GCSE Maths Practice: standard-form

Question 3 of 10

This Higher-tier question explores population change using subtraction in standard form.

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\( \begin{array}{l}\text{A city has } 4.6 \times 10^6 \text{ people.} \\ \text{After relocation, } 2.3 \times 10^5 \text{ people leave.} \\ \text{Find the new population in standard form.}\end{array} \)

Choose one option:

Match powers of ten first. Subtract only the coefficients and ensure the result remains in proper standard form.

Subtracting Numbers in Standard Form – Population Scenario

Subtraction in standard form is useful whenever you compare large real-world quantities such as city populations, national budgets, or scientific measurements. Expressing both numbers using powers of ten makes differences easier to calculate and understand.

Key Concept

To subtract values written as \( a \times 10^n \), both must share the same power of ten. Once they do, only the coefficients (the numbers before the powers) are subtracted. The power remains the same unless renormalising is needed.

\[ a_1 \times 10^{n} - a_2 \times 10^{n} = (a_1 - a_2) \times 10^{n}. \]

If the exponents differ, rewrite one number so that both match. This may involve moving the decimal in the coefficient and adjusting the exponent accordingly.

Step-by-Step Approach

  1. Ensure both numbers are written in standard form.
  2. Rewrite one value so the powers of ten match.
  3. Subtract the coefficients directly.
  4. Check that the resulting coefficient lies between 1 and 10; if not, adjust it and modify the exponent.
  5. Express the final result clearly in standard form.

Worked Example (Different Data)

A country’s population was \( 7.8 \times 10^6 \) and decreased by \( 3.5 \times 10^5 \) people. To find the new population, rewrite \( 3.5 \times 10^5 = 0.35 \times 10^6 \). Subtracting gives \( (7.8 - 0.35) \times 10^6 = 7.45 \times 10^6 \). The new population is \( 7.45 \times 10^6 \) people.

Common Pitfalls

  • Forgetting to match powers: You cannot subtract until both numbers use the same exponent.
  • Changing exponents incorrectly: Exponents are only added or subtracted when multiplying or dividing, not when adding or subtracting.
  • Coefficient outside range: If the coefficient becomes smaller than 1 or larger than 10, shift the decimal and adjust the power of ten.

Real-Life Relevance

Standard form is common in social statistics, economics, and environmental studies. Population changes, carbon emissions, or annual energy outputs can vary by millions. Recording such quantities as \( 3.4 \times 10^6 \) or \( 8.1 \times 10^7 \) instead of writing long strings of zeros prevents calculation errors and makes comparisons quick and reliable.

FAQs

Q1: Why do we keep the same exponent when subtracting?
A: Because the power of ten acts as a common scale factor; only the coefficients represent the differing amounts.

Q2: When would the exponent change after subtraction?
A: Only if the subtraction causes the coefficient to fall below 1 or exceed 10, requiring renormalisation.

Q3: Is rounding required?

A: Unless the question specifies, express the answer to the same number of significant figures as given data—typically 2 or 3 s.f.

Study Tip

Always line up exponents before operating. Writing each step avoids confusion between coefficient subtraction and exponent adjustment. Remember, subtraction in standard form keeps the scale (the power) constant until the very end.