RATIO

The comparison between two quantities of the same kind of unit is the Ratio of one quantity to another. The ratio of a and b is usually written as a : b or a/b, where a is called the antecedent (numerator) and b the consequent (denominator).

1. \(a : b = ka : kb\) where k is a constant

2. \(a : b = \frac{a}{k} : \frac{b}{k}\)

3. \(a : b : c = X : Y : Z\) is equivalent to \(\frac{a}{X} = \frac{b}{Y} =\frac{ c}{Z}\)

4. If \(\frac{a}{b} = \frac{c}{d}\), then

(i) \(\frac{(a + b)}{b} = \frac{(c + d)}{d}\)
(ii) \(\frac{(a – b)}{b} = \frac{(c – d)}{d}\)
(iii) \(\frac{(a + b)}{(a – b)} =\frac{ (c + d)}{(c – d)}\)

5. If \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = ...... = k\), then \(\frac{a + c + e +}{b + d + f +} .... .... = k\).

Also note that : \(k = \frac{a}{b }= \frac{Xc }{ Xd} = \frac{–5e }{ –5f}\) ⇒ Each ratio \(= \frac{(a + Xc – 5e) }{(b + Xd – 5f)} = k\). Here, we have randomly taken X and –5. You can take any factor.

6. If \(\frac{a}{b} > 1\) or \(a > b\) then \(\frac{(a + X) }{ (b + X)} <\frac{ a}{b}\) a, b, X are natural numbers

7. If \(\frac{a}{b} < 1\) or \(a < b\) then \(\frac{(a + X) }{ (b + X )} > \frac{a}{b}\) a, b, X are natural numbers

VARIATION

1. Direct proportion :

If two quantities X & Y are related such that any increase or decrease in ‘Y’ produces a proportionate increase or decrease in ‘X’ or vice versa, then the two quantities are said to be in direct proportion.
In other words \(X : Y = \frac{X}{Y} = k\) (a constant) or X = KY or Y = K’X (where K and K’ are constants)
X is directly proportional to Y is written as X ∝ Y or X = K Y

2. Inverse proportion :

Here two quantities X & Y are related such that, any increase in X would lead to a decrease in Y or any decrease in X would lead to an increase in Y. Thus the quantities X & Y are said to be inversely related and X is inversely proportional to Y is written as \(\frac{X ∝ 1}{Y}\) or \(X = \frac{k}{Y}\) or XY = k (constant) or the product of two quantities remains constant.


MIXTURES FORMULA

This rule enables us to find the proportion in which two or more ingredients at the given price must be mixed to produce a mixture at a given price. The C.P. of unit quantity of the mixture is called the MEAN PRICE. The rule says :
If two ingredients are mixed in a ratio, then

\(\frac{Quantity of Cheaper}{Quantity of Dearer} = \frac{CP of Dearer – Mean Price}{Mean Price – CP of Cheaper}\).