Quantitative Aptitude :: Ratio & Proportion

The ratio of two quantities of the same kind and in the same unit is a comparison by division of the measure of two quantities.
It determines how many times one quantity is greater or lesser than the other quantity.

The ratio of two quantities a and b in the same units, is the fraction \(\frac{a}{b}\) and we write it as a : b.

In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.

Ex. The ratio 4 : 9 represents \(\frac{4}{9}\) with antecedent = 4, consequent = 9.
9

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Eg. 2 : 7 = 4 : 14 = 8 : 56. Also, 9 : 27 = 1 : 3.

The equality of two ratios is called proportion.

If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.

Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a : b :: c : d \( \Leftrightarrow \) (b x c) = (a x d).

If a : b = c : d, then d is called the fourth proportional to a, b, c.

Third Proportional:

a : b = c : d, then c is called the third proportion to a and b.

Mean Proportional:

Mean proportional between a and b is \(\sqrt{ab}\).

We say that (a : b) > (c : d) \( \Leftrightarrow \) \(\frac{a}{b}\) > \(\frac{c}{d}\).

Compounded Ratio:

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate ratio of (a : b) is (\(a^{2}\) : \(b^{2}\)).

Sub-duplicate ratio of (a : b) is (\(\sqrt{a}\) : \(\sqrt{b}\)).

Triplicate ratio of (a : b) is (\(a^{3}\) : \(b^{3}\)).

Sub-triplicate ratio of (a : b) is (\(a^{\frac{1}{3}}\) : \(b^{\frac{1}{3}}\)).

If \(\frac{a}{b}\) = \(\frac{c}{d}\), then \(\frac{a + b}{a - b}\) = \(\frac{c + d}{c - d}\). [componendo and dividendo]

We say that x is directly proportional to y, if x = ky for some constant k and we write, \(x \ \alpha \ y\) .

We say that x is inversely proportional to y, if xy = k for some constant k and
we write, \(x \ \alpha \ \frac{1}{y}\)