Decimal Fractions are among the most scoring and formula-heavy topics in arithmetic. Yet many students find them confusing because decimals behave differently from vulgar fractions, recurring decimals introduce patterns, and conversions require precision.
If you understand these three ideas clearly:
✔ Decimal place value
✔ Converting decimals ↔ fractions
✔ Operations on decimals
Then you can solve ANY question effortlessly.
This Decimal Fractions Guide covers all formulas, rules, operations, shortcuts, examples, FAQs, and exam-level tips, making it the perfect one-stop reference for all competitive exams.
Quick Overview: Decimal Fraction Formulas
| Concept / Situation | Value / Distance Considered | Operation / Rule Used | Formula (With Meaning Inside Row) |
|---|---|---|---|
| Decimal fraction | Denominator is power of 10 | Place-value rule | a/10, a/100, a/1000 etc. represent tenths, hundredths, thousandths |
| Decimal → Fraction | Count decimal places | Power-of-10 conversion | 0.25 = 25/100 → denominator has two zeros |
| Fraction → Decimal | Divide numerator by denominator | Long division | 7/8 = 0.875 |
| Annexing zeros | Value unchanged | Decimal shifting | 0.8 = 0.80 = 0.800 |
| Removing decimals | Same decimal places | Multiply or reduce | 1.84/2.99 → 184/299 |
| Multiply decimal × 10ⁿ | Shift decimal right | Place value shift | 0.073×1000 → shift 3 places → 73 |
| Multiply decimals | Add total decimal places | Standard multiplication | 0.2×0.02×0.002 → 8 with 6 decimal places → 0.000008 |
| Decimal ÷ whole number | Divide normally | Reinsert decimal | 0.0204÷17 = 0.0012 |
| Decimal ÷ decimal | Make divisor whole | ×10,100… both sides | 0.00066÷0.11 = 0.066÷11 = 0.006 |
| Pure recurring decimal | Repeating digits only | 9-rule | 0.3̅ = 3/9, 0.53̅ = 53/99 |
| Mixed recurring decimal | Non-repeat + repeat digits | 9s + 0s rule | 0.16̅ = (16−1)/90 |
| Comparing fractions | Convert to decimal | Compare values | 3/5=0.6; 6/7=0.857; 7/9=0.777 |
Formulas for Decimal Fractions
Decimal formulas form the backbone of questions related to conversions, multiplication, division, recurring decimals, and comparisons. Before solving decimal-based problems, understand these systems clearly.
1. Converting a Decimal into a Fraction
Why this formula is important?
Exams often ask to convert decimals into lowest-term fractions. Understanding decimal place-value ensures 100% accuracy.
Formula
Decimal → Fraction = (Number without decimal) / (1 followed by number of decimal places)
Example:
0.25 = 25/100
2.008 = 2008/1000
Why this works?
Decimal places represent powers of 10:
- 0.1 → 1 tenth
- 0.01 → 1 hundredth
- 0.001 → 1 thousandth
Removing the decimal and dividing by 10ⁿ keeps the value unchanged.
2. Annexing Zeros Without Changing Decimal Value
Concept
Adding zeros to the right does NOT change value.
Formula
0.8 = 0.80 = 0.800
Why?
Because:
0.8 = 8/10
0.80 = 80/100 = 8/10
0.800 = 800/1000 = 8/10
The fraction value remains identical.
3. Removing Decimal Signs in Fractions
Rule
If numerator and denominator contain the same number of decimal places → remove decimals.
Formula
1.84 / 2.99 = 184 / 299
Why?
Multiply numerator & denominator by 10ⁿ to eliminate decimal.
4. Multiplying a Decimal by a Power of 10
Rule
Shift decimal to the RIGHT by the number of zeroes.
Formula
5.9632 × 100 = 596.32
0.073 × 10000 = 730
Why?
10 shifts decimal 1 place, 100 shifts 2, 1000 shifts 3.
5. Multiplying Decimal Fractions
Process
- Multiply ignoring decimal points
- Count TOTAL decimal places
- Insert decimal in product
Formula
0.2 × 0.02 × 0.002
→ 2×2×2 = 8
→ Decimal places = 1+2+3 = 6
→ Product = 0.000008
6. Dividing Decimal Fraction by Whole Number
Steps
Divide normally → reinsert decimal equal to original decimal places.
Example
0.0204 ÷ 17
204 ÷ 17 = 12
Decimal places = 4
Answer = 0.0012
7. Dividing Decimal by Decimal
Rule
Make divisor a whole number by multiplying both numbers by 10ⁿ.
Example
0.00066 ÷ 0.11
×100 → 0.066 ÷ 11 = 0.006
8. Converting Pure Recurring Decimals into Fractions
Formula
Repeated digits / (as many 9s as repeating digits)
Examples
0.3̅ = 3/9
0.53̅ = 53/99
0.067̅ = 67/999
Why?
Repeating digits represent infinite geometric pattern → denominator becomes 9, 99, 999…
9. Converting Mixed Recurring Decimals into Fractions
Formula
(All digits − non-repeating digits) / (99...00...)
9’s = repeating digits
0’s = non-repeating digits
Example
0.16̅
Digits = 16
Non-repeat = 1
Difference = 15
Denominator = 90
= 15/90 = 1/6
10. Comparing Fractions Using Decimals
Rule
Convert each fraction into decimal and compare directly.
Example
3/5 = 0.6
6/7 = 0.857
7/9 = 0.777…
Order:
6/7 > 7/9 > 3/5
Smart Tips and Practical Tricks for Solving Decimal Fractions
Mastering Decimal Fractions becomes simple when you understand how place value, conversion, and operations work together. Most students make mistakes not because decimal formulas are difficult, but because they apply them without understanding how decimals behave during multiplication, division, and recurring patterns.
This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.
1. Understand Place Value Before Any Operation
Every digit after the decimal carries a value:
- 1st place → tenths (1/10)
- 2nd place → hundredths (1/100)
- 3rd place → thousandths (1/1000)
If you identify place value correctly, conversions and comparisons become effortless.
Example:
0.007 = 7 thousandths
This simple understanding prevents most concept errors.
2. Convert Decimals Into Fractions by Power-of-10 Method
The fastest way to avoid mistakes is to remember:
Decimal → remove decimal → divide by 10, 100, 1000…
Examples:
0.25 → 25/100
2.008 → 2008/1000
Always reduce to lowest terms for exam accuracy.
3. Annexing Zeros Does Not Change Value
Students often get confused when decimals appear with extra zeros.
0.6 = 0.60 = 0.600
Zeros only change presentation, not the value.
Recognizing this avoids unnecessary conversions.
4. In Multiplication, Multiply First → Insert Decimal Later
This is one of the most scoring shortcuts.
Rule: Multiply normally ignoring decimals.
Then count total decimal places and insert the decimal point.
Example:
0.2 × 0.02 × 0.002
→ 2 × 2 × 2 = 8
→ Decimal places = 1+2+3 = 6
→ 0.000008
Knowing this pattern saves a lot of time.
5. For Division, Make the Divisor a Whole Number First
When dividing a decimal by a decimal:
Multiply both numbers by a power of 10 until the divisor becomes a whole number.
Example:
0.00066 ÷ 0.11
→ multiply both by 100
→ 0.066 ÷ 11 = 0.006
This method reduces unnecessary confusion.
6. For Comparison, Convert All Fractions Into Decimals
Instead of taking LCM and cross-multiplying, convert fractions to decimals.
Example:
3/5 = 0.6
6/7 = 0.857
7/9 = 0.777…
Descending order becomes clear immediately.
This technique is extremely fast in exams.
7. For Recurring Decimals → Use the 9s & 0s Rule
Pure recurring → only 9s
Mixed recurring → 9s followed by 0s
Example:
0.53̅ = 53/99
0.16̅ = (16–1)/90
Mastering this rule helps you convert recurring decimals effortlessly.
8. Understand Patterns Instead of Memorizing Every Case
Most decimal questions follow repeated patterns:
- Decimal → fraction
- Fraction → decimal
- Multiply decimals
- Divide decimals
- Recurring → fraction
- Comparison using decimals
All these follow the same principle:
decimal places determine the denominator.
Once you identify the pattern, solving becomes automatic.
9. Double-Check Decimal Places Before Final Answer
A common mistake is placing the decimal incorrectly in the final step.
Always re-check:
✔ Are total decimal places counted correctly?
✔ Did you remove decimal from divisor correctly?
✔ Did you reduce fractions to lowest form?
These checks prevent last-second mistakes.
10. Practice Standard Exam-Level Patterns
Decimal questions appear frequently in:
- SSC CGL & CHSL
- Banking (SBI, IBPS)
- RRB Exams
- Defence exams
- Management and campus aptitude tests
Practicing typical patterns builds speed and accuracy, making decimal-based questions high-scoring.
FAQs About Decimal Fractions
Q1. Why do we convert decimals into fractions using powers of 10?
Because each decimal place corresponds to a fixed power of 10 (tenth, hundredth, thousandth), making conversion straightforward and accurate.
Q2. Why does adding zeros after a decimal not change its value?
Because 0.8, 0.80, and 0.800 represent the same number, only expressed with different decimal places.
Q3. Why do we remove decimals from both numerator and denominator?
If both have equal decimal places, the decimals cancel out, making simplification easier.
Q4. Why do we shift the decimal point when multiplying by powers of 10?
Each zero shifts place value one step to the right, increasing the number by powers of 10.
Q5. Why multiply both numbers by the same power of 10 during decimal division?
To make the divisor a whole number, ensuring normal division rules can be applied.
Q6. What is the difference between pure and mixed recurring decimals?
Pure recurring decimals repeat from the first digit after decimal; mixed recurring decimals repeat after some non-repeating digits.
Q7. Why do recurring decimals convert using 9s and 0s?
Because each repeating digit forms a cycle of 9, and non-repeating digits form a cycle of 0 before repetition starts.
Q8. Why are decimals easier to compare than fractions?
Decimal values show magnitude directly, unlike fractions that require LCM or cross-multiplication.
Q9. Why do students make mistakes in decimal multiplication?
They multiply correctly but misplace the final decimal because they forget to count total decimal places.
Q10. Why is practice important in decimal problems?
Because accuracy depends on place value awareness, which improves only through repeated exposure to standard patterns.