GCSE Maths Practice: standard-form

Question 2 of 10

This question combines exponential growth and standard form, both key concepts in GCSE Maths higher tier.

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\( \begin{array}{l}\text{A bacteria culture starts at } 3.5 \times 10^4. \\ \text{It doubles each hour. Find its size after 3 hours in} \\ \text{standard form.}\end{array} \)

Choose one option:

Break the problem into two parts: apply doubling with powers, then convert to standard form carefully.

Understanding Exponential Growth

In GCSE Maths, questions involving repeated doubling or growth follow the rules of exponential growth. Exponential growth means a quantity increases by the same factor at regular intervals. This idea is widely used in science, population studies, computing, and finance.

When something doubles, it grows by a factor of 2. After each equal period, the total amount is multiplied by 2 again. After n such periods, the multiplier becomes 2^n. This pattern leads to very rapid increases, and standard form is a neat way to represent these large results clearly.

Why Standard Form Matters

Standard form (scientific notation) helps to express very large or very small numbers compactly. For example, instead of writing thousands of digits, we write a number between 1 and 10 multiplied by a power of 10. This allows for easier comparison, calculation, and presentation of data.

Step-by-Step Method

  1. Identify the starting value (initial quantity).
  2. Recognize the growth pattern — doubling means ×2 each time.
  3. Calculate how many times the growth happens.
  4. Use the formula: \text{Final} = \text{Initial} \times 2^n.
  5. Express the final answer in standard form by moving the decimal until the first number is between 1 and 10.

Example 1: Cell Division

If one cell divides every 30 minutes, after 4 cycles there will be 2^4 = 16 times more cells. So, starting with 500 cells gives 500 \times 16 = 8000 cells after two hours, written as 8.0 \times 10^3.

Example 2: Data Transmission

A file doubles in size every second due to repeated compression errors. Starting with 2.5 \times 10^3 KB, after 5 seconds the total size is multiplied by 2^5 = 32. The answer can again be expressed in standard form.

Common Mistakes

  • Adding instead of multiplying: Each doubling multiplies, not adds. Using addition gives a linear pattern, not exponential.
  • Incorrect power of 10: Always ensure the number before the power is between 1 and 10. Adjust both parts correctly.
  • Confusing power and multiplier: The power represents the number of doubling periods, not the number of times you multiply by 2 manually.

Real-Life Applications

Exponential growth appears in many real-world contexts:

  • Population biology: estimating bacteria, viruses, or human population increase.
  • Finance: compound interest grows exponentially when interest is added repeatedly.
  • Technology: Moore’s law describes how transistor counts double approximately every two years.
  • Environmental science: certain processes, such as radioactive decay (the opposite of growth), follow exponential decay patterns.

FAQ

Q1: How can I tell if a problem involves exponential growth?
A: Look for words like “doubles”, “triples”, or “increases by a fixed percentage each time”.

Q2: Do negative powers appear in these problems?
A: Negative powers are used in exponential decay, where values decrease by the same factor each period (e.g., halving).

Q3: What is the difference between growth factor and rate?
A: The growth factor is the multiplier (like 2), while the rate is the percentage increase per period (like 100% for doubling).

Study Tip

When solving exponential problems, always calculate the power expression first, then apply the multiplier. Use a calculator carefully and remember to check your answer’s format — the first number should be between 1 and 10 in standard form. This topic is essential in the GCSE Maths syllabus under Standard Form and Indices.