53. _Comparison, Addition and Subtraction of Fractions._--The quantities
3/4 of A and 5/7 of A are expressed in terms of different units. To
compare them, or to add or subtract them, we must express them in terms
of the same unit. Thus, taking 1/28 of A as the unit, we have (S 51)
3/4 of A = 21/28 of A; 5/7 of A = 20/28 of A.
Hence the former is greater than the latter; their sum is 41/28 of A;
and their difference is 1/28 of A.
Thus the fractions must be reduced to a _common denominator_. This
denominator must, if the fractions are in their lowest terms (S 54), be
a multiple of each of the denominators; it is usually most convenient
that it should be their L.C.M. (S 47).
54. _Fraction in its Lowest Terms._--A fraction is said to be _in its
lowest terms_ when its numerator and denominator have no common factor;
or to be reduced to its lowest terms when it is replaced by such a
fraction. Thus 8/22 of A is said to be reduced to its lowest terms when
it is replaced by 4/11 of A. It is important always to bear in mind that
4/11 of A is not the _same_ as 8/22 of A, though it is _equal_ to it.
+------+-----------+
| 1 | 7d. |
+------+-----------+
| 10 | 5s.10d. |
+------+-----------+
| 24 | 14s. |
+------+-----------+
55. _Diagram of Fractional Relation._--To find 10/24 of 14s. we have to
take 10 of the units, 24 of which make up 14s. Hence the required amount
will, in the multiple-table of S 36, be opposite 10 in the column in
which the amount opposite 24 is 14s.; the quantity at the head of this
column, representing the unit, will be found to be 7d. The elements of
the multiple-table with which we are concerned are shown in the diagram
in the margin. This diagram serves equally for the two statements that
(i) 10/24 of 14s. is 5s. 10d., (ii) 24/10 of 5s. 10d. is 14s. The two
statements are in fact merely different aspects of a single relation,
considered in the next section.
A
+------+-----------+
| 10 | 5s. 10d. |
+------+-----------+
| 24 | 14s. |
+------+-----------+
B
+------+-----------+
| 5 | 5s. 10d. |
+------+-----------+
| 12 | 14s. |
+------+-----------+
56. _Ratio._--If we omit the two upper compartments of the diagram in
the last section, we obtain the diagram A. This diagram exhibits a
relation between the two amounts 5s. 10d. and 14s. on the one hand, and
the numbers 10 and 24 of the standard series on the other, which is
expressed by saying that 5s. 10d. is to 14s. in the _ratio_ of 10 to 24,
or that 14s. is to 5s. 10d. in the ratio of 24 to 10. If we had taken
1s. 2d. instead of 7d. as the unit for the second column, we should have
obtained the diagram B. Thus we must regard the ratio of a to b as being
the same as the ratio of c to d, if the fractions a/b and c/d are equal.
For this reason the ratio of a to b is sometimes written a/b, but the
more correct method is to write it a:b.
If two quantities or numbers P and Q are to each other in the ratio of p
to q, it is clear from the diagram that p times Q = q times P, so that Q
= q/p of P.
57. _Proportion._--If from any two columns in the table of S 36 we
remove the numbers or quantities in any two rows, we get a diagram such
as that here shown. The pair of compartments on either side may, as
here, contain numerical quantities, or may contain numbers. But the two
pairs of compartments will correspond to a single pair of numbers, e.g.
2 and 6, in the standard series, so that, denoting them by M, N and P, Q
respectively, M will be to N in the same ratio that P is to Q.
+----------+---------------+
| 2s. 10d. | 7 yds. 1 ft. |
+----------+---------------+
| 8s. 6d. | 22 yds. |
+----------+---------------+
This is expressed by saying that M is to N as P to Q, the relation being
written M : N :: P : Q; the four quantities are then said to be _in
proportion_ or to be _proportionals_.
+---+---+
| M | P |
+---+---+
| N | Q |
+---+---+
This is the most general expression of the relative magnitude of two
quantities; i.e. the relation expressed by proportion includes the
relations expressed by multiple, submultiple, fraction and ratio.
If M and N are respectively m and n times a unit, and P and Q are
respectively p and q times a unit, then the quantities are in proportion
if mq = np; and conversely.