RATIO PROPORTION AND UNITARY METHODS
Ratio: The relation between two quantities of the same kind is called the ratio if a and b are the numbers (b ¹ 0). then a/b is called the ratio of a to b and is written as a : b, a and b are the terms known as antecedent (first term) and consequent (second term) .
Important points about ratio:
(1) If G.C.D. of a and b is 1, then a: b is said to be in the simplest form.
(2) The ratio of two numbers is usually expressed in the simplest form.
(3) Ratio has no units in itself.
(4) The two quantities compared in a ratio must be expressed in the same unit. (5) Ratio is commensurable.
Incommensurable:
If the ratio of two quantities cannot be expressed as the ratio of two integers are said to be incommensurable.
Example: Ratio of a side of square to its diagonal 1.21/2.
Properties 0' ratios':
(1) If both the quantities x and y of a ratio are multiplied or divided by the same quantity or by the same number, the value of their ratio does not change.
(2) Two or more ratios can be compared by making their denominator SAME.
(3) Convert each ratio having the same second term and compare the first term.
(4) Convert each ratio having the same first term and compare the second term, but the pattern of the terms will be reversed.
(5) (a : b) > (c : d), (a : b) < (c : d) or (a : b) = (c : d)
according as ad > bc, ad < bc or ad = bc respectively
Ratios of various kinds:
(1) Ratio of greater Inequality, less inequality and equality: A ratio a : b is called a ratio of greater inequality, of less inequality or the equality, according as a > b, a < b or a = b respectively.
(2) Compounded ratio of a: b and c : d is ac :: bd.
(3) Duplicate ratio of a : b is a2 : b2.
(4) Triplicate ratio of a : b is a3 : b3.
(5) Sub-duplicate ratio of a : b is a1/2 : b 1/2
(7) Reciprocal ratio of a : b is or b : a.
Proportion: a, b, c and d are said to in proportion if a/b = c/d and it is written as a : b :: c : d, then a and d are called the extreme terms and band c are called the means or the middle terms. We say, a, b, c, d are in proportional, the fourth term d, is called the fourth proportional to a, b, c. If a : b :: c : d, then we have the following three proportions.
(1) a : c :: b: d
(2) b : a :: d : c
(3) c : a :: d : b
Important points about proportions:
(1) Product of means = Product of extremes.
(2) The mean proportional between a and b is (ab)1/2
VARIOUS KINDS OF PROPORTIONS
Direct proportion: When two quantities are so related that an increase in one cause a corresponding increase in the other, it is called the direct proportion.
Inverse proportion: When two quantities are so related that an increase in one cause a corresponding decrease in the other, it is called the inverse proportion.
Continued proportion: Quantities are said to be in continued proportion, when the first is to the second as the second is to the third, as the third is to the fourth, and so on, thus, if
a/b = b/c = c/d = d/e = e/f = -------
The quantities a, b, c, d, e, f are said to be in continued proportion.
Theorems:
(1) Four quantities are in proportion if and only if, product of the extremes is equal to the product of means and conversely.
(2) If three quantities are in continued proportion then the product of the extremes is equal to the square of the mean.
(3) Fundamental theorem: If three quantities are in continued proportion then the ratio to first of third is the duplicate ratio of the first to second.
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