GCSE Maths Practice: integers-and-directed-numbers

Question 9 of 10

This Higher-tier BIDMAS problem includes a fraction and multiple negative signs, requiring careful order of operations.

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\( \begin{array}{l}\text{Evaluate } 8 \times (-2) + \frac{-6}{3} - 4.\end{array} \)

Choose one option:

Treat fractions as division; simplify them before continuing with other operations.

Working with Fractions and Negatives in BIDMAS

Fractions with negative signs are one of the most common causes of mistakes in GCSE arithmetic questions. This problem combines multiplication, division, and addition within a single expression, requiring full command of BIDMAS. The challenge lies in applying division first when a fraction appears and ensuring all negatives are handled correctly.

Breaking Down the Expression

The expression can be written as: 8 × (−2) + (−6 ÷ 3) − 4. It contains multiplication, division, and addition/subtraction. Following BIDMAS means completing the fraction (division) and the multiplication before dealing with the addition or subtraction.

  1. Division (fraction) step: (−6 ÷ 3) = −2.
  2. Multiplication step: 8 × (−2) = −16.
  3. Add/subtract in order: (−16) + (−2) − 4 = −22.

By rearranging brackets or fractions incorrectly, many students lose track of signs. Always write the negative sign clearly before dividing or multiplying—it belongs to the numerator, not the denominator, unless stated otherwise.

Key Sign and Fraction Rules

  • When dividing negatives: Negative ÷ Positive = Negative; Positive ÷ Negative = Negative.
  • When multiplying or dividing two negatives, the result is positive.
  • Adding a negative is the same as subtracting its positive version.
  • Fractions act as division: \(\frac{-6}{3}\) simply means −6 ÷ 3.

Worked Examples (Different from This Question)

  • Example 1: 10 × (−3) + (−8 ÷ 4) = −30 − 2 = −32.
  • Example 2: (−5 ÷ 2) + 6 × (−1) = −2.5 − 6 = −8.5.
  • Example 3: 12 − (−9 ÷ 3) = 12 − (−3) = 15.

Each example involves fractions or divisions that interact with negative numbers. These follow the same sequence: division or multiplication first, then any additions or subtractions.

Common Mistakes

  • Performing addition before division.
  • Dropping negative signs during simplification.
  • Interpreting −6 ÷ 3 as 6 ÷ −3 (they’re equivalent but easy to confuse mid-calculation).

Conceptual Understanding

Fractions don’t change BIDMAS—they are simply division in disguise. When a fraction contains a negative numerator, that sign affects the entire result. Recognising that connection ensures smoother transitions later to algebraic fractions and rational expressions.

Real-Life Applications

Fractions and negatives model real-world changes such as temperature drops per hour or financial losses per unit sold. For example, losing £6 for every 3 products is mathematically the same as −6 ÷ 3 = −2 per item. This problem mirrors that reasoning process but embeds it within a full expression.

FAQs

Q1: What happens if both numerator and denominator are negative?
A: The result becomes positive, since two negatives cancel out.

Q2: Should I simplify the fraction first or multiply first?
A: Simplify the fraction (division) first, as per BIDMAS, then multiply and finally add or subtract.

Q3: Why are brackets useful around fractions?

A: They make it clear which numbers the negative applies to, preventing sign confusion in multi-step problems.

Study Tip

Always rewrite fraction-based problems using division signs before beginning. Write one line per operation and label each with M (multiplication), D (division), A (addition), or S (subtraction). This keeps your process structured, reduces sign mistakes, and helps you develop the clarity expected in Higher GCSE answers.