CA Foundation Mathematics Summary Notes on Ratios, Proportions, Indices and Logarithms applicable for November 2018 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. a2 : b2 is the duplicate ratio of a b. Similarly, the triplicate ratio of a : b is a3 : b3.
- 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 a1/3: b1/3.
- 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. b2 = 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 b2 = ac i.e. b = ac.
INDICES
- am × an = am + n (base must be same), Ex. 23 × 22 = 23 + 2 = 25
- am × an = am–n , Ex. 25 × 23 = 25 – 3 = 22
- (am) n = amn , Ex. (25) 2 = 25 × 2 = 210
- ao = 1 , Ex. 20 = 1, 30 = 1
- a–m = 1/am and 1/a–m = am ,Ex. 2–3 = 1/23 and 1/2–5 = 25
- If ax = ay, then x=y
- If xa = ya , then x=y
- m√ a = a1/m , √x = x½ , √4 = (2)1/2 = 21/2 x 2 = 2 , Ex. 3√8 = 81/3 = (23)1/3 = 23×1/3 = 2
LOGARITHMS
- logamn = logam + logan , Ex. log (2 × 3) = log 2 + log 3
- loga(m/n) = logam – logan , Ex. log (3/2) = log3 – log2
- logamn = n logam , Ex. log 23 = 3 log 2
- logaa = 1, a = 1 , Ex. log1010 = 1, log22 = 1, log33 = 1 etc.
- loga1 = 0 , Ex. log21 = 0, log101 = 0 etc.
- logba × logab = 1 , Ex. log32 × log23 = 1
- logba × logcb = logca , Ex. log32 × log53 = log52
- logba = log a/log b , Ex. log32 = log2/log3
- logba = 1/logab
- 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 a1 = a, logaa = 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}.
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