Use of Brackets in Expressions
What is the answer to 2 + 3 × 4? Is it 20 or 14? If you add first, you get 5 × 4 = 20. If you multiply first, you get 2 + 12 = 14. The correct answer is 14, because multiplication is done before addition.
But what if you WANT to add first? That is where brackets come in. Writing (2 + 3) × 4 tells us to add first, giving 5 × 4 = 20. Brackets change the order of operations.
In this chapter, we will learn about the different types of brackets, the BODMAS rule, and how to simplify expressions correctly.
What is Use of Brackets in Expressions - Grade 6 Maths (Whole Numbers)?
Definition: Brackets are symbols used in maths to group numbers and operations together. Whatever is inside the brackets must be done first.
Types of brackets:
- ( ) — Round brackets or parentheses. Used most often.
- { } — Curly brackets or braces.
- [ ] — Square brackets.
When multiple brackets are used, solve from the innermost bracket outward:
- First solve ( ) round brackets.
- Then solve { } curly brackets.
- Finally solve [ ] square brackets.
BODMAS Rule:
BODMAS tells us the correct order to do operations:
- B — Brackets (do these first)
- O — Orders (powers and roots)
- D — Division (left to right)
- M — Multiplication (left to right)
- A — Addition (left to right)
- S — Subtraction (left to right)
Note: Division and Multiplication have the same priority — do them left to right. Same for Addition and Subtraction.
Use of Brackets in Expressions Formula
BODMAS Rule:
B → O → D/M (left to right) → A/S (left to right)
Order of brackets:
( ) first → { } next → [ ] last
Important rules:
- Always solve the innermost brackets first.
- Within brackets, follow BODMAS (division/multiplication before addition/subtraction).
- Work from left to right when operations have the same priority.
Derivation and Proof
Let us simplify step by step:
Example: 5 + [3 × {2 + (4 − 1)}]
- Innermost bracket ( ): 4 − 1 = 3.
- Expression becomes: 5 + [3 × {2 + 3}]
- Next bracket { }: 2 + 3 = 5.
- Expression becomes: 5 + [3 × 5]
- Outermost bracket [ ]: 3 × 5 = 15.
- Expression becomes: 5 + 15
- Final answer: 5 + 15 = 20.
Why brackets matter:
Without brackets: 2 + 3 × 4 = 2 + 12 = 14 (multiply first by BODMAS).
With brackets: (2 + 3) × 4 = 5 × 4 = 20 (brackets force addition first).
The brackets change the answer from 14 to 20.
Types and Properties
Types of problems with brackets:
- Type 1: Simple brackets — Solve the bracket, then complete the operation. Example: (7 + 3) × 2 = 10 × 2 = 20.
- Type 2: Nested brackets — Multiple layers of brackets. Solve innermost first. Example: [2 + {3 × (4 − 1)}].
- Type 3: BODMAS without brackets — Follow the order of operations. Example: 8 + 6 ÷ 2 = 8 + 3 = 11 (not 7).
- Type 4: Inserting brackets to get a target answer — "Put brackets in 3 + 4 × 2 to get 14." Answer: (3 + 4) × 2.
- Type 5: Word problems — Translate a situation into an expression with brackets. Example: "Buy 3 packs of 5 pencils and 2 erasers each" = 3 × (5 + 2).
Solved Examples
Example 1: Example 1: Simple Brackets
Problem: Simplify (8 + 4) × 3.
Solution:
- Brackets first: 8 + 4 = 12.
- Then multiply: 12 × 3 = 36.
Answer: 36.
Example 2: Example 2: BODMAS Without Brackets
Problem: Simplify 12 − 4 + 3 × 2.
Solution:
- No brackets. Follow BODMAS.
- Multiplication first: 3 × 2 = 6.
- Then left to right: 12 − 4 + 6 = 8 + 6 = 14.
Answer: 14.
Example 3: Example 3: Nested Brackets
Problem: Simplify [20 − {5 + (3 × 2)}].
Solution:
- Innermost ( ): 3 × 2 = 6.
- Expression: [20 − {5 + 6}].
- Next { }: 5 + 6 = 11.
- Expression: [20 − 11].
- Outermost [ ]: 20 − 11 = 9.
Answer: 9.
Example 4: Example 4: Division Before Addition
Problem: Simplify 15 + 20 ÷ 5.
Solution:
- Division first (BODMAS): 20 ÷ 5 = 4.
- Then addition: 15 + 4 = 19.
Answer: 19 (not 7).
Example 5: Example 5: Brackets Change the Answer
Problem: Compare 6 + 4 × 2 and (6 + 4) × 2.
Solution:
- Without brackets: 6 + 4 × 2 = 6 + 8 = 14.
- With brackets: (6 + 4) × 2 = 10 × 2 = 20.
Answer: The brackets change the answer from 14 to 20.
Example 6: Example 6: Complex Expression
Problem: Simplify 48 ÷ [2 × {3 + (9 − 5)}].
Solution:
- ( ): 9 − 5 = 4.
- Expression: 48 ÷ [2 × {3 + 4}].
- { }: 3 + 4 = 7.
- Expression: 48 ÷ [2 × 7].
- [ ]: 2 × 7 = 14.
- Final: 48 ÷ 14 = 24/7.
Wait, let us recheck. 48 ÷ 14 does not divide evenly. Let me recalculate.
Actually, 48 ÷ 14 = 48/14 = 24/7 ≈ 3.43.
Let us use a cleaner expression instead:
Revised Problem: Simplify 56 ÷ [2 × {3 + (9 − 5)}].
- ( ): 9 − 5 = 4.
- Expression: 56 ÷ [2 × {3 + 4}].
- { }: 3 + 4 = 7.
- Expression: 56 ÷ [2 × 7].
- [ ]: 2 × 7 = 14.
- Final: 56 ÷ 14 = 4.
Answer: 4.
Example 7: Example 7: Inserting Brackets
Problem: Put brackets in the expression 4 + 5 × 3 − 1 to get 26.
Solution:
- Without brackets: 4 + 15 − 1 = 18. Not 26.
- Try (4 + 5) × 3 − 1 = 9 × 3 − 1 = 27 − 1 = 26. Yes!
Answer: (4 + 5) × 3 − 1 = 26.
Example 8: Example 8: Word Problem with Brackets
Problem: A shopkeeper sells 4 boxes. Each box has 6 apples and 3 oranges. Write an expression for the total number of fruits and simplify.
Solution:
- Each box has 6 + 3 = 9 fruits.
- 4 boxes → 4 × (6 + 3).
- = 4 × 9 = 36 fruits.
Note: Without brackets, 4 × 6 + 3 = 24 + 3 = 27. This would be wrong because it adds only 3 oranges total, not 3 per box.
Example 9: Example 9: Multiple Operations
Problem: Simplify 100 − 3 × (4 + 6) + 8.
Solution:
- Brackets: 4 + 6 = 10.
- Expression: 100 − 3 × 10 + 8.
- Multiplication: 3 × 10 = 30.
- Expression: 100 − 30 + 8.
- Left to right: 100 − 30 = 70. Then 70 + 8 = 78.
Answer: 78.
Example 10: Example 10: True or False
Problem: True or false: 8 × 2 + 3 = 8 × (2 + 3).
Solution:
- Left side: 8 × 2 + 3 = 16 + 3 = 19.
- Right side: 8 × (2 + 3) = 8 × 5 = 40.
- 19 ≠ 40.
Answer: False.
Real-World Applications
Where do we use brackets?
- Maths calculations — Any time you want to change the normal order of operations, use brackets. They tell the reader (and calculators) what to do first.
- Calculators — Pressing the bracket buttons on a calculator ensures the correct answer. Without them, the calculator follows BODMAS and may give a different result.
- Formulas — Many formulas use brackets. Perimeter of rectangle = 2 × (length + width). Without brackets, 2 × length + width would give the wrong answer.
- Programming — All computer programs use brackets to group operations and ensure correct calculation order.
- Shopping — "3 packets of (2 biscuits + 1 chocolate)" means 3 × 3 = 9 items, not 3 × 2 + 1 = 7 items.
- Science — Physics and chemistry formulas frequently use brackets for grouping.
Key Points to Remember
- BODMAS gives the order: Brackets, Orders, Division/Multiplication, Addition/Subtraction.
- Always solve the expression inside brackets first.
- With nested brackets, work from the innermost to the outermost: ( ) → { } → [ ].
- Division and multiplication have the same priority — do them left to right.
- Addition and subtraction have the same priority — do them left to right.
- Brackets can change the answer: (2 + 3) × 4 = 20, but 2 + 3 × 4 = 14.
- Without brackets, always follow BODMAS — do not just go left to right.
- Formulas like perimeter = 2(l + w) use brackets to ensure correct grouping.
- On calculators, use bracket buttons to get the correct answer.
- When writing expressions, use brackets to make your meaning clear.
Practice Problems
- Simplify: (9 + 3) × 5.
- Simplify: 24 − 4 × 3 + 2.
- Simplify: [30 − {10 + (7 − 3)}].
- Which is greater: 5 × 3 + 2 or 5 × (3 + 2)?
- Put brackets in 2 + 6 × 3 to get 24.
- Simplify: 100 ÷ [5 × {2 + (8 − 6)}].
- A family buys 5 meals. Each meal has 2 rotis and 1 dal. Write an expression for total items using brackets and simplify.
- True or false: 10 − 3 + 2 = 10 − (3 + 2).
Frequently Asked Questions
Q1. What is BODMAS?
BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction. It tells us the order in which to do mathematical operations. Brackets are always done first, then powers, then multiplication/division (left to right), then addition/subtraction (left to right).
Q2. Why do we need brackets in maths?
Brackets tell us which part of an expression to solve first. Without brackets, we follow BODMAS, which means multiplication happens before addition. If we want addition to happen first, we put it in brackets.
Q3. What is the difference between ( ), { }, and [ ]?
They are different types of brackets used when expressions have multiple layers. Solve round brackets ( ) first, then curly brackets { }, then square brackets [ ]. In simple expressions, only round brackets ( ) are used.
Q4. Do multiplication and division have the same priority?
Yes. When both appear in an expression (without brackets), do them from left to right. For example, 12 ÷ 3 × 2 = 4 × 2 = 8 (NOT 12 ÷ 6 = 2).
Q5. What does 'left to right' mean?
When two operations have the same priority (like + and −, or × and ÷), do the one that comes first from left to right. For example, 10 − 4 + 3 = 6 + 3 = 9 (do subtraction first because it is on the left).
Q6. Is BODMAS the same as PEMDAS?
Yes, they are the same thing with different names. BODMAS is used in India, UK, and Australia. PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is used in the USA. The order of operations is identical.
Q7. Can brackets be used in word problems?
Yes. For example, 'Buy 5 packs of 3 pens and 2 pencils each' is 5 × (3 + 2) = 25. Without brackets, 5 × 3 + 2 = 17, which would be wrong.
Q8. What happens if I forget to use BODMAS?
You will get the wrong answer. For example, 2 + 3 × 4: with BODMAS, multiply first: 2 + 12 = 14 (correct). Without BODMAS, adding first: 5 × 4 = 20 (wrong).
