Look at 30 + 5 × 4. Is the answer 140 or 50? Both come from the same symbols. Only one is right. The rules of simplification settle the matter, and they give 50.
This article shows the order of work, a few shortcuts for squares and products, and five solved examples. Each example says why the method works.
- BODMAS: Brackets, Orders, Division and Multiplication, Addition and Subtraction.
- Same rung: work left to right.
- Brackets: open ( ) first, then { }, then [ ].
- Missing number: undo the operations in reverse order.
- Squares: (a + b)² = a² + 2ab + b². And (a − b)² = a² − 2ab + b².
- Difference of squares: a² − b² = (a + b)(a − b).
- Near 100: use 100 as a base to multiply numbers like 97 × 94 fast.
In what order do you work an expression?
What does BODMAS mean?
BODMAS tells you which step to do first. The letters stand for Brackets, Orders, Division, Multiplication, Addition and Subtraction.
Do brackets first. Then do powers and roots. Then divide and multiply. Last, add and subtract.
Sources: NCERT Class 7 Maths: Simple Expressions, order of operations; Practice question: first step in BODMAS.
Which comes first, multiplication or addition?
Multiplication comes first. So 26 + 13 × 10 − 13 means 26 + 130 − 13, which is 143.
Then go left to right: 26 + 130 = 156, and 156 − 13 = 143.
Sources: NCERT Class 7 Maths: Simple Expressions, evaluating terms; Practice question: 26 + 13 × 10 − 13.
What if division and multiplication both appear?
They are on the same rung. So go left to right.
For 48 ÷ 6 × 2, first divide to get 8. Then multiply to get 16. If you multiply first, you get 4, which is wrong.
Sources: NCERT Class 7 Maths: Simple Expressions, evaluating terms.
How do you open brackets of different types?
Start with the innermost bracket. The usual order is ( ) first, then { }, then [ ].
Each time, replace the bracket with its value. Then move to the next bracket outward.
Sources: NCERT Class 7 Maths: Simple Expressions, using brackets; Practice question: simplify with brackets.
What does a bracket with a minus sign in front do?
The minus applies to the whole bracket. So 100 − (15 + 56) = 100 − 71 = 29.
Never subtract just the first number. The bracket is a single block.
Sources: NCERT Class 7 Maths: Simple Expressions, using brackets.
Which shortcuts make simplification fast?
How do you find a missing number in an expression?
Work backward. Undo the last operation first, then the one before it.
For (? − 158) ÷ 4 × 7 = 910, undo × 7 to get 130. Undo ÷ 4 to get 520. Undo − 158 to get 678.
Sources: NCERT Class 7 Maths: Simple Expressions, unknown in an expression; Practice question: missing number with (? − 158).
How do you find an unknown inside nested brackets?
Peel the layers from the outside in. At each layer, subtract to find what the rest must be.
In 39 − [18 − {42 − (5 − x)}] = 34, the rest after 39 is 5. Then 18 − 5 = 13. Then 42 − 13 = 29. Last, 5 − x = 29, so x = −24.
Sources: NCERT Class 7 Maths: Simple Expressions, unknown in an expression; Practice question: unknown in nested brackets.
How do you square a sum or a difference quickly?
Use the identities. (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
For 63², write (60 + 3)². Then 3600 + 360 + 9 = 3969.
Sources: NCERT Class 9 Maths: Algebraic Identities, squares of a sum and difference.
How do you simplify the difference of two squares?
Use a² − b² = (a + b)(a − b). Add the numbers, subtract the numbers, and multiply the two answers.
So 52² − 8² = 60 × 44 = 2640.
Sources: NCERT Class 9 Maths: Algebraic Identities, difference of squares; Practice question: 52² − 8².
How do you multiply two numbers close to 100?
Use 100 as the base. Write each number as 100 minus a small gap. Then use the rule (100 − x)(100 − y) = 10000 − 100(x + y) + xy.
For 99 × 96, the gaps are 1 and 4. So 10000 − 500 + 4 = 9504.
Sources: NCERT Class 9 Maths: Algebraic Identities, product of two binomials; Practice question: 99 × 96.
Check 1
The value of 30 + 5 × 4 is 50.
Check 2
In 48 ÷ 6 × 2, you must multiply 6 × 2 first.
Check 3
(a − b)² equals a² − 2ab + b².
Check 4
(a + b)² equals a² + b².
How do you solve five typical problems step by step?
Example 1: Simplify 48 ÷ 6 × 2 + 9 − 4 × 3.
Step 1. Divide and multiply from the left: 48 ÷ 6 = 8, and 8 × 2 = 16. Also 4 × 3 = 12.
Step 2. The expression is now 16 + 9 − 12.
Step 3. Add and subtract from the left: 16 + 9 = 25, and 25 − 12 = 13.
Check. Terms are 16, +9 and −12. Their sum is 13.
Why this works. Each term is a block of multiplication and division. The blocks are added at the end.
Example 2: Simplify 120 − [30 − {18 − (9 − 4)}].
Step 1. Innermost: (9 − 4) = 5.
Step 2. Next: {18 − 5} = 13.
Step 3. Next: [30 − 13] = 17.
Step 4. Last: 120 − 17 = 103.
Check. Rebuild from 103: 120 − 103 = 17. Then 30 − 17 = 13, 18 − 13 = 5, and 9 − 5 = 4.
Why this works. Each bracket is a single number. So we find the inner value before we can use it outside.
Example 3: Find ? in (? + 45) ÷ 3 × 4 = 100.
Step 1. Undo × 4: 100 ÷ 4 = 25.
Step 2. Undo ÷ 3: 25 × 3 = 75.
Step 3. Undo + 45: 75 − 45 = 30.
Check. (30 + 45) = 75. Then 75 ÷ 3 = 25, and 25 × 4 = 100.
Why this works. The expression was built in steps. Reverse the steps, with opposite operations, to get back to the start.
Example 4: Find 97 × 94.
Step 1. The gaps from 100 are 3 and 6.
Step 2. Cross-subtract: 97 − 6 = 91. This gives the first part, 91.
Step 3. Multiply the gaps: 3 × 6 = 18. This is the last two digits.
Step 4. Join the parts: 9118.
Check. 97 × 94 = 97 × 100 − 97 × 6 = 9700 − 582 = 9118.
Why this works. (100 − 3)(100 − 6) = 10000 − 900 + 18. The 100 × 91 gives the front, and the 18 sits at the end.
Example 5: Find 73² − 27², and then 63².
Step 1. For the first part, a + b = 100 and a − b = 46. So 73² − 27² = 100 × 46 = 4600.
Step 2. For 63², split 63 as 60 and 3. Then 3600 + 2 × 60 × 3 + 9.
Step 3. That is 3600 + 360 + 9 = 3969.
Check. 73² = 5329 and 27² = 729. The difference is 4600. Also 63 × 63 = 3969.
Why this works. Both identities are just the expanded form of a product. They turn a hard multiplication into easy ones.
How do you avoid careless mistakes?
What habits keep answers right?
Rewrite the expression after every step. Do not do two steps in your head.
Use a quick estimate at the end. If your answer is far from the estimate, recheck the order.
Check 5
The value of 20 − 192 ÷ 12 + 10 is −10.
Check 6
The value of (25 − 16 + 5) × 7 is 98.
Exam traps
Left-to-right trap. Do not force multiplication before division. Move from the left.
Minus-bracket trap. A minus before a bracket applies to the whole bracket.
Bracket-order trap. Open ( ) before { } and { } before [ ].
Identity trap. (a + b)² is not a² + b². The middle term 2ab is needed.
Backward trap. When you undo steps, use the opposite operation in the reverse order.
MCQ on Simplification
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