It is not our intention to write a treatise on the part of arithmetic
which stands at the head of this article, or to enter into the reasons
why so many persons, who can solve a simple question in which there are
nothing but whole numbers, are puzzled by anything which contains
fractions. Our object is, to give some slight notions on this part of
the subject to those who are already able to work the four rules in
whole numbers.
When we add any two numbers together, it is understood that both of them
have the same unit, or that both are some number of times the same
thing. Thus, that two and three make five, means that two _yards_ and
three _yards_ make five _yards_, or that two _pounds_ and three _pounds_
make five _pounds_, and so on. We do not in that case say anything of
two _yards_ and three _feet_, or of two _pounds_ and three _shillings_.
The following questions might arise:--If we have a distance which is
neither six yards nor seven yards, but something between the two, how
are we to represent this in numbers, and form rules for adding and
subtracting this length to or from others of the same kind, without
introducing a new measure, or talking of any other length except a yard?
The answer to this will bring us, as we shall see, to the common meaning
of the word _fraction_, and the way of representing a fraction. As we
cannot measure anything exactly, we must first decide what degree of
accuracy is necessary. This will vary in different operations, but we
will suppose, for example’s sake, that a line may be rejected as
insignificant, of which it would take more than a hundred to make a
yard. If then we divide a yard into one hundred equal parts, and first
remove the six whole yards which the above-mentioned distance contains,
we have a remainder which does not contain all the hundred parts just
mentioned, since it is less than one yard. Suppose that, on measuring
the remainder, we find it to contain more than 53 and less than 54 of
the hundred parts: if then we call it 53 parts out of a hundred of a
yard, the error committed will be less than one part out of a hundred;
that is, by what was supposed above, it will be sufficient to say that
the length of the whole is 6 yards and 53 of the hundred equal parts
which would compose another yard, or 53 hundredths of a yard. If we were
inventing a system of arithmetic, we might choose among many different
ways of representing this. For example, 6 yards 53_{100} of a yard; 6
yards and 53÷100 of a yard; and so on. The common method is the
following, 6-53/100 yards, it being always understood that when we write
two numbers under one another with a line between, the unit of which we
speak, be it a yard, pound, acre, or any other, is cut into as many
equal parts as are shown by the lower number, and as many of them are
taken as is shown by the higher number. Thus, ⅞ of a mile is the length
obtained by cutting a mile into 8 equal parts, and taking 7 of them,
being of course less than the whole mile by one of these parts.
Such a fraction as we have described is less than the unit of which it
is a part; but a whole number of units and a fraction may be represented
together by the same method. If, in the preceding example, we had
divided each of the six yards into 100 parts, there would have been 600
such parts, which, with the 53 parts furnished by the fraction, would
have made 653, not of yards, but of the hundredth parts of yards. This
we should represent by 653/100, denoting that each of a succession of
yards has been divided into 100 parts, out of which collection of parts
653 have been taken. The term fraction is applied equally to all cases;
and with this extension of meaning, the unit itself maybe represented as
a fraction, for one yard is 2/2 yards, or 3/3 yards, or 4/4 yards, and
so on.
The lower line of a fraction is called the _denominator_, and the upper
the _numerator_: these are Latin words, which may be literally
translated by the _namer_ and the _numberer_; the first tells _what
sort_ of parts is taken, and the second _how many_ of them are taken.
The following propositions will serve for consideration, and also to
familiarize the reader with the use of these terms. When the numerator
is less than the denominator, the fraction is less than a unit. When the
numerator is greater than the denominator, the fraction is greater than
the unit. Of two fractions which have the same denominator, that is the
greater which has the greater numerator. Of two fractions which have the
same numerator, that is the greater which has the less denominator. It
is usual to distinguish fractions which are less than the unit from
those which are greater by calling the former _proper_, and the latter
_improper_, fractions.
As yet we have only considered fractions of the unit; and it is always
understood that a simple fraction, such as ⅞, is a fraction of the unit,
or it is _one_ yard or _one_ pound which is divided into 8 parts.
Fractions of other numbers are written by placing the number to be
divided after the fraction of it which is to be taken, thus--¾ of 7,
which means that 7 is to be divided into 4 parts, of which parts, 3 are
taken. We now ask, what fraction of the unit is ¾ of 7, or into how many
parts must _one_ yard be cut, and how many times must one of those parts
be repeated, so as to give the same length which arises from cutting
_seven_ yards into 4 parts, and taking 3 of them? It is obvious that ¾
of 7 yards is 7 times as much as ¾ of 1 yard, or simply ¾ and 3 quarters
of a yard repeated 7 times is 21 quarters or 21/4. Similarly ⅖ of 8 is
16/5 of 1, or 16/5. Hence it follows that ¼ of 3 is ¾, ⅑ of 13 is 13/9,
and so on. If therefore we take the eighth part of nine, we get the same
as if we had repeated the eighth part of the unit nine times. We may
therefore consider a fraction, such as ⅚, in two ways, either as the
sixth part of five, or as the sixth part of unity repeated 5 times. It
may sometimes be necessary to take a fraction of a fraction, such as ⅖
of ⅞, or having found ⅞ of 1, to divide it into five parts, and take two
of them. We ask, what fraction of the unit would the result of this
double operation give? The answer is multiply the two numerators
together, and also the two denominators, which gives 14/40, or
two-fifths of seven-eighths of a yard is fourteen parts out of forty. To
see the reason, let us first take the more simple case ⅕ of ⅛. It is
plain that if we divide one yard into eight equal parts, and afterwards
divide each of these parts into 5 equal parts, we have divided the whole
yard into 8 times 5, or 40 equal parts. Consequently the fifth part of
an eighth part is one fortieth of the whole, or ⅕ of ⅛ is 1/40. But one
fifth of _seven_ eighths will be 7 times as much as ⅕ of _one_ eighth,
and will therefore be 7/40; again, _two_ fifths of ⅞ will be twice as
much as _one_ fifth of ⅞, and will therefore be 14/40, or ⅖ of ⅞ is
14/40, according to the rule. In the same way 3/7 of 11/10 is 33/70.
This rule corresponds to the multiplication of whole numbers, and is
therefore called multiplication of fractions. The connexion is not
obvious at first, owing to a little difference in our manner of speaking
about whole numbers and fractions. But if we were in the habit of saying
that 2 multiplied by 6 is six _of_ 2, in the same way as we say “six of
them,” “six of his men,” it would appear natural to call those rules
which tell us how many units there are in six _of_ two, and what
fraction of unit there is in ⅖ _of_ ⅞, by the same name. By this rule
all questions of fractions are solved, which would have required
multiplication if they had been in whole numbers. For example, if 1
pound cost 2 shillings, 6 pounds will cost 6 times 2 shillings;
similarly, if 1 pound costs ⅞ of a shilling, ⅖ of a pound will cost ⅖ of
⅞ of a shilling.
The most important proposition relating to fractions, being the one on
which the rules most materially depend, is the following: If the
numerator and denominator be either both multiplied or both divided by
the same number, the value of the fraction is not altered. For example,
take ⅗ and multiply its numerator and denominator by 4, which gives
12/20. In the second fraction we cut the unit into four times as many
parts as in the first, consequently each part of the unit signified in
the second fraction is the fourth of that signified in the first. But in
the second fraction, four times as many parts are taken as in the first,
by which the balance is restored. Let us suppose that two yards of cloth
are to be measured by a foot measure. The foot being ⅓ of the unit, and
6 of these being necessary, 6/3 will be the fraction in yards,
representing not only the number of yards measured, but in what parts of
yards they were measured. No one would object to an inch measure, which
is 1/12 of a foot, provided 12 times as many inches were given as there
were feet in the first case. But one inch is 1/36 of a yard, and 12
times 6 is 72; and in this way of measuring 72/36 would represent the
number of yards given, which is derived from 6/3 by multiplying the
numerator and denominator by 12. Similarly, one shilling, the unit being
a pound, is 1/20; and 12 pence, the unit being also a pound, is 12/240;
and 1/20 and 12/240 only differ in that the numerator and denominator of
the first must be multiplied by 12 in order to make the second.
Hence it is allowable to multiply the numerator and denominator of a
fraction by any number which is convenient, and which is called
multiplicand, since that operation does not alter its value. Thus, ⅔,
4/6, 6/9, 8/12, &c. are all of the same value, when the unit is the same
in all: in common language, we should say, that two out of three is the
same as four out of six, six out of nine, and so on. We are now able to
remove two fractions which have different denominators, and substitute
others of the same value with the same denominator. Take the fractions ⅔
and ⅘. If we ask which is the greater, no answer can at first be given,
for though the second, is 4 and the first 2, yet the second is four of
the _fifth_ parts only of unity, while the first is 2 of the _third_
parts. But if we multiply the numerator and denominator of each fraction
by the denominator of the other, the results will be 10/15 and 12/15,
which have the same value ⅔ and ⅘, and also have the same denominator as
each other. Hence we see that, 12/15 being greater than 10/15, ⅘ is
greater than ⅔. The sum of the two is the fifteenth part of unity
repeated 22 times or 22/15; the difference is two parts out of fifteen
or 2/15. Hence follow the common rules for addition and subtraction of
fractions.
We now come to the reverse of multiplication. We have shown how to find
the value of one fraction of another, such as ⅗ of 6/11; we now ask,
what fraction of ⅞ must be taken, to give ⅔ of 1 or simply ⅔? Into how
many parts must we cut ⅞, and how many times must we repeat one of those
parts, in order that the result may be the same as if we had cut unity
into three parts, and taken 2 of them? Reduce the fractions ⅞ and ⅔ to
other equivalent fractions having the same denominator, which are 21/24
and 16/24. If we cut 21/24, which is ⅞, into _twenty-one_ equal parts,
each of these parts is 1/24; if we repeat 1/24 _sixteen_ times, the
result is 16/24, which is ⅔: hence, if ⅞ be cut into 21 equal parts and
16 of these parts be taken, the resulting fraction is ⅔, or if we ask,
what fraction of ⅞ is ⅔? the answer is 16/21 of ⅞. By our former rule
16/21 of ⅞ is 112/168, which does not appear at first sight to be the
same as ⅔, but if we examine its terms, we shall find that on dividing
the numerator and denominator by 56 (which does not alter its value) it
is reduced to ⅔. This rule being the reverse of multiplication is called
division; the fraction which is to be cut into parts is called the
_divisor_, that which is to be produced from it the _dividend_, and the
fraction of the first, which it is necessary to take, in order to
produce the second, is called the _quotient_. Thus, 16/21 is the
quotient of ⅔ divided by ⅞. The rule deduced from this reasoning is:
Reverse the divisor, that is, for ⅞ write 8/7, and proceed as in
multiplication with the reversed divisor and the dividend. Thus, 8/7 of
⅔ is 16/21. This rule is used in every question where division would
have been used, if whole numbers only had been given. Thus if 4 pounds
cost 20 shillings, the price of one pound is found by dividing 20 by 4,
and is 5 shillings. If ⅞ of a pound cost ⅔ of a shilling, the price of
one pound is found by dividing ⅔ by ⅞ and is 16/21 of a shilling. This
might be established by independent reasoning as follows: As ⅞ of a
pound costs ⅔ of a shilling, and 7 pounds cost 8 times as much as ⅞ of a
pound, 7 pounds will cost 16/3 of a shilling. But as the price of one
pound is one-seventh of that of 7 pounds, for every third of a shilling
which 7 pounds cost, one pound will cost the twenty-first part of a
shilling. Hence the price of one pound is 16/21 as before.
We shall proceed in a future number to the explanation of Decimal
Fractions.
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