College practice · Worked answers
Bacterial growth and generation time
Bacterial growth problems mix two jobs: recognizing a culture’s growth phase and calculating doublings. Do the first job before the math. A constant generation time is a model for exponential growth under fixed conditions, not a promise that a culture doubles forever.
Read the growth curve first
| Phase | Population pattern | Interpretation |
|---|---|---|
| Lag | Little net increase | Cells adapt; metabolism can be active. |
| Log or exponential | Rapid repeated doubling | Use the constant-generation-time model here. |
| Stationary | Viable count roughly levels off | Division and death are approximately balanced. |
| Death | Viable count declines | Cells lose viability faster than they are replaced. |
On a graph with a logarithmic count axis, ideal exponential growth appears as a straight rising segment. On a linear count axis it curves upward. Read the axis labels before judging the shape.
Keep the three quantities separate
N = N₀ × 2ⁿ; n = t/g; g = t / log₂(N/N₀)
N₀ is the starting count, N is the later count, n is the number of generations, t is elapsed time, and g is generation time. Use the same units for t and g. The ratio N/N₀ has no units. A tenfold increase is not ten generations: it is log₂(10), about 3.32 generations.
A colony-forming-unit count estimates viable units that can form colonies under the chosen culture conditions. A clump can form one colony, so CFU and individual-cell counts need not be identical. Use comparable measurements at the beginning and end.
Work through two examples
Predict a count from time
- Start with 300 cells, generation time 25 minutes, and 100 minutes of exponential growth.
- Find generations: n = 100/25 = 4.
- Calculate N = 300 × 2⁴ = 4,800 cells. The units of time cancel before you apply the exponent.
Find an unknown generation time
- A population rises from 500 to 8,000 cells in 120 minutes during exponential growth.
- The growth factor is 8,000/500 = 16 = 2⁴, so there are 4 generations.
- Generation time is 120/4 = 30 minutes per generation. This estimate applies to the observed interval and conditions.
Try the doubling model
Assume exponential growth with no cell loss. Enter elapsed time and generation time in minutes.
Try it yourself
Choose an answer, then check your reasoning. You can retry any question or open its solution.
Show solution 1
3. n = t/g = 60/20 = 3 generations. That is an eightfold increase in population.
Why the other choices are wrong:
- 2: Count the intervals: 0–20, 20–40, and 40–60 make three generations, not two.
- 1/3: That divides generation time by elapsed time. Flip it: 60/20 = 3.
- 8: 8 is the growth factor, 2³. The number of generations is the exponent, 3.
Show solution 2
3,200. N = 200 × 2⁴ = 200 × 16 = 3,200 cells.
Why the other choices are wrong:
- 800: That is 200 × 4. Use 2 raised to the number of generations: 2⁴ = 16.
- 204: That adds 4 cells. Each generation doubles the count.
- 1,600: That is three doublings. Four generations go 200 → 400 → 800 → 1,600 → 3,200.
Show solution 3
30 minutes. The factor of 8 means 3 doublings, so g = 90/3 = 30 minutes.
Why the other choices are wrong:
- 11.25 minutes: 90/8 divides by the growth factor. Turn the factor into doublings first: 8 = 2³, so divide by 3.
- 90 minutes: 90 minutes is the whole interval, and it holds three doublings.
- 45 minutes: That assumes two doublings. 1,000 → 2,000 → 4,000 → 8,000 is three.
Show solution 4
Exponential. The model assumes repeated exponential doubling under stable conditions. Lag, stationary, and death phases violate that assumption.
Why the other choices are wrong:
- Stationary: In stationary phase the count is roughly flat, so there is no steady doubling.
- Death: In death phase the viable count is falling.
- Lag: In lag phase cells are adjusting and the count barely rises, so there is no steady doubling yet.
Show solution 5
Division and death approximately balance. A flat net count does not mean zero activity. New cells and lost viable cells can approximately balance.
Why the other choices are wrong:
- Every cell is inactive: A flat count doesn’t mean nothing is happening. Cells can still divide and die.
- No cells can divide: Some cells still divide; their gains are offset by cells that die.
- All the cells have died: If the cells had all died, the viable count would drop. That is the death phase.
Show solution 6
About 3.32. Generations n = log₂(10) ≈ 3.32, because 2 raised to the power 3.32 is about 10. Each doubling is one generation, and 2³ = 8 falls just short of tenfold.
Why the other choices are wrong:
- 10: Tenfold isn’t ten doublings. Ten doublings would be 2¹⁰ = 1,024-fold.
- 5: 2⁵ = 32, which is more than tenfold.
- 1: log₁₀(10) = 1 uses the wrong base. Count doublings with log base 2.
Show solution 7
6,400. Convert first: 2 hours = 120 minutes, so n = 120/30 = 4. Then N = 400 × 2⁴ = 6,400 cells.
Why the other choices are wrong:
- 1,600: That is 400 × 4. Four generations means multiplying by 2⁴ = 16.
- About 419: That uses t = 2 with g = 30 without converting hours to minutes. 2 hours is 120 minutes.
- 3,200: That is only three doublings. 120/30 = 4 generations.
Show solution 8
A straight rising line. On a log axis, multiplying by 2 each generation adds the same amount each time, so steady doubling plots as a straight rising segment.
Why the other choices are wrong:
- An upward curve that keeps getting steeper: That is how exponential growth looks on a linear count axis. A log axis straightens it out.
- A flat line: A flat line means no net change, like stationary phase.
- A straight falling line: A falling line means the viable count is dropping, like death phase.
Answer key
Open all answers
- 3. n = t/g = 60/20 = 3 generations. That is an eightfold increase in population.
- 3,200. N = 200 × 2⁴ = 200 × 16 = 3,200 cells.
- 30 minutes. The factor of 8 means 3 doublings, so g = 90/3 = 30 minutes.
- Exponential. The model assumes repeated exponential doubling under stable conditions. Lag, stationary, and death phases violate that assumption.
- Division and death approximately balance. A flat net count does not mean zero activity. New cells and lost viable cells can approximately balance.
- About 3.32. Generations n = log₂(10) ≈ 3.32, because 2 raised to the power 3.32 is about 10. Each doubling is one generation, and 2³ = 8 falls just short of tenfold.
- 6,400. Convert first: 2 hours = 120 minutes, so n = 120/30 = 4. Then N = 400 × 2⁴ = 6,400 cells.
- A straight rising line. On a log axis, multiplying by 2 each generation adds the same amount each time, so steady doubling plots as a straight rising segment.
Common questions
Does every bacterium have the same generation time?
No. Generation time depends on the organism and conditions such as temperature and nutrients.
Can the calculator predict growth overnight?
Only if its exponential-growth assumptions remain appropriate. A closed culture eventually runs short of resources or accumulates wastes.
Why use log base 2?
Each binary-fission generation multiplies the population by 2. Log base 2 turns the overall growth factor into a number of doublings.
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Sources and fact checks
Definitions and methods checked against these textbook references. Practice scenarios and worked numbers were written for this guide.