¾ 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 (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (§ 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.
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 § 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.
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.