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Grade 12 · lesson 2 of 4

Calculate a Remaining Fraction

Separate fractional decay from a fixed subtraction.

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Supplied materials

  • Use the supplied fictional observations or model. Paper and pencil are optional; spoken answers are welcome. No equipment or real experiment is needed.

Learn the idea

For a fixed-half-life exponential model, N(t) = N0 × (1/2)^(t/h), where h is the half-life in the same time unit as t. Dividing N(t) by N0 gives the remaining fraction. After three half-lives the fraction is 1/8, regardless of the initial amount. This model assumes one isotope, a constant rate, no added parent isotope, and no removal other than decay.

A fictional model begins with 160 parent-isotope units and has h = 2 hours. After 6 hours, t/h = 3, so 20 parent units remain and 140 parent units have transformed. “Transformed” does not mean the total matter simply vanished; this ledger concerns the original parent isotope. We do not use the amount to infer exposure, safety, or how to handle material.

  1. Calculate the number of half-lives, 6/2.
  2. Compute remaining fraction (1/2)³.
  3. Multiply 160 by the fraction for remaining parent amount.
  4. Subtract remaining from initial to find the transformed parent amount.
Worked example

See the method in action

At 6 h, fraction remaining is 1/8 = 12.5%. Amount remaining is 160/8 = 20 parent units; 160 − 20 = 140 parent units transformed.

Your turn

Check your reasoning

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With an initial 240 units and half-life 4 h, what remains at 12 h under the same model assumptions?

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See the answer and explanation

30 units

12/4 = 3 half-lives, so 240×(1/2)³ = 30 units.

Take it further

Change elapsed time to 16 h, keeping 240 initial and 4 h half-life. Four half-lives leave 240/16 = 15 units, or 6.25% of the initial parent amount.

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