GCSE Maths Practice: decimals

Question 6 of 10

This question pushes decimal multiplication into a compound context, combining it with area and scale factor reasoning — a common Higher GCSE challenge involving multi-step logic.

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\( \begin{array}{1}\textbf{A scale model of a garden measures } 0.4\,\text{m by } 0.2\,\text{m.}\\\text{The model is built at a scale of } 1:100. \\\text{ Calculate the actual area of the garden in square metres.}\end{array} \)

Choose one option:

Estimate first — the model area is under 1 m² and scaled by 1:100, so the actual area should be hundreds of times larger. Then confirm with squared scale reasoning.

This Higher-tier GCSE Maths question combines decimal multiplication with unit conversions and scale reasoning. Such multi-step problems test both numerical fluency and conceptual understanding of proportion.

Scenario Context

A scale model of a rectangular garden measures 0.4 m by 0.2 m. The model is built at a scale of 1:100. Calculate the actual area of the garden in square metres.

This task requires two layers of reasoning: finding area using decimals and interpreting the scale factor correctly in two dimensions.

Step-by-Step Method

  1. Find the model area.
    \(0.4 \times 0.2 = 0.08\,\text{m}^2.\)
  2. Apply the scale factor.
    Since the model is 1:100, every dimension of the real garden is 100 times larger. For area, scale factors are squared: \(100^2 = 10{,}000.\)
  3. Calculate actual area.
    \(0.08 \times 10{,}000 = 800\,\text{m}^2.\)
  4. Optional conversion back.
    Some questions may ask for scaled-down or converted areas, e.g. \(800 \div 100{,}000 = 0.008\,\text{km}^2.\)

Mathematical Principles

Multiplying decimals here forms the base calculation, but applying scale requires understanding powers of ten. In GCSE Maths, scale factors appear in geometry, similarity, and real-world modelling problems.

Common Mistakes

  • Forgetting to square the scale factor when converting areas.
  • Misplacing the decimal point when multiplying by large powers of ten.
  • Confusing linear and area scaling (100 vs 100²).
  • Failing to write units clearly, leading to incorrect interpretation (m vs cm²).

Worked Example

Example: A floor plan shows a room measuring 0.35 m by 0.28 m at a scale of 1:50. Find the actual area in m².

Step 1: \(0.35 \times 0.28 = 0.098\,\text{m}^2.\)
Step 2: Multiply by \(50^2 = 2{,}500\): \(0.098 \times 2{,}500 = 245\,\text{m}^2.\)

Thus, the real room area is 245 m² — showing how quickly values grow under scaling.

Real-Life Applications

  • Architects and engineers use this principle when interpreting blueprints and models.
  • Map reading and geography problems apply similar reasoning using scale ratios.
  • Design and printing industries rely on proportional enlargement and reduction.

FAQ

Q1: Why is the scale factor squared for area?
A1: Because area = length × width. If each side increases by 100×, the total area increases by 100 × 100 = 10,000×.

Q2: How can I check the reasonableness of my answer?
A2: Estimate first: the model is small (under 1 m²), and a 1:100 enlargement should produce hundreds of square metres — the answer 800 m² fits that logic.

Q3: What if the question used centimetres instead of metres?
A3: You would convert units before multiplying. Mixing m and cm causes huge errors in area problems.

Study Tip

When problems combine decimals with scale, write all values in scientific notation. For instance, \(0.4 = 4.0 \times 10^{-1}\), \(0.2 = 2.0 \times 10^{-1}\). Multiplying gives \(8.0 \times 10^{-2}\), or 0.08 m² — an efficient method to avoid counting zeros manually.

Advanced decimal reasoning like this connects Number and Geometry topics and builds precision for complex modelling and exam questions involving units, scale, and powers of ten.