Charles Davies.

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25. Bought 48 yards of cloth at 125 cents a yard : how
many bushels of potatoes are required to pay for it at 150
cents a bushel ?

26. Mr. Butcher sold 342 pounds of beef at 6 cents a
pound, and received his pay in molasses at 36 cents a gallon :
how many gallons did he receive ?

27. Mr. Farmer sold 1263 pounds of wool at 5 cents a
pound, and took his pay in cloth at 421 cents a yard : how
many yards did he take ?

28. How many firkins of butter, each containing 56 pounds,
at 18 cents a pound, must be given for 3 barrels of sugar,
each containing 200 pounds, at 9 cents a pound ?

29. How many boxes of tea, each containing 24 pounds,
worth 5 shillings a pound, must be given for 4 bins of wheat,
each containing 145 bushels, at 12 shillings a bushel ?

30. A worked for B 8 days, at 6 shillings a day, for which
he received 12 bushels of corn : how much was the corn
worth a bushel ?

31. Bought 15 barrels of apples, each containing 2 bushels
at the rate of 3 shillings a bushel : how many cheeses, each
weighing 30 pounds, at 1 shilling a pound, will pay for the
apples ?

10



14:6 COMMON FRACTIONS.



COMMON FRACTIONS.

144. The unit 1 denotes an entire thing, as 1 apple,
1 chair, 1 pound of tea.

If the unit 1 be divided into two equal parts, each part
is called one-half.

If the unit 1 be divided into three equal parts, each part
is called one-third.

If the unit 1 be divided into four equal parts, each part
is called one-fourth.

If the unit 1 be divided into twelve equal parts, each part
is called one-twelfth ; and if it be divided into any number
of equal parts, we have a like expression for each part.

The parts are thus written :

is read, one-half. -f is read, one-seventh,

one-third | - - one-eighth,

one-fourth. . T\T ~ - one-tenth.

- one-fifth. T ^ - - one-fifteenth,

one-sixth. ^ - - one-fiftieth.

The i, is an entire half; the J, an entire third ; the J, an
entire fourth ; and the same for each of the other equal parts :
hence, each equal part is an entire thing, and is called a frac-
tional unit.

The unit 1 , or whole thing which is divided, is called the
unit of the fraction.

NOTE. In every fraction let the pupil distinguish carefully
between the unit of the fraction and the fractional unit. The first
is the whole thing from which the fraction is derived ; the second,
one of the equal parts into which that thing is divided.

145. Each fractional unit may become the base of a col-
lection of fractional units : thus, suppose it were required to
express 2 of each of the fractional units : we should then write

144. What is a unit ? What is each part called when the unit 1 is
divided into two equal parts ? When it is divided into 3 ? Into 4? Into
5? Into 12?

How may the one-half be regarded ? The one-third ? The one-fourth ?
What is each part called ?

What is the unit of a fraction ? What is a fractional unit ? How do
you distinguish between the one and the otlu-r ?



COMMON FRACTIONS.

which is read 2 halves = J x 2
" " " 2 thirds =Jx2
2fourths=Jx2
2 fifths =x2
&c., &c., &c. f &c.

If it were required to express 3 of each of the fractional
units, we should write %

-| which is read 3 halves =^ x 3

f " 3 thirds =4x3

" " " 3 fourths =1x3

J " " " 3 fifths =1x3

&c., &c., &c., &c. ; hence,

A FRACTION is one of the equal parts of the unit 1, or a
collection of such equal parts.

Fractions are expressed by two numbers, the one written
above the other, with a line between them. The lower num-
ber is called the denominator, and the upper number the
numerator.

The denominator denotes the number of equal parts into
which the unit is divided ; and hence, determines the value
of the fractional unit. Thus, if the denominator is 2, the
fractional unit is one-half; if it is 3, the fractional unit is one-
third ; if it is 4, the fractional unit is one-fourth, &c., &c.

The numerator denotes the number of fractional units taken.
Thus, -f denotes that the fractional unit is ^, and that 3 such
units are taken ; and similarly for other fractions.

In the fraction f , the base of the collection of fractional
units is , but this is not the primary base. For, is one-
fifth of the unit 1 ; hence, the primary base of every fraction
is the unit 1.

145. May a fractional unit become the base of a collection ? What is
a fraction ? How are fractions expressed ? What is the lower number
called ? What is the upper number called ? What does the denomina-
tor denote? What does the numerator denote? In the fraction
3 fifths, what is the fractional base ? What is the primary base ? What
is the primary base of every fraction ?



148 COMMON FRACTIONS.

146. If we take other units 1, each of the same kind, and
divide each into equal parts, such parts may be expressed
in the same collection with the parts of the first : thus,

f is read 3 halves.

I " " ? fourths.

i/- " " 16 fifths.

V " . *' 18 sixths.

2j&- 25 sevenths.

147. A whole number may be expressed fractionally by
writing 1 below it for a denominator. Thus,

3 may be written -f- and is read, 3 ones.
5 - - { - - 5 ones.
6 - - f - - - 6 ones.

8 - - - -f- - - - 8 ones.

But 3 ones are equal to 3, 5 ones to 5, 6 ones to 6, and
8 ones to 8 ; hence, the value of a number is not changed by
placing 1 under it for a denominator.

148. If the numerator of a fraction be divided by its de-
nominator, the integral part of the quotient will express the
number of entire units used in forming the fraction ; and the
remainder will show how many fractional units are over.
Tims, JyL are equal to 3 and 2 thirds, and is written -V- 3 I :
hence,

A fraction has the same form as an unexecuted division.

From what has been said, we conclude that,

1st. A fraction is one or more of the equal parts of the
unit 1.

2d. The denominator shows into how many equal parts
the unit is divided, and hence indicates the value of the
fractional unit :

146. If a second unit be divided into equal parts, may the parts be
expressed with those of the first? How many units have been divided
to obtain 6 thirds ? To obtain 9 halves ? 12 fourths ?

147. How may a whole number be expressed fractionally? Does
this change the value of the number?

148. If the numerator be divided by the denominator, what docs the
quotient show? What does the remainder show? What form has a
fraction ? What are the seven principles which follow ?



COMMON FRACTIONS. 149

3d. The numerator shows how many fractional units are
taken :

4th. The value of every fraction is equal to the quotient
arising from dividing the numerator by the denominator.

5th. When the numerator is less than the denominator,
the value of the fraction is less than 1.

6th. When the numerator is equal to the denominator,
the value of the fraction is equal to 1.

7th. When the numerator is greater than the denomina-
tor, the value of the fraction is greater than 1

EXAMPLES IN WRITING AND READING FRACTIONS.

1. Read the following fractions ;

T 5 u, f , , T 7 o, f , 5 9 o, TT.

What is the unit of the fraction, and what the fractional unit,
in each example ? How many fractional units are taken in each?

2. Write 12 of the 17 equal parts of 1.

3. If the unit of the fraction is 1, and the fractional unit
one-twentieth, express 6 fractional units. Express 12, 18,
16, 30, fractional units.

4. If the fractional unit is one 36th, express 32 fractional
units ; also, 35, 38, 54, 6, 8.

5. If the fractional unit is one-fortieth, express 9 fractional
units ; also, 16, 25, 69, 75.

DEFINITIONS.

149. A PROPER FRACTION is one whose numerator is less
than the denominator.

Tue following are proper fractions :

i i i I f J, A, t, *

150. An IMPROPER FRACTION is one whose numerator is
equal to, or exceeds the denominator.

NOTE. Such a . fraction is called improper because its value
equals or exceeds 1.

149. What is a proper fraction ? Give examples.

150. What is an improper fraction ? Why improper ? Give exam-
ples.



150 PROPOSITIONS IN

The following are improper fractions :

4, 4, 4, 4, f , 4, , , V-

151. A SIMPLE FRACTION is one whose numerator and de-
nominator are both whole numbers.

NOTE. A simple fraction may be either proper or improper.
The following are simple fractions :

i f , *, f , 4, 4, 4. *

152. A COMPOUND FRACTION is a fraction of a fraction, or
several fractions connected by the word of, or x .

The following are compound fractions :

Jofi iofiofj, x3, ixJx-4.

153. A MIXED NUMBER is made up of a whole number and
a fraction.

The following are mixed numbers :

3i, 41, 6f, 54, 6|, 3f

154. A COMPLEX FRACTION is one whose numerator or de-
nominator is fractional ; or, in which both are fractional.

The following are complex fractions :

j 2 f 45t

5 191' *' 69V

155. The numerator and .denominator of a fraction, taken
together, are called the terms of the fraction : hence, every
fraction has two terms.

FUNDAMENTAL PROPOSITIONS.

156. By multiplying the unit 1, we form all the whole
numbers,

151. What is a simple fraction ? Give examples. May it be proper
or improper ?
153. What is a compound fraction ? Give examples.

153. What is a mixed number ? Give examples.

154. What is a complex fraction ? Give examples.

155. How many terms has every fraction ? What are they ?

156. How may all the whole numbers be formed? How may the
fractional units be formed ? How many times is one-half less than 1 ?
How many times is any fractional unit less than 1 ?



COMMON FRACTIONS. 151

2, 3, 4, 5, 6, 1, 8, 9, 10, &c. ;
and by dividing the unit 1 by these numbers we form all the
fractional units,

i' 4' I* i> i' I' i> I' A &c -

Now, since in 1 unit there are 2 halves, 3 thirds, 4
fourths, 5 fifths, 6 sixths, &c., it follows that the fractional
unit becomes less as the denominators are increased : hence,

The fractional unit is such a part of I, as I is of the
denominator of the fraction.

Thus, J is such a part of 1, as 1 is of 2 ; J is such a part of
1, as 1 is of 3-; J is such a part of 1 as 1 is of 4, &c. &c.
157. Let it be required to multiply by 3.

ANALYSIS. In f there are 5 fractional OPERATION-.

units, each of which is ^, and these are to 4 x 3^-5-vp- J^A
be taken 3 times. But 5 things taken 3

times, gives 15 things of the same kind ; that is, 15 sixths : hence,
the product is 3 times as great as the multiplicand : therefore, we
have

PROPOSITION I. If the numerator of a fraction be multi-
plied by any number, the value of the fraction will be in-
creased as many times as there are units in the multiplier.



4. Multiply T V by 14.

5. Multiply % by 20.

6. Multiply Jj&z- by 25



EXAMPLES.

1. Multiply -3 by 8.

2. Multiply I by 5.

3. Multiply \ by 9.

158. Let it be required to multiply by 3.

ANALYSIS. In there are 4 fractional OPERATION.

units, each of which is . If we divide 4- X 3 4 .
the denominator by 3, we change the frac- 6 ~ 3

tional unit to \, which is 3 times as great as , since the first is
contained in 1, 2 times, and the second 6 times. If we take this
fractional unit 4 times, the result , is 3 times as great as $:
therefore, we have

PROPOSITION II. If the denominator of a fraction be divi-
ded by any number, the value of the fraction will be in-
creased as many times as there are units in that number.

157. What is proved in Proposition I. ?



152 PROPOSITIONS IN



EXAMPLES.



4. Multiply H by 2, 4, 6.

5. Multiply by 2, 6, 7.

6. Multiply $fo by 5, 10.



1. Multiply | by 2, by 4.

2. Multiply Jf by 2, 4, 8.

3. Multiply ^ by 2, 4, 6.

159. Let it be required to divide fa by 3.

ANALYSIS. In -ft, there are 9 fractional OPERATION.

units, each of which is -, 1 ,-, and these are s ' -f-3 9-3 -3

to be divided by 3. But 9 things, divided 1 1

by 3, gives 3 things of the same kind for a quotient ; hence, the
quotient is 3 elevenths, a number one-third as great as -ft ; hence,
we have

PROPOSITION III. If the numerator of a fraction be divi-
ded by any number, the value of the fraction will be dimin-
ished as many times as there are units in the divisor.



EXAMPLES.



1. Divide ff by 2, by 7

2. Divide $J by 56.



3. Divide f by 25, by 8.

4. Divide ff by 8, 16, 10.

1GO. Let it be required to divide fa by 3.

ANALYSIS. In -ft-, there are 9 fractional OPERATION.

units, each of which is -ft-. Now. if we $ -^-3=^ *-r.

multiply the denominator by 3 it becomes

33, and the fractional unit becomes -^-j, which is only ^ of -, 1 ,-, be-
cause 33 is 3 times as great as 11. If we take this fractional
unit 9 times, the result, -,-, is exactly ^ of -ft : hence, we
have

PROPOSITION IY. If the denominator of a fraction be
multiplied by any number, the value of the fraction will be
diminished as many times as there are units in that number.



EXAMPLES.



1. Divide \ by 2.

2. Divide by 1.

3. Divide -^ by 4.



4. Divide f by 8.

5. Divide fj- by 17.

6. Divide T V% by 45.



158. What is proved in proposition II. ?

159. What is proved in proposition III. ?
100. What is proved in proposition IV. ?



COMMON FRACTIONS. 153

161. Let it be required to multiply both terms of the frac-
tion f by 4.

ANALYSIS. In f, the fractional unit is , and it OPERATION.
is taken 3 times. By multiplying the denominator ?lf -JL2..

by 4, the fractional unit becomes ^7, the value of 5x4~~^o

which is ^ times as as great as i. By multiplying the numerator
by 4, we increase the number of fractional units taken, 4 times,
that is, we increase the number just as many times as we decrease
the value ; hence, the value of the fraction is not changed ; there-
fore, we have

PROPOSITION Y. If both terms of a fraction be multiplied
by the same number, the value of the fraction will not be
changed.

EXAMPLES.

1. Multiply the numerator and denominator of -f- by 7 :
this gires ^HM.

7 X 7 49

2. Multiply the numerator and denominator of -fa by 3, by
4, by 5, by 6, by 9.

3. Multiply each term of | by 2, by 3, by 4, by 5, by 6.

162. Let it be required to divide the numerator and de-
nominator of T 6 3- by 3.

ANALYSIS. In -rV, the fractional unit is -fa, and OPERATION.
is taken 6 times. By dividing the denominator 6 -r-3__2
by 3, the fractional unit becomes i, the value of T^_^o 7"*
which is 3 times as great as -fa. By dividing the
numerator by 3, we diminish the number of fractional units taken
3 times : that is, we diminish the number just as many times as we
increase the value : hence, the value of the fraction is not changed :
therefore we have

PROPOSITION YI. If both terms of a fraction be divided
by the same number, the value of the fraction will not be
changed.

EXAMPLES.

1. Divide both terms of the fraction ^ by 2 : this gives
= Ans.



161. What is proved hy proposition V. ?

162. What is proved by proposition VI. ?



154 REDUCTION OF

2. Divide both terms by 8 : this gives ^ f = J.

3. Divide both terms of the fraction -j 3 ^- by 2, by 4, by 8,
by 16.

4. Divide both terms of the fraction T ^j by 2, by 3, by 4,
by 5, by 6, by 10, by 12.

REDUCTION OF FRACTIONS.

163. REDUCTION OF FRACTIONS is the operation of changing
the fractional unit without altering the value of the fraction.

A fraction is in its lowest terms, when the numerator and
denominator have no common factor.

CASE i.

164. To reduce a fraction to its lowest terms.
1. Reduce T W to its lowest terms.

ANALYSIS. By inspection, it is seen that 5

is a common factor of the numerator and IST OPERATION.
denominator. Dividing by it, we have if. 5) T 7 -*y r 4-i.

We then see that 7 is a common factor of 14
and 35: dividing by it, we have . Now, fr\i 4 _ 2

there is no common factor to 2 and 5 : hence, 'So t'

is in its lowest terms.

The greatest common divisor of 70 and 175 2D OPERATION.
is 35, (Art. 136); if we divide both terms of 35) TJ^ .2.
the fraction by it, we obtain . The value of
the fraction is not changed in either operation, since the numera-
tor and denominator are both divided by the same number (Art.
162): hence, the following

RULE. Divide the numerator mid denominator by any
number that will divide them both without a remainder, and
divide the quotient, in the same manner until they have no
common factor.

Or : Divide the numerator and denominator by their great-
est common divisor.

163. What is reduction of fractions ? When is a fraction in its lowest
terms ?

164. How do you reduce a fraction to its lowest terms ?



COMMON FRACTIONS. 155

EXAMPLES.

Reduce the following fractions to their lowest terms.



1. Reduce -ff.

2. Reduce ff.

3. Reduce f.

4. Reduce

5. Reduce

6. Reduce
V. Reduce
8. Reduce



9. Reduce

10. Reduce

11. Reduce

12. Reduce

13. Reduce

14. Reduce

15. Reduce

16. Reduce



CASE II.

165. To reduce an improper fraction to its eouivalent
whole or mixed number.

1. In $/ how many entire units ?

ANALYSIS. Since there are 8 eighths in 1 unit, OPERATION.
in * there are as many units as 8 is contain- 8)59

ed times in 59, which is 7| times. = -

Hence, the following

RULE. Divide the numerator by the denominator, and the
result ivill be the whole or mixed number.

EXAMPLES.

1. Reduce & and fy to their equivalent whole or mixed
numbers.

OPERATION. OPERATION.

4)84 9)67

2. Reduce sg. to a whole or mixed number.

3. In I? 9 - yards of cloth, how many yards ?

4. In -^L of bushels, how many bushels ?

165. How do you reduce an improper fraction to a whole or mixed
number ?



156 REDUCTION OF

5. If I give I of an apple to each one of 15 children, how
many apples do I give ?

6. Reduce ffj, 3ff, JtfffiL, *fj#f. t to their whole or
mixed numbers.

7. If I distribute 878 quarter-apples among a number of
boys, how many whole apples do I use ?

8. Reduce % 5 T 8 ^, \W, WsWeS to tneir whole or mixed
numbers.

9. Reduce JLt^ffi^ J^\^a, 2^p } to t h e i r w hole
or mixed numbers.

CASE III.

160. To reduce a mixed number to its equivalent improper
fraction.

1. Reduce 4f- to its equivalent improper fraction.

ANALYSis.-Since ' in any number OPERATION.

there are 5 times as many fifths as A .. r O n Gp^ n

units, in 4 there will be 5 times 4 fifths,

or 20 fifths, to which add 4 fifths, and add 4 fifths.

we have 24 fifths. gives %* = 24 fifths.

Hence, the following

RULE. Multiply the whole number by the denominator of
the fraction : to the product add the numerator, and place the
sum over the given denominator.

EXAMPLES.

1. Reduce 47f to its equivalent fraction.

2. In It yards, how many eighths of a yard?

3. In 42 -/^ rods, how many twentieths of a rod ?

4. Reduce 625-^- to an improper fraction.

5. How many 112ths in 205 T 4 T % ?

6. In 84^ days, how many twenty-fourths of a day ?

7. In 15J$| years, how many 365ths of a year ?

8. Reduce 916-{} to an improper fraction.

9. Reduce 25 T %-, 156f^, to their equivalent fractions.

100. How do you reduce a mixed number to its equivalent improper
fraction.



COMMON FRACTIONS. 157

CASE IV.

167. To reduce a whole number to a fraction having a
given denominator.

1. Reduce 6 to a fraction whose denominator shall be 4.

ANALYSIS. Since in 1 unit there are 4 fourths, OPERATION.
it follows that in 6 units there are 6 times 4 fourths, 6x4 24.
or 24 fourths: therefore, 6=Y hence, .gjt

RULE. Multiply the whole number and denominator
together, and write the product over the required denomi-
nator.

EXAMPLES.

1. Reduce 12 to a fraction whose denominator shall be 9.
2 Reduce 46 to a fraction whose denominator shall be 15.



3. Change 26 to 7ths.

4. Change 178 to 40ths.

5. Reduce 240 to IHths.



6. Change $54 to quarters.

7. Change 96?/<^. to quarters.

8. Change 426/6. to 16ths.



CASE V.

168. To reduce a compound fraction to a simple one.
1. What is the value of of f?

ANALYSIS. Three-fourths of f is 3 times 1 fourth OPERATION.
of $ ; 1 fourth of f is & (Art. 160) ; 3 fourths of f is 3x5 15
3 times &, or if : therefore, f of $=i : hence, -= =

4x7 2o

RULE. Multiply the numerators together for a new
numerator, and the denominators together for a new de-
nominator.

NOTE. If there are mixed numbers, reduce them to their equiv-
alent improper fractions.

EXAMPLES.

P*educe the following fractions to simple ones.



1. Reduce J of J of f.

2. Reduce of of f.

3. Reduce f of f o



4. Reduce 2J of 6J of 7.

5. Reduce 5 of \ of | of 6.

6. Reduce 6^ of 7} of 6ff.



158 REDUCTION OF

METHOD BY CANCELLING.

169. The work may often be abridged by cancelling com-
mon factors in the numerator and denominator (Art. 143).

In every operation in fractions, let this be done whenever
it is possible.

EXAMPLES.

1. Reduce f of f of -f to a simple fraction.

5



Here,



7 | 5=f



NOTE. The divisors are always written on the left of the
vertical line, and the dividends on the right.



2



2. Reduce of f of T ^ to its simplest terms.

! * * 2 * *

-rr V V r ^

TT ^-o __- NX V ^^ T- . rv-t*

xi ere, i A A x vet F; UI R



5 | 2=2.

NOTE. Besides cancelling the like factors 8 and 8, and 9 and 9>
we also cancel the factor 3, common to 15 and 6, and write ovei
them, and at the left and right, the quotients 5 and 2.

3. Reduce | of -f of of -fife of T 5 ^ to its simplest terms.

4. Reduce -f-fc of T \ of T % of f to its simplest terms.

5. Reduce 3|- of f of ^ of 49 to its simplest terms.



CASE TI.

170. To reduce fractions of different denominators to
fractions having a common denominator.

1. Reduce \, % and 4 to a common denominator.

167. How do you reduce a whole number to a fraction having a
given denominator?

168. How do you reduce a compound fraction to a simple one ?

169. How is the reduction of compound fractions to simple ones
abridged by cancellation.



COMMON FRACTIONS. 159

ANALYSIS. If both terms of the OPERATION.

first fraction be multiplied by 15, 1x3x5=15 1st num.
the product of the other denomina- 7x2x5 = 70 2d num.
tors, it will become ft. If both i v 3v9 24- 3r1 nnm
terms of the second fraction be mul-
tiplied by 10, the product of the 2x3x5 = dO clenom.
other denominators, it will become $. If both terms of the
third be multiplied by 6, the product of the other denominators,
it will become f . In each case, we have multiplied both terms
of the fraction by the same number ; hence, the value has not
been altered (Art. 161) : hence, the following

RULE. Eeduce to simple fractions when necessary ; then
multiply the numerator of each fraction by all the denomi-
nators except its own, for the new numerators, and all the
denominators together for a common denominator.

NOTE. When the numbers are small the work may be per-
formed mentally. Thus,

i- \f *=

EXAMPLES.

Reduce the following fractions to common denominators.



1. Reduce f, f, and -$-.

2. Reduce f , -f^-, and f .

3. Reduce 4f-, |, and $.

4. Reduce 2J, and J of -f.

5. Reduce 5 J,f of J, and 4.



6. Reduce 3 of J and f .

7. Reduce ,Y/, and 37.



8. Reduce 4, fj, and .

9. Reduce 7J, ffr, 6J.

10. Reduce 4, 8|, and 2|.



NOTE. We may often shorten the work by multiplying the nu-
merator and denominator of each fraction by such a number as
will make the denominators the same in all.

10. Reduce J and J to a common denominator.

OPERATION.

ANALYSIS. Multiply both terms of the first by 1=4

3, and both terms of the second by 2. ls.

3 <



11. Reduce and J.

12. Reduce , ^, and }.

13. Reduce -.



14. Reduce f , 3, and |.

15. Reduce 6^, 9J,and5.

16. Reduce 7f,f, J, and.



170. How do you reduce fractions of different denominators to frac-
tions having a common denominator ? When the numbers are small,
how may the work be performed ?



160 REDUCTION OF

CASE VII.

171. To reduce fractions to their least common denominator.

The least common denominator is the number which con-
tains only the prime factors of the denominators.

1. Reduce J, f , and |, to their least common denominator.

OPERATION.

(12-=-3)xl = 4 1st Numerator. 3)3 . 6 . 4
(12-^-6) x 5 = 10 2d " 2)1 . 2 .~4~

(12-T-4)x3= 9 3d " 1.1.2

3x2x2 = 1 2, least com. denom.

Therefore, the fractions J, f, and f, reduced to their least
common denominator, are T %, -ff, and T \.
Hence, the following

RULE, I. Find the least common multiple of the denomi-
nators (Art. 140), which will be the least common denominator
of the fractions.

II. Divide the least common denominator by the denomina-
tors of the given fractions separately, and multiply the nume-
rators by the corresponding quotients, and place the products
over the least common denominator.

NOTES. 1. Before beginning the operation, reduce every frac-
tion to a simple fraction and to its lowest terms.


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Online LibraryCharles DaviesSchool arithmetic. Analytical and practical → online text (page 11 of 24)