Elements of Algebra

162 ELEMENTS SECT. III. which gives the weight of the ball B: and for another ball C, whose diameter is 15 inches, we should have, $2^3 : 15^3 :: 5 : 2109\frac{3}{8}$lb. 498. When the ratio of two fractions, as $\frac{a}{b} : \frac{c}{d}$, is required, we may always express it in integer numbers; for we have only to multiply the two fractions by $bd$, in order to obtain the ratio $ad : bc$, which is equal to the other; and from hence results the proportion $\frac{a}{b} : \frac{c}{d} :: ad : bc$. If, therefore, $ad$ and $bc$ have common divisors, the ratio may be reduced to fewer terms. Thus $\frac{15}{24} : \frac{25}{36} :: (15 \cdot 36) : (24 \cdot 25) :: 9 : 10$. 499. If we wished to know the ratio of the fractions $\frac{1}{a}$ and $\frac{1}{b}$, it is evident that we should have $\frac{1}{a} : \frac{1}{b} :: b : a$; which is expressed by saying, that two fractions, which have unity for their numerator, are in the reciprocal, or inverse ratio of their denominators: and the same thing is said of two fractions which have any common numerator; for $\frac{c}{a} : \frac{c}{b} :: b : a$. But if two fractions have their denominators equal, as $\frac{a}{c} : \frac{b}{c}$, they are in the direct ratio of the numerators; namely, as $a : b$. Thus, $\frac{3}{5} : \frac{6}{5} :: 3 : 6$, or $6 : 3 :: 2 : 1$, and $\frac{10}{7} : \frac{15}{7} :: 10 : 15$, or $2 : 3$. 500. It has been observed, in the free descent of bodies, that a body falls about 16 English feet in a second, that in two seconds of time it falls from the height of 64 feet, and in three seconds it falls 144 feet. Hence it is concluded, that the heights are to each other as the squares of the times; and, reciprocally, that the times are in the subduplicate ratio of the heights, or as the square roots of the heights *. If, therefore, it be required to determine how long a stone will be in falling from the height of 2304 feet; we have $16 : 2304 :: 1 : 144$, the square of the time; and consequently the time required is 12 seconds. 501. If it be required to determine how far, or through * The space, through which a heavy body descends, in the latitude of London, and in the first second of time, has been found by experiment to be $16\frac{1}{12}$ English feet; but in calculations where great accuracy is not required, the fraction may be omitted.