Fractions and Decimals: Practice Questions & Notes

Fractions and decimals are two different representations of non-whole numbers. While fractions express a part of a whole in terms of two integers, decimals express it in base-10 using a decimal point. They are essential for calculations involving percentages, averages, ratios, and comparisons. Mastery of this topic makes mental math quicker and enhances calculation efficiency across all aptitude areas.

Fractions

A fraction represents part of a whole. It's written in the form:

\frac{a}{b}, \quad b \ne 0

$a$ : numerator (part taken)
$b$ : denominator (total parts)

Example: $\frac{3}{4}$ means 3 parts out of 4 equal parts.

Decimals

Decimals are a way of expressing numbers in base-10 format. The decimal point separates the whole number from the fractional part.

Example:

0.75 = \frac{75}{100} = \frac{3}{4}

Both represent the same value in different formats.

Key Concepts and Classifications

Types of Fractions

Type	Description	Example
Proper	Numerator < Denominator	$\frac{2}{5}$
Improper	Numerator ≥ Denominator	$\frac{7}{4}$
Mixed	Whole + Proper Fraction	$1\frac{3}{4}$
Like Fractions	Same denominator	$\frac{2}{9}, \frac{5}{9}$
Unlike Fractions	Different denominators	$\frac{1}{2}, \frac{1}{3}$
Equivalent Fractions	Same value, different form	$\frac{1}{2} = \frac{2}{4}$

Types of Decimals

Type	Example	Characteristics
Terminating	0.5, 0.25	Finite digits after decimal
Recurring (Repeating)	0.666… = $0.\overline{6}$	Infinite repetition
Non-Terminating, Non-Repeating	$\pi$ , $\sqrt{2}$	Irrational (discussed in Number System)

Key Formulas and Shortcuts

Fraction Operations

Addition/Subtraction

Like Fractions:
$\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$
Unlike Fractions:
Find LCM of denominators, convert both to equivalent fractions, then add/subtract.

Multiplication

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Division

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Mixed to Improper Fraction

x\frac{y}{z} = \frac{xz + y}{z}

Decimal Operations

Addition/Subtraction: Align decimal points.

Multiplication: Multiply normally, then count total decimal digits.

Division: Move the decimal point in both numbers to make the divisor a whole number.

Shortcut: Repeating Decimal to Fraction

Pure Recurring:
Let $x = 0.\overline{3}$

10x = 3.\overline{3}, \quad \Rightarrow 10x - x = 9x = 3 \Rightarrow x = \frac{1}{3}

Mixed Recurring:
Let $x = 0.16\overline{3}$

1000x = 163.333..., \quad 10x = 1.633... \Rightarrow 990x = 161.7 \Rightarrow x = \frac{1617}{9900}

Conceptual Tips and Common Mistakes

Mistake	Correct Approach
Adding fractions directly without LCM	Always use LCM for unlike denominators
Assuming recurring decimals terminate	Know the difference between 0.33 and $0.\overline{3}$
Forgetting decimal place rules in multiplication	Count total decimal digits correctly
Converting mixed numbers incorrectly	Always convert to improper before operation

Examples

Example 1:

\frac{2}{3} + \frac{5}{6}

Solution:
LCM of 3 and 6 = 6

\Rightarrow \frac{4}{6} + \frac{5}{6} = \frac{9}{6} = \frac{3}{2}

Example 2:

Convert $0.\overline{81}$ into a fraction.

Let $x = 0.\overline{81}$

100x = 81.\overline{81}, \quad x = 0.\overline{81} \Rightarrow 99x = 81 \Rightarrow x = \frac{81}{99} = \frac{9}{11}

Example 3:

Convert $\frac{11}{16}$ to a decimal.

11 \div 16 = 0.6875