Ratios and Proportions Questions

FACTS  AND  FORMULAE  FOR  RATIO  AND  PROPORTION  QUESTIONS

1. RATIO: The ratio of two quantities a and b in the same units, is the fraction 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 5: 9 represents 5/9 with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.

2. PROPORTION : 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 <=> (b x c) = (a x d).

 3. (i) Fourth Proportional : If a : b = c: d, then d is called the fourth proportional to a, b, c.

     (ii) Third Proportional : If a: b = b: c, then c is called the third proportional to a and b.

     (iii) Mean Proportional : Mean proportional between a and b is $\sqrt{ab}$

 4. (i) COMPARISON OF RATIOS : We say that (a: b) > (c: d) $\iff$ (a/b)>(c /d).

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

5. (i) Duplicate ratio of (a : b) is $\left(a^2:b^2\right)$.

    (ii) Sub-duplicate ratio of (a : b) is (√a : √b).

    (iii)Triplicate ratio of (a : b) is $\left(a^3:b^3\right)$.

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

    (V) $If \frac{a}{b}=\frac{c}{d}, then\frac{a+b}{a-b}=\frac{c+d}{c-d}$ (Componendo and dividendo)

 6. VARIATION:

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

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