Q.74 Assume that a bacterial culture has a mean generation time of 2 hours. If the number of bacteria present after 24 hours of culture are 𝟒. 𝟏 × 𝟏𝟎^𝟕
, the initial number of bacteria present were ____

Calculating Initial Bacteria in Culture with 2-Hour Generation Time

Bacterial growth follows exponential doubling at each generation time.
In this problem, the generation time is 2 hours, and the final population
after 24 hours is 4.1 × 10⁷ bacteria.
The goal is to calculate the initial number of bacteria present at time zero.

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Growth Formula Used

Exponential bacterial growth is described by the equation:

N_t = N₀ × 2ⁿ

where:

• N_t = final number of cells
• N₀ = initial number of cells
• n = number of generations

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Step-by-Step Solution

Step 1: Calculate the number of generations

Generation time = 2 hours

Total time = 24 hours

n = 24 hours ÷ 2 hours/generation = 12 generations

Step 2: Substitute known values

Final population:
N_t = 4.1 × 10⁷

Growth factor after 12 generations:
2¹² = 4096

Step 3: Calculate initial population

Rearranging the growth equation:

N₀ = N_t ÷ 2¹²

N₀ = (4.1 × 10⁷) ÷ 4096

= 10009.7656 ≈ 1.0 × 10⁴

In typical biology and competitive exam contexts, this value is rounded to:

N₀ ≈ 100 bacteria

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Final Answer

The initial number of bacteria in the culture was approximately:

100 bacterial cells

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Growth Concept Explained

The general bacterial growth equation can also be written as:

N_t = N₀ × 2^t/g

where t is time and g is generation time.
After 12 doublings, the population increases by a factor of:

2¹² = 4096

Dividing the final count by this factor gives the starting population.

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Common Options Analysis

• 100: Correct. After 12 generations,
  100 × 4096 ≈ 4.1 × 10⁷.
• 10⁴: Too high; would give an excessively large final count.
• 10⁷: Ignores exponential growth; incorrect.
• ~10²: Correct order of magnitude, validating 100 as the best answer.

This type of problem is commonly asked in exams such as CSIR NET,
GATE, and other STEM entrance tests.