Order and Inverse Operations
This topic covers the fundamental rules of arithmetic, ensuring calculations are performed in a consistent, correct order. Mastering this is crucial for nearly all numerical problems in the ESAT, preventing simple errors and enabling faster problem-solving through simplification.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The order of operations follows the BIDMAS/PEMDAS hierarchy: Brackets, Indices (powers/roots), Division/Multiplication, Addition/Subtraction.
- Division and Multiplication have equal priority; work from left to right.
- Addition and Subtraction also have equal priority; work from left to right.
- Inverse operations are pairs that 'undo' each other (e.g., addition and subtraction, multiplication and division), which is key for rearranging equations.
- Always simplify calculations by cancelling common factors in fractions before multiplying or dividing.
- A negative number raised to an even power becomes positive, e.g., (-4)2 = 16. A negative number raised to an odd power remains negative, e.g., (-2)3 = -8.
Formulae
B → I → D/M → A/S This represents the order of operations (BIDMAS). Use it to decide the sequence for any multi-step calculation. Brackets first, then Indices (powers, roots), then Division and Multiplication (left-to-right), and finally Addition and Subtraction (left-to-right).
Definitions
- Order of Operations
- The universally agreed sequence in which to perform mathematical operations to ensure a consistent and correct result. Commonly remembered by the acronym BIDMAS or PEMDAS.
- Inverse Operation
- An operation that reverses the effect of another. For example, subtraction is the inverse of addition, and taking a square root is the inverse of squaring.
- Cancellation
- The process of simplifying a fraction or expression by dividing both the numerator and denominator (or different parts of an expression) by a common factor.
Worked example
Without a calculator, evaluate: 6 + (30 × 14) / (5 × 42) - 32
- 1
Follow the order of operations (BIDMAS).
Start with any Brackets.
The brackets contain (30 × 14) / (5 × 42).
- 2
Next, handle Indices.
Calculate 32 = 9.
The expression is now 6 + (30 × 14) / (5 × 42) - 9.
- 3
Now, evaluate the Division/Multiplication within the brackets.
Instead of multiplying out large numbers, use cancellation to simplify the fraction.
- 4
Cancel common factors:
30/5 = 6The fraction becomes (6 × 14) / 42.
- 5
Further cancellation:
42 is 6 × 7.
So, the fraction is (6 × 14) / (6 × 7).
The 6s cancel, leaving 14/7.
- 6
Calculate the final division:
14 / 7 = 2 - 7
Substitute this back into the expression:
6 + 2 - 9.
- 8
Finally, perform the Addition and Subtraction from left to right.
6 + 2 = 8Then 8 - 9 = -1
Answer: -1
Common mistakes
- ×Performing addition or subtraction before multiplication or division. For example, calculating 5 + 2 × 3 as 7 × 3 = 21 instead of the correct 5 + 6 = 11.
- ×When division and multiplication appear together, working from right-to-left instead of left-to-right. For example, calculating 12 / 2 × 3 as 12 / 6 = 2 instead of the correct 6 × 3 = 18.
- ×Multiplying out large numbers in fractions instead of cancelling first, which creates complex arithmetic and increases the chance of errors.
- ×Incorrectly handling powers with negative numbers, for example evaluating (-5)2 as -25 instead of 25.
No-calculator tips
- ✓Before multiplying terms in a fraction's numerator and denominator, always check for common factors to cancel. It's easier to simplify 24/16 by dividing both by 8 than to work with larger numbers.
- ✓Break down numbers into their prime factors to make spotting common factors for cancellation easier. For example, 54/36 becomes (2*33)/(22*32), which simplifies to 3/2.
- ✓When faced with a complex expression, rewrite it step-by-step as you evaluate each part according to BIDMAS. This avoids mental overload and reduces errors.