Decimal Fractions questions with answers and concepts

Learn decimal fractions with concept notes and worked examples, then practise focused questions with explained answers. Build accuracy for placement and competitive aptitude tests before trying timed mode.

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Decimal fractions represent parts of a whole using base-10 notation, where each digit after the decimal point represents tenths, hundredths, thousandths, etc. Understanding decimal operations, conversions between fractions and decimals, and recurring decimals is essential for solving quantitative aptitude problems efficiently.

Key formulas and rules

8 rules

Decimal to Fraction: Write digits as numerator and powers of $10$ as denominator, e.g. $0.75 = \frac{75}{100} = \frac{3}{4}$

Fraction to Decimal: Divide numerator by denominator, e.g. $\frac{3}{4} = 0.75$

Terminating Decimal condition: A fraction $\frac{p}{q}$ in lowest terms terminates iff $q = 2^{a} \times 5^{b}$ for non-negative integers $a, b$

Pure Recurring Decimal: $0. \overline{a} = \frac{a}{9}$ , $0. \overline{ab} = \frac{ab}{99}$ , $0. \overline{ab c} = \frac{ab c}{999}$

Mixed Recurring Decimal: $0. a \overline{b} = \frac{ab - a}{90}$ , $0. ab \overline{c} = \frac{ab c - ab}{900}$

Comparing Decimals: Compare whole numbers first, then tenths, hundredths, and thousandths from left to right

Decimal Multiplication: Multiply ignoring decimals, then count total decimal places across factors to place the point

Decimal Division: Multiply numerator and denominator by $1 0^{k}$ to make the divisor a whole number before dividing

Key concepts

Types of Decimals

Terminating decimals end after finite digits (e.g., 0.5, 0.25). Non-terminating decimals continue infinitely: pure recurring where all digits repeat (0.333...), and mixed recurring where some digits don't repeat (0.1666...). A fraction gives terminating decimal only when denominator (in lowest terms) has prime factors 2 and/or 5 only.

Converting Recurring Decimals to Fractions

For pure recurring: 0.a = a/9, 0.ab = ab/99, 0.abc = abc/999. For mixed recurring like 0.1a b : subtract non-repeating part from number with bar, divide by 9s for repeating digits followed by 0s for non-repeating. Example: 0.16 = $(16 - 1) /90 = 15/90 = 1/6$ . The general formula is (entire number - non-repeating part) / (9s for repeating digits, 0s for non-repeating digits).

Decimal Operations

Addition/Subtraction: Line up decimal points. You may add trailing zeros to equalize places. Multiplication: Ignore decimals, multiply as integers, then count total decimal places in factors for the result. Division: Convert divisor to whole number by shifting decimal points equally in both numbers, then divide normally.

Comparing and Ordering Decimals

To compare decimals, first write them with equal decimal places by adding trailing zeros. Then compare digit by digit from left. For example, comparing 0.3, 0.29, 0.305: write as 0.300, 0.290, 0.305. Comparing: 0.305 > 0.300 > 0.290. When arranging in ascending/descending order, this method ensures accuracy.

Rounding and Approximation

To round to n decimal places: look at digit at (n+1)th place. If it's 5 or more, round up the nth digit; otherwise keep it. For significant figures, count from first non-zero digit. Rounding is crucial for quick estimations in multiple-choice questions where exact calculation isn't necessary.

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Worked examples

Example 1

Convert 0.125 to a vulgar fraction in lowest terms.

Solution

Step 1: Express with power-of-10 denominator.

0.125 = \frac{125}{1 , 000}

Step 2: Divide numerator and denominator by $g cd (125, 1, 000) = 125$ .

\frac{125 \div 125}{1 , 000 \div 125} = \frac{1}{8}

Answer: $\frac{1}{8}$ .

Example 2

Convert the pure recurring decimal $0. \overline{45}$ into a vulgar fraction.

Solution

Step 1: Apply pure repeating decimal conversion ( $k = 2$ repeating digits).

0. \overline{45} = \frac{45}{1 0 ^{2} - 1} = \frac{45}{99}

Step 2: Reduce to lowest terms by dividing by 9.

\frac{45 \div 9}{99 \div 9} = \frac{5}{11}

Answer: $\frac{5}{11}$ .

Example 3

Convert the mixed recurring decimal $0.2 \overline{35}$ into a vulgar fraction.

Solution

Step 1: Apply the mixed recurring decimal formula.

0. a \overline{b c} = \frac{ab c - a}{990}

0.2 \overline{35} = \frac{235 - 2}{990} = \frac{233}{990}

Answer: $\frac{233}{990}$ .

Example 4

Evaluate the product $0.25 \times 0.04$ .

Solution

Step 1: Multiply whole numbers ignoring decimal places.

25 \times 4 = 100

Step 2: Count total decimal places ( $2 + 2 = 4$ ).

0.25 \times 0.04 = 0.0100 = 0.01 = \frac{1}{100}

Answer: 0.01 (or $\frac{1}{100}$ ).

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

Express 0.1666... (recurring) as a fraction.

Answer: A. 1/6

Explanation

Let $x = 0$ .1666...
$10 x = 1.666...$
$100 x = 16.666...$
$100x - 10x = 16.666... - 1.666... = 15$
$90 x = 15$
$x = 15/90 = 1/6$

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What is 0.272727... (recurring) as a fraction in simplest form?

3/11
27/100
3/10
7/33

Answer: A. 3/11

Explanation

Let $x = 0$ .272727...
$100 x = 27.272727...$
$100 x - x = 27$
$99 x = 27$
$x = 27/99 = 3/11$ (dividing by 9)
Answer: 3/11

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What is 0.454545... (recurring) as a fraction in simplest form?

5/11
9/20
9/22
5/9

Answer: A. 5/11

Explanation

Let $x = 0$ .454545...
$100 x = 45.454545...$
$100 x - x = 45$
$99 x = 45$
$x = 45/99 = 5/11$ (dividing by 9)
Answer: 5/11

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Memorize common fraction-decimal conversions: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/5=0.2, 1/8=0.125, 1/3~=0.333, 1/6~=0.167, 1/7~=0.143, 1/9~=0.111
For recurring decimals, count the length of the repeating cycle - this determines the denominator pattern (9, 99, 999, etc.)
When adding/subtracting decimals, always align decimal points vertically to avoid place value errors
A fraction has terminating decimal expansion only if denominator (in lowest terms) = 2^m x 5^n for non-negative integers m, n
For quick comparison, convert all decimals to same number of places by adding trailing zeros
In mixed recurring decimals like 0.ab c , the numerator is (abc - a) and denominator is 990 (two 9s for repeating digits, one 0 for non-repeating)
Use estimation to check answers: 0.4 x 0.5 should be around 0.2, not 2 or 0.02

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Next topic: Simplification

Key concepts

Types of Decimals

Converting Recurring Decimals to Fractions

Decimal Operations

Comparing and Ordering Decimals

Rounding and Approximation

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

Convert 0.125 to a vulgar fraction in lowest terms.

Solution

Step 1: Express with power-of-10 denominator.

0.125 = \frac{125}{1 , 000}

Step 2: Divide numerator and denominator by $g cd (125, 1, 000) = 125$ .

\frac{125 \div 125}{1 , 000 \div 125} = \frac{1}{8}

Answer: $\frac{1}{8}$ .

Example 2

Convert the pure recurring decimal $0. \overline{45}$ into a vulgar fraction.

Solution

Step 1: Apply pure repeating decimal conversion ( $k = 2$ repeating digits).

0. \overline{45} = \frac{45}{1 0 ^{2} - 1} = \frac{45}{99}

Step 2: Reduce to lowest terms by dividing by 9.

\frac{45 \div 9}{99 \div 9} = \frac{5}{11}

Answer: $\frac{5}{11}$ .

Example 3

Convert the mixed recurring decimal $0.2 \overline{35}$ into a vulgar fraction.

Solution

Step 1: Apply the mixed recurring decimal formula.

0. a \overline{b c} = \frac{ab c - a}{990}

0.2 \overline{35} = \frac{235 - 2}{990} = \frac{233}{990}

Answer: $\frac{233}{990}$ .

Example 4

Evaluate the product $0.25 \times 0.04$ .

Solution

Step 1: Multiply whole numbers ignoring decimal places.

25 \times 4 = 100

Step 2: Count total decimal places ( $2 + 2 = 4$ ).

0.25 \times 0.04 = 0.0100 = 0.01 = \frac{1}{100}

Answer: 0.01 (or $\frac{1}{100}$ ).

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

Express 0.1666... (recurring) as a fraction.

Answer: A. 1/6

Explanation

Let $x = 0$ .1666...
$10 x = 1.666...$
$100 x = 16.666...$
$100x - 10x = 16.666... - 1.666... = 15$
$90 x = 15$
$x = 15/90 = 1/6$

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

What is 0.272727... (recurring) as a fraction in simplest form?

3/11
27/100
3/10
7/33

Answer: A. 3/11

Explanation

Let $x = 0$ .272727...
$100 x = 27.272727...$
$100 x - x = 27$
$99 x = 27$
$x = 27/99 = 3/11$ (dividing by 9)
Answer: 3/11

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

What is 0.454545... (recurring) as a fraction in simplest form?

5/11
9/20
9/22
5/9

Answer: A. 5/11

Explanation

Let $x = 0$ .454545...
$100 x = 45.454545...$
$100 x - x = 45$
$99 x = 45$
$x = 45/99 = 5/11$ (dividing by 9)
Answer: 5/11

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Memorize common fraction-decimal conversions: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/5=0.2, 1/8=0.125, 1/3~=0.333, 1/6~=0.167, 1/7~=0.143, 1/9~=0.111
For recurring decimals, count the length of the repeating cycle - this determines the denominator pattern (9, 99, 999, etc.)
When adding/subtracting decimals, always align decimal points vertically to avoid place value errors
A fraction has terminating decimal expansion only if denominator (in lowest terms) = 2^m x 5^n for non-negative integers m, n
For quick comparison, convert all decimals to same number of places by adding trailing zeros
In mixed recurring decimals like 0.ab c , the numerator is (abc - a) and denominator is 990 (two 9s for repeating digits, one 0 for non-repeating)
Use estimation to check answers: 0.4 x 0.5 should be around 0.2, not 2 or 0.02

Key formulas and rules

Key concepts

Types of Decimals

Converting Recurring Decimals to Fractions

Decimal Operations

Comparing and Ordering Decimals

Rounding and Approximation

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Types of Decimals

Converting Recurring Decimals to Fractions

Decimal Operations

Comparing and Ordering Decimals

Rounding and Approximation

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.