GCSE Maths Practice: integers-and-directed-numbers
This problem tests your ability to apply BIDMAS when multiple negatives and nested brackets are involved.
Always work from the innermost brackets outward and apply sign rules carefully.
Advanced Use of BIDMAS with Negative Integers
At the Higher GCSE level, questions involving negative numbers and multiple operations often contain brackets within brackets, or mixed positive and negative multipliers. These questions are designed to test whether you truly understand the logic of BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction) rather than simply memorising it.
Whenever brackets and negative signs appear together, treat each sign carefully. The position of a minus sign before a bracket, for instance, changes the signs of every term inside that bracket when expanded. In arithmetic expressions, however, the same principle applies conceptually: you must handle the negative before performing any addition or subtraction.
Step-by-Step Strategy
- Identify and simplify inner brackets first. If you see nested brackets, always begin with the innermost part. Replace it with its simplified result.
- Handle multiplication or division next. These take priority over addition or subtraction even when negatives are present.
- Finally, process additions and subtractions in order from left to right. This ensures that your result is consistent and logical.
Take care not to mix up subtraction with negative signs. Subtraction is an operation; a negative sign is a property of a number. They often look identical but play different roles depending on placement.
Worked Examples (Different from the Question)
- Example 1: (−6) + 4×(−2 + 3) → work out the bracket, then multiply, then add.
- Example 2: −5×(−3 + 1) − 7 → apply multiplication before the final subtraction.
- Example 3: (−8) − [2×(−5 + 4)] → evaluate the inner bracket first, multiply, then subtract.
Each example requires following the same logic chain. Even one misplaced operation can completely change the final result.
Common Mistakes to Avoid
- Ignoring bracket order. Solving the outer operation first often leads to errors.
- Incorrect sign handling. Two negatives multiplied make a positive, but a negative added to a negative makes the value smaller.
- Expanding too early. Always simplify before expanding unless algebraic simplification demands it.
Real-Life Applications
Understanding multi-step integer operations applies to many real-world contexts. For example, in finance, profit and loss calculations often mix increases and decreases. In physics, direction-sensitive quantities such as velocity or temperature changes behave similarly—positive meaning upward or increase, negative meaning downward or decrease. The same rules of addition, subtraction, and multiplication apply behind the scenes.
FAQs
Q1: What happens if two minus signs appear side by side?
A: When two negatives meet, they effectively cancel to form a positive. This rule applies both in multiplication and in addition contexts such as −(−4) = +4.
Q2: How can I check if my final sign makes sense?
A: Estimate roughly: if most numbers are negative and you are adding or subtracting more negatives, the result should remain negative.
Q3: How do brackets affect negative values in multiplication?
A: Brackets indicate that the number inside must be treated as a whole. For example, −2×(3−5) means multiply −2 by the result of (3−5), not by each term individually.
Study Tip
When practising higher-tier problems, write each step explicitly in your working. This habit ensures you never lose track of a sign or skip a bracket operation. Over time, your accuracy and speed improve together. Understanding these patterns will also make future topics like algebraic simplification, gradient calculations, and vector addition far easier to master.