How do you convert a pure recurring decimal like 0.777... to a fraction?

0.777... = 7/9. The digit that repeats goes over as many 9s as there are repeating digits.

What is the fraction form of 0.142857142857...?

0.142857142857... = 142857/999999 = 1/7.

How do you convert a mixed recurring decimal like 0.16̄ (0.1666...) to a fraction?

0.1666... = (16 - 1)/90 = 15/90 = 1/6. Formula: (all digits after decimal - non-recurring part) / (9s for recurring × 10s for non-recurring).

Which fraction is larger: 3/7 or 4/9?

Cross multiply: 3×9=27 and 4×7=28. Since 28>27, we have 4/9 > 3/7.

Decimal Fractions — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

A decimal fraction is a fraction whose denominator is a power of 10 (10, 100, 1000, ...). Every decimal number — terminating or recurring — can be expressed as a fraction. Terminating decimals (like 0.75) are straightforward: 0.75 = 75/100 = 3/4. Recurring (repeating) decimals are more interesting — a digit or group of digits repeats infinitely, such as 1/3 = 0.333... or 1/7 = 0.142857142857...

There are two types of recurring decimals: pure recurring (the recurring part starts immediately after the decimal point, e.g., 0.636363...) and mixed recurring (there is a non-recurring part first, e.g., 0.1666... where 1 is non-recurring and 6 is recurring). Each type has a formula for direct conversion to fractions — these formulas appear in TNPSC questions and are well worth memorising.

Ordering fractions is another key skill. When comparing two fractions p/q and r/s, the fastest method is cross multiplication: compare p×s with r×q. The fraction whose numerator gives the larger product when cross-multiplied is the larger fraction. For multiple fractions, convert to a common denominator or convert to decimals for comparison. This skill is tested directly ("arrange in ascending order") and indirectly in ratio and proportion problems.

Key Formulas

Formula	Description	Example
0.aaa... = a/9	Single digit pure recurring	0.777... = 7/9
0.ababab... = ab/99	Two-digit pure recurring	0.363636... = 36/99 = 4/11
0.abcabc... = abc/999	Three-digit pure recurring	0.142857... = 142857/999999 = 1/7
0.abcc... = (abc - ab)/990	Mixed recurring (1 non-rec, 1 rec)	0.1666... = (16-1)/90 = 1/6
0.abcdd... = (abcd - abc)/9900	Mixed recurring general	0.4833... = (483-48)/900 = 435/900 = 29/60
General rule: (all digits - non-recurring part) / (9s × 10s)	9s = count of recurring digits; 10s = count of non-recurring	See examples below
To compare p/q and r/s: compare p×s vs r×q	If p×s > r×q then p/q > r/s	3/7 vs 4/9: 27 vs 28 → 4/9 > 3/7

Worked Examples

Example 1: Convert terminating decimal to fraction

Problem: Express 0.4375 as a fraction in its lowest terms.

Step 1: Write the decimal over a power of 10.
        0.4375 = 4375/10000

Step 2: Find HCF of 4375 and 10000.
        4375 = 5⁴ × 7 = 625 × 7
        10000 = 10⁴ = 2⁴ × 5⁴
        HCF = 5⁴ = 625

Step 3: Divide numerator and denominator by HCF.
        4375/625 = 7
        10000/625 = 16

        0.4375 = 7/16

Answer: 0.4375 = 7/16

Verification: 7 ÷ 16 = 0.4375 ✓

Example 2: Convert pure recurring decimal to fraction

Problem: Express 0.727272... (i.e., 0.72̄) as a fraction.

Step 1: Identify the recurring part.
        The recurring group is "72" (two digits).

Step 2: Apply the pure recurring decimal formula.
        Two-digit recurring → denominator = 99

        0.727272... = 72/99

Step 3: Simplify the fraction.
        HCF(72, 99):
        99 = 72 × 1 + 27
        72 = 27 × 2 + 18
        27 = 18 × 1 + 9
        18 = 9 × 2 + 0
        HCF = 9

        72/9 = 8
        99/9 = 11

        0.727272... = 8/11

Answer: 0.727272... = 8/11

Verification: 8 ÷ 11 = 0.727272... ✓

Example 3: Convert mixed recurring decimal to fraction

Problem: Express 0.23̄ (= 0.2333...) as a fraction.

Step 1: Identify non-recurring and recurring parts.
        0.2333...
        Non-recurring digit after decimal: 2 (one digit)
        Recurring digit: 3 (one digit)

Step 2: Apply the mixed recurring formula.
        Formula: (all digits after decimal - non-recurring part) / (9s × 10s)
        Number of 9s = number of recurring digits = 1
        Number of 0s = number of non-recurring digits = 1

        Denominator = 90

        Numerator = (23 - 2) = 21
        (Take all digits "23", subtract only the non-recurring part "2")

        0.2333... = 21/90

Step 3: Simplify.
        HCF(21, 90) = 3
        21/3 = 7
        90/3 = 30

        0.2333... = 7/30

Answer: 0.2333... = 7/30

Verification: 7 ÷ 30 = 0.2333... ✓

Example 4: Comparing and ordering fractions

Problem: Arrange the following fractions in ascending order: 5/8, 7/12, 11/18, 2/3.

Step 1: Find the LCM of all denominators.
        Denominators: 8, 12, 18, 3
        8  = 2³
        12 = 2² × 3
        18 = 2 × 3²
        3  = 3
        LCM = 2³ × 3² = 72

Step 2: Convert all fractions to the common denominator 72.
        5/8  = 5×9  / 72 = 45/72
        7/12 = 7×6  / 72 = 42/72
        11/18= 11×4 / 72 = 44/72
        2/3  = 2×24 / 72 = 48/72

Step 3: Compare numerators.
        42/72 < 44/72 < 45/72 < 48/72
        7/12  < 11/18 < 5/8   < 2/3

Answer: Ascending order: 7/12 < 11/18 < 5/8 < 2/3

Example 5: Operations with decimals — multiplication and simplification

Problem: Evaluate: (0.9 × 0.9 × 0.9 + 0.1 × 0.1 × 0.1) / (0.9 × 0.9 − 0.9 × 0.1 + 0.1 × 0.1)

Step 1: Recognise the algebraic identity.
        The expression has the form (a³ + b³) / (a² − ab + b²)
        where a = 0.9 and b = 0.1

Step 2: Apply the identity: a³ + b³ = (a + b)(a² − ab + b²)
        (a³ + b³) / (a² − ab + b²) = (a + b)(a² − ab + b²) / (a² − ab + b²)
                                   = (a + b)

Step 3: Calculate.
        a + b = 0.9 + 0.1 = 1.0

Answer: The expression = 1.0

Key insight: Spotting algebraic identities in decimal/fraction expressions
saves enormous calculation time in exams.

Shortcuts and Tricks

Trick 1: Quick recurring decimal conversions to memorise

Decimal	Fraction	Decimal	Fraction
0.333...	1/3	0.666...	2/3
0.1666...	1/6	0.8333...	5/6
0.1428...	1/7	0.2857...	2/7
0.125	1/8	0.375	3/8
0.111...	1/9	0.222...	2/9
0.0909...	1/11	0.1818...	2/11

Trick 2: Fast fraction comparison (cross multiply)

To compare 5/9 and 7/13:

Cross: 5 × 13 = 65 and 7 × 9 = 63
65 > 63 → 5/9 > 7/13

Never actually divide — cross multiply is always faster.

Trick 3: "Which is closest to 1?" type questions

Convert each fraction to see how far it is from 1. Alternatively compute 1 − each fraction and find the smallest gap.

Example: Which is closest to 1 among 7/8, 11/12, 13/15?

Closest is 11/12.

1 − 7/8 = 1/8 = 0.125
1 − 11/12 = 1/12 ≈ 0.083
1 − 13/15 = 2/15 ≈ 0.133

Trick 4: Simplifying complex fractions instantly

For expressions like (x³ − y³)/(x² + xy + y²), factor and cancel immediately.

x³ − y³ = (x−y)(x² + xy + y²)
Result = x − y

Similarly (x³ + y³)/(x² − xy + y²) = x + y.

Trick 5: Converting fractions to decimals quickly

For denominators that are factors of a power of 10:

/8 = /1000 equivalent → multiply top and bottom to get 1000: 3/8 = 375/1000 = 0.375
/4 = /100 equivalent → 3/4 = 75/100 = 0.75
/5 = /10 → 3/5 = 6/10 = 0.6
/25 = /100 → 7/25 = 28/100 = 0.28

TNPSC Exam Pattern

Frequency: 1–3 questions per TNPSC Group II/IV paper on decimal fractions. The topic often merges with simplification questions.

Common Question Types: 1. "Express the following recurring decimal as a fraction" (direct formula application) 2. "Arrange in ascending/descending order" (4 fractions to compare) 3. Arithmetic operations on decimals (addition, subtraction, multiplication) 4. "Which of the following is the largest/smallest fraction?" 5. Simplification using algebraic identities hidden in decimal form

Common Traps:

Mixed recurring vs pure recurring: students apply the wrong formula. Always identify the non-recurring part first.
Ordering fractions: candidates convert to decimals but make rounding errors. LCM method is safer when fractions have integer denominators.
Operations: misaligning decimal points during addition/subtraction in a hurry
The fraction 1/7 = 0.142857142857... has a 6-digit recurring block; candidates often mistake it for a terminating decimal

Marks Weightage: Typically 1 mark each. In SSC and Bank PO, ordering fractions and recurring decimal conversion are separate question types worth 1–2 marks each.

Practice Questions

Q1. Express 0.454545... as a fraction in lowest terms.

Q2. Convert 0.38̄ (= 0.3888...) to a fraction.

Q3. Which is larger: 8/15 or 11/21?

Q4. Arrange in descending order: 3/4, 5/6, 7/9, 11/15.

Q5. What is 3.6̄ (= 3.6666...) as a mixed number (whole number + fraction)?

Q6. Evaluate: (1.5³ − 0.5³) / (1.5² + 1.5 × 0.5 + 0.5²)

Q7. Which fraction is closest to 3/4: (a) 7/10 (b) 11/15 (c) 13/17 (d) 19/25?

Q8. The value of 0.001 ÷ 0.0001 is:

Q9. How many recurring digits does 1/7 have in its decimal representation?

Q10. Express the fraction 5/16 as a decimal.

Answers

Q	Answer	Key Working
Q1	5/11	0.454545... = 45/99 = 5/11 (HCF of 45 and 99 is 9)
Q2	7/18	(38−3)/90 = 35/90 = 7/18
Q3	11/21	Cross: 8×21=168, 11×15=165. 168>165 → 8/15 > 11/21... wait: 8/15 vs 11/21. Cross: 8×21=168, 11×15=165. Since 168>165, 8/15 > 11/21. Answer: 8/15 is larger
Q4	5/6 > 7/9 > 3/4 > 11/15	LCM(4,6,9,15)=180. Values: 3/4=135, 5/6=150, 7/9=140, 11/15=132. Descending: 5/6 > 7/9 > 3/4 > 11/15
Q5	3 and 2/3	0.6666... = 6/9 = 2/3. So 3.666... = 3 + 2/3 = 11/3 as improper fraction
Q6	1	(a³−b³)/(a²+ab+b²) = a−b = 1.5−0.5 = 1
Q7	19/25	3/4=0.75. Options: 7/10=0.7, 11/15=0.733, 13/17≈0.765, 19/25=0.76. Closest to 0.75 is 19/25 (diff=0.01) vs 13/17 (diff≈0.015)
Q8	10	0.001/0.0001 = 1/0.1 = 10
Q9	6	1/7 = 0.142857142857... The block "142857" has 6 digits
Q10	0.3125	5/16: multiply by 625/625 → 3125/10000 = 0.3125