive divisors 3 and 7.
{b) Explain by reference to your work the usual process of finding the
full remainder by means of the two partial remainders.
Paper II.
Questions on Articles 1 to 20.
1. In reducing £7 to shillings what multiplier, strictly considered, do
we employ? Explain.
EXAMINATION-PAPERS. 170
2. In dividing a concrete quantity by an abstract number, as for
example in finding the 8th pr.rt of £3 7s. 6d. (Colenso, p. 24), which cf
.the expressions is properly the quotient? and ^Yhy?
3. How would you reduce crowns to guineas? florins to crowns?
sovereigns to guineas? yards to English elle? lbs. Avoirdupois to lbs.
Troy?
4. {a) Under what conditions may one concrete quantity be added
to another? subtracted from another? divided by another?
(5) Why cannot one concrete quantity be multiplied by another? ■
5. (a) How is the square measure of a rectangular surface found
from its length and breadth? If the length be 5 feet, and breadth 4 feet,
is the area = 5 ft. x 4 ft. ? Explain.
(6) How is the widih of a rectangular space found when the length
and area are given?
6. (a) How is the cubic measure of a rectangular solid found from
its length, breadth, and heiglit? Suppose the dimensions are 8, 6, and
2 feet: -explain the process of finding the solidity.
(6) How is the height or the thickness of a rectangular solid found,
when its cubic content and its length and breadth are givsn?
Paper III.
Questions for Illustration of Ex. 17.
1. (a) A man's yearly income is known:— How would you find the
sum he must spend weekly, so as to lay by a given sum at the year's
end?
(b) Given, a man's daily income and his yearly expenditure: — How
do we find his weekly saving?
2. The sum of 3 crowns, 3 florins, and 3 pence, is equal to 3 times the
sum of a crown, a florin, and a penny, that is, 3 times 85d. — Apply this
consideration to the solution cf Exs. 50, 61, and 62, in Set 17.
3. If £342 is to bo multiplied by 242, r.nd the product divid^^d ly
11, 8, and 4, successively, the effect of the whole may bo symbolically
expressed thus, :^—^-^^^-, which, by cancelling, becomes ^-
11x8x4 ox^
cndbyfurther cancelling becomes ^^^i^=^f-= £235 25. Gd.
8 8
—Apply this mode of treatment to the solution of Exs. 46, 55, and G7,
in Set 17.
4. How do you find the average value per yard of a quaatity cf
goods, consisting of 20 yards at 125. 6d. and 35 yards at 95. 10(^.?—
Would tho result be alTcctcd by the alteration of taking one-iif tli of c-ch
180 EXAMINATION-PAPERS.
of the given quantities, making them together «= 11 yards? — Solve Exs.
42 and 61, in Set 17.
6 In Ex. 63, Set 17, show that the result equals li^^ years, or
'70 ^'J
_— , or _, of a ycar, = 2ii years; and explain the following process:—
12)365 da. 6 hrs.
30 IQl
Ans. 2 yrs. 334 da. 19^ hrs.
6. (a) Hcduce 4 men 7 boys to an equivalent number of boys, sup-
posing a man equivalent to 3 boys.
(b) Reduce 7 men 12 women 5 children to an equivalent number of
children, supposing 2 women equivalent to a man, and 3 children equi-
valent to a woman.
(c) Apply the above species of reduction to the solution of Exs, 58
and 65, in Set 17.
7. (a) Tf the number 365 is to be divided into four parts, three of
them equal, and the fourth 95 less than each of the others; how many
times the first part would make 365 + 95?
Apply a similar mode of inquiry in the solution of Ex. 64, Set 17.
(b) If the sum of two numbers is 135 and their difference is 95, show
how each number may be found.
Divide a sovereign between Harry and George, giving George 20(1.
less than Harry.
8. Explain the following method of solving the latter part of Ex. G8,
in Set 17 ; and find the first answer similarly : —
3 yrs. profit on 500 ac. @ £4 25. 4c/. = £2058 6 8
Tithes = produce of 50 ac. @ £27 45. Od.= 1360 0 0
2nd A71S. Gain in the three years £698 65. Sc/.
Paper IV.
Questions on Chapters II, III, and IV.
1. What is meant by a common measure of two or more numbers?
IIow is their g.c.m. ascertained?
2. "What is meant by a multiple of a number? How do you find
the L.c.M. of two or more numbers?
3. Show that the product of two numbers divided by their g.cm.
gives their l.c.m.
4. Find that the g.c.m. of 11310, 12354, and 64090, is 58.
5. How do you find the g.c.m. of numbers all or partly fractional?
Fmd the G.C.M. of 261, 28|, and 29I-/5.
I
EXAMINATION-PAPERS. 181
6. How do you find the L.c.sr. of numbers all or partly fractional?
Find the l.c.m. of IQi, 6|, and 4fjy = 4042^.
7. AVhat is a fraction? Is 3 farthings an integral or a fractional
quantity ? Define a concrete fraction.
8. What arithmetical operation is signified by the line separating
the terms of a fraction? -What is an improper fraction, and how is it
reduced to a proper form?
9. What rule of fractions is anticipated in reducing a mixed frac-
tion to an improper one?
10. Why is it necessary that fractions should be of one common
denominator for addition or subtraction?
11. (o) Show that multiplying the numerator of a fraction is equi-
valent to dividing the denominator, and that dividing the numerator is
equivalent to multiplying the denominator.
(b) Hence show that the value of a fraction is not changed by mul-
tiplying or dividing both its terms by any one number.
12. What name is given to a fractional expression of the form 3 of |?
Which quantity is thus denoted to be a multiplier of the other?
13. (a) Prove the rules for multiplication and division of fractions:
exemplify with f and |.
(t) What does multiplication by a fraction strictly mean ?
14. Explain the meaning of such a fraction as — ; — ^ — ^
15. (a) A certain quantity, A, is given: — If it be ^ of another quan-
tity B, how would you find B ? If it be half as much again as B, how
would you find B?
(b) A number increased by its 5lh part amounts to 30: how would
you find the number?
(c) A number diminished by its 5th part becomes 24: how would
you find the number?
16. (a) Distinguish between decimal and vulgar fractions. What
is the special utility of decimal fractions?
(b) Compare the metrical, or French, scale of lineal measure with
the English,
17. (a) State and prove the rule for pointing in multiplication of
decimals. (6) How do you determine the local values of the quotient
figures in division of decimals?
18. (a) What arc circulating decimals? (t) Distinguish those
vulgar fractions that are convertible into terminating decimals; and
show that all others are convertible into recurring decimals.
182
EXAMINATION- PAPERS.
Paper V.
Supplementary Questions in Beduclion of Measures,
Reduce 22870062 square inches to acres, &c.
12)22870062
rl2)2287'
144 <^ [■
1 12) 1905838..
9) 158819. ..10
(-
126 in.
30|) 17646... 5 ft.
4 4
121 r 11)122^ q^*-y<is.
lll)_6416.. 8^41 qr. yds. = 10 yds. 2 ft. 3G in.
40)583... 3j 5 126
4) 14.. .23 pp. '
Ans. 3 ac. 2 ro. 23 po. 10 yds. 8 ft. 18 in.
Reduce the preceding result to square inches.
8 ac. 2 ro. 23 po. 10 yds. 8 ft. 18 in.
_4
14 ro.
40
583 po.
145|
17500
17645^ yds.
1588195 ft.
12
22870062 in. Ans.
^ 3. Reduce 1254492 sq. in. to sq. poles, &c.
4. Reduce 1 ac. 3 ro. 39 po. 14 yd. 5 ft. to sq. inches.
5. Reduce 123456789 sq. inches to acres, &c.
6. Reduce 2 ac. 3ro. 13 po. 14 yd. 5 ft. 100 in. to sq. inches.
7. Reduce 9532482 sq. inches to acres, &c.
8. Reduce 2 ro. 22 po. 14^ yd. to sq. feet
9. Express 22 sq. po. 2 yd. 4 ft. 72 in. in the denomination of sq.
yards.
10. An imperial gallon measures 277.274 cubic inches; how many
gallons would a vessel contain of which the capacity is 196i cub. feet?
11. The length of a wall, according to the French metrical system,
is 9 metres 4 decimetres 8 centimetres; reduce this to English feet, the
leogth of the metre being 39.371 inches.
EXA^illNATION-rAPERS. 183
12. Reduce 13 feet to metres.
13. How many decametres correspond to 1760 yards?
14. A chain 66 feet long is divided into 100 equal parts called links.
Kcducc an acre to square links.
15. A rod of brickwork, viz. a square pole, or 272;^ square feet, has
a standard thickness of a brick and a half :— If a piece of brickwork be
43 feet long and 22 feet high, and 2| bricks thick, to how many rods of
standard thickness is it equivalent?
Paper VI.
Questions on Ratio. (See Art. 73.)
1. If the ratio of Z to M is 5 : 8, and that of ilf to iV is 6 : 7) what
is the simplest form of the ratio of L to iV?
Here Z is - of 31, and ilf is | of i^/;
8 /
/. Z is ^ of ^ of iV=V^ of N. Ans.
8 7 28 ^
Or, Z is to iVas 15 : 28. Ans.
2. M buys 15 cows and 130 sheep for a certain sum, and N buys
9 cows and 175 sheep, at the same rates as M, for the same sum. Com-
pare the values of a sheep and a cow.
Since N has 6 cows fewer than My
but has 45 sheep more than M^
and both persons pay the same amount,
it is evident that 6 cows are worth 45 sheep,
fi 2
or 1 sheep worth — ,' or — , of a cow,
^ 45 15
or the Tallies of a sheep and a cow are as 2 .' 15. Ans.
3. One vessel contains a mixture of 16 pints of brandy and ii of
■water; another coi. tains 24 pints of brandy with 11 of water. Com-
pare the strengths of the two mixtures.
1st mixture 21 pints, 16 of which are brandy,
2nd „ 35 „ 24 „ „
.'. the strengths are — and ~,
21 3p
2 3
cr r.s - to -, or as 10 : 9. Ans.
184 EXAMINATIOX-PAPERS.
^ 4. A boat whose speed was 9^ miles an Iiour sailed from A to B,
a distance of G5 miles; and a second boat, which left A 2\ hours after
the first, arrived at B 5 minutes before the first. Compare the rates of
sailing.
5. A and B buy oranges at 10 for a shilling; A retails them at 9 for
a shilling, and B at 17 d. for a dozen. Compare their gains on selling the
same number of oranges.
6. If A's rate of profit is -of J5's, and for every guinea that/? gains
C gains a sovereign, compare the profits of A and C.
7. A sum of money is so divided among Roger, Henry, William,
and Thomas, that R. gets 3d. as often as II. gets 2^(/., H. gets 3^. as
often as W. gets 4|(/., and W. gets 4d. as often as T. gets 3|rf. Find the
direct proportion of the four shares.
8. If 3 men and 11 boys, working together, can do 5 times as much
work per hour as a man and a boy together, compare the work of a boy
with that of a man.
9. One vessel M contains a mixture of 27 gallons of wine and 11 of
spirits; another vessel iV contains a mixture of 43 gallons of wine and
14 of spirits. Compare the strengths of the two mixtures, supposing
the strength of spirits to be three times that of wine.
Paper VII.
Questions on Averages.
1. In a school register of daily attendance the numbers for a
certain week were— Monday 83, Tuesday 80, Wednesday 75, Thurs-
day 80, Friday 77, Saturday 72. What was the average daily attend-
ance?
2. A tradesman's receipts of money in one week were — Mon.
33/lOi, Tues. 26/6, Wednes. nothing, Thurs. 10/81, Fri. 43/111, Satur-
day 30/10. What was the average daily receipt?
3. The quantities of maize raised in the United States, in three suc-
cessive years, were— 494618200, 421953000, and 417S99000 bushels.
What, in British currency, was the value of the average yearly produce,
rating it at 25 cents per bushel, and reckoning the dollar of 100 cents
to be worth 4s. ?
4. Required the mean of the following observations of temperature:
—41° 29', 41° 271', 39° 13', 41° 33', 37° 471', 44^^ 28', and 40'' 13'.
5. If 3 quarts of stout at 9d. a quart are mixed with 10 jjints of ale
at 2l(f. a pint, what is the worth of a pint of the mixture?
6. At a competitive examination there were 4 candidates at the age of
19, 3 at 20, 2 at 21, and 3 at 23. Find the average age.
EXAMINATION-rArERS. 185
7. IIow many square feet are ia a reguhirly tapering plauk 10 ft.
6 in. long, the width being 9 inches at one end and 7 inches at iho
other?
8. The average of twenty-one results is 61, that of the first eight
being 64, and of the next eleven 59. Required the average of the last
two.
9. Three quantities of tea, at 3/S, 4/2, and 4/4 per lb., respectively,
make a mixture of 136 lbs., there being 5 lbs. more of the first kind than
of the second, and 6 lbs, more of the third than of the first and second
together. What is the worth of the mixture per lb.?
10. The average of ten results was 17^; that of the first three was
16i, andof the next four I6|; the eighth was 3 less than the ninth, and
4 less than the tenth. What was the last result?
11. If 9 gallons of spirits at 18/6 arc mingled with 7 gallons at 21/,
bow much water must be added to reduce the value to 16/6 a gallon?
Paper VIII.
Questions on the Relation between Time and Tower.
1. M can do a piece of work in 20 days of 7 hours, and iVcan do it
in 14 days of 8 hours. For how many hours a day should i^/and N be
engaged together, that the work may be done in 10 days?
AT docs 1 measure of work per hour;
140 such measures = the whole work.
iVcan do, per hour, the 112th of the whole,
"viz. 140-4-112, or \\ measures;
.*. M and iV together do 2^ meas. per hour;
or the whole work in 140-f2}= 62| hrs.
62| hrs. = 10 days, is 6g hrs. a day. Aiis.
2. A cistern is filled by two pipes, A and B, in 20 and 24 minutes
respectively, and is emptied by a tap C in 30 minutes. What part of it
will be filled in 1.5 minutes, \i A, B, and C are all turned on togctlier?
If A runs 1 measure per minute, 20 measures would fill
the cistern; then J5 would run, per minute, the 24th of 20,
viz. - of a measure, and Cthe 30th of 20, viz. -of a measure;
6 3
and A, B, and C being all opened, the cistern would gain
1 + ^ _r. , or li mcas, per minute, and in 15 min. would gain
6 3
lixl5 = 17|meas.,
which is 17:> twentieths =- - of the cistern. Aji^,
186 EXAMINATION-PAPEKS.
3. J' and G together reap a field in 8| days, and F alone can reap
as much in 3^ days as G can do in 5. In what time could each by him-
self reap the field?
F'm 1 day does 1 measure, G - of 3^ mc:.s.^-imeas.
the whole work is 1^ x 8? = 14?
]
14^ ^ 1 = 2 U da. by G alone, ^
4. y and Z began together a piece of work which they could have
done singly in 34 and 38 days, respectively. Y continued till the work
was finished; but Z had left him 4 days before its completion. In what
time was the work done?
y did 1 measure per day, and the v/hole work was 34 measures;
so that Z did, per dav, the 38th of 34 = - of a measure.
•^ • 19
Now, if Zhad continued the whole time of Y, 4 times --, or
3|^ extra measures of work would have been done, viz. 37jg
meas. by both agents in Y's time; therefoi'e
37U^1|I = 714^36 = 19| da. Ans.
5. A cistern has two supj)lying pipes, A and B, and a tap C. When
the cistern is empty, A and B are turned on, and it is filled in 4 hours;
then B is shut and C turned on, and the cistern is quite emptied in 40
hours; when, lastly, A is shut and B turned on, and in 60 hours after-
wards the cistern is again filled. In what time could the cistern be
filled by each of the pipes A and B, singly?
-4 and ^ together supply 1 measure per hour,
and the whole content of the cistern is 4 measures.
B runs, per hour, more than C, — of the 4 meas.
' ^ ' '60
C runs, per hour, more than A, ^ of the 4 meas.
^ 40
/, B runs _ + — , or - meas. per hour more than A.
60 40 6 ^
/. A and B together, in I hour, run - meas. more than A
6
runs in 2 hours;
but A and B together run 1 measure per hour;
1 5
,*. A runs 1 — -, or - meas. in 2 hours,
6 6
5 7
B runs 2 — -, oi* -^ meas. in 2 hours:
bo '
EXAMINATION-PAPERS. 187
- meas. : 4 meas. : : 2 hrs. : 92 lirs. by A,
6
I meas. : 4 meas. : : 2 hrs. : 6f hrs. by B,
6
Ans.
9^ 6. A can do a piece of work iu 2.5 days, B can do it in 20 days,
and C in 24. The three work together for 2 days, and then A and B
leave; but C continues, and, after 8| days, is rejoined by A, who brings
D along with him, and these three finish the remainder of the work in
3 days more. In what time would D alone have done the whole work ?
7. A piece of work can be done by A and B together in 14 hours,
or by B and C in 10| hours, or by A and C in 12 l:ours. In wha^
time could each person do it by himself?
8. To complete a certain work, B would take twice as long as A and
C together, and C thrice as long as A and B together; and A, B, and
C, by their united exertions can do it in 5 days. In what time could
each do it by himself?
9. A can do a piece of work in 10 days, J5 in 9, C in 12. They
all begin it together ; but only C continues till the work is finished,— .4
leaving it 3^ days, and B 2§ days before its completion. In what time
is it performed?
10. A cistern has two pipes, A and B, which singly could fill it in 9
hours and 10 hours, respectively. It has also two taps, C and Z>, which
singly could empty it in 12 hours and 8 hours, respectively. Suppose
that when the cistern stands half-full of water, A and D are turned on
for 3 hours; that then B is also turned on for the next 2 hours; that
then A and D are turned off, and C is turned on for the next £, hours;
after which all are shut, and the cistern is found to contain 95 gallons
more than its half content : — Find the content of the cistern. Find also
how much per hour the cistern Avould lose or gain, if all the pipes were
set open at once.
V^VEVi IX.
Questions on Exchange. (See Note XI.)
1. Reduce 396 dollars 53 cents American to British money, at
4s. 6c?. per dollar.
2. Convert 1206.70 American dollars into French money, at 5 francs
45 centimes per dollar.
3. Reduce £3758 16s. 6^. to francs, at 25.35 francs per £.
4. Find the value, in British money, of goods sold for 7889 francs
90 centimes,— exchange, 24 fr. 41 i cts. per £.
188 EXAMlNATIOX-rAPERS.
5. What in Eoglisli money is the value of the franc, at the exchange
of 25.57 francs per £ sterling?
6. How many pence per milree ( = 1000 rces) is the exchange
between Portngal and Britain, -when £823 5s. 6d. worth of wine costs
3161 mih-ees 375 rees?
7. If, when the course of exchange between England and Spain is
38|(/. per dollar of 20 reals, a merchant in Liverpool draws a bill of
£354 16.S. 3d. on Madrid, how many dolhirsand reals will pay the draft?
8. What is the arbitrated rate of exchange between London and
Lisbon, wlien bills on Paris, bought in London at 25.65 francs per £,
are sold in Lisbon at 525 rees per 3 francs?
9. If 11.65 Dutch florins are given for 24.89 francs, 383 florins for
437 marks Ilambro', and 6%^ marks for 32 silver rubles of Petersburgh;
how many francs should be given for 932 silver rubles?
10. Reckoning a Roman scudo worth 5| francs, and a shilling worth
1^- franc, what amount of discount do I allow by accepting £10 in
excliange for 45 scudi and 12 francs? And if I were to allow 4 per
cent, discount, how many francs along with 50 scudi should I give for
£12?
11. A merchant in London owes to one in Amsterdam 350.75
florins, which must be remitted through Paris. The quotations being,
for London on Paris 25 fi'ancs 30 cents, per £, and for Amsterdam on
Paris 451 florins per 100 francs, the London merchant delays remitting
till the rates are 25.45 francs per £, and 11 florins per 24 francs.
What docs he gain or lose by the delay?
12. £1000 sterling is due from London to Portugal, when the
exchange is 61|r/. per milree. Whether is it better, for Portugal, to
draw directly on London, or circuitously, at an expense of 1^ per cent.,
through Holland and France; — exchange between Britain and Holland
11.90 florins per £ sterling, between Holland and France 10 florins for
21 francs, and between France and Portugal 480 rees for 3 francs?
13. When English money bears a premium of 5 per cent, in America,
how much sterling should be given for 750 dollars, each worth 4s. 6d. at
par?
14. A rupee contains 16 annas each 12 pice: — Find, in French
money, the annual interest, at 3^ per cent., on 5217 rup. 3 an. 6 pi.,
exchange 2.63 francs per rupee.
1.5. If goods bought in London at a guinea be exported to New
York, at how many dollars should they be sold there, in order to cover
all expenses; estimating the export charges to be 7| per cent., and ihe
sale charges 5 per cent. ; the course of exchange beisig 6 per cent, prc-
ipium for bills on London?
EXAMINATION-rAPERS. 180
16. At what price in Company's rupees (each = 16 annas) was
indigo purchased in Calcutta, if the sale of it in London at 5.?. per lb.
yielded a profit of 20 per cent.; the shipping charges in Calcutta being
6 per cent,, sale charges in London 9 per cent., and loss of weight \\
per cent.:— exchange 25d. per rupee?
17. Given — that 1 ounce Troy equals 31.1 grammes; that 10
grammes of French standard gold are worth 31 francs; and that the
worth of a given weight of English standard gold is to that of the same
weight of French standard as 3151 to 3100; —
(i.) To what number of Troy ounces of English standard gold is
the franc equivalent, and what is the fixed number of francs equivalent
to £l?— the English mint price for standard gold being 77s. IQid per
ounce.
(ii.) How many francs are equivalent to £l, when gold purchased in
London at 77s. lOld. is sold in Paris at 14i per mille (i.e. per 1000)
premium on the fixed price? and how many, when gold is at 1 per millo
discount?
(iii.) Find that the results are correctly stated in the following
newspaper reports; and give the percentage results more nearly : —
a. The premium of gold at Paris is 7^ per mille, which, at the
English mint price of £3 17s. 10|i/. per ounce for standard gold, gives
exchange 25.35f; and the exchange at Paris on London, at short,*
being 25.33^, it follows that gold is about 0.09 per cent, dearer in Paris
than in London.
b. The quotation of gold at Paris is about ^ per mille premium, and
the short exchange on London is 25. 27^. On comparing these rates
wiih the English mint price of £3 17s. lO^i. per ounce for standard
gold, it appears that gold is nearly 4~10th3 per cent, dearer in London
than in Paris.
Papek X.
Qucitions on the uniform consumption of uniformiy growing produce.
1. Suppose that in a meadow of 20 acres the grass grows at a
uniform rate, and that 133 oxen could consume the whole of the grass
in 13 days, or that 28 of the oxen could eat up 5 acres of it in 16 daysj
how many of the oxen could eat up 4 acres of it in 14 days?
133 ox. to 20 ac. is 26? ox. to 4 ac.
28 ox. to 5 ac. is 22f ox. to 4 ac_>
* Tliat is by bills payable at short sight, as 3 day,' sight, and therefore immediately
worth their :;moiint in cash.
K
190 EXAMINATIOX-PAPEKS.
16 da.
22.4 ox, : 26.60X. ::13cla. : 15^ da.
3 days' growth eaten by 22.4 ox. in ^da.
q
— da. : 16 da. ::3 da. growth : 85| da. growth.
16
.'. the original grass is = 69i da. growth.
691 691
16 U_
851 da, growth : 831 da. growth! J^^ -^ o^-
da. : 16 da. J
Kote. \n explanation of the above form of solution, it may be observed that as the
orig. grass+ 13 da. growth of the 4 acres is eaten by 2G.6 ox. in 13 da.
.*. orig. grass+13 da. growth is eaten by 22.4 ox. in l!)J- da.
but, orig. grass -l-lfi da. growth is eaten by 22.4 ox. in IG da.
.•. 3 da, growth is eaten by 22.4 ox. in ^ da.,
which amounts to 85l da. growth in the whole 10 da.j
10 that the quantity of grass in the meadow at first must have been GQI days' growth ;
md we havenow given, orig. grass + lG da growth eaten by 22.4 ox. m IG da., to find
how many ox. would eat orig. grass+14 da. growth in 14 da.
For another manner of solving problems of this kind see Hunter's Art of Teaching
Arithmetic, p, lOo, and Examination Questions on ' Colenso's Algebra,' p, 62.
2. If 133 oxen consume the grass of a meadow in 13 days, and
112 of the oxen could consume the grass of the same meadow in
16 days, — the grass growing uniformly; in what time could 125 of the
oxen do it?
Here, as in the preceding solution, the original grass will be
found =» 69| days' growth; and now, 16 + 69_i da. growth being
eaten by 112 oxen in 16 da., the time is required in which 125
oxen would cat what grows in the required time + 691 da. growth.
112 : 125 ox. ) .. gji j^_ jjj . 5|^ ^^_ growth.
16 : 1 da. 3 ^ ^ 21 b
or, 125 oxen eat 5|f da. growth in 1 day,
1
thus consuming 4|2 da. growth of the orig. grass per day,
or the whole in 69^
^ 3. If 29 oxen would cat up a field of grass in 7 weeks, or 25 oxen
would eat up the same field in 9 weeks,— the grass growing uniformly;
how many oxen would do it in 6 weeks?
4. Suppose that a tank receives a regular and continual supply of
EXAMINATIOI^ -PAPERS. 191
water, and that, when it contains a certain quantity, 12 equal taps being
set open would empty it in 7^ minutes, or 7 of the same taps would
empty it in 16 minutes; how many of the taps Avould empty it in 50
mmutcs?
5. Suppose that in a certain meadow the grass is of uniform quality
and growth, and that 20 oxen would exhaust the grass in 12| days, or
21 oxen would do so in 12 days; in what time Avould 26 oxen do it?
6. I find that I can engage 15 workmen for 11 weeks, or 31 work-
men for 5 weeks, at uniform wages, and in either case pay the wages
exactly by means of the interest now accumulated on a certain sum of
money and that which will arise during the particular period of engage-
ment:—For how long could I engage 9 workmen on the same prin-
ciple?
7. If 23 oxen consume 8 acres of pasture in 26 days, and 25 oxen
consume 7 acres of the same in 20 days, — the grass growing uniformly;
how many acres of it would 33 oxen consume in 5^ days?
8. Suppose that 17 oxen in 30 days, or 19 oxen in 24 days, could
consume a field of uniformly growing pasture; find what number of
oxen, diminished by the removal of 4 at the end of 6 days, would eat up
the same field in 8 days.
9. In a field in which grass grows uniformly, suppose that 31 oxen
can consume Sf acres in 2 of the time in which 15 oxen would consume
5^ acres, and that 22 oxen would require 3 days longer to consume 7^
acres than 20 oxen would require for 6^ acres: — In what time would
the 31 oxen eat up the 8| acres?
10. An empty cistern has two supplying pipes A and ^, and two
taps C and D A would fill the cistern in 42| minutes, and ^ in 46
minutes; and D can carry off per minute half as much again as C,
After A and B, running together, have supplied a certain quantity, C is
allowed to run with them, and takes 51 minutes to empty the cistern;
but had D been turned on along with C, the two Avould have taken only
5| minutes to empty it. In what time would the cistern have been
emptied if D had been turned on instead of C ? and how much of the
cistern was filled when C was set open?
Paper XI.
Questions similar to Concluding Misc. Examp. 134 ^ 194.
1. A certain number is divided into two parts, such that 10 times the
first added to 18 times the second gives 15 times the entire number;
what fraction of the whole is each of the parts?
Questions of this kind closely resemble Examp. 2 in
Paper VI., and may be solved similarly; thus, since we have
k2
ll.)2 EXAMINATION-PAPERS,
10 times the first part and 18 times the second together equal
to 15 times the first and 15 times the second, it is evident that
(15 — 10) times the first compensates or equals (18 — 15) times
the second ; i.e. 5 of the lst = 3 of the 2nd; or, 1 of the lst =
- of the 27id; or, 1st : 2nd *.: 3 : 5; so that the parts are -
5 b
5
8
Otherwise.
10 times the 1st with 18 times the 2nd=l5 times both;
10 10 10
.*. 8 times the 2nd= 5 times both;
or, the 2nd is - of the whole
8
Q
and the 1st is - of do.
8
2. Divide the quantity 520 into two parts, such that 118 times one
part added to 128 times the other shall give 63700.
Here, we have 63700 [-4-520] = 122i times the entire na
/. 118 times the \st with 128 times the 2w(/=122| times both;
.'. 10 times the 2nd-
or, the 2nd is ~ of the whole, = 234
, yAns.
and the 1st is — of do. = 286
20
3. A person borrows £618 in two separate sums, at the respective
rates of 3^ and 5 per cent, per annum; and he repays the two loans at
tiie end of 10 months, with interest amounting to £22 10s. Required
the amount of each loan.
The respective interests are
^ of -^ of Is< loan, and 1 of -A of 2nd:
6 100 6 100
and these together are equal to ^^ or iL of both loans;
618 412
i.e. -^ of \st with ^ of 2nJ=il of both;
240 240 412
/~of2nd={l^
240 Ul2 2^0j
716
of 2ncf=/i^ L\ of loth
\4
412x240
of £618 =
412x3
of £618.
2nc?=-^l^of £618=£358l ,
vAns,
Uf = £260j
EXAMINATION-PAPERS. 193.
% 4. Sold 449 yards of cloth, part at 125. a yard, and the remainder
at 17s., and for the whole received £315 135. How many yards were
sold at each rate ?
5. A woman sold 7^ dozen apples for 6s. 2t/., some at the rate of
3 for 2^(/., and the rest at 8 for Q^d. How many were sold at each
rate?
6. I gave Ss. for a basket of oranges and lemons, buying the former
at the rate of 2 for 3d., and the latter at 5 for 4d. I then sold all at the
uniform rate of 5 for 6c/., and gained 6^ per cent. How many had 1
of each kind?
7. 12 lbs. of tea and 25 lbs. of coffee together cost £i 6s. 8c/.; but if
tea were to rise 2i per cent, and coffee to fall 4| per cent., the same
quantities would cost £4 5s. lie/. Required the prices of tea and coffee
per lb.
8. If the increase in the number of male and female criminals be 1.8
per cent., wliile the decrease in the number of males alone is 4.6 per
cent, and the increase in the number of females is 9.8; compare the
antecedent numbers of male and female criminals.
Paper XH.
Questlotis on Involution and Ecoluliott,
1. Simplify the expression - of ^ x /->
7 VS Nr 3
To remove surd denominators, multiply the numerator and denomin*
ator of the second fraction by ^' 5, and those of the third fraction by
V3, which gives
^-o{'''^^'\^ = t^V30.Ans.
- 7 5 3 7
2. Which is the greater quantity, a/2 or ^3?
2^ and .35 = 2« and 3« = 8« and 9«;
.*. ^3 is the greater.
3. Find the diagonal of a rectangular space, 792 feet long and 406
feet broad.
The length and breadth form with the diagonal a right-
angled triangle, of which the two perpendicular sides arc
given, to find the third or longest side. Now, in every right-
angled triangle, the sum. of the squares of the perpendicular
Bides is equal to the square of the longest side; therefore,
793^ + 406- = 792 100, square of diag.
V792l00^890 ft., the diagonal. Ans.
194
fiXAMlNAtiON-PAr-ERS.
4. Show that the length of the edge of a cube multiplied by VS
gives the diagonal of the cube.
If AB and Bc, edges of a cube, be each n
represented by 1, then the square of AC, the
diagonal of a superficial side, is evidently
12+ 12 = 2, and the square of the cube's dia-
gonal ad is = AC- + CD2 = 2+ 1=3; therefore
AD= VS when the edge of the cube is 1 ; or,
by similar triangles, the diagonal of every
cube is the product of the length of the edge by v/3.
5. The tip of a reed was 8 inches above the surface of a lake; but,
forced by the wind, it gradually advanced, and was
submerged at a distance of 28 in. Find the depth of
the water.
Let AD, 2= DC, represent the reed; EC the'
lake's surface; bd the depth.
Given ab = 8, ec = 28, to find bd.
The- right-angled triangles ABC, AeD,
having the acute angle a common to both,
are similar; hence, da : ac :: ca : ab; or,
smce Ae is | ac.
— =*^^;or, 2 daxab
CA AB
*Ca2 = 8H28- = 848; or, DA x 16 = 848; or DA = 53 inches.
Hence bd = 53 - 8 = 45 inches. Ans.
6. "What quantity is — of its reciprocal?
The quantity -fits reciprocal is = p- j but the quotient of any
quantity -r its reciprocal is the square of that quantity j
^, or -^ of a/1 15 = .9325. Ans.
23 23
^ 7. A square space contains 1056 sq. yards: Express the length
of its side as the decimal of — of a mile.
11
8. Find the side of a square field containing 2 ae. 3 ro. 17 po.
30 yds.
9. A square space contains 38 sq. poles 6 yds. 4 ft. 72 in.; find
the length of its side.
10. A rectangular field is 190 yds. long and 123 yds. wide; find the
side of a square field of half the area; find also the length of a field
twice as large as the first, and twice as long as it is broad.
11. Show that 10t-'v/2 is = 5 X a/2.
EXAMINATION-FAPERS. 195
12. Multiply ^112 by Vi7o.
13. If the perpendicular sides of a right-angled triangle are 13.02
and 5.2 feet, what is the third side?
14. If the town A is 72 miles west of B and 135 south of C, what
is the distance from B io C?
15. Which is the greater of the two quantities ^9 and :^19? and
which of the two V3 and v^l5?
16. If the diagonal of a rectangular surface is 3.4061 inches, and
the length 3.406 inches, what is the width?
17. The diagonal of a square is 353.55; find the length of its side.
18. The members of a party being solicited for contributions to a
charitable object, each person gave a number of half-pennies equal to
the number of members, and thus made up a sum total of 126". O^d.
What sum was ccntributed by each ?
19. Suppose the top of a straight ladder, 18| feet long, to rest
against a building at the height of 13^ feet from the ground; at what
horizontal distance from the bottom of the building is the foot of the
ladder placed ?
20. The edge of a cube is 250; what is its diagonal?
21. Find the edge, and also the surface, of a cube of wood, the
diagonal of which is 3 ft. 9 in.
22. Of what sum of money is £28 the same fraction that the sum
itself is of 60 guineas ?
23. If the compound interest of £250 for 2 years be £20 8*., what
is the rate per cent, per annum ?
24. The capacity of a cistern is 478.4 gallons : — Required (a) the
length equal to the breadth of a cistern of the same capacity 2| feet
deep ; and (i) the breadth equal to twice the depth of a cistern of the
same capacity 6 feet long:— a gallon being = 277.274 cub. inches.
25. What fraction of ( \/4050 x .0OS-^.2O + v'1458)-v- V.02 is
a/(6.008-j-.3042) + ^/(116.6 x .046)?
26. A can excavate 14.2884 cubic yards per day; how many can B
do per day, if A could do jB's daily quantity in - of the time that B
would take to do ^'s daily quantit} ?
27. The original cost of a pipe of port is £55, and it is sold to A at
a certain loss per cent.; then A sells it to B at the same losing rate; but
B sells it to C, at a profit of 12 per cent., for the original cost. What
Wcis the loss per cent, at which ^4 and B sold the wine?
196 EXAMINATION-PAPERS.
Paper XIII.
Supplementary Miscellaneous Questions. [A.]
1. What is the greatest unit of time with which 15 ho. 12 min. and
1 da. 3 hr. 33 min. can be both represented by integers?
2. How many times can .0087 be subtracted from 2.291, and what
will the remainder be?
3. What is the greatest number by which 2500 and 3300 can be
divided, so as to leave remainders 4 and 36, respectively ?
4. Define Proportion. — Can the quantities 2 yds. 2 ft. 10^ in,,
£24 3s., £12 lis. 6|rf., and 5 yds. 2 ft., be formed into a pioportion?
Give the reason.
5. State the distinction (i) between simple and compound division,
(ii) between simple and compound proportion, and (iii) between simple
and compound interest.
6. Distinguish mercantile from true discount; and show that the
difference between the interest and the true discount on the same sura
is the interest of the discount.
7. Find by duodecimal multiplication the product of 13 ft. 5 in.
7 pts. by 3 ft. 5 in.
8. Multiply, by the method of duodecimals, 29 ft. 7 in. by 9 ft.
B in. 6 pts.
9. Express the results of the two preceding questions in sq^uarc
/eet, square inches, and a fraction of a square inch.
10. Find, by duodecimal multiplication, that the product of 26 ft.
8 in. by 5 in. 9 pts. is 12 sq. ft. 9' 4'' ; and calculate by Practice the
value of the latter quantity at 15s. 9ld. per square foot.
11. What two quantities have for their sum 9 guineas and 9 shil-
lings, and for their difference 10 crowns and 10 pence,'
12. A offers to J5 6 cwt. 2 qrs. 7 lbs. of sugar, worth 2Ss. per cwt.,
for 24 yds. of cloth, worth 8s. S^d. per yard. How much per cent,
would B gain or lose by accepting the -offer?
13. If one man can plough a quarter of an acre in 2 hrs. 23 min.,
and another can do it in 2 hrs. 34 min., what fraction of an acre could
they together plough in an hour?
2 4 7
14. What sum of money increased by - of- of i of itself amounts
5 5 8
to 3s. id.?
15. Wh:.t decimal fraction diminished by .037 of itself becomes
.6955?
16. Show that the amount of £7 for 3 years, at 5 per cent, per
annum, compound interest, h = £7 x 1.05\
EXAMINATION-PAPEKS.
197
17. If 35 per cent, is lost by selling steel nibs at 3s. 6d. a gross, how
much would be gained or lost per cent, by selling them at 2s. dgd. a
hundred ?
18. A fruiterer by selling apples at the rate of 8 for 6|(/. gains 17
percent.; at what rate should he sell them per dozen to gain 20 per
cent. ?
19. If by selling cloth at 28s. 6c?. for 5 yards my gain would be 6|
per cent., what should I gain or lose per cent, by selling it at 37s. 6d.
for 7 yards?
20. The population of a town is 3370; what was its population a
year ago, if in the interval there has been an increase of about
2. Go per cent. ?
21. The amounts £210 and £155 are payable 2 years and 5 years
hence, respectively ; assign the mean period, or equated time, at the end
cf which, according to mercantile discount, these two amounts might be
paid at onco?
22. The sum of £434 is due as follows:— § of it in 4 months, i in
5 months, and the remainder in 7 months. Find the equated time
for one payment of £434, according to mercantile discount.
23. Eind the value of
^ ^-i of iUlfL^L of ^^^y^^-Q'Q^^- of 13 days 3 hrs.
3_7_i| of 4i £2 17s. 2 yds. 1.7 ft.
Invent a question to which the last three factors in this expression may
be the answer; and show how they are so.
24. Divide 99 into four parts, so that the first shall contain 3 for
every 4 in the third and every 5 in the fourth, and so that | of the second
may be I of the sum of all the rest.
25. Divide 8s. among A, B, C, so that A may receive 8J. as often
as B receives 3t/,, and B may receive 5c/. as often as C receives Zd.
26. Express in lowest terms the product of
9 25 49 81 11 59 181
27. The sum of 1\d. was divided among A, B, C, in such proportion
that A received ^d. more than C, and B 2|c/. less than C: Suppose a
sovereign had been divided among them in the same proportion, what
would each have received?
28. What half-yearly dividend is derived from an investment of
£1000 in the 3 per cents, at 87^, after deducting for income-tax 7d, in
ihe£?
29. AVhat interest does a person obtain for his money, who invests
in the 3^ per cents, at 91?
3 1 - 8 5 f
30. How many acres, roods, &c* are equal to - ^^ a^^o^S
K 3
198
EXAJIINATIOX-PAPERS.
Jj nf ^^ ^^- of ^ ^^- '* oz- ^^ <^^^'t- 12 srs. ^77 da. 4 ho. 30 m. ^
.026 Hi'. 3(^. 2 IbsTToz.'^vorrcr) 6 da. 12 ho.
518 sq. ft. 28 in. ?
31. Find the true discount on £100 10*. 10c?. payable in 4 years, in-
terest being at 3| per cent, per annum.
32. Wliat sum of money improved by simple interest, at 3i per cent.
per anmim, for half a year, will amount to £14 16s.?
33. What would be the true present worth of £294 25. 6d., for 3^^tj
years, reckoning simple interest at the yearly rate of 4.027 guineas per
£100?
34. If the simple interest of £162.871 for 148 days were £2.8142
what would be the rate per cent, per annum?
Paper XIV.
Supplementary Miscellaneous Questions. [B. {
1. Two numbers have for their greatest common measure 537 and
for their least common multiple 18795. What must the greater ri° be,
if the less is =105 times ?| of ^^MZ?
n 8.4
2. The circumference of the fore wheel of a carriage is 6| feet,
and that of the hind wheel is 12§ feet. How many feet must the car-
riage pass over before both wheels shall have made a complete number of
revolutions?
3. The diameter of the fore wheel of a carriage is f of that of the
hind wheel, and the former makes 528 revolutions in passing over | of
a mile. How many revolutions does the hind wheel make in passing
over a mile? and what is the circumference of each wheel?
4. In what proportion must water be mingled with spirits worth
lO.s. 6c/. a gallon, to reduce the value to 9i'. \\d. per gallon ?
5. IIow much ore must one raise, that on losing — in roasting
40
and -— of the residue in smelting, there may result 506 tons of pure
metal?
6. £-225 9s. is due in 48 days, and £599 8s. in 26 days:— What
sum paid at present would discharge both these debts? and how many
days would be the equated time for one payment of the £824 17s.?^
interest being reckoned at 6 per cent.
1. A cubic foot of water weighs 1000 oz. avoirdupois ; a pipe
whose bore is 3^ square inches discharges 252 lbs. per minute; find the
velocity per hour of the issuing water.
EXAMINATION-PAPERS. ^*"^^
8. If when corn is 15s. Od. a quarter, and hay 5^^/. per stone,
7 horses can be kept 8 days for £4 Is. 3(1; how many weeks ca;} 16
horses be kept for £95, Avhen corn is 2s. a busliel, and hay 705. a ton,
supposing that 126 lbs. of hay are consumed with 1 bushel of corn?
9. An analysis of the Board of Trade returns for 1861, respecting
shipwrecked lives, gave the following results: — Saved by life-boats, 13^
per cent. ; by rocket and mortar apparatus, 8 per cent,; by ships' boats,
&c., 62 per cent. ; by individual exertion | per cent.: lost, 16 per cent.
Determine the number of lives saved, by the several means enumerated,
corresponding to an excess of 2619 rescues by ships' boats over those
by life- boats.
10. Find two decimal fractions together equal to -r, and such that
one may be — of the other.
11. A stationer by selling quills at a guinea a thousand, gained f of
what they cost him. What was the prime cost?
12. A ring weighs 1 dwt. 4 grs., and is worth £l 2s. If 1050 of
such rings be packed in a box weighing 3| lbs., what Avould it cost to
convey them 144 miles, at the rate of 55. per ton per mile, insurance
being demanded at the rate of | per cent. ?
13. A monolith of red granite m the Isle of Mull is said to be about
108 feet in length, and to have an average transverse section of 113
square feet. If shaped for an obelisk, it would probably lose one-third
of its bulk, and then weigh about 600 tons. Determine the number of
cubic yards in such an obelisk, and the weight in pounds of a cubic foct
of granite.
14. Show that, in comparing the rates of two locomotive bodies, A
and B, if the distance passed over per unit of time by ^ is | of that by
B, then A's time per unit of distance is ^ of J5's.
15. A has 38 florins and a sovereign; B has 61 half-sovereigns and
11 florins. What sum transferred by 5 to -4 would make B have
exactly 6 times as much money as ^ ?
16. The difference of two numbers is 477^, and oneof thera is to
the other as - of 2| of 1.53 is to 5^^ x 4i. Find the two numbers.
17. With what capital did a tradesman commence business, if at the
end of 12 months his nett gain amounted to £210 14s.] a certain
portion only of that gain being accounted trade profit, the remainder,
viz. 5 shillings for every 9 shillings of the trade profit, being legal
interest of capital?
18. The sum of £100 has been accumulating at compound interest
200
EXAMIXATIOK rAPKRS.
for 125 years jit S per c-nt. : the amount is now invested in 3 per cent,
consols at 95. What will be the annual income therefrom ?
N. B. LOS"'" = 4.383906; and only four places of decimals
need be retained in the result.
19. If the discount on £5G7 be £34 145. 3fi., simple interest being
reckoned at 4^ per cent., when is the sum due?
20. A narrow rectangular field, ABCD, has its length AB 160
yds. and breadth BC^\^- yards. To what point E m the side AB must
a straight line from C be drawn, so that AECD may contain an acre.'
21. A person ir.vcsts £6200 in the 3 per cents, at 89-J-, and pays
income-tax lOt/. in the pound; on the stock rising to 92 he sells cut,
and invests the proceeds in £50 railway shares which yield an annual
dividend of 3| per cent., clear of income-tax. Find the alteration in
his ineome.
22. Certain railway shares pay an annual dividend of £3 10s. A
person having bought 12 shares, at such a price that they yielded 5§ per
cent, on his investment, sold them when the price had risen £5, and
invested the proceeds in 3| per cent, stock at 85. Find the alteration
in his income.
23. What fraction of ^.0135 is ^.004.
24. From 1 of V5.92 subtract -1 of V61.77.
Papjer XV.
Supplemeniary Miscellaneous Questions. [C]
1. A corn merchant having bought 1300 quarters of wheat, sold
one-fifth of it at a profit of 5 per cent., one-third at a profit of 8 per
cent., and the remainder at a profit of 12 per cent.; but had he sold all
at a profit of 10 per cent., his gain would have been £16 135. 8 J. more.
\Yliat did the wheat cost him ?
l-I-l = - sold at 12 p. c. profit.
o o 15
.', the several quantities are as 3, 5, and 7.
£3x 1.05 = £3.15
5x1.08= 5.40
7_xl.l2=.- 7.84
16.39
15x1.10=^16.50
.11
EXAMIKATlOX-rArEIlG. 2Ul
That is, on every £15 of llic uliolc prime cost the gnia
would have been £.11 more ; liencc,
£.11 : £iG 135. 8c/. :: £15 : £2275. Aus.
2. The gross receipts of a railway company in a certain year are
apportioned thus : — 40 per cent, to pay the working expenses, 54 per
cent, to give the shareholders a dividend at the rate of 3| per cent, on
their sliares, and the remainder, £28350, is reserved. Find the paid-up
capital of the company.
100-40-54= 6 p. c. of gross receipts is reserved.
/. 6 : 54 :: £25350 : £255150 amt. of dividends.
3| : 255150 :: £100 : £7290000. Ans.
3. What is the exact time between 5 and 6 o'clock when the hour
and mmute hands of a watch should be at right angles to each other?
and what, when they should be coincident?
Call the hour hand H, and the minute hand M. At
5 o'clock, H is 5 twelfths of the circumference in advance of
M; and it is required to find at what time after 5 o'clock the
interval between // and M will be 3 twelfths.
Now, as (5 — 3) twelfths and (5 + 3) twelfths are both proper
fractions, there will be two occurrences of the interval.
In the first instance, M has to gain 2 twelfths on H, and in
the second instance 8 twelfths; and, as M goes 12 times as
fast as //, and gains 11 twelfths of the circumference per hour,
we have
11 tw. : 2 tw. :: go min. : lOif min. past 5;
11 tw. : 8 tw. :: eo min. : 43^ min. past 5;
which are the times when the hands intercept a fourth of the
circumference, or are at right angles.
Similarly, to find when the hands are coincident is to find
when M will have gained 5 twelfths of the circumf. on H.
11 tw. : 5 tw. :: eo min. : 27^ min. past 5;
which is the time when H and M point in one direction.
Note. The third answer might have been found thus:
(10i^ + 43^)-h2=27fimin. past 5.
4. At what rate must I sell sherry that cost me 40s. a dozen, if I
am to gain on every £100 of outlay the selling price of 5 dozen?
* £ 1 00 4- £2 = 50 dozen bought for £100;
and I am to sell (50-5) or 45 dozen for the prime cost of
50 dozen, viz. for £100 ;
.'. £100-^45 = 44s. by. per doz. Ans.
5. ^'s present age is to B's as 9 to 7; and 34 years ago the pro-
puriion was 5 to 2. Find the present age of each.
202 EXAMINATION-PAPERS.
In solving such problems it is borne in mind that the
difference of the ages of two persons is always the same, tlioiigh
the ratio of the ages is ala-aijs varying.
Here, then, we have A& present age to i^'s as 9 : 7 ; an:] 9
is 4^ times (9 — 7). Similarly, ^^'s former age -was to 2?'s as
5:2; and 5 is 1| times (5 — 2).
Therefore, ^'s present age is 4^ times the difference of ^'s
and J5's ages ; and his former age was 1| times the same dif-
ference ; so that we have
1-10
^'s former age = — ^" or - , of his present age ;
.'. — of ^'s present age = 34
.'. ^'s present age = 54,1
»,, 7 n vAns.
J5's-of54, =42.
9 * }
6. A boatman rows 5 miles with the tide in the time he would take
to row 3 miles against it ; but if the hourly velocity of the current were
\ a mile more, he would move twice as rapidly Avith the tide as against
it. What is his power of rowing in still water ?
If 5 represent his rate with the tide, then 3 represents his
rate against the tide, and the average of these, viz. ^(5 + 3), or
4, represents his rate in still water ; also 5 — 4, or 4 — 3, viz. 1,
represents the velocity of the current, = j of his rate in still
water.
Again, if 2 be his rate with the tide, and 1 his rate against
it, then ^2 + 1), or 1 \, is his rate in still water ; also 2 - 1 ^, or
1^ — 1, viz. i, is the velocity of the current, = i of his rate in
still water.
.*, ._ _ _ , or - of his rate in still water is = -^ a mile per
3 4 IJ 2 F
hour ; and hence his rate in still water is ^ a mile x 12 = 6 mi,
an hour. Ans.
7. A contractor engages what he considers a sufficient number of
men to execute a piece of work in 84 davs ; but he ascertains that three
of his men do, respectively, - ^ , and ^, less than an average day's
6' 7 9
work, and two others - and — more ; and in order to complete the
8 10 ' ^
work in the 14 weeks, he procures the help of 17 additional men for the
84th day. How much less or more than an average day's work on the
part of these l7 men is required?
EXAMIN.VTION-PAPERS.
203
Here, instead of 5 men working with ordinary ability,
during the 84 days, there are
•1 + !+?. + ^ + 1^ =r4mi ordinary men;
G 7 9 8 10 ^•'^" ^
so that the deficiency to be made up is equal to the work of 1
ordinary workman for 84 times — ^ da.
2520
493
» 1 ordinar}"- workman for -^ dnys,
30
= 17 ordinary workmen for — of a day,
or, 17 men each doing -— less than an average day's work. Arts.
30
8. A farmer gave for a horse a bill of £73 due in 1 month, nnd sold
him at once for a bill of £87 at 4 months. Required the farmer's gain
per cent , reckoning interest at 4^ per cent.
lOOi : 100 :: £73 : £— , Pres. Worth of £73;
loU : 100 :: £87 : £
600
Do.
800
11
of £87i
:117f;
GOO .. inn • inn 6 11
— i: 100 . 100 X - X • —
7 7 8
or, 1 7f per cent. gain. Ans.
9. Divide the number 237 into three parts such that 3 times the
first may be equal to 5 times the second and to 8 times the third.
Since 5 times the 2nd = 3 times the 1st,
.-. the 2nd = - of the 1st;
5
similarly, the 3rd=l of the 2nd;
8
and - of -,
8 5'
and the three parts arc as 1 , - ,
5
or, as 40, 24, and 15 j
40
79
24
of 237 = 120, the 1st,
~ of 237= 72, the'2nd,
"9 ' '
15
79
of 237= 45, the 3rd.
Ans.
10. Divide £5433 185. into three sums, such that their amounts by
compound interest at 5 per cent* per annum, for 20j 23> and 27 years,
respectively, shall be equal.
20i EXAMIXAriOX-PAPERS.
The 1st X 1.05-'' = tlic 3rd x 1.05",
/. the 1st = the 3nl x 1.05^;
The 2nd x 1.05-^- the 3rd x 1.05",
.*. the 2nd = the 3rd x 1.05*.
Thus, the three required parts of the given sum will be as
1.05^ 1.05\ and 1 ; or, as 1.4071, 1.2155, and 1 ;
or as 14071, 12155, and 10000;
accordingly, the 36226th part of the given sum, viz. 3s., mul-
tiplied by these proportional numbers gives £2100 135.,
£1823 5s., and £1500. Ans.
^ 11. Suppose 9 men or 15 women to earn 255. a day at reaping,
when they work 9j\ hours a day; how many men with 4 women would
earn 355. a day at the same employment, if the duration of daily work
were an eighth less than in the former case ?
12. Thirteen horses do the same Avork as twenty ponies, and 12
horses can just draw a certain load on level ground; how many ponies
o
along with 5 horses could draw a load - as heavy up a gradual slope
which makes the traction more laborious by - for ascent and — for
•^ 8 10
roughness?
13. What must a person have invested in the 3 per cents, at 90|, if a
transfer of - of his capital to the 4 per cents, at 115 would increase his
5
income by £7 ?
14. Suppose that from an official return of the arrivals of oxen,
calves, sheep, pigs, and horses, in the port of London, from the conti-
nent, in a certain week, it appears that there were 3 times as many sheep
as oxen, that the number of pigs was 13^ per cent, of the number of
sheep, that for every 28 pigs there were 25 calves, that the horses were
— per cent, of the whole, and that the horses and oxen together were
3587: — What was the number of oxen?
15. A merchant has three qualities of whisky, viz. at 185., 165., and
155. a gallon, and in quantities, respectively, as 3, 4, 5 ; and with these
he mingles such a quantity of water as makes the average value
155. 6 J. a gallon. How much per cent, of the mixture is water?
16. Suppose that 15 men would be necessary to excavate 966 cubic
yards in 8 days of 10| hours each: — How many men did a contractor
engage for 12 days of 7^ hours, to excavate 575 cubic yards, if he
found it requisite to engage 4 additional men during the last 4 days,
in order to complete the work in the 12 days?
EXAMINATIOK-PAPERS. ^6B
17. I bought 128 yards of cloth for £100, and am now obliged to
sell it at a loss of as much monc^^ as I shall receive for a dozen yards.
At SThat do I sell it per yard ?
18. I bought paper at the rate of 35. 7id. for 5 quires, and sold it so
as to gain as much on the cost of 32 quires as 3 quires were sold for.
At what rate did I sell it per quire ?
19. I gave 3 sovereigns for two dozen of wine, at different rates per
dozen ; and by selling the cheaper kind at a profit of 15 per cent., and
the dearer at a loss of 8 per cent., I obtained a uniform price for both.
\\ hat did each dozen cost me?
20. F and G are partners in trade ; F contributes - of the joint capital
5
for 10| months, and G receives - of the gain. Required G's period of
8
investment.
21. At what time between 11 and 12 o'clock will the hour and
minute hands of a clock make with each other an angle intercepting 27
of the minute divisions?
22. A merchant buys two pipes of wine, one for £112, one for £120,
and he also buys a third pipe; on mixing the three, he sells his wine at
50s. per dozen, gaining 25 per cent, on his outlay; what was the price
of the third pipe? — The n° of dozens in a pipe is 56.
23. My age is 62, and my son's age 30 ; how long ago was my age
5 times that of my son ? and how many years hence (if we are both
alive) will my age be a third of 5 times his age?
24. My age was 24 when my eldest son was born, and when I attain
to twice my present age he will be 8 times as old as he is now. What is
his age?
25. A boatman rowing against the tide passes a body floating with
the tide, and in 9 minutes afterwards is a mile distant from it; in 35
minutes more he rows 2^ miles, and then returns. At what rate per
hour does he return, supposing the tide to flow uniformly in one direc-
tion?
26. A corn merchant bought 121 quarters of wheat, and he sells it
so as to gain 17| per cent, on 26 quarters, and 13 per cent, on the
remaining quantity, having previously tried to sell the whole at a uniform
advance of 15 per cent., which would have brought him £4 5s. more
than he actually received. What did the wheat cost him per quarter?
27. A watch that gains 24 seconds per hour is set to right time at a
quarter to 5 p.m. What will be the right time between 8 and 9 o'clock
the same evening, when the hour and minute hands of the watch point
in exactly opposite directions?
20C) EXAMIXATION-PAPEKS.
28. Of the whole cost of constructing a railway, f is held in shares,
and the remainder, £400000, was borrowed on mortgage at 5 per cent.
Find what amount of gross annual receipts, — of which 40 per cent,
will be requh'cd for the working expenses of the Ime, and 8 per cent. fc\
a reserve fund,— w^ill yield to the shareholders a dividend of 4ipcr cent,
on their investments?
29. A dealer buys 18 cwt. 3 qrs. at Is. 3d. a lb., which, to obtain a
fair profit, he should retail at 8^ per cent, above cost price. Bnt, while
he professes to sell at the rate of 3 lbs. for 3s. lOrf., he serves his cus-
tomers, to his own advantage, with a false balance, in which 10 Ib^
weighs 10| lb., and at the same time he uses a false lb. of 6S60 grainSc
How much does he make beyond the fair profit?
30. I have this day paid £2180, being repayment, with interest, of
two loans, both contracted by me at one time, viz. of £1163 borrowed
at 4 per cent, per annum, and £994 at 4i per cent. How long is it
since the sums were borrowed ?
31. A person borrowed £272 65. 6c/. at 5 per cent, per annum, and
repaid the loan by yearly instalments of £100, that sura including the
year's interest; how much of the debt was discharged in 3 years.'
32. What must be the gross rental of an estate, so that, after deduct-
ing 7d. in the £ income-tax, and 4| per cent, on the remainder fur
expenses of collecting, there may be left a nett rental of £1000 ?
33. I sold an amount of railway stock at 104, and invested the pro-
ceeds in the 3 per cents at 91; I then sold out the 3 per cent, stock at
95, and re-purchasing the railway stock at 105, I found myself a gainer
of £50 by the whole transaction. Eequired the amount of railway
stock.
34. The interest on a certain sum of money for 2 years is
£71 16s. 7|f/., and the discount on the same sum, for the same time, is
£63 17s., simple interest being reckoned in both cases. Find the rate
"^er cent, per annum, aud the sum.
35. At what rate per cent, per annum, compound interest, would a
sum of money in 2 years amount to the same as at 3^ per cent, per
annum simple interest?
33. If a publisher, in selling a book for cash, rates it at 25 per cent,
below publishing price, and then charges for IS copies as 12, how long
credit could he allow, so that, on the principle of true discount at 4 per
cent, per annum, the sum to be received for a book should be just 29 per
cent, below publishing price?
37. The external length, breadth and height of a rectangular v/oodea
fiXAMit^ATION-PAPERS. 207
closed box are 18, 10, and 6, inches, respectively, and the thickness of
the wood is half an inch. When the box is empty it weighs 15 lbs,,
and when filled with sand, 100 lbs. Compare the weights of equal
bulks of wood and sand.
38. I bought goods at 235. 9(7. with 4 months' credit, and sold them
forthwith at 25s. 6d. with such allowance of credit as made my gain f>|
per cent. How long credit did I give, reckoning interest at 4 per cent,
per annum ?
39. If I am allowed 1^ per cent, discount on an amount charged to
me for goods, and give my acceptance at five months for the nett sum;
and if by selling the goods forthwith for a bill of £162 12^. 2d., payable
in 7 months, my present gain is ll^ per cent.; what is the amount
originally charged to me, interest being reckoned at 5 per cent, per
annum?
40. The present income of a railway company would justify a
dividend of 4 per cent., if there were no preference shares; but as
£200000 of the stock consists of such shares, which are guaranteed 5
per cent, per annum, the ordinary shareholders receive only 3^ per cent.
What is the whole amount of stock?
41. A man bought a house, which cost him 4 per cent, upon the
purchase money to put into repair; it then stood empty for a year, during
which time he reckoned he was losing 5 per cent, upon his total outlay.
He then sold it again for £1192, by which means he gained 10 per cent,
upon the original purchase-money. What did he give for the house?
42. (a) Show that if 5 times A, 6 times B, and 7^ times C, are equal
11 2
quantities, then A^ B, and Care in the proportion of -, -, and —
5 6 15
(i) What is meant by the reciprocal of a number ? What fraction
divided by its reciprocal gives a quotient equal to ^ ?
43. Divide 33 cwt. 2 qr 22 lb. into three such parts that 6 times the
first, 9 times the second, and 10 times the third may be equal amounts.
44. Divide £36 Ss. into four parts such that their simple interests for
4, 6, 7, and 10 months, and at 3, 4, 5, and 6 per cent, per annum,
respectively, shall be all equal.
45. Divide £3010 into three sums, so that if the first be put out at
.'imple interest for 3 years at 4 per cent, the second for 5 years at 3 per
cent., and the third for 2 years at 2| per cent., the amount of the second
shall be double that of the first, and the amount of the third treble that
of the second.
46. By the sale of g3ods Avhich cost mc £3 195. 2d. I lost a sura
208 EXAMINAtlOX-rAt»ERS.
equal to 5§ per cent, of the piocceds; and by the sale of another
quantity which cost me £o I gained a sum equal to 31^ per cent, of the
proceeds. What did I gain per cent, on the wliole ?
47. If 9 oxen arc kept for the same money as 7 horses (for any given
time), and a team of oxen arc - as long again in ploughing 97 acres
5
as tlie same number of horses arc in ploughing 90 acres, and a field
costs as much whether piouglicd by oxen or horses, viz. £7 5s. Gd. ; the
same men being required in both cases, and being paid by the time, wliat
is du3 to them ?
48. If 28 men can excavate 750 cubic yards in 4 days, working 62
hours a day ; what uniform length of day will 24 men require, to
excavate 615 cubic yards in 3i days, supposing that any 5 of the latter
party can do as much in 4 hours as any 6 of the former can do in 3i
hour?, and that 2 men will be withdrawn from the latter party after
2i days' work ?
49. In a certain manufactory, 158 men of ordinary ability, and
working the same number of hours each day, execute a certain piece of
work in a week ; but if the abilities of 2 of them had been, re^pectivcly,
11 3 3
-■ and - less than ordinary, and the abilities of 2 others - and - more,
7 9 •' 6 8
the work could have been finished '- of an hour sooner. How many
83 ^
hours a day did the men work ?
50. The interval between the firing of two guns, at a railway station,
was 6 minutes, and a passenger in a train, approaching the station at a
Uniform rate, heard the second report 5 min. 51 sec. after hearing the
first. Now, suppose the sound of the train's approach to have become
audible at the station when the train was 2 miles off, how soon after
that did the train pass the station, — sound travelling 1125 feet per
second ?
209
ANSWEES TO THE EXAMPLES.
1. 492480; 161280.
4. 3021; 3300.
7. 51520; 206080.
10. 996528; 73029.
13. 92160; 25200."
16. 3816; 21607.
19. 44160; 324003.
22. 1132; 37584.
25. 1096; 440.
28. 3936} 188.
2. 16000; 84000.
5. 45647 ; 40821.
8. 6912; 394240.
11. 10708; 408584.
14. 13200 ; 733.
17. 126060; 15620.
20. 1180; 716.
23. 351; 361152.
26. 1088; 7040.
29. 9855; 2030400.
3. 6600; 842.
6. 14161 ; 164760.
9. 21728; 84624.
12. 26921 ; 1741872
15. 4750; 16820.
18. 2S624; 45780.
21. 8760; 23184.
24. 1074088; 599616,
27. 1158; 1032.
30. 3960; 16815600.
1.
3.
5.
1.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23,
24.
2.
3751916; 3752. 2. 7329 ; 29316.
1429 ; £208 65. 8 J. 4. £295 175. U\d.', £458 7s. 8o?.
400^. 17s. 6(/. ; £l28 8o>. %\d. 6. £364 lis. 8c?. ; 1167^/. 13s. \ld
16 tons 15 cwt. 1 qr. 20 lbs. ; 3 cwt. 3 qrs. 2 lbs. 9 oz. 14 dis.
4 tons 1 cwt. 3 qrs. 7 lbs, 5 oz. 12 drs. ; 60 cwt. 1 qr. 16 lbs. 10 oz.
2 tons 15 cwt. 3 qrs. 6 lbs. ; 1 qr. 22 lbs. 1 oz. 5 drs.
6 tons 8 cwt. 14 lbs. 1 oz. ; 10 cwt. 3 qrs. 25 lbs. 6 oz. 15 drs.
5 cwt. 1 qr. 23 lbs. 7 drs. ; 28 tons 2 cwt. 2 qrs. 1 oz.
6 tons 12 cwt. 1 qr. 1 lb. 15 oz. ; 12 cwt. 3 qrs. 22 lbs. 5 oz. 3 drs.
2 lbs. 3 oz. 8 dwts. 20 grs. ; 125 lbs. 3 oz. 6 dwts.
2 lbs. 11 oz. 11 dwts. 9 grs. ; 2 lbs. 1 oz. 13 dw^is. 15 grs.
18 lbs. 11 oz. 10 grs. ; 32 lbs. 9 oz. 18 dwts. 9 grs.
47 lbs. 4 oz. 7 dwts. 13 grs. ; 22 lbs. 1 oz. 3 dwts.
6 m. 6 fur. 150 yds. ; 43 lea. 2 m. 2 fur. 31 yds.
15 fur. 56 yds. 1 ft. 7 in. ; 71 m. 4 fur. 205 yds.
8 m. 1 fur. 86 yds. 4 in. ; 11 lea. 1 m. 6 fur. 110 yds.
849 yds. 3 na. ; 9098 ells 2 qrs. 2 na.
758a. Ik. 1 p. ; 25 sq. yds. 6 ft. 69 in.
125 A.; 15sq. yds. 3 ft. 128 in.
4 cub. yds. 7 ft. 1280 in. ; 2 cub. yds. 26 ft. 57 in.
2 cub. yds. 7 ft. 1513 in. ; 3 r3- yds. 23 ft. 1 1 19 is*
210 ANSWERS TO THE EXAHPLES.
25. 2273 gals. 3 qts. 1 pt. ; 968 gals. 1 pt. 3 giils.
26. 22 Ids. 2 qrs. 1 bus. 1 pk. 1 gal. ; 178 qvs. 3 bus. 1 pk. 1 gal. 2 |tg.
27. 561 Ids. 1 bus. 1 pk. ; 22 Ids. 7 bus. 1 pk. 2 qts. 1 pt.
28. 278 Ids. 1 qr. 2 bus. 3 pks. 3 qts. ; 9354 qrs. 7 bus.
29. 377 yrs. 214 days ; 5 w. 6 d. 5 lirs. 23 m. 49 s.
30. 1404 w. 3 d. 23 h. ; 2 yrs. 101 d. 20 h. 25 m.
1.
£ s. d.
12 8 1 2.
£ s. d. £ s.
140 18 J.0 3. 207 12
d.
n
£ 8. rf.
4. 162 14 11
5.
120 1 8 6.
87 1 0 7. J14 12
loi
8. 169 19 Oi
9.
110 17 5f 10.
82 1 10 11. 172 2
li
12. 193 2 2i
13.
lbs. oz. dr.
47 1 11 14.
qrs. lbs. oz. cwt. qrs.
8 18 12 1.5. 61 3
lbs.
0
qrs. lbs. oz.
16. 80 15 0
17.
qr. lb. oz. dr.
12 11 5 9
cwt. qr. lb. oz.
18. 120 2 0 2
tons cwt. qr. lb.
19. 43 9 2 17
20.
oz. dut, gr.
31 1 14 21.
lb. oz. dut. oz. dwt.
84 7 9 22. 34 15
11
lb. oz. dwt.
23. 133 5 10
24.
lb. oz. dwt. gr.
116 6 2 23
lb. oz. dwt. gr.
25. 107 1 10 1^
lb. oz. dwt. gr.
26. 73 2 0 I
27.
dr. scr. gr. oz dr. scr. dr. scr.
22 2 16 28. 36 4 2 29. 37 0
. gr.
7
oz. dr, scr.
30. 39 6 1
31.
yds. ft. in.
58 0 3 32.
fur. po. yds. m, fur,
24 34 4 33. 21 0
. yds.
54
lea. m. fur.
34. 27 0 6
35.
fur. po. yds.
22 10 4| 36
po. yd. ft. yds. ft.
. 102 0 1 37. 30 1
in.
2
po. yds. ft, in.
38. 28 4 2 11
39.
po. yds. ft. in.
32 4 0 7
m. fur. po. yds.
40. 119 2 27 2
41.
m. fur. vds. ft.
27 0 i33 2
42.
yd 5. qrs. na.
167 0 1 43.
vds. qrs. na. ells qrs.
984 0 0 44. 328 3
na,
1
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45. 142 0 1
46.
s.yds. s.ft. s.in.
115 3 44 47.
R. P. S.vds. A. R.
30 9 18 48. 131 0
p.
21
A, R. P.
49. 162 2 23
50.
p. s.yds. s.ft. s.in.
16 24 3 101
A. R. P. S.yds.
51. 98 2 18 23 52,
, 103
p. s.yds, ft, in.
9 25i 3 23
53.
[^.yds. eft. c.in.
92 9 429
cvds. eft. c.in,
,54. 106 10 8
c.yds. eft. c.in.
55. 95 11 108
56.
gals. qts. pt.
150 3 1 57.
gals, qts, pt. pks. gal,
, 103 3 1 58. 21 1
qt.
bus. pk. gal.
59. 115 1 1
60.
qrs, bus. pks.
119 2 2 61.
Ids. qrs. bus. bus. gal,
119 4 4 62. 124 5
. qt.
bus. pks. gal.
63. 168 3 1
64.
gal. qt. pt. gills
93 1 0 3
bus. pks. gal. qts.
65. 155 3 1 2
66
qrs. bus. pks. gal.
:. 150 0 3 1
67.
d. h. m. s.
22 2 28 59
mo. w. d. li.
68. 115 1 1 14
69.
d, h. m. s.
20 21 49 48
70.
y. d. li. ni.
32 114 21 3
V. w. d. h.
71. 94 41 6 11
72.
V. d. h. m.
28 184 4 0
ANSWERS TO THE EXAMPLES. 211
£ s. d. £ s. d. £ s. d. £ s. d.
1. 10 3 3 2. 33 7 2^- 3. 60 12 2i 4. 15 3 10
5. 55 9 10 C. 8 7 6 7. 2 18 1| 8. 187 1 21
9. 25 17 2i 10. 38 2 Qi 11. 77 15 1| 12. 215 2 3i
lbs. oz. drs. qrs. lbs. oz. cwt. qrs. lbs. qrs. lbs. oz.
13. 14 4 2 14. 7 10 3 15. 20 2 15 16. 0 25 7
qrs. lbs. oz. ton cwt. qrs. cwt. lbs. oz. qrs. lbs. oz.
;7. 8 U 4 18. 1 G 2 19. 14 27 12 20. 3 21 6
oz. dwt. gr. , oz. dwt. gr. lbs. oz. dwt. oz. dwt. gr.
21. 3 4 10 22. 13 17 23 23. 6 7 17 24. 8 1 2
oz. dwt. gr. oz. dwt. gr. oz. dwt. gr. oz. dwt. gr.
25. 21 4 8 26. 36 8 11 27. 8 10 15 28. 14 6 6
dr. scr. gr. oz. dr. scr. ll)s. oz. dr. dr. scr. g-.
29. 3 0 19 30. 2 2 1 31. 17 7 7 32. 1 0 16
yd. ft. in. po. yds. ft. fur. po. yds. m. fur. yds.
33. 1 1 9 34. 9 3 2 35. 5 21 3 36. 4 6 124
m. fur. po. fur. po. yds. lea. m. fur, fur. po. yds.
37. 12 2 29 38. 1 18 5 39. 18 2 6 40. 0 27 4
po. yds. ft. yds. ft. in. yds. qrs. na. ells qrs. n.i.
41. 7 4 1 42. 7 0 5 43. 4 3 1 44. 4 4 2
s.yds. s.ft. s.in. p. s.yds. s.ft. R. p. s.yds. A, n. p.
45. 6 2 86 46. 8 22 6 47. 0 6 27 48. 13 2 34
A. n. r. R. p. s.yds. R.s.vds. s.ft. s.yds. s.ft. s.in.
49. 25 2 36 50. 1 13 22 51. 2 2^ 6 52. 3 3 27
c.yds, eft. c.'n. c yds. eft. c. in. c.yds. eft. c.iii. c.yds. eft. c.in.
53. 12 14 1071 54. 29 4 655 55. 33 4 1385 56. 13 16 999
gals. qts. pt. gals. qt. pt. pks. gal. qt. bus. pks. gal.
57. 2 2 1 58. 5 1 .1 59. 3 11 60. 18 2 1
qrs. bus. pks. Ids. qri. bus. bus. pk. gal. Ids. qr. bus.
61. 5 3 3 62. 12 4 6 63. 17 1 1 64. 2 1 4
hrs. in. s. d. hrs. m. w. d. brs. mo. w. d,
05. 13 57 49 66. 7 19 45 67. U 5 13 68. 3 2 6
vrs. d. lirs. yrs. w. d. yrs. w. d. yrs. d. hrs.
09. "12 196 9 70. 8 39 5 71. 10 43 4 72. 6 346 14
5.
£
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212 ANSWERS TO THE EXAMPLES.
6.
£ s. d. £ s. d. £ a. d.
1.
358
1 lOl
2.
1435 7 4|
3. 1961 14 li
4.
1389
14 8
5.
2392 18 lOi
6. 4
703 10 0
7.
5581
13 4
8.
6481 8 4
9. 3050 9 101
10.
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12. 5
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6. 1532 4 91
7.
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8.
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9. 2536 3 2i
10.
2318 16 9i
11.
6 cwt.
Iqr. 26lb,
15(
3Z. 8 dr.
12. 41 tons 18 cwt. Iqr.
181b. 10 oz.
13.
159 tons 1 cwt. 10 lb.
13 oz.
14. 314 tons 10 lbs.
15.
31 tons 19 cwt. 1 qr. 6lb. 11 oz.
16. 811 tons 15 cwt. 3 qrs. 3 lb. 4 oz.
12 dr.
9 dr.
17.
1821b.
lOoz. 1 dwt. 13 gr
18. 131 lb. :
2oz.
15 dwt.
20 gr.
19.
12 lea.
1 m. 4 fur.
16 yds. i
Bin.
20: 19 lea. :
2 m.
1 fur. 98 yds. 8 in.
21.
414 A.
IR. lOP.
22. 1255 a.
3r. 32 p.
23.
319 sq.
yds. 1ft. 112 in.
24. 1493 cub. yd
s. 11 ft.
1332 in.
25.
7908 gals. 3 qts.
26. 3612 gals.
27.
96 Ids.
1 qr. 2 bus.
28. 79 Ms. ;
3 qrs.
2 bus.
29.
I'yr. 3:
£ s.
23 d. 6 Ii. 40 m.
d. £
30. 2491 yr
s. 247 d. 2 h.
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16 m. 48 s.
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ANSWERS TO THE EXAMPLES.
213
11.
£ s. d.
1. 28 17 Hi
5. 12 15 111
0, 11 3 8i-
2. 17
6. 0
10. 0
£ s. d.
3. 1 7 2
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7. 9.
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11. 43.
12.
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29.
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157^79.
16.
26.
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1. 876 & 15*.; 1024 & 8s.
3. 154 & 85.; 1062 & 3*. 4
5. 147 & 2s. 6c/.; 1090 & Ad. 6
7. 138 lb. 6 oz. 10 d\Yt. ; 6 dr. 1 scr. 4 gr.
8. 24 IE). 3 oz. 13 dwt. 8 gr. ; 12 dwt. 12 gr.
9. 597 & 2 qr. ; 4 & 8 in. 10. 1000 j 550,
2. 90 & 105. ; 2814.
4. 1250 & 2s. ; 27750.
3150; 285 & 5*.
1ft.
1.
s.y. s.f. s.in.
11 3 30
s.y. s.f. s.in,
2. 8 6 84
s.y. s.f. s.in.
3. 3 0 72
4.
1 6 60
g. 20 5 101
6. 22 3 108
7.
56 8 0
8. 92 4 0
9. 1 1 34
10.
241 8 112
11. 55 2 142
12. 68 8 72
15.
1.
2 ft. 9 in.
2.
12
yds. 1 ft. 5 in.
3.
4 yds. 1 ft. 8 in.
4.
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9.
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13 ft. 1 in.
130.
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The other side is 26 yd
6.
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341yds. 1ft
11.
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12.
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16.
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4.
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7 ft.
8.
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120 ft.
9.
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15 2 1673
7783600 c. ft-
214
ANSWERS TO THE EXAMPLES.
1.
3.
6.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
36.
37.
39.
41.
43.
45.
47.
49.
61.
53.
64.
66.
68.
69.
61.
63.
65.
66.
67.
17.
2. 29 da. 12 hrs. 44 min. C ssc.
4. £16 45. Ud.
6. £21 195. 6d.
8. 250 ft.
10. £91 105. 6^.
12. £387 l5. ly.
14. £19 45. Old.
16. 15 cwt. 7 lbs. 8 oz»
18. £9895 165. Sd.
20. 91717720 mi.
22. £13069 05. 7d.
24. 105. i^d.
26. 13.
28. 114 lbs. 15 (Iwt. ; £3437500.
30. 11 da. 17 brs. 43 min. 20 sec.
£10110 Us.9d. 32. 20833i lbs.
34. 648.
36. £664.
88. 355 sq. yds. 7 ft. 126 in.
40. £22 7s. 6d.
ft. 1152 in. 42. 5ld.
44. 168 tons 7| cwt.
46. Is. U^d.
3 qrs. 12 lbs. 48. 7 mi. 2 fur. 120 yds.
60. £4 145. 7ld.; £5 85. 2d.
62. 63. Sd.
13 ac. 2957 sq. yds. 7 ft. ; 10 ac. 1477 sq. yds. 7 ft.
353571 tons 8 c-\rt. 2 qrs. 8 lbs. 65. 58| yds.
725 gal. 67. 5044.
A man, £16 105. ; a "woman, £5 105.
7. 60. 3 ac. 584 sq. yds. ; 10 ac.
20. 62. 750 bu.
2 yrs. 334 da. 19 hrs. 30 min. 64. £9 35. id. ; £5 85. id. ; £5 85. id.
A man, £66 05. 4^^. ; a woman, £33 05. 2\d. ; a child, £11 05. 0|d
A, 7s. 3^. ; 2i, Ids. U^d. : C, 27s. Ud.
10240.
Loss in one year, £122 105. ; gain in three years, £698 Qs, Sd,
21 lbs. 4 oz. 16dwt.
£94 195. 2d.
24857 mi. 1680 yds.
1907314.
6 da. 22 hrs. 40 min.
132 yds. 2 ft. 7 in.
976 ducats.
365 da. 5 hrs. 48 min. 48 sec.
44 tons 12 cwi:. 3 qrs. 12 lbs.
1 mi. 4 fur. 20 yds.
3 mi. 3 fur. 60 yds.
£7670.
63 yds.
£193 155.; 60 minse.
£1919 55. 5d.
£12389 15. 3ff,
37 oz.
50606 gal.
26 }-ds. 2 ft.
£33 25. 6ld.
102700 cub. yd
175. id.
£148 10^
1607 tons 2 cwt.
235. ild.
215.
3. 1(
ANSWERS TO THE EXAMPLES. 215
ISi 1. 112. 2, 4. 3. 1. 4. 25 5, 101.
6. 143. 7. 377. 8. 11. 9. 18. 10. 7.
11. 1. 12. 77. 13. 133. 14. 49. 15. 213.
16. 25. 17. 336. 18. 57. 19. 3. 20. 15.
19. 1. 60. 2. 42. 3. 16. 4. 198. 5. 240.
6. 80. 7. 180. 8. 144. 9. 120. 10. 68.
11. 144. 12. 216. 13. 240. 14. 2520. 15. 7560.
16. 1008. 17. 12G0. 18. 10500. 19. 7200. 20. 10800.
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ANSWEBS TO THE EXAMPLES.
21?
1.
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2. £2 65. 8 J. ; £2 45. lllJ. ; £3 25. Slid
3. £13 85. ll^d; £22 I5. 9^6?.; £39 lis. ll^c?.
4. £63 135. bd.; £91 175. IQlf/.; £9 95. 7^.
5. £176 135. lid.', £49 35. lOfJ.; £46 45. Ollc?.
6. 14 cwt. 1 qr. 4 lbs.; 3 oz. 8 dwts. 13f grs. ; 2 cwt. 2 qrs. 6 lb«.;
£4 l5. %\d.
7. 8 vr. 4 d. 10 h. 40 m. ; 39 A. 1 R. lip ; 35.
8. 25. 3y. ; 5 cwt. 2 qrs. 9|^lbs. ; £100 85. 4d
9. £4 l5.; £8 145. e^d. ; 45. 8d.
10. 2 fur. 124 yds. 2 ft. j £4 25. 2d
218 ANSWERS TO THE EXAMPLES.
11. 165. ll|i. ; 2 qrs. 17 lbs. l^oz. ; 5 d. 38 m. 20 sec.
12. IGs. ll|d 13. £3 Is. eld. 14. £l 2>.
15. 3s. 7ld. 16. £7 i7s
34.
*> .19.. JJL fi Jil-' 51 7 «jJ5 . 5 Q KinoS . II
^' 448 » TT2' O* TG0» ^4* '• ^63 » IT' «»• l^^^HJlS'
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•10 /'^9 .04 IJ. 100 . 12 IK 1 1 • 123 ic 4. ol4
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5 11 • .§_ 6 8^ • 8^ 7 194. 13 o IS. 119
9. i_. _21_, 10 18^' lOl n 1719. _1- 19 131. 17
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13 IM.. 9±2 14 414. 107 IK 43 . 12 ip:
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36.
1- :^ greatest, ^ least. 2. 14|. 3. 1^.
^ 7 ^ni T.„ 1 n 20002,400 ^ ^ oj
4. I of li by ^. 5 — ^yyg — . 6. 5s. 3d.
7. i- 8. ^. 9. 26fft. 10. 3^.
1 1. 9 oz. 3 dwt. 8 gr. ; 14*. 3d. 12. 3^ ; 108^ sq. in.
13. £5 6s.8d. 14. 31; li|. 15. |.
16. ^. 17. 1. 18. }. 19. fM.
20. 5^;2e. 21. £67 4s. 3ld. 22. 1^;11;2^.
23. £85 14s. 3f(f.; 4s. 7c?. 24. 4| ; 42. 25. £21 8s. l^^.
26. £9973 6s. 8d. 27. ^'^^ 21, 24^ 28. 4||. 29. £90.
20 ^*
30. 19 dwt. 9 gr. 31. 14^;^. 32. 240, 2^8^b, 303^
33. 1^; £7 16s. 5lc?. 34. 21s. 35. 23 lbs. 17 dwt. 5igr.
36. Iff. 37. £4 16s. 38. £l 13s. 7|(/.
39. 59yds.; £11 Is. 3c/. 40. ^;£3125. 41. JL^;680flbs.
42. f 43. 99. 44. 9.
45. 2 oz. 8 dwts. 8 gr. Troy ; 2 oz. 10^^ dr. A v. 46. 81.
47. 12li. 48. UOiyds.; £1 6s. 3^f/.
49- 1 7 cwt. 2 qrs. 5 lbs. ; £32 16a. 4d. 50. £333 6s. 8d ; i
ANSWERS TO THE EXAMPLES.
219
1,
3.
7.
9.
11.
.7, 11.7, .33, 1.015.
.230037. 4. 1.11111.
_2Z_ 1 1 3
1000» 5000' 4' 8
1^1 5- 23
ICOOO' 3200'
3, 300 : .03,
KiOO"
0003.
37.
2.
.5.
8.
10.
.01, .0021, .0117, .0000003.
13.003005. 6. 10.110101.
.125, 12.5 ; .0000125, .000000125 ; 5387340, .0538734.
12. 1100, 1100000; .0011, .0000011; 11025, 1102500; .011025,
.00011025; 213012000; .000213012.
1. 34.62156.
4. 2492.2622123.
7. 19.0002 : 1.0013.
9. 1.33678 ; 2.7486.
1. 723.6 : 146.4561.
3. .07504 : .000«i02.
5. 5.31441 : 4.096.
38.
2. 782.8594. 3. 420.615973.
5. 19.002 : 3.44902. 6. 21.1335 : .41213.
8. .0000013 : 23.016484.
10. .003213 : .34235.
39.
2. .0000001 : 74.151.
4. .0013014 : 1.5.
6. .0001234321 : .00044-108.
40.
1. 6.25 : .000625. 2. 6250000 : .0000625. 3. 490000 : 6.3.
4. 185:30. 5. 4000:4.8828125. 6. 2.4:1200.
7. .00015625 : 7118580. 8. .0122699 &c. : l.')G8.627 &c.
9. .3388278 &c. : .00383177 &C. 10. 290 : .014974 &c.
41.
.04 : .052 : 5.25 : 1.6. 2. .848 : 11.0136 : 15.625 : 5.1875.
7.203125 : .1328125 : .00015625 : 11.001696.
.001953125 : 1.0009765625; .008125; .0013671875.
.1705: .00216; .32.
42.
1. 1.4 : .57^ : 2.345 : .01236. 2. 2.9285714
3. .0085 : 5,?61904 : 17.12931 : .12345.
4. .0364S : .1003378 I .40864 I .0^050^.
5. .058823.5294117647.
.0434782608695652173913.
.0344S27586206S96551724I37931,
.03225806451612^.
5.045 : .ci3^ ; 23.i56.
220
ANSWEFvS TO THE EXA^IPLES.
43.
11 • 37'
^55 • 3ooo • •'ss • luaoo*
K O 101 . Ill . 1 3
44.
1.
47.411455286.
2. 168.7023511456.
4.
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5. 9.92S : 2.29?.
7.
3.6 : .051
1 . o 107,
148 • -2474'
/< r.n , 7145 . gl9 . _41_
^TTTO ' ' 222 * "55 » 440'
9 JL . A)!! . 11 19
3. .24 : .0327116.
6. 31.?9i : 3.52.08564.
8. 49 : 1.145.
95. : 135. 7|(/. : £2 65. 6(1.
1.
3c 135. ly. : Is. 6ld.
5. 23 d. 10 h. 4 m. 48 sec. : lA
6. £1 Us. 3d. : £47 55. 7^^.
8. £1 11 5. eld. : 105. lid.
9. 13 r. 2 yds. 1 ft. 4 in. : 21 lbs. 12 oz. 7.68 drs.
10. 3 sq. ft. 67| in. : 102 m. 875 yds. 5.76 in.
11. £78 35. 1.8645c/. : £120 55. 9.3125d
12. £2 Is. 3.50G25J. : 65. 6J. 13.
14. 125. 1^(Z. 15. lOs.lld. 16.
17. 85 m. 7 p. l|yd. : 73 a. 2r. 20^ yds. 18.
19. 75. llld. : 85. 7ld. 20.
45.
2. £8 25. 6c/. : 6s. 2(/. :£l ll5. 8d:
4. £18 25, 3d. : 9cwt. 3 qrs.
iR. 35 r.
7. £8 95. 3|c/. : £125 135. lOlJ.
£1 35. Old.
155. 4d. : 175. 3ld.
£7 13s. l^d. : 125. 3|c/.
16 lbs. : 1 qr. 4 lbs.
46.
1.
.475 : .021875. 2. .375 : 1.72.5.
4.
.125; 27.5. 5. .3125; .196875.
7.
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9.
.19453125; .03625. 10.
11.
2.6; 1.424. 12. .00022005; .924.
14.
97.6 ; .377083.
16.
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3. 1.125 : .2625.
6. .5703125 ; .30375.
.777587890625; .0.5.
.039375 ; .046875.
13. 1.86; ,859375.
15. 4,00; 4.2083.
1.
5.
6.
8.
10.
47.
1|. 2. Id. 3. 3956 miles nearly. 4. 3^ days.
.0273437.5, 36.671428 ; 3|, 3||, .0004935, .282.
.375; £2 135. 3d.; y^d. 7. 165, ll?,c/.
.136, 4.214285? ; ^ffo ; 530, .00341. 9. IO5. 3^(/.
.3571428; 8.75. 11. 7 n. 13 m.; lA. 3r. 13 r. 22 yds
ANSWERS TO THE EXAMPLES.
221
12,
14.
16.
18.
20.
22.
23.
25.
27.
28.
29.
31.
33.
34.
36.
39.
41.
42.
43.
46.
11^11 = 11.8208.
4s. 9(/. = 1.9 of 2s. 6d.
£2 lis. id.
13.
15.
17.
£9 4s. 8l</.
£463 16s. lie?
lis. 3d.
:gl27 9s. 6d.
.06640625, .0090 ; ^, iffs ; £3 13s. lid.
19. 35. lUd.
21. £3 2s. 11^.
21^ = 2.59375.
16 ft. 104§|m.; 20 ft. 1486^ in.
.18988 ; .025 ; £4 4s. 4lc?. 24.
£4 4s. 9ld. 26.
£2 Os. 3|(/.; £6 6s. 6|J. ; 4.78125.
8.175 ; .816; 27; .75; 135.1940625.
Is. d^d. 30. £21 3s,
.109375, .1076923; 1§. ^; .54140625.
2.625, .036, 2i ^; 3.971875.
7 cwt. 3 qrs. 8| lbs. ; £8 13s. 7d.
2.140625. 37. .03.
59.0625. 40.
£32 15s.; 41.92; 1250.
.021484375, .06; 2^,^; .0009765625.
.0875 ; 4.6?. 44.
£25 17s. 2|1J.; 7s. 2|(/.
751 yds.
ll^c/.
32. £15 14s. 1 Of J.
35. £81.
38. 1.1457; 423; 18s. 6l^.
£410 lis. 9l§£(/.; £41 lis. lOld
£2 4s
3.14159. 49. £3 6s. 6|c?. ; 6.65625.
45. 9.
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50. 2.7182818.
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£ 5. d. £ s. d. £ s. d.
2. 1486 6 8 3. 1452 5 0 4. 2606 8 0
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9. 377 12 6
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2. 24 1 1^
6. 143 15 2^
3. 179 18 11
4. 85 13 3
8. 361 15 8
11, 284 6 41 12. 448 11 7^
222
ANSWERS TO THE EXAMPLES.
1. 400 4
4. 248G 15
7. 2542 0
366 13
10.
11.
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1?
51.
£ s.
1059 9
125 16 8|
2696 5 10
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201 14 91
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7.
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9. 27 19 11|^
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£4713 Is. 6ld.
£22 19^. llifZ.
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£10369 05. lOd.
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6. £48 6s. lOld.
9. 313a. 1r. 18p.
10.
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26 lbs. 2 oz. 11 dwts. 16,
£1912 14s.
grs.
13. £223 lis. Zd.
14.
£80 17s. 2l(^.
15.
£26 65. 6d.
16.
£155 95. 2d. ; £2 135. 3c/.
17. £273 65. 6d.
18.
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19.
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20. £6 105. 9^.
21.
£173 95. 4ld.
22.
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23. £25 05. Oii^^.
24.
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5ft.
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If, 2|, 2f, 311
2.
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3. 31,414171.
6. 21 6|, 6|, 10.
8. 5, 5, 5, 91.
55.
1.
£10. 2.
207.
3. £72.
4. 30.
6.
35. 6.
£00.
7. 210.
8. 378 yds.
9.
£50. 10.
£10.
11. 395 qii.
12. £4 6s. 3d.
ANSWERS TO THE EXAMPLES. 223
56.
1. £58 35. U. 2.
£5 195. 2i§c?. 3. 176 m
4. 1 b. 14 m.
5. 75a. 2k. IOp. 6.
135. Zd. 7. £1 05.
l\\d. 8. £4 155. md.
9. 15. \^^d.
10. £11 95. ^yi.
'
57.
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6 mo. 3. 12
mo. 4. 171.
5. 4. 6.
8^^. 7. 622fA. 8. 8foz.
58.
1. l5. \\\d. 2.
£37 125. U. ;
3. hs. 4. 135^ bu.
5. £19 125. 6.
165 cwt. 191? lbs.
7. 35. Gd. 8. 17>
9. 2 cwt. 2 qrs. 15 lbs. 5 oz.
10. 2 lbs. 10|oz.
11. £2094155 165. lOffc^.
12. £79 l5. 1\d.
13. £7144 75. 6c?.
14. £26 185. im.
15. 540^ yds.
16. £11 lis. 1]^.
17. £3.
18. 65. 3|p.
19. £1451 175. ^d.
20. £450.
21. 85 days.
22. 178 ft. 11^ in.
23. 6|hrs. 24. 12
800. 25. 72.
26. 286^ m.
27. £79 105.
28. £8 35. 8|fc?.
29. £33 185. U.
30. £1 165. U.
31. 85. 5|fcf.
32. 10,\ G^^d.
33. 115. ^\d.
34. 4| yds.
35. £270.
36. 7722.
37. 32 ft. ; 152 ft.
38. £5 175. llxia^.
39. £13 95. 0|cf.
40. 26^ lbs.
59.
1. 44 da. 2.
27. 3. 16.
4. 15. 5. 12.
6. 3121 i|3s, 7.
125 rms. 8. £194 85. 9. 14 wks. 2 da.
10. £114 Gs. 11.
45. 12. 112.
13. £520 14. 9.
15. 6l5. 10^. 16.
6|da. 17. £545 65.
, U. 18. 3 wks. 6 da.
19. 34 mi. 20.
8. 21. 2401
22. 132.
23. 2i days. 24.
6 tons 17 c^-t. 16 lbs.
25. 4.
26. 10| lirs. 27.
182. 28. 8. 29
. 121 days. 30. 50.
60.
1. £125. 2.
£45. 3. £1260.
4. £2673 25. U.
5. £247 165. 7|c?.
6. £2857 105.
7. £744 16s. \\d.
8. £71 12:f. 1\d,
9. £37 17s. 3^(f.
10. £20 10s.
224 ANSWERS TO THE EXAMPLES.
61.
£ s. d. £ s. d. £ s. d.
T. 519 19 Ifl . 2. 7612 7 5i§ 3. 1196 19 6
4. 19 10 11^ 5. 492 0 4| 6. 284 6 U
62.
1.
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2. 57 17 7
1
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4.
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5. 26 5 5
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6.
24 12 lOffl
63.
1.
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£42 55. \Qd 3.
125 days.
4.
6.
5.
25yrs. 6.
£39
75. 6c?. 7.
2i.
8.
£1043 15s.
9.
3§yrs. 10.
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£8 85. S
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12.
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£ 5. d.
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2.
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3. 199
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8. 17 6
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2. £44.
3.
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4.
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5. £90.
6.
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7.
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8. £771 75. 6(f.;
£10 125.
6^.
9.
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lOo £25.
11.
Increase of £10.
12.
Increase of £20.
13. £16 135. Ad.
14.
£53 65. %d
15.
100§.
16. The 3i per ccnis.
17.
£1
7 35. 6c/.
18.
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£4241 175. 6(/.
19.
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20. 93|.
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1. £1 Is. U. 2. 22f per cent. 3. £2 45. 6^.
4. 12 per cent. 5. 17 per cent. 6. £30 165. ; Of.
7. 11^;£210 85. 7lf/. 8. £1 13?. ll^Z. 9. £1 05. 2H
10. £93 6s. 8f7.; lU per cent. 11. 8 p. c. gain. 12. £82 10s.
13. £44 155. 14. 40. 15. 9s. 2^^. 16. 2| p. c. loss.
17. £4 9s. Vrd. 18. bl p. c. gain. 19. 25. 20. 63^.
ANSWERS TO THE EXAMPLES.
225
1.
3.
4.
6.
8.
10.
11.
13.
15.
16.
10.
68.
213, 355, 497; 625, 315, 225. 2. £72, £99, £108.
C, 15 cwt. 0 qrs. 20 lis. ; T. 1 c\\'t. 2 qrs. 19 lbs.
£46 135. id., £35, £28, £23 6s. 8d., £20. 5. 14, 112, 378, 896
0. 889 oz. ; H. Ill oz. 7. £6Q Ids. id. ; £33 6s. 8d. ; £200
£6 175. 3d.; £4 155. Z^d. 9.
N. 1702f lbs. ; S. 212| lbs. ; C. 324| lbs.
1 lb. iToz. 10 dwt. 20if§ grs. 12.
£160, £175. 14.
£28 25. 6d. ; £35 35. li^. ; £11 145. 4|c?.
12 carats. 17» 15 carats; 15 oz. 18. 15 carats.
20. £100; £300.
3 oz. 7 dwt. 6ji grs.
2 oz. 4 dwt. 14 grs.
£102; £104; £78.
-
69.
1.
5. 2. 358|.
3. 3. lbs. 2 oz.
4. li|. 5. A shilling.
6.
6Mf.
7. £96.10561.
8. 15 sacks 59| lbs.
9.
98||.
10. 86.186 : 1.
70.
1.
73; 94.
2.
185; 371.
3.
729; 592.
4.
309 ; 499.
5.
590; 80700.
6.
6123; 4117.
7.
1880; 8097.
8.
9998; 4908.
9.
345761; 607002
10.
2.828427+ ; 4.472136- ;
19.052559-.
11.
187.403308+ ; 94.005319-
-. 12.
367.3; 806.64.
13.
81.6279-; 6.270009 + .
14.
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15.
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16.
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17.
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. 18.
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19.
16.9595+ ; 78i:
; 19.1647-
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20.
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I-. 21. 4^; 1.103026.
22.
925 links.
23. 38 ft. 9 in. nearly, 24. 225. lO^d.
71.
1.
57; 74.
2.
28; 190.
3.
163; 328.
4.
456; 9870.
5.
809; 4812.
6.
6397; 5608.
7.
7099; .369.
8.
36.8403+ ; 8.081 + .
9.
20.03909+ ; 17.84109 + .
10.
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11.
1ft. 10.624 + in.
12.
852300 miles.
226
ANSWERS TO THE EXAMPLES.
MISCELLANEOUS.
1.
4.
7.
10.
13.
17.
20.
23.
25.
28.
30.
34.
36.
37.
39.
42.
44.
47.
49.
62.
66.
67.
60.
64.
67.
71.
74.
75.
76.
79.
82.
85.
88.
90.
93.
95.
97.
99.
102.
104.
18880. 2. £345.
£1 17s. 7if?. 5. £1492 135. 1^.
411 ft. 8. £318 155.
2a. 1r. 5i§p. 11. 609 ; 85 ft. 10 in
31 14. 20. 15. \2^.
45.8^.; 11; ^. 18. £3 25.01^.
3. I; le; U IK; 3^.
6. 1811 qu.
9. 3l5. Qd.
16. i5. iiyt.
19. 309.76: 45.78082
£150, £180, £240, £300. 21. 5i§ clays. 22. 063^.
fl^; 162^; lifl; JjV 5 2308. 24. A, 8O5. Zd.) B, lis. 9d.
£11 55., £20, £29 5s, 26. £70 II5. 9^. 27. 23i days.
29. £6 5s. ; £4 35. id. ; £3 25. ed. ; £2 105.
31. 3 b. 20 min. 32. if. 33. 105 da.
960.040103 + . 35. 6|| da., or 5 da. 7^ brs.
00256256, 256.256, .0256256.
38. 59 min. 8l^ sec.
40. 4r. 41. £14 135. 6l^.
43. £3200, £4800, £6000, £7000.
45. 45. 46. £11 195. Hd.
48. £4 75. S^l.
50. Upiirs. 51. 1811.
64. £18 105. 4|cZ. 65. 1500.
510.9-.
884, 153.
3.035913 +
.68125; 1;
45. lOlc?.
13s. 2|f^., 65. 7^^., 35. 3fic?.
£1 19s. 6§f^.
3 tons 17 cwt. 2 qrs. 261 lbs.
.25298 &c. ; 61.
69^ degrees = 76ig grades.
f of a da. 63.215.8^.
.3(5; 25 lbs. 15 oz. 11.904 drs,
4 miles 30| vds.
£759 5s. 7^1^. 58. 6.25; 12.84. 59. £3499; £874 15s.
lOifp. 61. 111104. ■ 62. £3 7s. 2d. 63. .057 &c.
£34 105. I0\^d. 65. 3|| ; if. 66. ^ in.
10s. 5d. 68. 245 : 243. 69. 638^. 70. 25. 71^.
£49 95. 4cf. 72. ^; .315625; £2000. 73. £49.
6315 dollars 55^ cents.
£33 6s. 3|g^., £66 125. 7^^., £99 18s. llf^., £133 5s. S^d.
42 m. ; lOi m. 77. 6 per cent. 78. 10c?., Is. 41^., Is. Ud., &c.
678||. 80. 105. 81. Is. 9d., Is. 2d., and 7d.
35. 511^. 83. 660.22 &c. 84. 11 min. ; 4271 190.
£907 lOs. 86. ^0; .69140625. 87. £6 25. 2ld,
£15. 89. 107 yds. 2 ft. 11 in. ; £6 14s. U{^.
Zd. 91. 99^; £176 4s. 2^d. 92. £123 lis. id.
67iL. ^1 id.c n-irJ . «o . .06515625.
£1 Us. Old.
96. 6fll.
£3 45. Ud.
1 lb. 3 oz. 7 dwt. 4i§ grs
£315.
79.0079+ ; 37.9241- ; .069; 30.02. 100. 8 days. 101. 15 hrs.
6 min. 17^ sec. a.m. 103. 6001f§ yds.
S, 6d. ; C, 25. 6d
AilSVVEES TO THE EXAMPLES.
22?^
106.
110.
113.
116.
118.
121.
124.
127.
130.
132.
134.
136.
138.
112.
144.
146.
148.
151.
154.
156.
158.
162.
165.
168.
170.
172.
174.
176.
179.
181.
184.
186.
187.
189.
191.
192.
193.
196.
199.
201.
204.
206.
209.
£840, £795. 107. 89^.
£18668 2s. 7f^.
1 hr. 51| min. 114. 15 CTvt.
A, £16 Is. 8d.; B, £8 5^.
.095178+ ; 21|; |.
122. 9 days.
0^
£300.
7 ft. 4f in.
£1 8s. 6^.
£36893 6s. 8d.
3s. id. ; 5d.
.45.
125. 2133i.
128. 165. 3d.
131. £410 Us.
108. £1050. 109. £10560.
111. 65. 112. 13.S. dd.
115. 3 ft. 9.02221 &c. in.
117. £11 165. 8d.
119. £1706 135. id.
123. A, 264; B, 198:
126. £2771 75. O^d.
129. £22 135. 2|c?.
911^; £41 Us.lOld.
133. £6 85. lOld.
135. £3 155. 2M.
120. 3|.
a 308.
137. 18| per cent
£2027 l5. 7^^^^^. 139. 3i 140. 1.
£8 75. ^ 143. .05099902- ;
3| ft. ; 8 tons 3 cwt. 3 qrs. 1^ lbs.
£595 05. O^d. 147. 8 lirs. 30 min
141. 35. id.
0155048+ ; .9615-.
145. 1 per cent.
10 hrs. 22i min.
121i 149. £3 105. 9^. ; .77. 150. 63.
£4957 6s. 8d. 152. £1 35. d-^^. 153. 87|
£220, £6 l5. lid.
305., 155., 105., 75. 6d., 6s., 5s.
600. 159. £12800.
ff, .00390625, 8^, 1^.
£245 185. U^d. ^. 166. 15.
£292 45.
1.45, 6.485; 2.49, 8.57.
185. 2ld., lis. Q^d.
6384, 7695, 8321 ; 2^ da.
„|j. 177. f, .9147916.
.5.
273.649. 182. £520.
Tho 3i per cents.
4 lbs. 11 oz. 19 dwts., .165234375, £-^, ^, ^.
188. 176a. 640 sq. yds
155. 105. 8|(?.
157. 415.8, 356.4, 226.8.
160. 26^. 161. 4^ lbs.
163. 121i. 164. £130.
167. 4^, £293 6s. 8d.
169. £62 35. 8^d., 34733.92.
171. 550 tons, 68|.
173. £65 155. 9^.
175. £5 135. 0^.
178. 12 hrs. 8 min.
180. 9ii.
183. 2880, .009943i§, |f§.
185. £187 105., £312 105., £500.
190. £270, £11 85. S^S^J.
266 tons, 16i\j cwt.
75. 0^^.
2400, 1800, 1600, 1500.
£5 145. Old., £182 105., £6 165. lO^d.
£3250, £1560, £1440. 194. 80 and 160. 195. 66.286.
£211 195. 3^. 197. The 3 per cents. ; 195. 7^d. 198. 8^.
5|, If, l|f. 200. £532 45., £100 165., £492.
.45593- ; 70.61. 202. £94 105., £7 85. 10^. 203. 51, 1|^.
£127 55. 5^d., £127 125. Id. 205. 20.7846 &c., 203.646 &c.
£196, £304. 207. 491 j^. 208. £6 95, ll|c^.
£2 55. 210. £320, £293 6s. 8d., £110, £201 135. id.
228
ANSWERS TO THE EXAMPLES.
211. 45. ^Id. 212. 68715. 213. £10 85.
215. £1832 195. 6^d 216. £29 17s. 2^.
218. 1 ton 12 cwt. 2 qrs. 3 lb. 5 oz. j £8 145. 6|d
219. 12 lirs. 48 min. ; 4± 5^. 220. 224 miles 64 yds.
2U.
217.
35. Iflc?.
12 days.
221. £3829 85. 9i|V^.
222. £10278 95. 5^.
223. £51 8s. 4f,^., £129 17s. 2^d.
EXAMINATION-PAPERS.
Paper V.
31 sq. po. 30 yd. 2 ft.
19 ac. 2 ro. 29 po. 2 yd. 5 ft. 81 in.
1 ac. 2 ro. 3 po. 4 yd. 5 ft. 6 in.
10. 1224.6 gall.
9. 668 sq. yds.
12. 3.962 met.
4. 12524940 in.
6. 17778376 in.
8. 278971 ft.
11. 31.103 ft.
13. 160.93 decam. 14. 100000.
4. 13 : 20.
7. 96 : 80 : 120 : 105.
Paper VI.
5. 8 : 13. 6. 7 : 15.
8. 1 : 3ori 9. Mtoi^aslS: 17.
1. 771. 2. 24/3|.
6. 31c?. 6. 20^.
9. 4/11
Paper V2I.
3. £22241170. 4. 40° 53'.
7. 7 sq. ft. 8. 60.
10. 21|. 11. 3 gall.
6. 221 da.
8. A 12, B 15, C 20 da.
0. 360 gall. ; 1 gall, per
Pap6r VIII,
7. A, 33f hrs. ; B, 24 firs. ; C, 18| lirs.
9. 5|da.
hr. gained.
4. £323 3s. l^d.
7. 2211 dol. 16^ re.
10. 45. ; 421 francs. 11. Gains 85. nearly.
Paper IX.
2. 6576 fr. 511 cts.
5. 9.386c^.
8. 53|<^. per milree, nearly.
3. 95286.21 fr.
6. 621c?. nearly.
9. 3722.07 fr.
12. Circuitonsly, by 35.985 milrces.
13. £160 145. ZU.
ANSWERS TO EXAMINATION-PAPERS. 220
U. 480 fr. 2i}j cents. 15. 5 doll. 59^ cents..
16. 1 rupee 11.13 annas per lb. 17. (i.) .0102045 oz. ; 25.17 francs.
17. (ii.) 25 fr. 53^ cts. ; 25 fr. 141 cts. 17. (iii.) «• .088 p. c. dearer,
17. (iii.) b. .367 p. c dearer.
Paper X.
3. 32 oxen. 4. 4. 5. 9^ da. 6. 20 wks.
7. 3 ac. 8. 40 oxen. 9. 21 days.
10. 14.076 min. ; i||^ of the cist.
?aper 2£Z.
4. 264 at 12s. &c. 5. 42 and 48. 6. 40 or. ; 45 lem.
7. T. 35. 9(7., C. Is. Sd. 8. 5 : 4.
Paper XIZ.
7. .2031. 8. 21 po. 21 yd. 9. 6 po. 1 yd.
10. 108.097 yd.; 3052 yd. 12."^ 140. 13. 14.02 ft.
14. 153 mi. 15. 4/19; VS. 16. .0261 in.
17. 250. 18. 8ld. 19. 12 ft. 20. 433 iiearlf/.
21. 2 ft. 2 in. 7icarl?/; 28^ sq. ft. 22. £42. 23. 4.
24. 5.5413 ft.; 5.058 ft. 25. ^ .
26. 13.6801 cub. yds. 27. 5.51 p. c. iieaHj/.
Paper XXZZ.
1. 57 min. 2. 263 times ; .0029 rem. 3. 192.
7. 46 sq. ft. 0' 0" U'-'. 8. 287 sq. ft. 2' 5" 6"\
9. 46 sq. ft. 0^1 in. ; 287 sq. ft. 29i in. 10. £10 U. 9^.
11. £6 45. 5d., £3 13s. 7d. 12. Gain 25 p. c. 13. --^V_ ac.
14. 2s. 7M. 15. .75. 17. § per cent, gained.
19. Nothhig. 20. 3283.
22. 5|mths. 23. Yaluo = 242i da.
25. A OS., B Is. 10i(?., C Is. ^d.
27. A Us. U., B Is. 4c?., C Is. 4c?.
28. £16 13s. 4cf. 29. IS'early £3 16s. \\d. p. c.
30. 40 ac. 9 po. 10 yds. 32f in.
32. £14 10^. IQYid. 33. £260. 34. 4||.
M
18.
10c?.
21.
3 yrs. 1 00 da.
24.
18, 27, 24, 30.
26.
17
315*
230
ANSWERS TO EXAMINATIOX-rAPEES.
Paper XIV.
1. 3759. 2. 192|ft.
4. 1 gall, water to 17 spirits.
6. £821 5s.; 32 days.
8. 12 weeks.
10. .00116 and .0625.
13. 301ic. yds. ; 165.19 lbs.
17. £1505.
19. 1^ yr., or 1 }t. 164 da.
21. £24 increase.
23. -.
3. 39 li rev. ; 7^ acd 13| ft. circiimf.
5.
1520 tons.
7.
1 mile, 1557^ yds.
nearly.
9.
729, 432, 3348, 27
.
11.
16s. 4d. 12.
31s. 41^.
15.
4 florins. 16.
SlifandfiOQ^.
18.
£127 Is. 5d.
20.
12 yards from B.
22.
£10 165. decrease.
24.
.008.
11.
15.
19.
21.
23.
25.
27.
29.
SI.
S3.
S5.
SB.
Paper XV.
12 men. 12, 2 pon. 13. £6947 18s. ^d. 14. 3570.
3.627 p. e. 16. 7 men. 17. 14s. 2>^d. 18. ^d.
26s. 8c?., 33s. 4fZ. 20. 11| mths.
At 24 min. and at 30^ min. past 11. 22. £104.
22 yrs. ago ; 18 yrs. hence.
9^ mi. an hour.
Oj^ min. past 8.
£1 lis. Zd.
The whole.
£1400.
3.4408.
6 mths.
41. £1000.
S9. £147.
42. h. I
24. 4.
26. 68s.
28. £125000.
30. 92 days.
32. £1078 lis. 'id. nearly.
34. 6ip. c. ; £574 135.
37. 3 : 7.
40. £600000.
43. 14 cwt. 3 qr. 13 lb., &c.
44. £17 16s. 4^fZ., £8 18s. 2^d. &c.
45. £322, £627 As., £2060 16s. 46. 23i| p. c.
47. For ploughing the field with oxen, £4 7s. Of?. ; For ploughing it
with horses, £3 18s. 9|i?.
48. 11 hours. 49. 10^ hrs. 50. 0 min. 6.08 sec.
lOITDOIf : PnilTTED BT
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AND PARLIAMEST STEEET
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Rowton's Debater, or Art of Public Speaking, fcp 6a.
Twells's Poetry for Repetition, 200 short Pieces and Extracts, 18mo 2a. 6d.
Hughes's Select Specimens of English Poetry, 12mo 8a. 6d.
Bilton'a Repetition and Reading Book, crown 8vo 2a. 6d.
Arithmetic,
Hunter's New Shilling Arithmetic, 18mo la. Key 2a.
Colenso's Arithmetic designed for the use of Schools, 12mo 4a. 6d.
Key to Colenso's Arithmetic for Schools, by Rev. J. Hunter, M.A. 12mo 5a.
Colenso's Shilling Elementary Arithmetic, 18mo. la. with Answers la. 6d.
— Aritnmetic for National, Adult, and Commercial Schools :—
1. Text-Book, 18mo 6d. I 3. Examples, Part II. Compound Arithmetic 4d.
2 Examples, Part I. Simple 4. Examples, Part III. Fractions, Decimals
Arithmetic 4d. | Duodecimals 4d.
5. Answers to Examples, with Solutions of the difficult Questions ... la.
Colenso's Arithmetical Tables, on a Card Id.
Lupton's Arithmetic for Schools and Candidates for Examination, 12mo.
2a. 6d, or with Answers to the Questions, 3«. 6d. the Answers separately la.
— Examination-Papers in Arithmetic, crown 8vo \$,
Hunter's Modern Arithmetic for School Work or Private Study, 12mo.3«.6d.Key, 5a.
Combes and Hines' Standard Arithmetical Copy-Books, in Nine Books, id. each.
Combes and Hines' Complete Arithmetical Copy-Books. Complete in Nine
Books, on Fine Paper, 4d. to 6d. each. Price 4«. 6d. per dozen to Teachers.
M'Leod's Manual of Arthmetic, containing 1,750 Questions, 18mo „. 9d.
Hiley's Recapitulatory Examples in Arithmetic, 12mo. - ...^ \g. 6d.
Moffatt's Mental Arithmetic, 12mo. la. or with Key, la. 6d.
Anderson's Book of Arithmetic for the Army, 18mo .. i».
Mi'Leod's Mental Arithmetic, I.Whole Numbers, II. Fractions .. each la.
— Extended Multiplication and Pence Tables, 18mo...>.- 2d.
Johnston's Civil Service Arithmetic, 12mo 8a. 6d. Key 4a.
— Civil Service Tots, with Answers and Cross-Tots la.
London, LONGMANS & CO.
I
Thomson's Treatise on Arithmetic, 12mo „.^ St. 6d. Key 5«.
Tate's First Principles of Arithmetic, r2mo ^.„ „.^. 1$. 6d.
Fix's Miscellaneous Examples in Arithmetic, 12mo is. 6d,
Stevens and Hole's Arithmetical Examination Cards, in Eight Sets, each
Set consisting of Twenty-Four Cards. Price 1«. per Set.
A. Simple Addition and Subtraction. | E. Practice and Bills of Parcels.
B. Simple Multiplication and Division, i F. Vulgar and Decimal Fractions,
C. Compound Rules (Money). I G. Simple and Compound Proportion.
D. Compound Rules (Weights andMea- H. Interest, Stocks, and Miscellaneous
sures). i Problems.
Isbister's High School Arithmetic, 12mo. 1«. or with Answers „.... 1». 6d.
Calder's Familiar Arithmetic, 12mo. 4«. 6d. or with Answers, 5«. 6d. the
Answers separately, 1«. the Questions in Part II. separately ...„. 1$.
Calder's Smaller Arithmetic for Schools, 18mo is.M,
LiddeU's Arithmetic for Schools, 18mo. 1«. cloth ; or in Two Parts, Sixpence
each. The Answers separately, price Threepence.
Harris's Graduated Exercises in Arithmetic and Mensuration, crown 8vo.
2«.6d. or with Answers, 8s. the Answers separately, 9d Full Key 6».
Merrifield's Technical Arithmetic and Mensuration, small Svo. 3«.6d. Key 8«. 6d.
JBook-keepinff.
Isbister's Book-keeping by Single and Double Entry, 18mo 9<J.
— Set of Eight Account Books to the above each M.
Hunter's Exercises in Book-keeping by Double Entry, 12mo. ...1«. 6d. Key 2«. 6d,
— Examination- Questions in Book-keeping by Double Entry, 12mo. 2«. 6d.
— Exam ination-Questions &c. as above, separate from the Answers 1».
— Ruled Paper for Forms of Account Books, 5 sorts ... per quire, 1». 6d!.
— Self-Instruction in Book-keeping, 12mo 2«.
Mensuration.
Merrifield's Technical Arithmetic & Mensuration, small 8vo ~.~. Z». 6d.
Hunter's Elements of Mensuration, 18mo 1». Key 9d.
Hiley's Explanatory Mensuration, 12mo it. 6d.
Boucher's Mensuration, Plane and Solid, 12mo 8«.
Nesbit's Treatise on Practical Mensuration, by Hunter, 12mo. St. 6dt.g Key 5».
Algebi-a.
Colenso and Hunter's Introductory Algebra, 18mo 2«. Gd. Key it. 6d.
Griffin's Algebra and Trigonometry, small 8vo 8«. 6d.
— Notes on Algebra and Trigonometry, small 8vo 8«. 6d.
Colenso's Algebra, for National and Adult Schools, 18mo 1«. 6d. Key it. 6d.
— Algebra, for the use of Schools, Part 1. 12mo 4«. 6d. Key 5«.
— Elementsof Algebra, for the use of Schools, Part II. 12mo.6«. Key 5«.
— Examples and Equation Papers, with the Answers, 12mo it. 6d.
Tate's Algebra made Easy, 12mo .' 2». Key St.Gd,
Reynolds's Elementary Algebra for Beginners, 18mo. 9d. Answers, 8d. Key It.
Thomson's Elementary Treatise on Algebra, 12mo 5«. Key 4«. 6d.
Lund's Short and Easy Course of Algebra, crown Svo it. 6d. Key it. 6d.
Wood's Algebra, modernised by Lund, crown Svo 7». 6d.
MacCoU's Algebraical Problems, with Elliptical Solutions, 12mo it. 6d.
Geometry and Trigonometry.
Hawtrey's Introduction to Euclid cloth 2«. 6d.
Thomson's Euclid, Books I. to VI. and XI. &XII. 12mo 5».
— Plane and Spherical Trigonometry, Svo 4». 6d.
— Differential and Integral Calculus, 12mo 6«. 6d.
London, LONGMANS & CO.
6 Oeneral Lists of School-Books
Watson's Plane and Solid Geometry, small 8vo 8«. 6d.
Wright's Elements of Plane Geometry, crown 8vo 5«.
Willock's Elementary Geometry of the Right Line, crown 8vo 5».
Potts's Euchd, University Edition, 8vo 10«.
— — Intermediate Edition, Books I. to IV. 3«. Books I, to III. Is. 6d.
Books I. II. 1«. M. Book I. 1«.
— Enunciations of Euclid. 12mo 6d.
Tate's Practical Geometry, with 261 Woodcuts, 18mo 1«.
— Geometry, Mensuration, Trigonometry, &c. 12mo 3«. M.
Isbister's School Euclid, the First Two Books, 12mo 1». 6d.
Tate's First Three Books of Euclid, 18mo 9d.
Colenso's Elements of Euclid, 18mo is. 6d. or with Key to the Exercises 6«. 6d.
— Geometrical Exercises and Key 8«. 6d.
— Geometrical Exercises, separately, 18mo 1«.
— Trigonometry, 12mo. Part 1. 3». Cd. Key 3«. 6d. Part II. 2«. 6d. Key 5».
Hunter's Plane Trigonometry, for Beginners, 18mo 1«. Key 9d.
Booth's New Geometrical Methods, 2 vols. 8vo 36«.
Hymers's Differential Equations and Calculus, 8vo 12«.
Williamson on Differential Calculus, cro\\'n8vo 10«. 6d.
— on Integral Calculus, crown 8vo 10«. 6d.
Hunter's Treatise on Logarithms, 18mo 1». Key 9d.
Jeans* Plane and Spherical Trigonometry, 12mo. Is. 6d. or 2 Parts, each i».
— Problems in Astronomy &c. or Key to the above, 12mo 6«.
Land Surveying, Drmoing, and Practical Mathematics.
Nesbit's Practical Land Surveying, 8vo 12«.
Binns's Orthographic Projection and Isometrical Drawing, 18mo 1«.
Winter's Mathematical Exercises, postSvo 6«, 6d.
Winter's Elementary Geometrical Drawing, Part I. post 8vo. S».6d. Part II. 6«. 6d.
Pierce's Sohd or Descriptive Geometry, post 4to 12«. 6d.
Ember's Mathematical Course for the University of London, Svo 12».
Part I. for Matriculation, separately, 1«. 6d. Key, in 2 Parts, 5». each.
Salmon's Treatise on Conic Sections, Svo „.12«.
Wrigle.v's Examples in Pure and Mixed Mathematics, Svo 8». 6d.
Works hy John HuUah, Professor of Vocal Music in King's
College^ in Queen^s College, and in Bedford College, London.
Hullah's Manual of Singing. Parts I. and II. 2«. 6d. ; or together 5«.
Exercises and Figures contained in Parts I. and II. of the Manual,
Books I. and II. each 8d.
Large Sheets, containing the Figures in Part I. of the Manual. Nos. 1
to 8 in a Parcel 6s.
Large Sheets, containing the Exercises in Part I. of the Manual. Nos. 9
to 40, in Four Parcels of Eight Nos. each ,. per Parcel 6#.
Large Sheets, the Figures in Part II. Nos. 41 to 52 in a Parcel „ 9«.
Rudiments of Musical Grammar, royal Svo 3«.
Grammar of Musical Harmony, royal Svo. Two Parts „.each 1«. 6d.
Exercises to Grammar of Musical Harmony ...„. l«.
Grammar of Counterpoint. Part I. super-royal Svo 2». 6d.
Infant School Songs 6d.
School Songs for 2 and 3 Voices. 2 Books, Svo each 6d.
Hymns for the Young, set to Music, royal Svo Sd.
Old English Songs for Schools, Harmonised 6d.
Exercises for the Cultivation of the Voice. For Soprano or Tenor Is, 6d.
Time and Tune in the Elementary School, crown Svo 2«. 6<J.
Exercises and Figures in the same, crown Svo. 1«. or 2 Parts, 6d. each.
London, LONGMANS & CO.
Chromatic Scale, with the Inflected Syllables, on Large Sheet 1«. 6d.
Card of Chromatic Scale, price Id,
Notation, the Musical Alphabet, crown 8vo M,
Political and Historical Geography.
Thomson's Introduction to Modem Geography, New Edition in the press.
Hiley's Child's First Geography, 18mo „ 9d.
— Elementary Geography for Beginners, 18mo „.^.^. U. 6d.
— Compendium of European Geography and History, 12mo ^.„.„. St. 6d.
— Asiatic, African, American and Austrahan Geography, 12mo 8«.
Bnrbury's Mary's Geography, ISmo. 2«. 6d, „ „. Questions 1«.
The Stepping-Stone to Geography, 18mo. „ , „ 1».
Hughes's Child's First Book of Geography, 18mo „ ,..„.^.»,„.„.^. 9d.
— Geography of the British Empire, for Beginners, 18mo. ...^....„,^. 9d.
— General Geography, for Beginners, ISmo. „ ,,.„.,..^.„.^.„.„. 9d.
Questions on Hughes's General Geography, for Beginners, 18mo. ~., ..„,„.„. 9d.
Lupton's Examination-Papers in Geography, crown 8vo „.„.^.,. 1«.
Hughes's Geography of British History, fcp. 8vo 5«.
— Manual of Geography, with Six Coloured Maps, fcp. 8vo 7«. 6d.
Or in Two Parts .—1. Europe, Sg. 6d. II. Asia, Africa, America, &c 4«.
Hughes's Manual of British Geography, fcp 2t.
Sullivan's Geography Generalised, fcp. 2«. or with Maps, 2«. 6d.
— Introduction to Ancient and Modem Geography, 18mo Ig.
Maunder's Treasury of Geography, fcp 6«.
Keith Johnston's Gazetteer, or Geographical Dictionary, 8vo 42«.
Butler's Ancient andModem Geography, post 8vo 7g. 6cf.
— Sketch of Modem Geography, post 8vo it.
— Sketch of Ancient Geography, post 8to 4a.
M'Leod's Geography of Palestine or the Holy Land, 12mo 1«. 6d.
Physical Geography and Geology.
Proctor's Elementary Physical Geography, fcp 1«. 6d.
Hughes's (W.) Physical Geography for Beginners, 18mo 1».
Maury's Physical Geography for Schools and General Readers, fcp Ig. 6d.
Hughes's (E.) Outlines of Physical Geography, 12mo 8«. 6d. Questions 6d.
Keith's Treatise on the Use of the Globes. 12mo 6«. 6d. Key 2«.6d.
Butler's Text Book of Physical Geography In the press.
Woodward's Geology of England and Wales, crown 8vo 14«.
Nicols's Puzzle of Life (Elementary Geology), crownBvo 5,.
Evans's Petit Album de I'Age du Bronze, crown 8vo \%s.
School Atlases and Maps.
Pubhc Schools Atlas of Modem Geography, 31 entirely New Coloured Maps,
imperial 8vo. or imperial 4to. 5s. cloth.
Public Schools Atlas of Ancient Geography, 25 entirely New Coloured Maps,
imperial 8vo. or imperial 4to. 78. 6d. cloth.
Butler's Atlas of Modem Geography, royal 8vo 10«. 6d.
— Junior Modern Atlas, comprising 12 Maps, royal 8vo 4«. M.
— Atlas of Ancient Geography, royal 8vo 12».
— Junior Ancient Atlas, comprising 12 Maps, royal 8vo 4«. 6d.
— General Atlas, Modem & Ancient, royal 4to 22«.
M'Leod's Pupil's Atlas of Modem Geography, 4to 1».
Natural History and Botany,
The Stepping-Stone to Natural History, 18mo 2«.6d.
Or in Two Parts.— I. Mammalia, 1«. II. Birds, Reptiles, and Fixhes 1«.
Owen's Natural History for Beginners, 18mo. Two Parts 9d. each, or 1 vol. 2«.
London, LONGMANS & CO.
Mannder's Treasury of Natural History, revised by Holdsworth, fcp 6«.
Lindley and Moore's Treasury of Botany, Two Parts, fcp 12«.
Wood's Bible Animals, 8vo lis.
— Homes without Hands, 8vo 14«.
— Insects at Home, 8vo 14«.
— Insects Abroad, 8vo 14*.
Out of Doors, crown 8vo 7«. 6d.
— Strange Dwellings, crown 8vo 7«. 6d.
Chemistry and Telegraphy,
Tilden's Theoretical and Systematic Chemistry, small 8vo 8s. 6d.
Armstrong's Organic Chemistry, small 8vo S«. 6d
Miller's Elements of Chemistry, 8 vols. 8vo.
Part I.— Chemical Physics, Fifth Edition, 15«,
Part II.— Inorganic Chemistry, Fifth Edition, 21«.
Part III.— Organic Chemistry, Fifth Edition in the press.
— Introduction to Inorganic Chemistry, small 8vo 8». 6d.
Tate's Outlines of Experimental Chemistry, 18mo 9d.
Odling's Course of Practical Chemistry, for Medical Students, crown 8vo.., 6».
Thorpe's Quantitative Chemical Analysis, small 8vo ig. M.
Thorpe and Muir's Qualitative Chemical Analysis, small 8vo 8«, 6df.
Crookes's Select Methods in Chemical Analysis, crown 8vo 12«. 6d.
Preece and Sivewright's Telegraphy, crown 8vo ^ 3«. 6d.
Culley'B Practical Telegraphy, 8vo 16«.
Natural Philosophy and Natural Science,
Bloxam's Metals, their Properties and Treatment, small 8vo. „ 8«.6d.
Ganot's Physics, translated by Prof. E. Atkinson, post 8vo 15«.
— Natural Philosophy, translated by the same, crown 8vo 7«. 6d.
Helmholtz' Popular Lectures on Scientific Subjects, «vo 12«. &d.
Weinhold's Introduction to Experimental Physics, 8vo 81«. 6d.
Jenkin's Electricity & Magnetism, small 8vo Zs.&d.
Maxwell's Theory of Heat, small 8vo 8». 6d.
Marcet's Conversations on Natural Philosophy, fcp Is. 6d.
Irving's Short Manual of Heat, small 8vo. .„ 2s. 6d.
Day's Numerical Examples in Heat, crown 8vo la. 6d.
— Electrical & Magnetic Measurement, 16mo 2«. 6d.
Downing's Practical Hydraulics, Part 1. 8vo 5«. 6d.
Tate's Light & Heat, for the use of Beginners, 18mo 9d.
— Hydrostatics, Hydraulics, & Pneumatics, 18mo 9d.
— Electricity, explained for the use of Beginners. 18mo 9d.
— Magnetism, Voltaic Electricity, & Electro- Dynamics, 18mo 9d.
Tyndall's Lesson in Electricity, with 58 Woodcuts, crown 8vo ^ 2«. 6d.
— Notes of Lectures on Electricity, 1«. sewed, 1«. 6d. cloth.
— Notes of Lectures on Light, 1«. sewed, 1«. 6d. cloth.
Text'Books of Science, Mechanical and Physical, adapted for
• the use of Artisans, and of Students in Public and
Science Scliools.
Anderson's Strength of Materials, small Svo 3«. 6d.
Armstrong's Organic Chemistry 3g. &d.
Barry's Railway Appliances 3«. 6d.
Bloxam's Metals 8«. 6d,
(Joodeve's Elements of Mechanism 8». 6d.
— Principles of Mechanics 8«. 6d.
Griffin's Algebra and Trigonometry 88. 6d.
Jenkin's Electricity and Magnetism 8«. 6d.
London, LONGMANS & CO.
Maxwell's Theory of Heat S«. 6d.
Merrifield's Technical Arithmetic and Mensuration 8«. 6d.
Miller's Inorganic Chemistry 3«. 6d.
Preece & Sivewright's Telegraphy 8«.6d.
Shelley's Workshop Appliances 3«. 6d.
Thome's Structural and Physiological Botany „ 6».
Thorpe's Quantitative Chemical Analysis 4«. 6d.
Thorpe & Muir's Qualitative Analysis 8*. 6d.
Tilden's Chemical Philosophy S».6d.
Watson's Plane and Solid Geometry 3<. 6<{.
*«* Other Text-Books in active preparation.
Mechanics and Mechanism.
€k)odeve's Elements of Mechanism, small Svo 8«. M,
— Principles of Mechanics, small Svo ^.„ S«. Cd.
Magnus's Lessons in Elementary Mechanics, small Svo 8». 6d.
Tate's Exercises on Mechanics and Natural Philosophy, 12mo it. Key 8«. 6d.
— Mechanics and the Steam-Engine, for beginners, ISmo 9d.
— Elements of Mechanism, with many Diagrams, 12mo St. 6d,
Haughton's Animal Mechanics, Svo ^ 21«.
Twisden's Introduction to Practical Mechanics, crown Svo 10«. 6d,
— First Lessons in Theoretical Mechanics, crown Svo. 8«. M.
Willis's Principles of Mechanism, Svo 18«.
Barry's Railway Appliances, small Svo. Woodcuts„ 3». 6d.
Shelley's Workshop Appliances, small Svo. Woodcuts 8«. 6<2.
Engineering, Architecture, &c.
Anderson on the Strength of Materials and Structures, small Svo 3«. 6(1.
Bourne's Treatise on the Steam-Engine, 4to 42«.
— Catechism of the Steam-Engine, fcp 6«.
— Recent Improvements in the Steam-Engine, fcp 6«.
— Handbook of the Steam-Engine, fcp 9«.
Main and Brown's Marine Steam-Engine, Svo 12«. 6d.
— — Indicator & Dynamometer, Svo 4<. 6(2.
— — Questions on the Steam-Engine, Svo 5«. 6d.
Fairbaim's Useful Information for Engineers. 3 vols, crown Svo 81«. 6d.
— Treatise on Mills and Millwork, 2 vols. Svo 82«.
Mitchell's Stepping-Stone to Architecture, ISmo. Woodcuts 1«.
— Rudimentary Manual of Architecture, crown Svo 10«. 6(1.
Gwilt's Encyclopaedia of Architecture, Svo bit. 6(1.
Downing's Elements of Practical Construction, Part I. Svo. Plates lit,
Moseley's Mechanical Principles of Engineering and Architecture, 8vo....24».
Popular Astronomy and Navigation.
The Stepping-Stone to Astronomy, ISmo 1«.
Tate's Astronomy and the use of the Globes, for Beginners, ISmo M.
Proctor's Lessons in Elementary Astronomy, fcp. Svo 1«. M.
Brinkley's Astronomy, by Stubbs & Briinnow, crown Svo 6».
Herschel's Outlines of Astronomy, Twelfth Edition, square crown Svo 12#.
Webb's Celestial Objects for Common Telescopes, 16mo 1t.6d.
j Proctor's Library Star Atlas, foUo 25«.
— New Star Atlas for Schools, crown Svo 5«,
— Handbook for the Stars, square fcp, Svo 5».
' Evers's Navigation & Great Circle Sailing, ISmo 1«.
I Jeans's Handbook for the Stars, royalSvo it.Qd.
i — Navigation and Nautical Astronomy, Part 1. Practical, I'tmo 5«.
— — — Part II. TAeoreWca^, royal Svo. 7».6(i.
Merrifield's Magnetism & Deviation of the Compass, ISmo It. 6(1.
Laughton's Nautical Surveying, small Svo ,. 6#.
London, LONGMANS & CO.
10 General Lists of School-Books
Animal Physiology and Preservation of Health,
Bnckton's Health in the House, small 8vo „. 2».
House I Live In ; Structure and Functions of the Human Body, 18mo. 2«. 6<f.
Bray's Education of the Feelings, crown 8vo 28.6(1.
— Physiology and the Laws of Health, 11th Thousand, fcp. ...„ 1«. 6d.
— Diagrams for Class Teaching per pair 6«. 6d.
Marshall's Outlines of Physiology, Human and Comparative, 2 vols. or. Svo. 82«.
Mapother's Animal Physiology, 18mo 1».
Hartley's Air and its Relations to Life, small 8vo 6».
General Knowledge.
Sterne's Questions on Generalities, Two Series, each 2«. Keys each it.
The Stepping-Stone to Knowledge, 18mo \t.
Second Series of the Stepping-Stone to General Knowledge, 18mo \$.
Chronology and Historical Genealogy,
Gates and Woodward's Chronological and Historical Encyclopaedia, Svo. ...42«."^
Slater's Sententice ChronologicoB, the Original Work, 12mo 1«. %d.
— — — improved by M. SeweU, 12mo 8». 6d.
Orook's Events of England in Rhyme, square 16mo 1«.
Mythology and Antiquities.
Cox'b Manual of Mythology, in Question and Answer, fcp S».
— Mythology of the Aryan Nations, 2 vols. Svo ...„....2S».
— Tales of Ancient Greece, crown Svo 6«. 6d.
Hort's New Pantheon, 18mo. with 17 Plates 2«. M,
Becker's Ga//M«, Roman Scenes of the Time of Augustus, post Svo 7«. 6d.
— Charicles, illustrating the Private Life of the Ancient Greeks ... 7«. 6d.
Rich's Illustrated Dictionary of Roman and Greek Antiquities, post 8vo.... 7«. 6d.
Ewald's Antiquities of Israel, translated by Solly, Svo ., 12«. 6d,
Goldziher's Mythology among the Hebrews, translated by Martineau, Svo. 16«.
Biography.
The Stepping-Stone to Biography, ISmo Is,
Maunder's Biographical Treasury, re-written by W. L. R. Gates, fcp 6«.
Gates's Dictionary of General Biography, Svo 25».
Pattison's Isaac Casaubon, 1559-1614, Svo 18».
Epochs of Modern History,
Church's Beginning of the Middle Ages ., Nearly ready,
Gordery's French Revolution to the Battle of Waterloo Nearly ready.
Cox's Crusades, fcp. Svo. Maps 2«. 6d.
Creighton's Age of Elizabeth, fcp. Svo. Maps „... 2«.6d.
Gairdner's Houses of Lancaster & York, fcp. Svo. Maps 2«, 6d!.
Gardiner's Thirty Years' War, 1618-1618, fcp. Svo. Maps 2» M.
Gardiner's First Two Stuarts and the Puritan Revolution, fcp. Svo. Maps 2». 6d.
Hale's Fall of the Stuarts, fcp. Svo. Maps „ ^ 2«. 6d.
Johnson's Normans in Europe, fcp. Svo Nearly ready.
— Lawrence's Early Hanoverians In the press.
— Longman's Frederick the Great and the 7 Years' War In the press.
Ludlow's War of American Independence, fcp. Svo. Maps „ 2«. 6d.
Morris's Age of Anne In the press.
Seebohm's Protestant Revolution, fcp. Svo. Maps 2t.6d.
Stubbs's Early Plantagents, fcp. Svo. Maps 2«. 6d.
Warburton's Edward the Third, fcp. Svo. Maps 2«.6d.
London, LONGMANS & CO.
Epochs of English History.
Powell's Early England up to the Norman Conquest, fcp. 8vo. Maps Is.
Creighton's England a Continental Power. 1066-1216, fcp. Maps 9d.
Kowley's Riee of the People and Growth of Parliament, 1215-1485, fcp. Maps. 9<J.
Creighton's Tudors and the Reformation, 1485-1603, fcp. 8vo. Maps 9d.
Cordery's Struggle against Absolute Monarchy, 1603-1688, fcp. Maps.... 9d.
Rowley's Settlement of the Constitution, 1688-1778 Nearly ready.
Tancock's England during the Revolutionary Wars, 1778-1820 In the press.
Browning's Modern England, from 1820 to 1876 In the press.
British History.
Armitage's Childhood of the English Nation, fcp. 8vo 2«. 6d!.
Catechism of English History, edited by Miss Sewell, 18mo 1». 6d.
Turner's Analysis of English and French History, fcp 2a. 6d.
OutUnes of the History of England, 18mo 1«.
Morris's Class-Book History of England, fcp S«. 6d.
Cantlay's English History Analysed, fcp „. 2s.
The Stepping-Stone to EngUsh History, 18mo Is.
The Stepping-Stone to Irish History, 18mo Is.
Lupton's Examinatlon-Papers in History, crown 8vo 1».
— Enghsh History, revised, crown 8vo 7s. 6d.
Gleig'8 School History of England, abridged, 12mo .'. 6».
— First Book of History— England, 18mo. 2«, or 2 Parts each 9d.
— British Colonies, or Second Book of History.lSmo 9d.
— British India, or Third Book of History, 18mo 9d.
Historical Questions on the above Three Histories, 18mo 9d.
Littlewood's Essentials of English History, fcp 8*.
Bartle's Synopsis of English History, fcp, 8vo 8#. 6d.
Epochs of Ancient History.
Beesly's Gracchi, Marius and Sulla, fcp. 8vo. Maps 2s. 6d.
Capes'sAgeof the Antonines, fcp. 8vo. Maps 28. 6d.
— Early Roman Empire, fcp. 8vo. Maps 2«. 6d.
Cox's Athenian Empire, fcp. 8vo. Maps 2s. 6d.
— Greeks & Persians, fcp. 8vo. Maps „ 2«. 6d.
Curteis's Rise of the Macedonian Empire, fcp. 8vo, Maps .\ „. 2«. 6d.
Ihne's Rome to its Capture by the Gauls, fcp. Svo.Maps ...„ 2s. 6d.
Merivale's Roman Triumvirates, fcp. 8vo. Maps „ „ 2». 6d.
Sankey's Spartan and Theban Supremacy Nearly ready.
Smith's Rome and Carthage, the Punic Wars In the press.
History, Ancient and Modem,
Sewell's Popular History of France, crovra Svo. Maps.. ., 7«. 6d,
Gleig's History of France, 18mo is.
Maunder's Historical Treasury, with Index, fcp 6«.
Mangnall's Historical and Miscellaneous Questions, 12mo is. 6d.
Taylor's Student's Manual of the History of India, crown Svo 7«. 6d.
Marshman's History of India, 3 vols, crown 8vo 22». 6d,
Sewell's Ancient History of Egypt, Assyria, and Babylonia, fcp 6s.
The Stepping-Stone to Grecian History, 18mo is.
Browne's History of Greece, for Beginners, 18mo 9d,
Sewell's First History of Greece, fcp 8». 6d.
Cox's History of Greece, Vols. I. & II. Svo 86«.
— General History of Greece, crown 8vo. Maps ...„.., 7s. 6d.
— School History of Greece, fcp. Svo. Maps Nearly ready.
Puller's School History of Rome, abridged from Merivale, fcp. Maps. In thepress.
London, LONGMANS & CO.
Taylor's Student's Mannal of Ancient History, crown 8vo Is. 6d.
— Student's Manual of Modem History, crown 8vo 7». 6d.
Turner's Analysis of the History of Greece, fcp Zs. M.
Sewell'b Catechism of Grecian History, 18mo „ „. 1«. 6d.
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\
WYCLIFFE COLLEGE
Dl
LEOIMARD LIBRARY
ANNOTATED POEMS of ENGLISH AUTHORS.
The following may now he had: —
GOLDSMITH'S TRAVELLER, 9d. sewed, Is. cloth.
GOLDSMITH'S DESERTED VILLAGE, id. sewed, or 6d. cloth.
GRAY'S ELEGY, Ad. sewed, or Qd. cloth,
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MILTON'S IL PENSEROSO, id. sewed, or 6d. cloth.
EDITED AlfD ANNOTATED FOR THE USE of CANDIDATES BY the
Rev. E. T. STEVENS, M.A. & the Bev. DAVID MORRIS, B.A.
OPINIONS of the PRESS.,
' An excellent series, carefully edited,
and fumislied with explanatory foot-
notes by the Rev. E. T. Stevens and the
Rev. D. Morris. These handy little
volumes are well adapted for the guidance
of young students of English literature.'
Rock.
• This series is intended to meet the
requirements of elementary and second
gi-ade schools, and of youthful students
of our English literature in general.
Each work selected is one of classical
and standard merit, and is prefaced with
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writer, including an account of the stj'le
and design of his work. Allusions are
carefully explained, and all gi-ammatical
difficulties are removed in the'notes.'
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taining poems by standard English
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plained to be of service to boys and girls
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tary and second grade schools.'
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• So many sets of English classics pass
through our hands that their identity is
occasionally lost. This is a new series,
and one to which we can give a cordial
welcome, both for its selection of subjects
and for the style in which it is produced.
Each volume contains a short account of
the Author whose poem follows, and of
the poem itself. Unlike other annotated
editions for schools, the notes are in this
placed beneath the text, and not rele-
gated to the end. This plan adds to the
usefulness of each volume as a school
text-book. The notes are purely expla-
natory, the etymological and gramma-
tical explanations are of sufficient worth
to deserve permanent record. A good
example of this may be found in Gray's
Etec/t/ (14), on the expression, " Many a."
The same expression occurs in the
L'Alleffro, and the same note is repeated.
This is treatment which a really good
poem will bear, and which is of the
greatest service to the student cf Eng-
lish ; and the less opportunity he has of
a classical training the more he needs
that which this affords him.'
Nonconformist.
' Tiny books got up with great taste.
The works of standard poets having come
to be considered a necessary part of the
education of certain pupils, these books
have been prepared for their use. Each
book contains one poem, prefaced by a
sketch of the Author's life, and at the
foot of each page copious notes and
grammatical hints are given. The books
are well printed and capitally edited.'
Literary World.
' An excellent series of choice composi-
tions, selected from standard English
authors, for the use of pupils of the
second grade and elementary schools.
Each poem is prefaced with a brief sketch
of the Author s life, and a short criticism
on the poem, accompanied by numerous
explanatory and critical foot-notes. With
the aid of these annotations, written as
they are in simple language, there is no
reason why these great poems should not
be read with intelligence and apprecia-
tion.' Lancet.
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and accompanied with copious notes on
the meaning and derivation of words, and
such other subjects as are necessary to
comprehension of the text without re-
ference to dictionai-ies, or other sources
of information, on the part of the student.
Not only will these little books give
material aid to the understanding of our
best poets, but they offer a better insight
into the refinements of our language
than can be had from ordinary means.
The little books are very nicely got up,
and oifered to the public at the lowest
possible price.' Queen.
The following are nearly ready :-
Bloomfield's Farmer's Boy.
Burns' Cotter's Sainrdaij Night, and
other Poems.
Campbell's Gertrude of Wvomino.
Coleridge's Rime of the Ancient Mariner .
Cowper On His Mother's Picture.
Cowper's Task,
Longfellow's Evangeline.
Milton's Lpeidas.
Scott's Ladv of the Lake.
Scott's Lay of the Last Minstrel.
Shakespeare's Julius Coesar.
Wordsworth's Exairsion (Selection)
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