Ratio

QUIZ

Ratio is a comparison of two or more quantities of the same kind, expressed in the form
\(a:b \quad \text{or} \quad \frac{a}{b}\)

It forms the foundation for problems involving numbers, mixtures, populations, committees, and value comparisons.

1. Basic Properties of Ratio

  • If $a:b = c:d$, then
    \(ad = bc \quad \text{(cross multiplication)}\)

  • Ratios remain unchanged if both terms are multiplied or divided by the same non-zero number.

  • A ratio has no unit.

2. Ratio in Numbers

Digits of a Number

For a two-digit number with digits $x$ (tens) and $y$ (units):

\[\text{Number} = 10x + y\]

This idea is frequently used in ratio-based number problems.

3. Ratio as Fractions

A ratio can be converted into a fraction and vice versa.

Example: \(a:b = \frac{a}{b}\)

Many exam questions convert verbal statements into fractions and then equate ratios.

4. Ratio in Populations / Committees

When total members are unknown, assume the total to be $x$.

Fractions of groups are then expressed as parts of $x$, which makes calculations systematic.

5. Ratio in Mixtures

If ingredients are mixed in given ratios, the final proportion depends on:

  • Quantity taken from each mixture
  • Relative proportions of components

Equality of components often leads to equating algebraic expressions.

6. Ratio and Value Proportionality

Sometimes value is not proportional to weight, but to:

  • Square of weight
  • Cube of weight

This is common in questions involving precious stones, wires, or metals.

Exercise Problems (Based on NET Pattern)

Q1. (NET DEC-2016)

The sum of digits of a two-digit number is 9.
The fraction formed by taking 9 less than the number as numerator and 9 more than the number as denominator is $ \frac{3}{4} $.
Find the number.

(a) 36
(b) 63
(c) 45
(d) 54

Q2. (NET DEC-2016)

Nine-elevenths of the members of a parliamentary committee are men.
Of the men, two-thirds are from the Rajya Sabha.
Further, $ \frac{7}{11} $ of the total committee members are from the Rajya Sabha.
What fraction of the total members are women from the Lok Sabha?

(a) $ \frac{1}{11} $
(b) $ \frac{6}{11} $
(c) $ \frac{2}{11} $
(d) $ \frac{3}{11} $

Q3. (NET JUNE-2018)

Two solutions $X$ and $Y$ contain ingredients $A, B, C$ in the ratios
\(a:b:c \quad \text{and} \quad c:b:a\) respectively.
They are mixed so that the resultant mixture contains equal proportions of $A, B,$ and $C$.
Find the correct relation.

(a) $ b = \frac{c-a}{2} $
(b) $ c = \frac{a+b}{2} $
(c) $ c = \frac{a-b}{2} $
(d) $ b = \frac{a+c}{2} $

Q4. (NET JUNE-2019)

A precious stone breaks into four pieces having weights in the ratio $1:2:3:4$.
The value of the stone is proportional to the square of its weight.
Find the percentage loss in value due to breaking.

(a) 0
(b) 30
(c) 70
(d) 90

Solutions (Concise and Concept-Oriented)

Solution 1

Let the number be $10x + y$.
Given: \(x + y = 9\) \(\frac{10x + y - 9}{10x + y + 9} = \frac{3}{4}\) Solving gives $y = 6,\; x = 3$.
Number = 63

✅ Answer: (b)

Solution 2

Let total committee members = $x$.

  • Men = $ \frac{9x}{11} $
  • Men from Rajya Sabha = $ \frac{2}{3} \times \frac{9x}{11} = \frac{6x}{11} $
  • Total Rajya Sabha members = $ \frac{7x}{11} $

Women from Rajya Sabha: \(\frac{7x}{11} - \frac{6x}{11} = \frac{x}{11}\)

Lok Sabha members = $ x - \frac{7x}{11} = \frac{4x}{11} $

Women from Lok Sabha: \(\frac{4x}{11} - \frac{3x}{11} = \frac{x}{11}\)

Fraction = $ \frac{1}{11} $

✅ Answer: (a)

Solution 3

Let equal quantities of $X$ and $Y$ be mixed.

Equality of components implies: \(a + c = 2b \Rightarrow b = \frac{a+c}{2}\)

✅ Answer: (d)

Solution 4

Weights = $k, 2k, 3k, 4k$

Original value: \(\propto (10k)^2 = 100k^2\)

Value after breaking: \(\propto (1^2 + 2^2 + 3^2 + 4^2)k^2 = 30k^2\)

Loss: \(\frac{100 - 30}{100} \times 100 = 70\%\)

✅ Answer: (c)