Decimal Fractions

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What: Decimal Fractions covers conversion between fractions and decimals, terminating vs non-terminating expansions, recurring decimals, decimal arithmetic, and comparing fractions in decimal form.
Why it matters: CDS averages 2–4 questions per paper on this chapter, often blended with percentages, ratios or simplification problems.
Key fact: A rational number $p/q$ (in lowest terms) has a terminating decimal expansion iff the prime factorisation of $q$ is of the form $2^a \cdot 5^b$ — nothing else. Anything else gives a non-terminating repeating decimal.

Decimal Fractions is a quick-yield chapter — its rules are short, its tricks are uniform, and CDS recycles the same patterns year after year. Get fluent with the place-value structure, terminating-decimal test, and bar-notation arithmetic, and you bank 2–4 marks per paper without breaking stride.

This page is grounded in CDS Previous Year Questions from 2007 to 2022 plus NCERT Class 8 Fractions in Disguise. Pair with Number System (for prime factorisation) and Percentage (decimals are the bridge to percentages).

What This Topic Covers

The CDS scope covers four blocks: (1) representation — place value, fraction-to-decimal and decimal-to-fraction conversion; (2) classification — terminating, non-terminating recurring, non-terminating non-recurring; (3) arithmetic — addition, subtraction, multiplication, division of decimals, including recurring decimals; and (4) comparison — ordering fractions via their decimal expansions and simplifying mixed decimal expressions.

Why This Topic Matters

The terminating-decimal test (denominator = $2^a 5^b$) appears every other CDS sitting.
Recurring decimal to fraction conversion is a 30-second trick that saves time in simplification problems.
Decimal arithmetic blends seamlessly with Percentage and Ratio and Proportion — sharpening it lifts your speed everywhere.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-I/II	3	Fraction-to-decimal, decimal multiplication, ordering
2008-I	2	Recurring decimals, simplification
2009-II	2	Place-value reasoning, comparison
2010-I/II	3	Recurring to fraction, decimal arithmetic
2011-I	2	Terminating decimal test, simplification
2012-II	2	Decimal-to-fraction, ordering
2013-I	2	Bar notation arithmetic, simplification
2015-II	2	Mixed recurring decimals
2016-II	2	Decimal multiplication, division
2017-I/II	3	Ordering, terminating test, conversion
2018-II	2	Recurring decimals, simplification
2022-I	2	Mixed conversion, decimal placement

⚡ CDS Alert

When asked whether $p/q$ (in lowest terms) terminates, factor the denominator. If $q = 2^a 5^b$ only, the expansion terminates. If any other prime appears in $q$, the expansion is non-terminating recurring. This single rule appears almost every CDS sitting.

Core Concepts

Place Value and Notation

In a decimal like $345.678$, the digits after the point have place values of $\tfrac{1}{10}, \tfrac{1}{100}, \tfrac{1}{1000}$ and so on. So $345.678 = 345 + \tfrac{6}{10} + \tfrac{7}{100} + \tfrac{8}{1000}$.

Fraction-to-Decimal Conversion

Just divide numerator by denominator. The result either terminates or eventually settles into a repeating block (overlined with a bar).

Terminating Decimal Test $$\frac{p}{q} \text{ (in lowest terms) terminates} \iff q = 2^a \cdot 5^b \text{ for some } a, b \geq 0$$

Example: $\tfrac{7}{40}$ terminates because $40 = 2^3 \cdot 5$. $\tfrac{1}{6}$ does not — denominator has a factor of 3.

Decimal-to-Fraction Conversion

For a terminating decimal, count decimal places $n$; place the digits over $10^n$ and simplify. Example: $0.625 = \tfrac{625}{1000} = \tfrac{5}{8}$.

For a pure recurring decimal with repeating block of length $k$: place the block over $\underbrace{99\ldots9}_{k}$.

Pure Recurring Decimal $$0.\overline{abc} = \frac{abc}{999}, \qquad 0.\overline{3} = \frac{3}{9} = \frac{1}{3}$$

For a mixed recurring decimal (some non-repeating digits before the repeating block):

Mixed Recurring Decimal $$0.a\overline{bc} = \frac{abc - a}{990}$$

Subtract the non-repeating part from the whole digit string, divide by 9s (one per repeating digit) followed by 0s (one per non-repeating decimal digit).

Decimal Arithmetic

For addition and subtraction, line up the decimal points. For multiplication, ignore the decimal points, multiply as integers, then place the decimal so total decimal places = sum of decimal places of inputs. For division, multiply both numerator and denominator by a power of 10 to clear the divisor's decimals; then proceed as integer division.

⚠ Common Trap

$0.1 \times 0.1 = 0.01$, not $0.1$. Each factor has one decimal place, so the product has two. CDS plants this trap regularly in simplification problems.

Ordering Decimals

Pad all decimals to the same number of places by appending zeros, then compare digit by digit. Example: order $0.5, 0.55, 0.505, 0.55\overline{5}$. Pad: $0.5000, 0.5500, 0.5050, 0.5555$. Order: $0.5 < 0.505 < 0.55 < 0.5\overline{5}$.

Worked Examples

Example 1 — Recurring to Fraction (2010-I)

Q: Express $0.\overline{37}$ as a fraction.

The repeating block is "37", length 2.
Apply the pure-recurring formula: $0.\overline{37} = \tfrac{37}{99}$.
Check: $\gcd(37, 99) = 1$, so it's already in lowest terms. Answer: $\tfrac{37}{99}$.

Example 2 — Terminating Test (2011-I)

Q: Which of $\tfrac{7}{20}, \tfrac{5}{12}, \tfrac{11}{40}, \tfrac{4}{15}$ has a terminating decimal expansion?

Factor each denominator: $20 = 2^2 \cdot 5,\; 12 = 2^2 \cdot 3,\; 40 = 2^3 \cdot 5,\; 15 = 3 \cdot 5$.
Terminating ⇔ denominator has only 2 and 5 as primes.
$\tfrac{7}{20}$ ✓ and $\tfrac{11}{40}$ ✓. The other two have a 3 in the denominator → non-terminating.
Answer: $\tfrac{7}{20}$ and $\tfrac{11}{40}$.

Example 3 — Mixed Recurring to Fraction (2015-II)

Q: Convert $0.5\overline{27}$ to a fraction.

Identify non-repeating digit: 5 (one digit). Repeating block: 27 (two digits).
Apply mixed-recurring formula: $0.5\overline{27} = \tfrac{527 - 5}{990} = \tfrac{522}{990}$.
Simplify: $\gcd(522, 990) = 18$, so $\tfrac{522}{990} = \tfrac{29}{55}$.
Answer: $\tfrac{29}{55}$.

Example 4 — Decimal Multiplication (2016-II)

Q: Compute $0.12 \times 0.05$.

Multiply as integers: $12 \times 5 = 60$.
Count total decimal places: $2 + 2 = 4$.
Place the decimal: $0.0060 = 0.006$.
Answer: $0.006$.

Example 5 — Ordering Decimals (2017-I)

Q: Arrange in ascending order: $0.6, 0.\overline{6}, 0.66, 0.66\overline{6}$.

Convert each to a sufficient decimal expansion: $0.6 = 0.6000,\; 0.\overline{6} = 0.6666...,\; 0.66 = 0.6600,\; 0.66\overline{6} = 0.6666...$.
Note that $0.\overline{6} = \tfrac{6}{9} = \tfrac{2}{3}$ and $0.66\overline{6} = \tfrac{6}{10} + \tfrac{6}{90} = \tfrac{60+6}{90} \neq \tfrac{2}{3}$ — they differ in the 4th decimal place.
Compute carefully: $0.\overline{6} = 0.666666...$; $0.66\overline{6}$ means the 6 after the initial "66" repeats forever, so it's $0.666666...$ — same value!
So the order is $0.6 < 0.66 < 0.\overline{6} = 0.66\overline{6}$.

Example 6 — Decimal Division (2012-II)

Q: Compute $\tfrac{0.0625}{0.25}$.

Multiply top and bottom by 10000 to clear all decimals: $\tfrac{625}{2500} = \tfrac{1}{4}$.
Convert back to decimal: $\tfrac{1}{4} = 0.25$.
Answer: $0.25$.

Example 7 — Place Value (2009-II)

Q: In the decimal 24.5083, what is the place value of the digit 8?

From the decimal point, the digits 5, 0, 8, 3 occupy places $\tfrac{1}{10},\, \tfrac{1}{100},\, \tfrac{1}{1000},\, \tfrac{1}{10000}$.
The digit 8 is at the thousandths place.
Place value = $8 \times \tfrac{1}{1000} = 0.008$ (or $\tfrac{8}{1000} = \tfrac{1}{125}$ as a fraction).

How CDS Tests This Topic

Five archetypes recur: (1) "does this fraction terminate?" — apply the $2^a 5^b$ test, (2) "convert this recurring decimal to a fraction" — pure or mixed formula, (3) "arrange in ascending/descending order" — pad and compare, (4) "compute this decimal expression" — basic arithmetic with care on decimal-place counting, and (5) "place value of a digit" — fundamental notation.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — The "9s and 0s" Rule for Recurring Decimals

For a recurring decimal, the denominator of the equivalent fraction is built from 9s (one per repeating digit) and 0s (one per non-repeating digit after the decimal point). The numerator is "whole digit string − non-repeating part".

General Recurring → Fraction $$0.\underbrace{a_1 a_2 \cdots a_n}_{\text{non-rep}}\overline{b_1 b_2 \cdots b_k} = \frac{a_1 a_2 \cdots a_n b_1 \cdots b_k - a_1 a_2 \cdots a_n}{\underbrace{9 \cdots 9}_{k}\,\underbrace{0 \cdots 0}_{n}}$$

Shortcut 2 — Quick Reference for Common Fractions

Memorise these — they appear in every CDS paper:

Common Conversions $\tfrac{1}{8} = 0.125,\; \tfrac{1}{4} = 0.25,\; \tfrac{3}{8} = 0.375,\; \tfrac{1}{2} = 0.5,\; \tfrac{5}{8} = 0.625,\; \tfrac{3}{4} = 0.75,\; \tfrac{7}{8} = 0.875,\; \tfrac{1}{3} = 0.\overline{3},\; \tfrac{2}{3} = 0.\overline{6},\; \tfrac{1}{6} = 0.1\overline{6},\; \tfrac{1}{7} = 0.\overline{142857}$

Shortcut 3 — Decimal Places in Multiplication

Total decimal places in the product = sum of decimal places of inputs. So $0.025 \times 0.4 = ?$ has $3 + 1 = 4$ decimal places. Compute $25 \times 4 = 100$, so answer is $0.0100 = 0.01$.

Shortcut 4 — Comparing Fractions Without Full Conversion

To compare $\tfrac{a}{b}$ and $\tfrac{c}{d}$, cross-multiply: $ad$ vs $bc$. Whichever is larger corresponds to the larger fraction. Much faster than converting both to decimals.

Shortcut 5 — Spot the Denominator Type Instantly

If the denominator (in lowest terms) is 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, ... — only powers of 2 and 5 — it terminates. Anything else (3, 6, 7, 9, 11, 12, 13, ...) is non-terminating recurring.

Common Question Patterns

Pattern 1 — Identify Terminating Decimals

Given four fractions, pick the ones that terminate. Apply the $2^a 5^b$ rule on each denominator. Fast, mechanical.

Pattern 2 — Convert Recurring to Fraction

Use the 9s-and-0s formula. CDS occasionally adds a layer: "convert $0.\overline{142857}$ and simplify" — the answer is $\tfrac{1}{7}$, a famous repeating expansion.

Pattern 3 — Arrange Decimals or Fractions in Order

Pad to common decimal length, or cross-multiply pairwise. Watch out for visually similar decimals like $0.\overline{6}$ and $0.66\overline{6}$ — they are equal.

Pattern 4 — Decimal-to-Fraction Simplification

Multi-step simplification expressions like $\tfrac{0.5 \times 0.25 \times 0.4}{0.04 \times 0.2}$ — clear decimals by multiplying numerator and denominator by appropriate powers of 10, then simplify the integer ratio.

Pattern 5 — Place Value Reasoning

"In 4567.0023, what is the difference in place values of the two 0s?" Each occupies a different power-of-ten position; subtract.

Preparation Strategy

Week 1. Memorise the common fraction-to-decimal conversions (eighths, fifths, thirds, sevenths). Drill 20 problems on the terminating-decimal test. Practice the pure and mixed recurring-to-fraction formulae until they are reflex.

Week 2. Practice decimal arithmetic — especially multiplication and division, where decimal-place counting is the most common error source. Drill comparison and ordering problems. Mix in Percentage problems, since most percentages start as decimals.

Mock testing. Tag every decimal-related question in your timed papers. The chapter is short — once you have the rules cold, the only way to lose marks here is arithmetic slips. Sharpen reflexes with CDS mock tests.

Cross-train with Number System (prime factorisation underlies the terminating-decimal test) and Ratio and Proportion (cross-multiplication is the fastest fraction-compare).

Practice Decimal Fractions in Real Time

Take CDS mock papers with decimal-fraction questions, recurring conversions and simplification expressions. Reflex speed is everything in this chapter.

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Frequently Asked Questions

How do I know if a fraction has a terminating decimal expansion?

Reduce the fraction to lowest terms, then factorise the denominator. If the only prime factors are 2 and/or 5, the decimal terminates. Any other prime (3, 7, 11, …) and the expansion is non-terminating recurring. Example: $\tfrac{7}{40}$ terminates because $40 = 2^3 \cdot 5$; $\tfrac{1}{6}$ does not because $6 = 2 \cdot 3$.

How do I convert a recurring decimal to a fraction?

For a pure recurring decimal like $0.\overline{37}$: put the repeating block over a string of 9s of the same length, giving $\tfrac{37}{99}$. For a mixed recurring decimal like $0.5\overline{27}$: numerator = whole digit string minus non-repeating part $= 527 - 5 = 522$; denominator = 9s (one per repeating digit) followed by 0s (one per non-repeating decimal digit) $= 990$. So $\tfrac{522}{990} = \tfrac{29}{55}$.

Are $0.\overline{6}$ and $0.66\overline{6}$ the same number?

Yes. Both equal $0.666666...$ — the repeating block is "6" in each case, the position of the bar only changes notation. Convert both to fractions and you get $\tfrac{2}{3}$. CDS occasionally hides this equality in ordering problems.

How do I count decimal places when multiplying?

Total decimal places in the product = sum of decimal places of the factors. Example: $0.025 \times 0.4$. Multiply integers: $25 \times 4 = 100$. Total decimal places: $3 + 1 = 4$. Place the decimal: $0.0100 = 0.01$. Always count carefully — this is the most common slip.

What is the fastest way to compare two fractions?

Cross-multiply. To compare $\tfrac{a}{b}$ and $\tfrac{c}{d}$, compare $ad$ with $bc$. Whichever product is larger corresponds to the larger fraction (assuming positive denominators). No decimal conversion required.

Why is $0.999... = 1$?

Apply the recurring-decimal formula: $0.\overline{9} = \tfrac{9}{9} = 1$. They are two notations for the same number. CDS rarely tests this directly but the identity is useful for sanity checks on conversion problems.

Which CDS Maths topics connect to Decimal Fractions?

Number System supplies the prime-factorisation framework. Percentage is decimals × 100, so fluency here lifts speed there. Ratio and Proportion uses cross-multiplication, the same shortcut as fraction comparison.