1. Basic Rules for Operations on Fractions
Fractions follow the ordinary rules of arithmetic, but the numerator and denominator must be handled correctly. Addition and subtraction require compatible denominators, whereas multiplication and division follow different rules.
Four Basic Operations:
Addition → a/b + c/d
Subtraction → a/b − c/d
Multiplication → a/b × c/d
Division → a/b ÷ c/d
Important:
The denominator of every valid fraction must be non-zero.
Exam Strategy:
Before performing a long calculation, check whether fractions can be simplified or cancelled.
2. Addition and Subtraction of Like Fractions
Like fractions have the same denominator, so only the numerators need to be added or subtracted.
Rule:
a/c + b/c = (a+b)/c
a/c − b/c = (a−b)/c
Example:
5/11 + 3/11
=(5+3)/11
=8/11.
Example:
9/13 − 4/13
=(9−4)/13
=5/13.
Common Mistake:
Do not add the denominators when they are already equal.
5/11+3/11 is not 8/22.
3. Addition and Subtraction of Unlike Fractions
Unlike fractions have different denominators. Convert them to equivalent fractions having a common denominator before adding or subtracting.
General Rule:
a/b + c/d = (ad+bc)/(bd)
a/b − c/d = (ad−bc)/(bd)
Better Exam Method:
Instead of automatically using bd as the denominator, use LCM(b,d) when it makes the numbers smaller.
Example:
3/4 + 5/6
LCM(4,6)=12.
3/4=9/12
5/6=10/12.
Sum=19/12
=1 7/12.
Example:
7/8 − 1/3
=21/24−8/24
=13/24.
Shortcut:
If one denominator is already a multiple of the other, use the larger denominator directly.
4. Addition and Subtraction of Mixed Fractions
Mixed fractions may be handled either by converting them to improper fractions or, in simple addition/subtraction, by separately working with whole-number and fractional parts.
Safest General Method:
Convert every mixed fraction into an improper fraction before calculation.
Example:
2 1/3 + 1 3/4
=7/3 + 7/4
=28/12+21/12
=49/12
=4 1/12.
Example:
5 1/4 − 2 2/3
=21/4−8/3
=63/12−32/12
=31/12
=2 7/12.
Shortcut:
For easy mixed-fraction addition, whole parts and fractional parts may be grouped separately.
Exam Tip:
For multiplication, division or complicated expressions, improper fractions are usually safer and faster.
5. Multiplication of Fractions
Rule:
a/b × c/d = ac/bd
Unlike addition and subtraction, multiplication does not require a common denominator.
Example:
5/12 × 18/25
Cancel before multiplying:
5 with25 → 1 and5
18 with12 → 3 and2.
Result=(1×3)/(2×5)
=3/10.
Best Shortcut:
Cross-cancel common factors before multiplying. This keeps numbers small and reduces calculation errors.
Common Mistake:
Do not find the LCM of denominators before multiplication. It is unnecessary.
6. Reciprocal and Division of Fractions
The reciprocal of a non-zero fraction is obtained by interchanging its numerator and denominator.
Reciprocal:
Reciprocal of a/b = b/a, provided a≠0.
Examples:
Reciprocal of3/7 = 7/3.
Reciprocal of5 = 1/5.
Reciprocal of2 1/2 = reciprocal of5/2 = 2/5.
Important:
Zero has no reciprocal.
Division Rule:
a/b ÷ c/d
=a/b × d/c
Example:
7/9 ÷ 14/27
=7/9×27/14
=3/2.
Memory Rule:
Keep the first fraction, change ÷ into ×, and invert the second fraction.
7. Fraction of a Fraction
In arithmetic problems, the word “of” generally represents multiplication.
Rule:
a/b of c/d = a/b × c/d
Example:
3/5 of 10/21
=3/5×10/21
=2/7.
Example — Fraction of a Quantity:
3/8 of 240
=3/8×240
=90.
Exam Tip:
In BODMAS-based aptitude questions, “of” is normally handled as multiplication and is often evaluated before ordinary addition or subtraction.
8. Complex and Compound Fractions
A complex fraction is a fraction whose numerator, denominator or both contain fractions.
Example Form:
(a/b)/(c/d)
Method:
1. Simplify the numerator completely.
2. Simplify the denominator completely.
3. Divide numerator by denominator using reciprocal multiplication.
Example:
(5/6−1/4) ÷ 7/8
5/6−1/4
=10/12−3/12
=7/12.
Now:
7/12 ÷ 7/8
=7/12×8/7
=2/3.
Common Mistake:
Do not invert every fraction in a complex expression. Only the divisor is replaced by its reciprocal during division.
9. BODMAS with Fractions
Fractions do not change the order of operations. Apply the usual BODMAS rules.
Order:
Brackets → Orders → Division/Multiplication → Addition/Subtraction
Important:
Division and multiplication have equal priority and are evaluated from left to right when they occur at the same level. Addition and subtraction are handled similarly.
Example:
1/2 + 3/4 × 2/3
First multiplication:
3/4×2/3=1/2.
Then:
1/2+1/2=1.
Exam Trap:
Do not perform addition first merely because it appears first from the left.
10. Cancellation and Calculation Shortcuts
Competitive exams often test speed. Cancellation can substantially shorten multiplication and division calculations.
Cross Cancellation:
In a product of fractions, any numerator may be cancelled against any denominator when they contain a common factor.
Example:
2/3 × 3/4 × 8/5
Cancel3.
Cancel2 with4.
Cancel8 with remaining2.
Result=4/5.
Whole Number Shortcut:
Treat a whole number n as n/1 whenever it helps cancellation.
Exam Tip:
Cancel factors, not terms separated by addition or subtraction.
Wrong Cancellation:
In (2+3)/3, the3 in the numerator term cannot simply be cancelled with the denominator because the numerator is a sum.
11. Sign Rules with Fractions
Equivalent Sign Forms:
−a/b = (−a)/b = a/(−b)
Multiplication/Division Signs:
Same signs → Positive
Different signs → Negative
Example:
(−3/5)×(−10/9)
=+30/45
=2/3.
Example:
(−7/8)÷(14/5)
=−7/8×5/14
=−5/16.
12. Common Exam Traps
Trap 1:
Adding denominators directly during fraction addition.
Trap 2:
Finding LCM unnecessarily before multiplying fractions.
Trap 3:
Forgetting to invert the second fraction during division.
Trap 4:
Inverting the first fraction instead of the divisor.
Trap 5:
Trying to take the reciprocal of zero.
Trap 6:
Using a mixed fraction directly in multiplication without understanding its value.
Trap 7:
Cancelling terms across addition or subtraction signs.
Trap 8:
Ignoring BODMAS in complex fractional expressions.
13. Quick Revision
Remember:
• Like fractions → add/subtract numerators directly.
• Unlike fractions → first obtain a common denominator.
• LCM of denominators is usually preferable to their product.
• Mixed fractions can always be converted to improper fractions.
• Multiplication does not require a common denominator.
• Multiply numerators together and denominators together.
• Cancel common factors before multiplication whenever possible.
• Reciprocal of a/b is b/a, provided a≠0.
• Zero has no reciprocal.
• Division by a fraction means multiplication by its reciprocal.
• “Of” means multiplication in ordinary aptitude problems.
• In complex fractions, simplify the top and bottom separately first.
• Apply BODMAS normally.
• Never cancel terms separated by + or −.
• Same signs give positive; different signs give negative in multiplication/division.
14. Verified Previous-Year Questions
RRB Group D PYQ
Mixed Fraction Operations
5 September 2022 · Shift II
Q1. Evaluate:
[(1 2/3 + 2 1/2 × 2/3)] ÷ [5 − (4 1/2 ÷ 3/2)]
A. 3 1/3
B. 4 2/3
C. 2 1/3
D. 1 2/3
Correct Answer: D. 1 2/3
Convert mixed fractions:
1 2/3=5/3, 2 1/2=5/2, 4 1/2=9/2.
Numerator:
5/3+(5/2×2/3)
=5/3+5/3
=10/3.
Denominator:
5−(9/2÷3/2)
=5−3
=2.
Result:
10/3÷2=5/3=1 2/3.
RRB Group D PYQ
BODMAS with Fractions
29 August 2022 · Shift I
Q2. Evaluate:
1/4 + 2/5 ÷ [{2 1/5 − 2} × 5] − 2/3 × 3/5
A. 2/3
B. 3/5
C. 5/6
D. 1/4
Correct Answer: D. 1/4
2 1/5−2
=11/5−10/5
=1/5.
Therefore:
(1/5)×5=1.
Expression becomes:
1/4+2/5−2/5
=1/4.
RRB NTPC PYQ
Mixed Fraction Multiplication
21 June 2025 · Graduate CBT-I · Shift II
Q3. Simplify:
2 2/9 × 2 2/3 × 567 + 8 3/4
A. 3372.15
B. 3364.35
C. 3368.75
D. 3347.25
Correct Answer: C. 3368.75
2 2/9=20/9
2 2/3=8/3
8 3/4=35/4.
20/9×8/3×567
=3360.
Therefore:
3360+35/4
=3360+8.75
=3368.75.
SSC GD PYQ
Mixed Fraction Addition
29 April 2026 · Shift I
Q4. Find the sum:
4 2/3 + 44 2/3 + 444 2/3 + 4444 2/3
A. 4936 2/3
B. 4938 1/3
C. 4935 2/3
D. 4938 2/3
Correct Answer: D. 4938 2/3
Whole-number parts:
4+44+444+4444=4936.
Fractional parts:
2/3+2/3+2/3+2/3
=8/3
=2 2/3.
Total:
4936+2 2/3
=4938 2/3.
15. Practice MCQs
Practice MCQ
Q1. Find 3/4 + 5/6.
A. 17/12
B. 19/12
C. 21/12
D. 23/12
Correct Answer: B. 19/12
3/4=9/12 and5/6=10/12. Sum=19/12.
Practice MCQ
Q2. Find 7/8 − 1/3.
A. 11/24
B. 13/24
C. 15/24
D. 17/24
Correct Answer: B. 13/24
21/24−8/24=13/24.
Practice MCQ
Q3. Find 5/12 × 18/25.
A. 2/5
B. 3/10
C. 4/15
D. 5/18
Correct Answer: B. 3/10
Cancel before multiplication. Result=3/10.
Practice MCQ
Q4. Find 7/9 ÷ 14/27.
A. 2/3
B. 3/4
C. 3/2
D. 7/2
Correct Answer: C. 3/2
7/9×27/14=3/2.
Practice MCQ
Q5. Find (2/3 + 5/8) × 4/7.
A. 29/42
B. 31/42
C. 33/42
D. 35/42
Correct Answer: B. 31/42
2/3+5/8=31/24. Then31/24×4/7=31/42.
Practice MCQ
Q6. Evaluate 1/2 + 3/4 × 2/3.
A. 3/4
B. 5/6
C. 1
D. 7/6
Correct Answer: C. 1
Multiplication first:3/4×2/3=1/2. Then1/2+1/2=1.
Practice MCQ
Q7. Evaluate (5/6 − 1/4) ÷ 7/8.
A. 1/2
B. 2/3
C. 3/4
D. 4/5
Correct Answer: B. 2/3
5/6−1/4=7/12. Then7/12÷7/8=2/3.
Practice MCQ
Q8. Evaluate (3/5) ÷ (9/10 − 3/10).
A. 1/2
B. 3/5
C. 1
D. 3/2
Correct Answer: C. 1
9/10−3/10=6/10=3/5. Therefore3/5÷3/5=1.
Practice MCQ
Q9. Find 2/3 × 3/4 × 8/5.
A. 3/5
B. 4/5
C. 5/6
D. 8/15
Correct Answer: B. 4/5
After cancellation, the product is 4/5.
Practice MCQ
Q10. Find 7/6 + 5/4 − 2/3.
A. 5/4
B. 3/2
C. 7/4
D. 11/6
Correct Answer: C. 7/4
Common denominator12 gives14/12+15/12−8/12=21/12=7/4.
1. Fractions पर Operations के Basic Rules
Fractions पर addition, subtraction, multiplication एवं division सामान्य arithmetic rules के अनुसार किए जाते हैं, लेकिन numerator और denominator को सही तरीके से handle करना आवश्यक है।
चार Basic Operations:
Addition → a/b + c/d
Subtraction → a/b − c/d
Multiplication → a/b × c/d
Division → a/b ÷ c/d
महत्वपूर्ण:
किसी valid fraction का denominator कभी zero नहीं हो सकता।
Exam Strategy:
Long calculation शुरू करने से पहले simplification और cancellation की possibility अवश्य देखें।
2. Like Fractions का Addition एवं Subtraction
Like fractions के denominators समान होते हैं, इसलिए numerator को direct add या subtract किया जाता है।
Rule:
a/c + b/c = (a+b)/c
a/c − b/c = (a−b)/c
उदाहरण:
5/11+3/11
=8/11।
उदाहरण:
9/13−4/13
=5/13।
Common Mistake:
Same denominator होने पर denominators को add न करें।
5/11+3/11 का answer 8/22 नहीं है।
3. Unlike Fractions का Addition एवं Subtraction
Different denominators वाले fractions को पहले common denominator वाले equivalent fractions में बदलें।
General Rule:
a/b+c/d=(ad+bc)/(bd)
a/b−c/d=(ad−bc)/(bd)
Better Exam Method:
Denominators के product के बजाय उनका LCM लेना अक्सर calculation को छोटा बनाता है।
उदाहरण:
3/4+5/6
LCM=12।
3/4=9/12
5/6=10/12।
Sum=19/12=1 7/12।
उदाहरण:
7/8−1/3
=21/24−8/24
=13/24।
Shortcut:
यदि एक denominator दूसरे का multiple है, तो larger denominator को common denominator बनाया जा सकता है।
4. Mixed Fractions का Addition एवं Subtraction
Safest General Method:
Mixed fractions को improper fractions में convert करके calculation करें।
उदाहरण:
2 1/3 + 1 3/4
=7/3+7/4
=49/12
=4 1/12।
उदाहरण:
5 1/4−2 2/3
=21/4−8/3
=31/12
=2 7/12।
Shortcut:
Easy addition में whole-number एवं fractional parts को separately group किया जा सकता है।
Exam Tip:
Multiplication, division एवं complex expressions में mixed fractions को improper fractions में बदलना अधिक सुरक्षित है।
5. Fractions का Multiplication
Rule:
a/b × c/d = ac/bd
Multiplication के लिए common denominator की आवश्यकता नहीं होती।
उदाहरण:
5/12×18/25
Cancellation करने पर:
5 और25 →1 एवं5
18 और12 →3 एवं2।
Result=3/10।
Best Shortcut:
Multiply करने से पहले cross-cancellation करें।
Common Mistake:
Fraction multiplication से पहले denominators का LCM निकालना आवश्यक नहीं है।
6. Reciprocal एवं Fractions का Division
Reciprocal:
a/b का reciprocal=b/a, जहाँ a≠0।
उदाहरण:
3/7 का reciprocal=7/3।
5 का reciprocal=1/5।
2 1/2=5/2, इसलिए reciprocal=2/5।
महत्वपूर्ण:
Zero का reciprocal नहीं होता।
Division Rule:
a/b ÷ c/d
=a/b × d/c
उदाहरण:
7/9÷14/27
=7/9×27/14
=3/2।
Memory Rule:
पहला fraction same रखें, ÷ को × में बदलें और दूसरे fraction को उलट दें।
7. Fraction of a Fraction
Aptitude mathematics में “of” सामान्यतः multiplication दर्शाता है।
Rule:
a/b of c/d = a/b × c/d
उदाहरण:
3/5 of 10/21
=3/5×10/21
=2/7।
Quantity का Fraction:
3/8 of240
=3/8×240
=90।
Exam Tip:
BODMAS questions में “of” को multiplication के रूप में solve करें।
8. Complex एवं Compound Fractions
ऐसा fraction जिसके numerator या denominator में स्वयं fractions हों, complex fraction कहलाता है।
Method:
1. Numerator को completely simplify करें।
2. Denominator को completely simplify करें।
3. अंत में numerator को denominator से divide करें।
उदाहरण:
(5/6−1/4)÷7/8
5/6−1/4=7/12।
अब:
7/12÷7/8
=7/12×8/7
=2/3।
Common Mistake:
Complex expression में सभी fractions को invert नहीं किया जाता। केवल divisor का reciprocal लिया जाता है।
9. Fractions के साथ BODMAS
Order:
Brackets → Orders → Division/Multiplication → Addition/Subtraction
महत्वपूर्ण:
Division एवं multiplication की priority समान होती है और same level पर left-to-right solve होते हैं। Addition एवं subtraction भी इसी प्रकार handle होते हैं।
उदाहरण:
1/2+3/4×2/3
पहले multiplication:
3/4×2/3=1/2।
फिर:
1/2+1/2=1।
Exam Trap:
Expression में addition पहले दिखाई देने के कारण उसे multiplication से पहले solve न करें।
10. Cancellation एवं Calculation Shortcuts
Cross Cancellation:
Fraction multiplication में किसी numerator के common factor को किसी denominator के common factor से cancel किया जा सकता है।
उदाहरण:
2/3×3/4×8/5
Cancellation के बाद:
=4/5।
Whole Number Shortcut:
Whole number n को जरूरत पड़ने पर n/1 के रूप में लिखें।
Exam Tip:
Factors को cancel करें, addition या subtraction से जुड़े terms को नहीं।
गलत Cancellation:
(2+3)/3 में numerator के3 को denominator के3 से direct cancel नहीं कर सकते।
11. Fractions के Sign Rules
Equivalent Sign Forms:
−a/b=(−a)/b=a/(−b)
Multiplication/Division:
Same signs → Positive
Different signs → Negative
उदाहरण:
(−3/5)×(−10/9)
=2/3।
उदाहरण:
(−7/8)÷(14/5)
=−5/16।
12. Common Exam Traps
Trap 1:
Addition में denominators को direct add करना।
Trap 2:
Multiplication से पहले unnecessarily LCM निकालना।
Trap 3:
Division में second fraction का reciprocal लेना भूल जाना।
Trap 4:
Divisor के बजाय first fraction को invert कर देना।
Trap 5:
Zero का reciprocal निकालने का प्रयास करना।
Trap 6:
Complex multiplication/division में mixed fraction को बिना conversion के गलत तरीके से use करना।
Trap 7:
Addition या subtraction signs के across cancellation करना।
Trap 8:
Complex fractions में BODMAS ignore करना।
13. Quick Revision
एक नज़र में:
• Like fractions → numerators direct add/subtract करें।
• Unlike fractions → common denominator बनाएँ।
• Denominators का LCM सामान्यतः best common denominator है।
• Mixed fractions को improper fractions में बदला जा सकता है।
• Multiplication के लिए common denominator की आवश्यकता नहीं।
• Numerators को numerators तथा denominators को denominators से multiply करें।
• Multiply करने से पहले cancellation करें।
• a/b का reciprocal=b/a, यदि a≠0।
• Zero का reciprocal नहीं होता।
• Fraction से division → reciprocal से multiplication।
• “Of” सामान्यतः multiplication है।
• Complex fractions में numerator और denominator पहले separately simplify करें।
• BODMAS follow करें।
• + या − के across cancellation न करें।
• Multiplication/division में same signs positive और different signs negative result देते हैं।
14. Verified Previous-Year Questions
RRB Group D PYQ
Mixed Fraction Operations
5 सितंबर 2022 · Shift II
प्रश्न 1. निम्न expression का मान ज्ञात करें:
[(1 2/3 + 2 1/2 × 2/3)] ÷ [5 − (4 1/2 ÷ 3/2)]
A. 3 1/3
B. 4 2/3
C. 2 1/3
D. 1 2/3
सही उत्तर: D. 1 2/3
Mixed fractions को improper fractions में बदलने पर numerator=10/3 तथा denominator=2 मिलता है।
Result:
10/3÷2
=5/3
=1 2/3।
RRB Group D PYQ
BODMAS with Fractions
29 अगस्त 2022 · Shift I
प्रश्न 2. निम्न expression का मान ज्ञात करें:
1/4 + 2/5 ÷ [{2 1/5 − 2} × 5] − 2/3 × 3/5
A. 2/3
B. 3/5
C. 5/6
D. 1/4
सही उत्तर: D. 1/4
2 1/5−2=1/5।
(1/5)×5=1।
Expression:
1/4+2/5−2/5
=1/4।
RRB NTPC PYQ
Mixed Fraction Multiplication
21 जून 2025 · Graduate CBT-I · Shift II
प्रश्न 3. Simplify करें:
2 2/9 × 2 2/3 × 567 + 8 3/4
A. 3372.15
B. 3364.35
C. 3368.75
D. 3347.25
सही उत्तर: C. 3368.75
2 2/9=20/9
2 2/3=8/3
8 3/4=35/4।
पहला product=3360।
3360+35/4
=3360+8.75
=3368.75।
SSC GD PYQ
Mixed Fraction Addition
29 अप्रैल 2026 · Shift I
प्रश्न 4. निम्न का sum ज्ञात करें:
4 2/3 + 44 2/3 + 444 2/3 + 4444 2/3
A. 4936 2/3
B. 4938 1/3
C. 4935 2/3
D. 4938 2/3
सही उत्तर: D. 4938 2/3
Whole-number parts का sum=4936।
Fractional parts:
4×2/3=8/3=2 2/3।
Total=4938 2/3।
15. Practice MCQs
Practice MCQ
प्रश्न 1. 3/4+5/6 ज्ञात करें।
A. 17/12
B. 19/12
C. 21/12
D. 23/12
सही उत्तर: B. 19/12
9/12+10/12=19/12। Practice MCQ
प्रश्न 2. 7/8−1/3 ज्ञात करें।
A. 11/24
B. 13/24
C. 15/24
D. 17/24
सही उत्तर: B. 13/24
21/24−8/24=13/24। Practice MCQ
प्रश्न 3. 5/12×18/25 ज्ञात करें।
A. 2/5
B. 3/10
C. 4/15
D. 5/18
सही उत्तर: B. 3/10
Cancellation के बाद answer=3/10। Practice MCQ
प्रश्न 4. 7/9÷14/27 ज्ञात करें।
A. 2/3
B. 3/4
C. 3/2
D. 7/2
सही उत्तर: C. 3/2
7/9×27/14=3/2। Practice MCQ
प्रश्न 5. (2/3+5/8)×4/7 का मान ज्ञात करें।
A. 29/42
B. 31/42
C. 33/42
D. 35/42
सही उत्तर: B. 31/42
2/3+5/8=31/24। फिर31/24×4/7=31/42। Practice MCQ
प्रश्न 6. 1/2+3/4×2/3 का मान ज्ञात करें।
A. 3/4
B. 5/6
C. 1
D. 7/6
सही उत्तर: C. 1
पहले3/4×2/3=1/2। फिर1/2+1/2=1। Practice MCQ
प्रश्न 7. (5/6−1/4)÷7/8 का मान ज्ञात करें।
A. 1/2
B. 2/3
C. 3/4
D. 4/5
सही उत्तर: B. 2/3
5/6−1/4=7/12। फिर7/12÷7/8=2/3। Practice MCQ
प्रश्न 8. (3/5)÷(9/10−3/10) का मान क्या है?
A. 1/2
B. 3/5
C. 1
D. 3/2
सही उत्तर: C. 1
9/10−3/10=3/5। इसलिए3/5÷3/5=1। Practice MCQ
प्रश्न 9. 2/3×3/4×8/5 ज्ञात करें।
A. 3/5
B. 4/5
C. 5/6
D. 8/15
सही उत्तर: B. 4/5
Cancellation के बाद product=4/5। Practice MCQ
प्रश्न 10. 7/6+5/4−2/3 का मान ज्ञात करें।
A. 5/4
B. 3/2
C. 7/4
D. 11/6
सही उत्तर: C. 7/4
14/12+15/12−8/12=21/12=7/4।