1. Converting a Fraction into a Decimal
A fraction a/b represents the division a÷b. Therefore, the most direct method of converting a fraction into a decimal is to divide its numerator by its denominator.
Basic Rule:
a/b = a ÷ b
Example:
3/4
=3÷4
=0.75.
Example:
7/8
=7÷8
=0.875.
Shortcut:
If the denominator can easily be converted into 10, 100, 1000, etc., convert the fraction into an equivalent decimal fraction instead of using long division.
Example:
17/20
Multiply numerator and denominator by5:
=85/100
=0.85.
2. Converting a Terminating Decimal into a Fraction
Method:
1. Remove the decimal point and write the resulting integer as the numerator.
2. In the denominator, write1 followed by as many zeros as there are decimal places.
3. Reduce the fraction to its lowest terms.
Example:
0.375
=375/1000
=3/8.
Example:
2.45
=245/100
=49/20.
Memory Rule:
1 decimal place → denominator10
2 decimal places → denominator100
3 decimal places → denominator1000
Common Mistake:
Do not forget to simplify the fraction after removing the decimal point.
3. When Does a Fraction Give a Terminating Decimal?
The nature of the decimal expansion can be predicted from the denominator without actually performing long division.
Fundamental Rule:
First reduce the fraction p/q to lowest terms.
Its decimal expansion terminates if and only if the prime factorisation of q contains no prime factors other than2 and5.
Required Denominator Form:
q = 2m5n
where m,n are non-negative integers.
Examples:
3/8 → denominator8=2³ → Terminating
7/20 →20=2²×5 → Terminating
11/125 →125=5³ → Terminating
13/40 →40=2³×5 → Terminating
Non-terminating Recurring Case:
If the reduced denominator contains any prime factor other than2 or5, the decimal expansion is non-terminating recurring.
Examples:
1/3 → denominator3 → recurring
5/6 → denominator2×3 → recurring
7/11 → denominator11 → recurring
Exam Trap:
Always reduce the fraction to lowest terms before applying this test.
Example:
6/15 reduces to2/5.
Although the original denominator15 contains3, the reduced denominator is5.
Therefore the decimal expansion is terminating:
2/5=0.4.
4. Number of Decimal Places in a Terminating Decimal
Rule:
After reducing the fraction to lowest terms, suppose:
Denominator=2m5n.
Then the terminating decimal has at most:
max(m,n) decimal places.
Example:
7/40
40=2³×5.
max(3,1)=3.
7/40=0.175, which has3 decimal places.
Example:
13/250
250=2×5³.
max(1,3)=3.
13/250=0.052.
Reason:
The denominator is effectively converted into 10k, where k=max(m,n), by supplying the missing powers of2 or5.
5. Pure Recurring Decimals
A pure recurring decimal begins repeating immediately after the decimal point.
Examples:
0.333...
0.777...
0.252525...
0.473473473...
One Repeating Digit:
0.aaa... = a/9
Examples:
0.333...=3/9=1/3
0.777...=7/9
Two Repeating Digits:
0.abab... = ab/99
Example:
0.272727...
=27/99
=3/11.
Three Repeating Digits:
0.abcabc... = abc/999
Example:
0.125125125...
=125/999.
Memory Rule:
1 recurring digit →9
2 recurring digits →99
3 recurring digits →999
6. Algebraic Method for Pure Recurring Decimals
The shortcut above comes from a simple multiplication-and-subtraction method.
Example:
Let x=0.555...
10x=5.555...
x=0.555...
Subtract:
9x=5
x=5/9.
Example with Two Recurring Digits:
Let x=0.252525...
100x=25.252525...
x=0.252525...
99x=25
x=25/99.
Exam Tip:
Multiply by10r, where r is the number of digits in the repeating block.
7. Mixed Recurring Decimals
A mixed recurring decimal contains one or more non-repeating digits before the recurring part begins.
Examples:
0.1666... →1 is non-recurring,6 repeats.
0.53232323... →5 is non-recurring,32 repeats.
Shortcut Rule:
Fraction =
(All digits up to one complete recurring block − Non-recurring digits)
÷
(9s for recurring digits followed by0s for non-recurring decimal digits)
Example:
0.1666...
Numerator=16−1=15
Denominator=90
=15/90
=1/6.
Example:
0.53232323...
Non-recurring digit=5
Recurring block=32
Numerator=532−5=527
Denominator=990
Fraction=527/990.
Denominator Shortcut:
Write one9 for every recurring digit, followed by one0 for every non-recurring decimal digit.
8. Algebraic Method for Mixed Recurring Decimals
Example:
Let x=0.2666...
10x=2.666...
100x=26.666...
Subtract:
100x−10x=24
90x=24
x=24/90
=4/15.
Exam Tip:
First multiply enough to move past the non-recurring portion, then multiply again to move by one complete recurring block.
9. Recurring Decimals with a Whole-Number Part
The presence of a whole-number part does not change the basic recurring-decimal method.
Example:
6.555...
=6+0.555...
=6+5/9
=(54+5)/9
=59/9.
Example:
22.121212...
=22+12/99
=22+4/33
=730/33.
Shortcut:
For simple pure recurring decimals with an integer part, separating the integer part is often faster than using the full algebraic method.
10. Rational Numbers and Decimal Expansions
Important Property:
Every rational number has a decimal expansion that is either:
1. Terminating, or
2. Non-terminating recurring.
Conversely:
Every terminating decimal and every recurring decimal represents a rational number.
Non-terminating Non-recurring Decimals:
Such decimals cannot be expressed as p/q with integers p,q and q≠0. They represent irrational numbers.
Exam Tip:
“Non-terminating” alone does not mean irrational. A non-terminating recurring decimal is rational.
11. Important Fraction-Decimal Values
Frequently Useful Values:
1/2=0.5
1/4=0.25
3/4=0.75
1/5=0.2
2/5=0.4
3/5=0.6
4/5=0.8
1/8=0.125
3/8=0.375
5/8=0.625
7/8=0.875
1/10=0.1
Recurring Values:
1/3=0.333...
2/3=0.666...
1/9=0.111...
2/9=0.222...
4/9=0.444...
5/9=0.555...
7/9=0.777...
8/9=0.888...
Exam Shortcut:
Memorising these standard values is useful for comparison, simplification and approximation questions.
12. Common Exam Traps
Trap 1:
Checking the denominator before reducing the fraction to lowest terms.
Trap 2:
Assuming every non-terminating decimal is irrational.
Trap 3:
Using100 instead of99 for a two-digit pure recurring block.
Trap 4:
Using1000 instead of999 for a three-digit pure recurring block.
Trap 5:
Forgetting zeros corresponding to non-recurring decimal digits in a mixed recurring decimal.
Trap 6:
Using the terminating-decimal rule when the reduced denominator contains primes other than2 and5.
Trap 7:
Forgetting to simplify the final fraction obtained from a decimal.
Trap 8:
Confusing the repeating block with the entire decimal portion.
13. Quick Revision
Remember:
• Fraction to decimal → numerator ÷ denominator.
• Terminating decimal to fraction → remove decimal point and use10n as denominator.
• Always simplify the fraction.
• In lowest terms, denominator containing only2 and/or5 → terminating decimal.
• Any other prime factor in the denominator → non-terminating recurring decimal.
• If denominator=2m5n, maximum decimal places=max(m,n).
• Pure recurring1 digit → denominator9.
• Pure recurring2 digits → denominator99.
• Pure recurring3 digits → denominator999.
• Mixed recurring → subtract the non-recurring part.
• Denominator in mixed recurring form =9s followed by0s.
• Every terminating decimal is rational.
• Every recurring decimal is rational.
• Non-terminating non-recurring decimals are irrational.
14. Verified Previous-Year Questions
RRB NTPC PYQ
Terminating vs Recurring
7 April 2021 · CBT-I · Shift I
Q1. Which of the following fractions gives a non-terminating recurring decimal?
A. 13/4
B. 100/11
C. 7/5
D. 3/40
Correct Answer: B. 100/11
For a terminating decimal, the reduced denominator must contain only prime factors2 and5.
11 contains a prime factor other than2 and5.
Therefore 100/11 is non-terminating recurring.
RRB Group D PYQ
Pure Recurring Decimal
28 September 2022 · Shift II
Q2. Express 0.252525... as a fraction p/q.
A. 25/99
B. 25/90
C. 25/10000
D. 25/100
Correct Answer: A. 25/99
The recurring block contains two digits.
Therefore denominator=99.
0.252525...=25/99.
RRB NTPC PYQ
Mixed Recurring Decimal
4 February 2021 · CBT-I · Shift II
Q3. Convert 0.53232323... into a fraction.
A. 32/99
B. 527/990
C. 572/990
D. 537/990
Correct Answer: B. 527/990
Non-recurring digit=5.
Recurring block=32.
Fraction=(532−5)/990
=527/990.
RRB NTPC PYQ
Recurring Decimal to Fraction
18 June 2025 · Graduate CBT-I · Shift I
Q4. Find the fraction equivalent to 6.555...
A. 56/13
B. 59/9
C. 53/17
D. 61/3
Correct Answer: B. 59/9
6.555...
=6+0.555...
=6+5/9
=(54+5)/9
=59/9.
15. Practice MCQs
Practice MCQ
Q1. Convert 7/20 into decimal form.
A. 0.25
B. 0.35
C. 0.45
D. 0.70
Correct Answer: B. 0.35
7/20=35/100=0.35.
Practice MCQ
Q2. Express 0.625 in simplest fractional form.
A. 3/5
B. 5/8
C. 7/10
D. 8/13
Correct Answer: B. 5/8
0.625=625/1000=5/8.
Practice MCQ
Q3. Which fraction has a terminating decimal expansion?
A. 5/12
B. 7/15
C. 11/40
D. 13/21
Correct Answer: C. 11/40
40=2³×5 contains only2 and5. Hence 11/40 terminates.
Practice MCQ
Q4. Which fraction gives a non-terminating recurring decimal?
A. 3/16
B. 7/25
C. 9/40
D. 5/18
Correct Answer: D. 5/18
18=2×3² contains prime factor3. Therefore 5/18 is recurring.
Practice MCQ
Q5. How many decimal places are required for the terminating decimal form of 7/80?
A. 1
B. 2
C. 3
D. 4
Correct Answer: D. 4
80=2⁴×5. max(4,1)=4, and7/80=0.0875.
Practice MCQ
Q6. Convert 0.777... into a fraction.
A. 7/10
B. 7/9
C. 77/90
D. 77/99
Correct Answer: B. 7/9
One digit repeats, so denominator=9. Therefore 0.777...=7/9.
Practice MCQ
Q7. Convert 0.363636... into a fraction.
A. 4/11
B. 36/100
C. 18/55
D. 3/8
Correct Answer: A. 4/11
0.363636...=36/99=4/11.
Practice MCQ
Q8. Convert 0.1666... into a fraction.
A. 1/5
B. 1/6
C. 1/7
D. 1/8
Correct Answer: B. 1/6
(16−1)/90=15/90=1/6.
Practice MCQ
Q9. Which of the following is rational?
A. A non-terminating non-recurring decimal
B. 0.272727...
C. √2
D. π
Correct Answer: B. 0.272727...
Every recurring decimal is rational. Here0.272727...=3/11.
Practice MCQ
Q10. Convert 3.444... into a fraction.
A. 29/9
B. 31/9
C. 32/9
D. 34/9
Correct Answer: B. 31/9
3.444...=3+4/9=(27+4)/9=31/9.
1. Fraction को Decimal में बदलना
Fraction a/b का अर्थ a÷b होता है। इसलिए fraction को decimal में बदलने का direct method numerator को denominator से divide करना है।
Basic Rule:
a/b = a ÷ b
उदाहरण:
3/4=3÷4=0.75।
उदाहरण:
7/8=0.875।
Shortcut:
यदि denominator को आसानी से10,100,1000 आदि में convert किया जा सकता है, तो long division के बजाय equivalent decimal fraction बनाएँ।
उदाहरण:
17/20
=85/100
=0.85।
2. Terminating Decimal को Fraction में बदलना
Method:
1. Decimal point हटाकर प्राप्त integer को numerator बनाएं।
2. जितने decimal places हों, denominator में1 के बाद उतने zeros लिखें।
3. Fraction को lowest terms में simplify करें।
उदाहरण:
0.375
=375/1000
=3/8।
उदाहरण:
2.45
=245/100
=49/20।
Memory Rule:
1 decimal place →10
2 decimal places →100
3 decimal places →1000
Common Mistake:
Final fraction को lowest terms में reduce करना न भूलें।
3. Fraction कब Terminating Decimal देता है?
Fundamental Rule:
पहले fraction p/q को lowest terms में reduce करें।
Decimal expansion तभी terminating होगा जब q के prime factors केवल2 और/or5 हों।
Required Denominator Form:
q=2m5n
उदाहरण:
3/8 →8=2³ → Terminating
7/20 →20=2²×5 → Terminating
11/125 →125=5³ → Terminating
13/40 →40=2³×5 → Terminating
Non-terminating Recurring:
Reduced denominator में2 एवं5 के अलावा कोई अन्य prime factor हो तो decimal expansion non-terminating recurring होगा।
उदाहरण:
1/3 → recurring
5/6 → denominator2×3 → recurring
7/11 → recurring
Exam Trap:
Denominator test लगाने से पहले fraction को lowest terms में reduce करना आवश्यक है।
उदाहरण:
6/15=2/5।
Original denominator15 में3 होने के बावजूद reduced denominator5 है।
इसलिए decimal terminating है:
0.4।
4. Terminating Decimal में Decimal Places की संख्या
Rule:
Lowest terms में denominator यदि:
2m5n
है, तो decimal expansion में maximum:
max(m,n) decimal places होंगे।
उदाहरण:
7/40
40=2³×5।
max(3,1)=3।
7/40=0.175।
उदाहरण:
13/250
250=2×5³।
max=3।
13/250=0.052।
Reason:
Missing powers of2 या5 supply करके denominator को10k बनाया जाता है।
5. Pure Recurring Decimals
Pure recurring decimal में repeating pattern decimal point के immediately बाद शुरू हो जाता है।
उदाहरण:
0.333...
0.777...
0.252525...
0.473473473...
One Repeating Digit:
0.aaa...=a/9
उदाहरण:
0.333...=1/3
0.777...=7/9
Two Repeating Digits:
0.abab...=ab/99
उदाहरण:
0.272727...
=27/99
=3/11।
Three Repeating Digits:
0.abcabc...=abc/999
उदाहरण:
0.125125125...=125/999।
Memory Rule:
1 recurring digit →9
2 recurring digits →99
3 recurring digits →999
6. Pure Recurring Decimal का Algebraic Method
उदाहरण:
x=0.555... मानें।
10x=5.555...
x=0.555...
Subtract करने पर:
9x=5
x=5/9।
Two-Digit Recurring Example:
x=0.252525...
100x=25.252525...
x=0.252525...
99x=25
x=25/99।
Exam Tip:
Repeating block में r digits हों तो number को10r से multiply करें।
7. Mixed Recurring Decimals
Mixed recurring decimal में repeating part शुरू होने से पहले कुछ non-repeating digits होते हैं।
उदाहरण:
0.1666... →1 non-recurring,6 recurring।
0.53232323... →5 non-recurring,32 recurring।
Shortcut Rule:
Fraction =
(All digits up to one complete recurring block − Non-recurring digits)
÷
(Recurring digits के लिए9s, फिर non-recurring decimal digits के लिए0s)
उदाहरण:
0.1666...
Numerator=16−1=15
Denominator=90
=15/90
=1/6।
उदाहरण:
0.53232323...
Numerator=532−5=527
Denominator=990
Fraction=527/990।
Denominator Shortcut:
हर recurring digit के लिए एक9 तथा हर non-recurring decimal digit के लिए उसके बाद एक0 लिखें।
8. Mixed Recurring Decimal का Algebraic Method
उदाहरण:
x=0.2666... मानें।
10x=2.666...
100x=26.666...
Subtract करें:
90x=24
x=24/90
=4/15।
Exam Tip:
पहला multiplication non-recurring part को पार करने के लिए और दूसरा recurring block को align करने के लिए किया जाता है।
9. Whole-Number Part वाले Recurring Decimals
उदाहरण:
6.555...
=6+0.555...
=6+5/9
=59/9।
उदाहरण:
22.121212...
=22+12/99
=22+4/33
=730/33।
Shortcut:
Simple pure recurring decimal में integer part को अलग करना अक्सर full algebraic method से faster है।
10. Rational Numbers एवं Decimal Expansions
Important Property:
हर rational number का decimal expansion या तो:
1. Terminating होगा, या
2. Non-terminating recurring होगा।
Conversely:
हर terminating decimal तथा हर recurring decimal rational number होता है।
Non-terminating Non-recurring Decimal:
ऐसा decimal p/q form में express नहीं किया जा सकता और irrational number represent करता है।
Exam Tip:
केवल “non-terminating” देखकर irrational conclude न करें। Non-terminating recurring decimal rational होता है।
11. Important Fraction-Decimal Values
Frequently Used:
1/2=0.5
1/4=0.25
3/4=0.75
1/5=0.2
2/5=0.4
3/5=0.6
4/5=0.8
1/8=0.125
3/8=0.375
5/8=0.625
7/8=0.875
1/10=0.1
Recurring Values:
1/3=0.333...
2/3=0.666...
1/9=0.111...
2/9=0.222...
4/9=0.444...
5/9=0.555...
7/9=0.777...
8/9=0.888...
Exam Shortcut:
इन standard values को याद रखना comparison, simplification एवं approximation questions में useful है।
12. Common Exam Traps
Trap 1:
Fraction को lowest terms में reduce किए बिना denominator test लगाना।
Trap 2:
हर non-terminating decimal को irrational मान लेना।
Trap 3:
Two-digit pure recurring decimal के denominator में99 के बजाय100 लेना।
Trap 4:
Three-digit recurring block के लिए999 के बजाय1000 लेना।
Trap 5:
Mixed recurring decimal में non-recurring digits के corresponding zeros भूल जाना।
Trap 6:
Reduced denominator में2 और5 के अलावा prime factors होने के बावजूद decimal को terminating मानना।
Trap 7:
Decimal से प्राप्त fraction को simplify न करना।
Trap 8:
Recurring block की सही पहचान न करना।
13. Quick Revision
एक नज़र में:
• Fraction → Decimal: numerator ÷ denominator।
• Terminating decimal → Fraction: decimal हटाकर denominator10n लें।
• Final fraction simplify करें।
• Lowest terms में denominator में केवल2 और/or5 → terminating।
• किसी अन्य prime factor की presence → non-terminating recurring।
• Denominator=2m5n → maximum decimal places=max(m,n)।
• Pure recurring1 digit → denominator9।
• Pure recurring2 digits →99।
• Pure recurring3 digits →999।
• Mixed recurring में non-recurring part subtract करें।
• Mixed recurring denominator →9s followed by0s।
• हर terminating decimal rational है।
• हर recurring decimal rational है।
• Non-terminating non-recurring decimal irrational है।
14. Verified Previous-Year Questions
RRB NTPC PYQ
Terminating vs Recurring
7 अप्रैल 2021 · CBT-I · Shift I
प्रश्न 1. निम्न में से कौन-सा fraction non-terminating recurring decimal देता है?
A. 13/4
B. 100/11
C. 7/5
D. 3/40
सही उत्तर: B. 100/11
11,2 या5 के अलावा prime factor है।
इसलिए 100/11 का decimal expansion non-terminating recurring है।
RRB Group D PYQ
Pure Recurring Decimal
28 सितंबर 2022 · Shift II
प्रश्न 2. 0.252525... को p/q form में व्यक्त करें।
A. 25/99
B. 25/90
C. 25/10000
D. 25/100
सही उत्तर: A. 25/99
Recurring block में2 digits हैं। इसलिए denominator=99।
Answer=25/99।
RRB NTPC PYQ
Mixed Recurring Decimal
4 फरवरी 2021 · CBT-I · Shift II
प्रश्न 3. 0.53232323... को fraction में बदलें।
A. 32/99
B. 527/990
C. 572/990
D. 537/990
सही उत्तर: B. 527/990
Non-recurring digit=5 तथा recurring block=32।
Fraction=(532−5)/990
=527/990।
RRB NTPC PYQ
Recurring Decimal to Fraction
18 जून 2025 · Graduate CBT-I · Shift I
प्रश्न 4. 6.555... के equivalent fraction को ज्ञात करें।
A. 56/13
B. 59/9
C. 53/17
D. 61/3
सही उत्तर: B. 59/9
6.555...
=6+5/9
=(54+5)/9
=59/9।
15. Practice MCQs
Practice MCQ
प्रश्न 1. 7/20 को decimal में बदलें।
A. 0.25
B. 0.35
C. 0.45
D. 0.70
सही उत्तर: B. 0.35
7/20=35/100=0.35। Practice MCQ
प्रश्न 2. 0.625 को simplest fraction में बदलें।
A. 3/5
B. 5/8
C. 7/10
D. 8/13
सही उत्तर: B. 5/8
625/1000=5/8। Practice MCQ
प्रश्न 3. कौन-सा fraction terminating decimal देगा?
A. 5/12
B. 7/15
C. 11/40
D. 13/21
सही उत्तर: C. 11/40
40=2³×5। केवल2 एवं5 factors हैं। इसलिए 11/40 terminating है। Practice MCQ
प्रश्न 4. कौन-सा fraction non-terminating recurring decimal देगा?
A. 3/16
B. 7/25
C. 9/40
D. 5/18
सही उत्तर: D. 5/18
18=2×3² में prime factor3 है। इसलिए recurring होगा। Practice MCQ
प्रश्न 5. 7/80 के terminating decimal में कितने decimal places होंगे?
A. 1
B. 2
C. 3
D. 4
सही उत्तर: D. 4
80=2⁴×5। max(4,1)=4 तथा7/80=0.0875। Practice MCQ
प्रश्न 6. 0.777... को fraction में बदलें।
A. 7/10
B. 7/9
C. 77/90
D. 77/99
सही उत्तर: B. 7/9
एक digit repeat है, denominator9 होगा। Answer=7/9। Practice MCQ
प्रश्न 7. 0.363636... को fraction में बदलें।
A. 4/11
B. 36/100
C. 18/55
D. 3/8
सही उत्तर: A. 4/11
36/99=4/11। Practice MCQ
प्रश्न 8. 0.1666... को fraction में बदलें।
A. 1/5
B. 1/6
C. 1/7
D. 1/8
सही उत्तर: B. 1/6
(16−1)/90=15/90=1/6। Practice MCQ
प्रश्न 9. निम्न में से कौन rational number है?
A. Non-terminating non-recurring decimal
B. 0.272727...
C. √2
D. π
सही उत्तर: B. 0.272727...
हर recurring decimal rational होता है। 0.272727...=3/11। Practice MCQ
प्रश्न 10. 3.444... को fraction में बदलें।
A. 29/9
B. 31/9
C. 32/9
D. 34/9
सही उत्तर: B. 31/9
3.444...=3+4/9=(27+4)/9=31/9।