Ratios, Proportions, Indices and Logarithms Summary

CA Foundation Mathematics Summary Notes on Ratios, Proportions, Indices and Logarithms applicable for November 2018 Exams.

RATIOS

1. A ratio is a comparison of the sizes of two or more quantities of the same kind by division
2. 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.
3. 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.
4. The ratio compounded of the two ratios a : b and c : d is ac : bd.
5. A ratio compounded of itself is called its duplicate ratio. a² : b² is the duplicate ratio of a b. Similarly, the triplicate ratio of a : b is a³ : b³.
6. For any ratio a : b, the inverse ratio is b : a
7. 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.
8. 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

1. p : q = r : s => q : p = s : r (Invertendo)(p/q = r/s) => (q/p = s/r)
2. a : b = c : d => a : c = b : d (Alternendo)(a/b = c/d) => (a/c = b/d)
3. a : b = c : d => a + b : b = c + d : d (Componendo)(a/b = c/d) => (a + b)/b = (c + d)/d
4. a : b = c : d => a – b : b = c – d : d (Dividendo)(a/b = c/d) => (a – b)/b = (c – d)/d
5. a : b = c : d => a + b : a – b = c + d : c – d (Componendo & Dividendo)(a + b)/(a – b) = (c + d)/(c – d)
6. a : b = c : d = a + c : b + d (Addendo)
7. (a/b = c/d = a + c/b + d)
8. a : b = c : d = a – c : b – d (Subtrahendo)(a/b = c/d = a – c/b – d)
9. If a : b = c : d = e : f = ………… then each of these ratios = (a – c – e – …….) : (b – d – f – …..)
10. 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).
11. 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.
12. Three quantities a, b, c of the same kind (in same units) are said to be in continuousproportion
13. if a : b = b : c i.e. a/b = b/c i.e. b² = ac
14. 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.
15. Thus, if b is mean proportional between a and c, then b² = ac i.e. b = ac.

 

INDICES

1. a^m × aⁿ = a^{m + n} (base must be same),  Ex. 2³ × 2² = 2^{3 + 2} = 25
2. a^m × aⁿ = a^m–n , Ex. 2⁵ × 2³ = 2^{5 – 3} = 22
3. (a^m) ⁿ = a^mn  , Ex. (2⁵) ² = 2^{5 × 2} = 210
4. a^o = 1 , Ex. 2⁰ = 1, 3⁰ = 1
5. a^–m = 1/a^m and 1/a^–m = a^m ,Ex. 2^–3 = 1/2³ and 1/2^–5 = 25
6. If a^x = a^y, then x=y
7. If x^a = y^a , then x=y
8. ^m√ a = a^1/m ,  √x = x^½ ,  √4 = (2)^1/2 = 2^{1/2 x 2} = 2 , Ex. ³√8 = 8^1/3 = (2³)^1/3 = 2^3×1/3 = 2

 

LOGARITHMS

1. log_amn = log^am + log_an  , Ex. log (2 × 3) = log 2 + log 3
2. log_a(m/n) = log_am – log_an  , Ex. log (3/2) = log3 – log2
3. log_amⁿ = n log_am  , Ex. log 2³ = 3 log 2
4. log_aa = 1, a = 1  , Ex. log₁₀10 = 1, log₂2 = 1, log₃3 = 1 etc.
5. log_a1 = 0  , Ex. log₂1 = 0, log₁₀1 = 0 etc.
6. log_ba × log_ab = 1  , Ex. log₃2 × log₂3 = 1
7. log_ba × log_cb = log_ca  , Ex. log₃2 × log₅3 = log₅2
8. log_ba = log a/log b  , Ex. log₃2 = log2/log3
9. log_ba = 1/log_ab
10. a ^log_ax = x (Inverse logarithm Property)
11. 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 a₁ = 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}.