GCSE Maths Practice: integers-and-directed-numbers

Using BIDMAS in Contextual Problems

Mathematics is not just about numbers on paper—it represents real situations. In GCSE questions, negative values often describe quantities that move in the opposite direction, such as losses, decreases, or drops in temperature. Applying BIDMAS helps us handle these correctly in everyday contexts like finance or science.

Scenario

Imagine a shop owner reviewing sales for the week. On Monday, the business had a debt of £6. The owner divides that debt evenly over three suppliers, so each supplier is responsible for one-third of the total amount. The next day, a customer makes a purchase that brings in £8 profit. The question asks what the overall financial position becomes after these two events. This can be represented mathematically by the expression (−6 ÷ 3) + 8. Solving it correctly means interpreting both the division and addition steps using the BIDMAS rule.

Step-by-Step Thinking

1. Identify the operations. There is a division followed by an addition. Division must happen first.
2. Apply the sign rule. A negative divided by a positive gives a negative value, because the signs are different.
3. Perform the final operation. Once the division result is known, add the second value to determine the new total or final position.

Following this clear order avoids confusion and reflects how transactions, measurements, or data adjustments work in the real world.

Worked Examples (Different from This Question)

• Example 1: A company reports a loss of £12 spread equally among four departments and then gains £10 in new sales. Represented as (−12 ÷ 4) + 10.
• Example 2: The temperature is −15 °C in the morning and increases by half of that value later in the day. Represented as (−15 ÷ 2) + (−15).
• Example 3: A swimmer dives 9 metres underwater, comes up evenly over 3 equal moves, and then rises another 5 metres above the previous level. Represented as (−9 ÷ 3) + 5.

Each example involves a negative quantity being shared or divided, followed by an addition that changes the overall outcome. The method remains identical regardless of the numbers or setting.

Common Mistakes

• Adding before dividing, which breaks the BIDMAS rule.
• Assuming negative ÷ positive = positive (it is not—different signs make the result negative).
• Mixing up subtraction and a negative sign; subtraction is an operation, not a sign property.

Real-World Application

This concept appears frequently in personal finance, temperature tracking, and even physics. For instance, dividing a total loss among different months before adding new income follows exactly the same order of logic. In science, dividing a drop in temperature across several time intervals and then adding a rise models the same mathematical reasoning.

FAQs

Q1: Why must division happen before addition?
A: Because division simplifies grouped quantities. If you add first, you would mix totals before they are properly shared, producing an incorrect result.

Q2: What is the easiest way to spot BIDMAS in a word problem?
A: Look for words such as 'shared equally' (division), 'increase by' (addition), or 'reduced by' (subtraction). These tell you the operation type and order.

Q3: How can you double-check your signs?
A: Quickly reason whether the situation represents a loss (negative) or gain (positive). If two negatives interact, the result becomes positive; if the signs differ, the result stays negative.

Study Tip

When solving mixed-operation problems, write down each step separately. Highlight all negative signs before calculating. Doing so reduces careless mistakes and builds accuracy for algebraic operations later on. Practise by turning story problems into equations and vice versa—this skill ensures deep understanding and prepares you for higher-level problem-solving in GCSE Maths.