1. Why Brackets Matter in Simplification
Brackets are grouping symbols. They indicate that certain terms of an expression must be treated together before the surrounding operations are performed.
A change in bracket placement can completely change the value of an expression. Therefore, identifying the grouping structure correctly is more important than performing the arithmetic quickly.
Example:
Compare:
8 + 4 × 3 = 8 + 12 = 20
and
(8 + 4) × 3 = 12 × 3 = 36.
The numbers and operations are the same, but the bracket changes the answer.
Exam Tip:
Before solving a long expression, first identify which terms belong together. This prevents many sign and order-of-operation errors.
2. Types of Brackets and Grouping Symbols
The most common grouping symbols used in competitive mathematics are:
| Symbol | Name | Example |
|---|
| ( ) | Parentheses / Round Brackets | (8+3) |
| { } | Braces / Curly Brackets | {12−(5+2)} |
| [ ] | Square Brackets | [20−{8−3}] |
| a+b | Vinculum / Bar | A bar placed over grouped terms |
Important Observation:
The shape of a bracket does not itself make one operation mathematically more important than another. In a nested expression, the fundamental rule is to solve the innermost grouped expression first and then move outward.
3. Innermost-to-Outermost Principle
When one bracket lies inside another bracket, begin with the deepest or innermost grouping and work outward step by step.
Key Rule:
For nested brackets:
Innermost grouping → next outer grouping → outermost grouping
Example:
Simplify:
25 − [12 − {8 − (6−3)}]
Innermost bracket:
6−3 = 3
Then:
8−3 = 5
Then:
12−5 = 7
Finally:
25−7 = 18.
Shortcut:
Do not try to open all nested brackets simultaneously. Reduce one complete layer at a time. This greatly reduces sign errors.
4. Round, Curly and Square Brackets in Nested Expressions
Competitive-exam expressions are often written in the visual pattern:
[ ... { ... ( ... ) ... } ... ]
In such a structure, the round bracket is innermost, followed by the curly bracket and then the square bracket. The calculation therefore naturally proceeds from inside to outside.
Example:
Simplify:
40 − [15 − {10 − (7−2)}]
7−2 = 5
10−5 = 5
15−5 = 10
40−10 = 30.
Common Mistake:
Do not blindly memorise “round bracket before curly bracket before square bracket” as a universal independent rule. The correct principle is innermost first. The usual round-curly-square order occurs because exam setters commonly nest them in that arrangement.
5. Vinculum or Bar Bracket
A vinculum is a horizontal bar written over two or more terms to show that those terms form one grouped expression.
It behaves like an especially tight grouping symbol and should be simplified before the surrounding expression.
Key Rule:
Terms lying under the same vinculum must be treated as one grouped expression.
Example:
Suppose the expression contains:
20 − 8−3
First evaluate the barred portion:
8−3 = 5
Therefore:
20−5 = 15.
Exam Tip:
A vinculum can be visually easy to miss in a printed question paper. Before calculating, check whether a bar extends over one term or several terms.
6. Vinculum Inside a Bracket
If a vinculum appears within another bracket, simplify the vinculum first and then continue outward.
Example:
Simplify:
30 − (12 − 7−4)
Vinculum:
7−4 = 3
Round bracket:
12−3 = 9
Therefore:
30−9 = 21.
Observation:
Thinking of the vinculum as an invisible pair of brackets around the barred terms is often the safest approach.
7. Minus Sign Before a Bracket
A negative sign placed immediately before a bracket applies to the complete value of that bracket.
Sign Rules:
−(a+b) = −a−b
−(a−b) = −a+b
−(−a+b) = a−b
Example:
Simplify:
25 − (8−13)
Inside bracket:
8−13 = −5
Therefore:
25−(−5)
= 25+5
= 30.
Exam Trap:
Subtracting a negative quantity produces addition. The expression a−(−b) becomes a+b.
8. Opening Brackets Safely
Sometimes a bracket can be removed directly by applying the sign outside it.
Rules:
+(a+b−c) = a+b−c
−(a+b−c) = −a−b+c
Example:
48 − (16+9−5)
Opening the bracket:
48−16−9+5
= 32−9+5
= 23+5
= 28.
Shortcut:
A plus sign before a bracket leaves all internal signs unchanged. A minus sign before a bracket reverses every internal + and − sign.
9. Number Multiplied by a Bracket
If a number is directly written before a bracket, it multiplies every term inside that bracket.
Distributive Rule:
a(b+c) = ab+ac
a(b−c) = ab−ac
Example:
5(12−7)
Using the bracket first:
12−7 = 5
5×5 = 25.
Alternatively:
5×12 − 5×7
= 60−35
= 25.
Common Mistake:
Do not multiply only the first term inside the bracket. The outside multiplier applies to every term.
10. Negative Multiplier Outside a Bracket
A negative multiplier affects both the numerical value and the signs of the terms inside the bracket.
Example:
−3(8−5)
Method 1:
8−5 = 3
−3×3 = −9.
Method 2:
−3×8 + 3×5
= −24+15
= −9.
Exam Tip:
When a negative number is outside a bracket, simplify inside first whenever possible. It is usually safer than opening the bracket mentally.
11. Brackets in Division and Fractions
A bracket appearing in the denominator or as the divisor must be completely simplified before the division is performed.
Example:
Simplify:
96 ÷ [18−(9−3)]
9−3 = 6
18−6 = 12
96÷12 = 8.
Exam Trap:
Never divide by only the first term of a grouped denominator. The entire grouped quantity forms the divisor.
12. Complex Nested Expressions
Long expressions become much easier when each completed bracket is replaced by its numerical value before moving to the next level.
Example:
Simplify:
50 − [20 − {12 − (9−6)}]
9−6 = 3
12−3 = 9
20−9 = 11
50−11 = 39.
Exam Method:
Write one simplified line after every bracket layer. Avoid doing three or four sign changes mentally in one step.
13. Fast Method for Nested-Bracket Questions
Step-by-Step Method:
1. Locate the innermost bracket or vinculum.
2. Simplify it completely.
3. Replace that grouped part with its value.
4. Move one level outward.
5. Watch carefully for a minus sign before a bracket.
6. After all brackets are removed, apply ordinary BODMAS to the remaining expression.
Exam Tip:
If the options differ widely, estimate the sign and approximate size of intermediate results. This can help detect a sign mistake before the final calculation.
14. Common Bracket and Vinculum Traps
Trap 1 — Solving the outer bracket first:
Always begin with the innermost grouped expression.
Trap 2 — Missing the vinculum:
All terms covered by the bar belong to the same group.
Trap 3 — Forgetting double negative:
a−(−b)=a+b.
Trap 4 — Changing only one sign:
When a minus bracket is opened, every +/− sign inside is affected.
Trap 5 — Multiplying only one bracket term:
An outside multiplier applies to every term inside the bracket.
Trap 6 — Assuming bracket shapes alone decide priority:
Nesting determines the order. Solve the deepest group first.
15. Quick Revision
Remember:
• Brackets create grouped expressions.
• In nested expressions, solve the innermost group first.
• A vinculum groups all the terms under the bar.
• Solve a vinculum before the surrounding bracket.
• Minus before a bracket reverses signs when the bracket is opened.
• An outside multiplier applies to every term inside the bracket.
• A grouped denominator must be simplified completely before division.
• Reduce one bracket layer at a time to minimise errors.
16. Verified Previous-Year Questions
SSC CPO PYQ
Vinculum & Brackets
4 October 2023 · Paper-I · Shift II
Q1. Simplify: 19 ÷ 5 of (27 − 15−21) + 37
A. 6124/165
B. 19/165
C. 6142/165
D. 37/165
Correct Answer: A. 6124/165
First simplify the vinculum:
15−21 = −6.
Then:
27−(−6) = 33.
The conventional exam treatment gives:
5 of 33 = 165.
So:
19÷165 + 37
= 19/165 + 6105/165
= 6124/165.
RPF Constable PYQ
Vinculum & Nested Brackets
4 March 2025 · Shift I
Q2. Simplify: 5 1/2 + 6 1/2 − 8 1/4 + 24 − [9 − {6 − (10 − 4−3)}]
A. 12 1/4
B. 15 3/4
C. 12 3/4
D. 15 1/4
Correct Answer: B. 15 3/4
Vinculum first:
4−3 = 1.
Then:
10−1 = 9
6−9 = −3
9−(−3) = 12.
Therefore:
5 1/2 + 6 1/2 − 8 1/4 + 24 − 12
= 12 − 8 1/4 + 12
= 24 − 8 1/4
= 15 3/4.
RRB Group D PYQ
Nested Brackets
2 February 2026 · Shift II
Q3. Simplify: 18 − [6 − {4 − (8−6−3)}]
A. 17
B. 9
C. 19
D. 16
Correct Answer: A. 17
Innermost bracket:
8−6−3 = 2−3 = −1.
Then:
4−(−1) = 5
6−5 = 1
18−1 = 17.
RRB NTPC PYQ
Brackets & Negative Numbers
7 May 2026 · CBT-I · Shift III
Q4. Simplify: 30 + [{−9 × (26−11+6)}]
A. −162
B. −160
C. −159
D. −161
Correct Answer: C. −159
Inside the bracket:
26−11+6 = 15+6 = 21.
Then:
−9×21 = −189.
Finally:
30+(−189)
= −159.
17. Practice MCQs
Practice MCQ
Q1. Simplify: 30 − [12 − {8 − (5−2)}]
A. 21
B. 23
C. 25
D. 27
Correct Answer: B. 23
5−2=3; 8−3=5; 12−5=7; 30−7=23.
Practice MCQ
Q2. Simplify: 40 − [15 − {9 − (7−4)}]
A. 29
B. 30
C. 31
D. 32
Correct Answer: C. 31
7−4=3; 9−3=6; 15−6=9; 40−9=31.
Practice MCQ
Q3. Simplify: 25 − (8−13)
A. 20
B. 25
C. 30
D. 38
Correct Answer: C. 30
8−13=−5. Therefore 25−(−5)=25+5=30.
Practice MCQ
Q4. Simplify: 60 − [25 − (18−12)]
A. 39
B. 41
C. 43
D. 45
Correct Answer: B. 41
18−12=6; 25−6=19; 60−19=41.
Practice MCQ
Q5. Simplify: −4(9−6)+20
A. 6
B. 8
C. 10
D. 12
Correct Answer: B. 8
9−6=3. Then −4×3+20=−12+20=8.
Practice MCQ
Q6. Simplify: 72 ÷ [15−(8−5)]
A. 4
B. 5
C. 6
D. 8
Correct Answer: C. 6
8−5=3; 15−3=12; 72÷12=6.
Practice MCQ
Q7. Simplify: 5[8−{6−(5−3)}]
A. 15
B. 20
C. 25
D. 30
Correct Answer: B. 20
5−3=2; 6−2=4; 8−4=4; 5×4=20.
Practice MCQ
Q8. Simplify: 50 − {20 − [12 − (9−5)]}
A. 36
B. 38
C. 40
D. 42
Correct Answer: B. 38
9−5=4; 12−4=8; 20−8=12; 50−12=38.
Practice MCQ
Q9. Simplify: 45 − [18 − {7 − (4−9)}]
A. 37
B. 38
C. 39
D. 40
Correct Answer: C. 39
4−9=−5; 7−(−5)=12; 18−12=6; 45−6=39.
Practice MCQ
Q10. Simplify: 100 ÷ [25 − {15 − (8−3)}]
A. 5
B. 20/3
C. 10
D. 25/2
Correct Answer: B. 20/3
8−3=5; 15−5=10; 25−10=15. Therefore 100÷15=20/3.
1. Simplification में Brackets क्यों महत्वपूर्ण हैं?
Brackets अर्थात grouping symbols। ये बताते हैं कि expression के कुछ terms को बाकी operations से पहले एक group के रूप में solve करना है।
Bracket की position बदलने से expression का पूरा value बदल सकता है। इसलिए calculation की speed से पहले grouping को सही पहचानना आवश्यक है।
उदाहरण:
8 + 4 × 3 = 8 + 12 = 20
लेकिन:
(8 + 4) × 3 = 12 × 3 = 36।
Numbers और operations समान हैं, लेकिन bracket ने answer बदल दिया।
Exam Tip:
Long expression solve करने से पहले पहचानें कि कौन-से terms एक ही group का हिस्सा हैं।
2. Brackets एवं Grouping Symbols के प्रकार
| Symbol | नाम | उदाहरण |
|---|
| ( ) | Round Brackets / Parentheses | (8+3) |
| { } | Curly Brackets / Braces | {12−(5+2)} |
| [ ] | Square Brackets | [20−{8−3}] |
| a+b | Vinculum / Bar | Terms के ऊपर horizontal line |
महत्वपूर्ण Observation:
Bracket का shape अपने-आप mathematical priority तय नहीं करता। Nested expression में fundamental rule है—सबसे अंदर वाला grouped expression पहले solve करें।
3. Innermost-to-Outermost Principle
जब एक bracket दूसरे bracket के अंदर हो, तो सबसे अंदर वाले bracket से शुरू करके धीरे-धीरे बाहर की ओर बढ़ें।
मुख्य नियम:
Nested brackets में:
Innermost grouping → अगला outer grouping → outermost grouping
उदाहरण:
25 − [12 − {8 − (6−3)}]
6−3 = 3
8−3 = 5
12−5 = 7
25−7 = 18।
Shortcut:
सभी brackets को एक साथ खोलने की कोशिश न करें। एक बार में केवल एक complete layer solve करें।
4. Round, Curly एवं Square Brackets
Competitive examinations में expressions अक्सर इस pattern में दिखाई देते हैं:
[ ... { ... ( ... ) ... } ... ]
इस arrangement में round bracket सबसे अंदर है, उसके बाद curly और फिर square bracket है। इसलिए calculation naturally inside-to-outside होती है।
उदाहरण:
40 − [15 − {10 − (7−2)}]
7−2 = 5
10−5 = 5
15−5 = 10
40−10 = 30।
सामान्य गलती:
“Round bracket हमेशा curly से पहले और curly हमेशा square से पहले” को independent universal rule की तरह याद न करें। सही rule है innermost first।
5. Vinculum या Bar Bracket
Vinculum एक horizontal bar है जो दो या अधिक terms के ऊपर लगाई जाती है और बताती है कि वे terms एक single group हैं।
मुख्य नियम:
एक ही vinculum के नीचे आने वाले सभी terms को एक grouped expression की तरह solve करें।
उदाहरण:
20 − 8−3
पहले barred portion:
8−3 = 5
अब:
20−5 = 15।
Exam Tip:
Printed question paper में vinculum आसानी से miss हो सकती है। ध्यान दें कि bar केवल एक term के ऊपर है या कई terms के ऊपर।
6. Bracket के अंदर Vinculum
यदि vinculum किसी bracket के अंदर हो, तो पहले vinculum solve करें और फिर बाहर की grouping पर जाएँ।
उदाहरण:
30 − (12 − 7−4)
Vinculum:
7−4 = 3
Bracket:
12−3 = 9
30−9 = 21।
Observation:
Vinculum के नीचे के terms के चारों ओर imaginary bracket मानना एक सुरक्षित तरीका है।
7. Bracket के पहले Minus Sign
Bracket के तुरंत पहले दिया negative sign पूरे bracket के value पर लागू होता है।
Sign Rules:
−(a+b) = −a−b
−(a−b) = −a+b
−(−a+b) = a−b
उदाहरण:
25 − (8−13)
8−13 = −5
25−(−5)
= 25+5
= 30।
Exam Trap:
Negative quantity को subtract करना addition बन जाता है:
a−(−b)=a+b।
8. Brackets को Safely Open करना
Rules:
+(a+b−c) = a+b−c
−(a+b−c) = −a−b+c
उदाहरण:
48 − (16+9−5)
Bracket खोलें:
48−16−9+5
= 32−9+5
= 23+5
= 28।
Shortcut:
Bracket के पहले plus हो तो internal signs unchanged रहते हैं। Minus हो तो सभी +/− signs reverse हो जाते हैं।
9. Bracket के बाहर Multiplication
यदि कोई number सीधे bracket के पहले लिखा हो, तो वह bracket के प्रत्येक term से multiply होता है।
Distributive Rule:
a(b+c) = ab+ac
a(b−c) = ab−ac
उदाहरण:
5(12−7)
12−7 = 5
5×5 = 25।
या:
5×12 − 5×7
= 60−35
= 25।
सामान्य गलती:
Outside multiplier को केवल bracket के first term से multiply न करें। वह प्रत्येक term पर लागू होता है।
10. Bracket के बाहर Negative Multiplier
उदाहरण:
−3(8−5)
8−5 = 3
−3×3 = −9।
या:
−3×8 + 3×5
= −24+15
= −9।
Exam Tip:
यदि संभव हो तो negative multiplier वाले bracket को पहले अंदर से solve करें। इससे sign error की संभावना कम होती है।
11. Division एवं Fractions में Brackets
यदि divisor या denominator grouped expression है, तो पहले पूरे group को simplify करें और उसके बाद division करें।
उदाहरण:
96 ÷ [18−(9−3)]
9−3 = 6
18−6 = 12
96÷12 = 8।
Exam Trap:
Grouped divisor के केवल first term से divide न करें। पूरा bracket divisor है।
12. Complex Nested Expressions
उदाहरण:
50 − [20 − {12 − (9−6)}]
9−6 = 3
12−3 = 9
20−9 = 11
50−11 = 39।
Exam Method:
हर bracket layer के बाद नया simplified expression लिखें। कई sign changes को एक साथ mentally करने से बचें।
13. Nested-Bracket Questions का Fast Method
Step-by-Step Method:
1. Innermost bracket या vinculum पहचानें।
2. उसे पूरी तरह solve करें।
3. उस group को उसके numerical value से replace करें।
4. अगली outer layer पर जाएँ।
5. Minus sign before bracket को carefully check करें।
6. सभी brackets हटने के बाद remaining expression पर सामान्य BODMAS लागू करें।
Exam Tip:
Intermediate answer का sign और approximate size देखते रहें। इससे final answer से पहले sign mistake पकड़ने में मदद मिलती है।
14. Common Bracket एवं Vinculum Traps
Trap 1:
Outer bracket पहले solve करना गलत है। Innermost group से शुरू करें।
Trap 2:
Vinculum के नीचे आने वाले सभी terms एक ही group हैं।
Trap 3:
a−(−b)=a+b को भूलना common error है।
Trap 4:
Minus bracket खोलते समय केवल एक sign नहीं, बल्कि सभी internal +/− signs प्रभावित होते हैं।
Trap 5:
Outside multiplier bracket के प्रत्येक term को multiply करता है।
Trap 6:
Bracket का shape नहीं बल्कि nesting actual order निर्धारित करती है।
15. Quick Revision
एक नज़र में:
• Brackets grouped expressions बनाते हैं।
• Nested expressions में innermost group पहले solve करें।
• Vinculum के नीचे के सभी terms एक group हैं।
• Vinculum को surrounding bracket से पहले solve करें।
• Minus bracket खोलने पर signs reverse होते हैं।
• Outside multiplier प्रत्येक internal term पर लागू होता है।
• Grouped denominator/divisor को पहले पूरी तरह solve करें।
• एक बार में केवल एक bracket layer reduce करना safest method है।
16. Verified Previous-Year Questions
SSC CPO PYQ
Vinculum & Brackets
4 अक्टूबर 2023 · Paper-I · Shift II
प्रश्न 1. सरल कीजिए: 19 ÷ 5 of (27 − 15−21) + 37
A. 6124/165
B. 19/165
C. 6142/165
D. 37/165
सही उत्तर: A. 6124/165
पहले vinculum:
15−21 = −6।
फिर:
27−(−6) = 33।
Exam convention के अनुसार:
5 of 33 = 165।
अब:
19÷165 + 37
= 19/165 + 6105/165
= 6124/165।
RPF Constable PYQ
Vinculum & Nested Brackets
4 मार्च 2025 · Shift I
प्रश्न 2. सरल कीजिए: 5 1/2 + 6 1/2 − 8 1/4 + 24 − [9 − {6 − (10 − 4−3)}]
A. 12 1/4
B. 15 3/4
C. 12 3/4
D. 15 1/4
सही उत्तर: B. 15 3/4
Vinculum:
4−3 = 1।
फिर:
10−1 = 9
6−9 = −3
9−(−3) = 12।
इसलिए:
5 1/2 + 6 1/2 − 8 1/4 + 24 − 12
= 24 − 8 1/4
= 15 3/4।
RRB Group D PYQ
Nested Brackets
2 फरवरी 2026 · Shift II
प्रश्न 3. सरल कीजिए: 18 − [6 − {4 − (8−6−3)}]
A. 17
B. 9
C. 19
D. 16
सही उत्तर: A. 17
8−6−3 = 2−3 = −1।
फिर:
4−(−1) = 5
6−5 = 1
18−1 = 17।
RRB NTPC PYQ
Brackets & Negative Numbers
7 मई 2026 · CBT-I · Shift III
प्रश्न 4. सरल कीजिए: 30 + [{−9 × (26−11+6)}]
A. −162
B. −160
C. −159
D. −161
सही उत्तर: C. −159
26−11+6 = 15+6 = 21।
−9×21 = −189।
अब:
30+(−189)
= −159।
17. Practice MCQs
Practice MCQ
प्रश्न 1. सरल कीजिए: 30 − [12 − {8 − (5−2)}]
A. 21
B. 23
C. 25
D. 27
सही उत्तर: B. 23
5−2=3; 8−3=5; 12−5=7; 30−7=23।
Practice MCQ
प्रश्न 2. सरल कीजिए: 40 − [15 − {9 − (7−4)}]
A. 29
B. 30
C. 31
D. 32
सही उत्तर: C. 31
7−4=3; 9−3=6; 15−6=9; 40−9=31।
Practice MCQ
प्रश्न 3. सरल कीजिए: 25 − (8−13)
A. 20
B. 25
C. 30
D. 38
सही उत्तर: C. 30
8−13=−5। इसलिए 25−(−5)=25+5=30।
Practice MCQ
प्रश्न 4. सरल कीजिए: 60 − [25 − (18−12)]
A. 39
B. 41
C. 43
D. 45
सही उत्तर: B. 41
18−12=6; 25−6=19; 60−19=41।
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प्रश्न 5. सरल कीजिए: −4(9−6)+20
A. 6
B. 8
C. 10
D. 12
सही उत्तर: B. 8
9−6=3। फिर −4×3+20=−12+20=8।
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प्रश्न 6. सरल कीजिए: 72 ÷ [15−(8−5)]
A. 4
B. 5
C. 6
D. 8
सही उत्तर: C. 6
8−5=3; 15−3=12; 72÷12=6।
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प्रश्न 7. सरल कीजिए: 5[8−{6−(5−3)}]
A. 15
B. 20
C. 25
D. 30
सही उत्तर: B. 20
5−3=2; 6−2=4; 8−4=4; 5×4=20।
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प्रश्न 8. सरल कीजिए: 50 − {20 − [12 − (9−5)]}
A. 36
B. 38
C. 40
D. 42
सही उत्तर: B. 38
9−5=4; 12−4=8; 20−8=12; 50−12=38।
Practice MCQ
प्रश्न 9. सरल कीजिए: 45 − [18 − {7 − (4−9)}]
A. 37
B. 38
C. 39
D. 40
सही उत्तर: C. 39
4−9=−5; 7−(−5)=12; 18−12=6; 45−6=39।
Practice MCQ
प्रश्न 10. सरल कीजिए: 100 ÷ [25 − {15 − (8−3)}]
A. 5
B. 20/3
C. 10
D. 25/2
सही उत्तर: B. 20/3
8−3=5; 15−5=10; 25−10=15। इसलिए 100÷15=20/3।