## Ratios, Proportions, Indices and Logarithms Summary for CA Foundation November 2020 Exams

### RATIOS

- A ratio is a comparison of the sizes of two or more quantities of the same kind by division
- If a and b are two quantities of the same kind (in same units), then the fraction a/b is called the ratio of a to b. It is written as a : b. Thus, the ratio of a to b = a/b or a : b.
- The quantities a and b are called the terms of the ratio, a is called the first term or antecedent and b is called the second term or consequent.
- The ratio compounded of the two ratios a : b and c : d is ac : bd.
- A ratio compounded of itself is called its duplicate ratio. a
^{2}: b^{2}is the duplicate ratio of a b. Similarly, the triplicate ratio of a : b is a^{3}: b^{3}. - For any ratio a : b, the inverse ratio is b : a
- The sub-duplicate ratio of a : b is a : b and the sub-triplicate ratio of a : b is a
^{1/3}: b^{1/3}. - Continued Ratio is the relation (or compassion) between the magnitudes of three or more Quantities of the same kind. The continued ratio of three similar quantities a, b, c is written as a : b : c.

### PROPORTIONS

- p : q = r : s => q : p = s : r (Invertendo)(p/q = r/s) => (q/p = s/r)
- a : b = c : d => a : c = b : d (Alternendo)(a/b = c/d) => (a/c = b/d)
- a : b = c : d => a + b : b = c + d : d (Componendo)(a/b = c/d) => (a + b)/b = (c + d)/d
- a : b = c : d => a – b : b = c – d : d (Dividendo)(a/b = c/d) => (a – b)/b = (c – d)/d
- a : b = c : d => a + b : a – b = c + d : c – d (Componendo & Dividendo)(a + b)/(a – b) = (c + d)/(c – d)
- a : b = c : d = a + c : b + d (Addendo)
- (a/b = c/d = a + c/b + d)
- a : b = c : d = a – c : b – d (Subtrahendo)(a/b = c/d = a – c/b – d)
- If a : b = c : d = e : f = ………… then each of these ratios = (a – c – e – …….) : (b – d – f – …..)
- The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).
- If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc i.e. product of extremes =product of means. This is called cross product rule.
- Three quantities a, b, c of the same kind (in same units) are said to be in continuousproportion
- if a : b = b : c i.e. a/b = b/c i.e. b
^{2}= ac - If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional.
- Thus, if b is mean proportional between a and c, then b
^{2}= ac i.e. b = ac.

### INDICES

- a
^{m}× a^{n}= a^{m + n}(base must be same), Ex. 2^{3}× 2^{2}= 2^{3 + 2}= 25 - a
^{m}× a^{n}= a^{m–n}, Ex. 2^{5}× 2^{3}= 2^{5 – 3}= 22 - (a
^{m})^{n}= a^{mn }, Ex. (2^{5})^{ 2}= 2^{5 × 2}= 210 - a
^{o}= 1 , Ex. 2^{0}= 1, 3^{0}= 1 - a
^{–m}= 1/a^{m}and 1/a^{–m}= a^{m},Ex. 2^{–3}= 1/2^{3 }and 1/2^{–5}= 25 - If a
^{x}= a^{y}, then x=y - If x
^{a}= y^{a}, then x=y ^{m}√ a = a^{1/m}, √x = x^{½}, √4 = (2)^{1/2 }= 2^{1/2 x 2}= 2 , Ex.^{3}√8 = 8^{1/3}= (2^{3})^{1/3}= 2^{3×1/3}= 2

### LOGARITHMS

- log
_{a}mn = log^{a}m + log_{a}n , Ex. log (2 × 3) = log 2 + log 3 - log
_{a}(m/n) = log_{a}m – log_{a}n , Ex. log (3/2) = log3 – log2 - log
_{a}m^{n}= n log_{a}m , Ex. log 2^{3}= 3 log 2 - log
_{a}a = 1, a = 1 , Ex. log_{10}10 = 1, log_{2}2 = 1, log_{3}3 = 1 etc. - log
_{a}1 = 0 , Ex. log_{2}1 = 0, log_{10}1 = 0 etc. - log
_{b}a × log_{a}b = 1 , Ex. log_{3}2 × log_{2}3 = 1 - log
_{b}a × log_{c}b = log_{c}a , Ex. log_{3}2 × log_{5}3 = log_{5}2 - log
_{b}a = log a/log b , Ex. log_{3}2 = log2/log3 - log
_{b}a = 1/log_{a}b - a
^{logax}= x (Inverse logarithm Property) - The two equations ax= n and x = logan are only transformations of each other and should be remembered to change one form of the relation into the other.Since a
_{1 }= a, log^{a}_{a}= 1

Notes:

(A) If base is understood, base is taken as 10

(B) Thus log 10 = 1, log 1 = 0

(C) Logarithm using base 10 is called Common logarithm and logarithm using base e is called Natural logarithm {e = 2.33 (approx.) called exponential number}.