GATE 2024 BT Q.58: solution and answer

GATE 2024 · BTQ.58 · 2 marks · NAT

E. coli is inoculated in a shake flask containing nutrient rich medium. The initial number of viable cells in the medium is $1 0^{2}$ . After few hours, the number of viable cells is $1 0^{6}$ . Assuming cell divides by binary fission, the number of generations that have taken place is ________ (rounded off to the nearest integer).

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Answer and solution

ANSWER

About 13 generations have taken place ( $n = 4/ lo g_{10} 2 \approx 13.29$ ).

Official answer: 13.00 to 14.00Source: Official answer key, GATE 2024 BT (IISc Bengaluru)

In binary fission every cell splits into two, so each generation doubles the number of viable cells. After $n$ generations: $N = N_{0} \cdot 2^{n}$ , where $N_{0}$ is the initial and $N$ the final number of viable cells (no cell death is assumed, since both counts are of viable cells).
Take base-10 logarithms of both sides and solve for $n$ : $n = \frac{lo g _{10} N - lo g _{10} N _{0}}{lo g _{10} 2} = \frac{lo g _{10} ( N / N _{0} )}{lo g _{10} 2}$ .
Substitute $N_{0} = 1 0^{2}$ and $N = 1 0^{6}$ : $\frac{N}{N _{0}} = \frac{1 0 ^{6}}{1 0 ^{2}} = 1 0^{4}$ , so $lo g_{10} (N / N_{0}) = 4$ .
$n = \frac{4}{lo g _{10} 2} = \frac{4}{0.30103} \approx 13.29$
Rounded to the nearest integer: $n = 13$ generations. (Sanity check: $2^{13} = 8192$ and $2^{14} = 16384$ , so a $1 0^{4}$ -fold increase lies between 13 and 14 doublings, closer to 13.)

KEY IDEA

Binary fission doubles the population every generation, so $N = N_{0} \cdot 2^{n}$ and $n = lo g_{10} (N / N_{0}) / lo g_{10} 2 \approx 3.32 lo g_{10} (N / N_{0})$ . Only the fold-increase $N / N_{0}$ matters, not the absolute counts.

WATCH OUT FOR

Using the final count alone ( $n = lo g_{2} 1 0^{6} \approx 19.9$ ) and forgetting the initial $1 0^{2}$ cells. Other slips are mixing logarithm bases (for example $ln 1 0^{4} / lo g_{10} 2$ , or $4/ ln 2 \approx 5.8$ ), and dividing the fold-increase by 2 instead of taking $lo g_{2}$ of it.

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GATE 2024 BT Q.58: solution and answer

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