Chapter focus
An arithmetic expression is a structured calculation. Parentheses and operation rules tell us which step to do first so that everyone evaluating the same expression obtains the same value.
These original notes support practice; complete the examples and investigations in your assigned Ganita Prakash Part I.
1. Order of operations
When an expression has several operations, use this order:
- Evaluate expressions inside grouping symbols first, working from the innermost group.
- Evaluate powers where they occur.
- Multiply and divide from left to right.
- Add and subtract from left to right.
Multiplication does not always come before division, and addition does not always come before subtraction. Operations at the same level are handled from left to right.
Example: 24 ÷ 3 × 2 = 8 × 2 = 16, not 24 ÷ 6 = 4.
2. Brackets change the grouping
Parentheses make a group explicit. Calculate the group before using its result in the rest of the expression.
Worked example: 36 ÷ (3 × 2) + 4
- Inside brackets: 3 × 2 = 6.
- Divide: 36 ÷ 6 = 6.
- Add: 6 + 4 = 10.
Without those brackets, 36 ÷ 3 × 2 + 4 = 12 × 2 + 4 = 28. The expressions are different, so grouping matters.
3. Use expressions to describe a sequence of actions
Translate a written instruction one operation at a time. “Add 5 to 3, then multiply the result by 4” is 4 × (3 + 5) = 32. “Multiply 3 by 4, then add 5” is (4 × 3) + 5 = 17. The order in the words determines the grouping.
To check an expression, estimate its size and use the reverse operation. If an answer to 8 × (20 − 3) is larger than 8 × 20, recheck the subtraction step.
4. Compare two ways of calculating
Some expressions are equal because multiplication distributes over addition: 8 × 19 = 8 × (20 − 1) = 160 − 8 = 152. The rewritten form is useful when it makes mental arithmetic easier.
Keep the equality sign between quantities with the same value. It does not mean “write the answer next”; each side must balance.
Common errors to avoid
- Do not always calculate from left to right; first respect grouping and operation priority.
- At the same priority, multiplication/division and addition/subtraction proceed left to right.
- Do not insert or remove brackets without checking whether they change the value.
- Write enough intermediate steps to make your order clear.
Practice: show the order of steps
1. Evaluate 18 − 6 ÷ 3.
Answer: Divide first: 6 ÷ 3 = 2; then 18 − 2 = 16.
2. Evaluate (18 − 6) ÷ 3.
Answer: Brackets first: 12 ÷ 3 = 4.
3. Evaluate 24 ÷ 3 × 2 and explain why.
Answer: 24 ÷ 3 = 8, then 8 × 2 = 16; division and multiplication have equal priority and are done left to right.
4. Write “subtract 4 from 15, then multiply by 3” as an expression and find its value.
Answer: 3 × (15 − 4) = 3 × 11 = 33.
5. Calculate 25 × 16 by rewriting 16 as 4 × 4.
Answer: 25 × 16 = 25 × 4 × 4 = 100 × 4 = 400.
Chapter summary
Expressions encode a sequence of operations. Grouping symbols, powers and operation priority determine the order; operations at the same level proceed left to right. Show each step, estimate and check the result.