GCSE Maths Practice: decimals

Question 8 of 10

This GCSE Higher-level question involves dividing decimals within a real-world rate context. It tests both place-value precision and the ability to interpret results meaningfully (e.g. as speed or cost per unit).

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\( \begin{array}{1}\textbf{A cyclist travels } 0.3\,\text{km in } 0.25\,\text{hours.}\\ \text{Calculate the cyclist's average speed in kilometres per hour.}\end{array} \)

Choose one option:

Estimate first: since 0.3 ÷ 0.25 is slightly more than 1, expect an answer just above 1. Checking with multiplication (1.2 × 0.25 = 0.3) confirms accuracy.

This Higher-tier GCSE Maths question applies decimal division to a real-life context — calculating a rate (speed). It strengthens understanding of ratio, proportional reasoning, and place value accuracy when decimals appear in both numbers.

Scenario Context

A cyclist travels 0.3 km in 0.25 hours. To find the speed in kilometres per hour, divide the distance by the time:

\(\text{speed} = \dfrac{0.3}{0.25} = 1.2\,\text{km/h}.\)

This type of problem integrates arithmetic with the formula triangle used widely in science and physics (distance = speed × time).

Step-by-Step Reasoning

  1. Identify the operation. Division is used because we’re finding a rate per unit (kilometres per hour).
  2. Remove decimals. Multiply both by 100 → \(30 \div 25\).
  3. Simplify as a fraction. \(\frac{30}{25} = \frac{6}{5}\).
  4. Convert to decimal. \(\frac{6}{5} = 1.2\).
  5. Interpret the result. The cyclist’s average speed is 1.2 km/h.

Why This Works

Multiplying both numbers by the same power of ten (here, 100) doesn’t change the quotient — it simply moves the decimal points to make division easier. This method underlies calculator and long-division techniques alike.

Common Errors

  • Forgetting to move the decimal point in both numbers equally.
  • Misinterpreting units (e.g. thinking 1.2 means 1.2 minutes instead of km/h).
  • Writing 0.25 ÷ 0.3 instead — reversing the order.
  • Rounding too early, especially in multi-step problems.

Advanced Example

Example: A car travels 0.48 km in 0.4 minutes. Find its speed in km/min, then convert to km/h.

\(0.48 \div 0.4 = 1.2\,\text{km/min}.\) Multiply by 60 → 72 km/h.

This illustrates how one accurate decimal division can feed into another operation (unit conversion) — a frequent requirement in Higher papers.

Real-Life Applications

  • Calculating rates: speed, flow, density, population growth, or chemical concentration.
  • Converting currency or scaling recipes.
  • Finding unit prices in financial contexts (e.g. £0.30 ÷ 0.25 kg = £1.20 per kg).

FAQ

Q1: How do I decide how many zeros to move?
A1: Count the decimal places in the divisor and multiply both numbers by 10, 100, or 1000 until the divisor becomes a whole number.

Q2: Why estimate before dividing?
A2: Estimation checks that your decimal point is in the right place. Here, since 0.3 ÷ 0.25 is just over 1, an answer near 1.2 is reasonable.

Q3: Can decimals be divided without a calculator?
A3: Yes — by converting both to integers using powers of ten, then performing long division or simplifying fractions.

Study Tip

Always write division problems as fractions — it clarifies which number is the divisor and supports later algebraic manipulation. Practice estimating before computing; this habit prevents misplaced decimals and builds confidence for non-calculator exams.

Decimal division links directly to proportional reasoning, speed–distance–time relationships, and ratio scaling — all vital skills for Higher GCSE Maths and science problem-solving.