FRACTIONS.
104. Suppose it required to divide 49 yards into five equal parts, or,
as it is called, to find the fifth part of 49 yards. If we divide 45 by
5, the quotient is 9, and the remainder is 4; that is (72), 49 is made
up of 5 times 9 and 4. Let the line A B represent 49 yards:
A----------------------------------------B
C-------------- I --
D-------------- K --
E-------------- L --
F-------------- M --
G-------------- N --
I K L M N
H +-+-+-+-+-+
| | | | | |
Take 5 lines, C, D, E, F, and G, each 9 yards in length, and the line
H, 4 yards in length. Then, since 49 is 5 nines and 4, C, D, E, F, G,
and H, are together equal to A B. Divide H, which is 4 yards, into five
equal parts, I, K, L, M, and N, and place one of these parts opposite
to each of the lines, C, D, E, F, and G. It follows that the ten lines,
C, D, E, F, G, I, K, L, M, N, are together equal to A B, or 49 yards.
Now D and K together are of the same length as C and I together, and
so are E and L, F and M, and G and N. Therefore, C and I together,
repeated 5 times, will be 49 yards; that is, C and I together make up
the fifth part of 49 yards.
105. C is a certain number of yards, viz. 9; but I is a new sort of
quantity, to which hitherto we have never come. It is not an exact
number of yards, for it arises from dividing 4 yards into 5 parts, and
taking one of those parts. It is the fifth part of 4 yards, and is
called a FRACTION of a yard. It is written thus, ⁴/₅(23), and is what
we must add to 9 yards in order to make up the fifth part of 49 yards.
The same reasoning would apply to dividing 49 bushels of corn, or 49
acres of land, into 5 equal parts. We should find for the fifth part
of the first, 9 bushels and the fifth part of 4 bushels; and for the
second, 9 acres and the fifth part of 4 acres.
We say, then, once for all, that the fifth part of 49 is 9 and ⁴/₅, or
9 + ⁴/₅; which is usually written (9⁴/₅), or if we use signs, 49/5 =
(9⁴/₅).
EXERCISES.
What is the seventeenth part of 1237?--_Answer_, (72-¹³/₁₇).
10032 663819 22773399
What are -----, ------, and -------- ?
1974 23710 2424
162 23649 2343
_Answer_, (5 ----), (27 -----), (9394 ----).
1974 23710 2424
106. By the term fraction is understood a part of any number, or the
sum of any of the equal parts into which a number is divided. Thus,
⁴⁹/₅, ⁴/₅, ²⁰/₇, are fractions. The term fraction even includes whole
numbers:[14] for example, 17 is ¹⁷/₁, ³⁴/₂, ⁵¹/₃, &c.
[14] Numbers which contain an exact number of units, such as 5, 7,
100, &c., are called _whole numbers_ or _integers_, when we wish to
distinguish them from fractions.
The upper number is called the _numerator_, the lower number is
called the _denominator_, and both of these are called _terms_ of the
fraction. As long as the numerator is less than the denominator, the
fraction is less than a unit: thus, ⁶/₁₇ is less than a unit; for 6
divided into 6 parts gives 1 for each part, and must give less when
divided into 17 parts. Similarly, the fraction is equal to a unit when
the numerator and denominator are equal, and greater than a unit when
the numerator is greater than the denominator.
107. By ⅔ is meant the third part of 2. This is the same as twice the
third part of 1.
To prove this, let A B be two yards, and divide each of the yards A C
and C B into three equal parts.
|--|--|--|--|--|--|
A D E C F G B
Then, because A E, E F, and F B, are all equal to one another, A E is
the third part of 2. It is therefore ⅔. But A E is twice A D, and A
D is the third part of one yard, or ⅓; therefore ⅔ is twice ⅓; that
is, in order to get the length ⅔, it makes no difference whether we
divide _two_ yards at once into three parts, and take _one_ of them,
or whether we divide _one_ yard into three parts, and take _two_ of
them. By the same reasoning, ⅝ may be found either by dividing 5 into
8 parts, and taking one of them, or by dividing 1 into 8 parts, and
taking five of them. In future, of these two meanings I shall use that
which is most convenient at the time, as it is proved that they are
the same thing. This principle is the same as the following: The third
part of any number may be obtained by adding together the thirds of all
the units of which it consists. Thus, the third part of 2, or of two
units, is made by taking one-third out of each of the units, that is,
⅔ = ⅓ × 2.
This meaning appears ambiguous when the numerator is greater than the
denominator: thus, ¹⁵/₇ would mean that 1 is to be divided into 7
parts, and 15 of them are to be taken. We should here let as many units
be each divided into 7 parts as will give more than 15 of those parts,
and take 15 of them.
108. The value of a fraction is not altered by multiplying the
numerator and denominator by the same quantity. Take the fraction ¾,
multiply its numerator and denominator by 5, and it becomes ¹⁵/₂₀,
which is the same thing as ¾; that is, one-twentieth part of 15 yards
is the same thing as one-fourth of 3 yards: or, if our second meaning
of the word fraction be used, you get the same length by dividing a
yard into 20 parts and taking 15 of them, as you get by dividing it
into 4 parts and taking 3 of them. To prove this,
[Illustration]
let A B represent a yard; divide it into 4 equal parts, A C, C D, D E,
and E B, and divide each of these parts into 5 equal parts. Then A E is
¾. But the second division cuts the line into 20 equal parts, of which
A E contains 15. It is therefore ¹⁵/₂₀. Therefore, ¹⁵/₂₀ and ¾ are the
same thing.
Again, since ¾ is made from ¹⁵/₂₀ by dividing both the numerator
and denominator by 5, the value of a fraction is not altered by
dividing both its numerator and denominator by the same quantity. This
principle, which is of so much importance in every part of arithmetic,
is often used in common language, as when we say that 14 out of 21 is 2
out of 3, &c.
109. Though the two fractions ¾ and ¹⁵/₂₀ are the same in value,
and either of them may be used for the other without error, yet the
first is more convenient than the second, not only because you have a
clearer idea of the fourth of three yards than of the twentieth part
of fifteen yards, but because the numbers in the first being smaller,
are more convenient for multiplication and division. It is therefore
useful, when a fraction is given, to find out whether its numerator
and denominator have any common divisors or common measures. In (98)
was given a rule for finding the greatest common measure of any two
numbers; and it was shewn that when the two numbers are divided by
their greatest common measure, the quotients have no common measure
except 1. Find the greatest common measure of the terms of the
fraction, and divide them by that number. The fraction is then said to
be _reduced to its lowest terms_, and is in the state in which the best
notion can be formed of its magnitude.
EXERCISES.
With each fraction is written the same reduced to its lowest terms.
2794 22 × 127 22
---- = ---------- = ----
2921 23 × 127 23
2788 17 × 164 17
---- = ---------- = ----
4920 30 × 164 30
93208 764 × 122 764
----- = ---------- = ---
13786 113 × 122 113
888800 22 × 40400 22
-------- = ------------ = -----
40359600 999 × 40400 999
95469 121 × 789 121
------ = ----------- = ---
359784 456 × 789 456
110. When the terms of the fraction given are already in factors,[15]
any one factor in the numerator may be divided by a number, provided
some one factor in the denominator is divided by the same. This follows
from (88) and (108). In the following examples the figures altered by
division are accented.
[15] A factor of a number is a number which divides it without
remainder: thus, 4, 6, 8, are factors of 24, and 6 × 4, 8 × 3, 2 × 2 ×
2 × 3, are several ways of decomposing 24 into factors.
12 × 11 × 10 3′ × 11 × 10 1′ × 11 × 5′
------------ = ------------- = ------------- = 55.
2 × 3 × 4 2 × 3 × 1′ 1′ × 1′ × 1′
18 × 15 × 13 2′ × 3′ × 1′ 1′ × 1′ × 1′
------------ = ------------- = ------------- = ¹/₁₆.
20 × 54 × 52 4′ × 6′ × 4′ 2′ × 2′ × 4′
27 × 28 3′ × 4′ 3′ × 2′
------- = --------- = -------- = ⁶/₅.
9 × 70 1′ × 10′ 1′ × 5′
111. As we can, by (108), multiply the numerator and denominator of a
fraction by any number, without altering its value, we can now readily
reduce two fractions to two others, which shall have the same value as
the first two, and which shall have the same denominator. Take, for
example, ⅔ and ⁴/₇; multiply both terms of ⅔ by 7, and both terms of
⁴/₇ by 3. It then appears that
2 × 7
⅔ is ------- or ¹⁴/₂₁
3 × 7
4 × 3
⁴/₇ is ------- or ¹²/₂₁.
7 × 3
Here are then two fractions ¹⁴/₂₁ and ¹²/₂₁, equal to ⅔ and ⁴/₇, and
having the same denominator, 21; in this case, ⅔ and ⁴/₇ are said to be
_reduced to a common denominator_.
It is required to reduce ⅒, ⅚, and ⁷/₉ to a common denominator.
Multiply both terms of the first by the product of 6 and 9; of the
second by the product of 10 and 9; and of the third by the product of
10 and 6. Then it appears (108) that
1 × 6 × 9
⅒ is ----------- or ⁵⁴/₅₄₀
10 × 6 × 9
5 × 10 × 9
⅚ is ------------ or ⁴⁵⁰/₅₄₀
6 × 10 × 9
7 × 10 × 6
⁷/₉ is ------------ or ⁴²⁰/₅₄₀.
9 × 10 × 6
On looking at these last fractions, we see that all the numerators and
the common denominator are divisible by 6, and (108) this division will
not alter their values. On dividing the numerators and denominators of
⁵⁴/₅₄₀, ⁴⁵⁰/₅₄₀, and ⁴²⁰/₅₄₀ by 6, the resulting fractions are, ⁹/₉₀,
⁷⁵/₉₀, and ⁷⁰/₉₀. These are fractions with a common denominator, and
which are the same as ⅒, ⅚, and ⁷/₉; and therefore these are a more
simple answer to the question than the first fractions. Observe also
that 540 is one common multiple of 10, 6, and 9, namely, 10 × 6 × 9,
but that 90 is _the least_ common multiple of 10, 6, and 9 (103). The
following process, therefore, is better. To reduce the fractions ⅒,
⅚, and ⁷/₉, to others having the same value and a common denominator,
begin by finding the least common multiple of 10, 6, and 9, by the rule
in (103), which is 90. Observe that 10, 6, and 9 are contained in 90 9,
15, and 10 times. Multiply both terms of the first by 9, of the second
by 15, and of the third by 10, and the fractions thus produced are
⁹/₉₀, ⁷⁵/₉₀, and ⁷⁰/₉₀, the same as before.
If one of the numbers be a whole number, it may be reduced to a
fraction having the common denominator of the rest, by (106).
EXERCISES.
Fractions proposed reduced to a common denominator.
2 1 1 | 20 6 5
--- --- --- | ---- --- ---
3 5 6 | 30 30 30
|
|
1 2 3 12 3 | 28 24 18 48 63
--- --- --- ---- --- | --- --- --- --- ---
3 7 14 21 4 | 84 84 84 84 84
|
4 5 6 | 3000 400 50 6
3 --- ---- ---- | ---- ----- ----- ----
10 100 1000 | 1000 1000 1000 1000
|
33 281 | 22341 106499
---- ---- | -------- ------
379 677 | 256583 256583
112. By reducing two fractions to a common denominator, we are able
to compare them; that is, to tell which is the greater and which the
less of the two. For example, take ½ and ⁷/₁₅. These fractions reduced,
without alteration of their value, to a common denominator, are ¹⁵/₃₀
and ¹⁴/₃₁. Of these the first must be the greater, because (107) it may
be obtained by dividing 1 into 30 equal parts and taking 15 of them,
but the second is made by taking 14 of those parts.
It is evident that of two fractions which have the same denominator,
the greater has the greater numerator; and also that of two fractions
which have the same numerator, the greater has the less denominator.
Thus, ⁸/₇ is greater than ⁸/⁹, since the first is a 7th, and the
last only a 9th part of 8. Also, any numerator may be made to belong
to as small a fraction as we please, by sufficiently increasing the
denominator. Thus, ¹⁰/₁₀₀ is ¹/₁₀, ¹⁰/₁₀₀₀ is ¹/₁₀₀, and ¹⁰/₁₀₀₀₀₀₀ is
¹/₁₀₀₀₀₀₀ (108).
We can now also increase and diminish the first fraction by the second.
For the first fraction is made up of 15 of the 30 equal parts into
which 1 is divided. The second fraction is 14 of those parts. The sum
of the two, therefore, must be 15 + 14, or 29 of those parts; that is,
½ + ⁷/₁₅ is ²⁹/₃₀. The difference of the two must be 15-14, or 1 of
those parts; that is, ½-⁷/₁₅ = ¹/₃₀.
113. From the last two articles the following rules are obtained:
I. To compare, to add, or to subtract fractions, first reduce them to
a common denominator. When this has been done, that is the greatest of
the fractions which has the greatest numerator.
Their sum has the sum of the numerators for its numerator, and the
common denominator for its denominator.
Their difference has the difference of the numerators for its
numerator, and the common denominator for its denominator.
EXERCISES.
1 1 1 1 53 44 153 18329
--- + --- + --- - --- = ---- ---- - ----- = -------
2 3 4 5 60 3 427 1282
8 3 4 1834 1 12 253
1 + ---- + ---- + ---- = ---- 2 - --- + ---- = ----
10 100 1000 1000 7 13 91
1 8 94 3 163 97 93066
--- + --- + ---- = --- --- - ---- = -------
2 16 188 2 521 881 459001
114. Suppose it required to add a whole number to a fraction, for
example, 6 to ⁴/₉. By (106) 6 is ⁵⁴/₉, and ⁵⁴/₉ + ⁴/₉ is ⁵⁸/⁹; that is,
6 + ⁴/⁹, or as it is usually written, (6⁴/₉), is ⁵⁸/₉. The rule in this
case is: Multiply the whole number by the denominator of the fraction,
and to the product add the numerator of the fraction; the sum will be
the numerator of the result, and the denominator of the fraction will
be its denominator. Thus, (3¼) = ¹³/₄, (22⁵/₉) = ²⁰³/₉, (74²/₅₅) =
⁴⁰⁷²/₅₅. This rule is the opposite of that in (105).
115. From the last rule it appears that
907 17230907 225 667225
1723 ------ is --------, 667 ----- is ------,
10000 10000 1000 1000
99 2300099
and 23 ------ is -------.
10000 10000
Hence, when a whole number is to be added to a fraction whose
denominator is 1 followed by _ciphers_, the number of which is not less
than the number of _figures_ in the numerator, the rule is: Write the
whole number first, and then the numerator of the fraction, with as
many ciphers between them as the number of _ciphers_ in the denominator
exceeds the number of _figures_ in the numerator. This is the numerator
of the result, and the denominator of the fraction is its denominator.
If the number of ciphers in the denominator be equal to the number of
figures in the numerator, write no ciphers between the whole number and
the numerator.
EXERCISES.
Reduce the following mixed quantities to fractions:
23707 6 299 2210
1 ------, 2457 ---, 1207 --------, and 233 -----.
100000 10 10000000 10000
116. Suppose it required to multiply ⅔ by 4. This by (48) is taking ⅔
four times; that is, finding ⅔ + ⅔ + ⅔ + ⅔. This by (112) is ⁸/₃; so
that to multiply a fraction by a whole number the rule is: Multiply the
numerator by the whole number, and let the denominator remain.
117. If the denominator of the fraction be divisible by the whole
number, the rule may be stated thus: Divide the denominator of the
fraction by the whole number, and let the numerator remain. For
example, multiply ⁷/₃₆ by 6. This (116) is ⁴²/₃₆, which, since the
numerator and denominator are now divisible by 6, is (108) the same as
⁷/₆. It is plain that ⁷/₆ is made from ⁷/₃₆ in the manner stated in the
rule.
118. Multiplication has been defined to be the taking as many of one
number as there are units in another. Thus, to multiply 12 by 7 is to
take as many twelves as there are units in 7, or to take 12 as many
times as you must take 1 in order to make 7. Thus, what is done with 1
in order to make 7, is done with 12 to make 7 times 12. For example,
7 is 1 + 1 + 1 + 1 + 1 + 1 + 1
7 times 12 is 12 + 12 + 12 + 12 + 12 + 12 + 12.
When the same thing is done with two fractions, the result is still
called their product, and the process is still called multiplication.
There is this difference, that whereas a whole number is made by adding
1 to itself a number of times, a fraction is made by dividing 1 into
a number of equal parts, and adding _one of these parts_ to itself a
number of times. This being the meaning of the word multiplication,
as applied to fractions, what is ¾ multiplied by ⅞? Whatever is done
with 1 in order to make ⅞ must now be done with ¾; but to make ⅞, 1 is
divided into 8 parts, and 7 of them are taken. Therefore, to make ¾ ×
⅞, ¾ must be divided into 8 parts, and 7 of them must be taken. Now ¾
is, by (108), the same thing as ²⁴/₃₂. Since ²⁴/₃₂ is made by dividing
1 into 32 parts, and taking 24 of them, or, which is the same thing,
taking 3 of them 8 times, if ²⁴/₃₂ be divided into 8 equal parts, each
of them is ³/₃₂; and if 7 of these parts be taken, the result is ²¹/₃₂
(116): therefore ¾ multiplied by ⅞ is ²¹/₃₂; and the same reasoning
may be applied to any other fractions. But ²¹/₃₂ is made from ¾ and ⅞
by multiplying the two numerators together for the numerator, and the
two denominators for the denominator; which furnishes a rule for the
multiplication of fractions.
119. If this product ²¹/₃₂ is to be multiplied by a third fraction, for
example, by ⁵/₉, the result is, by the same rule, ¹⁰⁵/₂₈₈; and so on.
The general rule for multiplying any number of fractions together is
therefore:
Multiply all the numerators together for the numerator of the product,
and all the denominators together for its denominator.
120. Suppose it required to multiply together ¹⁵/₁₆ and ⁸/₁₀. The
product may be written thus:
15 × 8 120
-------, and is ----,
16 × 10 160
which reduced to its lowest terms (109) is ¾. This result might have
been obtained directly, by observing that 15 and 10 are both measured
by 5, and 8 and 16 are both measured by 8, and that the fraction may be
written thus:
3 × 5 × 8
-------------.
2 × 8 × 2 × 5
Divide both its numerator and denominator by 5 × 8 (108) and (87),
and the result is at once ¾; therefore, before proceeding to multiply
any number of fractions together, if there be any numerator and any
denominator, whether belonging to the same fraction or not, which have
a common measure, divide them both by that common measure, and use the
quotients instead of the dividends.
A whole number may be considered as a fraction whose denominator is 1;
thus, 16 is ¹⁶/₁ (106); and the same rule will apply when one or more
of the quantities are whole numbers.
EXERCISES
136 268 36448 18224
---- × --- = ------- = -------
7470 919 6864930 3432465
1 2 3 4 1 2 17 2
--- × --- × --- × --- = --- ---- × ---- = ----
2 3 4 5 5 17 45 45
2 13 241 6266 13 601 7813
--- × ---- × ---- = ------ ---- × ---- = -----
59 7 19 7847 461 11 5071
Fraction proposed. Square. Cube.
701 491401 344472101
----- ------- ---------
158 24964 3944312
140 19600 2744000
----- ------ --------
141 19881 2803221
355 126025 44738875
----- ------- ---------
113 12769 1442897
From 100 acres of ground, two-thirds of them are taken away; 50 acres
are then added to the result, and ⁵/₇ of the whole is taken; what
number of acres does this produce?--_Answer_, (59¹¹/₂₁).
121. In dividing one whole number by another, for example, 108 by 9,
this question is asked,--Can we, by the addition of any number of
nines, produce 108? and if so, how many nines will be sufficient for
that purpose?
Suppose we take two fractions, for example, ⅔ and ⅘, and ask, Can we,
by dividing ⅘ into some number of equal parts, and adding a number of
these parts together, produce ⅔? if so, into _how many parts_ must we
divide ⅘, and _how many of them_ must we add together? The solution of
this question is still called the division of ⅔ by ⅘; and the fraction
whose denominator is the number of parts into which ⅘ is divided, and
whose numerator is the number of them which is taken, is called the
quotient. The solution of this question is as follows: Reduce both
these fractions to a common denominator (111), which does not alter
their value (108); they then become ¹⁰/₁₅ and ¹²/₁₅. The question
now is, to divide ¹²/₁₅ into a number of parts, and to produce ¹⁰/₁₅
by taking a number of these parts. Since ¹²/₁₅ is made by dividing 1
into 15 parts and taking 12 of them, if we divide ¹²/₁₅ into 12 equal
parts, each of these parts is ¹/₁₅; if we take 10 of these parts, the
result is ¹⁰/₁₅. Therefore, in order to produce ¹⁰/₁₅ or ⅔ (108), we
must divide ¹²/₁₅ or ⅘ into 12 parts, and take 10 of them; that is, the
quotient is ¹⁰/₁₂. If we call ⅔ the dividend, and ⅘ the divisor, as
before, the quotient in this case is derived from the following rule,
which the same reasoning will shew to apply to other cases:
The numerator of the quotient is the numerator of the dividend
multiplied by the denominator of the divisor. The denominator of the
quotient is the denominator of the dividend multiplied by the numerator
of the divisor. This rule is the reverse of multiplication, as will be
seen by comparing what is required in both cases. In multiplying ⅘ by
¹⁰/₁₂, I ask, if out of ⅘ be taken 10 parts out of 12, how much _of a
unit_ is taken, and the answer is ⁴⁰/⁶⁰, or ⅔. Again, in dividing ⅔ by
⅘, I ask what part of ⅘ is ⅔, the answer to which is ¹⁰/₁₂.
122. By taking the following instance, we shall see that this rule can
be sometimes simplified. Divide ¹⁶/₃₃ by ²⁸/₁₅. Observe that 16 is 4 ×
4, and 28 is 4 × 7; 33 is 3 × 11, and 15 is 3 × 5; therefore the two
fractions are
4 × 4 4 × 7
------ and -----,
3 × 11 3 × 5
and their quotient, according to the rule, is
4 × 4 × 3 × 5
--------------,
3 × 11 × 4 × 7
in which 4 × 3 is found both in the numerator and denominator. The
fraction is therefore (108) the same as
4 × 5 20
------, or ----.
11 × 7 77
The rule of the last article, therefore, admits of this modification:
If the two numerators or the two denominators have a common measure,
divide by that common measure, and use the quotients instead of the
dividends.
123. In dividing a fraction by a whole number, for example, ⅔ by 15,
consider 15 as the fraction ¹⁵/₁. The rule gives ²/⁴⁵ as the quotient.
Therefore, to divide a fraction by a whole number, multiply the
denominator by that whole number.
EXERCISES.
Dividend. Divisor. Quotient.
41 63 41
---- ---- -----
33 11 189
467 907 47167
---- ---- -------
151 101 136957
7813 601 13
----- ---- ----
5071 11 461
¹/₅ × ¹/₅ × ¹/₅ - ²/₁₇× ²/₁₇ × ²/₁₇
What are -----------------------------------,
¹/₅ - ²/₁₇
and ⁸/₁₁ × ⁸/₁₁ - ³/₁₁ × ³/₁₁
----------------------- ?
⁸/₁₁ - ³/₁₁
559
_Answer_, ----, and 1.
7225
A can reap a field in 12 days, B in 6, and C in 4 days; in what time
can they all do it together?[16]--_Answer_, 2 days.
[16] The method of solving this and the following question may be shewn
thus: If the number of days in which each could reap the field is
given, the part which each could do in a day by himself can be found,
and thence the part which all could do together; this being known, the
number of days which it would take all to do the whole can be found.
In what time would a cistern be filled by cocks which would separately
fill it in 12, 11, 10, and 9 hours?--_Answer_, (2⁴⁵⁴/₇₆₃) hours.
124. The principal results of this section may be exhibited
algebraically as follows; let _a_, _b_, _c_, &c. stand for any whole
numbers. Then
_a_ 1 _a_ _ma_
(107) ---- = ---- × _a_ (108) ---- = ----
_b_ _b_ _b_ _mb_
_a_ _c_ _ad_ _bc_
(111) --- and --- are the same as ---- and ----
_b_ _d_ _bd_ _bd_
_a_ _b_ _a_ + _b_
(112) --- + --- = ---------
_c_ _c_ _c_
_a_ _b_ _a_ - _b_
--- - --- = ---------
_c_ _c_ _c_
_a_ _c_ _ad_ + _bc_
(113) --- + --- = -----------
_b_ _d_ _bd_
_a_ _c_ _ad_ - _bc_
--- - --- = -----------
_b_ _d_ _bd_
_a_ _c_ _ac_
(118) --- × --- = ----
_b_ _d_ _bd_
_a_ _c_ _a_/_b_ _ad_
(121) --- divᵈ. by --- or --------- = ----
_b_ _d_ _c_/_d_ _bc_
125. These results are true even when the letters themselves represent
fractions. For example, take the fraction
_a_/_b_
-------,
_c_/_d_
whose numerator and denominator are fractional, and multiply its
numerator and denominator by the fraction
_e_ _ae_/_bf_
---, which gives ----------,
_f_ _ce_/_df_
_aedf_
which (121) is -------,
_bfce_
which, dividing the numerator and denominator by _ef_ (108), is
_ad_
----.
_bc_
But the original fraction itself is
_ad_ _a_/_b_ (_a_/_b_) × (_e_/_f_)
----; hence ------- = ---------------------
_bc_ _c_/_d_ (_c_/_d_) × (_e_/_f_)
which corresponds to the second formula[17] in (124). In a similar
manner it may be shewn, that the other formulæ of the same article
are true when the letters there used either represent fractions, or
are removed and fractions introduced in their place. All formulæ
established throughout this work are equally true when fractions are
substituted for whole numbers. For example (54), (_m_ + _n_)_a_ = _ma_
+ _na_. Let _m_, _n_, and _a_ be respectively the fractions
_p_ _r_ _b_
---, ---, and ---.
_q_ _s_ _c_
Then _m_ + _n_ is
_p_ _r_ _ps_ + _qr_
--- + ---, or -----------
_q_ _s_ _qs_
and (_m_ + _n_)_a_ is
_ps_ + _qr_ _b_ (_ps_ + _qr_)_b_
----------- × ---, or ----------------
_qs_ _c_ _qsc_
_psb_ + _qrb_
or -------------.
_qsc_
_psb_ _qrb_ _pb_ _rb_
But this (112) is ----- + -----, which is ---- + ----,
_qsc_ _qsc_ _qc_ _sc_
_psb_ _pb_ _qrb_ _rb_
since ----- = ----, and ----- = ---- (103).
_qsc_ _qc_ _qsc_ _sc_
_pb_ _p_ _b_ _rb_ _r_ _b_
But ---- = --- × ---, and ---- = --- × ---.
_qc_ _q_ _c_ _sc_ _s_ _c_
Therefore (_m_ + _n_)_a_, or
(_p_ _r_ )_b_ _p_ _b_ _r_ _b_
(--- + --- )--- = --- × --- + --- × ---.
(_q_ _s_ )_c_ _q_ _c_ _s_ _c_
In a similar manner the same may be proved of any other formula.
[17] A formula is a name given to any algebraical expression which is
commonly used.
The following examples may be useful:
_a_ _c_ _e_ _g_
--- × --- + --- × ---
_b_ _d_ _f_ _h_ _acfh_ + _bdeg_
--------------------- = ---------------
_a_ _e_ _c_ _g_ _aedh_ + _bcfg_
--- × --- + --- × ---
_b_ _f_ _d_ _h_
1 _b_
---------- = ---------
1 _ab_ + 1
_a_ + ---
_b_
1 1 _bc_ + 1
--------------- = -------------- = -----------------
1 _c_ _abc_ + _a_ + _c_
_a_ + --------- _a_ + --------
1 _bc_ + 1
_b_ + ---
_c_
1 1 57
Thus, ------------ = --------- = ---
1 8 350
6 + ------- 6 + ---
1 57
7 + ---
8
The rules that have been proved to hold good for all numbers may be
applied when the numbers are represented by letters.