Scientific scale problems
Objective. Compare measurements spanning many orders of magnitude.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
Scientific scale problems ask pupils to compare measurements written in standard form that differ by many orders of magnitude, such as the width of a virus against the width of a human hair. To compare two numbers in standard form, compare the powers of 10 first; a larger power generally means a larger number, unless the coefficients differ enough to matter. To find how many times bigger one value is than another, divide the larger by the smaller, working with standard form division rules. For example, 2 × 10⁶ is 1000 times bigger than 2 × 10³, because 10⁶ ÷ 10³ = 10³ = 1000. These comparisons are common in science contexts describing cells, stars, or nanoparticles.
Worked example
A bacterium measures 4 × 10⁻⁶ m and a grain of sand measures 8 × 10⁻⁴ m. How many times larger is the grain of sand?
- 1Divide the sand's size by the bacterium's size: (8 × 10⁻⁴) ÷ (4 × 10⁻⁶).
- 2Divide coefficients: 8 ÷ 4 = 2.
- 3Subtract powers: −4 − (−6) = 2, giving 10².
- 4Combine: 2 × 10² = 200.
Answer: 200 times larger
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils often subtract the powers of 10 to compare sizes instead of dividing, concluding a difference of 3 in the power means '3 times bigger' rather than 1000 times bigger. Compare order-of-magnitude differences using division, not subtraction.
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